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Probability Theory and Related Fields (2020) 177:1137–1242https://doi.org/10.1007/s00440-020-00981-y

Infinite-dimensional stochastic differential equationsand tail �-fields

Hirofumi Osada1 · Hideki Tanemura2

Dedicated to Professor Shinzo Watanabe on his 80th birthday

Received: 18 March 2016 / Revised: 4 June 2020 / Published online: 4 July 2020© The Author(s) 2020

AbstractWe present general theorems solving the long-standing problem of the existence andpathwise uniqueness of strong solutions of infinite-dimensional stochastic differentialequations (ISDEs) called interacting Brownian motions. These ISDEs describe thedynamics of infinitely-many Brownian particles moving in R

d with free potential Φand mutual interaction potential Ψ . We apply the theorems to essentially all interac-tion potentials of Ruelle’s class such as the Lennard-Jones 6-12 potential and Rieszpotentials, and to logarithmic potentials appearing in random matrix theory. We solveISDEs of the Ginibre interacting Brownian motion and the sineβ interacting Brownianmotion with β = 1, 2, 4. We also use the theorems in separate papers for the Airy andBessel interacting Brownian motions. One of the critical points for proving the gen-eral theorems is to establish a new formulation of solutions of ISDEs in terms of tailσ -fields of labeled path spaces consisting of trajectories of infinitely-many particles.These formulations are equivalent to the original notions of solutions of ISDEs, andmore feasible to treat in infinite dimensions.

Keywords Interacting Brownian motions · Infinite-dimensional stochasticdifferential equations · Random matrices · Strong solutions · Pathwise uniqueness

Mathematics Subject Classification 60K35 · 60H10 · 82C22 · 60B20

B Hirofumi [email protected]

Hideki [email protected]

1 Faculty of Mathematics, Kyushu University, Fukuoka 819-0395, Japan

2 Department of Mathematics, Faculty of Science and Technology, Keio University,Yokohama 223-8522, Japan

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http://crossmark.crossref.org/dialog/?doi=10.1007/s00440-020-00981-y&domain=pdf

http://orcid.org/0000-0002-3321-0747

1138 H. Osada, H. Tanemura

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11382 Preliminary: logarithmic derivative and quasi-Gibbs measures . . . . . . . . . . . . . . . . . . 11453 The main general theorems: Theorems 3.1–3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 11504 Solutions and tail σ -fields: first tail theorem (Theorem 4.1) . . . . . . . . . . . . . . . . . . . . 11615 Triviality of Tpath(S

N): second tail theorem (Theorem 5.1) . . . . . . . . . . . . . . . . . . . 11756 Proof of Theorems 3.1–3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11897 The Ginibre interacting Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11908 Proof of Theorem 7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11959 Dirichlet forms and weak solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121510 Sufficient conditions of (SIN) and (NBJ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121911 Sufficient conditions of (IFC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122112 Sufficient conditions of (SIN) and (NBJ) for μ with non-trivial tails . . . . . . . . . . . . . . 122913 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1233Appendix: Tail decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1239References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1240

1 Introduction

We study infinite-dimensional stochastic differential equations (ISDEs) of

X = (Xi )i∈N ∈ C([0,∞);RdN), where RdN = (Rd)N,

describing infinitely-many Brownian particles moving in Rd with free potential Φ =

Φ(x) and interaction potential Ψ = Ψ (x, y). The ISDEs of X = (Xi )i∈N are givenby

dXit = dBi

t −β

2∇xΦ(Xi

t )dt −β

2

∞∑

j �=i∇xΨ (Xi

t , Xjt )dt (i ∈ N). (1.1)

Here B = (Bi )∞i=1, {Bi }i∈N are independent copies of d-dimensional Brownianmotions, ∇x = ( ∂

∂xi)di=1 is the nabla in x , and β is a positive constant called

inverse temperature. The process (X,B) is defined on a filtered probability space(Ω,F , P, {Ft }).

The study of ISDEs was initiated by Lang [16,17], and continued by Shiga [37],Fritz [5], and the second author [39], and others. In their respective work, the freepotential Φ is assumed to be zero and interaction potentials Ψ are of class C3

0(Rd)

or exponentially decay at infinity. This restriction on Ψ excludes polynomial decayand logarithmic growth interaction potentials, which are essential from the viewpointof statistical physics and random matrix theory. The following are examples of suchISDEs.

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Infinite-dimensional stochastic differential equations and tail σ -fields 1139

Sineβ interacting Brownian motions (Sect. 13.1)Let d = 1, Φ(x) = 0, Ψ (x, y) = − log |x − y|. We set

dXit = dBi

t +β

2limr→∞

∑

|Xit−X j

t |<r , j �=i

1

Xit − X j

t

dt (i ∈ N). (1.2)

This ISDE with β = 2 is called the Dyson model in infinite dimensions by Spohn[38].

Airyβ interacting Brownian motionsLet d = 1, Φ(x) = 0, Ψ (x, y) = − log |x − y|. We set

dXit = dBi

t +β

2limr→∞

⎧⎪⎨

⎪⎩

⎛

⎜⎝∑

j �=i, |X jt |<r

1

Xit − X j

t

⎞

⎟⎠−∫

|x |<r

ρ(x)

−x dx

⎫⎪⎬

⎪⎭dt (i ∈ N).

(1.3)

Here we set

ρ(x) = 1(−∞,0)(x)

π

√−x .

The function ρ is the scaling limit of the semicircle function

σsemi(x) = 1

2π

√4− x21(−2,2)(x)

with scaling n1/3σsemi(xn−2/3 + 2) associated with soft-edge scaling.We solve (1.3) for β = 1, 2, 4 in [30] using a result presented in this paper. As the

solutions X = (Xit )i∈N do not collide with each other, we label them in decreasing

order such that Xi+1t < Xi

t for all i ∈ N and 0 ≤ t < ∞. Let β = 2. The topparticle X1

t is called the Airy process and has been extensively studied by [11,12,33]and others. In [32], we calculate the space-time correlation functions of the unlabeleddynamics Xt =∑∞

i=1 δXitand prove that they are equal to those of [14] and others. In

particular, our dynamics for β = 2 are the same as the Airy line ensemble constructedin [3].

Besselα,β interacting Brownian motionsLet d = 1, 1 ≤ α <∞, Φ(x) = −α

2 log x and Ψ (x, y) = − log |x − y|. We set

dXit = dBi

t +β

2

⎧⎨

⎩α

2Xit+

∞∑

j �=i

1

Xit − X j

t

⎫⎬

⎭ dt (i ∈ N). (1.4)

Here particles move in (0,∞).

Ginibre interacting Brownian motions (Theorem 7.1)

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Let d = 2, Ψ (x, y) = − log |x − y|, and β = 2. We introduce the ISDE

dXit = dBi

t + limr→∞

∑

|Xit−X j

t |<r , j �=i

X it − X j

t

|Xit − X j

t |2dt (i ∈ N) (1.5)

and also

dXit = dBi

t − Xit dt + lim

r→∞∑

|X jt |<r , j �=i

X it − X j

t

|Xit − X j

t |2dt (i ∈ N). (1.6)

We shall prove that ISDEs (1.5) and (1.6) have the same strong solutions, reflecting thedynamical rigidity of two-dimensional stochastic Coulomb systems called the Ginibrerandom point field (see Sect. 7.1).

All these examples are related to random matrix theory. ISDEs (1.2) and (1.3)with β = 1, 2, 4 are the dynamical bulk and soft-edge scaling limits of the finite parti-cle systems of Gaussian orthogonal/unitary/symplectic ensembles, respectively. ISDE(1.4) with β = 1, 2, 4 is the hard-edge scaling limit of those of the Laguerre ensem-bles. ISDEs (1.5) and (1.6) are dynamical bulk scaling limits of the Ginibre ensemble,which is a system of eigen-values of non-Hermitian Gaussian random matrices.

The next two examples have interaction potentials of Ruelle’s class [35]. The pre-vious results also exclude these potentials because of the polynomial decay at infinity.We shall give a general theorem applicable to substantially all Ruelle’s class potentials(see Theorem 13.2).

Lennard-Jones 6-12 potentialLet d = 3 and Ψ6,12(x) = {|x |−12 − |x |−6}. The interaction Ψ6,12 is called theLennard-Jones 6-12 potential. The corresponding ISDE is

dXit = dBi

t +β

2

∞∑

j=1, j �=i

{12(Xi

t − X jt )

|Xit − X j

t |14− 6(Xi

t − X jt )

|Xit − X j

t |8

}dt (i ∈ N). (1.7)

Since the sum in (1.7) is absolutely convergent, we do not include prefactor limr→∞unlike other examples (1.2), (1.3), (1.5), and (1.6).

Riesz potentialsLet d < a ∈ N and Ψa(x) = (β/a)|x |−a . The corresponding ISDE is

dXit = dBi

t +β

2

∞∑

j=1, j �=i

X it − X j

t

|Xit − X j

t |a+2dt (i ∈ N). (1.8)

At first glance, ISDE (1.8) resembles (1.2) and (1.5) because (1.8) corresponds to thecase a = 0 in (1.2) and (1.5). However, the sums in the drift terms converge absolutelyunlike those in (1.2) and (1.5).

In the present paper, we introduce a new method of establishing the existence ofstrong solution and the pathwise uniqueness of solution of the ISDEs, including the

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ISDEs with long-range interaction potentials. Our results apply to a surprisingly widerange of models and, in particular, all the examples above (with suitably chosen β).

In the previous works, the Ito scheme was used directly in infinite dimensions. Thisscheme requires the “(local) Lipschitz continuity” of coefficients because it relies onthe contraction property of Picard approximations. Hence, a smart choice of stoppingtimes is pivotal but is difficult for long-range potentials in infinite-dimensional spaces.We do not apply the Ito scheme to ISDEs directly but apply it infinitely-many timesto an infinite system of finite-dimensional SDEs with consistency (IFC), which weexplain in the sequel.

Our method is based on several novel ideas and consists of three main steps. Thefirst step begins by reducing the ISDE to a differential equation of a random point fieldμ satisfying

∇x logμ[1](x, s) = −β

⎧⎨

⎩∇xΦ(x)+ limr→∞

∞∑

|si |<r

∇xΨ (x, si )

⎫⎬

⎭ . (1.9)

Here s =∑i δsi is a configuration, μ[1] is the 1-Campbell measure of μ as defined in

(2.1), and∇x logμ[1] is defined in (2.3).We call∇x logμ[1] the logarithmic derivativeof μ. Equation (1.9) is given in an informal manner here, and we refer to Sect. 2 forthe precise definition.

The first author proved in [26] with [25,27,28] that, if (1.9) has a solutionμ satisfy-ing the assumptions (A2)–(A4) in Sects. 9.2 and 10, then the ISDE (1.1) has a solution(X,B) starting at s = (si )i∈N. Here a solution means a pair of a stochastic process Xand R

dN-valued Brownian motion B = (Bi )i∈N satisfying (1.1). We thus see that thequartet of papers [25–28] achieved the first step. We note that such solutions in [26]are not strong solutions explained below and that the existence of strong solutions andpathwise uniqueness of solution of ISDEs (1.1) were left open in [26].

In the second step, we introduce the IFCmentioned above using the solution (X,B)in the first step. That is, we consider a family of the finite-dimensional SDEs ofYm = (Ym,i )mi=1, m ∈ N, given by

dYm,it = dBi

t −β

2∇xΦ(Ym,i

t )dt − β

2

m∑

j �=i∇xΨ (Ym,i

t ,Ym, jt )dt

− β

2limr→∞

∞∑

j=m+1, |X jt |<r

∇xΨ (Ym,it , X j

t )dt (1.10)

with initial condition

Ym0 = sm .

Here, for each m ∈ N, we set sm = (si )mi=1 for s = (si )∞i=1, and Bm = (Bi )mi=1denotes the (Rd)m-valued Brownian motions. Note that we regard X as ingredients

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of the coefficients of the SDE (1.10). Hence the SDE (1.10) is time-inhomogeneousalthough the original ISDE (1.1) is time-homogeneous.

Under suitable assumptions, SDE (1.10) has a pathwise unique, strong solutionYm .Hence, Ym is a function of s, B, and X:

Ym = Ym(s,B,X) = Yms,B(X).

As a function of (s,B), the process Ym depends only on the first m components(sm,Bm). Since we takem to go to infinity, the limit, if it exists, depends on the whole(s,B). As a function of X, the solution Ym depends only on Xm∗, where we set

Xm∗ = (Xi )∞i=m+1.

Hence, Ym(s,B, ·) is σ [Xm∗]-measurable. We therefore write

Ym = Ym(s,B, (0m,Xm∗)) = Yms,B((0

m,Xm∗)).

Here 0m = (0, . . . , 0) is the (Rd)m-valued constant path. The value 0 does not haveany special meaning. Just for the notational reason, we put 0m here.

With the pathwise uniqueness of the solutions of SDE (1.10), we see that Xm =(Xi )mi=1 is the unique strong solution of (1.10). Hence we deduce that

Xm = Yms,B((0

m,Xm∗)) for all m ∈ N. (1.11)

This implies that Xm becomes a function of s, B, and Xm∗. The dependence on Xm∗inherits the coefficients of the SDE (1.10).

Relation (1.11) provides the crucial consistency we use. From this we deduce thatthe mapsYm

s,B have a limitY∞s,B asm goes to infinity at least forX in the sense that the

first k components Ym,ks,B of Ym

s,B coincide with Ym′,ks,B for all k ≤ m,m′. Furthermore,

X is a fixed point of the limit map Y∞s,B:

X = Y∞s,B(X). (1.12)

Hence, the limit map Y∞s,B is a function of (s,B) and X itself through {Xm∗}m∈N. Thepoint is that, for each fixed (s,B), the limit Y∞s,B = Y∞(s,B, ·) as a function of Xis measurable with respect to the tail σ -field Tpath(R

dN) of the labeled path spaceW (RdN) := C([0,∞);RdN). Here we set Tpath(R

dN) such that

Tpath(RdN) =

∞⋂

m=1σ [Xm∗].

Let Ps be the distribution of the solution (X,B) of ISDE (1.1) starting at s. Bydefinition Ps is a probability measure on W (RdN) × W0(R

dN), where W0(RdN) =

{w ∈ W (RdN);w0 = 0}. We write (w,b) ∈ W (RdN) × W0(RdN). Let Ps,b be the

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distribution of the first component under conditioning the second component at b, thatis, Ps,b is the regular conditional distribution of Ps such that

Ps,b = Ps(w ∈ · |b).By construction Ps,b is the distribution of (X,B) starting at s conditioned at B = b.

BecauseY∞s,B is aTpath(RdN)-measurable function inX for fixed (s,b), we deduce

that, if Tpath(RdN) is trivial with respect to Ps,b, then Y∞s,B is a function of (s, B)

being independent of X for Ps,b-a.s. Hence, from the identity (1.12), the existence ofstrong solutions and the pathwise uniqueness of them are related to Ps,b-triviality ofTpath(R

dN).In Theorem 4.1 (First tail theorem), we shall give a sufficient condition of the

existence of the strong solutions and the pathwise uniqueness in terms of the propertyof Ps,b-triviality ofTpath(R

dN). This condition is necessary and sufficient as we see inTheorem 4.2. The critical point here is that, to some extent, we regard the labeled pathtail σ -field Tpath(R

dN) as a boundary of ISDE (1.1) and pose boundary conditions interms of Ps,b-triviality of it.

The formalism regarding a strong solution as a function F of path space is at the heartof the Yamada–Watanabe theory. They proved the equivalence between {the existenceof a weak solution + the pathwise uniqueness} and (the existence of) a unique strongsolution (see Theorem 1.1 [9, 163p]). Our main theorems, Theorems 4.1 and 4.3,clarify the relation between this property of solutions and tail triviality of the labeledpath space. We shall provide the existence of a strong solution and the pathwiseuniqueness of solutions of ISDEs. In this sense, this is a counterpart of Yamada–Watanabe’s result in infinite dimensions.Our argument is however completely differentfrom the Yamada–Watanabe theory. Indeed, the existence of a weak solution hasbeen established in the first step, and we shall prove the pathwise uniqueness andthe existence of strong solutions together using the analysis of the tail σ -field of thelabeled path space.

In the third step, we prove Ps,b-triviality of Tpath(RdN). Let S be the set of config-

urations on Rd . Then, by definition, S is the set given by

S ={s =∑

i

δsi ; s(K ) <∞ for all compact sets K ⊂ Rd

}. (1.13)

By convention, we regard the zero measure as an element of S. Each element s of S iscalled a configuration. We endow S with the vague topology, which makes S a Polishspace. Let T (S) be the tail σ -field of the configuration space S over R

d :

T (S) =∞⋂

r=1σ [πc

r ]. (1.14)

Here Sr = {|x | ≤ r} and πcr is the projection πc

r :S→S such that πcr (s) = s(· ∩ Scr ).

In Theorem 5.1 (Second tail theorem), we deduce Ps,b-triviality ofTpath(RdN) from

μ-triviality of S, where μ is the solution of equation (1.9). BecauseW (RdN) is a huge

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space and the tail σ -field Tpath(RdN) is not topologically well behaved, this step is

difficult to perform.The key point here is the absolute continuity condition (AC) given in Sect. 5,

that is, the condition such that the associated unlabeled process X = {Xt }, whereXt =∑∞

i=1 δXit, starting from λ satisfies

Pλ ◦ X−1t ≺ μ for each 0 < t <∞, (1.15)

where Pλ is the distribution of X such that Pλ ◦ X−10 = λ. Here we write m1 ≺ m2 ifm1 is absolutely continuous with respect to m2. This condition is satisfied if λ = μ

and μ is a stationary probability measure of the unlabeled diffusion X.Under this condition, Ps,b-triviality of Tpath(R

dN) follows from μ-triviality ofT (S).

One of the key points of the proof of the second tail theorem is Proposition 5.1,which proves that the finite-dimensional distributions of X satisfying (1.15) restrictedon the tail σ -field are the same as the restriction of the product measures of μ on thetail σ -field of the product of the configuration space S [see (5.13)].

The difficulty in controllingTpath(RdN) under the distribution given by the solution

of ISDE (1.1) is that the labeled dynamics X = (Xi )i∈N have no associated station-ary measures because they would be an approximately infinite product of Lebesguemeasures (if they exist). Instead, we study the associated unlabeled dynamics X asabove. We shall assume that X satisfies the absolute continuity condition. Then weimmediately see that its single time distributions Pλ ◦ X−1t (t ∈ (0,∞)) starting fromλ are tail trivial on S. From this, with some argument, we deduce triviality of the tailσ -field Tpath(S) of the unlabeled path space C([0,∞); S) =: W (S). We then furtherlift triviality of Tpath(S) to that of Tpath(R

dN). To implement this scheme, we employa rather difficult treatment of the map Y∞s,B in (1.12).

We write the second and third steps abstractly. This scheme is quite robust andconceptual, and can be applied to many other types of ISDEs involving stochasticintegrals beyond the Ito type. Indeed, we do not use any particular structure of Itostochastic analysis in these steps. Furthermore, even if μ is not tail trivial, we candecompose it to the tail trivial random point fields and solve the ISDEs in this case aswell.

In [5], Fritz constructed non-equilibrium solutions in the sense that the state space ofsolutionsX is explicitly given. In [5], it was assumed that potentials are ofC3

0(Rd) and

the dimension d is less than or equal to 4.Whilewewere preparing themanuscript, Tsai[40] constructed non-equilibrium solutions for the Dyson model with 1 ≤ β < ∞.His proof relies on the monotonicity specific to one-dimensional particle systems anduses translation invariance of the solutions. Hence it is difficult to apply his methodto multi-dimensional models, and Besselα,β and Airyβ interacting Brownian motionseven if in one dimension. Moreover, the reversibility of the unlabeled processes of thesolutions is unsolved in [40].

The present paper is organized as follows. In Sect. 2, we introduce notation usedthroughout the paper and recall some notions for random point fields.

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From Sects. 3 to 6 we devote to the general theory concerning on ISDEs. In Sect. 3,we state the main general theorems (Theorems 3.1–3.2). We explain the role of themain assumptions (IFC), (TT), (AC), (SIN), and (NBJ) in Sect. 3.4 and presenta list of conditions. In Sect. 4, clarify the relation between a strong solution and aweak solution satisfying (IFC) and triviality of Tpath(R

dN) in Theorem 4.1. We dothis in a general setting beyond interacting Brownian motions. We present and proveTheorem 4.1 (First tail theorem). In Sect. 5, we derive triviality of Tpath(R

dN) fromthat of T (S). We present and prove Theorem 5.1 (Second tail theorem). In Sect. 6,we prove the main theorems (Theorems 3.1–3.2).

From Sects. 7 to 8 we study the Ginibre interacting Brownian motion, which is oneof the most prominent examples of interacting Brownian motions with the logarithmicpotential. In Sect. 7.1,we present preliminary results. In Sect. 7.2,we state the result forthe Ginibre ensemble (Theorem 7.1). In Sect. 8, we prove Theorem 7.1 by employingTheorems 4.1 and 5.1.

FromSects. 9 to 13wedevote to applying the general theory to the class of the ISDEscalled interacting Brownian motions. We shall prepare feasible sufficient conditionsfor applications. In Sect. 9, we quote results onweak solutions of ISDEs and the relatedDirichlet form theory. In Sect. 10, we give sufficient conditions of assumptions (SIN)and (NBJ) used in Theorems 3.1–3.2. In Sect. 11, we give a sufficient conditionof assumption (IFC). In Sect. 12, we devote to the sufficient conditions of (SIN)and (NBJ) for μ with non-trivial tails. In Sect. 13, we give various examples andprove Theorems 13.1–13.2. In section (Appendix), we prove the tail decompositionof random point fields.

We shall explain the main assumptions in the present paper in Sect. 3.4 and presenta list of assumptions in Table 1.

2 Preliminary: logarithmic derivative and quasi-Gibbsmeasures

Let S be a closed set in Rd such that the interior Sint is a connected open set satis-

fying Sint = S and that the boundary ∂S has Lebesgue measure zero. Let S be theconfiguration space over S. The set S is defined by (1.13) by replacing R

d with S.A symmetric and locally integrable function ρn : Sn → [0,∞) is called the n-

point correlation function of a random point field μ on S with respect to the Lebesguemeasure if ρn satisfies

∫

Ak11 ×···×Akm

m

ρn(x1, . . . , xn)dx1 · · · dxn =∫

S

m∏

i=1

s(Ai )!(s(Ai )− ki )!dμ

for any sequence of disjoint bounded measurable sets A1, . . . , Am ∈ B(S) and asequence of natural numbers k1, . . . , km satisfying k1+ · · · + km = n. When s(Ai )−ki < 0, according to our interpretation, s(Ai )!/(s(Ai )− ki )! = 0 by convention.

Let μ[1] be the measure on (S × S,B(S)⊗B(S)) determined by

μ[1](A × B) =∫

Bs(A)μ(ds), A ∈ B(S), B ∈ B(S).

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The measure μ[1] is called the one-Campbell measure of μ. In case μ has one-correlation functionρ1, there exists a regular conditional probability μx ofμ satisfying

∫

Aμx (B)ρ1(x)dx = μ[1](A × B), A ∈ B(S), B ∈ B(S).

The measure μx is called the Palm measure of μ [13].In this paper, we use the probability measure μx (·) ≡ μx (· − δx ), which is called

the reduced Palm measure of μ. Informally, μx is given by

μx = μ(· − δx | s({x}) ≥ 1).

We consider the Radon measure μ[1] on S × S such that

μ[1](dxds) = ρ1(x)μx (ds)dx . (2.1)

In the present paper, we always use μ[1] instead of μ[1]. Hence we call μ[1] the one-Campbell measure of μ.

For a subset A, we set πA :S→S by πA(s) = s(· ∩ A). We say a function f on S islocal if f is σ [πK ]-measurable for some compact set K in S. For such a local functionf on S, we say f is smooth if f = fO is smooth, where O is a relative compact openset in S such that K ⊂ O . Moreover, fO is a function defined on

∑∞k=0 Ok such that

fO(x1, . . . xk) restricted on Ok is symmetric in x j ( j = 1, . . . , k) for each k suchthat fO(x1, . . . , xk) = f (x) for x =∑i δxi and that fO is smooth in (x1, . . . , xk) foreach k. Here, for k = 0, fO is a constant function on {s; s(O) = 0}. Because x is aconfiguration and O is relatively compact, the cardinality of the particles of x is finitein O . Note that fO has a consistency such that

fO(x1, . . . , xk) = fO ′(x1, . . . , xk) for all (x1, . . . , xk) ∈ (O ∩ O ′)k . (2.2)

Hence f (x) = fO(x1, . . . xk) is well defined.Let D◦ be the set of all bounded, local smooth functions on S. We set

C∞0 (S)⊗D◦ ={

N∑

i=1fi (x)gi (s) ; fi ∈ C∞0 (S), gi ∈ D◦, N ∈ N

}.

Let Sr = {s ∈ S ; |s| ≤ r}. We write f ∈ L ploc(S × S, μ[1]) if f ∈ L p(Sr × S, μ[1])

for all r ∈ N. For simplicity we set L ploc(μ

[1]) = L ploc(S × S, μ[1]).

Definition 2.1 An Rd -valued function dμ is called the logarithmic derivative of μ if

dμ ∈ L1loc(μ

[1]) and, for all ϕ ∈ C∞0 (S)⊗D◦,∫

S×Sdμ(x, y)ϕ(x, y)μ[1](dxdy) = −

∫

S×S∇xϕ(x, y)μ[1](dxdy). (2.3)

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If the boundary ∂S is nonempty and particles hit the boundary, then dμ wouldcontain a term arising from the boundary condition. For example, if the Neumannboundary condition is imposed on the boundary, then there would be a local time-type drift. In this sense, it would be more natural to suppose that dμ is a distributionrather than dμ ∈ L1

loc(μ[1]). Instead, we shall later assume that particles never hit

the boundary, and the above formulation is thus sufficient in the present situation. Itwould be interesting to generalize the theory, including the case with the boundarycondition; however, we do not pursue this here.

A sufficient condition of the explicit expression of the logarithmic derivative ofrandom point fields is given in [26, Theorem 45]. Using this, one can obtain the log-arithmic derivative of random point fields appearing in random matrix theory such assineβ , Airyβ, (β = 1, 2, 4), Bessel2,α (1 ≤ α), and the Ginibre random point field(see Lemma 13.2, [28,30], [8], Lemma 7.3, respectively). For canonical Gibbs mea-sures with Ruelle-class interaction potentials, one can easily calculate the logarithmicderivative employing Dobrushin-Lanford-Ruelle (DLR) equation (see Lemma 13.5).

Let Smr = {s ∈ S ; s(Sr ) = m}. Let Λr be the Poisson random point field whoseintensity is the Lebesgue measure on Sr and set Λm

r = Λr (· ∩ Smr ). We set mapsπr , π

cr :S→S such that πr (s) = s(· ∩ Sr ) and πc

r (s) = s(· ∩ Scr ).

Definition 2.2 A random point field μ is called a (Φ,Ψ )-quasi Gibbs measure if itsregular conditional probabilities

μmr ,ξ = μ(πr ∈ · |πc

r (x) = πcr (ξ), x(Sr ) = m)

satisfy, for all r ,m ∈ N and μ-a.s.ξ ,

c−11 e−Hr (x)Λmr (dx) ≤ μm

r ,ξ (dx) ≤ c1e−Hr (x)Λm

r (dx). (2.4)

Here c1 = c1(r ,m, ξ) is a positive constant depending on r , m, ξ . For two measuresμ, ν on a σ -field F , we write μ ≤ ν if μ(A) ≤ ν(A) for all A ∈ F . Moreover, Hr

is the Hamiltonian on Sr defined by

Hr (x) =∑

xi∈SrΦ(xi )+

∑

j<k, x j ,xk∈SrΨ (x j , xk) for x =

∑

i

δxi .

Remark 2.1 1. From (2.4), we see that for all r ,m ∈ N and μ-a.s. ξ , μmr ,ξ (dx) have

(unlabeled) Radon–Nikodym densities mmr ,ξ (dx) with respect to Λm

r . This fact isimportant when we decompose quasi-Gibbs measures with respect to tail σ -fieldsin Lemma 14.2. Clearly, canonical Gibbs measures μ with potentials (Φ,Ψ ) arequasi-Gibbs measures, and their densities mm

r ,ξ (dx) with respect to Λmr are given

by the DLR equation. That is, for μ-a.s.ξ =∑ j δξ j ,

mmr ,ξ (dx) =

1

Z mr ,ξ

exp

⎧⎨

⎩−Hr (x)−∑

xi∈Sr , ξ j∈ScrΨ (xi , ξ j )

⎫⎬

⎭ .

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HereZ mr ,ξ is the normalizing constant. For randompoint fields appearing in random

matrix theory, interaction potentials are logarithmic functions, where the DLRequations do not make sense as they are.

2. If μ is a (Φ,Ψ )-quasi Gibbs measure, then μ is also a (Φ + Φloc,bdd, Ψ )-quasiGibbsmeasure for any locally boundedmeasurable functionΦloc,bdd. In this sense,the notion of “quasi-Gibbs” is robust for the perturbation of free potentials. Inparticular, both the sineβ and Airyβ random point fields are (0,−β log |x − y|)-quasi Gibbs measures, where β = 1, 2, 4 (see [28,30]).

3. From (2.4) we see that Φ and Ψ are locally bounded from below with respect tothe L∞-norm by the Lebesgue measure.

Wecollect notationwe shall use throughout the paper: For a subset A of a topologicalspace, we shall denote by W (A) = C([0,∞); A) the set consisting of A-valuedcontinuous paths on [0,∞).

We set‖w‖C([0,T ];S) = supt∈[0,T ] |w(t)|.We then equipW (SN)withFréchetmetricdist(·, ∗) given by, for w = (wn)n∈N and w′ = (w′n)n∈N,

dist(w,w′) =∞∑

T=12−T{ ∞∑

n=12−n min{1, ‖wn − w′n‖C([0,T ];S)}

}.

We introduce unlabeling maps u[m] : Sm × S→S (m ∈ N) such that

u[m]((x, s)) =m∑

i=1δxi + s for x = (xi ) ∈ Sm, s ∈ S. (2.5)

By the same symbol u[m], we also denote the map u[m] : Sm→S such that u[m](x) =∑i δxi , where x = (xi )mi=1 and m ∈ N. Let u : SN→S such that

u((si )∞i=1) =

∞∑

i=1δsi . (2.6)

We often write s = (si )∞i=1 and s = ∑∞i=1 δsi . Thus (2.6) implies u(s) = s. For

w = {wt } = {(wit )i∈N}t∈[0,∞) ∈ W (SN), we set upath(w), called the unlabeled path of

w, by

upath(w)t = u(wt ) =∑

i

δwit. (2.7)

We note that upath(w) is not necessary an element of W (S). See Remark 3.10 forexample.

Let Ss be the subset of S with no multiple points. Let Ss.i. be the subset of Ssconsisting of an infinite number of points. Then by definition Ss and Ss.i. are given by

Ss = {s ∈ S ; s({x}) ≤ 1 for all x ∈ S}, Ss.i. = {s ∈ Ss ; s(S) = ∞}. (2.8)

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A Borel measurable map l :Ss.i.→ SN is called a label if u ◦ l(s) = s for all s ∈ Ss.i..Let W (Ss) and W (Ss.i.) be the sets consisting of Ss- and Ss.i.-valued continuous

path on [0,∞). Each w ∈ W (Ss) can be written as wt = ∑i δwit, where wi is an

S-valued continuous path defined on an interval Ii of the form [0, bi ) or (ai , bi ),where 0 ≤ ai < bi ≤ ∞. Taking maximal intervals of this form, we can choose[0, bi ) and (ai , bi ) uniquely up to labeling. We remark that limt↓ai |wi

t | = ∞ andlimt↑bi |wi

t | = ∞ for bi < ∞ for all i . We call wi a tagged path of w and Ii thedefining interval of wi . We set

WNE(Ss.i.) = {w ∈ W (Ss.i.) ; Ii = [0,∞) for all i}. (2.9)

We say tagged path wi of w does not explode if bi = ∞, and does not enter if Ii =[0, bi ), where bi is the right end of the interval where wi is defined. Thus WNE(Ss.i.)is the set consisting of non-explosion and non-entering paths. Then we can naturallylift each unlabeled path w ∈ WNE(Ss.i.) to the labeled path w = (wi )i∈N ∈ W (SN)using a label l = (li )i∈N such that w0 = l(w0). Indeed, we can do this because eachtagged particle can carry the initial label i forever. We write this correspondence bylpath(w) = (lipath(w))i∈N and set w as

w = lpath(w) with w0 = l(w0). (2.10)

Then wi = lipath(w) by construction. We set

wm∗ =∑

i>m

δwi ,

where∑

i>m δwi = {∑i>m δwit}t∈[0,∞). For an unlabeled path w, we call the path

w[m] =(l1path(w), . . . , l

mpath(w),

∑

i>m

δwi

)(2.11)

the m-labeled path. Similarly, for a labeled path w = (wi ) ∈ W (SN) we set w[m] by

w[m] =(w1, . . . , wm,

∑

i>m

δwi

). (2.12)

Remark 2.2 upath(w)t = u(wt ) by (2.7), whereas lpath(w)t �= l(wt ) in general.

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3 Themain general theorems: Theorems 3.1–3.2

3.1 ISDE

LetX = (Xi )i∈N be an SN-valued continuous process. We writeX = {Xt }t∈[0,∞) andXi = {Xi

t }t∈[0,∞). ForX and i ∈ N, we define the unlabeled processesX = {Xt }t∈[0,∞)

and Xi♦ = {Xi♦t }t∈[0,∞) as Xt =∑i∈N δXitand Xi♦t =∑ j∈N, j �=i δX j

t.

Let H and Ssde be Borel subsets of S such that

H ⊂ Ssde ⊂ Ss.i.⋂{s ; s(∂S) = 0}. (3.1)

Let u[1] be as in (2.5). Define Ssde ⊂ SN and S[1]sde ⊂ S × S by

Ssde = u−1(Ssde), S[1]sde = u−1[1] (Ssde). (3.2)

Let σ : S[1]sde→Rd2 and b : S[1]sde→R

d be Borel measurable functions, where d isthe dimension of the Euclidean space R

d that includes S. In infinite dimensions, itis natural to consider coefficients σ and b defined only on a suitable subset S[1]sde ofS × S. Let l : Ss.i.→ SN be the label introduced in Sect. 2. We consider an ISDE ofX = (Xi )i∈N starting from l(H) with state space Ssde such that

dXit = σ(Xi

t ,Xi♦t )dBi

t + b(Xit ,X

i♦t )dt (i ∈ N), (3.3)

X ∈ W (Ssde), (3.4)

X0 ∈ l(H). (3.5)

HereB = (Bi )i∈N is anRdN-valuedBrownianmotion; that is, {Bi }i∈N are independent

copies of a d-dimensional Brownian motion starting at the origin.

Remark 3.1 Note that l(H) ⊂ u−1(H) and that u−1(H) is much larger than l(H). Forexample, if we take l(s) = (si ) as |si | ≤ |si+1| for all i ∈ N, then Xt will soon exitfrom l(H). This is why we take Ssde = u−1(Ssde) in (3.2) rather than Ssde = l(Ssde).

From (3.4) the process X moves in the set Ssde where the coefficients σ and b arewell defined. Moreover, each tagged particle Xi of X = (Xi )i∈N never explodes . By(3.4), Xt ∈ Ssde for all t ≥ 0, and in particular the initial starting point s in (3.5) issupposed to satisfy s ∈ l(H) ⊂ Ssde, which implies u(s) ∈ H ⊂ Ssde.

By (3.1), H is a subset of Ssde. We shall take H in such a way that (3.3)–(3.5) has asolution for each s ∈ l(H). To detect a sufficiently large subset H satisfying this is animportant step to solve the ISDE.

The meaning of H to be large is however a problem at this stage because there isno natural measure on the infinite product space SN. In practice, we equip S with arandom point field μ such that μ(H) = μ(Ssde) = μ(S) = 1. We thus realize H as asupport of μ. We shall later assume (A1) in Sect. 9 to relate μ with (3.3) in such away that the unlabeled dynamics X of the solution X is μ-reversible. In this sense therandom point field μ is associated with ISDE (3.3).

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We remark that we can extend l(H) in (3.5) to u−1(H) by retaking other labels l.Because the coefficients of ISDE (3.3) have symmetry, this causes no problems.

Essentially, following [9, Chapter IV] in finite dimension,we present a set of notionsrelated to solutions of ISDE. In Definitions 3.1–3.8, (Ω,F , P, {Ft }) is a generalprobability space.

Definition 3.1 (weak solution) By a weak solution of ISDE (3.3)–(3.4), we mean anSN×R

dN-valued stochastic process (X,B) defined on a probability space (Ω,F , P)with a reference family {Ft }t≥0 such that

(i) X = (Xi )∞i=1 is an Ssde-valued continuous process. Furthermore, X is adaptedto {Ft }t≥0, that is, Xt isFt/Bt -measurable for each 0 ≤ t <∞, where

Bt = σ [ws; 0 ≤ s ≤ t, w ∈ W (SN)]. (3.6)

(ii) B = (Bi )∞i=1 is an RdN-valued {Ft }-Brownian motion with B0 = 0,

(iii) the family of measurable {Ft }t≥0-adapted processes Φ and Ψ defined by

Φ i (t, ω) = σ(Xit (ω),X

i♦t (ω)), Ψ i (t, ω) = b(Xi

t (ω),Xi♦t (ω))

belong to L 2 and L 1, respectively. Here L p is the set of all measurable{Ft }t≥0-adapted processes α such that E[∫ T0 |α(t, ω)|pdt] <∞ for all T . Herewe can and do take a predictable version of Φ i and Ψ i (see pp 45-46 in [9]).

(iv) with probability one, the process (X,B) satisfies for all t

Xit − Xi

0 =∫ t

0σ(Xi

u,Xi♦u )dBi

u +∫ t

0b(Xi

u,Xi♦u )du (i ∈ N).

Definition 3.2 (weak solution on A) We say the ISDE (3.3)–(3.4) has a weak solutionon a Borel set A if it has a weak solution for arbitrary initial distribution ν such thatν(A) = 1.

We say X is a weak solution if the accompanied Brownian motion B is obvious ornot important. A solution X staring at x means X is a solution such that X0 = x a.s.

Remark 3.2 In [9, Chap. IV], the state space and the set of the initial starting pointsof SDEs are the same and taken to be R

d . In the present paper, the set of the initialstarting points is l(H). l(H) is a subset of Ssde. So we introduced the notion “weaksolution on A” in Definition 3.2.

Definition 3.3 (uniqueness in law)We say that the uniqueness in law ofweak solutionson l(H) for (3.3)–(3.4) holds if whenever X and X′ are two weak solutions whoseinitial distributions coincide, then the laws of the processes X and X′ on the spaceW (SN) coincide. If this uniqueness holds for an initial distribution δs, then we say theuniqueness in law of weak solutions for (3.3)–(3.4) starting at s holds.

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Remark 3.3 For each s ∈ l(H) take δs as the initial law of the ISDE (3.3)–(3.4). Thenthe uniqueness in Definition 3.3 is equivalent to the uniqueness of the law of weaksolutions starting at each s ∈ l(H). We refer to Remark 1.2 in [9, 162 p] for thecorresponding result for finite-dimensional SDEs.

Definition 3.4 (pathwise uniqueness) We say that the pathwise uniqueness of weaksolutions of (3.3)–(3.4) on l(H) holds if whenever X and X′ are two weak solutionsdefined on the same probability space (Ω,F , P) with the same reference family{Ft }t≥0 and the sameR

dN-valued {Ft }-BrownianmotionB such thatX0 = X′0 ∈ l(H)a.s., then

P(Xt = X′t for all t ≥ 0) = 1.

Definition 3.5 (pathwise uniqueness of weak solutions starting at s) We say that thepathwise uniqueness of weak solutions of (3.3)–(3.4) starting at s holds if the conditionof Definition 3.4 holds for X0 = X′0 = s a.s.

We now state the definition of strong solution, which is analogous to Definition 1.6in [9, 163 p]. Let P∞Br be the distribution of an R

dN-valued Brownian motion B withB0 = 0. Let W0(R

dN) = {w ∈ W (RdN) ;w0 = 0}. Clearly, P∞Br (W0(RdN)) = 1.

Let Bt be as (3.6). Let Bt (P∞Br ) be the completion of σ [ws; 0 ≤ s ≤ t, w ∈W0(R

dN)] with respect to P∞Br . Let B(P∞Br ) be the completion of B(W0(RdN)) with

respect to P∞Br .

Definition 3.6 (a strong solution starting at s) A weak solution X of (3.3)–(3.4)with an R

dN-valued {Ft }-Brownian motion B is called a strong solution startingat s defined on (Ω,F , P, {Ft }) if X0 = s a.s. and if there exists a function Fs :W0(R

dN)→W (SN) such that Fs is B(P∞Br )/B(W (SN))-measurable, and that Fs isBt (P∞Br )/Bt -measurable for each t , and that Fs satisfies

X = Fs(B) a.s.

Also we call Fs itself a strong solution starting at s.

Definition 3.7 (a unique strong solution starting at s) We say (3.3)–(3.4) has a uniquestrong solution starting at s if there exists aB(P∞Br )/B(W (SN))-measurable functionFs :W0(R

dN)→W (SN) such that, for any weak solution (X, B) of (3.3)–(3.4) startingat s, it holds that

X = Fs(B) a.s.

and if, for any RdN-valued {Ft }-Brownian motion B defined on (Ω,F , P, {Ft })

with B0 = 0, the continuous process Fs(B) is a strong solution of (3.3)–(3.4) startingat s. Also we call Fs a unique strong solution starting at s.

We next present a variant of the notion of a unique strong solution.

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Definition 3.8 (a unique strong solution under constraint) For a condition (•), we say(3.3)–(3.4) has a unique strong solution starting at s under the constraint (•) if thereexists a B(P∞Br )/B(W (SN))-measurable function Fs :W0(R

dN)→W (SN) such thatfor any weak solution (X, B) of (3.3)–(3.4) starting at s satisfying (•), it holds that

X = Fs(B) a.s. (3.7)

and if for any RdN-valued {Ft }-Brownian motion B defined on (Ω,F , P, {Ft })

with B0 = 0 the continuous process Fs(B) is a strong solution of (3.3)–(3.4) startingat s satisfying (•). Also we call Fs a unique strong solution starting at s under theconstraint (•).

Remark 3.4 1. Themeaning of strong solutions is similar to the conventional situationin [9, pp 159–167]. The difference is that we consider solutions starting at a points. In [9], initial distributions are taken over all probability measures on the statespace.

2. Similarly as Definition 3.8 we can introduce the notions of constrained versionsof uniqueness in Definitions 3.3–3.5.

3.2 Main Theorem I (Theorem 3.1):�with trivial tail

Let (X,B) be an SN × RdN-valued continuous process defined on a filtered space

(Ω,F , P, {Ft }). We assume that (Ω,F , P) is a standard probability space. Thenthe regular conditional probability

Ps = P(·|X0 = s)

exists for P ◦X−10 -a.s. s. We investigate (X,B) under Ps, and thus regard (X,B) as astochastic process defined on the filtered space (Ω,F , Ps, {Ft }).

For X = (Xi )i∈N we set Xm∗t =∑∞i=m+1 δXi

tas before. Define

σmX : [0,∞)× Sm→R

d2 and bmX : [0,∞)× Sm→Rd

such that, for (u, v) ∈ Sm and v =∑m−1i=1 δvi ∈ S, where v = (v1, . . . , vm−1) ∈ Sm−1,

σmX (t, (u, v)) = σ(u, v+ Xm∗t ), bmX (t, (u, v)) = b(u, v+ Xm∗t ). (3.8)

We write l(s) = (si )i∈N = s and s∗m =∑∞

i=m+1 δsi . Recall X0 = l(s). We then haveXm∗0 = s∗m by construction. We remark that the coefficients σm

X and bmX depend on bothunlabeled path Xm∗ and the label l, although we suppress l from the notation. Let

Smsde(t,w) = {sm = (s1, . . . , sm) ∈ Sm ; u(sm)+ wm∗t ∈ Ssde}, (3.9)

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where wm∗t = ∑∞

i=m+1 δwitfor wt = ∑∞

i=1 δwit. By definition, Smsde(t,w) is a subset

of Sm depending on wm∗t . In particular, Smsde(t,w) is a time-dependent domain. Let

Ym = (Ym,i )mi=1, Ym,i♦ = (Ym, j )mj �=i , Ym,i♦t =

m∑

j �=iδYm, jt

.

We introduce the SDE with random environment X defined on (Ω,F , Ps, {Ft })describing Ym given by

dYm,it = σm

X (t, (Ym,it ,Ym,i♦

t ))dBit + bmX (t, (Y

m,it ,Ym,i♦

t ))dt, (3.10)

Ymt ∈ Smsde(t,X) for all t, (3.11)

Ym0 = sm, where sm = (s1, . . . , sm) for s = (si ) ∈ SN. (3.12)

A triplet of {Ft }-adapted, continuous process (Ym,Bm,Xm∗) on (Ω,F , Ps, {Ft })satisfying (3.10)–(3.12) is called a weak solution.

Remark 3.5 1. Equation (3.10) makes sense becauseXm∗,Bm , andYm are all definedon the same filtered space (Ω,F , Ps, {Ft }). We remark that (3.10) depends onXm∗, and that Xm∗ is regarded as a part of the coefficient of (3.10). We emphasize(X,B) is a priori given in SDE (3.10). We consider SDE (3.10) only for Bm , butnot for an arbitrary {Ft }-Brownian motion B

m.

2. TheSDE (3.10) is not a conventional type because the coefficient depends onX.Wecan regard X as a random environment, and call this SDE of random environmenttype. Random environment type SDEs had appeared in homogenization problem(see [22,34] for example). In this case, random environment and Brownian motionin SDEs are usually independent of each other. This is not the case in the presentsituation. If (B

m, X

m∗) is equivalent in law to (Bm,Xm∗), then we can replace

(Bm,Xm∗) by (Bm, X

m∗) in (3.10). The new SDE is equivalent to (3.10) in the

sense that the former has a weak solution if and only if the latter has one. Weemphasize that Bm and Xm∗ are {Ft }-adapted and can depend on each other.

3. The triplet (Xm,Bm,Xm∗)made of the original weak solution (X,B) of the ISDE(3.3)–(3.5) is a weak solution of (3.10)–(3.12). This yields the crucial identity(1.11).

We define the notion of strong solutions and a unique strong solution of (3.10)–(3.12). Let Pm be the distribution of (Bm,Xm∗) under (Ω,F , Ps, {Ft }):

Pm = Ps ◦ (Bm,Xm∗)−1.

Let W0(Rdm) = {w ∈ W (Rdm) ;w0 = 0} as before and set

C m = B(W0(Rdm)×W (RdN))Pm

,

C mt = Bt (W0(Rdm)×W (RdN))

Pm

.

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Here Bt (W0(Rdm) × W (RdN)) = σ [(vs,ws); 0 ≤ s ≤ t, (v,w) ∈ W0(R

dm) ×W (RdN)]. Let Bm

t = σ [ws; 0 ≤ s ≤ t, w ∈ W (Rdm)]. We state the definition ofstrong solution.

Definition 3.9 (strong solution for (X,B) starting at sm)Ym is called a strong solutionof (3.10)–(3.12) for (X,B) under Ps if (Ym,Bm,Xm∗) satisfies (3.10)–(3.12) and thereexists a C m-measurable function

Fms :W0(R

dm)×W (RdN)→W (Rdm)

such that Fms is C m

t /Bmt -measurable for each t , and Fm

s satisfies

Ym = Fms (Bm,Xm∗) Ps-a.s.

Remark 3.6 Our definition of a strong solution is different from that of Definition 1.6in [9, 163 p] with the following points: We consider solutions starting at a point sm

only. The main difference is that both the {Ft }-Brownian motion B and the processXm∗ are a priori given and fixed. Hence the solution Ym is a function of not onlyB but also Xm∗. This means, if we put an arbitrary {Ft }-Brownian motion B′ intoFms as Fm

s (B′,Xm∗), then Fms (B′,Xm∗) is not necessary a solution. We call Xm∗ an

environment processes.We note that there is no environment process in the frameworkof Definition 1.6 in [9, 163 p]. We shall take the limit m → ∞, and prove that theeffect ofXm∗ will vanish in the limit. As a result, the limit ISDE becomes conventional.Vanishing the effect of Xm∗ as m →∞ is a key to our argument. We will do this bythe second main theorem (Theorem 5.1).

Definition 3.10 (a unique strong solution for (X,B) starting at sm) The SDE (3.10)–(3.12) is said to have a unique strong solution for (X,B) under Ps if there exists afunction Fm

s satisfying the conditions in Definition 3.9 and, for any weak solution(Y

m,Bm,Xm∗) of (3.10)–(3.12) under Ps,

Ym = Fm

s (Bm,Xm∗) for Ps-a.s.

The function Fms in Definition 3.9 is also called a strong solution starting at sm .

The SDE (3.10)–(3.12) is said to have a unique strong solution Fms if Fm

s satisfies thecondition in Definition 3.10. We note that the function Fm

s is unique for Pm-a.s. inthis case.

We recall that these two notions are different from those of the infinite-dimensionalcounterparts Definitions 3.6 and Definition 3.7 because the SDE (3.10)–(3.12) is ofrandom environment type.

We introduce the IFC condition of (X,B) defined on (Ω,F , P, {Ft }) as follows.(IFC) The SDE (3.10)–(3.12) has a unique strong solution Fm

s (Bm,Xm∗) for (X,B)under Ps for P ◦ X−10 -a.s. s for all m ∈ N.For convenience we introduce a quenched version of (IFC):(IFC)s The SDE (3.10)–(3.12) has a unique strong solution Fm

s (Bm,Xm∗) for (X,B)under Ps for all m ∈ N.

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By definition (IFC) holds if and only if (IFC)s holds for P ◦ X−10 -a.s. s.The SDE (3.10)–(3.12) is time inhomogeneous and the state space of the solution

Ym given by (3.11) also depends on time t through Xm∗t . Because the SDE (3.10)

is finite-dimensional, one can apply the classical theory of SDEs directly. A feasiblesufficient condition of (IFC) is given in Sect. 11.

We remark that a continuous process (X,B) satisfying (IFC) is not necessary aweak solution. See Definition 4.3 and Remark 4.4.

Let T (S) be the tail σ -field of S defined as (1.14). A random point field μ on S iscalled tail trivial ifμ(A) ∈ {0, 1} for all A ∈ T (S). LetX = (Xi )i∈N be a continuousprocess defined on (Ω,F , P, {Ft }) and X be the associated unlabeled process suchthat Xt = ∑i δXi

t. Let WNE(Ss.i.) be as (2.9). Let mr ,T :W (SN)→N ∪ {∞} be such

that

mr ,T (w) = inf{m ∈ N ; mint∈[0,T ] |w

nt | > r for all n ∈ N such that n > m}. (3.13)

We make assumptions of μ and dynamics X under P .(TT) μ is tail trivial.(AC) P ◦ X−1t ≺ μ for all 0 < t <∞.(SIN) P(X ∈ WNE(Ss.i.)) = 1.(NBJ) P(mr ,T (X) <∞) = 1 for each r , T ∈ N.

We define the conditions (AC), (SIN), and (NBJ) for a probability measure P onW (RdN) by replacing X andX byw andw, respectively. We remark here (AC), (SIN),and (NBJ) are conditions depend only on the distribution of X.

We remark that, if (X,B) under P satisfies (SIN), then (X,B) under Ps satisfies(SIN) for P ◦ X−10 -a.s. s, where Ps = P(·|X0 = s). The same holds for (NBJ). Thisis however not the case for (AC). Similarly as (IFC)s, we write (SIN)s and (NBJ)swhen we emphasize dependence on s.

It is known that all determinantal random point fields on continuous spaces are tailtrivial [2,19,29]. These results are a generalization of that of determinantal randompoint fields on discrete spaces [1,18,36].

In Sect. 5, we deduce triviality of Tpath(SN) from that of T (S) through the tail σ -fieldTpath(S) of the unlabeled path space under these assumptions. We shall introducethe scheme carrying the tail σ -field of S to the tail σ -field of W (SN).

The assumption (NBJ) is crucial for the passage from the unlabeled dynamics X tothe labeled dynamics X. If P(mr ,T (X) = ∞) > 0, then we can not use this scheme.To catch the image for mr ,T (X) = ∞, we shall give an example of path w such thatmr ,T (w) = ∞ in Remark 3.10. This example indicates the necessity of (NBJ).

Let (X,B) be a weak solution of (3.3)–(3.4) defined on (Ω,F , P, {Ft }).If Ps = P(·|X0 = s) is a regular conditional probability, then (X,B) under Ps is a

weak solution of (3.3)–(3.4) starting at s for P ◦ X−10 -a.s. s.Conversely, suppose that {Ps} is a family of probability measures on (Ω,F , {Ft }),

given for m-a.s. s ∈ SN, such that (X,B) under Ps is a weak solution starting at s.If Ps(A) is B(SN)

m-measurable in s for any A ∈ B(W (SN)), then (X,B) under

P := ∫ Psm(ds) is a weak solution of (3.3)–(3.4) such that m = P ◦ X−10 .

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Taking these into account, we introduce the following condition for a family ofstrong solutions {Fs} of (3.3)–(3.4) given for P ◦ X−10 -a.s. s.

(MF) P(Fs(B) ∈ A) isB(SN)P◦X−10 -measurable in s for any A ∈ B(W (SN)).

For a family of strong solutions {Fs} satisfying (MF) we set

P{Fs} =∫

P(Fs(B) ∈ ·)P ◦ X−10 (ds). (3.14)

Let (X,B) be a weak solution under P . Suppose that (X,B) is a unique strongsolution under Ps for P ◦X−10 -a.s. s, where Ps = P(·|X0 = s). Let {Fs} be the uniquestrong solutions given by (X,B) under Ps. Then (MF) is automatically satisfied and

P{Fs} = P ◦ X−1. (3.15)

Indeed, B is a Brownian motion under both P and Ps. Then for P ◦ X−10 -a.s. s

P(Fs(B) ∈ ·) = Ps(Fs(B) ∈ ·) = Ps(X ∈ ·). (3.16)

Hence we deduce (3.15) from (3.14) and (3.16).

Definition 3.11 For a condition (•), we say (3.3)–(3.4) has a family of unique strongsolutions {Fs} starting at s for P ◦X−10 -a.s. s under the constraints of (MF) and (•) if{Fs} satisfies (MF) and P{Fs} satisfies (•). Furthermore, (i) and (ii) are satisfied.

(i) For any weak solution (X, B) under P of (3.3)–(3.4) with

P ◦ X−10 ≺ P ◦ X−10

satisfying (•), it holds that, for P ◦ X−10 -a.s. s,

X = Fs(B) Ps-a.s.,

where Ps = P(·|X0 = s).(ii) For an arbitraryR

dN-valued {Ft }-BrownianmotionBdefinedon (Ω,F , P, {Ft })with B0 = 0, Fs(B) is a strong solution of (3.3)–(3.4) satisfying (•) starting ats for P ◦ X−10 -a.s. s.

Theorem 3.1 Assume (TT) for μ. Assume that (3.3)–(3.4) has a weak solution (X,B)under P satisfying (IFC), (AC) forμ, (SIN), and (NBJ). Then (3.3)–(3.4) has a familyof unique strong solutions {Fs} starting at s for P ◦ X−10 -a.s. s under the constraintsof (MF), (IFC), (AC) for μ, (SIN), and (NBJ).

The first corollary is a quench result.

Corollary 3.1 Under the same assumptions as Theorem 3.1 the following hold.

1. (X,B) under Ps := P(·|X0 = s) is a strong solution starting at s for P ◦X−10 -a.s. s.

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2. Let (X′,B′) be any weak solution of (3.3)–(3.4) defined on (Ω ′,F ′, P ′, {F ′t })

satisfying (IFC), (AC) forμ, (SIN), and (NBJ). Assume that P ′◦X′−10 is absolutelycontinuous with respect to P ◦ X−10 . Then (X′,B′) under P ′s becomes a strongsolution starting at s satisfying X′ = Fs(B′) for P ′ ◦ X′−10 -a.s. s. Furthermore,

P ′s ◦ X′−1 = Ps ◦ X−1

for P ′ ◦ X′−10 -a.s. s. Here P ′s = P ′(·|X′0 = s).3. For any Brownian motion B′′, (Fs(B′′),B′′) becomes a strong solution of (3.3)–

(3.4) satisfying (IFC)s, (SIN)s, and (NBJ)s starting at s for P ◦ X−10 -a.s. s.

The second corollary is an anneal result.

Corollary 3.2 Under the same assumptions as Theorem 3.1 the following hold.

1. The uniqueness in law of weak solutions of (3.3)–(3.4) holds under the constraintsof (IFC), (AC) for μ, (SIN), and (NBJ).

2. The pathwise uniqueness of weak solutions of (3.3)–(3.4) holds under the con-straints of (IFC), (AC) for μ, (SIN), and (NBJ).

Remark 3.7 Because we exclude t = 0 in (AC), Theorem 3.1 is valid even if P ◦ X−10is singular to μ.

Remark 3.8 We study ISDEs on SN. It is difficult to solve the ISDEs on SN directly.One difficulty in treating SN-valued ISDEs is that SN does not have any goodmeasures.To remedy this situation, we introduce the representation of SN as an infinite sequenceof infinite-dimensional spaces:

S, S × S, S2 × S, S3 × S, S4 × S, . . . . (3.17)

Each space in (3.17) has a good measure called them-Campbell measure (see (9.11)).Using (3.8), we can rewrite (3.10) as

dYm,it = σ(Ym,i

t , Ym,i♦t + Xm∗t )dBi

t + b(Ym,it , Ym,i♦

t + Xm∗t )dt . (3.18)

Thus (Ym,Xm∗) is an Sm × S-valued process, and the scheme of infinite-dimensionalspaces {Sm × S}∞m=0 in (3.17) is useful.

Remark 3.9 We call (NBJ) no big jump condition because for any path w = (wi )i∈Nsuch that mr ,T (w) = ∞

supi∈N

sup0≤s,t≤T

|wis − wi

t |1[0,r ](

min0≤u≤T |w

iu |)= ∞

and so for any δ > 0

supi∈N

sup0≤s,t≤T|s−t |≤δ

|wis − wi

t |1[0,r ](

min0≤u≤T |w

iu |)= ∞,

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which implies the existence of paths which visit Sr during [0, T ] and have modulusof continuity bigger than � for any � ∈ N.

Remark 3.10 An example of a path w = (wi )i∈N ∈ W (R2N) such thatmr ,T (w) = ∞is as follows. Let ti =∑i

j=1 2− j and

wit =

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(1, i) t ∈ [0, ti ] ∪ [ti+1,∞)

linear [ti , ti + 2−i−2](0, 0) t = ti + 2−i−2

linear [ti + 2−i−2, ti+1].

All particles sit on the vertical line {(1, y); y ∈ R+} at time zero. The i th particle sits

at (1, i), jumps off at time ti and touch the origin at time ti + 2−i−2. Then it springsup to the original position (1, i). We need (NBJ) to exclude this type of “big jump ”paths. We note that upath(w) /∈ W (S) although w ∈ W (R2N). We conjecture that, ifupath(w) ∈ WNE(Ss.i.), then mr ,T (w) <∞ is automatically satisfied.

3.3 Main theorem II (Theorem 3.2):�with non-trivial tail

In this section, we relax (TT) of μ by the tail decomposition of μ as follows.Let μa

Tail be the regular conditional probability of μ conditioned by T (S):

μaTail = μ( · |T (S))(a). (3.19)

Because S is a Polish space, such a regular conditional probability exists and satisfies

μ(A) =∫

SμaTail(A)μ(da). (3.20)

By construction, μaTail(A) is a T (S)-measurable function in a for each A ∈ B(S).

Let H be a subset of Ssde in (3.5) and μ(H) = 1. We assume there exists a versionof μa

Tail with a subset of H, denoted by the same symbol H, such that μ(H) = 1 andthat, for all a ∈ H,

μaTail is tail trivial, (3.21)

μaTail({b ∈ S;μa

Tail = μbTail}) = 1, (3.22)

μaTail and μb

Tail are mutually singular on T (S) if μaTail �= μb

Tail. (3.23)

Let ∼Tail

be the equivalence relation such that a ∼Tail

b if and only if

μaTail = μb

Tail.

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Let Ha = {b ∈ H; a ∼Tail

b}. Then H can be decomposed as a disjoint sum

H =∑

[a]∈H/ ∼Tail

Ha. (3.24)

From (3.22), we see that μaTail(Ha) = 1 for all a ∈ H.

For a labeled processX = (Xi ) on (Ω,F , P, {Ft })we set Xt =∑i δXitas before.

Let l be a label. We assume X0 = l ◦ X0 for P-a.s. We note that plural labels satisfythe relation X0 = l ◦ X0 for P-a.s. in general. For μ as above we assume

μ = P ◦ X−10 . (3.25)

We set Pa = P( · |X−10 (T (S)))|X0=a. Then by (3.19) and (3.20)

Pa =∫

P(·|X0 = s)μaTail(ds). (3.26)

We can rewrite Pa as

Pa =∫

P(·|X0 = s)μaTail ◦ l−1(ds). (3.27)

From (3.19) and (3.25) we easily see μaTail = Pa ◦ X−10 and

μaTail ◦ l−1 = Pa ◦ X−10 . (3.28)

Let Pas = Pa(·|X0 = s). Then Pa

s = P(·|X0 = s) for μaTail ◦ l−1-a.s. s and for μ-a.s.

a.

Theorem 3.2 Assume μ, μaTail, and H satisfy (3.21)–(3.23) for all a ∈ H ⊂ Ssde,

μ(H) = 1, and (3.25). Assume that (X,B) under P is a weak solution of (3.3)–(3.4) satisfying (IFC), (SIN), and (NBJ). Assume that, for μ-a.s. a, (X,B) under Pa

satisfies (AC) for μaTail. Then, for μ-a.s. a, (3.3)–(3.4) has a family of unique strong

solutions {Fas } starting at s for Pa ◦X−10 -a.s. s under the constraints of (MF), (IFC),

(AC) for μaTail, (SIN), and (NBJ).

Remark 3.11 1. We shall prove inLemma14.2 that a version {μaTail} satisfying (3.21)–

(3.23) exists if μ is a quasi-Gibbs random point field satisfying (A2) in Sect. 9.2.All examples in the present paper are such quasi-Gibbs random point fields.

2. The unique strong solution Fas in Theorem 3.2 yields the corollaries similar to

Corollary 3.1 and Corollary 3.2. In particular, for μ-a.s. a, (X,B) = (Fas (B),B)

under Pas for μa

Tail ◦ l−1-a.s. s. We note here μaTail ◦ l−1 = Pa ◦ X−10 by (3.28).

3. In Theorem 3.2, the assumption “(X,B) under Pa satisfies (AC) forμaTail forμ-a.s.

a” is critical, and does not hold in general. We shall give sufficient conditions inTheorems 12.1 and 12.2. From these theorems we deduce all the examples in thepresent paper satisfy the assumption.

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3.4 The role of the five assumptions: (IFC), (TT), (AC), (SIN), and (NBJ)

In Sect. 3.2, we introduced the five significant assumptions: (IFC), (TT), (AC), (SIN),and (NBJ). In this subsection, we explain the role of these assumptions in the proof ofTheorem 3.1. We also explain the role of other main assumptions used in the presentpaper.

To prove Theorem 3.1, we use the strategy introduced in Sect. 1. One of thecritical points of the strategy is the reduction of ISDE to the infinite system of finite-dimensional stochastic differential equations (SDEs). For this, we use the pathwiseunique strong solutions of the finite-dimensional SDEs associatedwith ISDE. The con-dition (IFC) claims the finite-dimensional SDEs have such a unique strong solution.So the (IFC) is pivotal to the reduction of ISDE to the IFC schemes.

Another critical point of the proof is tail triviality of the labeled path space underthe distribution of the weak solution. We shall deduce this from tail triviality of μ,denoted by (TT), in a general framework as the second tail theorem in Sect. 5.

The key idea for this is the passage from (TT) to that of the path space of thelabeled dynamics. Because of (AC), we deduce triviality of the tail σ -field of S underthe single time marginal distributions of X. From this we shall deduce tail triviality ofthe unlabeled path space under the finite-dimensional distributions of the unlabeleddynamics X, where the meaning of tail is spatial [see (5.7) for definition].

We next deduce tail triviality of the labeled path space from that of the unlabeledpath space. The map lpath in (2.10) from the unlabeled path space to the labeled oneplays an essential role in our argument. The assumption (SIN) is necessary for theconstruction of this map. Here (SIN) is an abbreviation of “unlabeled path spaces onsingle, infinite configurations with no explosion of tagged particles”. To carry out thepassage, we shall use (NBJ) in addition to (SIN).

In Sect. 4, we shall deduce the existence of a unique strong solution of ISDE fromtail triviality of labeled path space in Theorem 4.1. We call Theorem 4.1 the first tailtheorem. The conditions in Theorem 4.1 are denoted by (Tpath1)s and (Tpath2)s.

The assumptions (IFC), (TT), (AC), (SIN), and (NBJ) are used in the proof ofTheorem 5.1 (the second tail theorem) in Sect. 5. (TT) and (AC) are the assumptionsof Theorem 5.2, which claims triviality of the labeled path space at the cylindricallevel. (TT), (AC), (SIN), and (NBJ) are the assumptions of Theorem 5.3, whichproves (Cpath1) and (Cpath2).

Conditions (A1)–(A4) given in Sects. 9 and 10 are related to Dirichlet forms; theseare the conditions for random point fields μ from the viewpoint of Dirichlet formtheory. Recall that (IFC) is the most critical condition for the theory. We shall givethe feasible sufficient condition of (IFC) in terms of the assumptions (B1)–(B2) and(C1)–(C2) in Sect. 11. We shall present them in Theorems 11.1 and 11.2.

We present a table for these conditions in Table 1.

4 Solutions and tail �-fields: first tail theorem (Theorem 4.1)

This section proves the existence of a strong solution, the pathwise uniqueness ofsolutions, and that the ISDE (4.2)–(4.4) has a unique strong solution. The ISDEs

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Table 1 List of conditions

Assumption Place The main role of the assumption

(IFC), (IFC)s Section 3.2 Theorem 3.1: To construct a scheme of finite-dimensional SDEs

(TT) Section 3.2 Theorem 3.1 and the second tail theorem (Theorem 5.1)

(AC) Section 3.2 Theorem 3.1 and the second tail theorem (Theorem 5.1)

(SIN) Section 3.2 Theorem 3.1 and the second tail theorem (Theorem 5.1)

(NBJ) Section 3.2 Theorem 3.1 and the second tail theorem (Theorem 5.1)

(MF) Section 3.2 Theorem 3.1 and Definition 3.11: To construct P{Fs} from {Fs}(Tpath1)s Section 4.1 The first tail theorem (Theorem 4.1)

(Tpath2)s Section 4.1 The first tail theorem (Theorem 4.1)

(Cpath1) Section 5.2 Theorem 5.4, which yields the second tail theorem

(Cpath2) Section 5.2 Theorem 5.4, which yields the second tail theorem

(A1)–(A3) Section 9 To construct weak solutions of (ISDE) via Dirichlet forms

(A4) Section 10 To derive (SIN) and (NBJ)

(B1)–(B2) Section 11 To derive (IFC)

(C1)–(C2) Section 11.3 To derive (B2)

studied in this section are more general than those in Sects. 1 and 3. Naturally, theISDEs in the previous sections are typical examples that our results (Theorems 4.1,4.2, and 4.3) can be applied to.

Throughout this section, X = (Xi )i∈N is an SN-valued, continuous {Ft }-adaptedprocesses defined on (Ω,F , Ps, {Ft }) starting at s, which is indicated by the subscripts in Ps. B = (Bi )i∈N is an R

dN-valued, standard {Ft }-Brownian motion starting atthe origin. P∞Br is the distribution of B. Thus,

Ps(X0 = s) = 1, Ps(B ∈ ·) = P∞Br . (4.1)

We shall fix the initial starting point s throughout Sect. 4.

4.1 General theorems of the uniqueness and existence of strong solutions of ISDEs

In this subsection, we introduce ISDE (4.2)–(4.4) and state one of the main theorems(Theorem 4.1: First tail theorem).

Let Wsol be a Borel subset of W (SN). Let B(Wsol) be the Borel σ -field of Wsol.Let Bt (Wsol) be the sub σ -field of B(Wsol) such that

Bt (Wsol) = σ [wu; 0 ≤ u ≤ t , w ∈Wsol].

Following [9] in finite dimensions, we shall introduce SDEs in infinite dimensions.

Definition 4.1 A d,r is the set of all functions α : [0,∞)×Wsol→Rd ⊗R

r such that

1. α isB([0,∞))×B(Wsol)/B(Rd ⊗ Rr )-measurable,

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2. Wsol � w �→ α(t,w) ∈ Rd ⊗R

r isBt (Wsol)/B(Rd ⊗Rr )-measurable for each

t ∈ [0,∞).

Let σ i :Wsol→ W (Rd2) and bi :Wsol→ W (Rd) such that σ i (w)t ∈ A d,d andbi (w)t ∈ A d,1. We assume σ i ∈ L 2 and bi ∈ L 1, where L p is the same asDefinition 3.1. We introduce the ISDE on SN of the form

dXit = σ i (X)t d Bi

t + bi (X)t dt (i ∈ N), (4.2)

X ∈Wsol, (4.3)

X0 = s. (4.4)

Here (X,B) is defined on (Ω,F , Ps, {Ft }), and B = (Bi )i∈N is an RdN-valued,

{Ft }-Brownian motion as before.The definition of a weak solution and a strong solution, and the related notions are

similar to those of Sect. 3.We remark that, in this section, we do not assume Wsol = u−1path(W (W)) for some

W ⊂ S unlike the previous sections. This is because we intend to clarify the relationbetween the strong and pathwise notions of solutions of ISDE and tail triviality ofthe labeled path space. Indeed, our theorems (Theorems 4.1 and 4.2) clarify a generalstructure of the relation between the existence of a strong solution and the pathwiseuniqueness of the solutions of ISDE (4.2)–(4.4) and triviality of the tail σ -field of thelabeled path space Tpath(SN) defined by (4.9) below.

We make a minimal assumption for this structure. As a result, ISDEs in this sectionare much general than before. In Sect. 5, we return to the original situation, and deducetail triviality of the labeled path space from that of the configuration space.

The correspondence between ISDE (3.3)–(3.4) and (4.2)–(4.3) is as follows.

Wsol = {w ∈ W (Ssde);w0 ∈ l(H)},σ i (X)t = σ(Xi

t ,Xi♦t ), bi (X)t = b(Xi

t ,Xi♦t ).

Here H, σ , and b are given in ISDE (3.3) and (3.5). Moreover, Wsol corresponds toboth l(H) and u−1(Ssde).

The final form of our general theorems (Theorems 3.1–3.2) are stated in terms ofrandom point fields. We emphasize that there are many interesting random point fieldssatisfying the assumptions, such as the sine, Airy, Bessel, and Ginibre random pointfields, and all canonical Gibbsmeasures with potentials of Ruelle’s class (with suitablesmoothness of potentials such that the associated ISDEs make sense).

We take the viewpoint not to pose the explicit conditions of the coefficients σ i andbi to solve ISDE (4.2)–(4.4), but to assume the existence of a weak solution and thepathwise uniqueness of solutions of the associated infinite systemof finite-dimensionalSDEs instead.

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For a given (X,B) defined on (Ω,F , Ps, {Ft }) satisfying (4.1), we introduce theinfinite system of the finite-dimensional SDEs (4.5)–(4.7).

dYm,it = σ i (Ym ⊕ Xm∗)t d Bi

t + bi (Ym ⊕ Xm∗)t dt (i = 1, . . . ,m), (4.5)

Ym ⊕ Xm∗ ∈Wsol, (4.6)

Ym0 ⊕ Xm∗

0 = s, (4.7)

where Ym = (Ym,1, . . . ,Ym,m) is an unknown process defined on (Ω,F , Ps, {Ft }),and Ym ⊕ Xm∗ = (Ym,1, . . . ,Ym,m, Xm+1, Xm+2, . . .).

The processYm denotes a solution of (4.5)–(4.7) starting at sm = (s1, . . . , sm). Thenotion of a strong solution and a unique strong solution of (4.5)–(4.7) for (X,B) underPs is defined as Definition 3.9 and Definition 3.10 with an obvious modification. LetBm = (Bi )mi=1 be the firstm-components of the {Ft }-Brownian motion B = (Bi )∞i=1.The following assumption corresponds to (IFC)s in Sect. 3.2.(IFC)s SDE (4.5)–(4.7) has a unique strong solutionYm = Fm

s (Bm,Xm∗) for (X,B)under Ps for each m ∈ N.

Remark 4.1 The meaning of SDE (4.5)–(4.7) is not conventional because the coeffi-cients include additional randomness Xm∗, that is, Xm∗ is interpreted as ingredientsof the coefficients of the SDE (4.5). Furthermore, Bm and Xm∗ can depend on eachother. See Remark 3.5. Another interpretation of SDE (4.5)–(4.7) is that (Bm,Xm∗) isregarded as input to the system rather than the interpretation such thatXm∗ is regardedas a part of the coefficient. Solving SDE (4.5)–(4.7) thenmeans constructing a functionof (Bm,Xm∗). Thus a strong solution means a functional of (Bm,Xm∗).

We assume (X,B) under Ps is a weak solution of (4.2)–(4.4) satisfying (IFC)s.Then we obtain the crucial identity:

Ym = Xm . (4.8)

As we saw in Sect. 1, (4.8) plays an important role in the whole theory.We set wm∗ = (wi )∞i=m+1 for w = (wi )∞i=1 ∈ W (SN). Let

Tpath(SN) =

∞⋂

m=1σ [wm∗]. (4.9)

By definition, Tpath(SN) is the tail σ -field of W (SN) with respect to the label. For aprobability measure P on W (SN), we set

T {1}path(S

N; P) = {A ∈ Tpath(SN) ; P(A) = 1}. (4.10)

For the continuous process (X,B) given at the beginning of this section, we set

Ps = Ps ◦ (X,B)−1. (4.11)

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By definition Ps is a probability measure onW (SN)×W0(RdN). We denote by (w,b)

generic elements inW (SN)×W0(RdN). Let Ps,b be the probabilitymeasure onW (SN)

given by the regular conditional probability Ps,b of Ps such that

Ps,b(·) = Ps(w ∈ · |b). (4.12)

We introduce the conditions.(Tpath1)s Tpath(SN) is Ps,b-trivial for P∞Br -a.s.b.(Tpath2)s T

{1}path(S

N; Ps,b) is independent of the distribution Ps for P∞Br -a.s.b if (X,B)under Ps is a weak solution of (4.2)–(4.4) satisfying (IFC)s and (Tpath1)s.In other words, (Tpath2)s means(Tpath2)s’ If (X,B) under Ps and (X′,B′) under P ′s satisfy (IFC)s and (Tpath1)s, then

T {1}path(S

N; Ps,b) = T {1}path(S

N; P ′s,b) for P∞Br -a.s.b.

In this sense, (Tpath2)s is a condition for ISDE (4.2)–(4.4) rather than a specific solution(X,B) under Ps.

Remark 4.2 1. The conditions (Tpath1)s and (Tpath2)s depend on s, which is the initialstarting point of X. To indicate this we put the subscript s.

2. We emphasize that we fix s throughout Sect. 4, while we do not fix s in Sect. 5.We shall present a sufficient condition such that (Tpath1)s and (Tpath2)s hold fora.s. s with respect to the initial distribution of X0. We remark that, unlike Sect. 4,X does not necessarily start at a fixed single point in Sect. 5.

We note that (Tpath2)s implies T {1}path(S

N; Ps,b) depends only on b for P∞Br -a.s.b.We now state the main theorem of this section, which we shall prove in Sect. 4.2.

In Theorem 4.1 and Corollary 4.1 we consider ISDE (4.2)–(4.4). Recall the notionsof strong solutions starting at s given by Definition 3.6 and a unique strong solutionunder the constraint of a condition (•) given by Definition 3.8.

Theorem 4.1 (First tail theorem) The following hold.

1. Assume that (X,B) under Ps is a weak solution of (4.2)–(4.4) satisfying (IFC)sand (Tpath1)s. Then (X,B) under Ps is a strong solution of (4.2)–(4.4).

2. Make the same assumptions as (1). We further assume that ISDE (4.2)–(4.4) satis-fies (Tpath2)s. Then (4.2)–(4.4) has a unique strong solution Fs under the constraintof (IFC)s and (Tpath1)s.

Corollary 4.1 Make the same assumptions as Theorem 4.1 (2). Then the followinghold.

1. For any weak solution (X′,B′) of (4.2)–(4.4) satisfying (IFC)s and (Tpath1)s, itholds that X′ = Fs(B′).

2. For any Brownian motion B′′, Fs(B′′) is a strong solution of (4.2)–(4.4) satisfying(IFC)s and (Tpath1)s.

3. The uniqueness in law of weak solutions of (4.2)–(4.4) holds under the constraintof (IFC)s and (Tpath1)s.

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4.2 Infinite systems of finite-dimensional SDEs with consistency

In this section we prove Theorem 4.1. Let (X,B) be a pair of an {Ft }-adapted continu-ous processX and an {Ft }-BrownianmotionB defined on (Ω,F , Ps, {Ft }) satisfying(4.1) as before. We assume that Ps satisfies:

Ps((X,B) ∈Wsol ×W0(RdN)) = 1.

We denote by Ps,b the regular conditional probability of Ps conditioned by the randomvariable B such that

Ps,b(·) = Ps((X,B) ∈ · |B = b). (4.13)

By construction Ps,b(Wsol × {b}) = 1. This follows from the fact that the regularconditional probability in (4.13) is conditioned by the random variable B. We refer to[9, (3.1) 15p] for proof. We set

Ps(·) =∫

Ps,b(·)P∞Br (db). (4.14)

From (4.1), (4.13), and (4.14) we have a representation of Ps = Ps ◦ (X,B)−1 suchthat for any C ∈ B(Wsol)×B(W0(R

dN))

Ps(C) = Ps(C). (4.15)

Remark 4.3 The probability measure Ps,b in (4.12) resembles Ps,b in (4.13), and theyare closely related to each other. The difference is that Ps,b is a probability measureon W (SN), whereas Ps,b is on W (SN)×W0(R

dN).

In general,FPdenotes the completion of the σ -fieldF with respect to probability

P . Recall that Tpath(SN) = ∩m∈Nσ [wm∗]. Then it is easy to see that

Tpath(SN)×B(W0(R

dN)) ⊂⋂

m∈N

{σ [wm∗] ×B(W0(R

dN))}. (4.16)

Hence taking the closure of both sides in (4.16) with respect to Ps we have

Tpath(SN)×B(W0(RdN))Ps ⊂

⋂

m∈N

{σ [wm∗] ×B(W0(RdN))

}Ps

=⋂

m∈Nσ [wm∗] ×B(W0(RdN))

Ps. (4.17)

The equality in (4.17) follows from a general fact such that the completion ofthe countable intersection of σ -fields with decreasing property coincides with the

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countable intersection of the completion of σ -fields with decreasing property. Indeed,if

K ∈⋂

m∈N

{σ [wm∗] ×B(W0(RdN))

}Ps, (4.18)

then there exist A, B ∈ ⋂m∈N σ [wm∗] ×B(W0(RdN)) such that A ⊂ K ⊂ B and

that Ps(B\A) = 0. Then A, B ∈ σ [wm∗] ×B(W0(RdN)) for each m. This deduces

K ∈⋂

m∈Nσ [wm∗] ×B(W0(RdN))

Ps. (4.19)

Conversely, if (4.19) holds, then there exist Am, Bm ∈ σ [wm∗] ×B(W0(RdN)) such

that Am ⊂ K ⊂ Bm and that Ps(Bm\Am) = 0 for each m ∈ N. It is clear thatlim supm Am and lim infm Bm are tail events such that

Ps(lim infm

Bm\ lim supm

Am) ≤∞∑

m=1Ps(Bm\Am) = 0.

Hence (4.18) holds.We need the inverse inclusion of (4.17), which does not hold in general. Hence we

introduce further completions of Tpath(SN)×B(W0(RdN)) as follows. We set

K = Tpath(SN)×B(W0(R

dN)) (4.20)

and I intuitively given by

I =⋂

b

KPs,b

P∞Br. (4.21)

Here the intersection is taken over P∞Br -a.s.b. To be precise, we set forU ⊂ W0(RdN)

K [U ] =⋂

b∈UK

Ps,b.

Then the set I is defined by

I = {V ; there exist Vi ∈ K [U ] (i = 1, 2), U ∈ B(W0(RdN))

such that V1 ⊂ V ⊂ V2, Ps,b(V2\V1) = 0 for all b ∈ U , P∞Br (U ) = 1}.(4.22)

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Lemma 4.1 Let Wm = σ [wm∗] ×B(W0(RdN)). Then the following holds.

∞⋂

m=1Wm

Ps ⊂ I . (4.23)

Proof We recall Ps = Ps ◦ (X,B)−1 by (4.11). Similarly as the equality in (4.17),

∩∞m=1WmPs = ∩∞m=1Wm

Ps.

From this we deduce that (4.23) is equivalent to

∩∞m=1WmPs ⊂ I . (4.24)

Let V ∈⋂∞m=1WmPs. Then there exist Vi ∈ ∩∞m=1Wm (i = 1, 2) such that

V1 ⊂ V ⊂ V2, Ps(V2\V1) = 0. (4.25)

For b ∈ W0(RdN), we set

Wb := (W (SN),b) = {(w,b) ; w ∈ W (SN)}.

Note that Vi ∩ Wb = (Abi ,b) for a unique Ab

i ⊂ W (SN). Clearly, Wb ∈ ∩∞m=1Wm .From these and Vi ∈ ∩∞m=1Wm we see

(Abi ,b) = Vi ∩Wb ∈ ∩∞m=1Wm .

Hence (Abi ,b) ∈ Wm for all m ∈ N. Then Ab

i ∈ ∩∞m=1σ [wm∗]. This implies

Abi ∈ Tpath(S

N). (4.26)

Let Ps,b be as in (4.12). Let Ab be a unique subset of W (SN) such that (Ab,b) =V ∩ Wb. From (Ab

i ,b) = Vi ∩ Wb combined with (4.12), (4.25), and (4.26), thereexists a set U ∈ B(W0(R

dN)) such that P∞Br (U ) = 1 and that

Ab ∈ Tpath(SN)Ps,b

for all b ∈ U . (4.27)

Recall the relation between Ps,b and Ps,b given by Remark 4.3. Then (4.27) implies

(Ab,b) ∈ Tpath(SN)×B(W0(RdN))Ps,b

for all b ∈ U .

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This together with (Ab,b) = V ∩Wb yields

V ∩Wb ∈ Tpath(SN)×B(W0(RdN))Ps,b

for all b ∈ U . (4.28)

Recall that K = Tpath(SN)×B(W0(RdN)) by (4.20). Then we rewrite (4.28) as

V ∩Wb ∈ KPs,b for all b ∈ U .

Obviously, Wcb ∈ K

Ps,b . Hence we deduce

(V ∩Wb

) ∪Wcb ∈ K

Ps,b for all b ∈ U . (4.29)

Because Wb = W (SN)× {b} and Ps,b is concentrated on Wb, we deduce from (4.29)

V ∈ KPs,b for all b ∈ U .

Then we obtain

V ∈⋂

b∈UK

Ps,b. (4.30)

From (4.22), (4.30), and P∞Br (U ) = 1, we obtain (4.24). This completes the proof. ��

For (X,B) as above, we assume (X,B) satisfies (IFC)s with Fms . We set

Fms (X,B) = Fm

s (Bm,Xm∗)⊕ Xm∗. (4.31)

Then we see Fms (X,B) ∈Wsol. By construction, (Fm

s (X,B),B) satisfies the SDE inintegral form such that, for i = 1, . . . ,m,

Fm,is (X,B)t = si +

∫ t

0σ i (Fm

s (X,B))udBiu +∫ t

0bi (Fm

s (X,B))udu, (4.32)

where Fms = (Fm,1

s , . . . , Fm,ms , Xm+1, Xm+2, . . .) by construction. We write

F∞s (X,B) = limm→∞ Fm

s (X,B) inWsol under Ps (4.33)

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if F∞s (X,B) ∈ Wsol and limits (4.34)–(4.36) converge in W (S) for Ps-a.s. for alli ∈ N:

limm→∞ Fm,i

s (X,B) = F∞,is (X,B), (4.34)

limm→∞

∫ ·

0σ i (Fm

s (X,B))udBiu =∫ ·

0σ i (F∞s (X,B))udBi

u, (4.35)

limm→∞

∫ ·

0bi (Fm

s (X,B))udu =∫ ·

0bi (F∞s (X,B))udu. (4.36)

Lemma 4.2 Assume that (X,B) under Ps is a weak solution of (4.2)–(4.4) satisfying(IFC)s. Then the following hold.

1. The sequence of maps {Fms }m∈N is consistent in the sense that, for Ps-a.s.,

Fm,is (X,B) = Fm+n,i

s (X,B) for all 1 ≤ i ≤ m, m, n ∈ N. (4.37)

Furthermore, (4.33) holds and the map F∞s is well defined.2. (X,B) is a fixed point of F∞s in the sense that, for Ps-a.s.,

(X,B) = (F∞s (X,B),B). (4.38)

3. F∞s (·, b) is Tpath(SN)s,b-measurable for P∞Br -a.s.b, where

Tpath(SN)s,b = Tpath(SN)

Ps,b. (4.39)

F∞s is I -measurable, where I is given by (4.21) and (4.22).

Proof The consistency (4.37) is clear because (X,B) under Ps is a weak solution of(4.2)–(4.4) satisfying (IFC)s. (4.38) is immediate from the consistency (4.37). Wehave thus obtained (1) and (2).

LetWm be the completion ofWm with respect to Ps as before. From Definition 3.9and (4.31) we see Fm

s is Wm-measurable. Clearly,

Wm ⊃ Wn for m ≤ n.

Hence, Fns areWm-measurable for all n ≥ m. Combining this with (4.33), we see that

the limit function F∞s is Wm-measurable for each m ∈ N. Then F∞s is {∩∞m=1Wm}-measurable. Hence F∞s is I -measurable from Lemma 4.1. The first claim in (3)follows from the second and the definition of I . We have thus obtained (3). ��

The next theorem reveals the relation between the existence and pathwise unique-ness of strong solutions and tail triviality of the labeled path space W (SN). Thedefinition of the strong solution staring at s is given by Definition 3.6 with replacementof (3.3)–(3.5) by (4.2)–(4.4).

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Recall that (X,B) is a continuous process defined on (Ω,F , Ps, {Ft }) satisfying(4.1) introduced at the beginning of this section, and Ps is the distribution of (X,B)under Ps. From Ps, we set Ps,b as (4.12) and T {1}

path(SN; ·) as (4.10).

Theorem 4.2 1. Assume that (X,B) under Ps is a weak solution of (4.2)–(4.4) satis-fying (IFC)s. Then (X,B) under Ps is a strong solution of (4.2)–(4.4) if and onlyif (Tpath1)s holds.

2. Let X and X′ be strong solutions of (4.2)–(4.4) defined on (Ω,F , Ps, {Ft }) withthe same {Ft }-BrownianmotionB. Assume that (X,B) and (X′,B) under Ps satisfy(IFC)s. Then,

Ps(X = X′) = 1 (4.40)

if and only if for P∞Br -a.s.b

T {1}path(S

N; Ps,b) = T {1}path(S

N; P ′s,b). (4.41)

Here P ′s,b is defined by (4.11) and (4.12) by replacing X by X′.

Proof We prove (1). We note that by Lemma 4.2 (2)

(X,B) = (F∞s (X,B),B) for Ps-a.s. (4.42)

The fixed point property (4.42) is a key of the proof. We shall utilize the structureX = F∞s (X,B). By assumption (X,B) under Ps is a weak solution of (4.2)–(4.4).Then (F∞s (X,B),B) under Ps is a weak solution by (4.42).

Suppose (Tpath1)s. ThenTpath(SN) is Ps,b-trivial for P∞Br -a.s.b. By Lemma 4.2 (3),we see F∞s (·,b) is Tpath(SN)s,b-measurable for P∞Br -a.s. b. From these, we see thatF∞s (·,b) under Ps,b is a constant. Hence, F∞s (X,B) under Ps becomes a function inB. So we write, under Ps,

F∞s (X,B) = Fs(B). (4.43)

We next prove that Fs is a B(P∞Br )-measurable function. Let A ∈ B(W (SN)) andset A′ = (F∞s )−1(A). From (4.13), (4.42), and (4.43), we deduce that there existsNsuch that P∞Br (N ) = 0 and that

Ps,b(A′) = Ps((F

∞s (X,B),B) ∈ A′|B = b)

= Ps((Fs(B),B) ∈ A′|B = b)

= 1A′((Fs(b),b)) for all b /∈ N

= 1A(Fs(b)) = 1Fs−1(A)(b). (4.44)

Note that Ps,b(A′) isB(W0(RdN))-measurable in b because Ps,b is the regular condi-

tional probability of Ps induced by B. Hence from (4.44) we see Fs−1(A) ∈ B(P∞Br ).This implies Fs is a B(P∞Br )-measurable function.

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Similarly, we can prove Fs to be a Bt (P∞Br )/Bt -measurable function for each t .Indeed, let X[0,t] = {Xs}0≤s≤t and B[0,t] = {Bs}0≤s≤t be given by the restriction ofthe time parameter set [0,∞) to [0, t]. Then (X[0,t],B[0,t]) is a weak solution of (4.2)–(4.4) on [0, t]. The filtered space of (X[0,t],B[0,t]) is taken as (Ω,Ft , Ps, {Fs}0≤s≤t ).

We set F [0,t]s (B) = {Fs(B)s}0≤s≤t . Then from (4.42) and (4.43) we see under Ps

F [0,t]s (B) = X[0,t]. (4.45)

Applying the argument for (X,B) to (X[0,t],B[0,t]), we obtain the counterparts to(4.42) and (4.43) of (X[0,t],B[0,t]), that is, we have functions F∞,t

s of (X[0,t],B[0,t])and Ft

s of B[0,t] satisfying under Ps

(X[0,t],B[0,t]) = (F∞,ts (X[0,t],B[0,t]),B[0,t]), (4.46)

F∞,ts (X[0,t],B[0,t]) = Ft

s (B[0,t]). (4.47)

Hence from (4.45), (4.46), and (4.47) we have under Ps

F [0,t]s (B) = Fts (B

[0,t]). (4.48)

We set W0,t = {w ∈ C([0, t];RdN);w0 = 0}. We regard P∞Br as a probabil-ity measure on (W0,t ,B(W0,t )) and denote by B(W0,t ) the completion of B(W0,t )

with respect to P∞Br . We replace B(P∞Br ) and B(W (SN)) with B(W0,t ) and Bt ,respectively. Then, applying the argument above to (X[0,t],B[0,t]), we deduce Ft

s isB(W0,t )/Bt -measurable. Because we can naturally identifyB(W0,t ) withBt (P∞Br ),we see Ft

s isBt (P∞Br )/Bt -measurable. This and (4.48) imply F [0,t]s isBt (P∞Br )/Bt -measurable. Hence, we conclude Fs isBt (P∞Br )/Bt -measurable immediately.

Collecting these we see Fs(B) = F∞s (X,B) under Ps is a strong solution. Inparticular, the function Fs is a strong solution in the sense of Definition 3.6.

Suppose that F∞s (X,B) under Ps is a strong solution. Then by Definition 3.6 thereexists a function Fs such that F∞s (X,B) = Fs(B) for Ps-a.s. By (4.11) and (4.12) wehave Ps,b = Ps(X ∈ ·|B = b) for given b. Hence the distribution of F∞s (w,b) underPs,b is the delta measure δz concentrated at a non-random path z, say. Therefore, wededuce that Tpath(SN)s,b is Ps,b-trivial for P∞Br -a.s.b. We have thus obtained (1).

We proceed with the proof of (2).We suppose (4.40). Then the image measures Ps,b and P ′s,b defined by (4.12) for

P∞Br -a.s.b are the same. We thus obtain (4.41).We next suppose (4.41). By assumption (X,B) and (X′,B) under Ps are strong

solutions. Hence from (1) we see Tpath(SN) is trivial with respect to Ps,b and P ′s,b forP∞Br -a.s.b. Combining this with (4.41) we obtain for P∞Br -a.s.b

Ps,b|Tpath(SN) = P ′s,b|Tpath(SN) (4.49)

and in particular

Tpath(SN)s,b = Tpath(S

N)′s,b. (4.50)

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Here we set Tpath(SN)′s,b by (4.39) by replacing Ps,b with P ′s,b.Let Ps,b and P

′s,b be defined by (4.13) for (X,B) and (X

′,B) under Ps, respectively.Then from (4.49) we easily see for P∞Br -a.s.b

Ps,b|Tpath(SN)×{b} = P′s,b|Tpath(SN)×{b}. (4.51)

Here we set Tpath(SN)× {b} = Tpath(SN)× σ [{b}].Let I and I ′ be the σ -fields defined by (4.22) for (X,B) and (X′,B) under Ps,

respectively. Then we obtain from (4.49) and (4.51) combined with (4.14) and (4.15)

I = I ′ (4.52)

and for P∞Br -a.s. b

KPs,b = K

P′s,b . (4.53)

Let F∞s and F ′∞s be as Lemma 4.2 for (X,B) and (X′,B) under Ps, respectively.Then by (4.52) and Lemma 4.2 (3) both F∞s and F ′∞s areI -measurable. From (4.50)and Lemma 4.2 (3) we have both F∞s (·,b) and F ′∞s (·,b) areTpath(SN)s,b-measurablefor P∞Br -a.s. b.

LetU , V ∈ I be subsets with Ps(U ) = P ′s(V ) = 1 such that F∞s and F ′∞s satisfy(4.38) on U and V , respectively. From (4.38) we have F∞s (w,b) = w on U andPs(U ) = 1. Then we take a version of F∞s under Ps such that F∞s (w,b) = w onU ∪ V . Similarly, we take a version of F ′∞s under P ′s such that F ′∞s (w,b) = w onU ∪ V . We thus have

F∞s (w,b) = w = F ′∞s (w,b) for (w,b) ∈ U ∪ V .

Hence for Ps- and P ′s -a.s.

F∞s = F ′∞s . (4.54)

We then deduce from (4.54) for P∞Br -a.s. b

F∞s (·,b) = F ′∞s (·,b) for Ps,b- and P ′s,b-a.s. (4.55)

We recall that F∞s (·,b) and F ′∞s (·,b) are Tpath(SN)s,b-measurable for P∞Br -a.s. b,and that Tpath(SN)s,b is Ps,b- and P ′s,b-trivial for P∞Br -a.s.b. From these, we write

Fs(b) = F∞s (·,b), F ′s(b) = F ′∞s (·,b). (4.56)

Then the strong solutions X and X′ are given by X = Fs(B) and X′ = F ′s(B). Putting(4.49), (4.55), and (4.56) together, we obtain Fs(b) = F ′s(b) for P∞Br -a.s. b. ThisdeducesX = Fs(B) = F ′s(B) = X′ under Ps, which yields (4.40). This concludes (2).

��

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Let (X,B) be a continuous process defined on (Ω,F , Ps, {Ft }) satisfying (4.1) asbefore. Recall the notion of pathwise uniqueness starting at s under the constraint of(IFC)s given by Definition 3.5 and Remark 3.4 (2).

Theorem 4.3 Assume (Tpath2)s. Then the pathwise uniqueness of weak solutions start-ing at s under the constraint of (IFC)s and (Tpath1)s holds.

Proof Let (X,B) and (X′,B) be weak solutions of (4.2)–(4.4) satisfying (IFC)s and(Tpath1)s with the same Brownian motion B. Then (X,B) and (X′,B) are strongsolutions by Theorem 4.2 (1). By (Tpath2)s we see (4.41) holds. Hence applyingTheorem 4.2 (2) we obtain (4.40). From this we deduce the pathwise uniqueness ofweak solutions starting at s under the constraint of (IFC)s and (Tpath1)s. ��Proof of Theorem 4.1 The claim (1) follows from Theorem 4.2 (1) immediately.

Let (X, B) be any weak solution of (4.2)–(4.4) satisfying (IFC)s and (Tpath1)s.Then from (1) we deduce that (X, B) becomes a strong solution with Fs such thatX = Fs(B).

Because the distribution of B coincides with that of B, (Fs(B),B) and (Fs(B), B)have the same distribution. Hence (Fs(B), B) is a weak solution of (4.2)–(4.4) satis-fying (IFC)s. By assumption, (Tpath1)s and (Tpath2)s hold. Then Theorem 4.3 yields

Ps(Fs(B) = Fs(B)) = 1.

Thus, we deduce Fs = Fs for P∞Br -a.s. Hence we obtain

X = Fs(B) = Fs(B).

This implies (3.7).Let B′ be any {F ′

t }-Brownian motion defined on (Ω ′,F ′, P ′, {F ′t }). Because B′

and B have the same distribution, (Fs(B′),B′) and (Fs(B),B) have the same distri-bution. Hence (Fs(B′),B′) is a weak solution of (4.2)–(4.4) satisfying (IFC)s and(Tpath1)s. This implies Fs(B′) = F∞s (Fs(B′),B′) similarly as the argument at thebeginning of the proof. Then by Theorem 4.2 (1) we deduce that Fs(B′) is a strongsolution. We have thus completed the proof of (2). ��

We conclude this section with two notions of solutions.

Definition 4.2 (IFC solution) A continuous process (X,B) under Ps is called an IFCsolution of (4.2)–(4.4) if (X,B) under Ps is a weak solution of (4.2)–(4.4) satisfying(IFC)s. Also, the distribution of an IFC solution is called an IFC solution.

We introduce a new notion of solutions of ISDEs. The notion is an equivalent notionof the weak solution of (4.2)–(4.4) in terms of an asymptotic infinite system offinite-dimensional SDEs with consistency (AIFC). We do not use the notion of AIFCsolutions in the present paper; still, we expect that it will be useful to solve ISDEs.

Definition 4.3 A continuous process (X,B) under Ps is called an AIFC solution of(4.2)–(4.4) if (X,B) under Ps satisfies (IFC)s and (4.33). Also, the distribution of anAIFC solution is called an AIFC solution.

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Remark 4.4 We note that IFC solutions in Definition 4.2 are always AIFC solutions.Conversely, we can construct an IFC solution of (4.2)–(4.4) from an AIFC solution.Indeed, assuming that (X,B) under Ps is an AIFC solution of (4.2)–(4.4) and lettingF∞s be as (4.33), we see (F∞s (X,B),B) under Ps is a weak solution of (4.2)–(4.4).This is obvious from (4.32) and (4.34)–(4.36). Then (F∞s (X,B),B) under Ps is anIFC solution. Thus, constructing an IFC solution and, in particular, a weak solutionare reduced to constructing an AIFC solution.

5 Triviality ofTpath(SN): second tail theorem (Theorem 5.1)

Let Tpath(SN) be the tail σ -field of the labeled path space W (SN) introduced in (4.9).The purpose of this section is to prove triviality of Tpath(SN) under distributions ofIFC solutions, which is a crucial step in constructing a strong solution as we saw inTheorem 4.1. This step is very hard in general because W (SN) is a huge space and itstail σ -field Tpath(SN) is topologically wild. To overcome the difficulty, we introducea sequence of well-behaved tail σ -fields, and deduce triviality of Tpath(SN) from thatof T (S) under the stationary distribution of unlabeled dynamics X step by step alongthis sequence of tail σ -fields.

The space S is a tiny infinite-dimensional space compared with SN and W (SN).Hence, S enjoys a nice probability measure μ unlike SN andW (SN). We can take μ tobe associated with S-valued stochastic dynamics satisfying the μ-absolute continuitycondition (AC) and the no big jump condition (NBJ). This fact is important to thederivation.

Let λ be a probability measure on (S,B(S)). We write w = {wt } ∈ W (S). Let Pλbe a probability measure on (W (S),B(W (S))) such that

Pλ ◦ w−10 = λ. (5.1)

If necessary, we extend the domain of Pλ using completion of measures. We call Pλa lift dynamics of λ if Pλ and λ satisfy (5.1). For a given random point field λ thereexist many lift dynamics of it. We shall take a specific lift dynamics given by (5.2).

LetX be an SN-valued continuous process on (Ω,F , P, {Ft }). We set upath(X) =X =∑i∈N δXi as before. We assume that the associated unlabeled process upath(X) isan S-valued continuous process such that P ◦ u(X0)

−1 = λ and define Pλ by

Pλ := (P ◦ X−1) ◦ u−1path = P ◦ upath(X)−1 = P ◦ X−1. (5.2)

Let μ be a probability measure on (S,B(S)) and let WNE(Ss.i.) be as (2.9). We makethe following assumptions originally introduced in Sect. 3.2.(TT) μ is tail trivial.(AC) Pλ ◦ w−1t ≺ μ for all 0 < t <∞.(SIN) Pλ(WNE(Ss.i.)) = 1.(NBJ) P({mr ,T (X) <∞}) = 1 for all r , T ∈ N.

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We denote by Ps the regular conditional probability such that

Ps = P(·|X0 = s). (5.3)

Let B be a continuous process defined on the filtered space (Ω,F , P, {Ft }). Let Psdenote the distribution of (X,B) under Ps:

Ps = Ps ◦ (X,B)−1. (5.4)

Let Ps,b be the regular conditional probabilities of Ps such that

Ps,b = Ps(X ∈ · |B = b). (5.5)

Then we easily see

Ps = P((X,B) ∈ ·|X0 = s),

Ps,b = P(X ∈ · |X0 = s, B = b) = Ps(w ∈ ·|b).We assume (X,B) under (Ω,F , P, {Ft }) is a weak solution of ISDE (4.2)–(4.3).Then (X,B) under (Ω,F , Ps, {Ft }) is aweak solution of ISDE (4.2)–(4.4) for P◦X0-a.s. s. Hence we can apply the results in Sect. 4 to (X,B) under Ps.

We now state the main theorem of this section.

Theorem 5.1 (Second tail theorem) Assume that μ and Pλ satisfy (TT) and (AC) forμ, (SIN), and (NBJ). Assume that there exists a label l such that l ◦ u(s) = s forP ◦ X−10 -a.s. s. Assume that Ps is a weak solution of (4.2)–(4.4) satisfying (IFC)sfor P ◦ X−10 -a. s. s. Then Ps satisfies (Tpath1)s for P ◦ X−10 -a.s. s. Furthermore, ISDE(4.2)–(4.4) satisfies (Tpath2)s for P ◦ X−10 -a.s. s.

To explain the strategy of the proof, we introduce the notions of cylindrical tailσ -fields on W (SN) and W (S). We set

T = {t = (t1, . . . , tm) ; 0 < ti < ti+1 (1 ≤ i < m), m ∈ N}. (5.6)

We remark that we exclude t1 = 0 in the definition of T in (5.6).Let πc

r be the projection πcr : S→ S such that πc

r (s) = s(· ∩ Scr ). For w = {wt } ∈W (S) and t = (t1, . . . , tm) ∈ T we set πc

r (wt) = (πcr (wt1), . . . , π

cr (wtm )) ∈ Sm .

Let Tpath(S) be the cylindrical tail σ -field of W (S) such that

Tpath(S) =∨

t∈T

∞⋂

r=1σ [πc

r (wt) ]. (5.7)

Let Tpath(SN) be the cylindrical tail σ -field of W (SN) defined as

Tpath(SN) =

∨

t∈T

∞⋂

n=1σ [wn∗

t ]. (5.8)

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Here wn∗t = (wn∗

ti )mi=1 for t ∈ T, where wn∗

t = (wkt )∞k=n+1 for w = (wn)n∈N.

We shall prove Theorem 5.1 along with the following scheme:

T (S)(Step I)−−−−−−−→

Theorem 5.2(TT), (AC)

Tpath(S)(Step II)−−−−−−−−−−−−−−→

Theorem 5.3(TT), (AC), (SIN), (NBJ)

Tpath(SN)

(Step III)−−−−−−−→Theorem 5.4

(IFC)s

Tpath(SN)

μ Pλ = P ◦ X−1 P ◦ X−1 Ps,b.

We shall prove triviality of each tail σ -field in the scheme under the distribution putunder the tail σ -field. The theorems under the arrows correspond to each step and theconditions there indicate what are used at each passage. Our goal is to obtain trivialityof Tpath(SN) under Ps,b for P∞Br -a.s.b and for P ◦ X−10 -a.s. s.

Remark 5.1 Theorem 5.2 needs only (TT) and (AC) for μ. Theorem 5.3 needs only(TT) and (AC) for μ, (SIN), and (NBJ). In Theorems 5.2 and 5.3, we do not useany properties of ISDE. That is, Ps is not necessary a weak solution of (4.2)–(4.4)satisfying (IFC)s. Such generality of these theorems would be interesting and usefulin other aspects. Theorem 5.4 requires the property of weak solutions of (4.2)–(4.4)satisfying (IFC)s unlike Theorems 5.2 and 5.3. The map F∞s in (4.33) given by aweak solution of (4.2)–(4.4) satisfying (IFC)s plays an important role in the proof ofTheorem 5.4.

Remark 5.2 An example of unlabeled pathw such thatmr ,T (lpath(w)) = ∞ is given byRemark 3.10. Such a large fluctuationmr ,T (lpath(w)) = ∞ of unlabeled pathw yieldsdifficulty to control Tpath(SN) by the cylindrical tail σ -field Tpath(S) of unlabeledpaths. Hence we assume (NBJ).

5.1 Step I: FromT (S) to Tpath(S)

Let T (S) be the tail σ -field of S and λ a probability measure on S with lift dynamicsPλ as before. We shall lift μ-triviality of T (S) to Pλ-triviality of the cylindrical tailσ -field Tpath(S) of W (S). For a probability Q on B(W (S)), we set

T {1}path(S; Q) = {X ∈ Tpath(S) ; Q(X ) = 1}.

We state the main theorem of this subsection.

Theorem 5.2 Assume (TT) for μ and (AC) for μ and Pλ. The following then hold.

1. Tpath(S) is Pλ-trivial.

2. T {1}path(S; Pλ) depends only on μ.

Remark 5.3 Theorem 5.2 (2) means, if μ is invariant under the dynamics,

T {1}path(S; Pλ) = T {1}

path(S; Pμ).

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In case of Theorem3.1,we can takePμ as the distribution of the diffusion inLemma9.1starting from the stationary measure μ. We shall identify the distribution Pλ|Tpath(S)

in

Proposition 5.1. From this we can specify T {1}path(S; Pλ).

For a probability ν on B(S), we set T {1}(S; ν) = {A ∈ T (S) ; ν(A) = 1}.Lemma 5.1 Under the same assumptions as Theorem 5.2, the following hold.

1. T (S) is Pλ ◦ w−1t -trivial for each t > 0.2. T {1}(S; Pλ ◦ w−1t ) = T {1}(S;μ) for each t > 0.

Proof (1) is obvious. Indeed, let A ∈ T (S) and suppose Pλ ◦ w−1t (A) > 0. Thenμ(A) > 0 by (AC). Hence from (TT), we deduce μ(A) = 1. This combined with(AC) implies Pλ ◦w−1t (A) = 1. We thus obtain (1). We deduce from (TT), (AC), and(1) that A ∈ T {1}(S; Pλ ◦ w−1t ) if and only if A ∈ T {1}(S;μ). This implies (2). ��We extend Lemma 5.1 for multi-time distributions. LetT t

path(S) be the cylindrical tailσ -field conditioned at t = (t1, . . . , tn) ∈ T:

T tpath(S) =

∞⋂

r=1σ [πc

r (wt) ]. (5.9)

Using (5.9) we can rewrite Tpath(S) as Tpath(S) =∨t∈TT tpath(S).

We set Su = Sn , μ⊗u = μ⊗n , and |u| = n if u = (u1, . . . , un) ∈ T.

Lemma 5.2 (1) Let u = (ui )pi=1, v = (v j )

qj=1 ∈ T such that u p < v1 and set (u, v) =

(u1, . . . , u p, v1, . . . , vq) ∈ T. Then,

T upath(S)×T v

path(S) ⊂ T (u,v)path (S). (5.10)

(2) For each C ∈ T (u,v)path (S) and y ∈ Sv, the set C[y] is T u

path(S)-measurable. Here

C[y] = {x ∈ Su ; (x, y) ∈ C}.

Furthermore, μ⊗u(C[y]) is a T vpath(S)-measurable function in y.

Proof We easily deduce for each r ∈ N

σ [πcr (wu) ] × σ [πc

r (wv) ] = σ [πcr (w(u,v)) ]. (5.11)

Then, taking intersections in the left-hand side step by step, we obtain

∞⋂

s=1σ [πc

s (wu) ] ×∞⋂

t=1σ [πc

t (wv) ] ⊂ σ [πcr (w(u,v)) ]. (5.12)

Hence, taking the intersection in r on the right-hand side of (5.12), we deduce (5.10).

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Next, we prove (2). By assumption and from the obvious inclusion, we see that

C ∈ T (u,v)path (S) ⊂ σ [πc

s (w(u,v)) ] for all s ∈ N.

Then, from (5.11), we see that

C ∈ σ [πcs (wu) ] × σ [πc

s (wv) ] for all s ∈ N.

Hence, C[y] is σ [πcs (wu) ]-measurable for all s ∈ N. From this we deduce that C[y] is

T upath(S)-measurable. The second claim in (2) can be proved similarly. ��

To simplify the notation, we set Pwtλ = Pλ ◦ w−1t for t ∈ T.

Proposition 5.1 Under the same assumptions as Theorem 5.2, the following hold.For each t = (t1, . . . , tn) ∈ T,

Pwtλ |T t

path(S)= μ⊗t|T t

path(S). (5.13)

In particular, for any C ∈ T tpath(S), the following identity with dichotomy holds.

Pwtλ (C) = μ⊗t(C) ∈ {0, 1}. (5.14)

Proof We prove Proposition 5.1 by induction with respect to n = |t|.If n = 1, then the claims follow from Lemma 5.1. Here we used (TT) and (AC) to

apply Lemma 5.1.Next, suppose that the claims hold for n − 1. Let u and v be such that (u, v) = t

and that 1 ≤ |u|, |v| < n. We then see that |u| + |v| = |t| = n. We shall prove thatthe claims hold for t with |t| = n in the sequel.

Assume that C ∈ T tpath(S). Then we deduce from Lemma 5.2 that C[y] ∈ T u

path(S)

and thatμ⊗u(C[y]) is aT vpath(S)-measurable function in y. By the induction hypothesis,

we see that Pλ(wu ∈ C[y]) is T vpath(S)-measurable in y and that the following identity

with dichotomy holds.

Pλ(wu ∈ C[y]) = μ⊗u(C[y]) ∈ {0, 1} for each y ∈ Sv. (5.15)

By disintegration, we see that

Pλ(wu ∈ ·) =∫

SvPλ(wu ∈ ·|wv = y)Pwv

λ (dy). (5.16)

By the induction hypothesis, we see that Pλ(wu ∈ A) ∈ {0, 1} for all A ∈ T upath(S).

Hence let A ∈ T upath(S) and suppose that

Pλ(wu ∈ A) = a, (5.17)

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where a ∈ {0, 1}. We then obtain from (5.16) and Pλ(wu ∈ A) ∈ {0, 1} that

Pλ(wu ∈ A|wv = y) = a for Pwvλ -a.s. y. (5.18)

From (5.16), (5.17), and (5.18), we deduce that for each A ∈ T upath(S)

Pλ(wu ∈ A|wv = y) = Pλ(wu ∈ A) for Pwvλ -a.s. y. (5.19)

We next remark that

Pλ(wv = y|wv = y) = 1 for Pwvλ -a.s. y. (5.20)

We refer the reader to the corollary of Theorem 3.3 on page 15 of [9] for the generalresult from which (5.20) is derived.

For all A ∈ T upath(S) and B ∈ B(Sv) we deduce from (5.19) and (5.20) that

Pλ(wu ∈ A, wv ∈ B) =∫

SvPλ(wu ∈ A, wv ∈ B |wv = y) Pwv

λ (dy)

=∫

BPλ(wu ∈ A|wv = y) Pwv

λ (dy) by (5.20)

=∫

BPλ(wu ∈ A) Pwv

λ (dy) by (5.19)

= Pλ(wu ∈ A) Pwvλ (B). (5.21)

From (5.21) and the monotone class theorem, we deduce that Pwtλ = P(wu,wv)

λ

restricted on T upath(S)×B(Sv) is a product measure. We thus obtain

(Pwtλ ,T u

path(S)×B(Sv)) = (Pwuλ |T u

path(S)× Pwv

λ |B(Sv),Tupath(S)×B(Sv)). (5.22)

That is,

Pwtλ |T u

path(S)×B(Sv) = Pwuλ |T u

path(S)× Pwv

λ |B(Sv).

In particular, from (5.22), we deduce that for Pwvλ -a.s.y

Pλ(wu ∈ A |wv = y ) = Pλ(wu ∈ A ) for all A ∈ T upath(S). (5.23)

For any C ∈ B(St), we deduce that

Pλ(wt ∈ C) =∫

SvPλ(wu ∈ C[y], wv = y |wv = y) Pwv

λ (dy)

=∫

SvPλ(wu ∈ C[y] |wv = y ) Pwv

λ (dy). (5.24)

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Here, we used Pλ(wv = y|wv = y) = 1 for Pwvλ -a.s.y, which follows from (5.20).

Assume C ∈ T tpath(S). Then from Lemma 5.2 (2), we obtain

C[z] ∈ T upath(S) for all z ∈ Sv. (5.25)

Hence from (5.23) and (5.25), we see that for Pwvλ -a.s.y

Pλ(wu ∈ C[z] |wv = y ) = Pλ(wu ∈ C[z] ) for all z ∈ Sv. (5.26)

We emphasize that (5.26) holds for all z ∈ Sv. Hence we can take z = y in (5.26) forPwvλ -a.s.y. This yields for Pwv

λ -a.s.y

Pλ(wu ∈ C[y] |wv = y ) = Pλ(wu ∈ C[y] ). (5.27)

From (5.24), (5.27), and (5.15) we obtain

Pλ(wt ∈ C) =∫

SvPλ(wu ∈ C[y])Pwv

λ (dy) =∫

Svμ⊗u(C[y])Pwv

λ (dy). (5.28)

From Lemma 5.2 (2), we see that μ⊗u(C[y]) is a T vpath(S)-measurable function in y.

Combining this with the induction hypothesis (5.13) for |v| < n, we obtain

∫

Svμ⊗u(C[y])Pwv

λ (dy) =∫

Svμ⊗u(C[y])Pwv

λ |T vpath(S)

(dy)

=∫

Svμ⊗u(C[y])μ⊗v|T v

path(S)(dy)

=∫

Svμ⊗u(C[y])μ⊗v(dy) = μ⊗t(C). (5.29)

From (5.28) and (5.29), we obtain (5.13) and the equality in (5.14) for |t| = n.We deduce μ⊗t-triviality of T t

path(S) from Lemma 5.2 (2) and the equality

∫

Svμ⊗u(C[y])μ⊗v(dy) = μ⊗t(C) (5.30)

by induction with respect to n = |t|. Indeed, because μ⊗u(C[y]) ∈ {0, 1} by theassumption of induction and μ⊗u(C[y]) is T v

path(S)-measurable in y by Lemma 5.2

(2), we obtain μ⊗t-triviality of T tpath(S) from (5.30). Then from this we see that

μ⊗t(C) ∈ {0, 1} holds. This completes the proof. ��Proof of Theorem 5.2 For t = (t1, . . . , tn) ∈ T and Ai ∈ T (S), let

X = {w ∈ W (S) ; wti ∈ Ai (i = 1, . . . , n)}. (5.31)

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Let CTpath(S) denote the family of the elements of Tpath(S) of the form (5.31).CTpath(S) is then Pλ-trivial by Proposition 5.1. The first claim (1) follows from thisand the monotone class theorem. The second claim (2) immediately follows from theequality (5.14) in Proposition 5.1. ��

5.2 Step II: From Tpath(S) to Tpath(SN)

Let Tpath(S) and Tpath(SN) be as (5.7) and (5.8), respectively:

Tpath(S) =∨

t∈T

∞⋂

r=1σ [πc

r (wt) ], Tpath(SN) =

∨

t∈T

∞⋂

n=1σ [wn∗

t ].

This subsection proves the passage from Tpath(S) to Tpath(SN).For a given label l, we can define the label map lpath : WNE(Ss.i.)→ W (SN) by

(2.10). Let l be a label such that l ◦ u(w0) = w0 for P ◦ X−1-a.s. By (SIN) the maplpath for l is well defined for Pλ-a.s. Then lpath ◦ upath(w) = w holds for P ◦ X−1-a.s.Hence from (5.2) we deduce

Pλ ◦ l−1path = P ◦ X−1. (5.32)

Assumption (NBJ) is a key to constructing the lift from the unlabeled path spaceto the labeled path space. We use (NBJ) in Lemma 5.3 to control the fluctuation ofthe trajectory of the labeled path X. Indeed, we see the following.

Lemma 5.3 Assume (SIN) and (NBJ). Then

l−1path(Tpath(SN)) ⊂ Tpath(S) under Pλ. (5.33)

Here in general for sub σ -fields G and H on (Ω,F , P) we write G ⊂H under Pif G ⊂ H holds up to P , that is, for each A ∈ G there exists an A′ ∈ H such thatP(A" A′) = 0, where A" A′ = {A∪ A′}\{A∩ A′} denotes the symmetric difference.

Proof Let mr ,T be as (3.13). By (5.32) we can rewrite (NBJ) as

Pλ({w ; mr ,T (lpath(w)) <∞}) = 1 for all r , T ∈ N. (5.34)

From (5.34) we easily see

Pλ

( ∞⋂

r=1{w ; mr ,T (lpath(w)) <∞}

)= 1 for all T ∈ N. (5.35)

Let A ∈⋃t∈T⋂∞

n=1 σ [wn∗t ]. Then there exists a t ∈ T such that

A ∈∞⋂

n=1σ [wn∗

t ].

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Let T ∈ N such that tk < T , where t = (t1, . . . , tk). Then we deduce for each r ∈ N

l−1path(A) ∩ {w ; mr ,T (lpath(w)) <∞}∈ σ [πc

r (wt)] ∩ {w ; mr ,T (lpath(w)) <∞}, (5.36)

where F ∩ A = {F ∩ A; F ∈ F } for a σ -field F and a subset A.Combining (5.35) and (5.36), we obtain under Pλ

l−1path(A) = l−1path(A)⋂{ ∞⋂

r=1{w ; mr ,T (lpath(w)) <∞}

}by (5.35)

=∞⋂

r=1

{l−1path(A)

⋂{w ; mr ,T (lpath(w)) <∞}

}

∈∞⋂

r=1

{σ [πc

r (wt)]⋂{w ; mr ,T (lpath(w)) <∞}

}by (5.36)

=∞⋂

r=1σ [πc

r (wt)] by (5.35).

By (5.7)we see⋂∞

r=1 σ [πcr (wt)] ⊂ Tpath(S). Thus, for arbitraryA ∈⋃t∈T

⋂∞n=1 σ [wn∗

t ],we see l−1path(A) ∈ Tpath(S) under Pλ from the argument above. Hence we obtain

l−1path

(⋃

t∈T

∞⋂

n=1σ [wn∗

t ])⊂ Tpath(S) under Pλ. (5.37)

Applying the monotone class theorem, we then deduce (5.33) from (5.37). ��For a probability measure Q on W (SN), we set

T {1}path(S

N; Q) = {A ∈ Tpath(SN); Q(A) = 1}.

We set two conditions:(Cpath1) Tpath(SN) is P ◦ X−1-trivial.(Cpath2) T

{1}path(S

N; P ◦ X−1) depends only on μ ◦ l−1.Theorem 5.3 Assume (TT) for μ and (AC) for μ and Pλ. Assume (SIN) and (NBJ).Then (Cpath1) and (Cpath2) hold.

Proof LetA ∈ Tpath(SN). Then we see l−1path(A) ∈ Tpath(S) under Pλ from Lemma 5.3.

Theorem 5.2 (1) deduces that Tpath(S) is Pλ-trivial. Hence we have

Pλ(l−1path(A)

)∈ {0, 1}. (5.38)

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We see from (5.32)

Pλ(l−1path(A)

)= P ◦ X−1(A). (5.39)

Combining (5.38) and (5.39), we obtain (Cpath1).

From Lemma 5.3 we see that T {1}path(S

N; Pλ◦ l−1path) depends only on T{1}path(S; Pλ) and

lpath. From Theorem 5.2 (2) we see that T {1}path(S; Pλ) depends only on μ. Recall that

lpath(w) is defined for all w ∈ WNE(Ss.i.) and lpath is unique for a given l. Collectingthese, we have(Cpath2’) T {1}

path(SN; Pλ ◦ l−1path) depends only on μ and l.

For given μ ◦ l−1 and μ we see the label l is uniquely determined for μ-a.s. Thatis, if l is a label such that μ ◦ l−1 = μ ◦ l−1, then l(s) = l(s) for μ-a.s. s. It is clearthat μ is uniquely determined by μ ◦ l−1 because (μ ◦ l−1) ◦ u−1 = μ.

Combining this with (Cpath2’) we deduce

(Cpath2”) T {1}path(S

N; Pλ ◦ l−1path) depends only on μ ◦ l−1.From (5.32) we have Pλ ◦ l−1path = P ◦ X−1. Then (Cpath2”) is equivalent to (Cpath2).Thus we obtain (Cpath2). ��

5.3 Step III: From Tpath(SN) toTpath(SN): Proof of Theorem 5.1

Let Ps = Ps ◦ (X,B)−1 be as (5.4) and let (Cpath1) and (Cpath2) be as Theorem 5.3.We state the main result of this section.

Theorem 5.4 Assume that Ps is a weak solution of (4.2)–(4.4) satisfying (IFC)s forP ◦ X−10 -a. s. s.

1. Assume (Cpath1). Then Ps satisfies (Tpath1)s for P ◦ X−10 -a.s. s.2. Assume (Cpath1) and (Cpath2). Then (4.2)–(4.4) satisfies (Tpath2)s for P ◦X−10 -a.s.

s.

Triviality in (Cpath1) and (Cpath2) is with respect to the anneal probability measureP ◦X−1. The pair of assumptions (Tpath1)s and (Tpath2)s is its quenched version. SoTheorem 5.4 derives the quenched triviality from the annealed one. Another aspectof Theorem 5.4 is the passage of triviality of the cylindrical tail σ -field of the labeledpath space to that of the full tail σ -field of the labeled path space.

Let Ps,b = Ps(X ∈ · |B = b) be as (5.5). To simplify the notation, we set

Υ = P ◦ (X0,B)−1. (5.40)

Let F∞s be the map in (4.33). Such a map F∞s exists for P ◦X−10 -a.s.s because Ps is aweak solution of (4.2)–(4.4) satisfying (IFC)s for P ◦X−10 -a.s.s. Then forΥ -a.s. (s,b)

F∞s (w,b) = w for Ps,b-a.s.w (5.41)

and the map Fs in Theorem 4.1 is given by Fs(b) = F∞s (w,b). Recall that F∞sis I -measurable by Lemma 4.2 (3). F∞s is however not necessary Tpath(SN) ×

123


B(W0(RdN))-measurable. We define the Wsol-valued map F∞s,b on a subset of Wsol

as

F∞s,b(·) = F∞s (·,b). (5.42)

The map F∞s,b is defined for Ps,b-a.s. Let Wfixs,b = {w ∈Wsol ; F∞s,b(w) = w}.

Lemma 5.4 Make the same assumption as Theorem 5.4. Then, Ps,b(Wfixs,b) = 1 for

Υ -a.s. (s, b).

Proof We deduce Lemma 5.4 from (5.5), (5.41), and (5.42) immediately. ��We next recall the notion of a countable determining class.Let (U ,U ) be a measurable space and let P be a family of probability measures

on it. Let U0 be a subset of ∩P∈PU P , where U P is the completion of the σ -fieldU with respect to P . We call U0 a determining class underP if any two probabilitymeasures P and Q inP are equal if and only if P(A) = Q(A) for all A ∈ U0. Herewe extend the domains of P and Q to ∩P∈PU P in an obvious manner. Furthermore,U0 is said to be a determining class of the measurable space (U ,U ) ifP can be takenas the set of all probability measures on (U ,U ). A determining classU0 is said to becountable if its cardinality is countable.

It is known that a Polish space X equipped with the Borel σ -field B(X) has acountable determining class. If we replaceB(X)with a sub-σ -field G , the measurablespace (X ,G ) does not necessarily have any countable determining class in general.One of the difficulties to carry out our scheme is that measurable spaces W (S) andW (SN) equipped with tail σ -fields do not have any countable determining classes. Inthe sequel, we overcome this difficulty using F∞s,b and Wfix

s,b.

As SN is a Polish space with the product topology,W (SN) becomes a Polish space.Hence, there exists a countable determining class V of (W (SN),B(W (SN))). We cantake such a class V as follows. Let S1 be a countable dense subset of SN, and

U = A [{Ur (s); 0 < r ∈ Q, s ∈ S1}].

Here A [·] denotes the algebra generated by ·, and Ur (s) is an open ball in SN withcenter s and radius r . We also take a suitable metric defining the same topology of thePolish space SN. We note that U is countable because the subset {Ur (s); 0 < r ∈Q, s ∈ S1} is countable. Let

V =∞⋃

k=1{(wt)

−1(A) ; A ∈ U k, t ∈ (Q ∩ (0,∞))k}. (5.43)

We then see that V is a countable determining class of (W (SN),B(W (SN))).

Lemma 5.5 Make the same assumption as Theorem 5.4. Then for each V ∈ V and forΥ -a.s. (s, b)

(F∞s,b)−1(V)

⋂Wfix

s,b ∈ Tpath(SN)s,b. (5.44)

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Here Tpath(SN)s,b is the completion of the σ -field Tpath(SN) with respect to Ps,b.

Proof Let Tpath(SN)s,b be as (4.39). Then from Lemma 4.2 (3), we deduce that F∞s,bis a Tpath(SN)s,b-measurable function for Υ -a.s. (s,b). Then for Υ -a.s. (s,b)

(F∞s,b)−1(V) ∈ Tpath(SN)s,b.

Hence for Υ -a.s. (s,b)

(F∞s,b)−1(V)⋂

Wfixs,b ∈ Tpath(S

N)s,b⋂

Wfixs,b. (5.45)

Here for a σ -field F and a subset A, we set F ∩ A = {F ∩ A; F ∈ F } as before.Suppose w ∈ (F∞s,b)−1(V)

⋂Wfix

s,b. Then we see F∞s,b(w) ∈ V from w ∈(F∞s,b)−1(V) and w = F∞s,b(w) from w ∈ Wfix

s,b. Hence w ∈ V⋂

Wfixs,b. We thus

obtain

(F∞s,b)−1(V)⋂

Wfixs,b ⊂ V

⋂Wfix

s,b.

Suppose w ∈ V⋂

Wfixs,b. Then F∞s,b(w) = w ∈ V. Hence w ∈ (F∞s,b)−1(V)

⋂Wfix

s,b.Thus,

V⋂

Wfixs,b ⊂ (F∞s,b)−1(V)

⋂Wfix

s,b.

Combining these, we see that, for Υ -a.s. (s,b),

(F∞s,b)−1(V)⋂

Wfixs,b = V

⋂Wfix

s,b. (5.46)

Because V ∈ V , there exists a t ∈ (Q ∩ (0,∞))k such that V = (wt)−1(A) for some

A ∈ U k . From V = (wt)−1(A) we have V ∈ σ [wt]. Hence we obtain

V⋂

Wfixs,b ∈ σ [wt]

⋂Wfix

s,b. (5.47)

Combining (5.46) and (5.47), we obtain for Υ -a.s. (s,b)

(F∞s,b)−1(V)⋂

Wfixs,b ∈σ [wt]

⋂Wfix

s,b. (5.48)

From (5.45) and (5.48) we deduce for Υ -a.s. (s,b)

(F∞s,b)−1(V)⋂

Wfixs,b ∈

{Tpath(S

N)s,b⋂

σ [wt]}⋂

Wfixs,b. (5.49)

We easily see for Υ -a.s. (s,b)

Tpath(SN)s,b

⋂σ [wt] ⊂ Tpath(S

N)s,b.

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This together with (5.49) yields

(F∞s,b)−1(V)⋂

Wfixs,b ∈ Tpath(S

N)s,b⋂

Wfixs,b. (5.50)

By Lemma 5.4 we have Ps,b(Wfixs,b) = 1. Then (5.50) implies (5.44) immediately. ��

Lemma 5.6 1. Assume (Cpath1). Then, for each A ∈ Tpath(SN),

Ps,b(A) ∈ {0, 1} for Υ -a.s. (s, b). (5.51)

2. Assume (Cpath1) and (Cpath2). Then T{1}path, Υ (S

N) depends only on μ ◦ l−1, where

T {1}path, Υ (S

N) = {A ∈ Tpath(SN) ; Ps,b(A) = 1 for Υ -a.s. (s, b)

}.

Proof We first prove (1). By construction we have a decomposition of P such that

P ◦ X−1(A) =∫

SN×W0(RdN)

Ps,b(A) Υ (dsdb). (5.52)

Let BA = {(s,b) ; 0 < Ps,b(A) < 1}. Suppose that (5.51) is false. Then there existsan A ∈ Tpath(SN) such that Υ (BA) > 0. Hence we deduce that

0 <

∫

BAPs,b(A)Υ (dsdb) < 1. (5.53)

From (5.52) and (5.53) together with Υ (BA) > 0 we deduce 0 < P ◦ X−1(A) < 1.This contradicts (Cpath1). Hence we obtain (1).

We next prove (2). Suppose that A ∈ T {1}path(S

N; P ◦ X−1). Then from (5.51) and

(5.52), we deduce that Ps,b(A) = 1 forΥ -a.s. (s,b). Furthermore, T {1}path(S

N; P ◦X−1)depends only on μ ◦ l−1 by (Cpath2). Collecting these, we obtain (2). ��

We next prepare a general fact on countable determining classes.

Lemma 5.7 1. Let (U ,U ) be a measurable space with a countable determiningclass V = {Vn}n∈N. Let ν be a probability measure on (U ,U ). Suppose thatν(Vn) ∈ {0, 1} for all n ∈ N. Then, ν(A) ∈ {0, 1} for all A ∈ U . Furthermore,there exists a unique Vν ∈ U such that Vν ∩ A ∈ {∅, Vν} and ν(A) = ν(A ∩ Vν)for all A ∈ U .

2. In addition to the assumptions in (1), we assume {u} ∈ U for all u ∈ U. Thenthere exists a unique a ∈ U such that ν = δa.

Proof We first prove (1). Let N (1) = {n ∈ N; ν(Vn) = 1}. Then we take

Vν =⎛

⎝⋂

n∈N (1)

Vn

⎞

⎠⋂⎛

⎝⋂

n /∈N (1)

V cn

⎞

⎠ .

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Clearly, we obtain ν(Vν) = 1.Let A ∈ U . Suppose that Vν ∩ A /∈ {∅, Vν}. We then cannot determine the value

of ν(Vν ∩ A) from the value of ν(Vn) (n ∈ N). This yields a contradiction. Hence,Vν ∩ A ∈ {∅, Vν}. If Vν ∩ A = ∅, then ν(A) = 0. If Vν ∩ A = Vν , then ν(A) ≥ν(Vν) = 1. We thus complete the proof of (1).

We next prove (2). Since ν(Vν) = ν(Vν ∩U ) = ν(U ) = 1, we have #Vν ≥ 1.Suppose #Vν = 1. Then there exists a unique a ∈ U such that Vν = {a}. This

combined with (1) yields that ν(A) = ν(A ∩ Vν) = ν(A ∩ {a}) ∈ {0, 1} for allA ∈ U . Hence we see that ν = δa .

Next suppose #Vν ≥ 2. Then Vν can be decomposed into two non-empty mea-surable subsets Vν = V 1

ν + V 2ν because {u} ∈ U for all u ∈ U . From (1), we have

proved that ν(V 1ν ), ν(V

2ν ) ∈ {0, 1} and that Vν is unique. Hence such a decomposition

Vν = V 1ν + V 2

ν yields a contradiction. This completes the proof of (2). ��Let Ps and Ps,b be as (5.4) and (5.5), respectively.

Lemma 5.8 Make the same assumption as Theorem 5.4. Assume (Cpath1). Then Ps,bis concentrated at a single path w = w(s, b) ∈ W (SN), that is,

Ps,b = δw(s,b) (5.54)

In particular, w is a function of b under Ps for P ◦ X−10 -a.s. s.

Proof From (5.41) and (5.42), we deduce F∞s,b(w) = w for Ps,b-a.s.w forΥ -a.s. (s,b).Hence by (5.5) we deduce for Υ -a.s. (s,b)

Ps,b ◦ (F∞s,b)−1 = Ps,b. (5.55)

Let V be the countable determining class given by (5.43). Then, we deduce fromLemmas 5.4, 5.5, 5.6, and the definition of Ps,b that, for all V ∈ V ,

Ps,b ◦ (F∞s,b)−1(V) =Ps,b((F∞s,b)

−1(V))

=Ps,b((F∞s,b)−1(V)

⋂Wfix

s,b) ∈ {0, 1} for Υ -a.s. (s,b).

(5.56)

Because V is countable, we deduce from (5.56) that, for Υ -a.s. (s,b),

Ps,b ◦ (F∞s,b)−1(V) ∈ {0, 1} for all V ∈ V . (5.57)

We denote by B(W (SN))s,b the completion of B(W (SN)) with respect to Ps,b.From (5.57) and Lemma 5.7, we obtain for Υ -a.s. (s,b),

Ps,b ◦ (F∞s,b)−1(A) ∈ {0, 1} for all A ∈ B(W (SN))s,b.

Furthermore, for Υ -a.s. (s,b), there exists a unique Us,b ∈ B(W (SN))s,b such that

Us,b ∩ A ∈ {∅,Us,b}, Ps,b(A) = Ps,b(A ∩ Us,b)

123


for all A ∈ B(W (SN))s,b. Hence we deduce that the set Us,b consists of a singlepoint {w(s,b)} for some w = w(s,b) ∈ W (SN). Then the probability measure Ps,b ◦(F∞s,b)−1 is concentrated at the single path w = w(s,b) ∈ W (SN), that is,

Ps,b ◦ (F∞s,b)−1 = δw(s,b).

In particular, w is a function of b under Ps because of (5.3), (5.4), and (5.5). Thiscombined with (5.55) implies (5.54). ��Proof of Theorem 5.4 From (5.54), we immediately obtain (Tpath1)s. Hence we obtain(1). Let w(s,b) and Us,b be as Lemma 5.8. From Lemma 5.6 (2), we deduce that theset Us,b depends only on μ ◦ l−1. In particular, w(s,b) depends only on μ ◦ l−1. Thiscompletes the proof of (2). ��Proof of Theorem 5.1 By assumption, (TT) and (AC) for μ, (SIN), and (NBJ) aresatisfied, and furthermore, Ps is a weak solution of (4.2)–(4.4) satisfying (IFC)s forP ◦ X−10 -a. s. s. Then we apply Theorem 5.3 to obtain (Cpath1) and (Cpath2). Hencewe deduce from Theorem 5.4 that (Tpath1)s and (Tpath2)s hold for P ◦ X−10 -a.s. s. ��

6 Proof of Theorems 3.1–3.2

This section is devoted to proving Theorems 3.1–3.2. Throughout this section, S isRd or a closed set satisfying the assumption in Sect. 3, and S is the configuration

space over S. We have established two tail theorems: Theorems 4.1 and 5.1. ThenTheorems 3.1 and 3.2 are immediate consequences of them.

Proof of Theorem 3.1 By assumption, μ is tail trivial and (X,B) under P is a weaksolution satisfying (IFC), (AC) for μ, (SIN), and (NBJ). Hence we deduce (Tpath1)sand (Tpath2)s for P ◦ X−10 -a.s. s from the second tail theorem (Theorem 5.1). Wetherefore conclude from the first tail theorem (Theorem 4.1) that (3.3)–(3.4) has aunique strong solution Fs starting at s for P ◦ X−10 -a.s. s under the constraints of(IFC)s and (Tpath1)s.

Because the family of strong solutions {Fs} is given by a weak solution (X,B)under P , {Fs} satisfies (MF). Recall that P{Fs} = P ◦ X−1 by (3.15). Hence P{Fs}satisfies (IFC), (AC) for μ, (SIN), and (NBJ).

We next check (i) and (ii) in Definition 3.11.Let (X, B) under P be a weak solution in (i). Then (X, B) under P satisfies (IFC),

(AC) for μ, (SIN), and (NBJ) by assumption. This implies (X, B) under Ps satisfies

(IFC)s, (SIN)s, and (NBJ)s for P ◦ X−10 -a.s. s by Fubini’s theorem. Because Fs is aunique strong solution starting at s for P ◦X−10 -a.s. s under the constraints of (IFC)s

and (Tpath1)s and P ◦ X−10 ≺ P ◦ X−10 , we obtain (i). Condition (ii) is clear becauseFs is a unique strong solution. This completes the proof. ��Proof of Theorem 3.2 By (3.21) μa

Tail is tail trivial. Hence (TT) for μaTail is satisfied.

By assumption, we have (X,B) under Pa satisfies (AC) for μaTail for μ-a.s. a.

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We see that, forμ-a.s. a, (IFC), (SIN), and (NBJ) for (X,B) under Pa follow fromFubini’s theorem and those for (X,B) under P .

Indeed, from (3.20), (3.25), and (3.26) we obtain

P =∫

Paμ(da). (6.1)

Then from (SIN) and (NBJ) for (X,B) under P and (6.1) we deduce

1 = P(X ∈ WNE(Ss.i.)) =∫

SPa(X ∈ WNE(Ss.i.))μ(da),

1 = P(mr ,T (X) <∞) =∫

SPa(mr ,T (X) <∞)μ(da).

Hence Pa(X ∈ WNE(Ss.i.)) = Pa(mr ,T (X) < ∞) = 1 for μ-a.s. a. This implies(SIN) and (NBJ) for (X,B) under Pa holds for μ-a.s. a.

By disintegration of P ◦ X−10 ,

P ◦ X−10 =∫

Pa ◦ X−10 μ(da). (6.2)

We set Ps = P(·|X0 = s). By definition, (IFC) for (X,B) under P implies (IFC)sfor (X,B) under Ps for P ◦ X−10 -a.s. s. Then by (6.2) this implies (IFC)s for (X,B)under Ps holds for Pa ◦ X−10 -a.s. s and for μ-a.s. a. We easily see for μ-a.s. a

Ps = Pas for Pa ◦ X−10 -a.s. s.

Hence, for μ-a.s. a, we have (IFC)s for (X,B) under Pas for Pa ◦ X−10 -a.s. s. Then,

for μ-a.s. a, we deduce (IFC) for (X,B) under Pa.We have thus seen that, for μ-a.s. a, (X,B) under Pa fulfills the assumptions of

Theorem 3.1. Hence Theorem 3.2 follows from Theorem 3.1. ��

7 The Ginibre interacting Brownianmotion

In this section, we apply our theory to the special example of the Ginibre interactingBrownian motion and prove existence of strong solutions and pathwise uniqueness.Our proof is based on the idea explained in Introduction. Behind it there are two generaltheories called tail theorems. These two theories are robust and can be applied tovarious kinds of infinite-dimensional stochastic (differential) equationswith symmetrybeyond interacting Brownianmotions. The purpose of this section is to clarify the rolesof these two theories by applying them to the Ginibre interacting Brownian motion.

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7.1 ISDE related to the Ginibre random point field

In Sect. 7.1, we introduce the Ginibre interacting Brownian motion and prepare theresult of the first step.

The Ginibre random point field μGin is a random point field on R2 whose n-point

correlation function ρnGin with respect to the Lebesgue measure is given by

ρnGin(x1, . . . , xn) = det[KGin(xi , x j )]ni, j=1, (7.1)

where KGin :R2 × R2→C is the kernel defined by

KGin(x, y) = π−1e−|x |22 − |y|

2

2 · ex y . (7.2)

Here we identify R2 as C by the obvious correspondence R

2 � x = (x1, x2) �→x1+

√−1x2 ∈ C, and y = y1−√−1y2 is the complex conjugate in this identification.

It is known thatμGin is translation and rotation invariant. Moreover,μGin(Ss.i.) = 1and μGin is tail trivial [2,19,29].

We next introduce the Dirichlet form associated with μGin and construct S-valueddiffusion. Let (E μGin ,DμGin◦ ) be a bilinear form on L2(S, μGin) defined by

DμGin◦ = { f ∈ D◦ ∩ L2(S, μGin) ; E μGin( f , f ) <∞},E μGin( f , g) =

∫

SD[ f , g]μGin(ds),

D[ f , g](s) = 1

2

∑

i

(∂si f , ∂si g)R2 .

Here s =∑i δsi , ∂si = ( ∂∂si1

, ∂∂si2

), and (·, ·)R2 denotes the standard inner product in

R2. f is defined before (2.2).

Lemma 7.1 ([27, Theorem 2.3])

1. The Ginibre random point field μGin is a (|x |2,−2 log |x − y|)-quasi Gibbs mea-sure.

2. (E μGin ,DμGin◦ ) is closable on L2(S, μGin).3. The closure (E μGin ,DμGin) of (E μGin ,DμGin◦ ) on L2(S, μGin) is a quasi-regular

Dirichlet form.4. There exists a diffusion {Ps}s∈S associated with (E μGin ,DμGin) on L2(S, μGin).

A family of probability measures {Ps}s∈S on (W (S),B(W (S))) is called a diffusionif the canonical process X = {Xt } under Ps is a continuous process with the strongMarkov property starting at s. Here Xt (w) = wt for w = {wt } ∈ W (S) by definition.X is adapted to {Ft }, where Ft = ∩νF ν

t and the intersection is taken over all Borelprobability measures ν, F ν

t is the completion of F+t = ∩ε>0Bt+ε(S) with respect

to Pν =∫Psν(ds). The σ -fieldBt (S) is defined by

Bt (S) = σ [ws; 0 ≤ s ≤ t]. (7.3)

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Furthermore, {Ps}s∈S is called stationary if it has an invariant probability measure.We refer to [20] for the definition of quasi-regular Dirichlet forms. We also refer

to [6] for the Dirichlet form theory.We recall the definition of capacity [Chapter 2.1 in [6]]. Denote by O the family of

all open subsets of S. Let (E ,D) be a quasi-regular Dirichlet form on L2(S, μ), andE1( f , f ) = E ( f , f )+ ( f , f )L2(S,μ). For B ∈ O we define

Capμ(B) ={infu∈LB E1(u, u), LB �= ∅∞ LB = ∅,

where LB = {u ∈ D : u ≥ 1, μ-a.e on B}, and we let for all set A ⊂ S

Capμ(A) = infA⊂B∈O

Capμ(B).

We call this one-capacity of A or simply the capacity of A.We recall the notion of quasi-everywhere and quasi-continuity. Let A be a subset

of S. A statement depending on s ∈ A is said to hold quasi-everywhere (q.e.) on A ifthere exists a set N ⊂ A of zero capacity such that the statement is true for everys ∈ A\N . When A = S, quasi-everywhere on S is simply said quasi-everywhere. Letu be an extended real valued function defined q.e. on S. We call u quasi-continuous ifthere exists for any ε > 0 an open set G ⊂ S such that Capμ(G) < ε and the restrictionu|S\G of u on S \ G is finite continuous.

We set the probability measure PμGin by

PμGin(A) =∫

SPs(A)μGin(ds) for A ∈ F . (7.4)

Under PμGin , the unlabeled process X is a μGin-reversible diffusion. Let CapμGin be thecapacity on the Dirichlet space (E μGin ,DμGin , L2(S, μGin)). Let Ss.i. and WNE(Ss.i.)be as (2.8) and (2.9), respectively .

Lemma 7.2 ([24,26]) PμGin(WNE(Ss.i.)) = 1 and CapμGin(Scs.i.) = 0.

Proof PμGin(WNE(S)) = 1 follows from [26, (2.10)]. Let Ss be as (2.8). Then by [24,Theorem 2.1], we see CapμGin(Scs) = 0. Clearly, μGin(Ss.i.) = 1. Combining theseimply the unlabeled diffusion never hits the set consisting of the finite configurations.We thus see the capacity of this set is zero. Hence we obtain CapμGin(Scs.i.) = 0. Inparticular, PμGin(W (Ss.i.)) = 1, which together with PμGin(WNE(S)) = 1 implies thefirst claim. ��

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Lemma 7.3 ( [26, Theorem 61, Lemma 72]) The Ginibre random point field μGin hasa logarithmic derivative dGin. Furthermore, dGin has plural expressions:

dGin(x, s) = 2 limr→∞

∑

|x−si |<r

x − si|x − si |2 , (7.5)

dGin(x, s) = −2x + 2 limr→∞

∑

|si |<r

x − si|x − si |2 . (7.6)

Here the convergence takes place in L ploc(μ

[1]Gin) for any 1 ≤ p < 2.

From Lemma 7.2 the process X = (Xi )i∈N = lpath(X) ∈ W (R2N) is well defined,where R

2N = (R2)N. The unlabeled process X is defined on the canonical filteredspace as the evaluation map Xt (w) = wt . That is, (Ω,F , {Ps}, {Ft }) is given byΩ = W (Ss.i.), F = B(W (Ss.i.)), {Ps} is the family of diffusion measures given byLemma 7.1 (4). {Ps} can be regarded as diffusion on Ss.i. by Lemma 7.2. The labeledprocess X = lpath(X) is thus defined on (Ω,F , {Ps}, {Ft }). The ISDE satisfied by Xis as follows.

Lemma 7.4 ([26, Theorem21, Theorem22])LetX = lpath(X) be the stochastic processdefined on (Ω,F , {Ps}, {Ft }) as above. Then there exists a set H such that

μGin(H) = 1, H ⊂ Ss.i.

and that X under Ps satisfies both ISDEs on R2N starting at each point s = l(s) ∈ l(H)

dXit = dBi

t + limr→∞

∑

|Xit−X j

t |<r , j �=i

X it − X j

t

|Xit − X j

t |2dt (i ∈ N), (1.5)

dXit = dBi

t − Xit dt + lim

r→∞∑

|X jt |<r , j �=i

X it − X j

t

|Xit − X j

t |2dt (i ∈ N). (1.6)

Furthermore, X ∈ W (u−1(H)). The process B = (Bi )∞i=1 in (1.5) and (1.6) is thesame and is the R

2N-valued, {Ft }-Brownian motion given by the formula

Bit = Xi

t − Xi0 −∫ t

0limr→∞

∑

|Xis−X j

s |<r , j �=i

X is − X j

s

|Xis − X j

s |2ds, (7.7)

Bit = Xi

t − Xi0 +∫ t

0Xisds −

∫ t

0limr→∞

∑

|X js |<r , j �=i

X is − X j

s

|Xis − X j

s |2ds. (7.8)

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We note that (X,B) above is a solution of the two ISDEs (1.5) and (1.6). We referto Definition 3.1 for the definition of the solution in Lemma 7.4, which is often calleda weak solution. We also note that the identity between (7.7) and (7.8) follows fromthe plural expressions of dGin in Lemma 7.3.

Remark 7.1 1. We cannot replace W (u−1(H)) by W (l(H)) in Lemma 7.4. Indeed,l(H) ⊂ u−1(H) and Xt /∈ l(H) for some 0 < t <∞.

2. The Brownian motion B = (Bi )∞i=1 in Lemma 7.4 is given by a functional ofX. Hence we write B(X) = (Bi (X))∞i=1. It is given by the martingale part of theFukushima decomposition of the Dirichlet process Xi

t − Xi0. Indeed, for (1.5) and

(1.6), Bi is given by (7.7) and (7.8), respectively.We note that the coordinate function xi does not belong to the domain of theunlabeled Dirichlet form even if locally. Hence we introduced in [26] the Dirichletspace of them-labeled process (X1, . . . , Xm,

∑j>m δX j ) to apply the Fukushima

decomposition to the coordinate function xi , where i ≤ m. The consistency of theDirichlet spaces plays a crucial role for this argument [25,26]. See Sect. 9.3 for thedefinition of the Dirichlet space of the m-labeled process and their consistency,which is given under the general situation.

3. The function in the coefficient in (1.6) belongs to the domain of the m-labeledDirichlet space locally. Indeed, regarded as a function on R

2 × S, we prove that itbelongs to the domain of one-labeled Dirichlet form locally (see Lemma 8.4). Bythe same argument we can prove that it is in the domain of them-labeled Dirichletform if we regard as a function on (R2)m × S in an obvious manner. Hence,by taking a quasi-continuous version of the function, the drift term becomes aDirichlet process. We thus see that the process

−Xit + lim

r→∞∑

|X jt |<r , j �=i

X it − X j

t

|Xit − X j

t |2

in the drift term is a continuous process. This makes the meaning of the drift termmore explicit because we usually take a predictable version of the coefficients (seepp 45–46 in [9]). The key point here is that the coefficient is in the domain of theDirichlet space. All examples in the present paper enjoy this property. We shallassume this in (C1)–(C2) and use this in the proof of Proposition 11.2.

7.2 Main result for the Ginibre interacting Brownianmotion

All above results in Sect. 7.1 belong to the first step explained in Introduction (Sect. 1).Our purpose in Sect. 7.2 is to obtain the existence of strong solutions and the path-wise uniqueness of the solution, which is the main result for the Ginibre interactingBrownian motion.

Let (Ω,F , Ps, {Ft }) be as Lemma 7.4. Let (X,B) be the solution of ISDEs (1.5)and (1.6) given by Lemma 7.4. Recall that X = lpath(X), where X is the canonicalprocess such that Xt (w) = wt and B is the {Ft }-Brownian motion given by (7.8). LetPμGin be as (7.4).

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Theorem 7.1 1. (X,B) under PμGin satisfies the same conclusions as Theorem 3.1,Corollaries 3.1, and 3.2 for ISDE (1.5).

2. There exists a subset H of Ssde such that μGin(H) = 1 and that (X,B) is a strongsolution of ISDE (1.5) starting at s = l(s) ∈ l(H) defined on (Ω,F , Ps, {Ft }).

3. The same statements as (1) hold for ISDE (1.6).4. The same statements as (2) hold for ISDE (1.6).5. Let (X′,B′) and (X′′,B′) under P ′ be weak solutions of (1.5) and (1.6) satisfying

(IFC), (AC) forμGin, (SIN), and (NBJ), respectively. Suppose that both solutionsare defined on the same filtered space (Ω ′,F ′, P ′, {F ′

t }) with the same {F ′t }-

Brownian motion B′ and that X′0 = X′′0 a.s. Then P ′(X′t = X′′t for all t) = 1.

We remark that ISDEs (1.5) and (1.6) are different ISDEs in general. Theorem 7.1(5) asserts that, if the unlabeled particles start from a support Ssde of μGin and ifthe label l is common, then these two labeled dynamics are equal all the time. Theintuitive explanation of this fact is as follows. One may regard the set u−1(Ssde) as asub-manifold of R

2N and the drift terms b1 and b2

b1

⎛

⎝xi ,∑

j �=iδx j

⎞

⎠ = limr→∞

∑

|xi−x j |<r

xi − x j|xi − x j |2 ,

b2

⎛

⎝xi ,∑

j �=iδx j

⎞

⎠ = −xi + limr→∞

∑

|x j |<r

xi − x j|xi − x j |2

of each ISDE are regarded as “tangential vectors on u−1(Ssde)”. In [26], it was shownthat both drifts are equal on u−1(Ssde). This implies the coincidence of ISDEs (1.5)and (1.6) on u−1(Ssde). Since the drift terms b1 and b2 are tangential, the solutionsstay in u−1(Ssde) all the time, which combined with the pathwise uniqueness of thesolutions of ISDEs yields (4).

We note that the unlabeled dynamics X areμGin-reversible because X is given by thesymmetric Dirichlet form (E μGin ,DμGin) on L2(S, μGin) (see Lemma 7.1 (4)). Hence,the distribution of Xt with initial distributionμGin satisfiesμGin ◦X−1t = μGin for each0 < t <∞. In contrast, the labeled dynamics Xt are trapped on a very thin subset ofthe huge space R

2N. We conjecture that the distribution of Xt is singular to the initialdistribution μGin ◦ l−1 for some t > 0.

8 Proof of Theorem 7.1

In Sect. 8, we prove the main theorem (Theorem 7.1) using Theorem 3.1.

123


8.1 Localization of coefficients and Lipschitz continuity

Recall that the labeled processX = (Xi )i∈N = lpath(X) is obtained from the unlabeledprocess X under PμGin . Set the m-labeled process (Xm,Xm∗) by

(Xm,Xm∗

) =(X1, . . . , Xm,

∞∑

i=m+1δXi

).

This correspondence is similar to the correspondence ofm-labeled path in the sense of(2.11) and (2.12). This relationship between the unlabeled process and the m-labeledprocess is called consistency.

The m-labeled process is associated with the m-labeled Dirichlet space given inSect. 10 for the m-Campbell measure μ[m]Gin of μGin. Here

μ[m]Gin(dxdy) = ρm

Gin(x)μGin,x(dy)dx,

where ρmGin is the m-point correlation function of μGin with respect to the Lebesgue

measure dx on R2m , and μGin,x is the reduced Palm measure conditioned at x ∈ R

2m .The m-labeled Dirichlet form is given by

E μ[m]Gin( f , g) =

∫

R2m×S

{1

2

m∑

i=1

(∂ f

∂xi,∂g

∂xi

)

R2+ D[ f , g]

}μ[m]Gin (dxdy) , (8.1)

where ∂/∂xi is the nabla in R2. This coincides with the Dirichlet form (9.12) with

d = 2, S = R2, and μ[m] = μ

[m]Gin. Furthermore, a(x, y) in (9.12) is taken to be the

2 × 2 unit matrix. We denote by Capμ[m]Gin the capacity given by the Dirichlet form

E μ[m]Gin .Let a = {aq}q∈N be an increasing sequence of increasing sequences aq =

{aq(r)}r∈N such that aq(r) < aq+1(r) and aq(r) < aq(r + 1) for all q, r ∈ N

and that limr→∞ aq(r) = ∞ for all q ∈ N. We take aq(r) ∈ N.Let K[aq] = {s ; s(Sr ) ≤ aq(r) for all r ∈ N}. Then K[aq] ⊂ K[aq+1] for all q ∈ N.

It is easy to see that K[aq] is a compact set in S for each q ∈ N. Let

K[a] =∞⋃

q=1K[aq].

We take aq(r) = qr2. Then because μGin is translation invariant, we have

μGin(K[a]) = 1.

We introduce an approximation of R2m × S consisting of compact sets. Let

S[m]s.i. ={(x, s) ∈ R

2m × S ; u(x)+ s ∈ Ss.i.},

123


where x = (x1, . . . , xm) and u(x) = ∑mi=1 δxi . Let a+q such that a+q (r) = aq(r + 1)

and

Rp,r(s) ={x ∈ Smr ; min

j �=k |x j − xk | ≥ 2−p, infl,i|xl − si | ≥ 2−p

}, (8.2)

where j, k, l = 1, . . . ,m, s =∑i δsi , and Smr = {x ∈ R2; |x | ≤ r}m . We set

H[a]p,q,r ={(x, s) ∈ S[m]s.i. ; x ∈ Rp,r(s), s ∈ K[a+q ]

}, (8.3)

H[a] =∞⋃

r=1H[a]r, H[a]r =

∞⋃

q=1H[a]q,r, H[a]q,r =

∞⋃

p=1H[a]p,q,r. (8.4)

Although Rp,r(s),H[a]p,q,r,H[a]q,r,H[a]q, and H[a] depend on m ∈ N, we suppressm from the notation. To simplify the notation we set N = N1 ∪ N2 ∪ N3, where

N1 = {r ∈ N}, N2 = {(q, r) ; q, r ∈ N}, N3 = {(p,q, r) ; p,q, r ∈ N} (8.5)

and for n ∈ N we define n+ 1 ∈ N such that

n+ 1 =

⎧⎪⎨

⎪⎩

(p+ 1,q, r) for n = (p,q, r) ∈ N3,

(q+ 1, r) for n = (q, r) ∈ N2,

r+ 1 for n = r ∈ N1.

(8.6)

We shall take a limit in n along with the order n �→ n+1. We write H[a]n = H[a]p,q,rfor n = (p,q, r) ∈ N3. We set H[a]n for n = (q, r) ∈ N2 and n = r ∈ N1 similarly.

We remark that H[a]n is compact for n ∈ N3. This property has critical importancein the proof of Proposition 8.1.

We set Sm,◦r = {|x | < r, x ∈ R

2}m . Let R◦p,r(s) be the open kernel of Rp,r(s):

R◦p,r(s) =

{x ∈ Sm,◦

r ; infj �=k |x j − xk | > 2−p, inf

l,i|xl − si | > 2−p

}. (8.7)

For n = (p,q, r) ∈ N3 we set

H[a]◦n = H[a]◦p,q,r :={(x, s) ∈ S[m]s.i. ; x ∈ R

◦p,r(s), s ∈ K[a+q ]

}, (8.8)

H[a]◦ =∞⋃

r=1H[a]◦r , H[a]◦r =

∞⋃

q=1H[a]◦q,r, H[a]◦q,r =

∞⋃

p=1H[a]◦p,q,r. (8.9)

Then H[a]◦n ⊂ H[a]n and H[a]◦n ∪ ∂H[a]◦n = H[a]n. Note that H[a]◦n has a compactclosure for each n ∈ N3. The next lemma gives a localization of them-labeled process.

123


Lemma 8.1 For each m ∈ N the following holds:

Ps(limn→∞ τH[a]◦n(X

m, Xm∗) = ∞)= 1 for μGin -a.s. s. (8.10)

Here τA denotes the exit time from a set A, (Xm, Xm∗) is the m-labeled process givenby (X,B) in Theorem 7.1, and

limn→∞ = lim

r→∞ limq→∞ lim

p→∞ . (8.11)

Proof Let a+ = {a+q }q∈N. Then from [23, Lemma 2.5 (4)], we obtain

limq→∞CapμGin(K[a+q ]c) = 0.

Then we see for μGin-a.s. s

Ps

(limq→∞ τK[a+q ](X) = ∞

)= 1. (8.12)

By definition we deduce τK[a+q ](X) ≤ τK[a+q ](Xm∗). This combined with (8.12) yields

Ps

(limq→∞ τK[a+q ](X

m∗) = ∞)= 1. (8.13)

FromLemma 7.2 we have PμGin(WNE(Ss.i.)) = 1. Then tagged particles neither collidenor explode. Hence we deduce for μGin-a.s. s

Ps(τS[m]s.i.

(Xm,Xm∗) = ∞)= 1, (8.14)

Ps(limr→∞ τSm,◦

r(Xm) = ∞

)= 1. (8.15)

From (8.14) we have for μGin-a.s. s

Ps

(limp→∞ τH[a]◦p,q,r(X

m,Xm∗) = τH[a]◦q,r(Xm,Xm∗)

)= 1. (8.16)

By (8.13) we see for μGin-a.s. s

Ps

(limq→∞ τH[a]◦q,r(X

m,Xm∗) = τH[a]◦r (Xm,Xm∗)

)= 1. (8.17)

From (8.14) and (8.15) we see for μGin-a.s. s

Ps(limr→∞ τH[a]◦r (X

m,Xm∗) = ∞)= 1 (8.18)

Putting (8.16), (8.17), and (8.18) together we conclude (8.10). ��

123


Let bm = (bm,i )mi=1 be the drift coefficient of the SDE describing Xm and let Bm

be the 2m-dimensional Brownian motion. Then

dXmt = dBm

t + bm(Xmt ,X

m∗t )dt (8.19)

and bm,i is given by

bm,i (x, s) = 1

2dμGin

⎛

⎝xi ,m∑

j=1, j �=iδx j + s

⎞

⎠ , x = (x1, . . . , xm).

Let bm be a version of bm = (bm,i )mi=1 with respect to μ[m]Gin. We shall prove in

Lemma 8.4 that bm,i are locally in the domain of the m-labeled Dirichlet form, andwe shall take a (locally) quasi-continuous version of bm later.

Let Π :R2m × S→ S be the projection such that (x, s) �→ s. Let {Im}m∈N be anincreasing sequence of closed sets in R

2m × S. Then by definition

Π(H[a]◦n ∩ Im) ={s ∈ S ; H[a]◦n ∩ Im ∩(R2m × {s}) �= ∅}. (8.20)

We set

〈H[a]◦n ∩ Im〉 =⋃

s∈Π(H[a]◦n∩Im)R◦p,r(s)× {s}. (8.21)

For n = (p,q, r) ∈ N3, let c2(m,n) be the constant such that 0 ≤ c2 ≤ ∞ and that

c2 = sup

{ |bm(x, s)− bm(y, s)||x− y| ; x �= y, s ∈ Π(H[a]◦n ∩ Im),

(x, s), (y, s) ∈ R◦p,r(s), (x, s) ∼p,r (y, s)

}. (8.22)

Here (x, s) ∼p,r (y, s)means x and y are in the same connected component of R◦p,r(s).

Proposition 8.1 There exist a μ[m]Gin-version bm of bm and an increasing sequence of

closed sets {Im}m∈N such that for each m ∈ N and n ∈ N3

c2(m, n) <∞, (8.23)

Ps(limn→∞ lim

m→∞ τ〈H[a]◦n∩Im〉(Xm, Xm∗) = ∞

)= 1 for μGin-a.s. s. (8.24)

Here τ〈H[a]◦n∩Im〉(Xm, Xm∗) denotes the exit time of (Xm, Xm∗) from the set 〈H[a]◦n ∩ Im〉and (Xm, Xm∗) is given by (X,B) in Theorem 7.1 starting at l(s).

We shall prove Proposition 8.1 in Sect. 8.2.From c2(m,n) < ∞ we see bm(x, s) restricted on each connected component of

R◦p,r(s) is Lipschitz continuous in x for each fixed s and that the Lipschitz constant

123


is bounded by c2(m,n). Thus, Proposition 8.1 implies a local Lipschitz continuity ofthe coefficients of the m-labeled SDE (8.70). Using this we shall obtain the pathwiseuniqueness and the existence of a strong solution of finite-dimensional SDEs.

The ideas of the proof of Proposition 8.1 are twofolds. One is the property that bm

are in the domain of Dirichlet forms, hence we can take a quasi-continuous version ofthem, which enable us to control the maximal normwith suitable cut off due to Im. Thesecond is the Taylor expansion of bm using the logarithmic interaction potential. Wenote here that differential gains the integrability of coefficients at infinity, which is akey point of the proof of Proposition 8.1. We refer to Sect. 11.3 for Taylor expansion,and to Sect. 8.2 for a specific calculation in case of the Ginibre interacting Brownianmotion.

8.2 Proof of local Lipschitz continuity of coefficients: Proposition 8.1

This section proves Proposition 8.1 to complete the proof of Theorem 7.1. For sim-plicity we prove only for m = 1. Let N3 be as (8.5). Let χn (n ∈ N3) be the cut-offfunction defined on R

2 × S introduced by (11.14) with m = 1. Then by Lemma 11.4the function χn satisfies the following.

χn(x, s) ={0 for (x, s) /∈ H[a]n+11 for (x, s) ∈ H[a]n , χn ∈ D [1]

Gin,

0 ≤ χn(x, s) ≤ 1, |∇xχn(x, s)|2 ≤ c16, D[χn, χn](x, s) ≤ c17. (8.25)

Here c16(n) and c17(n) are positive constants independent of (x, s) in Lemma 11.4,and D [1]

Gin is the domain of the Dirichlet form of the 1-labeled process (X1,X1∗) givenby (8.1). Moreover, ∇x = ( ∂

∂x1, ∂∂x2

) for x = (x1, x2) ∈ R2.

We refine the result in Lemma 7.3 from L ploc(μ

[1]Gin) (1 ≤ p < 2) to L2(χ2

nμ[1]Gin).

Lemma 8.2 dGin ∈ L2(χ2nμ

[1]Gin) holds and the convergence (7.5) and (7.6) takes place

in L2(χ2nμ

[1]Gin) for each n ∈ N3.

Proof From [26, Lemma 72], we deduce the convergence in L2loc(μ

[1]Gin) of the series

dGin1+ (x, s) := 2 limR→∞

∑

1≤|x−si |<R

x − si|x − si |2 ,

dGin2+ (x, s) := −2x + 2 limR→∞

∑

1≤|si |<R

x − si|x − si |2 .

123


By the definition of χn, this yields the convergence in L2(χ2nμ

[1]Gin). Because the weight

χn cuts off the sum around x , we easily see that

dGin1− (x, s) := 2∑

|x−si |<1

x − si|x − si |2 ∈ L2(χ2

nμ[1]Gin),

dGin2− (x, s) := −2x + 2∑

|si |<1

x − si|x − si |2 ∈ L2(χ2

nμ[1]Gin).

As dGin = dGin1+ + dGin1− = dGin2+ + dGin2− , we conclude Lemma 8.2. ��Let ϕR ∈ C∞0 (R2) be a cut-off function such that

0 ≤ ϕR(x) ≤ 1, |∇ϕR(x)| ≤ 2, ϕR(x) = ϕR(|x |) for all x ∈ R2, (8.26)

where ϕR ∈ C∞0 (R) is such that

ϕR(t) ={1 for |t | ≤ R,

0 for |t | ≥ R + 1for all t ∈ R. (8.27)

We set

dGinR (x, s) = −2x + 2∑

i

ϕR(si )x − si|x − si |2 . (8.28)

We write dGinR = t (dGinR,1,dGinR,2) and ∂p = ∂

∂xp, where x = t (x1, x2) ∈ R

2. Then astraightforward calculation shows for j, k, l ∈ {1, 2}

∂ jdGinR,k(x, s) = −2δ jk + 2

∑

i

ϕR(si )A jk(x − si )

|x − si |4 , (8.29)

∂ j∂kdGinR,l (x, s) = 2

∑

i

ϕR(si )∂ j

{Akl(x − si )

|x − si |4}, (8.30)

where A :R2→R4 is the 2× 2 matrix-valued function defined by

A(x) = [Ai j (x)]2i, j=1 =(−x21 + x22 −2x1x2−2x1x2 x21 − x22

)for x = (x1, x2) ∈ R

2. (8.31)

We easily see there exist constants c3 and c4 such that for all x ∈ R2

|Akl(x)||x |4 ≤ c3

|x |2 , (8.32)∣∣∣∣∂ j{ Akl(x)

|x |4}∣∣∣∣ ≤

c4|x |3 . (8.33)

123


We write dGin = t (dGin1 ,dGin2 ) similarly as dGinR = t(dGinR,1,d

GinR,2

).

Lemma 8.3 For any n = (p, q, r) ∈ N3 and j, k, l ∈ {1, 2}

dGin = limR→∞ dGinR in L2(χ2

nμ[1]Gin) (8.34)

∂ jdGink = lim

R→∞ ∂ jdGinR,k weakly in L2(χ2

nμ[1]Gin), (8.35)

∂ j∂kdGinl (x, s) = lim

R→∞∑

|si |≤R

∂ j

{Akl(x − si )

|x − si |4}

for each (x, s) ∈ H[a]n. (8.36)

Here the sum converges absolutely and uniformly in H[a]n. In particular,

sup{|∂ j∂kdGinl (x, s)| ; (x, s) ∈ H[a]n} <∞ (8.37)

and ∂ j∂kdGinl (x, s) is continuous in x for each s on H[a]n.

Proof We deduce (8.34) from Lemma 8.2, (8.26), (8.27), and (8.28) immediately.We next prove (8.35). For this purpose it is enough to show the summation term

in (8.29) is bounded in L2(χ2nμ

[1]Gin) as R →∞ for each n = (p,q, r) ∈ N3. Indeed,

we deduce from this that the sequence {∂ jdGinR,k} is relatively compact under the weak

convergence in L2(χ2nμ

[1]Gin) and that the limit points are unique by (8.34).

Let ρ1Gin and ρ

1Gin,x be the one-point correlation functions of μGin and μGin,x with

respect to the Lebesgue measure, respectively. Then

ρ1Gin(x) =

1

π, ρ1

Gin,x (s) =1

π− 1

πe−|x−s|2 . (8.38)

Here ρ1Gin(x) = 1

πfollows from (7.1) and (7.2), and ρ1

Gin,x (s) = 1π− 1

πe−|x−s|2

follows from the formula due to Shirai-Takahashi [36] such that the determinantalkernel KGin,x of Palm measure μGin,x is given by

KGin,x (y, z) = {KGin(y, y)KGin(z, z)− KGin(y, z)KGin(z, y)}/KGin(x, x).

Let H(x,p) = {s ∈ R2; |x − s| ≥ 2−p}, where p ∈ N and x ∈ R

2. Then we see

∣∣∣EμGin,x

[⟨1H(x,p)ϕR

Akl(x − ·)|x − ·|4 , s

⟩] ∣∣∣

=∣∣∣∫

R21H(x,p)(s)ϕR(s)

Akl(x − s)

|x − s|4 ρ1Gin,x (s)ds

∣∣∣

≤∣∣∣∫

R21H(x,p)(s)ϕR(s)

Akl(x − s)

|x − s|41

πds∣∣∣+∫

R21H(x,p)(s)

c3|x − s|2

1

πe−|x−s|2ds.

(8.39)

123


Here we used (8.38) and (8.32) for the last line. Note that by (8.26) and (8.31)

∫

R21H(0,p)(s)ϕR(s)

Akl(0− s)

|0− s|41

πds = 0. (8.40)

Then we easily see from (8.40)

∣∣∣∣∫

R21H(x,p)(s)ϕR(s)

Akl(x − s)

|x − s|41

πds

∣∣∣∣

=∣∣∣∣∫

R2

{1H(x,p)(s)

Akl(x − s)

|x − s|4 − 1H(0,p)(s)Akl(0− s)

|0− s|4}ϕR(s)

1

πds

∣∣∣∣

≤∫

R2

∣∣∣1H(x,p)(s)Akl(x − s)

|x − s|4 − 1H(0,p)(s)Akl(0− s)

|0− s|4∣∣∣1

πds. (8.41)

We set I (x,p) = {s ∈ R2; |x − s| < 2−p}. Then I (x,p) = R

2\H(x,p). Hence

1H(x,p)(s)Akl(x − s)

|x − s|4 − 1H(0,p)(s)Akl(0− s)

|0− s|4= 1H(x,p)(s)1I (0,p)(s)

Akl(x − s)

|x − s|4 − 1I (x,p)(s)1H(0,p)(s)Akl(0− s)

|0− s|4

+ 1H(x,p)(s)1H(0,p)(s)

(Akl(x − s)

|x − s|4 − Akl(0− s)

|0− s|4). (8.42)

It is clear that

sup|x |≤r+1

∫

R2

∣∣∣1H(x,p)(s)1I (0,p)(s)Akl(x − s)

|x − s|4∣∣∣1

πds <∞,

sup|x |≤r+1

∫

R2

∣∣∣1I (x,p)(s)1H(0,p)(s)Akl(0− s)

|0− s|4∣∣∣1

πds <∞,

sup|x |≤r+1

∫

|s|≤2(r+1)1H(x,p)(s)1H(0,p)(s)

∣∣∣Akl(x − s)

|x − s|4 − Akl(0− s)

|0− s|4∣∣∣1

πds <∞.

(8.43)

We write x = (x1, x2) ∈ R2. Using (8.33), we see for all |x | ≤ r+1 and |s| > 2(r+1)

∣∣∣Akl(x − s)

|x − s|4 − Akl(0− s)

|0− s|4∣∣∣ =∣∣∣∫ 1

0

2∑

j=1x j∂ j{ Akl(·)| · |4

}(t x − s)dt

∣∣∣

≤∫ 1

0

2∑

j=1|x j | sup

|x |≤r+1

{ c4|t x − s|3

}dt by (8.33)

≤ 2(r+ 1){c423

|s|3}.

123


Here we used |s|/2 < |t x−s| in the last line. This follows from 0 ≤ t ≤ 1, |x | ≤ r+1,and |s| > 2(r+1). Note that 1H(x,p)(s)1H(0,p)(s) = 1 for |x | ≤ r+1 and |s| > 2(r+1).Hence we deduce

sup|x |≤r+1

∫

|s|>2(r+1)1H(x,p)(s)1H(0,p)(s)

∣∣∣Akl(x − s)

|x − s|4 − Akl(0− s)

|0− s|4∣∣∣1

πds

≤ c424(r+ 1)

∫

|s|>2(r+1)1

|s|31

πds <∞. (8.44)

Collecting (8.41)–(8.44) together, we obtain

lim supR→∞

sup|x |≤r+1

∣∣∣∣∫

R21H(x,p)(s)ϕR(s)

Akl(x − s)

|x − s|41

πds

∣∣∣∣ <∞. (8.45)

Clearly, we have

∫

R21H(x,p)(s)

c3|x − s|2

1

πe−|x−s|2ds <∞. (8.46)

Putting (8.45) and (8.46) into (8.39), we obtain

lim supR→∞

sup|x |≤r+1

EμGin,x

[⟨1H(x,p)ϕR

Akl(x − ·)|x − ·|4 , s

⟩]<∞. (8.47)

From the inequalityVar[〈 f , s〉] ≤ ∫ | f |2ρ1(s)ds valid for determinantal randompointfields with Hermitian symmetric kernel, we have

lim supR→∞

sup|x |≤r+1

VarμGin,x

[⟨1H(x,p)ϕR

Akl(x − ·)|x − ·|4 , s

⟩]

≤ lim supR→∞

sup|x |≤r+1

∫

R2

∣∣∣∣1H(x,p)(s)ϕR(s)Akl(x − s)

|x − s|4∣∣∣∣2

ρ1Gin,x (s)ds

<∞ by (8.32) and (8.38). (8.48)

Putting (8.47) and (8.48) together we immediately deduce

lim supR→∞

sup|x |≤r+1

EμGin,x

[∣∣∣〈1H(x,p)ϕRAkl(x − ·)|x − ·|4 , s〉

∣∣∣2]<∞. (8.49)

From (8.25), (8.38), (8.49) and recalling n = (p,q, r) ∈ N3 we easily obtain

lim supR→∞

∫ ∣∣∣2∑

i

ϕR(si )Akl(x − si )

|x − si |4∣∣∣2χ2ndμ

[1]Gin <∞. (8.50)

123


Then from (8.29) and (8.50) with a simple calculation we see

lim supR→∞

∫|∂ jdGinR,k |2χ2

ndμ[1]Gin <∞.

Hence {∂ jdGinR,k} is relatively compact in the weak topology in L2(χ2nμ

[1]Gin) for each

n = (p,q, r) ∈ N3. This and Lemma 8.2 yield (8.35).We next prove (8.36) and (8.37). We see from (8.33)

∑

i

∣∣∣∂ j{Akl(x − si )

|x − si |4}∣∣∣ ≤ c4

∑

i

1

|x − si |3 . (8.51)

We have for any (x, s) ∈ H[a]n∑

i

1

|x − si |3 =∑

|si |<r+1

1

|x − si |3 +∞∑

t=r+2

∑

t−1≤|si |<t

1

|x − si |3

≤23ps(Sr+1)+∞∑

t=r+2

∑

t−1≤|si |<t

1

(|si | − r)3. (8.52)

Here we used |x − si | ≥ 2−p and |x | ≤ r. By a straightforward calculation we have

∞∑

t=r+2

∑

t−1≤|si |<t

1

(|si | − r)3

≤∞∑

t=r+2

s(St )− s(St−1)(t − 1− r)3

≤ limR→∞

{ s(SR)(R − 1− r)3

+R∑

t=r+3s(St−1)

{1

(t − 2− r)3− 1

(t − 1− r)3

}}. (8.53)

Recall that s(St ) ≤ aq(t) by (x, s) ∈ H[a]n. Then (8.51)–(8.53) yield

∑

i

∣∣∣∂ j{ Akl(x − si )

|x − si |4}∣∣∣ ≤ c42

3paq(r+ 1)

+ c4 limR→∞

{aq(R)

(R − 1− r)3+

R∑

t=r+3aq(t − 1)

{ 1

(t − 2− r)3− 1

(t − 1− r)3

}}.

(8.54)

Because aq(t) = kt2, the sum in (8.54) converges in H[a]n uniformly. Hence, weobtain (8.36) and (8.37) from (8.26), (8.30), and (8.54) immediately. The last claimfollows from (8.36) and the uniform convergence of the series in (8.36). ��

123


Recall that σ(x, s) = E and b(x, s) = 12d

Gin(x, s). Let D [1]Gin be the domain of the

1-labeled Dirichlet form of the Ginibre interacting Brownian motion.

Lemma 8.4 χndGin, χn∂ jd

Gin, χn∂ j∂kdGin ∈ D [1]

Gin ( j, k ∈ {1, 2}) for all n ∈ N3.

Proof We only prove χndGin ∈ D [1]

Gin because the other cases can be proved in a similarfashion. We set D[ f ] = D[ f , f ]. Recall that ∇x = ( ∂

∂x1, ∂∂x2

) = (∂1, ∂2). Then

E μ[1]Gin(χnd

Gin, χndGin) =

∫

R2×S1

2|∇x (χnd

Gin)|2 + D[χndGin]dμ[1]Gin.

From (8.25), Lemma 8.2, and Lemma 8.3, we deduce that

∫

R2×S|∇x (χnd

Gin)|2dμ[1]Gin ≤ 2∫

H[a]n+1{χ2

n |∇xdGin|2 + |∇xχn|2|dGin|2}dμ[1]Gin <∞.

We set ∂∂s j= ( ∂

∂s j,k)k=1,2. From (8.28) and dGinR = t (dGinR,1,d

GinR,2) we have

∂

∂s j,kdGinR,l (x, s) = 2(∂kϕR)(s j )

(x − s j )l|x − s j |2 − 2ϕR(s j )

Akl(x − s j )

|x − s j |4 .

Then from (8.25) and (8.32), we similarly deduce that

∫

R2×SD[χndGin]dμ[1]Gin ≤ 2

∫

R2×S{χ2nD[dGin] + D[χn] |dGin|2

}dμ[1]Gin

≤ 2∫

H[a]n+1

{D[dGin] + c217|dGin|2

}dμ[1]Gin

≤ 2∫

H[a]n+1

{c5

(∑

i

1

|x − si |4)+ c217|dGin|2

}dμ[1]Gin < ∞.

Here c5 is a finite, positive constant and c17(n) is the positive constant in Lemma 11.4.The finiteness of the integral in the last line follows from the translation invariance ofμGin and Lemma 8.2.

Combining these, we obtain E μ[1]Gin(χnd

Gin, χndGin) < ∞. We proved χnd

Gin ∈L2(μ

[1]Gin) in Lemma 8.2. Hence, we see χnd

Gin ∈ D [1]Gin. This completes the proof. ��

Proof of Proposition 8.1 For simplicity we prove the case m = 1; the general casefollows from the same argument.

By Lemma 8.3 we can take a version of ∂ j∂kdGinl (x, s) which is continuous in x

for each s on H[a]n. We always take this version and denote it by the same symbol∂ j∂kd

Ginl . By (8.37) we have a finite constant c6(n) (n ∈ N3) such that

c6 = sup{|∂ j∂kdGinl (x, s)| ; (x, s) ∈ H[a]n, j, k, l ∈ {1, 2}} <∞. (8.55)

123


We denote by f a quasi-continuous version of f ∈ D [1]Gin. From Lemma 8.4 we take

a quasi-continuous version of χndGinl being commutative with ∂k :

∂k˜

(χndGinl ) = ˜

(∂kχndGinl ). (8.56)

Then by definition there exists an increasing sequence {Im}∞m=1 of closed sets such

that ˜χnd

Ginl and ˜

(∂kχndGinl ) are continuous on Im for each m and that

limm→∞Capμ

[1]Gin(H[a]n+1\Im) = 0 for each n ∈ N3.

We used here χn = 0 on H[a]cn+1 by (8.25). Because H[a]n ⊂ H[a]n+1, we have

limm→∞Capμ

[1]Gin(H[a]n\Im) = 0 for each n ∈ N3. (8.57)

Note that χn = 1 on H[a]n by (8.25), H[a]n ↑ H[a], and μ[1]Gin(H[a]c) = 0. We set

˜dGinl (x, s) = ˜χnd

Ginl (x, s) for (x, s) ∈ H[a]n.

Thus˜dGinl is a version of dGinl such that˜dGinl and ∂k˜dGinl are continuous on H[a]n ∩ Im

for each n ∈ N3. Let n ∈ N3 and m ∈ N. Let c7(n,m) be the constant such that

c7 = sup{|˜dGinl (ξ, s)|, |∂k˜dGinl (ξ, s)| ; (ξ, s) ∈ H[a]n ∩ Im, k, l ∈ {1, 2}

}. (8.58)

Note that H[a]n is a compact set because n ∈ N3. Recall that Im is a closed set. Then

H[a]n ∩ Im is compact. Because˜dGinl and ∂k˜dGinl are continuous on H[a]n ∩ Im, these

are bounded on H[a]n ∩ Im. Thus we have c7 <∞.We suppose

s ∈ Π(H[a]n ∩ Im). (8.59)

We write n = (p,q, r) as before. Let ∼p,r be same as (8.22). Fix m and n. Thenthere exists a positive constant c8 < ∞ depending only on m and n such that thefollowing holds: For any (x, s), (ξ, s) ∈ H[a]n such that (x, s) ∼p,r (ξ, s), there exist{x1, . . . , xq} with x1 = ξ and xq = x such that (8.60) and (8.61) hold.

[x j , x j+1] × {s} ⊂ H[a]n+1 for all 1 ≤ j < q, (8.60)

where [x j , x j+1] ⊂ R2 denotes the segment connecting x j and x j+1,

q−1∑

j=1|x j − x j+1| ≤ c8|ξ − x |. (8.61)

123


Using (8.56) and Taylor expansion we deduce that for each 1 ≤ j < q

∂k˜dGinl (x j+1, s)− ∂k

˜dGinl (x j , s)

=∫ 1

0

2∑

p=1(x j+1,p − x j,p)∂p∂k

˜dGinl (t(x j+1 − x j )+ x j , s)dt . (8.62)

Here we set x j = (x j,1, x j,2) ∈ R2. Taking the sum of both sides of (8.62) over

j = 1, . . . , q − 1 and recalling xq = x and x1 = ξ we see

∂k˜dGinl (x, s)− ∂k

˜dGinl (ξ, s) =q−1∑

j=1

∫ 1

0

2∑

p=1(x j+1,p − x j,p)∂p∂k

˜dGinl (t(x j+1 − x j )+ x j , s)dt .

This together with (8.60) and (8.61) yields

|∂k˜dGinl (x, s)− ∂k˜dGinl (ξ, s)|

≤q−1∑

j=1

∫ 1

0|

2∑

p=1(x j+1,p − x j,p)∂p∂k

˜dGinl (t(x j+1 − x j )+ x j , s)|dt

≤ √2c6c8|x − ξ |≤ √2c6c82r by x, ξ ∈ Sr. (8.63)

Hence for each s satisfying (8.59), ∂k˜dGinl (x, s) is bounded in x with bound

|∂k˜dGinl (x, s)| ≤ |∂k˜dGinl (ξ, s)| + √2c6c82r. (8.64)

Because of (8.59), we can take (ξ, s) ∈ H[a]n ∩ Im. Then by (8.58)

|∂k˜dGinl (ξ, s)| ≤ c7. (8.65)

From (8.64) and (8.65) we obtain

|∂k˜dGinl (x, s)| ≤ c7 +√2c6c82r =: c9

The bound c9 depends only on m,n appearing in (8.59). Then

sup{|∂k˜dGinl (x, s)| ; (x, s) ∈ H[a]n, s ∈ Π(H[a]n ∩ Im)} ≤ c9. (8.66)

123


Using (8.56) and Taylor expansion again we deduce that for each 1 ≤ j < q

˜dGinl (x j+1, s)−˜dGinl (x j , s) =∫ 1

0

2∑

p=1(x j+1,p − x j,p)∂p

˜dGinl (t(x j+1 − x j )+ x j , s)dt .

Then from this and (8.66) we deduce in a similar fashion as (8.63)

|˜dGinl (x, s)−˜dGinl (ξ, s)| ≤√2c8c9|x − ξ |. (8.67)

Note that (8.67) holds for all (x, s) ∼p,r (ξ, s) ∈ H[a]n with s satisfying (8.59). Here(x, s) is not necessary inH[a]n∩ Im, whereas (ξ, s) ∈ H[a]n∩ Im. The constant

√2c8c9

depends only on m and n. Hence we deduce c2(m,n) <∞.By definition H[a]◦n ⊃ H[a]n−1. Then we easily deduce from (8.20) and (8.21)

〈H[a]◦n ∩ Im〉 ⊃ 〈H[a]n−1 ∩ Im〉 ⊃ H[a]n−1 ∩ Im. (8.68)

Let (X1,X1∗) = (X1,∑∞

i=2 δXi ) be the one-labeled process. By (8.57) we see

Ps(τH[a]n(X1,X1∗) = lim

m→∞ τH[a]n∩Im(X1,X1∗)) = 1

for all n ∈ N3. Then this yields

Ps(limn→∞ τH[a]n(X1,X1∗) = lim

n→∞ limm→∞ τH[a]n∩Im(X1,X1∗)

) = 1. (8.69)

Combining (8.68), (8.69), and Lemma 8.1, we obtain

Ps(limn→∞ lim

m→∞ τ〈H[a]◦n∩Im〉(X1,X1∗) = ∞

)≥ Ps

(limn→∞ lim

m→∞ τH[a]n−1∩Im(X1,X1∗) = ∞)

= Ps(limn→∞ τH[a]n−1(X1,X1∗) = ∞

)

≥ Ps(limn→∞ τH[a]◦n−1(X

1,X1∗) = ∞)

= 1.

Indeed, the first line follows from (8.68). The second line follows from (8.69). Theinequality in the third line is clear by H[a]n−1 ⊃ H[a]◦n−1. The last line is immediatefrom Lemma 8.1. We have thus proved (8.24). ��

8.3 A unique strong solution of SDEs with random environment

Throughout Sect. 8.3, (X,B) is the weak solution of (1.6) starting at s = l(s) definedon (Ω,F , Ps, {Ft }) obtained in Lemma 7.4. We write X = (Xi )∞i=1 and denote byBm the first m-components of the Brownian motion B = (Bi )∞i=1.

123


For each m ∈ N, we introduce the 2m-dimensional SDE of Ym for (X,B) under Pssuch that Ym = (Ym,i )mi=1 is an {Ft }-adapted, continuous process satisfying

dYm,it = dBi

t − Ym,it dt +

m∑

j=1j �=i

Ym,it − Ym, j

t

|Ym,it − Ym, j

t |2dt

+ limr→∞

∞∑

j=m+1,|X j

t |<r

Ym,it − X j

t

|Ym,it − X j

t |2dt, (8.70)

Ps(Ymt ∈ Smsde(t,X) for all t) = 1, (8.71)

Ym0 = lm(s). (8.72)

Here Smsde(t,w) is the set given by (3.9) with replacement of S by R2. Furthermore,

we set lm(s) = (li (s))mi=1 for l(s) = (li (s))∞i=1. For convenience we introduce a variantof notation of solution. Let Xm∗ = ∑∞

i=m+1 δXi as before. We say (Ym,Bm,Xm∗) isa solution of (8.70) if and only if (Ym,Bm,Xm∗) is a solution of (8.70).

Lemma 8.5 1. For each m ∈ N, the SDE (8.70)–(8.72) has a pathwise unique, weaksolution for μGin-a.s. s. That is, for μGin-a.s. s, arbitrary solutions (Ym,Bm, Xm∗)and (Y

m,Bm, Xm∗) of (8.70)–(8.72) defined on (Ω,F , Ps, {Ft }) satisfy

Ps(Ym = Ym) = 1. (8.73)

In particular, (Ym,Bm, Xm∗) coincides with (Xm,Bm, Xm∗) under Ps.2. Let (Zm, B

m, X

m∗) and (Z

m, B

m, X

m∗) be weak solutions of the SDE (8.70)–(8.72)

defined on a filtered space (Ω ′,F ′, P ′, {F ′t }) satisfying

(Zm, B

m, X

m∗)law= (Xm,Bm, Xm∗).

Then, for Zm and Zmsatisfying Zm

0 = Zm0 a.s., it holds that for μGin-a.s. s

P ′(Zm = Zm) = 1. (8.74)

3. Make the same assumptions as (2) except that the filtrations of (Zm, Bm, X

m∗)

and (Zm, B

m, X

m∗), {F ′

t } and {F ′′t } say, are different (but on the same probability

space (Ω ′,F ′, P ′)). Then (8.74) holds for μGin-a.s. s.

Remark 8.1 In conventional situations, the pathwise uniqueness in Lemma 8.5 (3) iscalled the pathwise uniqueness in the strict sense [9, Remark 1.3, 162p].

Proof We first prove (1). Recall that (Xm,Bm,Xm∗) under Ps is a weak solution of(8.70)–(8.72). So it only remains to prove (8.73) for Y

m = Xm .Let n ∈ N3 with n = (p,q, r). Let R

◦p,r(s) be as (8.7).

123


For (u, v) ∈ W (R2m × S) and (u, v,w) ∈ W (R2m × S× R2m), let

θ1p,r(u, v) = inf{t > 0 ; ut /∈ R◦p,r(vt )}, (8.75)

θ2p,r(u, v,w) = inf{t > 0; (ut , vt ) �∼p,r (wt , vt )}. (8.76)

Here (x, s) ∼p,r (x′, s) is same as (8.22). (x, s) �∼p,r (y, s) means x and y are not inthe same connected component of R

◦p,r(s). Let Im be as Proposition 8.1. Let θ3m,n(v)

be the exit time of v from Π(H[a]◦n ∩ Im):

θ3m,n(v) = inf{t > 0 ; vt /∈ Π(H[a]◦n ∩ Im)}.

Then we easily see for n = (p,q, r) ∈ N3 and m ∈ N

min{θ1p,r(u, v), θ3m,n(v)} ≥ τ〈H[a]◦n∩Im〉(u, v). (8.77)

For n = (p,q, r) ∈ N3 and m ∈ N we set

ϑm,n(u, v,w) = min{θ1p,r(u, v), θ1p,r(w, v), θ2p,r(u, v,w), θ3m,n(v)}.

Let

τ(m,n) = ϑm,n(Xm,Xm∗,Ym).

Then from (8.19) and (8.22) and a straightforward calculation we see

|Xmt∧τ(m,n) − Ym

t∧τ(m,n)| =∣∣∫ t∧τ(m,n)

0bm(Xm

u ,Bmu ,X

m∗u )− bm(Ym

u ,Bmu ,X

m∗u )du

∣∣

≤ √mc2

∫ t∧τ(m,n)

0|Xm

u − Ymu |du.

From this we easily deduce

(Xmt ,B

mt ,X

m∗t ) = (Ym

t ,Bmt ,X

m∗t ) for all t ≤ τ(m,n). (8.78)

Suppose that τ(m,n) < θ3m,n(Xm∗). Note that R

◦p,r(s) is an open set. Combining this

with (8.75), (8.76), and (8.78) we obtain

θ1p,r(Xm,Xm∗) = θ1p,r(Y

m,Xm∗) = θ2p,r(Xm,Xm∗,Ym). (8.79)

Hence we deduce from the definition of τ(m,n) and (8.79)

τ(m,n) =min{θ1p,r(Xm,Xm∗), θ3m,n(Xm∗)}. (8.80)

123


From (8.80), (8.77), and Proposition 8.1 we deduce for μGin-a.s. s

Ps( limn→∞ lim

m→∞ τ(m,n) = ∞)

= Ps( limn→∞ lim

m→∞min{θ1p,r(Xm,Xm∗), θ3m,n(Xm∗)} = ∞)

≥ Ps( limn→∞ lim

m→∞ τ〈H[a]◦n∩Im〉(Xm,Xm∗) = ∞)

= 1.

We therefore deduce that the equality in (8.78) holds for all 0 ≤ t <∞. We thus seethat the SDE (8.70)–(8.72) has a pathwise unique weak solution.

By (Zm, B

m, X

m∗)law= (Xm,Bm,Xm∗) we can prove (2) similarly as (1).

We note that the coefficients of themartingale terms of the SDE are constant. Hencewe have not used any specific property of the filtrations in the proof of (1). Then wecan prove (3) similarly as (1). ��

Lemma 8.6 1. For each m ∈ N, the SDE (8.70)–(8.72) has a unique strong solutionFms for (X,B) under Ps for μGin-a.s. s, where s = l(s).

2. For μGin-a.s. s

Fms (Bm,Xm∗) = Xm under Ps.

Proof In Yamada–Watanabe theory the existence of weak solutions and the pathwiseuniqueness imply that the SDE (8.70)–(8.72) has a unique strong solution (see The-orem 1.1 in [9, 163p]). We modify it to prove (1). Indeed, we shall prove the generalresult Proposition 11.1 including (1). The appearance of new randomness X∗ requiresa substantial modification of the theory.

We next prove (2). From (1), the SDE (8.70)–(8.72) has a unique strong solutionFms . Because (Xm,Bm,Xm∗) under Ps is a weak solution, (2) follows from (1). ��

8.4 Completion of proof of Theorem 7.1

Let (X,B) and PμGin =∫PsμGin(ds) be as Sect. 7.2.

Lemma 8.7 (X,B) under PμGin satisfies (NBJ).

Proof Let R(t) = ∫∞t (1/√2π)e−|x |2/2dx be a (scaled) complementary error func-

tion. Let ρ1Gin(x) = 1/π be the one-point correlation function of μGin. Then

∫

R2R

( |x | − r

T

)ρ1Gin(x)dx <∞ for each r , T ∈ N. (8.81)

123


Let (E μ[1]Gin ,D [1]

Gin) be the Dirichlet form of the 1-labeled process given by (8.1). Let

P[1](x,s) be the associated diffusion measure starting at (x, s). We set

P[1]μ[1]Gin

=∫

R2×SP[1](x,s)μ

[1]Gin(dxds). (8.82)

Let (X1,X1∗) = (X1,∑∞

i=2 δXi ) denote the one-labeled process. Then applying theLyons–Zheng decomposition ( [6, Theorem 5.7.1]) to the coordinate function x , wehave for each 0 ≤ t ≤ T ,

X1t − X1

0 =1

2B1t +

1

2B1t for P[1]

μ[1]Gin

-a.e., (8.83)

where B1 is the time reversal of B1 on [0, T ] such that

B1t = B1

T−t (rT )− B1T (rT ). (8.84)

Herewe set rT :C([0, T ];R2×S)→C([0, T ];R2×S) such that rT (w)(t) = w(T−t).Because of the coupling in Lemma 9.3 ([25, Theorem 2.4]), the definition of the

one-Campbell measure, and (8.82), we see

∞∑

i=1PμGin

(inf

t∈[0,T ] |Xit | ≤ r

)=∫

S×Sμ[1]Gin(dxds)P

[1](x,s)

(inf

t∈[0,T ] |X1t | ≤ r

)

= P[1]μ[1]Gin

(inf

t∈[0,T ] |X1t | ≤ r

). (8.85)

Then we have, taking x = X10,

P[1]μ[1]Gin

(inf

t∈[0,T ] |X1t | ≤ r

)≤ P[1]

μ[1]Gin

(sup

t∈[0,T ]|X1

t − x | ≥ |x | − r

)

≤ P[1]μ[1]Gin

(sup

t∈[0,T ]|B1

t | ≥ |x | − r

)+ P[1]

μ[1]Gin

(sup

t∈[0,T ]| B1

t | ≥ |x | − r

)by (8.83)

= 2P[1]μ[1]Gin

(sup

t∈[0,T ]|B1

t | ≥ |x | − r

)

≤ 2∫

SR

( |x | − r

c10T

)μ[1]Gin (dxds)

= 2∫

R2R

( |x | − r

c10T

)ρ1Gin (x) dx < ∞ by (8.81), (8.86)

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where c10 is a positive constant. From (8.85) and (8.86) we deduce

∞∑

i=1PμGin

(inf

t∈[0,T ] |Xit | ≤ r

)<∞.

Hence from Borel–Cantelli’s lemma we have for each r , T ∈ N

PμGin

(lim supi→∞

{inf

t∈[0,T ] |Xit | ≤ r

})= 0. (8.87)

Then we deduce from (3.13) and (8.87) for each r , T ∈ N

PμGin(mr ,T (X) <∞) = 1− PμGin

(lim supi→∞

{inf

t∈[0,T ] |Xit | ≤ r

})= 1.

This completes the proof. ��Lemma 8.8 1. (X,B) under PμGin satisfies (IFC) for ISDE (1.6).2. (X,B) under PμGin satisfies (AC) for μGin.

Proof By Lemma 7.4, (X,B) under Ps is a weak solution of (1.6) starting at s = l(s)for μGin-a.s. s. By Lemma 8.6 we see (X,B) under Ps satisfies (IFC)s for μGin-a.s. s.Hence the first claim holds. The second claim is obvious because {Ps} is a μGin-stationary diffusion and PμGin =

∫PsμGin(ds). ��

Proof of Theorem 7.1 We use Theorem 3.1. For this, we check that μGin is tail trivialand that (X,B) under PμGin satisfies (IFC), (AC) for μGin, (SIN), and (NBJ).

Because μGin is a determinantal random point field, μGin satisfies (TT). (IFC)for (1.6) and (AC) for μGin follow from Lemma 8.8. (SIN) follows from Lemma 7.2.(NBJ) follows fromLemma 8.7. Thus all the assumptions of Theorem 3.1 are fulfilled.Hence we obtain (3) by Theorem 3.1. (4) is clear. Indeed, we can take a subset H suchthat μGin(H) = 1 and that the same conclusions of (3) hold for all s ∈ H.

If a weak solution (X, B) of (1.5) satisfies (AC) for μGin. Then (X, B) satisfies theISDE (1.6) because of the coincidence of the logarithmic derivatives (7.5) and (7.6)given by Lemma 7.3. Furthermore, the SDE (8.70) for (X, B) coincides with

dYm,it = d Bi

t +m∑

j=1j �=i

Ym,it − Ym, j

t

|Ym,it − Ym, j

t |2dt + lim

r→∞

∞∑

j=m+1,|Ym,i

t −X jt |<r

Ym,it − X j

t

|Ym,it − X j

t |2dt .

Thus (X, B) is a weak solution of (1.5) satisfying (IFC) if and only if (X, B) is a weaksolution of (1.6) satisfying (IFC). Hence (1) and (2) follow from (3) and (4).

We proceed with the proof of (5). Recall that both solutions satisfy (AC) for μGin.Then the drift coefficients are equal because of the coincidence of the logarithmicderivatives given by Lemma 7.3. Hence we deduce (5) from (1) and (3). ��

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9 Dirichlet forms and weak solutions

9.1 Relation between ISDE and a random point field

We shall deduce the existence of a strong solution and the pathwise uniqueness ofsolution of ISDE (3.3)–(3.4) from the existence of a random point field μ satisfyingthe assumptions in the sequel. Recall the notion of the logarithmic derivative dμ givenby Definition 2.1. To relate the ISDE (3.3) with the random point field μ, we makethe following assumption.(A1) There exists a random point field μ such that σ ∈ L∞loc(μ[1]) and b ∈ L1

loc(μ[1])

and that μ has a logarithmic derivative dμ = dμ(x, y) satisfying the relation

b(x, y) = 1

2{∇x · a(x, y)+ a(x, y)dμ(x, y)}. (9.1)

Here ∇x · a(x, y) = (∑d

q=1∂apq∂xq

(x, y))dp=1 and a(x, y) = {apq(x, y)}dp,q=1 is thed × d-matrix-valued function defined by

a(x, y) = σ(x, y)tσ(x, y). (9.2)

Furthermore, we assume that there exists a bounded, lower semi-continuous, non-negative function a0 :R+→R

+ and positive constants c11 and c12 such that

a0(|x |)|ξ |2 ≤ (a(x, y)ξ, ξ)Rd ≤ c11a0(|x |)|ξ |2 for all ξ ∈ Rd , (9.3)

c12 := supt∈R+

a0(t) <∞ (9.4)

and that a(x, y) is smooth in x for each y.

Remark 9.1 From (2.3), we obtain an informal expression dμ = ∇x logμ[1]. We theninterpret the relation (9.1) as a differential equation of random point fields μ as

b(x, y) = 1

2{∇x · a(x, y)+ a(x, y)∇x logμ

[1](x, y)}. (9.5)

That is, (9.5) is an equation of μ for given coefficients σ and b. Theorems 3.1–3.2deduce that, if the differential equation (9.5) of random point fields has a solution μ

satisfying the assumptions in these theorems, then the existence of a strong solutionand the pathwise uniqueness of a solution of ISDE (3.3)–(3.4) hold.

9.2 A weak solution of ISDE (First step)

In this subsection, we recall a construction of μ-reversible diffusion from [23,27].Recall the notion of quasi-Gibbs property given by Definition 2.2. We now make thefollowing assumption.

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(A2) μ is a (Φ,Ψ )-quasi Gibbs measure such that there exist upper semi-continuousfunctions (Φ, Ψ ) and positive constants c13 and c14 satisfying

c−113 Φ(x) ≤ Φ(x) ≤ c13Φ(x), c−114 Ψ (x, y) ≤ Ψ (x, y) ≤ c14Ψ (x, y)

and that Φ and Ψ are locally bounded from below.We refer to [27,28] for sufficient conditions of (A2). These conditions give us the

quasi-Gibbs property of the random point fields appearing in random matrix theory,such as sineβ , Airyβ (β = 1, 2, 4), and Bessel2,α (1 ≤ α), and the Ginibre randompoint fields [8,27,28,30].

Let σmr be the m-density function of μ on Sr with respect to the Lebesgue measure

dxm on Smr . By definition, σmr is a non-negative symmetric function such that

1

m!∫

Smr

f (xm)σmr (xm)dxm =

∫

Smr

f (x)μ(dx) for all f ∈ Cb(S).

Here we set xm = (x1, . . . , xm) and dxm = dx1 · · · dxm . Furthermore, we denoteby Smr the subset of S such that Smr = {s ∈ S ; s(Sr ) = m} as before. We make thefollowing assumption.(A3) μ satisfies for each m, r ∈ N

∞∑

k=m

k!(k − m)!μ(S

kr ) <∞. (9.6)

Clearly, (9.6) is equivalent to∫Smr

ρm(xm)dxm <∞ if them-point correlation functionρm of μ with respect to the Lebesgue measure exists. Under assumptions of (A2) and(A3) μ has correlation functions and Campbell measures of any order.

Let (E a,μ,Da,μ◦ ) be a bilinear form on L2(S, μ) with domain Da,μ◦ defined by

Da,μ◦ = { f ∈ D◦ ∩ L2(S, μ) ; E a,μ( f , f ) <∞}, (9.7)

E a,μ( f , g) =∫

SDa[ f , g]μ(ds), (9.8)

Da[ f , g](s) = 1

2

∑

i

(a(si , si♦)∂si f , ∂si g)Rd . (9.9)

Here s =∑i δsi and si♦ =∑ j �=i δs j , ∂si = ( ∂

∂si1, , . . . , ∂

∂sid) and (·, ·)Rd denotes the

standard inner product in Rd . When a is the unit matrix, we often remove it from the

notation; for example, E a,μ = E μ, Da,μ◦ = Dμ◦ , and Da = D.

Lemma 9.1 Assume (9.3), (9.4), (A2), and (A3). Then (E a,μ,Da,μ◦ ) is closable onL2(S, μ), and its closure (E a,μ,Da,μ) is a quasi-regular Dirichlet form on L2(S, μ).Moreover, the associated μ-reversible diffusion (X, {Ps}s∈S) exists.

Lemma 9.1 is a refinement of [23, 119p.Corollary 1] and can be proved similarly.

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In Theorem 3.2 we decompose μ by taking the regular conditional probability withrespect to the tail σ -field. The refinement above is motivated by this decomposition.Indeed, (9.6) is stable under this decomposition (see Lemma 12.1).

By construction, a diffusion measure Ps given by a quasi-regular Dirichlet formwith quasi-continuity in s is unique for quasi-everywhere starting point s. Equivalently,there exists a set H such that the complement of H has capacity zero, and the diffusionmeasure Ps associated with the Dirichlet space above with quasi-continuity in s isunique for all s ∈ H and Ps(Xt ∈ H for all t) = 1 for all s ∈ H. The set H is unique upto capacity zero.

Let {Ps}s∈S be as in Lemma 9.1. We set

Pμ =∫

SPs μ(ds) (9.10)

Let WNE(Ss.i.) be as in (2.9). We assume the following.(SIN) Pμ satisfies Pμ(WNE(Ss.i.)) = 1.From (SIN) we see lpath(X) is well defined under Pμ-a.s. We present an ISDE thatX = lpath(X) satisfies.

Lemma 9.2 ( [26, Theorem 26]) Assume (A1)–(A3). Assume that Pμ satisfies (SIN).Then there exists an H ⊂ Ssde satisfying (3.1) and (3.2) such that (lpath(X),B) definedon (Ω,F , Ps, {Ft }) is a weak solution of ISDE (3.3)–(3.5) for each s ∈ H and thatμ(H) = 1 and Ps(Xt ∈ H for 0 ≤ ∀t <∞) = 1.

Remark 9.2 1. The solution lpath(X) of the ISDE (3.3)–(3.4) is defined on the space(W (S),B(W (S))) with filtering Bt (S) given by (7.3).

2. In Lemma 9.2, Brownian motion B = (Bi )i∈N is given by the additive functionalof the diffusion (X, {Ps}). In particular, B is a functional of X. Hence we writeB(X) when we emphasize this. Summing up, the weak solution in Lemma 9.2 isgiven by (lpath(X),B(X)) and defined on the filtered space (W (S),B(W (S)))with{Bt (S)}.

3. A sufficient condition of (SIN) will be given in Sect. 10.

9.3 The Dirichlet forms of them-labeled processes and the coupling

Let (E a,μ,Da,μ) be the Dirichlet form on L2(S, μ) in (A3), and let (X, {Ps}s∈S) bethe associated diffusion.

Let l be a label. Assume that (SIN) holds. Then lpath makes sense and we constructthe fully labeled process X = (Xi )i∈N with X0 = l(X0) associated with the unlabeledprocess X by taking X = lpath(X), where X =∑i∈N δXi .

LetXm = (Xi )mi=1 and Xm∗ =∑m<i δXi . We call the pair (Xm,Xm∗) them-labeled

process. We shall present the Dirichlet form associated with the m-labeled process.We write xm = (xi )mi=1 ∈ (Rd)m . Let μ[m] be the (reduced) m-Campbell measure

of μ defined as

μ[m](dxmdy) = ρm(xm)μxm (dy)dxm, (9.11)

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whereρm is them-point correlation function ofμwith respect to theLebesguemeasuredx on Sm , and μxm is the reduced Palm measure conditioned at xm ∈ Sm . Let

E a,μ[m]( f , g) =∫

Sm×S

⎧⎨

⎩1

2

m∑

i=1

(a(xi , x

i♦ + y)∂ f

∂xi,∂g

∂xi

)

Rd+ D

a[ f , g]⎫⎬

⎭μ[m](dxmdy),

(9.12)

where ∂∂xi

is the nabla inRd , xi♦ =∑m

j �=i δx j , and a(x, y) is given by (9.2).Moreover,Da is defined by (9.9) naturally regarded as the carré du champ on Sm × S, and

Da,μ[m]◦ ={f ∈ C∞0 (Sm)⊗D◦ ; E a,μ[m]( f , f ) <∞, f ∈ L2(Sm × S, μ[m])

}.

When m = 0, we interpret E a,μ[0] and μ[0] as the unlabeled Dirichlet form E a,μ andμ, respectively.

The closablity of the bilinear form (E a,μ[m] ,C∞0 (Sm)⊗D◦) on L2(Sm × S, μ[m])follows from (9.3), (9.4), (A2), and (A3). We can prove this in a similar fashion as thecasem = 0 as Lemma 9.1 ( [23]).We then denote by (E a,μ[m] ,Da,μ[m]) its closure. Thequasi-regularity of (E a,μ[m] ,Da,μ[m]) is proved by [25] forμwith bounded correlationfunctions. The generalization to (A3) is easy, and we left its proof.

Let P[m](sm ,sm∗) denote the diffusion measures associated with the m-labeled Dirichlet

form (E a,μ[m] ,Da,μ[m]) on L2(Sm × S, μ[m]). (see [25]). We quote:

Lemma 9.3 ([25, Theorem 2.4]) Let s = u ((sm, sm∗)) =∑mi=1 δsi + sm∗. Then

P[m](sm ,sm∗) = Ps ◦ (Xm, Xm∗)−1.

Note that Ps in the right-hand side is independent of m ∈ N. Hence this gives asequence of coupled Sm × S-valued continuous processes with distributions P[m](sm ,sm∗).In this sense, there exists a natural coupling among the m-labeled Dirichlet forms(E a,μ[m] ,Da,μ[m]) on L2(Sm × S, μ[m]). This coupling is a crucial point of the con-struction of weak solutions of ISDE in [26].

Introducing the m-labeled processes, we can regard Xm as a Dirichlet process ofthe diffusion (Xm,Xm∗) associated with the m-labeled Dirichlet space. That is, onecan regard A[xm]t := Xm

t − Xm0 as a dm-dimensional additive functional given by the

composition of (Xm,Xm∗) with the coordinate function xm = (x1, . . . , xm) ∈ (Rd)m .Although Xm can be regarded as an additive functional of the unlabeled process X =∑

i δXi , Xm is no longer a Dirichlet process in this case. Indeed, as a function of X,we cannot identify Xm

t without tracing the trajectory of Xs =∑i δXisfor s ∈ [0, t].

Once Xm can be regarded as a Dirichlet process, we can apply the Ito formula(Fukushima decomposition), and Lyons–Zheng decomposition toXm , which is impor-tant in proving the results in the subsequent sections.

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10 Sufficient conditions of (SIN) and (NBJ)

The purpose of this section is to give sufficient conditions of (SIN) and (NBJ) forX = lpath(X) under Pμ. The following assumptions are related to Dirichlet formsintroduced in Sect. 9.2. So we named the series of them as (A4) followed by (A3) inSect. 9.2. Let R = R(t) be a (scaled) complementary error function:

R(t) =∫ ∞

t(1/√2π)e−|x |2/2dx .

We set 〈 f , s〉 =∑i f (si ) for s =∑i δsi as before.(A4) (1)

For each r , T ∈ N

Eμ

[⟨R

( | · | − r

T

), s⟩]

<∞, (10.1)

and there exists a T > 0 such that for each R > 0

lim infr→∞ R

( rT

)Eμ[〈1Sr+R , s〉

] = 0. (10.2)

(2) Each Xi neither hits the boundary ∂S of S nor collides each other. That is,

Capa,μ({s; s(∂S) ≥ 1}) = 0, (10.3)

Capa,μ(Scs) = 0. (10.4)

Here Capa,μ is the capacity of the Dirichlet form (E a,μ,Da,μ) on L2(S, μ).The condition (10.2) is an analogy of [6, (5.7.14)]. Unlike this, the carré du champ

has uniform upper bounds c11c12 by (9.2)–(9.4).We remark that (10.2) is easy to check. (10.2) holds if the 1-correlation function

of μ has at most polynomial growth at infinity. Obviously, condition (10.3) is alwayssatisfied if ∂S = ∅. We state sufficient conditions of (10.4).

Lemma 10.1 ([24, Theorem 2.1, Proposition 7.1]) Assume (A1)–(A3). Assume thatΦ and Ψ are locally bounded from below. We then obtain the following.

1. Assume that μ is a determinantal random point field with a locally Lipschitzcontinuous kernel with respect to the Lebesgue measure. (10.4) then holds.

2. Assume that d ≥ 2. (10.4) then holds.

Proof Claims (1) and (2) follow from [24, Theorem 2.1, Proposition 7.1]. ��We now deduce (SIN) from (A1)–(A4).

Lemma 10.2 Assume (A1)–(A4). Then X = lpath(X) under Pμ satisfies (SIN) and

Capa,μ(Scsde) = 0, Pμ ◦ l−1path(W (Ssde)) = 1. (10.5)

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Proof Applying the argument in [6, Theorem 5.7.3] to Xm of this Dirichlet form, wesee that the diffusion (Xm,Xm∗) is conservative. Because this holds for all m ∈ N,

Pμ( sup{|Xit |; t ≤ T } <∞ for all T , i ∈ N ) = 1. (10.6)

Because X ∈ W (Ss) Pμ-a.s. by (10.4), we write Xt = ∑i δXitsuch that Xi ∈

C(I i ; S), where I i = [0, b) or I i = (a, b) for some a, b ∈ (0,∞]. We shall proveI i = [0,∞) Pμ-a.s.. Suppose that I i = [0, b). Then, from (10.6), we deduce thatb = ∞. Next suppose that I i = (a, b). Then, applying the strong Markov propertyof the diffusion {Ps} at any a′ ∈ (a, b) and using the preceding argument, we deducethat b = ∞. As a result, we have I i = (a,∞). Because of reversibility, we see thatsuch open intervals do not exist. Hence, we obtain I i = [0,∞) for all i . From this,(10.4), and μ({s(S) = ∞}) = 1, we obtain Capa,μ({s ∈ S ; s(S) < ∞}) = 0. Fromthis and (A4) (2) we have

Capa,μ(Scs.i.) = 0. (10.7)

From (10.6), (10.7), and (10.3) we obtain Pμ(WNE(Ss.i.)) = 1. By Lemma 9.2 wesee the first claim in (10.5). The second claim is clear from the first. We have thuscompleted the proof. ��

We next deduce (NBJ) from (A1)–(A4).

Lemma 10.3 Assume (A1)–(A4). Then X = lpath(X) under Pμ satisfies (NBJ).

Proof Let ρ1 be the one-point correlation function of μ with respect to the Lebesguemeasure. Then (10.1) implies

∫

SR

( |x | − r√c11c12T

)ρ1(x)dx <∞. (10.8)

Here c12 = supt∈R+ a0(t) is finite by (9.4).

Let (E a,μ[1] ,Da,μ[1]) be the Dirichlet form of the 1-labeled process. Let P[1](x,s) bethe associated diffusion measure starting at (x, s) ∈ S × S. We set

P[1]μ[1] =

∫

S×SP[1](x,s)μ

[1](dxds). (10.9)

Let (X1,X1∗) = (X1,∑∞

i=2 δXi ) denote the one-labeled process. Then applying theLyons–Zheng decomposition ([6, Theorem 5.7.1]) to the coordinate function x , wehave a continuous local martingale M1 such that for each 0 ≤ t ≤ T ,

X1t − X1

0 =1

2M1

t +1

2M1

t for P[1]μ[1] -a.e., (10.10)

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where M1 is the time reversal of M1 on [0, T ] such that

M1t = M1

T−t (rT )− M1T (rT ). (10.11)

Herewe set rT :C([0, T ];Rd×S)→C([0, T ];Rd×S) such that rT (w)(t) = w(T−t).By (9.1), (9.2), and (9.12) the quadratic variation of M1 = (M1

1 , . . . ,M1d ) is given by

〈M1p,M

1q 〉t =

∫ t

0apq(X1u,

∞∑

i=2δXi

u

)du. (10.12)

We can prove Lemma 10.3 in a similar fashion as Lemma 8.7. Indeed, (8.81), (8.82),(8.83), and (8.84) correspond to (10.8), (10.9), (10.10), and (10.11), respectively.Although the continuous local martingale M1 in (10.10) is not Brownian motion, thequadratic variation process in (10.12) is controlled by (9.3) and (9.4). So the proof ofLemma 8.7 is still valid for Lemma 10.3. Hence we omit the detail. ��

11 Sufficient conditions of (IFC)

11.1 Localization of coefficients

Let a = {aq}q∈N be a sequence of increasing sequences aq = {aq(r)}r∈N of naturalnumbers such that aq(r) < aq+1(r) for all r ,q ∈ N. We set for a = {aq}q∈N

K[a] =∞⋃

q=1K[aq], K[aq] = {s ; s(Sr ) ≤ aq(r) for all r ∈ N}. (11.1)

By construction, K[aq] ⊂ K[aq+1] for all q ∈ N. It is well known that K[aq] is acompact set in S for each q ∈ N. To quantify μ by a we assume(B1) μ(K[a]) = 1.

We note that a sequence a with μ(K[a]) = 1 always exists for any random pointfield μ (see [23, Lemma 2.6]). We set a+q (r) = aq(r + 1) and a+ = {a+q }q∈N. Let

S[m]s.i. = {(x, s) ∈ Sm × S ; u(x)+ s ∈ Ss.i.},

where x = (x1, . . . , xm) and u(x) =∑mi=1 δxi . Similarly as (8.2)–(8.4) let

Rp,r(s) ={x ∈ Smr ; min

j �=k |x j − xk | ≥ 2−p, infl,i|xl − si | ≥ 2−p

},

where j, k, l = 1, . . . ,m, s = ∑i δsi , and Smr = {x ∈ S; |x | ≤ r}m , and forn = (p,q, r)

H[a]n = H[a]p,q,r :={(x, s) ∈ S[m]s.i. ; x ∈ Rp,r(s), s ∈ K[a+q ]

}.

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We set Sm,◦r = {|x | < r, x ∈ S}m . Let R

◦p,r(s) be the open kernel of Rp,r(s):

R◦p,r(s) =

{x ∈ Sm,◦

r ; infj �=k |x j − xk | > 2−p, inf

l,i|xl − si | > 2−p

}.

For n = (p,q, r) ∈ N3 we set

H[a]◦n = H[a]◦p,q,r :={(x, s) ∈ S[m]s.i. ; x ∈ R

◦p,r(s), s ∈ K[a+q ]

}.

Similarly as (8.4) and (8.9), we set H[a], H[a]q,r, H[a]◦, and H[a]◦q,r.Lemma 11.1 Assume (A1)–(A4) and (B1). For each m ∈ N the following then holds:

Ps( limn→∞ τH[a]◦n(X

m, Xm∗) = ∞) = 1 for μ-a.s. s. (11.2)

Here (Xm, Xm∗) is the m-labeled process given by (X,B) in Lemma 9.2, τH[a]◦n is theexit time from H[a]◦n, and limn→∞ is same as (8.11).

Proof The proof is same as Lemma 8.1. Hence we omit it. ��

11.2 A sufficient condition of IFC andYamada–Watanabe theory for SDE ofrandom environment type

We set x = (x1, . . . , xm) ∈ Sm and xi♦ =∑mj �=i δx j . Let (σ, b) be as (3.3). We set

σm(x, s) = (σ (xi , xi♦ + s))mi=1, bm(x, s) = (b(xi , xi♦ + s))mi=1.

Then the time-inhomogeneous coefficients ofSDE (3.18) are givenby (σm , bm)|s=Xm∗t ,where (σm, bm) is a version of (σm, bm) with respect to μ[m].

Let N and n be as (8.5) and (8.6). Let {Im}m∈N be an increasing sequence of closedsets in Sm × S. Let Π : Sm × S→S be the projection such that (x, s) �→ s. Then

Π(H[a]◦n ∩ Im) = {s ∈ S ; H[a]◦n ∩ Im ∩(Sm × {s}) �= ∅}.

For n = (p,q, r) ∈ N3, let c15(m,n) be the constant such that 0 ≤ c15 ≤ ∞ and that

c15 = sup{ |σm(x, s)− σm(y, s)|

|x− y| ,|bm(x, s)− bm(y, s)|

|x− y| ; x �= y,

s ∈ Π(H[a]◦n ∩ Im), (x, s), (y, s) ∈ R◦p,r(s), (x, s) ∼p,r (y, s)}. (11.3)

Here (x, s) ∼p,r (y, s)means x and y are in the same connected component of R◦p,r(s).

Let (X,B) = (lpath(X),B) be the weak solution of ISDE (3.3)–(3.5) given byLemma 9.2 defined on (Ω,F , Ps, {Ft }). We set similarly as (8.21)

〈H[a]◦n ∩ Im〉 =⋃

s∈Π(H[a]◦n∩Im)R◦p,r(s)× {s}.

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We assume the following.(B2) For eachm ∈ N, there exist aμ[m]-version (σm, bm) of (σm, bm) and an increas-ing sequence of closed sets {Im}m∈N such that for eachm ∈ N and n = (p,q, r) ∈ N3

c15(m,n) <∞, (11.4)

limm→∞Capa,μ

[m](H[a]◦n\〈H[a]◦n ∩ Im〉

)= 0. (11.5)

Remark 11.1 We shall later assume the coefficients are in the domain of them-labeledDirichlet form, and take (σm, bm) as a quasi-continuous version and {Im}∞m=1 as anest. We refer to [6, pp 67-69] for quasi-continuous version and nest. The sequenceof sets {Im} plays a crucial role in the proof of Lemma 13.1.

Lemma 11.2 Assume (A1)–(A4) and (B1)–(B2). Then the following hold.

1. For eachm ∈ N, the SDE (3.10)–(3.12) has a pathwise unique, weak solution start-ing at sm = lm(s) for μ-a.s. s in the sense that arbitrary solutions (Ym,Bm, Xm∗)and (Y

m,Bm, Xm∗) of (3.10)–(3.12) defined on (Ω,F , Ps, {Ft }) satisfy

Ps(Ym = Ym) = 1.

In particular, (Ym,Bm, Xm∗) coincides with (Xm,Bm, Xm∗) under Ps.2. Let (Zm, B

m, X

m∗) and (Z

m, B

m, X

m∗) be weak solutions of the SDE (3.10)–(3.12)

defined on a filtered space (Ω ′,F ′, P ′, {F ′t }) satisfying

(Zm, B

m, X

m∗)law= (Xm,Bm, Xm∗).

Then it holds that for μ-a.s. s

P ′(Zm = Zm) = 1. (11.6)

3. Make the same assumptions as (2) except that the filtrations of (Zm, Bm, X

m∗)

and (Zm, B

m, X

m∗), {F ′

t } and {F ′′t } say, are different (but on the same probability

space (Ω ′,F ′, P ′)). Assume that the coefficient σm is constant. Then (11.6) holdsfor μ-a.s. s.

Proof From (A1)–(A4) we can construct a weak solution (Xm,Bm,Xm∗) under Ps of(3.10)–(3.12). We remark that (A1)–(A4) and (B1)–(B2) yield the related claims ofProposition 8.1. Indeed, (11.4) corresponds to (8.23). From (A1)–(A4) and (B1)–(B2)we apply Lemma 11.1 to obtain (11.2). Then from (11.2) and (11.5) we easily see

Ps( limn→∞ lim

m→∞ τ〈H[a]◦n∩Im〉(Xm,Xm∗) = ∞) = 1 for μ-a.s. s. (11.7)

Here τ〈H[a]◦n∩Im〉(Xm,Xm∗) is the exit time of (Xm,Xm∗) from the set 〈H[a]◦n ∩ Im〉.

(11.7) corresponds to (8.24). The claims in Proposition 8.1 are used in the proof ofLemma 8.5. We do not need any other specific properties of the Ginibre interacting

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Brownian motion in the proof of Lemma 8.5. Hence the proof of Lemma 11.2 is sameas Lemma 8.5. ��Remark 11.2 In (3) the reference families {F ′

t } and {F ′′t } of SDEs are different

although the probability space and the Brownian motion is the same. The pathwiseuniqueness in (3) is called the pathwise uniqueness in the strict sense in [9, 162p]. Weshall use this refinement in the proof of Proposition 11.1. In the classical situation ofYamada–Watanabe theory, the pathwise uniqueness in the strict sense follows fromthe pathwise uniqueness and the existence of weak solutions as a corollary of theirmain result. In the current case, it has been not yet succeeded to generalize this partof the Yamada–Watanabe theory to SDEs of random environment type. So we add theadditional assumption in (3).

Recall that (X,B) is the weak solution of (3.3)–(3.5) defined on (Ω,F , Ps, {Ft })given by Lemma 9.2. Let (Xm,Bm,Xm∗) be the weak solution of (3.10) made of(X,B). To simplify the notation, we set

w = (b, x) ∈ W0(Rdm)×W (S).

Let Pw be the regular conditional probability such that

Pw = Ps(Xm ∈ · | (Bm,Xm∗) = w).

Let (Ym, Bm, X

m∗) be an independent copy of (Xm,Bm,Xm∗). Let P be the dis-

tributions of (Ym, Bm, X

m∗). Let Pw = P(Ym ∈ · | (Bm

, Xm∗

) = w). Let Q be thedistribution of (Bm,Xm∗). We set the probability measure R on

W • := W (Sm)×W (Sm)×W0(Rdm)×W (S)

by

R(dudvdw) = Pw(du)Pw(dv)Q(dw). (11.8)

We set G to be the completion of the topological σ -field B(W •) by R, and Gt =∩ε(Bt+ε(W •) ∨N ), where Bt = σ [(u(u), v(u), w(u)) ; 0 ≤ u ≤ t] and N is theset of all R-null sets.

Proposition 11.1 Assume (A1)–(A4) and (B1)–(B2). Assume that the coefficient σm

is constant. Then X = lpath(X) under Pμ satisfies (IFC).

Proof FromLemma11.2 (3)wehave apathwise unique,weak solution (Xm,Bm,Xm∗).Then from a generalization of the Yamada–Watanabe theory (see Theorem 1.1 in [9,163p]) to SDE with random environment, we shall construct a strong solution.

Under R, both (u,b, x) and (v,b, x) are weak solutions of (3.10)–(3.12). Thesesolutions are defined on Ξ = (W •,G , R, {Gt }) and SDEs (3.10) for (u,b, x) and

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(v,b, x) become as follows.

duit = σmdbit + bmx (t, (uit ,u

i♦t ))dt, (11.9)

dvit = σmdbit + bmx (t, (vit , v

i♦t ))dt . (11.10)

Here bmx is defined by (3.8) with replacement of Xm∗t by xm∗t =∑∞j=m+1 δx j

t.

Solutions of SDEs (11.9) and (11.10) are defined on Ξ = (W •,G , R, {Gt }).Although we do not need the fact that b under R is a {Gt }-Brownian motion in thepresent proof, we shall prove this here combinedwith Lemma11.3 below.By construc-tion b under R is a Brownian motion. So for this, it only remains to prove b(u)−b(t)is independent of Gt for all t < u under R, which we prove in Lemma 11.3.

Note that the distributions of both (u,b, x) and (v,b, x) under R coincide with thatof (Xm,Bm,Xm∗). Hence we obtain

R(u = v) = 1 (11.11)

from the pathwise uniqueness of weak solutions given by Lemma 11.2 (3). Let

Qw = R((Xm,Ym) ∈ ·|(Bm,Xm∗) = w). (11.12)

Then we deduce from (11.8)

Qw(dudv) = Pw(du)Pw(dv). (11.13)

The identity R(u = v) = 1 in (11.11) together with (11.12) implies Qw(u = v) = 1for Q-a.s.w. Meanwhile, from (11.13) we deduce that u and v under Qw are mutuallyindependent. Hence the distribution of (u, v) under Qw is δ(F(w),F(w)), where F(w) isa nonrandomelement ofW (Sm)depending onw. Thus F is regarded as a function fromW0(R

dm)×W (S) toW (Sm) byw �−→ F(w). The distributions of Pw(du) and Pw(dv)coincide with δF(w). We therefore obtain (Xm,Bm,Xm∗) = (F(Bm,Xm∗),Bm,Xm∗).

We easily see that F is B(W0(Rdm)×W (S))Q/B(W (Sm))-measurable. Indeed,

letting ι and κ be the projections such that ι(u, v, w) = u and κ(u, v, w) = w, we seeF = ι ◦ κ−1 R-a.s. Then κ−1(F−1(A)) = ι−1(A) R-a.s. for each A ∈ B(W (Sm)),that is,

R(κ−1(F−1(A))" ι−1(A)) = 0,

where " denotes the symmetric difference of sets. Note that

ι−1(A) ∈ G = B(W (Sm)×W (Sm)×W0(Rdm)×W (S))R.

Then we see κ(ι−1(A)) ∈ B(W0(Rdm)×W (S))Qby Q = R ◦ κ−1. This implies

F−1(A) ∈ B(W0(Rdm)×W (S))Q.

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Hence F isB(W0(Rdm)×W (S))Q/B(W (Sm))-measurable.

We can prove that F isBt (W0(Rdm)×W (S))Q/Bt (W (Sm))-measurable for each

t in a similar fashion. Here subscript t denotes the σ -field being generated until timet . Indeed, we can localize the ISDE and the solution in time to [0, T ] for all 0 <

T . The restriction of the original weak solution (Xm,Bm,Xm∗) on the time interval[0, T ] is also a solution of the ISDE. So with the same argument we can construct astrong solution FT defined on W0([0, T ];Rdm) × W ([0, T ]; S). The solution FT is

B(W0([0, T ];Rdm)×W ([0, T ]; S))Q / B(W ([0, T ]; Sm))-measurable. Because ofthe pathwise uniqueness of weak solutions, FT (b, x)(t) = F(b, x)(t) for all 0 ≤ t ≤T . We have natural identities

B(W0([0, T ];Rdm)×W ([0, T ]; S)) =BT (W0(Rdm)×W (S))

B(W ([0, T ]; Sm)) =BT (W (Sm)).

From this F isBT (W0(Rdm)×W (S))Q/BT (W (Sm))-measurable for each T .

Recall that s is suppressed from the notation of F . We set

Fms (Bm,Xm∗) = F(Bm, upath(Xm∗)).

Because lpath◦upath = id. forPμ◦l−1path-a.s. andupath◦lpath = id. forPμ-a.s., the functionFms inherits necessary measurabilities in Definition 3.9 from those of F . Hence we

see Fms is a strong solution of (3.10)–(3.12) for (X,B) under Ps for Pμ ◦ l−1-a.s. s. We

have already proved the pathwise uniqueness. Thus, Fms is a unique strong solution

for (X,B) starting at sm for Pμ ◦ l−1-a.s. s. Hence X = lpath(X) under Pμ satisfies(IFC). ��Lemma 11.3 For each t, {bu − bt } (t < u <∞) are independent of Gt under R.

Proof Note that Pw = R(u ∈ ·|w) and P ′w = R(v ∈ ·|w). Let F be the strong solution

obtained in the proof of Proposition 11.1. Note that F is Bt (W0(Rdm)×W (S))Q-

adapted. Then for any A1 × A2 × A3 ∈ Gt and θ ∈ Rdm

E R[e√−1〈θ,bu−bt 〉1A1×A2×A3] =

∫

A3

e√−1〈θ,bu−bt 〉Pw(A1)P

′w(A2)Q(dw)

=∫

A3

e√−1〈θ,bu−bt 〉1A1(F(w))1A2(F(w))Q(dw)

= e−|θ |2/2(u−t)∫

A3

1A1(F(w))1A2(F(w))Q(dw)

= e−|θ |2/2(u−t)R(A1 × A2 × A3).

This implies the claim. ��Recall that Pμ is given by (9.10).

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Theorem 11.1 Assume that μ and lpath(X) under Pμ satisfy (TT), (A1)–(A4), and(B1)–(B2). Assume that the coefficient σm is constant. Then ISDE (3.3)–(3.4) hasa family of unique strong solutions {Fs} starting at s = l(s) for μ-a.s. s under theconstraints of (MF), (IFC), (AC) for μ, (SIN), and (NBJ).

Proof We take (X,B) and P in Theorem 3.1 as X := lpath(X) and P := Pμ. We checkthis satisfies all the assumptions of Theorem 3.1. That is, we have to check (TT) forμ and that (X,B) under P is a weak solution satisfying (IFC), (AC) for μ, (SIN),and (NBJ).

We see (TT) for μ follows by assumption. By Lemma 10.2 lpath(X) under Pμsatisfies (SIN). Hence by Lemma 9.2 lpath(X) under Pμ is a weak solution. Note thatthe coefficient σm is constant and that (A1)–(A4) and (B1)–(B2) hold by assumption.Then the assumptions of Proposition 11.1 are fulfilled. Hence we obtain (IFC) fromProposition 11.1. Using Lemma 10.3 we obtain (NBJ). Because Pμ is μ-reversible,(AC) for μ is obvious. Thus, all the assumptions of Theorem 3.1 are fulfilled. Hencethe claim follows from Theorem 3.1. ��

Corollary 11.1 Under the same assumptions of Theorem 11.1, lpath(X) under Pμ is aweak solution of (3.3)–(3.4) satisfying (IFC), (AC) for μ, (SIN), and (NBJ). Fur-thermore, lpath(X) under {Ps} is a family of unique strong solutions starting at s = l(s)for μ-a.s. s under the constraints of (MF), (IFC), (AC) for μ, (SIN), and (NBJ).

11.3 A sufficient condition of (B2) and Taylor expansion of coefficients

In this section we give a sufficient condition of (B2). We begin by introducing thecut-off functions on Sm × S. Let ϕr ∈ C∞0 (Sm) be the cut-off function such that

0 ≤ ϕr(x) ≤ 1, |∇ϕr(x)| ≤ 2, ϕr(x) = ϕr(|x|) for all x ∈ Sm,

where ϕr ∈ C∞0 (R) is given by (8.27). Let hp ∈ C∞0 (R) (p ∈ {0} ∪ N) such that

hp(t) ={1 (t ≤ 2−p−2),0 (2−p−1 ≤ t)

, 0 ≤ hp(t) ≤ 1, |h′p(t)| ≤ 2p+3 for all t .

We write x = (xk)mk=1 ∈ Sm and s =∑i δsi ∈ S. Let h†p : Sm × S→R such that

h†p(x, s) =m∏

k=1

{ m∏

j �=k{1− hp(|xk − x j |)}

}{∏

i

{1− hp(|xk − si |)}}.

We label s =∑i δsi in such a way that |si | ≤ |si+1| for all i . We set

I (q, r , s) = {i ; i > aq(r), si ∈ Sr }.

123


Here aq = {aq(r)}r∈N are the increasing sequences in (11.1). We set

daq(s) =⎧⎨

⎩

∞∑

r=1

∑

i∈I (q,r ,s)(r − |si |)2

⎫⎬

⎭

1/2

,

χaq(s) = h0 ◦ daq(s).

We introduce the cut-off functions defined as

⎧⎪⎨

⎪⎩

χr(x, s) = ϕr(x),χq,r(x, s) = ϕr(x)χaq(s),χp,q,r(x, s) = ϕr(x)χaq(s)h

†p(x, s)

. (11.14)

We easily see that

limr→∞ lim

q→∞ limp→∞χp,q,r(x, s) = 1 for all (x, s) ∈ H[a].

Let N = {(p,q, r), (r,q),q ; p,q, r ∈ N}, N1, N2, and N3 be as (8.5). Recall that forn ∈ N we define n+ 1 ∈ N such that

n+ 1 =

⎧⎪⎨

⎪⎩

(p+ 1,q, r) for n = (p,q, r),

(q+ 1, r) for n = (q, r),

r+ 1 for n = r.

We set χn = χp,q,r. Then {χn} are consistent in the sense that χn(x, s) = χn+1(x, s)for (x, s) ∈ H[a]n. We suppress m from the notation of χn although χn depends onm ∈ N. By a direct calculation similar to that in [23, Lemma 2.5], we obtain thefollowing.

Lemma 11.4 For each m ∈ N, the functions χn (n ∈ N) satisfy the following.

χn(x, s) ={0 for (x, s) /∈ H[a]n+11 for (x, s) ∈ H[a]n , χn ∈ Da,μ[m] ,

0 ≤ χn(x, s) ≤ 1, |∇xχn(x, s)|2 ≤ c16, D[χn, χn](x, s) ≤ c17.

Here c16(n) and c17(n) are positive finite constants independent of (x, s), andDa,μ[m]

is the domain of the Dirichlet form of the m-labeled process (Xm, Xm∗).

We shall give a sufficient condition of (B2) using Taylor expansion. Let

J[l] ={j = ( jk,i )k=1,...,m, i=1,...,d ; jk,i ∈ {0} ∪ N,

m∑

k=1

d∑

i=1jk,i = l

}. (11.15)

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We set ∂j = ∏k,i (∂/∂xk,i )jk,i for j = ( jk,i ) ∈ J[l], where ((xk,i )di=1)

mk=1 ∈ R

dm and(∂/∂xk,i ) jk,i denotes the identity if jk,i = 0. For � ∈ N, we introduce the following:(C1) For each j ∈ ∪�l=0J[l] and n ∈ N3,

χn∂jσm, χn∂jb

m ∈ Da,μ[m] .

(C2) There exists a μ[m]-version {σm, bm} of {σm, bm} such that

sup{|∂jf(x, s)|; (x, s) ∈ H[a]n, f ∈ {σm, bm}} <∞

for each j ∈ J[�] and n ∈ N3.

Remark 11.3 Note that ϕr(x) and h†p(x, s) in (11.14) are smooth in x, and that ∂iϕr(x)

and ∂ih†p(x, s) are bounded. Hence (11.14) and (C1) with a straightforward calculation

show (∂iχn)(∂jσm) and (∂iχn)(∂jbm) belong to Da,μ[m] .

Proposition 11.2 Assume that (C1) and (C2) hold for some � ∈ N. Then (B2) holds.

Proof We omit the proof of Proposition 11.2 because it is same as Proposition 8.1.Indeed, in the proof of Proposition 8.1, we need Lemma 8.1, Lemma 8.4, and (8.55).These correspond to Lemma 11.1, (C1), and (C2), respectively. ��Theorem 11.2 Under the same assumptions as Theorem 11.1 with replacement (B2)by (C1)–(C2), the same conclusions as Theorem 11.1 holds.

Proof From Proposition 11.2 and Theorem 11.1 we obtain the claim. ��Corollary 11.2 Under the same assumptions of Theorem 11.2, the same conclusionsas Corollary 11.1 hold.

Remark 11.4 For sine2, Airy2, and Bessel2,α random point fields, there is anotherconstruction of stochastic dynamics based on space-time correlation functions [14,33].Theorem 11.2 combined with tail triviality obtained in [2,19,29] proves that these twodynamics are the same [30–32].

12 Sufficient conditions of (SIN) and (NBJ) for � with non-trivial tails

In this section, we deduce (SIN) and (NBJ) from the assumptions ofμ. We shall showthe stability of (SIN) and (NBJ) under the operation of conditioning with respect tothe tail σ -field T (S). Recalling the decomposition (3.20) of μ we have

μ(A) =∫

SμaTail(A)μ(da). (12.1)

Lemma 12.1 Assume (A1)–(A3) for μ. Then μaTail satisfies (A1)–(A3) for μ-a.s.a.

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Proof (A1)–(A2) forμaTail are clear from the definitions of logarithmic derivatives and

quasi-Gibbs measures combined with Fubini’s theorem, respectively. (A3) for μaTail

follows from (A3) for μ and Fubini’s theorem. Indeed, we have

∫

S

[ ∞∑

k=m

k!(k − m)! μ

aTail(S

kr )

]μ(da) =

∞∑

k=m

k!(k − m)!

∫

SμaTail(S

kr )μ(da)

=∞∑

k=m

k!(k − m)! μ(S

kr ) <∞.

Hence we see that μaTail satisfies (9.6) for μ-a.s.a, which implies (A3). ��

Let (E a,μaTail ,Da,μa

Tail) be the Dirichlet form given by (9.7)–(9.8) with replacementof μ by μa

Tail. By Lemma 9.1 and Lemma 12.1 (E a,μaTail ,Da,μa

Tail) are quasi-regularDirichlet forms on L2(S, μa

Tail) for μ-a.s. a. We easily see that

Da,μaTail ⊃ Da,μ for μ-a.s. a. (12.2)

Let Capa,μ and Capa,μaTail be the capacities associated with (E a,μ,Da,μ) on L2(S, μ)

and (E a,μaTail ,Da,μa

Tail) on L2(S, μaTail), respectively. Then by the variational formula

of capacity and (12.2), we easily deduce that for each A

Capa,μaTail(A) ≤ Capa,μ(A) for μ-a.s. a.

This implies

∫

SCapa,μ

aTail(A)μ(da) ≤ Capa,μ(A). (12.3)

Lemma 12.2 Assume (A4) for μ. Then μaTail satisfies (A4) for μ-a.s.a.

Proof We begin by proving (A4) (1) for μaTail. By (10.1) for μ and (12.1) we have

∫

SEμa

Tail

[〈R(

| · | − r

T), s〉]μ(da) = Eμ

[〈R(

| · | − r

T), s〉]<∞.

Then μaTail satisfies (10.1) by Fubini’s theorem. By Fatou’s lemma, (10.2), and (12.1)

∫

Slim infr→∞ R

( rT

)Eμa

Tail[〈1Sr+R , s〉]μ(da)

≤ lim infr→∞

∫

SR( rT

)Eμa

Tail[〈1Sr+R , s〉]μ(da) by Fatou’s lemma

= lim infr→∞ R

( rT

) ∫

SEμa

Tail[〈1Sr+R , s〉]μ(da)

= lim infr→∞ R

( rT

)Eμ[〈1Sr+R , s〉] = 0 by (12.1), (10.2). (12.4)

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Hence from (12.4) we see lim infr→∞R( rT

)Eμa

Tail[〈1Sr+R , s〉] = 0 for μ-a.s. a. Thisimplies (10.2) for μa

Tail. We have thus obtained (A4) (1) for μaTail.

From (12.3) and (A4) (2) for μ, we deduce

∫

SCapa,μ

aTail({s; s(∂S) ≥ 1})μ(da) ≤ Capa,μ({s; s(∂S) ≥ 1}) = 0

∫

SCapa,μ

aTail(Scs)μ(da) ≤ Capa,μ(Scs) = 0.

Hence we obtain (A4) (2) for μaTail for μ-a.s. a. This completes the proof. ��

We can regard (E a,μaTail ,Da,μa

Tail) as a Dirichlet form on L2(Ha, μaTail) from (3.22)–

(3.24). Let Pas be the distribution of the unlabeled diffusion starting at s associatedwith the Dirichlet form (E a,μa

Tail ,Da,μaTail) on L2(Ha, μ

aTail) given by Lemma 9.1. Let

Pa(·) =∫

Ha

Pas (·)μaTail(ds). (12.5)

Then Pa is a μaTail-stationary diffusion on Ha.

Lemma 12.3 Assume (A1)–(A4) forμ. Then the diffusion Pa satisfies (SIN) and (NBJ)for μ-a.s.a. Furthermore, for μ-a.s.a

Capa,μaTail(Scsde) = 0, Pa ◦ l−1path(W (Ssde)) = 1. (12.6)

Proof By Lemma 12.1 and Lemma 12.2 μaTail satisfy (A1)–(A4) for μ-a.s. a, which

are the assumptions of Lemma 10.2 and Lemma 10.3. ThenμaTail and P

a satisfy (SIN),(12.6), and (NBJ) for μ-a.s.a by Lemma 10.2 and Lemma 10.3. ��Lemma 12.4 Assume that μ and {Ps} satisfy (A1)–(A4) and (B1)–(B2). Assume thatthe coefficient σm is constant. Then X = lpath(X) under Pa satisfies (IFC) for μ-a.s.a.

Proof We use Proposition 11.1 to prove Lemma 12.4. Our task is to check μaTail and{Pas } fulfill the assumptions of Proposition 11.1: namely, (A1)–(A4) and (B1)–(B2).

We seeμaTail satisfies (A1)–(A4) forμ-a.s.a by Lemma 12.1 and Lemma 12.2. From

(B1) for μ and the tail decomposition (3.20) of μ combined with Fubini’s theorem,we obtain (B1) for μa

Tail for μ-a.s.a.We next proceed with (B2) for μa

Tail and Pa. Let Im be the increasing sequence ofclosed sets in (B2) forμ and {Ps}. Then (11.4) is satisfied. Indeed, the condition (11.4)is independent of the measure, so (11.4) for μ implies (11.4) for μa

Tail.

Let μa,[m]Tail be the m-Campbell measure of μa

Tail. From the analogy of (12.3) for them-labeled Dirichlet form, we deduce for all n ∈ N3 and m ∈ N

∫Capa,μ

a,[m]Tail

(H[a]◦n\〈H[a]◦n ∩ Im〉

)μ(da) ≤ Capa,μ

[m](H[a]◦n\〈H[a]◦n ∩ Im〉

).

(12.7)

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By (11.5) for all n ∈ N3

limm→∞Capa,μ

[m](H[a]◦n\〈H[a]◦n ∩ Im〉

)= 0. (12.8)

From (12.7), (12.8), and Fatou’s lemma we obtain for μ-a.s.a and for all n ∈ N3

limm→∞Capa,μ

a,[m]Tail

(H[a]◦n\〈H[a]◦n ∩ Im〉

)= 0.

We hence obtain (11.5) for μaTail. Then we have verified (B2) for μa

Tail for μ-a.s.a.We thus see that all the assumptions of Proposition 11.1 are fulfilled for μa

Tail andPa, which deduces that Pa satisfies (IFC) for μ-a.s.a. ��

The next theorem claims quenched results follows from anneal assumptions.

Theorem 12.1 Assume that μ and {Ps} satisfy (A1)–(A4) and (B1)–(B2). Let Pa as(12.5). Assume that the coefficient σm is constant. Then, for μ-a.s.a, (3.3)–(3.4) hasa unique strong solution starting at s = l(s) for μa

Tail-a.s. s under the constraints of(MF), (IFC), (AC) for μa

Tail, (SIN), and (NBJ).

Proof We use Theorem 3.1 for the proof. We take (X,B) and P in Theorem 3.1as X := lpath(X) and P := Pa for μ-a.s. a. We check this choice satisfies all theassumptions of Theorem 3.1, namely, lpath(X) under Pa is a weak solution satisfying(IFC), (AC) for μa

Tail, (SIN), and (NBJ), and that μaTail satisfies (TT) for μ-a.s. a.

Below we omit “for μ-a.s. a”. From Lemma 9.2, Lemma 12.1, and Lemma 12.3,we see that lpath(X) under Pa is a weak solution of (3.3)–(3.4).

(TT) for μaTail follows from (3.21). Obviously, lpath(X) under Pa satisfies (AC) for

μaTail because Pa is μa

Tail-reversible. (SIN), (NBJ), and (IFC) for lpath(X) under Pa

follow from Lemma 12.3 and Lemma 12.4.Thus, all the assumptions of Theorem 3.1 are fulfilled. ��

Theorem 12.2 Under the same assumptions of Theorem 12.1 with replacement (B2)by (C1)–(C2) the same conclusions as Theorem 12.1 hold.

Proof From Proposition 11.2, (B2) holds. Then from Theorem 12.1 we deduce theclaim. ��Corollary 12.1 Make the same assumptions of Theorem 12.1 or Theorem 12.2. Thenlpath(X) under Pas is a unique strong solution of (3.3)–(3.4) starting at s = l(s) underthe same constraints as Theorem 12.1.

Remark 12.1 1. We have two diffusions {(X, Ps)}s∈S and {{(X, Pas )}s∈Ha}a∈H. The for-mer is deduced from (A2) and (A3) for μ, and the associated Dirichlet form isgiven by (E μ,Dμ) on L2(S, μ). The latter is deduced from (A2) and (A3) forμa

Tail,and is a collection of diffusions whose Dirichlet forms are (E a,μa

Tail ,Da,μaTail) on

L2(Ha, μaTail). These two diffusions are the same (up to quasi-everywhere starting

points) when μ is tail trivial. We emphasize that Theorem 12.1 does not followfrom Theorem 3.2 because we do not know how to prove {(X, Ps)}s∈S satisfies(AC) for μa

Tail.

123


2. In Theorem 12.1, we have constructed the tail preserving solution. Its uniquenesshowever is under the constraint of (AC) for μa

Tail. Hence it does not exclude thepossibility that there exists a family of solutions X not satisfying this condition,and, in particular, a family of solutions whose distributions on the tail σ -fieldT (S)changing as t varies. We refer to [15] for a tail preserving property of the Dysonmodel.

13 Examples

Wedevote to examples and applications of Theorems 11.1, 12.1, and 12.2. Throughoutthis section, b(x, y) = 1

2dμ(x, y), where dμ is the logarithmic derivative of random

point field μ associated with ISDE.We present a simple sufficient condition of (C2) introduced before Proposition 11.2.

We write x = (x1, . . . , xm) ∈ Rdm and s = ∑i δsi as before. Let J

[�] be as (11.15)with l = �. Recall that we suppress m from the notation.

Lemma 13.1 Assume that for each f ∈ {σm, bm}, j ∈ J[�], and k ∈ {1, . . . ,m}, thereexists gj,k such that gj,k ∈ C(S2\{x = s}) and that

∂jf(x, s) =m∑

k=1

⎧⎨

⎩

m∑

p �=kgj,k(xk, xp)+

∑

i

gj,k(xk, si )

⎫⎬

⎭ for (x, s) ∈ H[a].

Assume (B1). Assume that there exist positive constants c18 and c19 such that

lim supr→∞

aq(r)

rc18<∞,

∞∑

r=1

aq(r)

rc18+1<∞, (13.1)

|gj,k(x, s)| ≤c19

(1+ |s|)c18 for all (x, s) ∈ Hp,r. (13.2)

Here Hp,r = {(x, s) ∈ S2; 2−p ≤ |x − s|, x ∈ Sr} for p, r ∈ N. We then obtain (C2).

Proof From (13.2), we deduce that for (x, s) ∈ H[a]n∑

i

|gj,k(xk, si )| ≤ c19

{( ∞∑

r=1

∑

si∈Sr \Sr−1

1

(1+ |si |)c18)+ s(S0)

}

≤ c19

{( ∞∑

r=1

∑

si∈Sr \Sr−1

1

(1+ r − 1)c18

)+ s(S0)

}

= c19 lim supR→∞

{s(SR)Rc18

+R∑

r=2s(Sr−1)

{1

(r − 1)c18− 1

rc18

}+ s(S0)

}.

The last line is finite by (11.1) and (13.1). This yields (C2). ��In the rest of this section, σ is the unit matrix.

123


13.1 sineˇ random point fields/Dysonmodel in infinite dimensions

Let d = 1 and S = R. Recall ISDE (1.2) and take β = 1, 2, 4.

dXit = dBi

t +β

2limr→∞

∞∑

|Xit−X j

t |<r , j �=i

1

Xit − X j

t

dt (i ∈ Z). (13.3)

Let μsinβ be the sineβ random point field [4,21] with β = 1, 2, 4. μsin2 is the randompoint field on R whose n-point correlation function ρn

sin2is given by

ρnsin2(x) = det[Ksin2(xi , x j )]ni, j=1. (13.4)

Here Ksin2(x, y) = sin π(x − y)/π(x − y) is the sine kernel. μsin1 and μsin4 arealso defined by correlation functions given by quaternion determinants [21]. μsinβ aresolutions of the geometric differential equations (9.5) with a(x, y) = 1 and

b(x, y) = β

2limr→∞

∑

|x−yi |<r

1

x − yiin L1

loc

(μ[1]sinβ

). (13.5)

Here “in L1loc(μ

[1]sinβ

)” means the convergence in L1(Sr × S, μ[1]sinβ) for all r ∈ N.

Unlike the Ginibre random point field, (13.5) is equivalent to

b(x, y) = β

2limr→∞

∑

|yi |<r

1

x − yiin L1

loc

(μ[1]sinβ

).

We obtain the following.

Theorem 13.1 1. The conclusions of Theorem 11.1 hold for μsin2 .2. Let β = 1, 4. Let μa

sinβ ,Tailbe defined as (3.19) for μsinβ . Then, for μsinβ -a.s. a,

the conclusions of Theorem 12.1 hold for μasinβ ,Tail

.

Remark 13.1 When β = 2, the solution of ISDE (13.3) is called the Dyson model ininfinite dimensions [38]. The random point fields μsinβ are constructed for all β > 0[41]. It is plausible that our method can be applied to this case. We also remark thatTsai [40] solved ISDE (13.3) for all β ≥ 1 employing a different method.

To prove Theorem 13.1 we shall check the assumptions of Theorem 11.1 (β = 2)and Theorem 12.1 (β = 1, 4) for μsinβ . Let χn be as in Lemma 11.4.

Lemma 13.2 Let β = 1, 2, 4. For each n ∈ N3 the following hold.

1. The logarithmic derivative dμsinβ of μsinβ exists in L2(χnμ

[1]sinβ

).

123


2. The logarithmic derivative dμsinβ has the expressions.

dμsinβ (x, s) = β limr→∞

∑

|x−si |<r

1

x − siin L2

(χnμ

[1]sinβ

), (13.6)

dμsinβ (x, s) = β limr→∞

∑

|si |<r

1

x − siin L2

(χnμ

[1]sinβ

). (13.7)

Proof (1) and (13.6) follow from [26, Theorem 82]. Set Sxr = {s; |x − s| < r} and letSr " Sxr be the symmetric difference of Sr and Sxr . Then we have

limr→∞

∑

si∈Sr"Sxr

1

x − si= 0 in L2

loc

(μ[1]sinβ

)(13.8)

because d = 1 and one- and two-point correlation functions of μsinβ are bounded.Hence, (13.7) follows from (13.6) and (13.8). ��

Take � = 1 in Theorem 11.1 and note that σ(x, s) = 1 and b(x, s) = 12d

μsinβ (x, s).

Let D [1]sinβ

be the domain of the Dirichlet form associated with the 1-labeled process.

Lemma 13.3 Let β = 1, 2, 4. Then χndμsinβ , χn∇xd

μsinβ ∈ D [1]sinβ

for all n ∈ N3. Inparticular, (C1) holds for � = 1.

Proof We only prove χndμsinβ ∈ D [1]

sinβbecause χn∇xd


can be provedsimilarly. We set D[ f ] = D[ f , f ] for simplicity. By definition,

Eμ[1]sinβ(χnd

μsinβ , χndμsinβ) =∫

S×S1

2|∇xχnd

μsinβ |2 + D[χndμsinβ ]dμ[1]sinβ. (13.9)

From Lemmas 11.4 and 13.2 (1), we deduce that

∫

S×S|∇xχnd

μsinβ |2dμ[1]sinβ≤2∫

H[a]n+1{χ2

n |∇xdμsinβ |2 + |∇xχn|2|dμsinβ |2}dμ[1]sinβ

<∞.

From the Schwarz inequality and Lemma 11.4, we deduce that

∫

S×SD[χndμsinβ ]dμ[1]sinβ

≤ 2∫

S×Sχ2nD[dμsinβ ] + D[χn] |dμsinβ |2 dμ[1]sinβ

≤ 2∫

H[a]n+1D[dμsinβ ] + c17|dμsinβ |2 dμ[1]sinβ

≤ 2∫

H[a]n+1β2

2

(∑

i

1

|x − si |4)+ c17|dμsinβ |2 dμ[1]sinβ

< ∞ .

Here the last line follows from a direct calculation and Lemma 13.2. Putting theseinequalities into (13.9), we obtain χnd


for all n ∈ N3. ��

123


Lemma 13.4 μsinβ satisfies (A1)–(A4) for β = 1, 2, 4.

Proof (9.1) in (A1) follows from [26, Theorem 82]. In [27, Theorem 2.2], it wasproved that μsinβ is a (0,−β log |x − y|)-quasi-Gibbs measure for β = 1, 2, 4. Thisyields (A2). (A3) is immediate from (13.4) for β = 2, and a similar determinantalexpression of correlation functions in [21] for β = 1, 4. We next verify (A4). (10.3)holds obviously. (10.4) is satisfied by Lemma 10.1. Hence we have (A4) (2). Becauseμsinβ is translation invariant, (A4) (1) holds We thus see (A4) holds. ��Proof of Theorem 13.1 We check the assumptions of Corollary 12.1. (A1)–(A4) followfrom Lemma 13.4. Let a = {aq} be as in (11.1). Take aq(r) = qr . Then (B1) holdsbecauseμsinβ is translation invariant. (C1) follows fromLemma13.2 andLemma13.3.From the Lebesgue convergence theorem, we obtain

∇xb(x, s) = 1

2∇xd

μsinβ (x, s) = −β

2

∑

i

1

(x − si )2∈ L∞(H[a]n, μ[1]sinβ

).

Hence we can apply Lemma 13.1 to obtain (C2).Assume β = 2. Then μsin2 satisfies (TT) because μsin2 is a determinantal random

point field. Hence applying Theorem 11.2 we obtain (1). Assume β = 1, 4. Thenapplying Theorem 12.2 we obtain (2). ��

13.2 Ruelle’s class potentials

Let S = Rd andΦ = 0. LetΨ be translation invariant, that is,Ψ (x, y) = Ψ (x− y, 0)

for all x, y ∈ Rd . We set Ψ (x) = Ψ (x, 0). Then (1.1) becomes

dXit = dBi

t −β

2

∞∑

j=1, j �=i∇Ψ (Xi

t − X jt )dt (i ∈ N).

Assume that Ψ is a Ruelle’s class potential, smooth outside the origin. That is, Ψ issuper-stable and regular in the sense of Ruelle [35]. Here we say Ψ is regular if thereexists a non-negative decreasing function ψ : [0,∞)→[0,∞) and R0 such that

Ψ (x) ≥ −ψ(|x |) for all x, Ψ (x) ≤ ψ(|x |) for all |x | ≥ R0,∫ ∞

0ψ(t) td−1dt <∞.

Let μΨ be a canonical Gibbs measure with interaction Ψ . We do not a priori assumethe translation invariance ofμΨ . Instead, we assume a quantitative condition in (13.10)below, which is obviously satisfied by the translation invariant canonical Gibbs mea-sures. Let ρm be the m-point correlation function of μΨ .

Theorem 13.2 Let S = Rd and β > 0. Let Ψ be an interaction potential of Ruelle’s

class smooth outside the origin. Assume that, for each p ∈ N, there exist positive

123


constants c20 and c21 satisfying

∞∑

r=1

∫Srρ1(x)dx

rc20+1<∞, lim sup

r→∞

∫Srρ1(x)dx

rc20<∞,

|∇Ψ (x)|, |∇2Ψ (x)| ≤ c21(1+ |x |)c20 for all x such that |x | ≥ 1/p. (13.10)

Assume either d ≥ 2 or d = 1 with μΨ such that there exist a positive constant c22and a positive function h : [0,∞)→[0,∞] depending on m, R ∈ N such that

∫

0≤t≤c221

h(t)dt = ∞,

ρm(x1, . . . , xm) ≤ h(|xi − x j |) for all xi �= x j ∈ SR . (13.11)

Then the conclusions of Theorem 12.1 hold for μΨ .

We begin with the calculation of the logarithmic derivative.

Lemma 13.5 Assumption (A1) holds and the logarithmic derivative dμΨ is given by

dμΨ (x, y) = −β∑

i

∇Ψ (x − yi ) (y =∑

i

δyi ). (13.12)

Proof This lemma is clear from the DLR equation. For the sake of completeness wegive a proof. We suppose μΨ (s(S) = ∞) = 1.

Let S[1],mr = Sr ×Smr form ∈ {0}∪N, where Smr = {s ∈ S; s(Sr ) = m}. Letμ[1]r ,η be

a conditional measure of the 1-Campbell measure μ[1]Ψ conditioned at πcr (y) = πc

r (η)

for (x, y) ∈ S × S. We normalize μ[1]Ψ (· ∩ S[1],mr ) and we denote by σ [1],mr ,η the density

function of μ[1]r ,η on S[1],mr . Then, by the DLR equation and the definitions of reducedPalm and Campbell measures we obtain

σ [1],mr ,η (x, y) = 1

Z mr ,η

e−β{∑m

i=1 Ψ (x−yi )+∑mi< j Ψ (yi−y j )+∑ηk∈Scr {Ψ (x−ηk )+∑m

i=1 Ψ (yi−ηk )}},

where y = (y1, . . . , ym) ∈ Smr , η =∑

k δηk , and Z mr ,η is the normalizing constant.

Then we see that

∇x log σ[1],mr ,η (x, y) = −β

{ m∑

i=1∇Ψ (x − yi )+

∑

ηk∈Scr∇Ψ (x − ηk)

}. (13.13)

For ϕ ∈ C0(S)⊗D◦ such that ϕ(x, y) = 0 on Scr × S, we have that

123


−∫

S×S∇xϕ(x, y)μ

[1]Ψ (dxdy)

= −∞∑

m=0

∫

Sr×Smr ×S

{∇xϕ

(x,

m∑

i=1δyi

)}σ [1],mr ,η (x, y) dxdyμ[1]Ψ ◦ (πc

r )−1(dη)

=∞∑

m=0

∫

Sr×Smr ×Sϕ(x,

m∑

i=1δyi ){∇x log σ

[1],mr ,η (x, y)

}σ [1],mr ,η (x, y) dxdyμ[1]Ψ ◦ (πc

r )−1(dη)

=∫

S×Sϕ(x, y)

{−β

∞∑

i=1∇Ψ (x − yi )

}μ[1]Ψ (dxdy) by (13.13).

From this, we obtain (13.12). ��

Lemma 13.6 Assume that d = 1 and also (13.11). Then (10.4) holds.

Proof We can prove Lemma 13.6 in a similar fashion as the proof of [24, Theorem2.1]. We easily deduce from the argument of [10] that [24, (4.5)] holds under theassumptions (13.11), and the rest of the proof is the same as that of [24, Theorem 2.1].

��

Proof of Theorem 13.2 We verify that μΨ satisfies the assumptions (A1)–(A4), (B1),and (C1)–(C2) with � = 1.

(A1) follows from Lemma 13.5. We obtain (A2) from the DLR equation and theassumption that Ψ is smooth outside the origin. We deduce (A3) and (A4) from(13.10), Lemma 13.6, and ∂S = ∅. Taking aq(r) = q

∫Srρ1(x)dx , we obtain (B1).

From (13.10) we can apply the Lebesgue convergence theorem to differentiate (13.12)to obtain (C1). (C2) follow from Lemma 13.1 and (13.10). ��

Remark 13.2 1. Inukai [10] proved that the assumption (13.11) is a necessary andsufficient conditions of the particles never to collide for finite particle systems.

2. One can easily generalize Theorem 13.2 even if a free potential Φ exists.3. For given potentials of Ruelle’s class Ψ , there exist translation invariant grand

canonical Gibbs measures associated with Ψ such that the m-point correlationfunction ρm with respect to the Lebesgue measure satisfies ρm(x1, . . . , xm) ≤ cm23for all (x1, . . . , xm) ∈ (Rd)m andm ∈ N (see [35]). Here c23 is a positive constant.

Acknowledgements H.O. is supported in part by a Grant-in-Aid for Scientific Research (Grant Nos.16K13764, 16H02149, 16H06338, and KIBAN-A, No. 24244010) from the Japan Society for the Pro-motion of Science. H.T. is supported in part by a Grant-in-Aid for Scientific Research (KIBAN-C, No.15K04916, Scientific Research (B), No. 19H01793) from the Japan Society for the Promotion of Science.

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Appendix: Tail decomposition

This section proves the tail decomposition of μ using [7]. The notation in this sectionare adjusted as much as possible with [7]. We begin by preparing several notions.

Definition 14.1 (probability kernels [7, 13p]) Let (X ,X ) and (Y ,Y ) be measurablespaces. A function π :X × Y→[0, 1] is called a probability kernel from Y toX if

(i) π(·|y) is a probability on (X ,X ) for all y ∈ Y .(ii) π(A|·) is Y -measurable for each A ∈X .

For a measurable space ∗, we denote by P(∗) the set of all probabilities on ∗.Let (Ω,F ) be a measurable space. Let A ⊂ F be a sub-σ -field. Let P be a

non-empty subset ofP(Ω,F ). We set ΩP = {ω ∈ Ω ; π(·|ω) ∈P}.Definition 14.2 [(P,A )-kernel [7, (7.21) Definition, 130p]] A probability kernelπ :F ×Ω→[0, 1] is called (P,A )-kernel if it satisfies the following:

(i) π(A|·) = μ(A|A ) μ-a.s. for all μ ∈P and A ∈ F .(ii) ΩP ∈ A .(iii) μ(ΩP ) = 1 for all μ ∈P .

Let (Ω,F ), A ⊂ F , and P be as in Definition 14.2. Let

PA = {μ ∈P ; μ(A) ∈ {0, 1} for all A ∈ A }.

We set e(PA ) = σ [eA; A ∈ F ], where eA :PA →[0, 1] such that eA(μ) = μ(A).

Lemma 14.1 ([7, (7.22) Proposition, 130p]) Assume that (Ω,F ) has a countabledetermining class (also called a countable core in [7]). Suppose that there exists a(P,A )-kernel π . Then the following holds:

1. PA �= ∅.2. For each μ ∈P there exists a unique w ∈P(PA , e(PA )) such that

μ =∫

PA

νw(dν).

Furthermore, w is given by w(M) = μ({ω ∈ Ω;π(·|ω) ∈ M}) for M ∈ e(PA ).3. PA satisfies the following:

PA = {μ ∈P ; μ({ω;π(·|ω) = μ}) = 1}. (14.1)

(4) μ({ω;π(·|ω) ∈PA }) = 1 for all μ ∈P .

Proof (1) and (2) follow from [7, (7.22) Proposition]. (3) follows from [7, (7.23)]. (4)follows from 2) of the proof of [7, (7.22) Proposition] (see 12-17 lines, 131p in [7]). ��

We introduce the notion of specifications γ . We slightly modify it according to thepresent situation. Let X = Y and Y ⊂ X in Definition 14.1. Then a probabilitykernel π is called proper if π(A ∩ B|·) = π(A|·)1B for all A ∈X and B ∈ Y .

123


Definition 14.3 (specification [7, 16p]) A family γ = {γr }r∈N of proper probabilitykernels γr from σ [πc

r ] to B(S) is called a specification if it satisfies the consistencycondition γsγr = γs when r ≤ s ∈ N, where for A ∈ B(S)

γsγr (A, s) =∫

Sγs(dt, s)γr (A, t).

The random point fields in the set

G (γ ) = {υ ∈P(S,B(S)) ; υ(A|σ [πcr ]) = γr (A, ·) υ-a.s.for all A ∈ B(S) and r ∈ N}

are said to be specified by γ .

With these preparations we recall the tail decomposition given by Georgii [7]. Letμ and μa

Tail = μ( · |T (S))(a) be as in Theorems 3.1–3.2.

Lemma 14.2 Assume (A2). Let H ∈ B(S) such that μ(H) = 1. There then exists asubset of H denoted by the same symbol H, such that μ(H) = 1 and that for all a ∈ H

μaTail(A) ∈ {0, 1} for all A ∈ T (S), (14.2)

μaTail({b ∈ S;μa

Tail = μbTail}) = 1, (14.3)

μaTail and μb

Tail are mutually singular on T (S)if μaTail �= μb

Tail. (14.4)

Proof We use Lemma 14.1 for the proof of Lemma 14.2. Let (Ω,F ) = (S,B(S)).Then (Ω,F ) has a countable determining class because S is a Polish space.

Let γr (·, a) = μ(·|σ [πcr ])(a), where μ(·|σ [πc

r ])(a) is the regular conditional prob-ability of μ. By (A2), we can take a version of μ(·|σ [πc

r ])(a) in such a way thatγ = {γr }r∈N becomes a specification and μ ∈ G (γ ). We set

π(·|a) = μaTail(·), P = G (γ ), A = T (S).

From [7, (7.25) Proposition, 132p] we see that π(·|a) = μaTail(·) becomes a (P,A )-

kernel. Let Ω1 = {a;π(·|a) ∈ G (γ )}. It is also proved in the proof of [7, (7.25)Proposition, 132p] that μ(Ω1) = 1 for all μ ∈ G (γ ). Then (14.2) follows from (4) ofLemma 14.1, and (14.3) follows from (14.1); moreover, (14.4) follows from [7, (7.7)Theorem (d), 118p]. ��Remark 14.1 In [7], a parameter is taken to be a countable infinite set and spin is ageneralmeasurable set. The argument in [7] is valid in the present paper by consideringparameters consisting of a countable partition of the continuous set S and taking a spinas a configuration space on each element of the partition.

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