dirichlet operators on loop spaces: essential self-adjointness and log-sobolev inequality

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Dirichlet operators on loop spaces: Essential self-adjointness and log-Sobolev inequality Yong Moon Park and Hyun Jae Yoo Citation: Journal of Mathematical Physics 38, 3321 (1997); doi: 10.1063/1.532054 View online: http://dx.doi.org/10.1063/1.532054 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/38/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Essential self-adjointness of Wick squares in quasi-free Hadamard representations on curved spacetimes J. Math. Phys. 53, 042502 (2012); 10.1063/1.3703516 Essential self-adjointness of the graph-Laplacian J. Math. Phys. 49, 073510 (2008); 10.1063/1.2953684 Operator domains and self-adjoint operators Am. J. Phys. 72, 203 (2004); 10.1119/1.1624111 A reverse log-Sobolev inequality in the Segal-Bargmann space J. Math. Phys. 40, 1677 (1999); 10.1063/1.532825 Dirichlet forms and Dirichlet operators for infinite particle systems: Essential self-adjointness J. Math. Phys. 39, 6509 (1998); 10.1063/1.532662 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.123.44.23 On: Thu, 18 Dec 2014 13:14:21

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Page 1: Dirichlet operators on loop spaces: Essential self-adjointness and log-Sobolev inequality

Dirichlet operators on loop spaces: Essential self-adjointness and log-SobolevinequalityYong Moon Park and Hyun Jae Yoo Citation: Journal of Mathematical Physics 38, 3321 (1997); doi: 10.1063/1.532054 View online: http://dx.doi.org/10.1063/1.532054 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/38/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Essential self-adjointness of Wick squares in quasi-free Hadamard representations on curved spacetimes J. Math. Phys. 53, 042502 (2012); 10.1063/1.3703516 Essential self-adjointness of the graph-Laplacian J. Math. Phys. 49, 073510 (2008); 10.1063/1.2953684 Operator domains and self-adjoint operators Am. J. Phys. 72, 203 (2004); 10.1119/1.1624111 A reverse log-Sobolev inequality in the Segal-Bargmann space J. Math. Phys. 40, 1677 (1999); 10.1063/1.532825 Dirichlet forms and Dirichlet operators for infinite particle systems: Essential self-adjointness J. Math. Phys. 39, 6509 (1998); 10.1063/1.532662

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Dirichlet operators on loop spaces: Essential self-adjointness and log-Sobolev inequality

Yong Moon Parka)Department of Mathematics and Institute for Mathematical Sciences, Yonsei University,Seoul 120-749, Korea

Hyun Jae Yoob)Institute for Mathematical Sciences, Yonsei University, Seoul 120-749, Korea and Institutfur Mathematik, Ruhr-Universita¨t-Bochum, D-44780 Bochum, Germany

~Received 24 June 1996; accepted 7 February 1997!

For eachgP@0,1# and potential functionV:Rd→R, we consider the Dirichlet formE m

(g) and the associated Dirichlet operatorHm(g) for the Gibbs measurem on the

loop spaceE5$vPC(@0,1#;Rd):v(0)5v(1)%. The Gibbs measurem is related tothe Gibbs state of the quantum anharmonic oscillator with the potentialV via theFeynman–Kac formula. We formulate Dirichlet forms in the framework of riggedHilbert spaces which are related to the loop spaceE. We then give an approximatecriterion for the essential self-adjointness of Dirichlet operators associated withDirichlet forms given by probability measures on Hilbert spaces. Under appropriateconditions on the potential, we apply the approximate criterion to show that theDirichlet operatorHm

(g) is essentially self-adjoint on the domain of smooth cylinderfunctions. In addition, if the potential satisfies a uniform convexity condition, weprove that the Dirichlet operatorHm

(g) has a gap at the lower end of spectrum. Wealso show that the Gibbs measurem satisfies the log-Sobolev inequality. We usethe approximation method developed by Albeverio, Kondratiev, and Ro¨ckner withnecessary modifications. ©1997 American Institute of Physics.@S0022-2488~97!03306-9#

I. INTRODUCTION

In this paper we consider one parameter family of Dirichlet forms and the associated Dirichletoperators for Gibbs measures on the loop spaceE5$vPC(@0,1#;Rd):v(0)5v(1)%. For eachgP@0,1# and potentialV:Rd→R, we define the Dirichlet formEm

(g) and the associated DirichletoperatorHm

(g) for the Gibbs measurem in the framework of rigged Hilbert spaces. The Gibbsmeasurem for the potentialV is related to the Gibbs state of the quantum anharmonic oscillatorwith the potentialV via the Feynman–Kac formula.1 If the potential is three times differentiableand if the potential and its Hessian are bounded below and satisfy an appropriate growth condition,we show that the Dirichlet operatorHm

(g) is essentially self-adjoint on the domain of smoothcylinder functions. Furthermore, if the potential satisfies a uniform convexity condition, we provethat the Dirichlet operatorHm

(g) has a gap at the lower end of spectrum. We also show that theGibbs measurem satisfies the log-Sobolev inequality.2 The main ingredient we use is the approxi-mation method developed in Refs. 3–5 with necessary modifications.

Dirichlet forms and the associated diffusion processes have been intensively investigated inconnection with their important applications to mathematical physics and theory of random pro-cesses~see Refs. 6–10 and references therein!. The theory of Dirichlet forms on finite dimensionalspaces is a well-known modern tool in the potential theory8,11 and quantum mechanics.12,13Therehave been many efforts to extend the general theory to the case where the state spaces are ofinfinite dimensional~e.g., Refs. 3–7, 9, 14–19, and references therein!. In all cases the forms are

a!Electronic-mail: [email protected]!Electronic-mail: [email protected] and [email protected]

0022-2488/97/38(6)/3321/26/$10.003321J. Math. Phys. 38 (6), June 1997 © 1997 American Institute of Physics

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given first on some minimal domains of smooth functions with compact support or cylinder ones.Most of results then touch upon the problems of the closability of the forms and the constructionof corresponding diffusion processes. The uniqueness problem of determining whether a givenclosable form possessing the contraction property has a unique closed extension has also beendiscussed in recent years. There are two kinds of uniqueness problems.

~i! ~Markov uniqueness!: Is there exactly one positive definite operatorHm(g ) that extends

(Hm(g) ,D(Hm

(g)))? @i.e., D(Hm(g )).D(Hm

(g)) and Hm(g )u5Hm

(g)u on D(Hm(g)) and exp(2tHm

(g)) issub-Markovian#.

~ii ! ~Essential self-adjointness!: Is Hm(g) essentially self-adjoint inL2(dm) with a core of

minimum definition?Obviously,~ii ! implies ~i!. For Markov uniqueness problem there is a good deal of work for

finite and infinite dimensional spaces.13,20–26The results have been applied in the examples in-cluding the Dirichlet forms associated with models of space–time quantum fields~resp., time zeroquantum fields! with space–time~resp. space! cut-off interaction.

For the essential self-adjointness problem, when a given measure has a smooth logarithmicderivative, the essential self-adjointness is proved in Ref. 27 extending a Kato inequality. Inapplications, including the case of this paper, the condition of having a smooth logarithmic de-rivative is hard to fulfill. On the other hand, in Refs. 3–5, there has been developed a criterion foressential self-adjointness of Dirichlet operators and a criterion for log-Sobolev inequalities formeasures. The method is, on its base, an application of Berezansky’s abstract parabolic criterion ofessential self-adjointness.28 It is to approximate the logarithmic derivative of a given measure bya sequence of mappings with nice behavior.

In applications, the presence of log-Sobolev inequality for the Gibbs measures is essential toprove theL2-ergodicity of the semi groupTt :5exp(2Hm

(g)t), t>0, and it has a wide range ofapplications.29 The log-Sobolev inequality was first proven by Gross for the Gaussian measures onRn,2 and then extended in many directions.5,29–34

Let us describe briefly the results and main ideas in this paper. Denote by (•,•) the innerproduct inL2(@0,1#):5L2(@0,1#;Rd,dt). Let Dp be the Laplacian operator onL2(@0,1#) withperiodic boundary conditions. Let us define

A:5~2Dp11!.

Let m0 be the Gaussian measure on the spaceL2(@0,1#) with mean zero and covariance operatorA21, i.e.,

E exp@ i ~h,v!#dm0~v!5expF21

2~h,A21h!G , hPL2~@0,1# !.

We notice thatm0 is supported on the loop spaceE. For any potentialVPC3(Rd;R) which isbounded below, the Gibbs measurem on E is given by

dm~v!51

ZexpH 2E

0

1

V~v~t!!dtJ dm0~v!, ~1.1!

whereZ is the normalization factor~partition function!, andV(x):5V(x)2 12uxu2. We notice that

m(E)51.For eachgP@0,1# we introduce real separable Hilbert spaces,H1

(g) , H (g), andH2(g) with

scalar products•,•&1(g) , ^•,•& (g), and^•,•&2

(g) , respectively, such thatE,H2(g) and

H1~g!,H~g!,H2

~g! ~1.2!

3322 Y. M. Park and H. J. Yoo: Dirichlet operators for Gibbs measures on loop spaces

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is a rigging ofH (g) by H1(g) andH2

(g) .3–5,28 See~2.12! and ~2.13! in Section II. The dualitybetweenH1

(g) andH2(g) is given by the scalar product inH (g) and is denoted by•,•& (g). Denote

by F Cb`(H2

(g)) the set of all smooth cylinder functions onH2(g) with all derivatives bounded.5

Define the bilinear form~pre-Dirichlet form! by

D~Em~g!!5F Cb

`~H2~g!!,

~1.3!

Em~g!~u,v !5

1

2EH2~g!

^¹u,¹v&~g!dm, u,vPF Cb`~H2

~g!!.

We define the Dirichlet operatorHm(g) associated toEm

(g) by

Hm~g!v~v!52 1

2 Dv~v!2 12 ^b~g!~v!,¹v~v!&~g!, vPF Cb

`~H2~g!!, ~1.4!

where

b~g!v~t!52Agv~t!2A2~12g!]V~v~t!!, ~1.5!

where]V denotes the gradient ofV. See Section II for the notation. It can be shown thatubu2(g)

PL2(H2(g) ,dm) ~Lemma 4.1!. Thus,Hm

(g) is a well defined symmetric operator. Due to theintegration by parts formula, the relation

Em~g!~u,v !5~u,Hm

~g!v !L2~m! , u,vPF Cb`~H2

~g!!,

holds. Thus, the pre-Dirichlet form (Em(g) ,F Cb

`(H2(g))) is closable and it also can be shown that

it is Markovian and quasi-regular in the sense of Ref. 9. Therefore, the closure is a Dirichlet formand there exists an associated diffusion process.9

In order to show the essential self-adjointness ofHm(g) onF Cb

`(H2(g)), we use an approximate

criterion~Theorem 3.1! which is a modification of Theorem 1 of Ref. 5. As in Ref. 5, we introducea sequence$bn :nPN%, bn :H2

(g)→H2(g) , which approximates the logarithmic derivativeb (g) of

the Gibbs measurem. The condition~iv! of Theorem 1 of Ref. 5 is that there exists a constantc>0 such that the bounds

^bn8~v!h,h&2<cuhu22 , hPH2 ,

hold uniformly innPN andvPH2 , wherebn8 is the derivative ofbn in the sense of Fre´chet. Inour case, the above bound does not hold in general. Instead, under Assumption 2.1 ford51 ~withan additional assumption ford>2), it can be shown that the bounds

^bn8~v!h,h&~g!<c~ uhu~g!!2, hPH~g! ~1.6!

hold uniformly in nPN andvPH2(g) . Thus, we need to develop an approximate criterion of

essential self-adjointness of Dirichlet operators which is applicable to our situation. See Section IIIand Section IV for the details. We plan to apply the criterion to prove the essential self-adjointnessof Dirichlet operators for quantum unbounded spin systems in a forthcoming paper.35

If the potentialV satisfies the uniform convexity condition~Assumption 5.1! the Gibbs mea-surem satisfies the uniform log-concavity condition,5 and so the existence of a gap at the lowerend of the spectrum ofHm

(g) follows, and, furthermore, the measurem satisfies the log-Sobolevinequality.2

We organize the paper as follows: In order to fix the notation, we recall the definition ofDirichlet operators associated with Dirichlet forms in the framework of rigged Hilbertspaces3–7,27,30in Section II. For given potentialV:Rd→R, we then introduce the Gibbs measure

3323Y. M. Park and H. J. Yoo: Dirichlet operators for Gibbs measures on loop spaces

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m on the loop space, and list basic assumptions on the potential~Assumption 2.1!. For givengP@0,1# and the potential, we define the pre-Dirichlet formEm

(g) and the associated DirichletoperatorHm

(g) . In Section III, we give an approximate criterion~Theorem 3.1! for essential self-adjointness of the Dirichlet operator which is a modification of Theorem 1 of Ref. 5. We thenproduce the proof of the theorem by using a method similar to that employed in Ref. 5. In SectionIV, we apply the approximate criterion to show that for eachgP@0,1# the Dirichlet operatorHm(g) is essentially self-adjoint on the smooth cylinder functions. For the proof we introduce a

sequence$bn :nPN% which approximates the logarithmic derivativeb (g) of the Gibbs measurem and show that the conditions in Theorem 3.1 are satisfied. In Section V, under the uniformconvexity condition on the potential, we show thatHm

(g) has a gap at the lower end of the spectrum~Theorem 5.1!. We also show that the Gibbs measure satisfies the log-Sobolev inequality~Theo-rem 5.2!.

II. DIRICHLET FORMS AND DIRICHLET OPERATORS FOR GIBBS MEASURES ONLOOP SPACE

In this section we introduce one parameter family of Dirichlet forms and the associatedDirichlet operators for a given Gibbs measure on the loop space,

E5$vPC~@0,1#;Rd!:v~0!5v~1!%.

The Gibbs measures we are dealing with are related to the Gibbs states of quantum anharmonicoscillators via the Feynman–Kac formula.1

In order to fix the notation we review briefly the general formalism of Dirichlet forms andDirichlet operators in the framework of a rigged Hilbert space.3–7,27,30LetH be a separable realHilbert space with scalar product^•,•& and normu•u, and let

H1,H,H2 ~2.1!

be a rigging ofH by the Hilbert spacesH1 andH2 with scalar products and norms^•,•&1 ,u•u1 , resp.,^•,•&2 , u•u2 . We suppose that the embeddings in~2.1! are everywhere dense andbelong to the Hilbert–Schmidt class. We also suppose thatu•u2<u•u<u•u1 . Otherwise it issufficient to renormH1 . The duality betweenH1 andH2 is given by the inner product inH and will also be denoted by•,•&. We also use the usual complexifications of the rigging~2.1!.The complexification of a real Hilbert spaceH will be denoted byHC” .

Denote byCk(H2 ,B) the set of all mappings fromH2 into a Banach spaceB that arek-times continuously differentiable in the sense of Fre´chet. DefineCb

k(H2 ,B) as the subset ofCk(H2 ,B) which are characterized by the condition of global boundedness in the usual operatornorms of derivatives

f ~ l !:H2→L~H2 ,L~H2 , . . . ,L~H2 ,B!••• !!, l50,1, . . . ,k.

For f :H2→C” identify f 8(•)PL(H2 ,C” ) with the vector f 8(•)PH1,C” and f 9(•)PL(H2 ,L(H2 ,C” )) with the operatorf 9(•)PL(H2 ,H1,C” ) by the formula

f 8~x!h5^ f 8~x!,h&, ~ f 9~x!h!g5^ f 9~x!h,g& ~h,gPH2!. ~2.2!

For the function fPC2(H2):5C2(H2 ,C” ) we make the convention that the symbols¹ f :5 f 85 f 8 and f 95 f 9 denote the realization of their first and second derivatives, respectively,in terms of the scalar product inH in the sense of~2.2!. In the spaceCb

2(H2),C2(H2) weintroduce the norm

3324 Y. M. Park and H. J. Yoo: Dirichlet operators for Gibbs measures on loop spaces

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i f iCb25 sup

xPH2

$u f ~x!u1u f 8~x!u11i f 9~x!iL~H2 ,H1,C” !%. ~2.3!

For fPCb2(H2) we denote byD f5TrH( f 9) the trace inH of the operatorf 9 and it is finite due

to the inclusionf 9PL(H2 ,H1,C” ).We denote byF Cb

`(H2) the set of all smooth cylinder functions onH2 with all derivativesbounded. That is, if fPF Cb

`(H2), there exist NPN, f1 ,f2 , . . . ,fN,H1 and f NPCb

`(RN) such that

f ~x!5 f N~^x,f1&, . . . , x,fN&!, xPH2 . ~2.4!

Introduce also the setCpolk (H2 ,B),Ck(H2 ,B) of all polynomially bounded mappings, i.e., any

fPCpolk (H2 ,B), satisfies

uu f ~ l !~x!uu<c~11uxu2!p, xPH2 ,

for some c.0 and pPN in the corresponding operator norms of the derivativesf ( l ),l50, . . . ,k.5 For example, for anyfPCpol

2 (H2 ,C” ), there existc.0 andpPN such that

u f ~x!u1u f 8~x!u11i f 9~x!iL~H2 ,H1,C” !<c~11uxu2!p, xPH2 .

Let m be a probability measure on the Borels-algebraB(H2) with suppm5H2 , i.e.,m(U).0 for every non-empty open setU. It can be shown thatF Cb

`(H2) is dense inL2(H2 ,m).9 Define the bilinear form~pre-Dirichlet form!

D~Em!5F Cb`~H2!,

~2.5!

Em~u,v !51

2EH2

^¹u~x!,¹v~x!&dm~x!.

Then (Em ,F Cb`) is a densely defined positive definite symmetric bilinear form onL2(H2 ,m).

Let m be quasi-invariant under translations by vectors inH1 . That is, for eachfPH1 themeasuredm(x) anddmf(x) are mutually absolutely continuous and the Radon–Nikodym deriva-tive,

rf~x!5dmf~x!

dm~x!PL1~H2 ,m!

is defined, wheredmf(•):5dm(•1f). We suppose that there exists the logarithmic derivativebf of the measurem in all directionsfPH1 in the sense of the equality

liml→0

EH2

f ~x!$rlf~x!21%dm~x!5EH2

f ~x!bf~x!dm~x!, fPCb2~H2!, ~2.6!

wherebf can be represented as

bf~x!5^b~x!,f& m2a.e.xPH2 ,fPH1 , ~2.7!

for some measurable mappingb:H2→H2 . We callb the logarithmic derivative of the measurem.

Let us introduce a differential operatorHm on the domainD(Hm)5F Cb`(H2) by the for-

mula

3325Y. M. Park and H. J. Yoo: Dirichlet operators for Gibbs measures on loop spaces

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~Hmv !~x!52 12 Dv~x!2 1

2 ^b~x!,¹v~x!&, vPF Cb`~H2!, xPH2 . ~2.8!

By the definition ofb, a direct computation yields

Em~u,v !5~u,Hmv !L2~m! , u,vPF Cb`~H2!. ~2.9!

The operatorHm is called the Dirichlet operator associated to the pre-Dirichlet form(Em ,F Cb

`(H2)). Notice thatHm is a symmetric operator onL2(m).We now introduce Dirichlet forms associated to Gibbs measures on the loop space

E5$vPC~@0,1#;Rd!:v~0!5v~1!%, ~2.10!

equipped with the uniform normuvuu :5suptP[0,1]$uv(t)u%. If one considers Gibbs measures atthe temperatureT, one has to replaceE by $vPC(@0,1/T#;Rd):v(0)5v(1/T)%. In order toavoid unnecessary notational complications we setT51 throughout this paper. Denote byDp theLaplacian operator onL2(@0,1#;Rd,dt) with periodic boundary conditions. Put

A:52Dp11. ~2.11!

For anygP@0,1#, letH1(g) , H (g), andH2

(g) be the real Hilbert spaces obtained by completionsof C`(@0,1#;Rd) with normsu•u1

(g) , u•u(g), andu•u2(g) induced by

^h,g&1~g!5~Ah,Ag!L2,

^h,g&~g!5~A~12g!/2h,A~12g!/2g!L2, ~2.12!

^h,g&2~g!5~A2gh,A2gg!L2,

where we have used the abbreviated notationL2:5L2(@0,1#;Rd,dt). Notice that for anygP@0,1#, A2(11g)/2 belongs to the Hilbert–Schmidt class and so

H1~g!,H~g!,H2

~g! ~2.13!

is a rigging ofH (g) byH1(g) andH2

(g) and the embeddings belong to the Hilbert–Schmidt class.Let m0 be the Gaussian measure on (E,B(E)) for which its characteristic functional is given

by

EEexp$ i ~v,h!L2%dm0~v!5expH 2

1

2~h,A21h!L2J , hPL2. ~2.14!

The above Gaussian measure can be expressed as a measure onH2(g) . For any givengP@0,1# let

(H2(g) ,B(H2

(g)),m0) be the Gaussian measure space for which its characteristic functional isgiven by

EH2

~g!exp$ i ^v,h&~g!%dm0~v!5expH 2

1

2^h,A2gh&~g!J , hPH1

~g! . ~2.15!

We remark thatm0(E)51 and~2.15! reduces to~2.14! on E.Next we consider the potentialV:Rd→R and the corresponding Gibbs measure. For

x5(x1,x2, . . . ,xd)PRd let uxu be the Euclidean norm ofx. For fPC1(Rd,R) denote by] i f ,i51,2, . . . ,d, the partial derivatives off .

Assumption 2.1: The potential V:Rd→R is a three times continuously differentiable functionsatisfying the following conditions.

3326 Y. M. Park and H. J. Yoo: Dirichlet operators for Gibbs measures on loop spaces

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(a) There exist positive constants a and b such that for somea>2 the bound

V~x!>auxua2b

holds.(b) For any positive real numberd.0, there exists positive constant D(d).0 such that the

bound

uV~x!u1(i51

d

u] iV~x!u1 (i , j51

d

u] i] jV~x!u<D~d!exp~duxu2!

holds.(c) There exists a constant MPR such that

Hess.V~x!>M1,

uniformly in xPRd, whereHess.V is the Hessian of V, i.e., the d3d matrix for which its i- jelement, i, j51,2, . . . ,d, is given by] i] jV(x).

For given potentialV we write

V~x!5V~x!2 12 uxu2. ~2.16!

We define the Gibbs measurem on (H2(g) ,B(H2

(g))) associated to the potentialV by

dm~v!51

ZexpH 2E

0

1

V~v~t!!dtJ dm0~v!, ~2.17!

whereZ is the normalization factor~the partition function! given by

Z5EH2

~g!expH 2E

0

1

V~v~t!!dtJ dm0~v!.

We remark that the functional*01V(v(t))dt is defined onE and so it is defined onH2

(g) m —a.e.By a of Assumption 2.1 and~2.16! the partition functionZ is finite.

As in ~2.5!, we define for eachgP@0,1# a pre-Dirichlet form,

D~Em~g!!5F Cb

`~H2~g!!,

~2.18!

Em~g!~u,v !5

1

2EH2~g!

^¹u~x!,¹v~x!&~g!dm~x!.

We next derive the Dirichlet operator associated toEm(g) . By the definition of the Gibbs measure

m in ~2.14! and ~2.17!, the logarithmic derivativebf(g) of the measurem in the directionf

PH1(g) is given by

bf~g!~v!52~v,Af!L22~]V~v!,f!L25^Agv,f&~g!2^A2~12g!]V~v!,f&~g!, m2a.e.,

where]V(x) is the gradient ofV(x), xPRd, and forvPE, ]V(v)PE is defined by

]V~v!~t!:5]V~v~t!!, tP@0,1#.

3327Y. M. Park and H. J. Yoo: Dirichlet operators for Gibbs measures on loop spaces

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In this paper we use the notation]g, instead of¹g, for the gradiant ofgPC1(Rd,R) to avoidunnecessary confusion with¹u for uPC1(H2). Thus, the logarithmic derivative ofm is givenby

b~g!~v!52Agv2A2~12g!]V~v!. ~2.19!

We considerb (g) as a mapping fromE(,H2(g)) to H2

(g) . Thus,b (g) is definedm2a.e. As in~2.8! we define the Dirichlet operatorHm

(g) associated toEm(g) by

~Hm~g!v !~v!52 1

2 Dv~v!2 12 ^b~g!~v!,¹v~v!&~g!, vPF Cb

`~H2~g!!, vPE. ~2.20!

We remark thatDv and¹v depend also ong which we suppressed in the notation. In Section IV,we will show that ubu2

(g)PL2(m). Thus, Hm(g) is a densely defined symmetric operator on

L2(m), and the relation

Em~g!~u,v !5~u,Hm

~g!v !L2~m! , u,vPF Cb`~H2

~g!!

holds.Remark 2.1: The most interesting Dirichlet operators would be Hm

(0) and Hm(1) which corre-

spond tog50 and g51, respectively. These Dirichlet operators and corresponding Dirichletforms have been considered in Ref. 36 and Ref. 37, respectively, for one site (one particle) spaces.

III. AN APPROXIMATE CRITERION OF ESSENTIAL SELF-ADJOINTNESS

In this section, we give a modified version of the approximate criterion given by Albeverio,Kondratiev, and Ro¨ckner3–5 for essential self-adjointness of Dirichlet operators associated withDirichlet forms given by probability measures.

We return to the general formalism of Dirichlet forms and Dirichlet operators in the frame-work of rigged Hilbert spaces. LetEm andHm be the Dirichlet form and the associated Dirichletoperator given by a probability measurem on H2 , respectively. See~2.5! and ~2.8! for thedefinitions. As in Ref. 5, we assume that for anypPN,

EH2

uxu2p dm~x!,` ~3.1!

and

EH2

ub~x!u22 uxu2

p dm~x!,`. ~3.2!

The condition~3.1! gives the embeddings

Cb2~H2!,Cpol

2 ~H2!,L2~m!:5L2~H2 ,B~H2!,m!.

Under the condition~3.1!, due to the Ho¨lder’s inequality a sufficient condition for~3.2! is thatthere existsd.0 such that

EH2

ub~x!u221ddm~x!,`. ~3.3!

By ~3.2! the Dirichlet operatorHm is defined onCpol2 (H2).

The following result is a modified version of Theorem 1 of Ref. 5:

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Theorem 3.1:Letm be a probability measure onB(H2) which satisfies the conditions (3.1)and (3.2). Letb be written asb5b11b2 . Suppose that there exists a sequence$bn :nPN%,bn :H2→H2 , bn5b1,n1b2,n , nPN, such that the following properties hold:

(i) For any nPN, bnPCpol2 (H2 ,H2).

(ii) For any nPN, there exists a constant c(n)>0 such that the following bound holds:

^bn~x!,x&2<c~n!~11uxu22 !, xPH2 .

(iii) For any nPN, there exists a constant M(n)>0 such that the bound

ibn8~x!iL~H2 ,H!<M ~n!

holds uniformly in xPH2 .(iv) For any nPN, there exists a constant c1(n) depending on nPN such that for any h

PH2 the bound

^h,bn8~x!h&2<c1~n!uhu22

holds uniformly in xPH2 .(v) There exists a constant c2>0 andN0PN, such that for any n>N0 and hPH the bound

^h,b2,n8 ~x!h&<c2uhu2

holds uniformly in xPH2 and n>N0 .(vi) There exists a sequence$an :nPN% of positive real numbers such that for the constants

c1(n), nPN, appeared in (iv),

an exp~c1~n!/2!→0, as n→`,

and such that for any nPN,

iub1,n2b1,nu2iL2~m!<an .

(vii) iub2,n2b2,nuiL2(m)→0 as n→`.Then, the Dirichlet operator Hm is essentially self-adjoint on Cb

2(H2).If one compares the conditions in the above theorem to those in Theorem 1 of Ref. 5, it looks

very complicated. Thus, a few comments are in order.Remark 3.1: (a) Suppose that one can choose the constants c1(n), nPN, in the condition (iv)

independently of nPN, i.e., there exists a constant c such that for any nPN and hPH2 thebound

^h,bn8~x!h&2<cuhu22 ~3.4!

holds uniformly in nPN and xPH2 . In this case, we do not need to split bn5b1,n1b2,n . Thecondition (iii) which will be used typically for the ‘‘quantum’’ cases (e.g., for loop spaces in thispaper or for quantum spin systems with a slightly different form35 [cf. (iii) 8 of Theorem 3.2)] is notneeded. See (3.10) and Lemma 3.1 below. The condition (vi) is then reduced to

iub2bnu2iL2→0, as n→`. ~3.5!

The case of classical spin systems is an example.5,38The conditions (i), (ii), (3.4), and (3.5) are theconditions in Theorem 1 of Ref. 5. Thus, Theorem 3.1 can be considered as an extension ofTheorem 1 of Ref. 5.

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(b) Consider the case in whichb150 and b25b. Choose b1,n50 and b2,n5bn for any nPN. Then, the condition (iv) and (vi) are not needed for Theorem 3.1 to hold. See the proof ofTheorem 3.1 below. The study for classcical spin systems belong to this case again.

(c) In the application in Section IV, we shall take the first term and the second term in theright hand side in (2.19) forb1 andb2 , respectively.

Proof of Theorem 3.1: We shall follow the method used in the proof of Theorem 1 of Ref. 5with suitable modifications. For anynPN, we define a differential operatorHn on the domainCpol2 (H2) by the formula

Hnu52 12 Du2 1

2 ^bn ,¹u&. ~3.6!

We shall use the following general parabolic criterion of essential self-adjointness~see Ref. 28,Chap. 5, Theorem 1.10!: Let us consider the Cauchy problems

d

dtun~ t !1Hnun50, un~0!5 f ,tP@0,1#, ~3.7!

where fPCb2(H2) is arbitrary. If one can prove the existence of strong solutions

un :@0,1#→L2~m!,

for ~3.7! such that

un~ t !PD~Hn!, for any tP@0,1# and nPN ~3.8!

and

E0

1i~Hm2Hn!un~ t !iL2~m!dt→0, n→`, ~3.9!

thenHm is essentially self-adjoint onCb2(H2).

The existence and the differentiability of a strong solution of~3.7! satisfying~3.8! are guar-anteed by the fact thatbnPCpol

2 (H2 ,H2), nPN. See Ref. 5 and the references therein. Thus,we only need to show~3.9!. It follows from ~2.8! and ~3.6! that

E0

1i~Hm2Hn!un~ t !iL2~m!dt5E

0

1i^b2bn ,¹un~ t !&iL2~m!dt

<E0

1i^b12b1,n ,¹un~ t !&iL2~m!dt1E

0

1i^b22b2,n ,¹un~ t !&iL2~m!dt

<E0

1i~ ub12b1,nu2!~ u¹un~ t !u1!iL2~m!dt

1E0

1i~ ub22b2,nu!~ u¹un~ t !u!iL2~m!dt. ~3.10!

Under the assumptions in Theorem 3.1, we shall show that asn goes to infinity, each term in theright hand side of~3.10! converges to zero.

In order to control¹un(t) in ~3.10!, we use the method similar to that in Ref. 5. As in Refs.3–5, let us consider the stochastic differential equation,

djn,x~ t !5 12bn~jn,x~ t !!dt1dw~ t !. ~3.11!

3330 Y. M. Park and H. J. Yoo: Dirichlet operators for Gibbs measures on loop spaces

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Here,w:@0, )→H2 is a standard Wiener process which corresponds to the Hilbert spaceH.That is, there is given a probability space (V,S,P) and a Gaussian processR13V{(t,v)°w(t,v)PH2 such that for anywPH1 and t,sPR1

E~^w~ t !,w&!50, E~^w~t!2w~s!,w&2!5ut2suuwu2,

whereE(•) denotes the expectation with respect to the measureP. Notice that the rigorous formof ~3.11! gives the integral equation

jn,x~ t !5x11

2E0t

bn~jn,x~t!!dt1w~ t !. ~3.12!

For the existence and various differentiability properties of the solution of~3.12!, we refer to Ref.5 and the references therein. For anyhPH2 , let hh(t) be the directional derivative ofjn,x(t) inthe directionh:

hh~ t !:5jn,x8 ~ t !h.

From ~3.12! one has

hh~ t !5h11

2E0t

bn8

~jn,x~t!!hh~t!dt. ~3.13!

See also the equation~18! of Ref. 5. For givenfPCb2(H2) andnPN, we introduce the function

un~ t,x!5E$ f ~jn,x~ t !!%, xPH2 ,tPR1 . ~3.14!

By Lemma 5 of Ref. 5,un(t) is continuously differentiable inL2(m) and is a solution of~3.6!.Lemma 3.1: (a) For any nPN and fPCb

2(H2), the bound

u¹un~ t !u1<i f iCb2ec1~n!t/2

holds, where c1(n), nPN, are the constants in the condition (iv) of Theorem 3.1.(b) For any fPCb

2(H2), the bound

u¹un~ t !u<i f iCb2ec2t/2

holds uniformly in n>N0 , where N0PN and c2>0 are the constants appeared in the condition(v) of Theorem 3.1.

Proof: ~a! This follows from the condition~iv! of Theorem 3.1 and the bound in~25! of Ref.5.

~b! Using ~3.13! and the condition~iii ! in Theorem 3.1, we obtain that forhPH,

uhh~ t !u<uhu11

2M ~n!E

0

t

uhh~t!u2dt, P2a.s.

We use the condition~iv! in Theorem 3.1. Then, by Lemma 2 of Ref. 5 it follows that

uhh~t!u2<uhu2exp~c1~n!t/2!, P2a.s.

Sinceu•u2<u•u<u•u1 , we see that for anyhPH, hh(t)PH, P2a.s. It follows from~3.14! thatfor any hPH andxPH2 ,

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^¹un~ t,x!,h&5un8~ t,x!h5E$^ f 8~jn,x~ t !!,hh~ t !&%

<$supxPH2u f 8~x!u%E~ uhh~ t !u!<i f iC

b2E~ uhh~ t !u!. ~3.15!

Here we have used the fact thatu f 8(x)u<u f 8(x)u1 to get the last inequality. Using the condition~v! in Theorem 3.1 and~3.13!, we conclude that forn>N0,

uhh~ t !u22uhu25E0

t d

dtuhh~t!u2dt5E

0

t

^hh~t!,bn8~jn,x~t!!hh~t!&t<c2E0

t

uhh~t!u2dt.

By the Gronwall’s inequality we obtain from the above bound that forn>N0 ,

uhh~ t !u2<uhu2ec2t. ~3.16!

Since

u¹un~ t,x!u5 suphPH:uhu51

u^¹un~ t,x!,h&u,

the part~b! of the lemma follows from~3.15! and the above bound. j

Let us return to the proof of Theorem 3.1. We use~3.10!, Lemma 3.1, and the conditions~v!– ~vii ! in Theorem 3.1 to conclude that

E0

1i~Hm2Hn!un~ t !iL2~m!dt<iub12b1,nu2iL2~m!$i f iC

b2ec1~n!/2%1iub22b2,nuiL2~m!$i f iC

b2ec2/2%

→0, as n→`.

This completes the proof of Theorem 3.1. j

Let us generalize Theorem 3.1 so that the result turns out to be useful in the study of the othersubjects.35 Let K1 andK2 be real separable Hilbert spaces with scalar products and norms^•,•&K1

, u•uK1, resp.,^•,•&K2

, u•uK2. Suppose that the inclusions

H1,K1,H,K2,H2 ~3.17!

hold and suppose that the duality betweenK1 andK2 is given by the scalar product inH. Wedo not assume that the embeddingsK1,H andH,K2 belong to the Hilbert–Schmidt class.The following is a generalization of Theorem 3.1.

Theorem 3.2: Suppose that all the conditions except the conditions (iii), (v), and (vii) inTheorem 3.1 are satisfied. Suppose that the following properties hold:

(iii) 8 For any nPN, there exists a constant M(n)>0 such that the bound

ibn8~x!iL~H2 ,K2!<M ~n!

holds uniformly in xPH2 .(v)8 There exists a constant c2>0 and N0PN such that for any n>N0 , xPH2 , and h

PK2 the bound

^h,bn8~x!h&K2<c2uhuK2

2

holds uniformly in xPH2 .(vii)8 iub22b2,nuK2

iL2(m)→0, asn→`.

Then, the Dirichlet operator Hm is essentially self-adjoint on Cb2(H2).

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Proof. We first assert that for anyfPCb2(H2) the bound

u¹un~ t !uK1<i f iC

b2ec2t/2 ~3.18!

holds uniformly inn>N0 . To prove the above bound, we use the methods similar to that em-ployed in the proof of Lemma 3.1~b!. Notice that the condition~iii !8 implieshh(t)PK2 for anyhPK2 . Using the duality betweenK1 andK2 ~with respect to the scalar product^•,•& inH), and the method similar to that used to obtain~3.16!, we prove the assertion.

As in ~3.10!, we have

E0

1i~Hm2Hn!un~ t !iL2~m!dt<E

0

1i~ ub12b1,nu2!~ u¹un~ t !u1!iL2~m!dt1E

0

1i~ ub22b2,nuK2

!

3~ u¹un~ t !uK1!iL2~m!dt.

We note that the conditions~iii !, ~v!, and ~vii ! in Theorem 3.1 were not used in the proof ofLemma 3.1~a!. Using the condition~vi! in Theorem 3.1, the condition~vii !8, Lemma 3.1~a!, andthe bound~3.18! to the above, we complete the proof of the theorem. j

The following result, which can be proven using Theorem 3.1, is Lemma 6 of Ref. 5 for thespaceF there being replaced byH1 .

Lemma 3.2 (Ref. 5, Lemma 6): Assume thatubu2PL2(m), so that Hm is well-defined on thedomain Cb

2(H2). Then, the closure of(Hm ,F Cb`(H2)) coincides with the closure of

(Hm ,Cb2(H2)).

We will apply the above lemma to show that for eachgP@0,1#, the Dirichlet operatorHm(g) is

essentially self-adjoint onF Cb`(H2

(g)).

IV. ESSENTIAL SELF-ADJOINTNESS OF DIRICHLET OPERATORS

For anygP@0,1# we will prove thatHm(g) defined in~2.19! and ~2.20! is essentially self-

adjoint onCb2(H2

(g)) @and also onF Cb`(H2

(g))] by showing that all the conditions in Theorem 3.1hold. In the case ford51, Assumption 2.1 is sufficient. Ford>2, we impose the sphericalsymmetricity to the potentialV.

Assumption 4.1: Consider the case d>2.The potential V:Rd→R is a three times continuouslydifferentiable function satisfying the following condition: There exists a function Q:R→R suchthat V(x)5Q(uxu) for any xPRd. That is, V is spherically symmetric.

Remark 4.1: The spherical symmetricity of the potential V for d>2 is introduced for technicalreasons. We believe that the restriction is unnecessary. At the beginning of Section VI, we shallgive a possible relaxation of the spherical symmetricity such that Theorem 4.1 and Theorem 5.1still hold.

The following is the main result in this section.Theorem 4.1: Let the potential V satisfy the conditions in Assumption 2.1. For d>2, we

further assume that V also satisfies the condition in Assumption 4.1. Then, for eachgP@0,1#, theDirichlet operator Hm

(g) is essentially self-adjoint on Cb2(H2

(g)). Here F Cb`(H2

(g)) is also adomain of essential self-adjointness of Hm

(g) .In the rest of this section we produce the proof of Theorem 4.1 by showing that all the

conditions in Theorem 3.1 are satisfied. In order to avoid the unnecessary notational complica-tions, we suppress the parametergP@0,1# in the notations if there is no confusion involved. Thus,in the rest of this sectiongP@0,1# is given ~fixed!, and we use the notationH2 :5H2

(g) ,^•,•&2 :5^•,•&2

(g) , u•u2 :5u•u2(g) , etc. The following lemma implies that the conditions in~3.1!

and ~3.2! are satisfied and also thatHm is well defined onCb2(H2).

Lemma 4.1: Under Assumption 2.1, the following results hold:(a) For any pPN,

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EH2

uvu2p dm~v!,`;

~b!

EH2

~ ub~v!u2!4dm~v!,`,

whereb(:5b (g)) is the logarithmic derivative ofm given in (2.19).Proof: ~a! We recall that any Gibbs measurem is supported onE, i.e.,m(E)51. We notice

that (m0 ,H(0),E) is an abstract Wiener space.39 By Fernique’s theorem~see e.g., Ref. 39, Chap.

III, Theorem 3.1! we see that there exists a constanta.0 such that

EEeauvuu

2dm0~v!,`, ~4.1!

whereu•uu is the uniform norm inE. Part~a! follows from ~4.1! and the fact thatuvu2<uvuu .~b! From ~2.12! and the definition ofb in ~2.19!, it follows that

ubu2<uvuu1u]V~v!uL2. ~4.2!

By Assumption 2.1~a! and ~b!, we see that for sufficiently smalld.0,

EEu]V~v!uL2

4 dm~v!<D~d!4EEH E

0

1

exp~4duv~t!u2!dtJ dm~v!<MEEexp~4duvuu

2!dm0~v!

,`,

where we have used~4.1! in the last inequality. j

In order to apply Theorem 3.1, we need to introduce a sequence$bn :nPN% which satisfies theconditions in Theorem 3.1. We first note that forg.0 the Hilbert spaceH25H2

(g) introduced in~2.12! consists of generalized functions and so*0

1V(v(t))dt, vPH2 , is not defined in general.Thus, we introduce a regularization. Let$el :el(t)5exp(2pilt),tP@0,1#,lPZ% be the orthonormalbasis for L2(@0,1#;C” ,dt) of eigenvectors of A52Dp11. For v5(v1,v2, . . . ,vd)PH,L2(@0,1#;Rd,dt), define a partial sum operatorSk , kPN, by

Sk~v!5S (l152k

k

~el1,v1!el1, . . . , (

l d52k

k

~eld,vd!eldD . ~4.3!

We define the mean operatorMn , nPN, by

Mn~v!51

n11(k50

n

Sk~v!, vPH. ~4.4!

Notice thatMnPL(H,H1). See Ref. 40, Chap. 8, for the detailed properties ofMn .Let g:R→R be aC`-function satisfying the following properties:38

~a! g~2t !52g~ t !;

~b! g~ t !5t for tP~21,1!;~4.5!

~c! g is monotonic increasing, i.e.,g8~ t !.0 and g8~ t !<1 for tPR;

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~d! g~ t !→2 as t→`.

For nPN and tPR we put

gn~ t !5ng~ t/n!. ~4.6!

We are ready to introduce a sequence$bn :nPN% of mappings which approximates the logarith-mic derivativeb of the Gibbs measurem. Recall the definition ofb(5b (g)) in ~2.19!. We write

b~v!5b1~v!1b2~v!, b1~v!52Agv, b2~v!52A2~12g!]V~v!. ~4.7!

For eachnPN, define a~bounded! linear operatorGn :H2→H1 by

Gnv5AMnA21v5~AMnA

21v1, . . . ,AMnA21vd!, v5~v1, . . . ,vd!PH2 . ~4.8!

Let $an :an.0,nPN% be a sequence of positive real numbers which satisfies the condition that foranyaPR1 ,

anexp$ean2%→0, as n→`. ~4.9!

Recall that ford>2, V(x)5Q(uxu) for xPRd. See Assumption 4.1. We write

Q~ uxu!5Q~ uxu!2 12 uxu2.

For given«P(0,1/4) and$an :nPN% as above, we define a sequence of mappings$bn :nPN%,bn :H2→H2 , by

bn~v!5b1,n~v!1b2,n~v!,

b1,n~v!52Ag~exp~2anA«!!v, ~4.10!

b2,n~v!5H 2A2~12g!GnV8~gn~Gnv!!, d51,

2A2~12g!Gn$Q8~gn~ uGnvu!!~Gnv/uGnvu!%, d>2.

Due to the definitions ofgn andGn in ~4.6! and~4.8!, respectively,bn is well-defined onH2 foreachnPN. Also notice thatbnPCpol

2 (H2 ,H2), nPN.We collect useful properties of the mean operatorMn , nPN, from Ref. 40, Chap. 8.Lemma 4.2: Let Mn , nPN, be the mean operator defined in (4.4). Then, the following

properties hold:(a) For vPC(@0,1#;Rd), uMnv2vuu→0 as n→`.(b) iMniL(L2,L2)<1 for any nPN.~c! iAMniL(L2,L2)<an , wherean511(2pn)2.~d! uMnvuu<Aduvuu for anyvPC(@0,1#;Rd).~e! Gnv5Mnv for vPL2.In the above we have used the abbreviated notation L2:5L2(@0,1#;Rd,dt).Proof: ~a! This is the content of Theorem 8.15 of Ref. 40.~b!, ~c!, and~e! are obvious from

the definition ofMn . For ~d!, let Kn(t) be the Feje´r’s kernel for the operatorMn , i.e.,

Kn~t!:51

n11(k50

n S (l52k

k

e2 i2p l tD , tP@0,1#. ~4.11!

Then, it is easily seen that

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Kn~t!>0, E0

1

Kn~t!dt51. ~4.12!

Since

~Mnv!k~ t !5E0

1

vk~ t2t!Kn~t!dt, k51, . . . ,d;

~d! follows from ~4.12! and the above relation. j

We next compute the derivative ofbn which will be used later. For anyf,cPH2 ,

^bn8~v!f,c&5 limt→`

1

t$^bn~v1tf!,c&2^bn~v!,c&%.

Consider the case ford51. It follows from ~4.10! that

bn8~v!5b1,n8 ~v!1b2,n8 ~v!, b1,n8 ~v!52Ag exp~2anA«!,

~4.13!

b2,n8 ~v!52A2~12g!GnV9~gn~Gnv!!g8~Gnv!Gn .

Next we consider the case ford>2. For anyx5(x1, . . . ,xd)PRd, denote byxWxW thed3d matrixwhosei - j th element is given by (xWxW ) i j5xixj , i , j51,2, . . . ,d. A direct computation yields

]]V~x!5Q9~ uxu!xWxW /uxu21Q8~ uxu!S 12xWxW

uxu2D /uxu, ~4.14!

whereV(x)5Q(uxu) andQ(x)5Q(x)2 12uxu2. For eachnPN, put

Rn~x!5Q9~gn~ uxu!!gn8~ uxu! S xWxWuxu2D 1Q8~gn~ uxu!!S 12xWxW

uxu2D /uxu. ~4.15!

By the continuity ofHess.V ~Assumption 2.1!, ~4.14! and ~4.15! are defined atx50. Recall thedefinitionb2,n(v) for d>2 in ~4.10!. A direct computation gives that ford>2,

bn8~v!5b1,n8 ~v!1b2,n8 ~v!, b1,n8 ~v!52Agexp~2anA«!,

~4.16!

b2,n8 ~v!52A2~12g!GnRn~Gnv!Gn .

From ~4.13! and ~4.16! we have the following results.Lemma 4.3: (a) For any nPN there exists positive constanta.0 such that the bound

ib2,n8 ~v!iL~H2 ,H2!<exp~an2!

holds uniformly invPH2 .(b) For any nPN, there exists a constant M(n) such that the bound

ibn8~v!iL~H2 ,H!<M ~n!

holds uniformly invPH2 .Proof: ~a! It follows from ~2.12! that for anyv,hPH2 ,

3336 Y. M. Park and H. J. Yoo: Dirichlet operators for Gibbs measures on loop spaces

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ub2,n8 ~v!hu25uA2gb2,n8 ~v!huL2<iA2gb2,n8 ~v!AgiL~L2,L2!uhu2 . ~4.17!

The following bounds can be obtained easily from Lemma 4.2 and the definition ofMn andgn in~4.4! and ~4.6!, respectively:

iApGniL~L2,L2!<a~n!p, an511~2pn!2, pPN,

ugn~ uGnvu!uu<2n, ugn8~ uGnvu!uuu<1, ~4.18!

uV8~gn~ uGnu!!uu1uV9~gn~ uGnu!!uu<exp~cn2!.

For the last inequality we have used Assumption 2.1~b!. Ford51, the lemma follows from~4.13!,~4.17!, and~4.18!.

Consider the case ford>2. By ~4.16!, ~4.17!, and the first bound in~4.18!, it is sufficient toshow that there exists a constantc1.0 such that the bound

supvPH2

suptP@0,1#

iRn~~Gnv!~t!!iL~Rd,Rd!<exp~c1n2! ~4.19!

holds. Recall the definition ofRn in ~4.15!. Sincegn(x)5x if uxu,n, it follows from ~4.14! and~4.15! that

Rn~x!5Hess.V~x!, uxu,n. ~4.20!

Let vPH2 andtP@0,1# be given. By the continuity ofHess.V, there exists a constantM1.0such that

iRn~~Gnv!~t!!i<M1 , if u~Gnv!~t!u,1. ~4.21!

On the other hand,ixWxW /uxu2i and uxu21 are bounded uniformly foruxu>1. Thus, we use~4.18! toconclude that

iRn~~Gnv!~t!!i<exp~c1n2!, if u~Gnv!~t!u>1, ~4.22!

for some constantc1.0. Thus,~4.19! follows from ~4.21! and~4.22!. This completes the proof ofthe part~a! of the lemma.

~b! Notice that for anyv,hPH2 ,

ubn8~v!hu5uA~12g!/2bn8~v!huL2<uA~12g!/2bn8~v!AguL~L2,L2!uhu2 . ~4.23!

By the factor exp(2anA«) in b1,n8 , it follows from ~4.13! and ~4.16! that

uA~12g!/2b1,n8 ~v!AguL~L2,L2!<M1~n!,

for some constantM1(n) depending onnPN. On the other hand the method used in the proof ofthe part~a! of the lemma gives the bound

uA~12g!/2b2,n8 ~v!AguL~L2,L2!<M2~n!,

for some constantM2(n). Thus, the part~b! of the lemma follows from~4.23! and the abovebounds. j

Lemma 4.4: There exists a constant c and N0PN such that for anyfPH, the bound

^f,bn8~v!f&<cufu2

3337Y. M. Park and H. J. Yoo: Dirichlet operators for Gibbs measures on loop spaces

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holds uniformly invPH2 and n>N0 .Proof:We first consider the case ford51. It follows from ~2.12! and ~4.13! that

^f,bn8~v!f&5^f,b1,n8 ~v!f&1^f,b2,n8 ~v!f&,

where

^f,b1,n8 f&52^f,Ag exp~2anA«!f&<0, ~4.24!

and

^f,b2,n8 ~v!f&52E0

1

~Gnf!~t!V9~gn~~Gnv!~t!!!gn8~~Gnv!~t!!~Gnf!~t!dt.

Notice that 0<gn8(x)<1 for any xPR. Assumption 2.1~c! implies that2V9(x)<c for any xPR for some constantc.0. Therefore, we have that

^f,b2,n8 ~v!f&<cuGnfuL22 .

SincefPH,L2, we use Lemma 4.2~e! and ~b! and the fact thatu•uL2<u•u in that order toconclude the proof ford51. Consider the cased>2. It follows from ~4.16! that for d>2@(•,•):5Rd—inner product#,

^f,b2,n8 ~v!f&52E0

1

~~Gnf!~t!,Rn~~Gnv!~t!!~Gnf!~t!!dt. ~4.25!

Let us put the functionQ(uxu) by

Q~ uxu!:5 12M «uxu21Q~x!, ~4.26!

whereM « :5M2«, «.0, andM is given in Assumption 2.1~c!. SinceHess.V>M1, we seethatHess.Q(uxu)>«1. Thus, we may assume that there existsR.0 such that

Q8~ uxu!>0, Q9~uxu!>0, if uxu>R. ~4.27!

Suppose thatn is sufficiently large so thatn>R. Then, we notice that for anyx,yPRd,

~y,Rn~x!y!5S y,H ~Q9~gn~ uxu!!1~M «21!!gn8~ uxu!xWxW

uxu2yJ D

1S y,H Q8~gn~ uxu!!1~M «21!gn~ uxu!uxu S 12

xWxW

uxu2D yJ D ,and so

~y,Rn~x!y!>~M «21!uyu2, if uxu<R,~4.28!

~y,Rn~x!y!>~M «21!S y,H gn8~ uxu!xWxW

uxu2y1

gn~ uxu!uxu S 12

xWxW

uxu2D yJ D>min$0,M «21%uyu2, if uxu.R.

3338 Y. M. Park and H. J. Yoo: Dirichlet operators for Gibbs measures on loop spaces

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In the above we have used the fact thatgn(uxu)5uxu, if uxu<R(<n) ~and hence the left hand sideof ~4.28! is just (y,Hess.V(x)y)>(M «21)uyu2) andgn(uxu)>R if uxu.R. We use the inequalityin ~4.28! to ~4.25! to obtain that~using u•uL2<u•u)

^f,b2,n8 ~v!f&<~ uM u11!ufuL22 <~ uM u11!ufu2, n sufficiently large. ~4.29!

The proof of the lemma is now completed. j

The following result implies that the conditions~vi!–~vii ! are satisfied.Lemma 4.5: Under the assumptions in Theorem 4.1 we have the following results:(a) There exists a constant c.0 such that for each nPN the bound

iub12b1,nu2iL2~m!<can

holds, where$an :nPN% is the sequence introduced in the definition of b1,n .(b)

iub22b2,nuiL2~m!→0, as n→`.

Proof: ~a! Due to the definitions ofb1 andb1,n in ~4.7! and ~4.10!, and the definition ofu•u2 in~2.12!, we have that forvPE,

ub1~v!2b1,n~v!u22 5~v,~12exp~2anA

«!!2v!L2,<an2~v,A2«v!L2,

where (•,•)L2 is the inner product ofL2(@0,1#;Rd,dt). Here, we have used the fact that12exp(2x)<x for x>0. Notice that for any«P(0,1/4), the operatorA2«21 is of the trace class.By Theorem 3.11 of Ref. 1 or Exercise 20 of Ref. 39, Chap. I, Sect. 4, we conclude that for given«P(0,1/4),

EE~v,A2«v!L2dm<constE

E~v,A2«v!L2dm05constE

E^v,A2«21v&H~0!dm0

5const TrH~0!~A2«21!,`.

This proves the part~a! of the lemma.~b! We first consider the case ford51. It follows from ~2.12! and the definitions ofb2 and

b2,n in ~4.7! and ~4.10! that for anyvPE

ub2~v!2b2,n~v!u25E0

1

$A~12g!/2~b2~v~t!!2b2,n~v~t!!!%2dt

<E0

1

$V~v~t!!2GnV8~gn~~Gnv!~t!!!%2dt. ~4.30!

By Lemma 4.2~d!–~e! and Assumption 2.1~b!, we obtain that for anyd.0,

ub2~v!2b2,n~v!u2<4D~d!exp~Adduvuu2!. ~4.31!

By ~4.1! we see that for sufficiently smalld.0,

EEexp~Adduvuu

2!dm~v!<constEEexp~Adduvuu

2!dm0~v!,`. ~4.32!

3339Y. M. Park and H. J. Yoo: Dirichlet operators for Gibbs measures on loop spaces

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The part~b! of the lemma follows from the dominated convergence theorem,~4.30!, and Lemma4.2 ~a!. Ford>2, it is easy to check that the bound~4.31! also holds, and so the lemma holds ford>2. This completes the proof of the lemma. j

Using Lemma 4.1–Lemma 4.5, we now prove the essential self-adjointness of the Dirichletoperator.

Proof of Theorem 4.1:Lemma 4.1 implies~3.1! and~3.2!. Recall the definition ofbn in ~4.10!.Sinceb2,nPCb

2(H2 ,H2), the conditions~i! and ~ii ! in Theorem 3.1 are satisfied. Lemma 4.3implies the conditions~iii ! and ~iv!. Obviously, Lemma 4.4 and Lemma 4.5 prove that the con-ditions ~v!–~vii ! are satisfied. Thus, Theorem 4.1 follows from Theorem 3.1 and Lemma 3.2.j

V. MASS GAP AND LOG-SOBOLEV INEQUALITY

Consider the case in which the potentialV(x) is strictly convex. In this case we will demon-strate that for eachgP@0,1#, the Dirichlet operatorHm

(g) has a mass gap, i.e., there exists a gap atthe lower end of the spectrum ofHm

(g) . We shall also show that the log-Sobolev inequality holds.Throughout this section we will also omit the superscript (g) in the notations and every

formulas are understood as being formulated with a fixedgP@0,1# if there is no special remark.We assume that the potentialVPC3(Rd;R) satisfies the strict convexity condition:

Assumption 5.1: The uniform convexity constant M in Assumption 2.1 (c) is strictly positive,i.e.,

Hess.V~x!>M1, M.0,

holds.We note that under Assumption 5.1,

V~x!>V~0!1~]V~0!,x!1M

2uxu2,

and so Assumption 2.1~a! holds.Let us define the spaceW2

1(m) as the closure ofCb2(H2) with respect to the norm

iuiW21~m!

2:5E

H2

$uu~v!u21uu8~v!u12 %dm~v!.

We can easily show that form-a.a. vPH2 ~e.g., ;vPE), there exists a linear operatorb8(v):H1→H2 such that;f,hPH1 ,

L2~m!2 limt→0

1

t$^b~v1tf!,h&2^b~v!,h&%5^b8~v!f,h&.

Let us putRm(v):52b8(v). We say that a measurem is unformly log-concave~ULC! if thereexists al.0 such that for allfPH1

^Rm~v!f,f&>lufu2, m2a.e. ~5.1!

Under Assumption 5.1, we have the following result~cf. Ref. 5, Theorem 2!.Theorem 5.1:Assume that the potential V satisfies the conditions in Assumption 2.1 (b) and

Assumption 5.1. For d>2, we also assume that V satisfies the conditions in Theorem 4.1. ForeachgP@0,1#, denote again by Hm

(g) the self-adjoint extension of the Dirichlet operator Hm(g) .

Then, the following results hold:(a) The point0PR is a simple eingenvalue of Hm

(g) .

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(b) There is a gap at the lower end of the spectrum of Hm(g) . Moreover, Hm

(g)>l/4 on theorthogonal complement to the constant functions in L2(m), wherel:5min$M ,Mg%.0.

Proof:We will omit the superscript (g) in the notation. We first assert that the Gibbs measurem corresponding to the potentialV isH1-ergodic, i.e., the only measurable subsets ofH2 whichareH1-invariant havem-measure zero or one. Here, theH1-invariantness of am-measurable setA,H2 is defined by5 ;fPH1 ,

m~~A\Af!ø~Af\A!!50,

Af5A1f5$v1f:vPA%.

Since m0 is H1-ergodic andm!m0 with a strict positive Radon–Nikodym derivative,H1-ergodicity ofm also follows.

We next show thatm is ~ULC! in the sense of~5.1!. From~2.19! and~2.12!, it follows that foranyfPH1 andvPE,

^Rm~v!f,f&52^b8~v!f,f&5E0

1

~f~t!,$A1Hess.V~v~t!!%f~t!!dt

>~f,~2Dp1M !f!L2

>min$M ,Mg%ufu2. ~5.2!

Here we have used the fact that forgP@0,1#,

~2Dp1M !A2~12g!>min$M ,Mg%1.

Now, the theorem follows from the exactly same method used in Theorem 2 of Ref. 5. See alsoRef. 30. j

We recall that a probability measurem on (H2 ,B(H2)) satisfies alog-Sobolev inequality~LS! if and only if there exists some constantcm.0 such that for anyfPW2

1(m) the inequality

EH2

u f ~v!u2 logu f ~v!udm~v!<cmEH2

u¹ f ~v!u2dm~v!1i f iL2~m!2 logi f iL2~m! ~5.3!

holds. The constantcm is called theSobolev coefficient.Theorem 5.2:Assume that the conditions listed in Theorem 5.1 hold. Then, the measurem

satisfies the log-Sobolev inequality (5.3) with a Sobolev coefficient cm51/l, l5min$1,M %.0.At first we notice that the integrand of the first integration in the right hand side of~5.3!

depends on the parametergP@0,1#, that isu¹ f (v)u25(u¹ (g) f (v)u(g))2, where¹ (g) is the gradi-ent operator in the riggingH1

(g),H (g),H2(g) . Let $en%nPN be an orthonormal basis ofH (0)

consisting of eigenvectors of A, i.e., Aen5anen ,nPN,an>1. Note that$en

(g)%nPN ,en(g) :5Ag/2en , is an orthonormal basis ofH (g). For uPF Cb

`(H2(g)) it holds that9

Em~g!~u,u!5

1

2(nPN

E U ]

]en~g! u~w!U2dm~w!,

where (]/]en(g)) u is the directional derivative ofu in the direction ofen

(g) . Using the definition ofdirectional derivatives9 and the fact thata>1 for anynPN, it is not hard to show that

Em~0!~u,u!<Em

~g!~u,u!, gP@0,1#.

Therefore, it is sufficient to show~LS! ~5.3! only for g50. From now on, we fixg50.

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Before proving the theorem we give the main idea of the proof. It follows from~2.19! that forg50,

b~v!5b1~v!1b2~v!, b1~v!52v, b2~v!52A21]V~v!. ~5.4!

Let «.0 be an arbitrary number such thatM « :5M2«.0. We put the potential functionV as@cf. ~4.26!#

V~x!:5 12M «uxu21V~x!

~V~x!5Q~ uxu! for some Q:R→R if d>2!. ~5.5!

Let us define for eachnPN,

Vn~x!:55 V~0!11

2M «uxu21E

0

x

V8~gn~y!!dy, d51,

V~0!11

2M «uxu21E

0

uxuQ8~gn~r !!dr, d>2.

~5.6!

For eachnPN, we also writeVn(x):5Vn(x)212uxu2 and define a probability measuremn onE by

dmn~v!51

ZnexpH 2E

0

1

Vn~v~t!!dtJ dm0~v!. ~5.7!

By a direct calculation we see that the logarithmic derivativebn of mn , nPN, is given by

bn~v!:5b1,n~v!1b2,n~v!, b1,n~v!52v, ~5.8!

b2,n~v!5H 2A21~~M «21!v1V8~gn~v!!!, d51,

2A21S ~M «21!v1Q8~gn~ uvu!!v

uvu D , d>2.

We will show that the sequence of measures$mn%nPN converges tom and that each measuremn , nPN, satisfies the log-Sobolev inequality for a uniform Sobolev coefficient. Let us denote byHmn

the Dirichlet operator for the measuremn @with respect to the rigging~2.13! for g50]. Wehave the following results.

Lemma 5.1: Let the assumptions in Theorem 5.1 be satisfied. For each nPN, the followingresults hold:

(a) Hmnis essentially self-adjoint onF Cb

`(H2).

(b) Let Tt(n)5exp(2tHmn

) be the corresponding semigroup in L2(mn). Then, Tt(n) , tPR1 ,

forms a positive preserving semigroup from Cpol2 (H2) into itself.

(c) For any «.0 with 0,«,M , there exists N(«)PN such that the measuremn ,n>N(«), satisfies the log-Sobolev inequality (5.3) with Sobolev coefficient cm(«)51/l(«),l(«)5min$1,M «%.0, uniformly in n>N(«), where M« :5M2«.

~d! *H2f dmn→*H2

f dm as n→`, for any fPF Cb`(H2).

Proof: ~a! We notice that sinceHess.V>M1, Hess.V>«1. From this and~5.6! we see thatthere exist positive constantsa1.0 andb1.0, not depending onnPN, such that

Vn~x!> 12M «uxu22a1uxu2b1 . ~5.9!

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By using the methd similar to that used in the proof of Lemma 4.1, it is easy to show that theconditions~3.1! and ~3.2! are satisfied. Sinceb2,nPCb

2(H2 ,H2), it is obvious that the condi-tions ~i! and ~ii ! in Theorem 3.1 are satisfied. Using the method similar to that employed in theproof of Lemma 4.3~a! and the definition ofbn in ~5.8!, one can check that the condition~iv! isalso satisfied. By Theorem 1 of Ref. 5@see Remark 3.1~a!#, the above results are sufficient toshow thatHmn

is essentially self-adjoint onCb2(H2). This fact and Lemma 3.2 imply the part~a!

of the lemma.~b! This follows from Lemma 4 of Ref. 5@cf. ~26! of Ref. 5#.~c! Notice that for anyfPH1 ,

2^f,b1,n8 f&52^f,b18f&5ufu2. ~5.10!

For the part2^f,b2,n8 f&, we separately consider ford51 andd>2. Ford51,

2^f,b2,n8 f&5~f,~M «21!f!L21~f,V9~gn~v!!gn8~v!f!L2>~M «21!ufuL2, d51,~5.11!

where we have used the fact thatV9>0 andgn8>0. Ford>2, we use the argument used in Lemma4.4. We see that ford>2,

2^f,b2,n8 ~v!f&5~M «21!ufuL22

1~f,Rn~v!f!L2, ~5.12!

whereRn(x), xPRd, is defined by@cf. ~4.15!#

Rn~x!5Q9~gn~ uxu!!gn8~ uxu!xWxW

uxu21Q8~gn~ uxu!!

uxu S 12xWxW

uxu2D .We can check by the same argument used in~4.28! that

~y,Rn~x!y!Rd>0, ;x,yPRd, n sufficiently large. ~5.13!

We use the inequality~5.13! to ~5.12! and see that

2^f,b2,n8 ~v!f&>~M «21!ufuL22 , n sufficiently large. ~5.14!

From ~5.10! – ~5.14! we see using the factufuL2<ufu that

^f,Rmn~v!~f!&>ufu21min$0,M «21%ufuL2

2 >min$1,M «%ufu2, n sufficiently large.~5.15!

Using~5.15! and the method employed in the proof of Theorem 3 of Ref. 5, we complete the proofof the part~c!.

~d! Using the inequality~5.9! and the fact thatVn(x)5Vn(x)212uxu2, we obtain that there exist

positive constantsm.0 andc.0 such that for anynPN andvPE,

E0

1

Vn~v~t!!dt>1

2~m21!uvuL2

22c,

wherem can be chosen such that 0,m,M « . Thus it follows from the above bound that

expH 2E0

1

Vn~v~t!!dtJ <expH 21

2~m21!uvuL2

21cJ , ~5.16!

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uniformly in nPN. Sincem.0, the right hand side of~5.16! is integrable with respect todm0(v).

1,39 Sincegn(x)→x asn→`,

E0

1

Vn~v~t!!dt→E0

1

V~v~t!!dt, as n→`. ~5.17!

Thus we use the Lebesgue dominated convergence therem and~5.17! to conclude that for anyfPF Cb

`(H2),

EEf ~v!expH 2E

0

1

Vn~v~t!!dtJ dm0~v!→EEf ~v!expH 2E

0

1

V~v~t!!dtJ dm0~v!, as n→`.

Due to the definitions ofm andmn in ~2.7! and~5.7! respectively, the above result implies the part~d! of the lemma. This completes the proof of Lemma 5.1. j

Now, we turn to the proof of Theorem 5.2. It is really a consequence of Lemma 5.1.Proof of Theorem 5.2. For a given«.0 with 0,«,M , we use Lemma 5.1~c! and ~d! and

take a limit asn→`. We then obtain that the log-Sobolev inequality~5.3! holds with a Sobolevcoefficientcm(«)51/l(«), l(«)5min$1,M «%.0, onF Cb

`(H2). Since«.0 is arbitrary, we seethat the log-Sobolev inequality holds with a Sobolev coefficientcm51/l, l5min$1,M %.0. Now,sinceF Cb

`(H2) is dense inW21(m), we complete the proof of Theorem 5.2. j

VI. CONCLUDING REMARKS

1. We give a possible relaxation of the spherical symmetricity of the potentialV for d>2~Assumption 4.1!. For d>2, the potentialVPC3(Rd;R) can be written as

V~x!5Q~ uxu!1W~x!, xPRd, ~6.1!

whereW is aC3-function satisfying the following bounds: there exists a constantK.0 such thatfor any xPRd,

u]W~x!u<K~11uxu!,~6.2!

uuHess.W~x!uuL~Rd,Rd!<K.

We also assume thatW(3)(x) is bounded~in the operator norm! uniformly in xPRd, whereW(3) is the third order derivative ofW. Let us replaceb2,n in ~4.10! by

b2,n~v!52A2~12g!Gn$Q8~gn~ uGn~v!u!!~Gn~v!/uGn~v!u!%2A2~12g!Gn]W~Gn~v!!.

Due to the bounds in~6.1!, it is easy to check that the results corresponding to Lemma 4.3 –Lemma 4.6 hold. Thus Theorem 4.1 and Theorem 5.1 hold for the potentialV satisfying~6.1! and~6.2!. We leave the details to the reader.

2. When we have finished the first version of this paper, we have come to know that at thesame time the log-Sobolev inequality for Gibbs measures on loop spaces was also shown in Ref.37. In Ref. 37, however, a different method~a finite dimensional approximation to the givenmeasure! was used. It may be worthwhile to compare the results. We emphasize that we havepresented the log-Sobolev inequality in~5.3! uniformly for any gP@0,1#, i.e., the gradientu¹ f (v)u2 in the integrand of the right hand side of~5.3! depends on gP@0,1#:u¹ f (v)u25(u¹ (g) f (v)u(g))2. However, in Ref. 37, it was presented forg51 ~Ref. 37, Theorem5.1!. Their result can be written down~in our notation! as

3344 Y. M. Park and H. J. Yoo: Dirichlet operators for Gibbs measures on loop spaces

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Page 26: Dirichlet operators on loop spaces: Essential self-adjointness and log-Sobolev inequality

E u f ~v!u2logu f ~v!udm~v!<1

a2E ~ u¹~1! f ~v!u~1!!2dm~v!1i f iL2~m!2 logi f iL2~m! , ~6.3!

wheneverV(x)5 12a

2x21V0(x) with V09(x)>0. ~They dealt with only the cased51. We have putW50 in Ref. 37. The case of bounded perturbationWÞ0 can be dealt with a perturbationtheorem for the log-Sobolev constant in Ref. 31 as mentioned in Ref. 37.! In order to compare theresult, we take the operator

A:52Dp1a21 ~6.4!

in ~2.11! and make the rigged Hilbert spacesH1(g),H (g),H2

(g) through the inner products in~2.12! with the new operatorA. We take, of course, the notationsV(x):5V(x)2 1

2a2x2, e.g. in

~2.16!. Then, as readily seen in~5.2!, we have that for anyfPH1 andvPE,

^Rm~v!f,f&>~f,~2Dp1a2!f!>a2gufu2.

Thus, forg50 we get a~ULC! constantl51 and the exact method used in the proof of Theorem5.2 provides that we have~LS! with cm51:

E u f ~v!u2logu f ~v!udm~v!<E ~ u¹~0! f ~v!u~0!!2dm~v!1i f iL2~m!2 logi f iL2~m! . ~6.5!

SinceA>a21, it is readily seen that~see the arguments below the statement of Theorem 5.2!

~ u¹~0! f ~v!u~0!!2<1

a2~ u¹~1! f ~v!u~1!!2. ~6.6!

We use the inequality~6.6! to ~6.5!. Then, the result is exactly the one in~6.3!.3. In Ref. 37, using~LS!, a uniqueness theorem for Gibbs measures of certain class of

potentials for spin systems was also proven. The method and the results in Ref. 37 can be extendedto a class of superstable and regular interactions.

4. For historical backgrounds of this study and for wide ranged applications we refer to Ref.14.

ACKNOWLEDGMENTS

This work was supported in part by the Korean Science Foundation and Basic ScienceReaserch Institute Program~BSRI 96-1421!, Korean Ministry of Education~1996 – 1998!. Theauthors would like to thank Professor S. Albeverio, Professor Yu. G. Kondratiev, and Professor M.Rockner for stimulating discussions and comments. H.J.Y. would like to express great thanks toProfessor S. Albeverio for his kind and warm hospitality during his stay at Ruhr-Universita¨t-Bochum. He also gratefully thanks KOSEF for financial support.

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