diffusive processes on the wasserstein space: coalescing ...marx/soutenance.pdf · soutenance de...

Diusive processes on the Wasserstein space:

Coalescing models, Regularization properties and

McKean-Vlasov equations

Soutenance de thèse de Victor Marx

Laboratoire J.A. Dieudonné, Nice (France)sous la direction de François Delarue

Vendredi 25 octobre 2019

Brownian motion on Rd

?Smoothing properties:

I Restoration of uniqueness

dxt = b(xt)dt

+ dBt .

There might be several solutions to the Cauchy problem, if b is not smoothenough (e.g. b(x) = sign(x)

√|x |).

uniqueness of a solution.→ Stroock-Varadhan.→ Zvonkin, Veretennikov, Krylov-Röckner.

I Associated semi-group: heat semi-group.

Ptφ(x) =1√2πt

∫R

φ(y)e−(x−y)2

2t dy .

The semi-group has a regularizing eect.

Our aim: obtain similar properties for a diusion on a space of probabilitymeasures.

2 / 39 Soutenance de thèse de Victor Marx Diusive processes on the Wasserstein space




dxt = b(xt)dt + dBt .


√|x |). uniqueness of a solution.

→ Stroock-Varadhan.→ Zvonkin, Veretennikov, Krylov-Röckner.


Ptφ(x) =1√2πt

∫R

φ(y)e−(x−y)2

2t dy .


Our aim: obtain similar properties for a diusion on a space of probabilitymeasures.





dxt = b(xt)dt + dBt .


√|x |). uniqueness of a solution.

→ Stroock-Varadhan.→ Zvonkin, Veretennikov, Krylov-Röckner.


Ptφ(x) =1√2πt

∫R

φ(y)e−(x−y)2

2t dy .


Our aim: obtain similar properties for a diusion on a space of probabilitymeasures.2 / 39 Soutenance de thèse de Victor Marx Diusive processes on the Wasserstein space

Interest of the space of probability measures:

Optimal transportation:Let us dene the L2-Wasserstein space of R:

P2(R) :=

µ probability measure on R:

∫R

x2dµ(x) < +∞,

and the Wasserstein distance W2 on P2(R):

W2(µ, ν) = infγ

(∫∫R2

|x − y |2 γ(dx , dy)

)1/2

where the inmum is taken over all probability measures γ on R2 havingmarginals µ and ν. → Monge-Kantorovich problem.

understand the geometric properties of the space of probability measures→ Benamou-Brenier, McCann.Gradient ows → Jordan-Kinderlehrer-Otto, Ambrosio-Gigli-Savaré.


Interest of the space of probability measures:

Systems of interacting particles: McKean-Vlasov equation

N (very large number) particles.Dynamics of particle i are inuenced by

∗ position of particle i ;∗ distribution of all particles;∗ noises: common noise and proper (idiosyncratic) noise.

Asymptotic behaviour when N → +∞: propagation of chaos→ Kac, McKean, Funika, Sznitman, ...

Convergence rate→ Bossy-Talay, Bossy-Jabir-Talay, Jourdain-Reygner, Fournier-Guillin, ...


Finite dimension

Brownian motionon Rd

Innite dimension

Construction of adiusion on the

L2-Wasserstein spacePart 1

?

in this thesis−→?

Smoothing properties

Restoration ofuniqueness

Estimates on thegradient of thesemi-group

Smoothing properties

Restoration ofuniqueness: Part 2

Estimates on thegradient of thesemi-group: Part 3


Construction of a diusion process on the Wasserstein space

Outline

1 Construction of a diusion process on the Wasserstein space

2 Restoration of uniqueness of McKean-Vlasov equations

3 Gradient estimates for a Wasserstein diusion on the torus



Wasserstein diusionThe following diusive properties of some stochastic processes (µt)t∈[0,T ] onP2(R) have already been studied:

large deviations in small time Varadhan formula: for any measurable subset A in P2(R),

ε lnP [µt+ε ∈ A |µt ] −→ε→0−W2(µt ,A)2

2.

Itô formula: for any smooth map φ : P2(R)→ R

dφ(µt) = martingale term + (second order derivative of φ)(µt)dt,

where the martingale term has the square of Wasserstein gradient of φ aslocal quadratic variation.

Examples:

I construction via Dirichlet forms and entropic measure on P2[0, 1]→ von Renesse-Sturm, Sturm, Andres-von Renesse

I construction via a system of coalescing particles→ Konarovskyi, Konarovskyi-von Renesse, M.



Konarovskyi's modelLet N > 1. For every i ∈ 1, . . . ,N, the particle (x it )t∈[0,T ] follows the equation:

x it = g( i

N

)+

∫ t

0

1

N

N∑j=1

1x is=x j

s

mis

√NdB j

s ,

where B1, . . . ,BN are N independent Brownian motions on [0,T ].

mis := 1

N

∑Nk=1 1x i

s=xks represents the mass of particle i at time s.



We associate a process (µNt )t∈[0,T ] on P2(R) by taking the empirical measure

µNt :=

1

N

N∑i=1

δx it.

The sequence of the laws of the processes (µNt )t∈[0,T ] is tight: there exists a limit

process (µt)t∈[0,T ]. → Konarovskyi

Two nice properties: for each time t > 0, µt supported by a nite set.Canonical representation as a quantile function (yt(u))t∈[0,T ],u∈[0,1] satisfying

(i) y0 = g ;

(ii) for each t ∈ [0,T ], u 7→ yt(u) is non-decreasing;

(iii) for each u ∈ [0, 1], (yt(u))t∈[0,T ] is an L2-martingale;

(iv) quadratic co-variation:

〈y·(u), y·(u′)〉t =

∫ t

0

1τu,u′6s

ms(u)ds,

where ms(u) = Lebv ∈ [0, 1] : τu,v 6 s.



Theorem 1

Let g : [0, 1]→ R be an increasing Lp-function, p > 2.Let y satisfy (i)− (iv).Then there exists a Brownian sheet w on [0, 1]× [0,T ] such that almost surely forall u ∈ (0, 1] and for all t ∈ [0,T ],

yt(u) = g(u) +

∫ t

0

∫ 1

0

1ys (u)=ys (u′)

ms(u)dw(u′, s).

Characterization of Brownian sheet:

- for each progressively measurable L2-function f on [0, 1]× [0,T ],

(∫ t

0

∫ 1

0f (u, s)dw(u, s))t∈[0,T ] is a local martingale;

- for each f1 and f2,

〈∫ ·0

∫ 1

0

f1(u, s)dw(u, s),

∫ ·0

∫ 1

0

f2(u, s)dw(u, s)〉t =

∫ t

0

∫ 1

0

f1(u, s)f2(u, s)duds.



Open problem: Uniqueness of a process (yt(u))t∈[0,T ],u∈[0,1] satisfying (i)− (iv).

yt(u) = g(u) +

∫ t

0

∫ 1

0

10(ys(u)− ys(u′))

ms(u)dw(u′, s).

Let us introduce a smooth approximation of the equation:

yσ,εt (u) = g(u) +

∫ t

0

∫ 1

0

ϕσ(yσ,εs (u)− yσ,εs (u′))

ε+ mσ,εs (u)

dw(u′, s), (SmoothDi)

where mσ,εs =

∫ 1

0ϕ2σ(yσ,εs (u)− yσ,εs (v))dv .

. The indicator function 10 is replaced by a smooth function ϕσ.

. The denominator is bounded by below by ε.



yσ,εt (u) = g(u) +

∫ t

0

∫ 1

0

ϕσ(yσ,εs (u)− yσ,εs (u′))

ε+ mσ,εs (u)

dw(u′, s), (SmoothDi)

where mσ,εs =

∫ 1

0ϕ2σ(yσ,εs (u)− yσ,εs (v))dv .

Theorem 2 - Tightness and characterization of the limit

Let g : [0, 1]→ R be an increasing Lp-function, p > 2.

For each σ > 0 and for each ε > 0,

- there is a unique solution yσ,ε to equation (SmoothDi) in L2([0, 1], C[0,T ]);

- almost surely for each time t ∈ [0,T ], u 7→ yσ,εt (u) is non-decreasing andcàdlàg.

Furthermore, there is a subsequence of (yσ,ε)σ,ε∈Q∩(0,+∞) that converges indistribution to a limit y in L2([0, 1], C[0,T ]).The limit process y

- satises properties (i)− (iv);

- belongs to D((0, 1), C[0,T ]);

- for each time t ∈ (0,T ], u 7→ yt(u) is a step function.



Key step in the proof

In order to prove tightness, we use the following uniform control on the inverse ofmass:

Let g ∈ Lp[0, 1], p > 2. Let β ∈(1, 32 −

1p

).

Then there is C > 0 independent of σ, ε such that for every t ∈ [0,T ],

E

∫ t

0

∫ 1

0

(mσ,ε

s (u)(ε+ mσ,ε

s (u))2)β

duds

6 C√t.

The termmσ,εs (u)(

ε+mσ,εs (u))2 is arising when computing the quadratic variation of the

martingale part of yσ,εt .


Restoration of uniqueness of McKean-Vlasov equations

Outline







Back to nite dimension.Let us consider a system of N random particles X 1, . . . ,XN interacting through:

dX it = b(t,X i

t , µNt )dt + σ(t,X i

t , µNt )dB i

t

+ σ0(t,X it , µ

Nt )dWt .

where µNt = 1

N

∑Nj=1 δX j

t.

(B i )16i6N : idiosyncratic noise.

W : common noise.

innite-dimensional equation describes asymptotical behavior:

dXt = b(t,Xt , µt)dt + σ(t,Xt , µt)dBt

+ σ0(t,Xt , µt)dWt .

where µt = Law(Xt).




Back to nite dimension.Let us consider a system of N random particles X 1, . . . ,XN interacting through:

dX it = b(t,X i

t , µNt )dt + σ(t,X i

t , µNt )dB i

t + σ0(t,X it , µ

Nt )dWt .

where µNt = 1

N

∑Nj=1 δX j

t.

(B i )16i6N : idiosyncratic noise. W : common noise.

innite-dimensional equation describes asymptotical behavior:

dXt = b(t,Xt , µt)dt + σ(t,Xt , µt)dBt + σ0(t,Xt , µt)dWt .

where(((((((µt = Law(Xt) µt = Law(Xt |σ((Ws)s6t)).

The process (µt)t∈[0,T ] is now random!



Assumptions on the drift function

Restoration of uniqueness by noise:

dXt = b(Xt , µt)dt + σ(Xt)dBt .

→ Jourdain, Mishura-Veretennikov, Lacker, Chaudru de Raynal-Frikha,Röckner-Zhangwhere the drift function b : R× P2(R)→ R satises

µ xed, x 7→ b(x , µ) bounded measurable,

x xed, µ 7→ b(x , µ) Lipschitz-continuous with respect to the total variationdistance dTV, dened by

dTV(µ, ν) := sup

∣∣∣∣∫ f dµ−∫

f dν

∣∣∣∣ ; f measurable, ‖f ‖L∞ 6 1

.

Idea: instead of a nite-dimensional Brownian motion, use our diusion on P2(R)to obtain a regularization result for a drift b with lower regularity than Lipschitz.



Innite-dimensional common noise

Let g : [0, 1]→ R be an initial quantile function.Let (yt)t∈[0,T ] be a process of quantile functions satisfying for each u ∈ [0, 1]:

dyt(u) = b(yt(u), µt)dt +1

mt(u)1/2

∫R

f (k)<(e−ikyt(u)dw(k, t));

µt = Leb |[0,1] y−1t ;

y0(u) = g(u),

(?)

where

mt(u) =∫ 1

0ϕ(yt(u)− yt(v))dv ;

w = w< + iw= is a complex-valued Brownian sheet

<(e−ikyt(u)dw(k , t)) = cos(kyt(u))dw<(k , t) + sin(kyt(u))dw=(k, t);

f (k) = fα(k) := Cα(1+k2)α/2

.

α ⇒ the map u 7→ yt(u) is more regular.

The process (µt)t∈[0,T ] is here random.



dyt(u) = b(yt(u), µt)dt +

1

mt(u)1/2

∫R

f (k)<(e−ikyt(u)dw(k, t));

µt = Leb |[0,1] y−1t ;

y0(u) = g(u).

(?)

Theorem 3 - Restoration of uniqueness

Let g be a strictly increasing C1-function.Let f : k 7→ 1

(1+k2)α/2with α > 3

2 .

Let b : R× P2(R)→ R be a bounded measurable function such that

- for each µ ∈ P2(R), x 7→ b(x , µ) is C2;- ∂xb and ∂

(2)x b are uniformly bounded on R× P2(R).

Then there is a unique weak solution to equation (?).



Theorem 3 - Restoration of uniqueness

Let g be a strictly increasing C1-function.Let f : k 7→ 1

(1+k2)α/2with α > 3

2 .

Let b : R× P2(R)→ R be a bounded measurable function such that

- for each µ ∈ P2(R), x 7→ b(x , µ) is C2;- ∂xb and ∂

(2)x b are uniformly bounded on R× P2(R).

Then there is a unique weak solution to equation (?).

Method of proofFind an L2(R;C)-valued process (ht)t∈[0,T ] such that

b(yt(u), µt) =1

mt(u)1/2

∫R

f (k)e−ikyt(u)ht(k)dk.

and apply Girsanov's Theorem.The solution h is of the form:

ht(k) =1

f (k)F−1(b(·, µt) · (ϕ ∗ µt))(k),

1

f (k)=

(1 + k2)α/2

Cα.

α ⇒ more dicult to obtain integrability of h.19 / 39 Soutenance de thèse de Victor Marx Diusive processes on the Wasserstein space


Gap

Velocity eld b : R× P2(R)→ R

(x , µ) 7→ b(x , µ)

Hypotheseson b

Jourdain, Mishura-Veretennikov,Lacker, ...

Theorem 4η > 3

2 (1− δ)

Theorem 3

Regularity in xbounded andmeasurable

Hη

C2

Regularity in µ(dTV distance)

Lipschitz

Cδ, δ ∈ [0, 1]

bounded andmeasurable

ExampleE [b(Xt)],b bounded

b(Xt ,E [Xt ])b(E [Xt ]),

b bounded orHölder



Gap

Velocity eld b : R× P2(R)→ R

(x , µ) 7→ b(x , µ)

Hypotheseson b

Jourdain, Mishura-Veretennikov,Lacker, ...

Theorem 4η > 3

2 (1− δ)Theorem 3

Regularity in xbounded andmeasurable

Hη C2

Regularity in µ(dTV distance)

Lipschitz Cδ, δ ∈ [0, 1]bounded andmeasurable

ExampleE [b(Xt)],b bounded

b(Xt ,E [Xt ])b(E [Xt ]),

b bounded orHölder



Diusion model with idiosyncratic noise

averaging over nite dimension with β;

no quantile representation anymore;

constant mass equal to one;

notion of weak compatible solution.

dzt = b(zt , µt)dt +

∫Rf (k)<(e−ikztdw(k , t)) + dβt ,

µt = LP(zt |Gµ,wt ), (µ,w) ⊥⊥ (β, ξ)

z0 = ξ, LP(ξ) = µ0.

(N)

For each t, (ξ,w , µ) and β conditionally independent given Gt .

Theorem 4

Let η > 0 and δ ∈ [0, 1] be such that η > 32 (1− δ) and let b be of class (Hη, Cδ).

Let f : R→ R be dened by f (k) := 1(1+k2)α/2

, with 32 < α 6 η

1−δ .

Then existence and uniqueness of a weak compatible solution to equation (N)hold.



Method of proof

Step 1: drift b ≡ 0.

dzt =∫Rf (k)<(e−ikztdw(k , t)) + dβt .

∃! strong solution zt = Zt(ξ,w , β) Yamada-Watanabe Theorem.Moreover, if (Ωi ,G i ,Pi , z i ,w i , βi , ξi ), i = 1, 2, are two solutions, then

LP1(z1,w1, β1) = LP2(z2,w2, β2). (Uniq 1)

Step 2: drift b such that there is C satisfying

- b uniformly bounded: ∀x , µ, |b(x , µ)| 6 C ;

- b uniformly Lipschitz-continuous: ∀x , µ, ν, |b(x , µ)− b(x , ν)| 6 CdTV(µ, ν) .dzt =

∫Rf (k)<(e−ikztdw(k, t)) + dβt + b(zt , µt)dt,

µt = LP(zt |Gµ,wt ).



Method of proof

Step 1: drift b ≡ 0.

dzt =∫Rf (k)<(e−ikztdw(k , t)) + dβt .

∃! strong solution zt = Zt(ξ,w , β) Yamada-Watanabe Theorem.Moreover, if (Ωi ,G i ,Pi , z i ,w i , βi , ξi ), i = 1, 2, are two solutions, then

LP1(z1,w1, β1) = LP2(z2,w2, β2). (Uniq 1)

Step 2: drift b such that there is C satisfying

- b uniformly bounded: ∀x , µ, |b(x , µ)| 6 C ;

- b uniformly Lipschitz-continuous: ∀x , µ, ν, |b(x , µ)− b(x , ν)| 6 CdTV(µ, ν) .dzt =

∫Rf (k)<(e−ikztdw(k , t)) + dβt + b(zt , µt)dt,




Step 2 (foll.):dzt =



∃! semi-strong solution, i.e. (µt)t∈[0,T ] is adapted with respect to (Gwt )t∈[0,T ].Moreover, if (Ωi ,G i ,Pi , z i ,w i , βi , ξi ), i = 1, 2, are two solutions, then

LP1(z1,w1) = LP2(z2,w2). (Uniq 2)

Idea: Existence: xed-point argument → Jourdain, Lacker, ...Girsanov argument with respect to β:

dPµ

dP= exp

(∫ t

0

b(zs , µs)dβs −1

2

∫ t

0

b(zs , µs)2ds

).

Solve the xed-point problem "LPµ(zt) = µ".Uniqueness: prove that each weak solution µ is solution to the xed-pointproblem µ adapted with respect to (Gwt )t∈[0,T ]. Use (Uniq 1).



Step 3: drift b of class (Hη, Cδ).dzt =



Let us write: b = b︸︷︷︸Lipschitz in µ w.r.t. dTV

+ b − b︸︷︷︸regular in x

low regularity in µ

.

dzt =

∫Rf (k)<(e−ikztdw(k , t)) + dβt + b(zt , µt)dt + (b − b)(zt , µt)dt,




Step 3 (foll.):dzt =

∫Rf (k)<(e−ikztdw(k , t)) + dβt + b(zt , µt)dt + (b − b)(zt , µt)dt,


Idea: Fourier inversion (as previously):

ht(k) =1

f (k)F−1((b − b)(·, µt))(k).

and Girsanov argument with respect to w

dQdP

= exp

(∫ t

0

∫R

<(ht(k)dw(k , t))− 1

2

∫ t

0

∫R

|ht(k)|2dkdt).

Prove that the solution (µt)t∈[0,T ] todzt =


µt = LP(zt |Gwt )

satises µt = LQ(zt |Gµ,wt ), where

w(k, t) = w(k , t) +

∫ t

0

∫ k

0

hs(l)dlds.



Open problem

Inuence of a non-constant mass:

I in Konarovskyi's model mass helps to obtain tightness.

I here, it could also bring some additional regularity to our system.


Gradient estimates for a Wasserstein diusion on the torus

Outline






Finite-dimensional caseLet X x

t be solution of dX x

t = b(t,X xt )dt + σ(t,X x

t )dWt ;

X x0 = x .

Associated semi-group

Ptφ(x) = E [φ(X xt )] .

Under some regularity assumptions on b and σ, Bismut-Elworthy-Li integration byparts formula yields

∇(Ptφ)x0(v0) =1

tE[φ(X x

t )

∫ t

0

〈Vs , σ(X xs )dWs〉

],

where (Vt)t∈[0,T ] is a certain adapted stochastic process starting at v0.→ Bismut, Elworthy, Elworthy-LiIt follows that

‖∇Ptφ‖L∞ 6C√t‖φ‖L∞ .



Semi-groups and applications

Applications in nite dimension:

Geometry: study of PDEs on manifolds. → Thalmaier, Thalmaier-Wang

Hypoelliptic diusions: e.g. Bakry-Baudoin-Bonnefont-Chafaï.

Finance: computing price sensitivies (Greeks).

Numerical analysis: Euler schemes for SDEs.

McKean-Vlasov equations:→ Crisan-McMurray, Baños

Further works related to the study of semi-groups for innite-dimensionalequations arising from particle systems:→ Kolokoltsov, Buckdahn-Li-Peng-Rainer, Chassagneux-Crisan-Delarue



Study of a diusion on the torus TTwo main reasons for working on the torus T:I to avoid problems of density at innity;

I to work only with measure-valued processes having a positive densityeverywhere on the state space.

Equation of the diusion:

dxgt (u) =∑k∈Z

fk<(e−ikxgt (u)dW k

t ) + dβt ; u ∈ [0, 1], t ∈ [0,T ].

Assumptions:

for each k , W k· = W<,k· + iW=,k· ;

((W<,k· )k∈Z, (W=,k· )k∈Z, β·) is a collection of independent real-valued

Brownian motions;

fk = fα,k := Cα(1+k2)α/2

.

Role of α: controlling the regularity of the solution u 7→ xgt (u) for each t.



Quantile functions on the torus

The initial condition g : R→ R is seen as a quantile function on the torus.It satises:

regularity: g ∈ C1;monotonicity: g ′(u) > 0 for each u ∈ R;pseudo-periodicity: g(u + 1) = g(u) + 2π for each u ∈ R.

By the previous assumption, it can be seen as a function g : [0, 1]→ T.

If g ∈ C1+θ, θ > 0:those properties are preserved almost surely for every t by the solution u 7→ xgt (u).



Role of β

dxgt (u) =∑k∈Z

fk<(e−ikxgt (u)dW k

t ) + dβt ; u ∈ [0, 1], t ∈ [0,T ].

As before, we study:

((((((((((νgt = Leb[0,1] (xgt )−1

µgt = (Leb[0,1]⊗Pβ) (xgt )−1.

µgt is also the conditional law of xgt given σ((W k

s )s6t)k∈Z.

Equation on the density pgt of µgt :

dpgt (v) = −∂v

(pt(v)

∑k∈Z

fk<(e−ikvdW kt )

)+ λp′′t (v)dt,

→ [Denis, Matoussi, Stoica]: threshold at λ >∑

k∈Z f 2k2 .

without β: λ =∑

k∈Z f 2k2 ; with β: λ =

∑k∈Z f 2k +1

2 .



Semi-group

Let φ : P2(R)→ R.We say that φ is T-stable if φ(µ) = φ(ν) if µ and ν have same trace on the torus.

Let us dene the semi-group:

Ptφ(µg0) = EW [φ(µg

t )]



Theorem 5 - Gradient estimate

Let φ : P2(R)→ R be T-stable and belong to C1,1b .Let θ ∈ (0, 1) and α > 7

2 + θ.Let g be an initial condition of class C3+θ.Let h be a 1-periodic perturbation of class C1.Then ρ 7→ Ptφ(µg+ρh

0 ) is dierentiable at ρ = 0 and there is C > 0 independentof h such that for every t ∈ (0,T ],∣∣∣∣ ddρ |ρ=0

Ptφ(µg+ρh0 )

∣∣∣∣ 6 CVW [φ(µg

t )]1/2

t2+θ‖h‖C1 .

Theorem 6

Let φ : P2(R)→ R be a bounded, uniformly continuous and T-stable function.If h belongs to C3+θ,then there is ρ0 > 0 and C > 0, depending on g and h such that for everyρ ∈ (−ρ0, ρ0) and every t ∈ (0,T ],∣∣∣Ptφ(µg+ρh

0 )− Ptφ(µg0)∣∣∣ 6 C |ρ| ‖φ‖L∞

t2+θ.



Method of proof → Thalmaier, Thalmaier-Wang

Expansion of the process (xg+ρtht )t∈[0,T ] → Kunita

xg+ρtht (u) = xg0 (u) +∑k∈Z

fk

∫ t

0

<(e−ikxgs (u)dW k

s ) + βt

+ρ

∫ t

0

∂uxgs (u)

g ′(u)h(u)ds +O(ρ2).

Solve an inversion problem (as previously by Fourier transform): nd acollection of adapted processes ((λkt )t∈[0,T ])k∈Z such that for all u ∈ [0, 1],s ∈ [0,T ],

∂uxgs (u)

g ′(u)h(u) =

∑k∈Z

fk<(e−ikxgs (u)λks ).

Apply Girsanov's Theorem:

EW[φ(µg+ρth

t )Eg ,ρt

]= EW [φ(µg

t )] .



Method of proof → Thalmaier, Thalmaier-Wang

We dierentiate with respect to ρ at point ρ = 0 the Girsanov equality:

EW[φ(µg+ρth

t )Eg ,ρt

]= EW [φ(µg

t )] .

?

d

dρ |ρ=0

td

dρ |ρ=0

Ptφ(µg+ρh0 )− EWEβ

[φ(µg

t )∑k∈Z

∫ t

0

<(λks dWks )

]= 0.

Bismut-Elworthy-Li integration by parts formula.



Problem:

∂uxgs (u)

g ′(u)h(u) =

∑k∈Z

fk<(e−ikxgs (u)λks ), fk =

Cα(1 + k2)α/2

.

Role of α:α ⇒ higher regularity on u 7→ ∂ux

gs (u) and more dicult to invert fk .

not possible to nd an appropriate exponent α.

Split the left hand side into two terms:

∂uxgs (u)

g ′(u)h(u) = Ag ,ε

s︸︷︷︸smooth

(u) + (Ags − Ag ,ε

s )︸︷︷︸remainder term

(u).

Apply Girsanov's Theorem with respect to the noise β to deal with theremainder term.

This leads to an integration by parts formula with remainder term and we get agradient estimate with a deteriorated explosion rate t−(2+θ).



Open problems

I improvement of the explosion rate t−(2+θ):

I Already done: improvement to t−(1+θ) by an interpolation argument, assumingmore regularity on g and the direction of perturbation h.

I Already done: Application: gradient estimate with a non trivial source term.

I improvement of the explosion rate get closer to t−1/2. Aim: obtain agradient estimate for a diusion with a non-smooth drift function b;

I convergence rate of the particle system associated to the diusion on thetorus.


Thank you for your attention.

Merci pour votre attention.

diffusive processes on the wasserstein space: coalescing ...marx/soutenance.pdf · soutenance de...

Documents