[PPT] - On a stochastic mass conserved Allen-Cahn equation with nonlinear PowerPoint Presentation

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On a stochastic mass conserved Allen-Cahn equation with nonlinear diffusion

Perla El Kettani1, Danielle Hilhorst2, Kai Lee3

1 University of Paris-Sud 2 CNRS and University of Paris-Sud 3 University of Tokyo, Japan

March 25th, 2019

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 1 / 35

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Two related topics

I - A stochastic mass conserved reaction-diffusion equation with nonlinear diffusion;

nonlinear diffusion, mass conservation, additive noise

II - Stochastic nonlocal Allen-Cahn equation with a multiplicative noise.

linear diffusion, no mass conservation, multplicative noise

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 2 / 35

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The mathematical problem

Prove the existence and uniqueness of the solution of the nonlocal stochastic reaction-diffusion equation with nonlinear diffusion (P)            ∂ϕ ∂t = div(A(∇ϕ)) + f(ϕ) − 1 |D|

D

f(ϕ)dx + ∂W ∂t , in D × (0, T) A(∇ϕ).ν = 0,

n ∂D × (0, T)

ϕ(x, 0) = ϕ0(x), x ∈ D

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 3 / 35

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Motivation

1

[Rubinstein and Sternberg, 1992] Deterministic mass conserved Allen-Cahn equation with linear diffusion. Binary mixture undergoing phase separation.

2

[Boussaïd, Hilhorst and Nguyen, 2015] proved the well-posedness and the stabilization of the solution for large times for the corresponding Neumann problem.

3

[Antonopoulou, Bates, Blömker and Karali, 2016]

use the stochastic mass conserved equation with linear diffusion to

describe the motion of a droplet;

They study the singular limit of the solution of this equation, letting

a small parameter tend to zero;

They consider the colored noise, namely a white noise in time

which also depends on space.

4

[Funaki and Yokoyama, 2016] study the singular limit of the solution of the stochastic mass conserved Allen-Cahn equation with a smoothened one dimensional white noise in time.

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 4 / 35

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Hypotheses on the domain D and on the function A

D open bounded set of Rn, with a smooth boundary ∂D;
A is such that

|A(a) − A(b)| ≤ C|a − b|;

A = ∇Ψ : Rn → Rn for some convex C1,1-function Ψ and A(0) = 0;
A is monotone

(A(a) − A(b)).(a − b) ≥ C0(a − b)2, C0 > 0; for all a, b ∈ Rn. [Funaki, Spohn,1997]. Remark: If A = I ⇒ div(A(∇u)) = ∆u.

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Usual diffusion

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Anisotropic diffusion

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The nonlinear function f

The nonlinear function f is given by f(s) =

2p−1

j=0

bjsj with b2p−1 < 0, p ≥ 2, which also includes the Allen-Cahn equation with f(s) = s(1 − s2).

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 8 / 35

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The Q-Brownian motion

The function W(x, t) is a Q-Brownian motion in L2(D).

W(x, t) =

∞

k=1

βk(t)Q

1 2 ek(x) =

∞

k=1
λkβk(t)ek(x),

where

1

{ek}k≥1 is an orthonormal basis in L2(D) diagonalizing Q;

2

{λk}k≥1 are the corresponding eigenvalues for all k ≥ 1;

3

Q is a nonnegative definite symmetric operator on L2(D) with Tr Q < +∞. Tr Q =

∞

k=1

Qek, ekL2(D) =

∞

k=1

λk ≤ Λ0;

∞

k=1

λkek2

L∞(D) ≤ Λ1;

4

{βk(t)}k≥1 is a sequence of independent (Ft)-Brownian motions defined on the probability space (Ω, F, P).

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 9 / 35

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The nonlinear stochastic heat equation

We consider the nonlinear stochastic heat equation (P1)            ∂WA ∂t = div(A(∇WA)) + ∂W ∂t , in D × (0, T) A(∇(WA)).ν = 0,

n ∂D × (0, T)

WA(x, 0) = 0, x ∈ D [Krylov, Rozovskii, 1981]

Definition

1

WA ∈ L∞(0, T; L2(Ω × D)) ∩ L2(Ω × (0, T); H1(D));

2

div(A(∇WA)) ∈ L2(Ω × (0, T); (H1(D))′);

3

WA satisfies a.s. for a.e. t ∈ (0, T) the problem    WA(t) = t div(A(∇WA(s)))ds + W(t) in (H1(D))′, A(∇WA(t)).ν = 0, in the sense of distributions on ∂D.

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 10 / 35

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A preliminary change of functions

We remark that WA ∈ L∞(0, T; Lq(Ω × D)) for all q ≥ 2, and define: u(t) := ϕ(t) − WA(t), (P2)                  ∂u ∂t = div(A(∇(u + WA)) − A(∇WA)) + f(u + WA) − 1 |D|

D

f(u + WA)dx, in D × (0, T) A(∇(u + WA)).ν = 0,

n ∂D × (0, T)

u(x, 0) = ϕ0(x), x ∈ D Remark: The conservation of mass property holds, namely

D

u(x, t)dx =

D

ϕ0(x)dx, for a.e. t ∈ (0, T).

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 11 / 35

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Definition of the solution

Definition

1

u ∈ L∞(0, T; L2(Ω×D))∩L2(Ω×(0, T); H1(D))∩L2p(Ω×(0, T)×D), div[A(∇(u + WA))] ∈ L2(Ω × (0, T); (H1(D))′);

2

u satisfies the integral equation a.s. for a.e. t ∈ (0, T) in the sense

f distributions in D

u(t) = ϕ0 + t div[A(∇(u + WA)) − A(∇WA)]ds + t f(u + WA) − t 1 |D|

D

f(u + WA)dxds

3

u satisfies the natural boundary condition in the sense of distributions on ∂D.

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 12 / 35

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Existence of a solution of Problem (P2)

Theorem

There exists a unique solution of Problem (P2). We work with the following spaces: H =

v ∈ L2(D),
D

v = 0

, V = H1(D) ∩ H and Z = V ∩ L2p

Proof: We apply the Galerkin method, and use the following notations

0 < γ1 < γ2 ≤ ... ≤ γk ≤ ... eigenvalues of −∆ with homogeneous

Neumann boundary conditions.

wk, k = 0, ... smooth unit eigenfunctions in L2(D).

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 13 / 35

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Existence of a solution of Problem (P2)

We look for an approximate solution of the form um(x, t) − M =

m

i=1

uim(t)wi, M = 1 |D|

D

ϕ0(x)dx which satisfies the equation:

D

∂ ∂t (um(x, t) − M)wj = −

D

[A(∇(um − M + WA)) − A(∇(WA))]∇wj +

D

f(um + WA)wj − 1 |D|

D

D

f(um + WA)dx

wjdx,

for all wj, j = 1, ..., m. um(x, 0) = M +

m

i=1

(ϕ0, wi)wi converges strongly in L2(D) to ϕ0 as m → ∞.

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 14 / 35

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Existence of a solution of Problem (P2)

Remark: The contribution of the nonlocal term vanishes !!

D

wj(x)dx = 0, for all j = 0 ⇓ − 1 |D|

D

D

f(um + WA)dx

wj = 0
D

∂ ∂t (um(x, t) − M)wj = −

D

[A(∇(um − M + WA)) − A(∇(WA))]∇wj +

D

f(um + WA)wj for all wj, j = 1, ..., m.

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 15 / 35

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A priori estimates

Lemma

There exists a positive constant C such that E

D

(um(t) − M)2dx ≤ C, for all t ∈ [0, T], E T

D

|∇(um − M)|2dxdt ≤ C, E T

D

(um − M)2pdxdt ≤ C, E T

D

(f(um + WA))

2p 2p−1

≤ C, E T divA(∇(um + WA))2

(H1(D))′

≤ C.

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 16 / 35

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Weak convergence properties

Hence there exist a subsequence which we denote again by {um − M} and functions u − M ∈ L2(Ω × (0, T); V) ∩ L2p(Ω × (0, T) × D) ∩ L∞(0, T; L2(Ω × D)), χ and Φ such that: um − M ⇀ u − M weakly in L2(Ω × (0, T); V) and L2p(Ω × (0, T) × D) um − M ⇀ u − M weakly star in L∞(0, T; L2(Ω × D)) f(um + WA) ⇀ χ weakly in L

2p 2p−1 (Ω × (0, T) × D)

div(A(∇(um + WA))) ⇀ Φ weakly in L2(Ω × (0, T); (H1)′) as m → ∞.

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 17 / 35

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Passing to the limit

It remains to prove that : Φ + χ, w = div(A(∇(u + WA))) + f(u + WA), w for all w ∈ V ∩ L2p(D).

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 18 / 35

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Monotonicity argument

Sketch of proof : [Marion,1987], [Krylov, Rozovskii, 1981] Let w be such that w − M ∈ L2(Ω × (0, T); V) ∩ L2p(Ω × D × (0, T)) Om = E T e−cs{2div

A(∇(um − M + WA)) − A(∇WA)
−div
A(∇(w − M + WA)) − A(∇WA)
, um − M − (w − M)Z ∗,Z

+2f(um + WA) − f(w + WA), um − M − (w − M)Z ∗,Z −cum − M − (w − M)2}ds]

Lemma

Om ≤ 0

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 19 / 35

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Uniqueness

Proof:

Let u1 and u2 be two solutions of Problem (P2)

u1(t) − u2(t) = t div(A(∇(u1 + WA) − A(∇(u2 + WA)) + t [f(u1 + WA) − f(u2 + WA)] − 1 |D| t [

D

f(u1 + WA) −

D

f(u2 + WA)dx].

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 20 / 35

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Uniqueness

Taking the duality product with u1 − u2
Same initial condition u1(x, 0) = u2(x, 0) = ϕ0(x) ⇒

− 1 |D| t [

D

f(u1 + WA) −

D

f(u2 + WA)]

D

(u1 − u2) = 0.

D

(u1 − u2)2(x, t)dx ≤ C6 t

D

(u1 − u2)2(x, t)dxds, By Gronwall’s Lemma u1 = u2 a.e. in Ω × D × (0, T).

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II- Stochastic nonlocal Allen-Cahn equation with a multiplicative noise

We consider stochastic nonlocal Allen-Cahn equation with multiplicative noise (P2)            ∂ϕ ∂t = ∆ϕ + f(ϕ) − 1 |D|

D

f(ϕ) + Φ(ϕ)∂W ∂t (t), in D × (0, T), ∂ϕ ∂ν = 0,

n ∂D × (0, T),

ϕ(x, 0) = ϕ0(x), x ∈ D, where

D is an open domain of Rn with smooth boundary such that

1 ≤ n ≤ 6;

ϕ0 ∈ H1(D) is a given function;
The nonlinear function f is given by f(s) = −s3 + s.

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The multiplicative term

We suppose that there exist positive constants C2, C3 and C4 such that

∞

l=1

Φl(x, u)2

H ≤ C2, ∞

l=1

Φl(x, u)p

V ≤ C3(1 + up V), ∞

l=1

Φl(x, u1) − Φl(x, u2)2

V ≤ C4u1 − u22 V,

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Definition of a martingale solution

Definition

A triple ((Ω, F, (Ft), P), ϕ, W) is called a strong Martingale solution of Problem (P2) with the inital condition ϕ0 if

1

(Ω, F, (Ft); P) is a stochastic basis and W is a Q-Wiener process;

2

ϕ : [0, T] × Ω → L2(D) is a progressively measurable process such that almost surely ϕ ∈ C([0, T]; V ′), and ϕ ∈ L2(Ω × (0, T); H2(D)) ∩ L4(Ω × (0, T) × D);

3

(ϕ, W) satisfies                ϕ(t) − ϕ0 = t ∆ϕ(s)ds + t f(ϕ(s)) − 1 |D| t

D

f(ϕ(s)) + t Φ(ϕ(s))dW(s), in Ω × D × (0, T), ∂ϕ ∂ν = 0,

n Ω × ∂D × (0, T).

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Definition of a pathwise solution

Definition

Let W be a Q-Wiener process on (Ω, F, (Ft), P). A solution ϕ is called a strong pathwise solution of Problem (P2) with the initial condition ϕ0 if

1

ϕ : [0, T] × Ω → L2(D) is a progressively measurable process such that almost surely ϕ ∈ C([0, T]; V ′), and ϕ ∈ L2(Ω × (0, T); H2(D)) ∩ L4(Ω × (0, T) × D);

2

ϕ satisfies                ϕ(t) − ϕ0 = t ∆ϕ(s)ds + t f(ϕ(s)) − 1 |D| t

D

f(ϕ(s)) + t Φ(ϕ(s))dW(s), in Ω × D × (0, T), ∂ϕ ∂ν = 0,

n Ω × ∂D × (0, T).

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Existence of a martingale solution

To prove the existence of a martingale solution, we have to

1

Obtain a priori estimates for solutions of the Galerkin approximation;

2

Deduce the tightness of the laws;

3

Identify the limit.

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Galerkin approximation

For each integer m we look for an approximate solution ϕm of the form : ϕm(x, t) =

m

i=0

ϕim(t)wi, which satisfies the equations

D

ϕm(t)wj −

D

ϕm(0)wj + t

D

∇ϕm∇wj = t

D

f(ϕm)wj −

D

1 |D| t

D

f(ϕm)wjdx + t

D

Φ(ϕm)wjdW(s) for all wj, j = 0, ..., m.

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 27 / 35

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A priori estimates

Lemma

There exist positive constants K,K1 and ˜ K such that E T

D

|∇ϕm|2dxds ≤ K, E T

D

ϕ4

m ≤ K,

E sup

t∈(0,T)

ϕm(t)p ≤ K, for all p ≥ 2, E T

D

|∆ϕm|2dxds ≤ K1, E∇ϕm2 ≤ K1, for all t ∈ [0, T], E

t

PmΦ(ϕm)dW

p

W α,p(0,T;L2(D))

≤ ˜

K, for p ≥ 2, α ∈ [0, 1 2), E

ϕm −

t PmΦ(ϕm)dW

2

H1(0,T;L2(D))

≤ ˜

K.

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Tightness

We consider the spaces: XS = L2(0, T; V) ∩ C([0, T]; V ′), XW = C([0, T]; H), and X = XS × XW. We then define the probability measures µϕm(.) = P(ϕm ∈ .), µm

W(.) = µW(.) = P(W ∈ .),

namely µϕm(A) = P(ϕm ∈ A), for all A ⊂ XS, and µW(B) = P(W ∈ B), for all B ⊂ XW. This defines a sequence of probability measures on X µm = µϕm × µW.

Lemma

The sequence µm is tight over X and hence weakly compact in X.

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From the tightness property and Prokhorov’s Theorem, there exists a subsequence µmk such that µmk ⇀ µ weakly where µ is a probability measure on X. We associate the corresponding distribution to the approximate solution of the Galerkin scheme as follows: Given a stochastic basis (Ω, F, P), let ϕm be the sequence defined above or a similar sequence. Then there exists a probability space (˜ Ω, ˜ F, ˜ P), a subsequence mk → ∞ and a sequence of X-valued random variables ( ˜ ϕmk, ˜ W mk) such that :

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The probability law of ( ˜

ϕmk, ˜ W mk) is µmk.

The probability law of ( ˜

ϕ, ˜ W) is µ.

( ˜

ϕmk, ˜ W mk) converges almost surely in the topology of X to an element ( ˜ ϕ, ˜ W) i.e. ˜ ϕmk → ˜ ϕ in L2(0, T; V) ∩ C([0, T]; V ′) a.s., ˜ W mk → ˜ W in C([0, T]; H) a.s.

Let ˜

Fmk

t

:= σ( ˜ W mk(s), ˜ ϕmk, s ≤ t) be the union of σ-algebras generated by a random variable ( ˜ ϕmk, ˜ W mk); then each ˜ W mk is a Q-Brownian motion process with respect to the filtration ˜ Fmk

t

.

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Each ( ˜

ϕmk, ˜ W mk) satisfies ˜ P- a.s.

D

˜ ϕmk(t)wj −

D

ϕ0,mk(x)wj = − t

D

∇ ˜ ϕmk∇wj + t

D

f( ˜ ϕmk)wj − 1 |D| t

D

{

D

f( ˜ ϕmk)}wj+ t

D

Φ( ˜ ϕmk)wjd ˜ W mk, Let ˜ Ft := σ( ˜ W(s), ˜ ϕ, s ≤ t), then ((˜ Ω, ˜ F, ( ˜ Ft)t≥0, ˜ P), ˜ ϕ, ˜ W) is a strong martingale solution of (P2).

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Pathwise uniqueness

Proposition

Suppose ϕ and ψ are martingale solutions of Problem (P2) relative to the same stochastic basis (Ω, F, (Ft), P, W). We also suppose that ϕ0 = ψ0. Then ϕ and ψ are respectively indistinguishable in the sense that P(ϕ(t) = ψ(t), ∀t ≥ 0) = 1. Having pathwise uniqueness of the martingale solution we infer the existence and uniqueness of the strong pathwise solution of Problem (P2) by the Gyöngy-Krylov theorem.

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Open problems and future research

Extend to the nonlinear diffusion case for Problem (P2);
Extend to the case of a general reaction term for Problem (P2);
Long time behaviour and invariant measures;
Extend the models to realistic biological situation.

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Thank you for your attention!

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CHAPITRE 1

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The nonlinear function f is a smooth function which satisfies the

following properties:

(F1) There exist positive constants C1 and C2 such that f(a + b)a ≤ −C1a2p + f2(b), |f2(b)| ≤ C2(b2p + 1), for all a, b ∈ R (F2) There exist positive constants C3 and ˜ C3(M) such that |f(s)| ≤ C3|s − M|2p−1 + ˜ C3(M) (F3) There exists a positive constant C4 such that f ′(s) ≤ C4. In particular, one could take f(s) =

2p−1

j=0

bjsj with b2p−1 < 0, p ≥ 2.

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Monotonicity argument

Lemma

Om ≤ 0

Proof.

For the nonlinear diffusion term we use monotonicity

J1 = −2E T e−cs

D

[A(∇(um − M + WA)) − A(∇(w − M + WA))] [∇(um − M + WA) − ∇(w − M + WA)] ≤ −2C0E T e−cs∇(um − w)2 ≤ 0

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Proof.

For the reaction term we use the property (F3)

J2 = E T e−cs2f(um + WA) − f(w + WA), um − wZ ∗,Zds ≤ E T e−cs2C4um − w2ds.

The nonlocal term vanishes

Choosing c ≥ 2C4, we conclude the result.

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Lq regularity

Theorem

Let WA be a solution of Problem (P1); then WA ∈ L∞(0, T; Lq(Ω × D)), for all q ≥ 2.

Proof.

For each positive constant k, denote by Φk : R → R the function Φk(ξ) =        |ξ|q, if |ξ| < k, q 2(q − 1)kq−2ξ2 − q(q − 2)kq−1|ξ| + (q 2 − 1)(q − 1)kq, if k ≤ |ξ|. Φk is a convex C2 function and Φ′

k is a Lipschitz-continuous function

with Φ′

k(0)=0. The function Φk satisfies the inequalities

0 ≤ Φ′

k(ξ) ≤ c(k)ξ and 0 ≤ Φk(ξ) =

ξ

0 Φ′ k(ζ)dζ ≤ c(k) 2 ξ2 for all

ξ ∈ R+.

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Inequalities for the test functions for Lq regularity

Lemma

1

One has 0 ≤ Φ′′

k(ξ) ≤ ck for all ξ ∈ R where ck is a positive

constant depending on k.

2

One has 0 ≤ Φ′′

k(ξ) ≤ q(q − 1)(1 + Φk(ξ)), for all ξ ∈ R.

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Applying Itô’s formula

D

Φk(WA(t))dx = t div(A(∇WA(s))), Φ′

k(WA(s))ds

+ t

D

Φ′

k(WA(s))dW(s)

+1 2

∞

l=1

t

D

Φ′′

k(WA)λl(el)2dxds

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 35 / 35

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Using the monotonicity property of A − t

D

Φ′′

k(WA(s))∇WA(s)A(∇WA(s))ds ≤ −C0E

t

D

Φ′′

k(WA)|∇WA|2

From (3) : 1 2

∞

l=1

t

D

Φ′′

k(WA)λl(el)2dxds ≤ 1

2

∞

l=1

λlel2

L∞

t

D

Φ′′

k(WA)dxds

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The solution of the stochastic heat equation

E

D

Φk(WA(t))dx ≤ −C0E t

D

Φ′′

k(WA)|∇WA|2

+1 2Λ1E t

D

Φ′′

k(WA)dxds

≤ 1 2q(q − 1)Λ1E t

D

(1 + Φk(WA))dxds ≤ C(q)Λ1t|D|eC(q)Λ1t Since Φk(WA(x, t)) converges to |WA(x, t)|q for a.e. x and t when k tends to infinity E

D

|WA(x, t)|qdx = E

D

lim

k→∞Φk(WA(x, t))dx ≤ C(q)Λ1t|D|eC(q)Λ1t

for all t > 0. Therefore, WA ∈ L∞(0, T; Lq(Ω × D)) for all q ≥ 2.

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 35 / 35

SLIDE 45

Uniqueness

Proof:

Let u1 and u2 be two solutions of Problem (P2)

u1(t) − u2(t) = t div(A(∇(u1 + WA) − A(∇(u2 + WA)) + t [f(u1 + WA) − f(u2 + WA)] − 1 |D| t [

D

f(u1 + WA) −

D

f(u2 + WA)dx].

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 35 / 35

SLIDE 46

Uniqueness

Taking the duality product with u1 − u2
Same initial condition u1(x, 0) = u2(x, 0) = ϕ0(x) ⇒

− 1 |D| t [

D

f(u1 + WA) −

D

f(u2 + WA)]

D

(u1 − u2) = 0.

Taking the expectation of the equation

E

D

(u1 − u2)2(x, t)dx ≤ C6E t

D

(u1 − u2)2(x, t)dxds, By Gronwall’s Lemma u1 = u2 a.e. in Ω × D × (0, T).

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 35 / 35

SLIDE 47

CHAPITRE 2

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 35 / 35

SLIDE 48

Proof: ϕ(t) − ψ(t) = t ∆(ϕ − ψ)ds + t (f(ϕ) − f(ψ))ds − 1 |D| t ds

D

dx(f(ϕ) − f(ψ)) + t [Φ(ϕ) − Φ(ψ)]dW We apply Itô’s formula F(s, X) = e−

s

0 B(τ)dτX2. Choosing ε small

enough, we have that Ee−

t

0 B(s)dsϕ(t) − ψ(t)2

≤ −E t B(s)e−

s

0 B(τ)dτϕ(s) − ψ(s)2

+(2 + C5)E

t e−

s

0 B(τ)dτϕ − ψ2

+˜

CεE t e−

s

0 B(τ)dτ

D

(ϕ4 + ψ4)ϕ − ψ2

.

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 35 / 35

SLIDE 49

Applying Gronwall Lemma yields Ee−

t

0 B(s)dsϕ(s) − ψ(s)2 = 0, for all t ∈ (0, T),

so that for all t ∈ (0, T) ϕ(t) = ψ(t) a.e. in D almost surely, which gives the pathwise uniqueness.

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 35 / 35

SLIDE 50

|Mm(t)|4 = |

ϕm(x, 0)dx|4 + 4

t (

ϕm(x, s)dx)3
(Φ(ϕm(x, s))
λl

+6 t

∞

l=1

(

ϕm(x, s)dx)2(
(Φ(ϕm(x, s))
λlel(x))dx)2ds

We bound the last term of (1), also using (1) t

∞

l=1

(

ϕm(x, s)dx)2(
(Φ(ϕm(x, s))
λlel(x))dx)2ds

= t (

ϕm(x, s)dx)2

∞

l=1

(

(Φ(ϕm(x, s))
λlel(x))dx)2ds

≤ 1 |D| t (

ϕm(x, s)dx)2

∞

l=1
D

(Φ(ϕm)

λlel)2dxds

≤ 1 |D| t (

ϕm(x, s)dx)2

∞

l=1

Φ(ϕm)

λlel2dxds

≤ C1 |D| t |Mm(t)|2ds ≤ ˜ C1 t |Mm(t)|4ds. (2)

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 35 / 35

SLIDE 51

Applying Burkholder with G = Pm(Φ(ϕm)√λlel) = PmΦl, X = L2(D) E

t

PmΦ(ϕm)dW

p

W α,p(0,T;L2(D))

≤

cpE T PmΦ(ϕm)p

L2,Q(H,L2(D))dt

≤

cpC3E T (1 + ϕmp

V)dt

Danielle Hilhorst (Université Paris-Sud)

A stochastic nonlocal parabolic equation March 25th, 2019 35 / 35

SLIDE 52

Definition

Suppose that X is a separable Hilbert space. Given p ≥ 2, α ∈ (0, 1), we define the fractional derivative space W α,p(0, T; X) as the Sobolev space of all u ∈ Lp(0, T; X) such that T T |u(t) − u(s)|p

X

|t − s|1+αp dtds < ∞, up

W α,p(0,T;X) =

T |u(t)|p

Xdt +

T T |u(t) − u(s)|p

X

|t − s|1+αp dtds. W 1,2(0, T; X) = H1(0, T; X) = {u ∈ L2(0, T; X) : du dt ∈ L2(0, T; X)}, u2

H1(0,T;X) =

T |u(s)|2

Xds +

T

du

dt (s)

2

X

ds. (3)

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 35 / 35

SLIDE 53

Note that for α ∈ (0, 1

2),

H1(0, T; X) ⊂ W α,2(0, T; X) uW α,2(0,T;X) ≤ CuH1(0,T;X). (4)

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 35 / 35

SLIDE 54

E

ϕm −

t PmΦ(ϕm(s))dW(s)

2

H1(0,T;L2(D))

=

E(ϕ0m + t ∆ϕm(s)ds + t Pmf(ϕm(s))ds − 1 |D| t

D

f(ϕm(s))dxds2

H1(0,T;L2(D)))

≤ ˜ c(T) E( ϕ02 + T ∆ϕm2dt + T ϕ3

m2dt +

T ϕm2 +2 T ϕ3

m2dt + 2

T ϕm2) ≤ ˜ c(T) E

ϕ02 +

T ∆ϕm2dt + 3 T ϕm6

L6(D)dt + 3

T ϕm2dt where we have used the a priori estimates in Lemma 0.1 and the fact that H2(D) is embedded in L6(D) for d ≤ 6.

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 35 / 35

SLIDE 55

1

B1

R := {u ∈ L2(0, T; H2(D)) ∩ W

1 5 ,2(0, T; L2(D)) :

u2

L2(0,T;H2(D)) + u2 W

1 5 ,2(0,T;L2(D)) ≤ R2},

We find that the embedding from L2(0, T; H2(D)) ∩ W

1 5 ,2(0, T; L2(D)) into L2(0, T; H1(D)) is

compact.

2

Let B2,1

R

and B2,2

R

be the balls of radius R > 1 in H1(0, T; L2(D)) and W α,p(0, T; L2(D)) with p > 2 and αp > 1, respectively. Then B2

2R := B2,1 R

+ B2,2

R

is compact in C([0, T]; (H1(D))′). Moreover, for p > 2 we choose α such that αp > 1. We deduce from Lemma that the embeddings W α,p(0, T; H) ⊂ C([0, T]; V ′), (5) is compact.

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 35 / 35

SLIDE 56

Step 1: µϕm((B1

R)c)

= P(ϕm ∈ (B1

R)c)

= P

ϕm2

L2(0,T;H2(D)) + ϕm2 W

1 5 ,2(0,T];H) > R2

≤

P

ϕm2

L2(0,T;H2(D)) > R2

2

+ P
ϕm2

W

1 5 ,2(0,T;H) > R2

2

Step2:

{ϕm− t PmΦ(ϕm)dW ∈ B2,1

R }∩{

t PmΦ(ϕm)dW ∈ B2,2

R } ⊂ {{ϕm} ⊂ B2 2R

By De Morgan’s law (A ∩ B)c = Ac ∪ Bc, the Tchebychev inequality, we infer that µϕm((B2

2R)c)

≤ P

ϕm −

t PmΦ(ϕm)dW2

H1(0,T;H) > R2

+P
t

PmΦ(ϕm)dWp

W α,p(0,T;H) > Rp

Danielle Hilhorst (Université Paris-Sud)

A stochastic nonlocal parabolic equation March 25th, 2019 35 / 35

SLIDE 57

Definition

Suppose (X, d) is a complete separable metric space with B(X) its associated Borel σ- algebra. Let Cb(X) be the set of all real-valued continuous bounded functions on X, and let Pr(X) be the set of all probability measures on (X, B(X)). A collection Λ ⊂ Pr(X) is tight if for every ε > 0 there exists a compact set Kε ⊂ X such that µ(Kε) ≥ 1 − ε, for all µ ∈ Λ. In particular, a sequence {µm}m≥0 ⊂ Pr(X) is tight if for every ε > 0 there exist m0 > 0 and a compact set Kε ⊂ X such that µm(Kε) ≥ 1 − ε, for all m ≥ m0. A sequence {µm}m≥0 ⊂ Pr(X) converges weakly to a probability measure µ if

X

fdµn →

X

fdµ, for all f ∈ Cb(X).

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 35 / 35

SLIDE 58

˜ ϕmk → ˜ ϕ in L2(0, T; V) a.s., we deduce that E   T ˜ ϕmk(t)2

V

2  ≤ T 2E sup

t∈(0,T)

˜ ϕmk4

V ≤ K.

By Lemma which is deduce from the Vitali theorem ( Lemma ??) we conclude that ˜ ϕmk → ˜ ϕ strongly in L2(˜ Ω; L2((0, T); V)). (6) Finally we conclude that there exists a subsequence of { ˜ ϕmk} which we denote again by { ˜ ϕmk} such that ˜ ϕmk − ˜ ϕ2

V → 0

for a.e. (ω, t) ∈ ˜ Ω × [0, T]. (7)

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 35 / 35

SLIDE 59

A typical application of Vitali’s lemma is provided by the next result.

Lemma

We suppose that (i) un → u a.e. in A; (ii) un is uniformly bounded in Lp(µ), for p > 1 . Then un → u in Lr(µ) for all r ∈ [1, p).

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 35 / 35

SLIDE 60

Lemma

Let (Ω, F, P) be a fixed probability space, X a separable Hilbert space. Consider a sequence of stochastic bases Sn = (Ω, F, {Fn

t }t≥0, P, W n),

where W n is a Q-Brownian motion over H with respect to {Fn

t } .

Assume that {Gn}n≥0 are a collection of X-valued {Fn

t } predictable

processes such that Gn ∈ L2(0, T; L2,Q(H, X)) a.s. Finally conisder S = (Ω, F, {Ft}t≥0, P, W) and G ∈ L2([0, T], L2,Q(H, X)) a.s., which is Ft predictable. If Gn → G in probability in L2(0, T; L2,Q(H, X)), (8) W n → W in probability in C([0, T]; H), (9) then t GndW n → t GdW in probability in L2(0, T; X). (10)

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 35 / 35

SLIDE 61

DIVERS

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 35 / 35

SLIDE 62

Brownian motion

Brownian motion is described by the Wiener process. The Wiener process Wt is characterised by four facts:

1

W0 = 0.

2

Wt is almost surely continuous.

3

Wt has independent increments, which means that if 0 ≤ s1 < t1 ≤ s2 < t2 then Wt1 − Ws1 and Wt2 − Ws2 are independent random variables.

4

Wt − Ws ∼ N(0, t − s)(for 0 ≤ s ≤ t). N(µ, σ2) denotes the normal distribution with expected value µ and variance σ2.

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 35 / 35

SLIDE 63

Itô’s Formula

X(t) = X(0) + t h(s)ds + t Φ(s)dW(s), t ∈ [0, T]. Assume a function F : H → R and its partial derivatives Fx, Fxx, are uniformly continuous on bounded subsets of H. Under the above conditions, P-a.s., for all t ∈ [0, T]. F(X(t)) = F(X(0)) + t

0Fx(X(s)), h(s)ds

+ t

0Fx(X(s)), Φ(s)dW(s)H

+ 1

2

t

0 Tr(Fxx(X(s))(Φ(s)

√ Q)(Φ(s) √ Q)∗)ds where Tr

FXX(s, X(s))(Φ(s)Q

1 2 )(Φ(s)Q 1 2 )∗

ds =

∞

k=1

FXX(s, X(s))(Φ(s)Q

1 2 ek), (Φ(s)Q 1 2 ek)H. Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 35 / 35

SLIDE 64

Applying It to an Allen-Cahn type equation with E = R, x ∈ D fixed and with X(t) = ϕm(x, t), F(X) = X 2, ∂F ∂X (X) = 2X, ∂2F ∂X 2 (X) = 2, h(s) = ∆ϕm + Pmf(ϕm) − ϕm + vm, G(s) = PmΦ1(ϕm, vm − ϕm)˜ Pm. We remark that here F does not depend on t. We obtain almost surely, for all t ∈ [0, T] and after integrating on D ϕm(t)2 = ϕ0m2 − 2 t

D

|∇ϕm|2 + 2 t

D

f(ϕm)ϕmdxds −2 t

D

ϕ2

mdxds + 2

t

D

vmϕmdxds +2 t

D

ϕmPmΦ1(ϕm, vm − ϕm)˜ PmdW1(s)dx + t

∞

l=1

PmΦ1(ϕm, vm − ϕm)˜ Pm

λ1

l el2ds.

(11)

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 35 / 35

SLIDE 65

Adapted process

In the study of stochastic processes, an adapted process (also referred to as a non-anticipating or non-anticipative process) is one that cannot "see into the future". An informal interpretation is that X is adapted if and only if, for every realisation and every n, Xn is known at time n. Definition Let

(Ω, F, P) be a probability space;
I be an index set with a total order ≤ (often, I is N, N0, [0, T] or

[0, +∞) ) ;

F· = (Fi)i∈Ibe a filtration of the sigma algebra F;
(S, Σ) be a measurable space, the state space;
X : I × Ω → S be a stochastic process.

The process X is said to be adapted to the filtration (Fi)i∈I if the random variable Xi : Ω → S is a (Fi, Σ)-measurable function for each i ∈ I.

Danielle Hilhorst (Université Paris-Sud) A stochastic nonlocal parabolic equation March 25th, 2019 35 / 35