The Obstacle Problem for Quasilinear Stochastic PDEs with non-homogeneous operator

# The Obstacle Problem for Quasilinear Stochastic PDEs with non-homogeneous operator

## Abstract

We prove the existence and uniqueness of solution of the obstacle problem for quasilinear Stochastic PDEs with non-homogeneous second order operator. Our method is based on analytical technics coming from the parabolic potential theory. The solution is expressed as a pair where is a predictable continuous process which takes values in a proper Sobolev space and is a random regular measure satisfying minimal Skohorod condition. Moreover, we establish a maximum principle for local solutions of such class of stochastic PDEs. The proofs are based on a version of Itô’s formula and estimates for the positive part of a local solution which is non-positive on the lateral boundary.

{frontmatter}\runtitle\thankstext

T1The work of the first and third author is supported by the chair risque de crédit, Fédération bancaire Française \addressUniversité d’Evry-Val-d’Essonne-FRANCE
\printeade1 \thankstextt2The research of the second author was partially supported by the Chair Financial Risks of the Risk Foundation sponsored by Société Générale, the Chair Derivatives of the Future sponsored by the Fédération Bancaire Française, and the Chair Finance and Sustainable Development sponsored by EDF and Calyon \address LUNAM Université, Université du Maine - FRANCE

Université d’Evry-Val-d’Essonne -FRANCE

\runauthor

L. Denis, A. Matoussi and J. Zhang

{keyword}\kwd

parabolic potential, regular measure, stochastic partial differential equations, non-homogeneous second order operator, obstacle problem, penalization method, Itô’s formula, comparison theorem, space-time white noise {keyword}[class=AMS] \kwd[Primary ]60H15; 35R60; 31B150

## 1 Introduction

In this paper we study the following SPDE with obstacle (in short OSPDE):

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩dut(x)=∂i(ai,j(t,x)∂jut(x))dt+∂igi(t,x,ut(x),∇ut(x))dt+f(t,x,ut(x),∇ut(x))dt      ++∞∑j=1hj(t,x,ut(x),∇ut(x))dBjt+ν(t,dx),ut≥St,u0=ξ. (1)

where is a time-dependant symmetric, uniformly elliptic, measurable matrix defined on some open domain , with null Dirichlet condition. The initial condition is given as , a valued random variable, and , and are non-linear random functions. Given an obstacle , we study the obstacle problem for the SPDE (1), i.e. we want to find a solution of (1) which satisfies ”″ where the obstacle is regular in some sense and controlled by the solution of a SPDE.
In recent work [10] we have proved in the homogeneous case, existence and uniqueness of the solution of equation (1) with Dirichlet boundary condition under standard Lipschitz hypotheses and -type integrability conditions on the coefficients. Moreover in [11], still in the homogeneous case, we have obtained a maximum principle for local solutions. In these papers we have assumed that does not depend on time and so many proofs are based on the notion of semigroup associated to the second order operator and on the regularizing property of the semigroup. The aim of this paper is to extend all the results to the non homogeneous case.
Let us recall that the solution is a couple , where is a process with values in the first order Sobolev space and is a random regular measure forcing to stay above and satisfying a minimal Skohorod condition. In order to give a rigorous meaning to the notion of solution, inspired by the works of M. Pierre in the deterministic case (see [24, 24]), we introduce the notion of parabolic capacity. We construct a solution which admits a quasi continuous version hence defined outside a polar set and use the fact that regular measures which in general are not absolutely continuous w.r.t. the Lebesgue measure, do not charge polar sets.

There is a huge literature on parabolic SPDE’s without obstacle. The study of the norms w.r.t. the randomness of the space-time uniform norm on the trajectories of a stochastic PDE was started by N. V. Krylov in [16] (see also Kim [14]), for a more complete overview of existing works on this subject see [8, 9] and the references therein. Let us also mention that some maximum principle have been established by N. V. Krylov [17] for linear parabolic spde’s on Lipschitz domain. Concerning the obstacle problem, there are two approaches, a probabilistic one (see [20, 15]) based on the Feynmann-Kac’s formula via the backward doubly stochastic differential equations and the analytical one (see [12, 22, 27]) based on the Green function.

The main results of this paper are first an existence and uniqueness Theorem for the solution with null Dirichlet condition and a maximum principle for local solutions. This yields for example:

###### Theorem 1.

Let be an Itô process satisfying some integrability conditions, and be a local weak solution of the obstacle problem (1). Assume that is Lipschitz and on , then for all :

 E∥∥(u−M)+∥∥p∞,∞;t≤k(p,t)C(S,f,g,h,M)

where depends only on the barrier , the initial condition , coefficients , the boundary condition and is a function which only depends on and , is the uniform norm on .

## 2 Hypotheses and preliminaries

### 2.1 Settings

Let be an open bounded domain in The space is the basic Hilbert space of our framework and we employ the usual notation for its scalar product and its norm,

 (u,v)=∫Ou(x)v(x)dx,∥u∥=(∫Ou2(x)dx)12.

In general, we shall extend the notation

 (u,v)=∫Ou(x)v(x)dx,

where , are measurable functions defined on such that .
The first order Sobolev space of functions vanishing at the boundary will be denoted as usual by Its natural scalar product and norm are

 (u,v)H10(O)=(u,v)+d∑i=1∫O∂iu(x)∂iv(x)dx,∥u∥H10(O)=(∥u∥22+∥∇u∥22)12.

We shall denote by the space of functions which are locally square integrable in and which admit first order derivatives that are also locally square integrable.
Another Hilbert space that we use is the second order Sobolev space of functions vanishing at the boundary and twice differentiable in the weak sense.

We consider a sequence of independent Brownian motions defined on a standard filtered probability space satisfying the usual conditions.

Let be a measurable and symmetric matrix defined on . We assume that there exist positive constants , and such that for all and almost all :

 λ|η|2≤∑i,jai,j(t,x)ηiηj≤Λ|η|2 and |ai,j(t,x)|≤M. (2)

Let . We denote by the weak fundamental solution of the problem

 ∂tG(t,x;s,y)−d∑i=1∂iai,j(t,x)∂jG(t,x;s,y)=0 (3)

with Dirichlet boundary condition , for all .

We consider the quasilinear stochastic partial differential equation (1) with initial condition and Dirichlet boundary condition .

We assume that we have predictable random functions

 f:R+×Ω×O×R×Rd→R, g=(g1,...,gd):R+×Ω×O×R×Rd→Rd, h=(h1,...,hi,...):R+×Ω×O×R×Rd→RN∗,

In the sequel, will always denote the underlying Euclidean or norm. For example

 |h(t,ω,x,y,z)|2=+∞∑i=1|hi(t,ω,x,y,z)|2.

Assumption (H): There exist non-negative constants such that for almost all , the following inequalities hold for all :

1. the contraction property:

###### Remark 1.

This last contraction property ensures existence and uniqueness for the solution of the SPDE without obstacle (see [9]).

Moreover for simplicity, we fix a terminal time , we assume that:
Assumption (I):

 ξ∈L2(Ω×O) is an F0−measurable random variable
 f(⋅,⋅,⋅,0,0):=f0∈L2([0,T]×Ω×O;R)
 g(⋅,⋅,⋅,0,0):=g0=(g01,...,g0d)∈L2([0,T]×Ω×O;Rd)
 h(⋅,⋅,⋅,0,0):=h0=(h01,...,h0i,...)∈L2([0,T]×Ω×O;RN∗).

We denote by the space of valued predictable continuous processes which satisfy

 ∥u∥T=Esupt∈[0,T]∥ut∥2+E∫T0∥∇ut∥2dt<+∞.

It is the natural space for solutions.
The space of test functions is denote by , where is the space of all real valued infinitely differentiable functions with compact support in and the set of -functions with compact support in .

#### Main example of stochastic noise

Let be a noise white in time and colored in space, defined on a standard filtered probability space whose covariance function is given by:

 ∀s,t∈R+, ∀x,y∈O,  E[˙W(x,s)˙W(y,t)]=δ(t−s)k(x,y),

where is a symmetric and measurable function.
Consider the following SPDE driven by :

 dut(x)=(d∑i,j=1∂iai,j(t,x)∂jut(x)+f(t,x,ut(x),∇ut(x))+d∑i=1∂igi(t,x,ut(x),∇ut(x)))dt+~h(t,x,ut(x),∇ut(x))W(dt,x), (4)

where and are as above and is a random real valued function.
We assume that the covariance function defines a trace class operator denoted by in . It is well known that there exists an orthogonal basis of consisting of eigenfunctions of with corresponding eigenvalues such that

 +∞∑i=1λi<+∞,

and

 k(x,y)=+∞∑i=1λiei(x)ei(y).

It is also well known that there exists a sequence of independent standard Brownian motions such that

 W(dt,⋅)=+∞∑i=1λ1/2ieiBi(dt).

So that equation (4) is equivalent to equation (1) without obstacle and with where

 ∀i∈N∗, hi(s,x,y,z)=√λi~h(s,x,y,z)ei(x).

Assume as in [26] that for all , and

 +∞∑i=1λi∥ei∥2∞<+∞.

Since

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satisfies the Lipschitz hypothesis (H)-(ii) if satisfies a similar Lipschitz hypothesis.

### 2.2 Parabolic potential analysis

In this section we will recall some important definitions and results concerning the obstacle problem for parabolic PDE in [24] and [24].
denotes equipped with the norm:

 ∥v∥2K = ∥v∥2L∞([0,T];L2(O))+∥v∥2L2([0,T];H10(O)) = supt∈[0,T[∥vt∥2+∫T0(∥vt∥2+∥∇vt∥2)dt.

denotes the space of continuous functions on compact support in and finally:

 W={φ∈L2([0,T];H10(O)); ∂φ∂t∈L2([0,T];H−1(O))},

endowed with the norm.
It is known (see [18]) that is continuously embedded in , the set of -valued continuous functions on . So without ambiguity, we will also consider , , .
We now introduce the notion of parabolic potentials and regular measures which permit to define the parabolic capacity.

###### Definition 1.

An element is said to be a parabolic potential if it satisfies:

 ∀φ∈W+T, ∫T0−(∂φt∂t,vt)dt+∫T0E(φt,vt)dt≥0.

We denote by the set of all parabolic potentials.

The next representation property is crucial:

###### Proposition 1.

(Proposition 1.1 in [24]) Let , then there exists a unique positive Radon measure on , denoted by , such that:

 ∀φ∈WT∩C, ∫T0(−∂φt∂t,vt)dt+∫T0E(φt,vt)dt=∫T0∫Oφ(t,x)dνv.

Moreover, admits a right-continuous (resp. left-continuous) version .
Such a Radon measure, is called
a regular measure and we write:

 νv=∂v∂t+Av.
###### Remark 2.

As a consequence, we can also define for all :

 vT=limt↑T¯vt ∈L2(O).
###### Definition 2.

Let be compact, is said to be superior than 1 on , if there exists a sequence with on a neighborhood of converging to in .

We denote:

 SK={v∈P; v is ν−superior to 1 on K}.
###### Proposition 2.

(Proposition 2.1 in [24]) Let compact, then admits a smallest and the measure whose support is in satisfies

 ∫T0∫OdνvK=infv∈P{∫T0∫Odνv; v∈SK}.
###### Definition 3.

(Parabolic Capacity)

• Let be compact, we define ;

• let be open, we define ;

• for any borelian , we define .

###### Definition 4.

A property is said to hold quasi-everywhere (in short q.e.) if it holds outside a set of null capacity.

###### Definition 5.

(Quasi-continuous)

A function is called quasi-continuous, if there exists a decreasing sequence of open subsets of with:

1. for all , the restriction of to the complement of is continuous;

2. .

We say that admits a quasi-continuous version, if there exists quasi-continuous such that .

The next proposition, whose proof may be found in [24] or [24] shall play an important role in the sequel:

###### Proposition 3.

Let a compact set, then

 cap({t}×K)=λd(K),

where is the Lebesgue measure on .
As a consequence, if is a map defined quasi-everywhere then it defines uniquely a map from into . In other words, for any , is defined without any ambiguity as an element in . Moreover, if , it admits version which is left continuous on with values in so that is also defined without ambiguity.

###### Remark 3.

The previous proposition applies if for example is quasi-continuous.

###### Proposition 4.

(Theorem III.1 in [24]) If , then it admits a unique quasi-continuous version that we denote by . Moreover, for all , the following relation holds:

 ∫[0,T[×O~φdνv=∫T0(−∂tφ,v)+E(φ,v)dt+(φT,vT).

We end this section by a convergence lemma which plays an important role in our approach (Lemma 3.8 in [24]):

###### Lemma 1.

If is a bounded sequence in and converges weakly to in ; if is a quasi-continuous function and is bounded by a element in . Then

 limn→+∞∫T0∫Oudνvn=∫T0∫Oudνv.
###### Remark 4.

For the more general case one can see [24] Lemma 3.8.

## 3 Quasi-continuity of the solution of SPDE without obstacle

We consider the SPDE without obstacle:

 dut(x) = ∂i(ai,j(t,x)∂jut(x)+gi(t,x,ut(x),∇ut(x)))dt+f(t,x,ut(x),∇ut(x))dt (5) ++∞∑j=1hj(t,x,ut(x),∇ut(x))dBjt,

As a consequence of well-known results (see for example [9], Theorem 11), we know that under assumptions (H) and (I), SPDE (5) with zero Dirichlet boundary condition, admits a unique solution in , we denote it by , moreover it satisfies the following estimate:

 E[∥u∥2T]≤cE[∥ξ∥2+∫T0(∥∥f0t∥∥2+∥∥|g0t|∥∥2+∥∥|h0t|∥∥2)dt] (6)

The main theorem of this section is the following:

###### Theorem 2.

Under assumptions (H) and (I), the solution of SPDE (5) admits a quasi-continuous version denoted by i.e. a.e. and for almost all , is quasi-continuous.

Before giving the proof of this theorem, we need the following lemmas. The first one is proved in [24], Lemma 3.3:

###### Lemma 2.

There exists such that, for all open set and with on :

 capϑ≤C∥v∥2K.

Let be defined as following

 κ=essinf{v∈P; v≥u a.e. and v(0)≥u+(0)}.

One has to note that is a random function. From now on, we always take for the following measurable version

 κ=supnvn,

where is the non-decreasing sequence of random functions given by

 ⎧⎪⎨⎪⎩∂vnt∂t=Lvnt+n(vnt−ut)−vn0=u+(0). (7)

From F.Mignot and J.P.Puel [21], we know that for almost all , converges weakly to in and that .

###### Lemma 3.

We have the following estimate:

 E∥κ∥2K≤C(E∥u+0∥2+E∥u0∥2+E∫T0∥f0t∥2+∥|g0t|∥2+∥|h0t|∥2dt),

where is a constant depending only on the structure constants of the equation.

Thanks to (2), the proof of Lemma 3 in [10] can be easily extended to the case of non-homogeneous operator.

Proof of Theorem 2: First of all, we remark that we only need to prove this result in the linear case, namely we consider that , and only depend on , and . Then, we approximate the coefficients, the domain and the second order operator in the following way:

1. We mollify coefficients and so consider sequences of functions such that for all , the matrix satisfies the same ellipticity and boundedness assumptions as and

 ∀1≤i,j≤d, limn→+∞ani,j=ai,j a.e.
2. We approximate by an increasing sequence of smooth domains .

3. We consider a sequence in which converges to in and such that for all , .

4. For each , we construct a sequence of predictable functions in
which converges in to such that for all , and

 ∀t≥0, E[∫t0∥hni,s∥2ds]≤E[∫t0∥hi,s∥2ds],

so that

 E[∫t0∥|hns|∥2ds]≤E[∫t0∥|hs|∥2ds]<+∞.
5. We consider a sequence of predictable functions in which converges in to and such that for all , .

6. Finally, let be a sequence in which converges in to and such that for all , .

For all , we put We denote by the weak fundamental solution of the problem (3) associated to and :

 ∂tGn(t,x;s,y)−d∑i,j=1∂iani,j(t,x)∂jGn(t,x;s,y)=0 (8)

with Dirichlet boundary condition .
In a natural way we extend on by setting: on .
We define the process by setting for all :

 unt(⋅)=∫OG(t,⋅,0,y)ξn(y)dy+∫t0∫OG(t,⋅,s,y)fns(y)dyds  +d∑i=1∫t0∫OG(t,⋅,s,y)∂i,ygni(y)dyds  ++∞∑j=1∫t0∫OG(t,⋅,s,y)hnj,s(y)dBjs. (9)

The main point is that there exists a subsequence of which converges everywhere to on , where still denotes the fundamental solution of (3), see Lemma 7 in [9]. From Proposition 6 in [9], we know that is the unique weak solution of (5). is uniformly continuous in space-time variables on any compact away from the diagonal in time ( see Theorem 6 in [1]) and satisfies Gaussian estimates (see Aronson [2]), this ensures that for all , is -almost surely continuous in .
Moreover, since sequences , and are uniformly bounded in -spaces, as a consequence of estimate (6), is bounded in hence in , so that we can extract a subsequence which converges weakly in and such that a sequence of convex combinations of the form

 ^un=Nn∑k=1αnkunk

converges strongly to in . It is clear that for all , is almost surely continuous in .

We consider a sequence of random open sets

 ϑn={|^un+1−^un|>ϵn},Θp=+∞⋃n=pϑn.

Let , from the definition of and the relation (see [24])

 κ(|v|)≤κ(v,v+(0))+κ(−v,v−(0)),

we know that satisfy the conditions of Lemma 2, i.e. et on , thus we get the following relation

 cap(Θp)≤+∞∑n=pcap(ϑn)≤+∞∑n=p∥κn∥2K.

Thus, remarking that , we apply Lemma 3 to and and obtain:

 E[cap(Θp)]≤+∞∑n=pE∥κn∥2K ≤ 2C+∞∑n=p1ϵ2n(E∥ξn+1−ξn∥2+E∫T0∥fn+1t−fnt∥2 + ∥|gn+1t−gnt|∥2+∥|hn+1t−hnt|∥2dt).

Then, by extracting a subsequence, we can consider that

 E∥ξn+1−ξn∥2+E∫T0∥fn+1t−fnt∥2+∥|gn+1t−gnt|∥2+∥|hn+1t−hnt|∥2dt≤12n.

Then we take