The Obstacle Problem for Quasilinear Stochastic PDEs with nonhomogeneous operator
Abstract
We prove the existence and uniqueness of solution of the obstacle problem for quasilinear Stochastic PDEs with nonhomogeneous 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 nonpositive on the lateral boundary.
T1The work of the first and third author is supported by the chair risque de crédit, Fédération bancaire Française
\addressUniversité
d’EvryVald’EssonneFRANCE
\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
\printeade2
Université
d’EvryVald’Essonne
FRANCE
\printeade3
L. Denis, A. Matoussi and J. Zhang
parabolic potential, regular measure, stochastic partial differential equations, nonhomogeneous second order operator, obstacle problem, penalization method, Itô’s formula, comparison theorem, spacetime 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):
(1) 
where is a timedependant 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
nonlinear 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 spacetime 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 FeynmannKac’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 :
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,
In general, we shall extend the notation
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
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 :
(2) 
Let . We denote by the weak fundamental solution of the problem
(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
In the sequel, will always denote the underlying Euclidean or norm. For example
Assumption (H): There exist nonnegative constants such that for almost all , the following inequalities hold for all :




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):
We denote by the space of valued predictable continuous processes which satisfy
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:
where is a symmetric and measurable function.
Consider the following SPDE driven by :
(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
and
It is also well known that there exists a sequence of independent standard Brownian motions such that
So that equation (4) is equivalent to equation (1) without obstacle and with where
Assume as in [26] that for all , and
Since
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:
denotes the space of continuous functions on compact support in and finally:
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:
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:
Moreover, admits a rightcontinuous (resp. leftcontinuous)
version .
Such a Radon measure, is called a regular measure and we write:
Remark 2.
As a consequence, we can also define for all :
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:
Proposition 2.
(Proposition 2.1 in [24]) Let compact, then admits a smallest and the measure whose support is in satisfies
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 quasieverywhere (in short q.e.) if it holds outside a set of null capacity.
Definition 5.
(Quasicontinuous)
A function is called quasicontinuous, if there exists a decreasing sequence of open subsets of with:

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

.
We say that admits a quasicontinuous version, if there exists quasicontinuous 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
where is the Lebesgue measure on .
As a consequence, if is a map defined quasieverywhere 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 quasicontinuous.
Proposition 4.
(Theorem III.1 in [24]) If , then it admits a unique quasicontinuous version that we denote by . Moreover, for all , the following relation holds:
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 quasicontinuous function and is bounded by a element in . Then
Remark 4.
For the more general case one can see [24] Lemma 3.8.
3 Quasicontinuity of the solution of SPDE without obstacle
We consider the SPDE without obstacle:
(5)  
As a consequence of wellknown 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:
(6) 
The main theorem of this section is the following:
Theorem 2.
Under assumptions (H) and (I), the solution of SPDE (5) admits a quasicontinuous version denoted by i.e. a.e. and for almost all , is quasicontinuous.
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 :
Let be defined as following
One has to note that is a random function. From now on, we always take for the following measurable version
where is the nondecreasing sequence of random functions given by
(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:
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 nonhomogeneous 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:

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

We approximate by an increasing sequence of smooth domains .

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

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

We consider a sequence of predictable functions in which converges in to and such that for all , .

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 :
(8) 
with Dirichlet boundary condition .
In a natural way we extend on by setting:
on .
We define the process by setting for all
:
(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 spacetime 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
converges strongly to in . It is clear that for all , is almost surely continuous in .
We consider a sequence of random open sets
Let , from the definition of and the relation (see [24])
we know that satisfy the conditions of Lemma 2, i.e. et on , thus we get the following relation
Thus, remarking that , we apply Lemma 3 to and and obtain:
Then, by extracting a subsequence, we can consider that
Then we take