Probabilistic representation of weak solutions to a parabolic boundary value problem on a non-smooth domain

Probabilistic representation of weak solutions to a parabolic boundary value problem
on a non-smooth domain

Masaaki Tsuchiya    Hajime Kawakami: Appendix B
Abstract

The probabilistic representation of weak solutions to a parabolic boundary value problem is established in the following framework. The boundary value problem consists of a second order parabolic equation defined on a time-varying Lipschitz domain in a Euclidean space and of a mixed boundary condition composed of a Robin and the homogeneous Dirichlet conditions. It is assumed that the time-varying domain is included in a fixed smooth domain and that a certain part of the boundary of the time-varying domain is also included in the boundary of the fixed domain, say the fixed boundary. The Robin condition is imposed on a part of the boundary included in the fixed one and the Dirichlet condition on the rest of the boundary. Such a parabolic boundary value problem always has a unique weak solution for given data; however it does not possess a classical or strong solution in general, even in the case of equations with constant coefficients. The stochastic solution to the boundary value problem is also considered and, by showing the equality between both the solutions, it is obtained the probabilistic representation for the weak solution. Furthermore, it is ensured that, for the weak solution, the stochastic solution gives a version which is continuous up to the lateral boundary of the domain except the border of the adjoining place imposed each of the boundary conditions. As an application, it is shown the continuity property of a functional (cost function) related to an optimal stopping problem motivated by an inverse problem determining the shape of a domain.

AMS Subject Classification: primary: 60G46, 60J60;
              secondary: 35D30, 35K20, 35R37, 60J55

Keywords: parabolic boundary value problem, weak solution, stochastic
       solution, oblique reflecting diffusion process, local time on the
       boundary, shape identification inverse problem

Kanazawa University Akita University

Probabilistic representation of weak solutions to a parabolic boundary value problem

on a non-smooth domain

()

1 Introduction

The probabilistic representation of solutions to second order parabolic equations is a useful tool to analyze these solutions (e.g., boundary sensitive analysis of solutions in [7], numerical analysis of solutions in [8]) and has several applications (e.g., the literature cited in [11], [24], [31], and [16]). For classical solutions to elliptic or parabolic equations, such probabilistic representation has been studied extensively in [11]. Parabolic equations encountered in applications have often no solutions with sufficient regularity. Several authors have studied the probabilistic representation of weak solutions to parabolic equations on a whole Euclidean space via backward stochastic differential equations (e.g., [24], [31]).

This paper concerns with the probabilistic representation of weak solutions to a parabolic equation with a mixed boundary condition on a time-varying Lipschitz domain in a Euclidean space: the boundary condition is composed of a Robin and the homogeneous Dirichlet conditions. Such a parabolic equation is treated in [19] to study an inverse problem determining the shape of a time-varying domain; in general, it does not possess a classical or strong solution, even if the equation has constant coefficients.

The probabilistic representation is considered in the following framework: The time-varying domain is included in a fixed smooth domain and a certain part of the boundary of the time-varying domain is also included in the boundary of the fixed domain, say the fixed boundary. The Robin condition is imposed on a part of the boundary included in the fixed boundary and the homogeneous Dirichlet condition is imposed on the rest of the boundary; we call the place setting the Robin condition (resp. Dirichlet condition) the Robin part (resp. Dirichlet part). Accordingly the Dirichlet part may be varied with time and both of the parts may be adjoining. As a basic stochastic process for the representation, we take the corresponding oblique reflecting diffusion process on the closure of the fixed domain to the parabolic equation.

Then the probabilistic representation of a weak solution is given by the stochastic solution, which is the expectation of the quantity obtained by applying Itô’s formula formally to the weak solution and the diffusion process. The equality within the domain is verified based on an appropriate approximation of the weak solution; the boundedness and regularity of the weak solution and a Poincaré type inequality with weight (see Lemma 4.1 of [19]) play key roles. To show the equality on the lateral boundary, we need to examine the continuity up to the boundary for the stochastic solution, since the boundary values of weak solutions are given by their traces on the boundary.

The continuity of the stochastic solution is shown through the coupled martingale formulation for the oblique reflecting diffusion process (see [14], [29], [18] for such martingale formulations), because we have to use the continuity of the local time with respect to random parameter and to treat the weak convergence property of functionals which contain the local time. Then we ensure that the stochastic solution is continuous up to the lateral boundary except the border between the Robin and Dirichlet parts: it is derived from the continuity property of the local time and the entrance time to the Dirichlet part and further from a certain estimate for the distribution of the entrance time in the case the process starts from a point near the Dirichlet part.

The paper is organized as follows. In Section 2, we provide a necessary setting and assumptions: they are concerned with the considered domain and the parabolic boundary value problem. The main result (Theorem 3.1 and Corollary 3.1) is stated with proof in Section 3 after describing the coupled martingale problem for the oblique reflecting diffusion process. In Section 4, as a simple application of the main result, we show the continuity of a functional whose argument is a domain belonging to an admissible class. In Appendix, we provide several technical details for the proof of the main result: the boundedness and regularity of weak solutions; no attainability of the process to closed null sets of the boundary; the continuity of the entrance time to the Dirichlet part; the estimate for the distribution of the entrance time; a variant of the mapping theorem in weak convergence of probability measures.

2 Preliminaries and a parabolic boundary value problem

We start with introducing necessary notations for describing the parabolic boundary value problem. It is treated in the backward form adapted to the probabilistic consideration. In what follows, we treat the time-varying domain in as a non-cylindrical domain in the time-space . For a subset of denote the –section of by

If necessary, we identify with the set in For a bounded open subset of we consider its parabolic boundary, lateral boundary and ceiling (the time reversed notion of the bottom in the forward form) in the backward form and denote by and respectively, which are subsets of the boundary in Here, for the precise definition (in the forward form), we refer to page 1787 in [6]. For and denote by the cylindrical set determined by and in particular, we put

Now we denote the usual Sobolev space on an open set in with a nonnegative integral order by ; that is,

where indicates the weak derivative of with respect to multi-index Moreover, we recall Sobolev spaces and another function space on a domain in the time-space For nonnegative integers and , we set

and, when , we set

where let These function spaces are equipped with the following norms, respectively:

For a Lipschitz domain in denote by the boundary trace operator from into it can be given by the following pointwise limit (see [10], p. 133): for

(2.1)

where  () and stands for the –dimensional Hausdorff measure (then is the surface measure on ). More generally, if is an open subset of is a neighborhood of in and , then the limit in (2.1) by replacing with exists –a.e. Hence, if necessary, we use the notion of boundary trace in this extended sense; the limit is also denoted by .

We need the notion of regularized distance for the Euclidean distance. Following Theorem 2 on page 171 of [25], we summarize the fundamental properties. For a closed set in a Euclidean space, there exists a function , the regularized distance for the Euclidean distance such that (i) there are positive constants , satisfying for every closed set , where is taken as an absolute constant and is taken as a constant depending only on and increasing with the dimension of the Euclidean space; (ii) is a function in and for any multi-index there is a positive constant satisfying for every closed set For a subset of the Euclidean space, let Then for every subset with ; so that, in what follows, we use instead of in and , where is taken as the constant in the time-space.

This paper concerns with a domain which varies with time of in a fixed bounded Lipschitz domain in Therefore the time-varying domain, say is described as a domain of in the cylindrical domain Moreover we take a Lipschitz dissection of the boundary of Here by a Lipschitz dissection of we mean that and are disjoint open subsets of and is an –dimensional closed Lipschitz surface in (see [23] for the precise definition). In the following, we further need the notion of Lipschitz lateral surface with extreme time edges in the time-space (see Condition 3.1 in [19] for the definition).

We first impose the following condition on the considered domain

Condition 2.1
  1. is a bounded Lipschitz domain in

  2. for each is a non-empty Lipschitz domain in and where and

  3. and

  4. the lateral boundary of is a Lipschitz lateral surface with extreme time edges and includes the set and further there is an open subset in satisfying and

Therefore the time-varying portion of the lateral boundary of is included in the set In the paper, we mainly treat the case where has no connected component included in and , although we can treat some of general types of division of into two parts: one includes the time-varying portion and the other does not. The case treated here is a typical one where each of both the parts has a component adjoining each other; such a case needs a complicated treatment to obtain the result.

Let and be an real matrix-valued function, -dimensional real vector-valued functions and a real-valued function defined on respectively, and a real-valued function defined on . Furthermore suppose that is symmetric and positive definite. For such , and , define a second-order differential operator on :

As mentioned in the beginning, we treat the parabolic equation in the backward form and hence we consider weak solutions to the terminal-boundary value problem for the parabolic equation. Define the parabolic operator on in the backward form, and the conormal derivative relative to and the boundary operator on by

where is the inward unit normal vector at the boundary of in We note that under Condition 2.1.

Now consider the following terminal-boundary value problem on the domain :

(2.2)

In what follows, assume that is extended onto with value zero outside and are extended onto with value zero outside On the problem (2.2) we impose the following assumption.

Assumption 2.1

The coefficients and source terms of the terminal-boundary value problem (2.2) satisfy the following conditions.

  1. There exists a constant such that for every and

  2. The coefficients of the parabolic operator and are bounded measurable on and respectively.

  3. Suppose that where (resp. ) stands for the space with respect to the measure (resp. , here is the surface measure on ). Moreover, is defined and it belongs to where for  

The notion of weak solutions to (2.2) is defined as follows.

Definition 2.1

A function is called a weak solution in to the terminal-boundary value problem (2.2) if it satisfies the following conditions:

  1. that is, on for almost every ;

  2. for every with and

    (2.3)

By the results in [19], we see that, under Assumption 2.1, there exists a unique weak solution in to the problem (2.2). Moreover, in the proof of the main theorem, the boundedness of the weak solution plays a key role. The boundedness result is shown in Appendix A.1. There we use the following norms: for a measurable function on or on for a measurable subset of and let

3 Main result

3.1 Coupled martingale problem

To state and prove the main result, we use the coupled martingale formulation for the oblique reflecting diffusion process associated with the parabolic and boundary operators. For this purpose, we need a further assumption on the domain the set and the coefficients and the source terms in (2.2).

Assumption 3.1
  1. is a bounded domain in for an

  2. Let and For denote by , and the boundary of , the interior of and the closure of in respectively, and let Suppose that

    where indicates the Euclidean distance.

  3. and are Lipschitz continuous in and their Lipschitz constants are uniformly bounded in ; furthermore, and are differentiable in and their derivatives are –Hölder continuous on . In addition, and are also –Hölder continuous on , and is Lipschitz continuous on .

  4. Suppose that , is continuous on , is Lipschitz continuous on and is Lipschitz continuous on .

In what follows, assume that such coefficients and source terms defined on in such a way as , and so on for and According to the extension of the coefficients and the source terms, we also extend the domain and a subset of the lateral boundary to and along the time axis as follows:

In addition, if necessary, and are appropriately extended onto with the same property and further extended onto in the same way as above. Under Assumption 3.1, the operator and the boundary operator are rewritable in the non-divergence and in the oblique derivative forms, respectively:

Further set

Then we consider the coupled martingale problem associated with as in [29], [18]. Let and We assume that , and each are equipped with the locally uniform convergence topology. Denote generic elements of , and by , and , respectively; that is, Then put

and define the fields and generated by and by , respectively. We now introduce two spaces of test functions for the coupled martingale problem: stands for the space of functions on having bounded continuous derivatives with and for the space of continuous functions on with . By [29], [18], under Assumptions 2.1 and 3.1, for each the coupled martingale problem associated with has a unique solution starting from , say that is, is a unique probability measure on satisfying the following conditions:

  1. ;

  2. ;

  3. for every the process defined by

    is a martingale on the filtered probability space

We know that, under the condition (i), the condition (iii) is equivalent to the fact: for every the process is a martingale on the filtered probability space ; that is, the filtration may be replaced with Moreover we note that the condition (ii) is equivalent to the condition , that is, for every , . This is also equivalent to the condition that, for every , the process is a –martingale, because is a –adapted continuous bounded variation process with . We also observe that, for , if we set , then for . Accordingly we have another equivalent formulation of the coupled martingale problem as in the following remark.

Remark 3.1

A probability measure on is called a solution to the coupled martingale problem associated with starting from if it satisfies the following conditions:

  1. ;

  2. for every and the process is a two-dimensional martingale on the filtered probability space

(There are various equivalent types of the condition in (ii) just above as in Theorem 4.2.1 of [27].) By this unified martingale formulation, the fundamental facts on the martingale problem described in §6.1 and §6.2 of [27] can be directly applied to the coupled martingale problem (see also [9], Chap. 4 for a general treatment of martingale problems and the Markov property of their solutions).

The coupled martingale formulation for diffusion processes with reflecting barrier is crucial for our treatments, since the continuity in of the canonical process for the local time on the boundary is essentially used. By virtue of the fundamental facts derived from the unified martingale formulation, noted in Remark 3.1, the uniqueness of solutions implies that the family is strong Markov in the sense as in Theorem 6.2.2 of [27].

In addition, we need the uniform exponential integrability of :

(3.1)

(see Proof of Theorem 4.3 in [18] and also Proposition 3.5 in [7]).

Moreover, under Assumptions 2.1 and 3.1, the transition probability has a transition density with respect to the –dimensional Lebesgue measure , which is a fundamental solution to the parabolic equation on with the boundary condition . In addition, the transition density and its first order derivatives in have the usual upper Gaussian bound: for every there exist positive constants and such that

(3.2)

for every and (see [12], [18]).

3.2 Statement and proof of the main result

For a Borel set in and we set

in particular, in the case and Whenever it holds that and that and for with (). Moreover, for a closed set is a stopping time relative to the filtration and, for an open set is a stopping time relative to the filtration

Let

Then, under Condition 2.1, Assumptions 2.1 and 3.1, we have the following probabilistic representation of the weak solution to the problem (2.2).

Theorem 3.1

The equality

(3.3)

holds in the following each case:

  1. –a.e.

  2. –a.e. , where indicates the surface measure on .

Furthermore, the right hand side of (3.3) is continuous on .

Proof. We first verify the equality (3.3) in the case (i). Throughout the proof, we extend onto with value zero outside and use the same symbol for the extension. Choose a increasing function on such that for for and for In the following, we take and satisfying the conditions:

  1. The set is contained in a tubular neighborhood of in (see [13], Chap. 14, Appendix for such a neighborhood).

  2. .

  3. .

For such and , let

Using the argument on page 27 in [19], we know that the family is uniformly integrable and that for a positive constant for sufficiently small Therefore

(3.4)

Moreover, we see that

(3.5)

which is verified in Appendix A.2. Hence, applying the smoothing procedure in Theorem 3 on page 127 in [10] separately to the time variable and the space variable, we can take an approximating sequence to the function such that

We recall briefly the smoothing procedure for functions to use again it later. It is based on a partition of unity for subordinate to a finite open covering of and on mollifying the localized ones of the functions on each open set consisting of the covering; in particular, in each open set including some part of the boundary, the mollifying is combined with a certain inward parallel displacement. Moreover, by Theorem A.1, we see that

then, from the smoothing procedure for it follows that

for and . For let

Then, for and sufficiently small and mentioned above, define

and, for