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

TL;DR

This paper establishes a probabilistic representation for weak solutions to a parabolic boundary value problem on non-smooth, time-varying domains with mixed boundary conditions, linking stochastic processes with PDE solutions.

Contribution

It introduces a probabilistic framework for weak solutions of parabolic PDEs on non-smooth, time-dependent domains with mixed boundary conditions, including continuity properties and applications to inverse problems.

Findings

Probabilistic representation of weak solutions established.

Weak and stochastic solutions are shown to coincide.

Continuity of the solution up to the boundary is proved.

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.