Probabilistic Solution of the General Robin Boundary Value Problem on Arbitrary Domains

TL;DR

This paper develops a probabilistic framework for solving the General Robin boundary value problem on arbitrary domains using capacity methods and measure perturbation of Dirichlet forms, leading to new insights into boundary phenomena.

Contribution

It introduces a novel probabilistic representation for Robin boundary problems on arbitrary domains, involving killing reflecting Brownian motion at random times.

Findings

Derived properties of the semigroup from probabilistic representation

Established convergence theorems for the process

Provided a probabilistic interpretation of boundary phenomena

Abstract

Using a capacity approach, and the theory of measure's perturbation of Dirichlet forms, we give the probabilistic representation of the General Robin boundary value problems on an arbitrary domain $\Omega$, involving smooth measures, which give arise to a new process obtained by killing the general reflecting Brownian motion at a random time. We obtain some properties of the semigroup directly from its probabilistic representation, and some convergence theorems, and also a probabilistic interpretation of the phenomena occurring on the boundary.