Probabilistic representations of solutions of elliptic boundary value problem and non-symmetric semigroups

TL;DR

This paper develops a probabilistic framework to establish the existence and uniqueness of bounded solutions for non-symmetric elliptic boundary value problems with singular coefficients, extending classical results without relying on the maximum principle.

Contribution

It introduces a probabilistic approach using Dirichlet forms and heat kernel estimates to handle general non-symmetric elliptic operators with singular coefficients.

Findings

Existence and uniqueness of bounded solutions for the boundary value problem.

Probabilistic representation of the associated non-symmetric semigroup.

Application of Dirichlet form theory to non-symmetric elliptic operators.

Abstract

In this paper, we use a probabilistic approach to show that there exists a unique, bounded continuous solution to the Dirichlet boundary value problem for a general class of second order non-symmetric elliptic operators $L$ with singular coefficients, which does not necessarily have the maximum principle. The theory of Dirichlet forms and heat kernel estimates play a crucial role in our approach. A probabilistic representation of the non-symmetric semigroup $\{T_t\}_{t\ge 0}$ generated by $L$ is also given.