Time-reversal and elliptic boundary value problems

TL;DR

This paper establishes the existence and uniqueness of bounded weak solutions for a class of elliptic boundary value problems with singular coefficients using probabilistic methods involving time reversal of Markov processes.

Contribution

It introduces a novel probabilistic approach leveraging time reversal and Dirichlet forms to solve elliptic boundary value problems without relying on the maximum principle.

Findings

Existence and uniqueness of solutions proven

Applicable to elliptic operators with singular coefficients

Method bypasses the need for maximum principle

Abstract

In this paper, we prove that there exists a unique, bounded continuous weak solution to the Dirichlet boundary value problem for a general class of second-order elliptic operators with singular coefficients, which does not necessarily have the maximum principle. Our method is probabilistic. The time reversal of symmetric Markov processes and the theory of Dirichlet forms play a crucial role in our approach.