The complement value problem for a class of second order elliptic integro-differential operators

TL;DR

This paper establishes existence, uniqueness, and probabilistic representation of solutions for a class of elliptic integro-differential equations with boundary conditions, using stochastic calculus and heat kernel estimates.

Contribution

It provides the first comprehensive analysis of the complement value problem for these operators, including explicit probabilistic solutions.

Findings

Unique bounded continuous weak solutions exist under mild conditions.

Explicit probabilistic representation of solutions is derived.

Stochastic calculus and heat kernel estimates are key tools used.

Abstract

We consider the complement value problem for a class of second order elliptic integro-differential operators. Let $D$ be a bounded Lipschitz domain of $\mathbb{R}^d$. Under mild conditions, we show that there exists a unique bounded continuous weak solution to the following equation $$ \left\{\begin{array}{l}(\Delta+a^{\alpha}\Delta^{\alpha/2}+b\cdot\nabla+c+{\rm div} \hat{b})u+f=0\ \ {\rm in}\ D,\\ u=g\ \ {\rm on}\ D^c. \end{array}\right.$$ Moreover, we give an explicit probabilistic representation of the solution. The recently developed stochastic calculus for Markov processes associated with semi-Dirichlet forms and heat kernel estimates play important roles in our approach.