Dirichlet problem for diffusions with jumps

TL;DR

This paper establishes the existence and uniqueness of solutions to the Dirichlet problem for a class of non-local operators with jumps, linking solutions to associated Feller processes and extending classical diffusion theory.

Contribution

The paper introduces a novel approach to solving the Dirichlet problem for non-local operators with jumps, demonstrating the existence of a unique Feller process and weak solutions.

Findings

Existence of a unique Feller process associated with the operator.

Unique bounded continuous weak solutions to the Dirichlet problem.

Representation of solutions via the associated Feller process.

Abstract

In this paper, we study Dirichlet problem for non-local operator on bounded domains in ${\mathbb R}^d$ $$ {\cal L}u = {\rm div}(A(x) \nabla (x)) + b(x) \cdot \nabla u(x) + \int_{{\mathbb R}^d} (u(y)-u(x) ) J(x, dy) , $$ where $A(x)=(a_{ij}(x))_{1\leq i,j\leq d}$ is a measurable $d\times d$ matrix-valued function on ${\mathbb R}^d$ that is uniformly elliptic and bounded, $b$ is an ${\mathbb R}^d$-valued function so that $|b|^2$ is in some Kato class ${\mathbb K}_d$, for each $x\in {\mathbb R}^d$, $J(x, dy)$ is a finite measure on ${\mathbb R}^d$ so that $x\mapsto J(x, {\mathbb R}^d)$ is in the Kato class ${\mathbb K}_d$. We show there is a unique Feller process $X$ having strong Feller property associated with ${\cal L}$, which can be obtained from the diffusion process having generator $ {\rm div}(A(x) \nabla (x)) + b(x) \cdot \nabla u(x) $ through redistribution. We further show that for any bounded connected open subset $D\subset{\mathbb R}^d$ that is regular with respect to the Laplace operator $\Delta$ and for any bounded continuous function $\varphi $ on $D^c$, the Dirichlet problem ${\cal L} u=0$ in $D$ with $u=\varphi$ on $D^c$ has a unique bounded continuous weak solution on ${\mathbb R}^d$. This unique weak solution can be represented in terms of the Feller process associated with ${\cal L}$.