Solvability of Dirichlet problem with Integro-differential Operator

TL;DR

This paper investigates the solvability of Dirichlet problems involving non-linear integro-differential operators, using probabilistic methods to construct solutions and analyze exit time operators.

Contribution

It introduces a probabilistic approach to construct continuous supersolutions for non-linear integro-differential Dirichlet problems, advancing theoretical understanding.

Findings

Established conditions for solvability of the Dirichlet problem

Developed a probabilistic construction of supersolutions

Analyzed the continuity set of exit time operators

Abstract

This paper studies the solvability of a class of Dirichlet problem associated with non-linear integro-differential operator. The main ingredient is the probabilistic construction of continuous supersolution via the identification of the continuity set of the exit time operators under Skorohod topology.