Exit problem as the generalized solution of Dirichlet problem

TL;DR

This paper links exit problems with Dirichlet problems through Feynman-Kac functionals, establishing conditions for solutions and applying them to non-stationary HJB equations with fractional Laplacian operators.

Contribution

It introduces new conditions for the continuity of exit operators and connects overfitting boundary issues with fine topology, advancing the understanding of Dirichlet problems and their solutions.

Findings

Established the continuity of exit operators under Skorokhod topology.

Connected overfitting Dirichlet boundary with fine topology.

Verified solvability of non-stationary HJB equations with fractional Laplacian.

Abstract

This paper investigates sufficient conditions for a Feynman-Kac functional up to an exit time to be the generalized viscosity solution of a Dirichlet problem. The key ingredient is to find out the continuity of exit operator under Skorokhod topology, which reveals the intrinsic connection between overfitting Dirichlet boundary and fine topology. As an application, we establish the sub and supersolutions for a class of non-stationary HJB (Hamilton-Jacobi-Bellman) equations with fractional Laplacian operator via Feynman-Kac functionals associated to $\alpha$-stable processes, which help verify the solvability of the original HJB equation.