On divergence of expectations of the Feynman-Kac type with singular potentials

TL;DR

This paper investigates conditions under which Feynman-Kac expectations with singular potentials diverge, focusing on Brownian motion and stable processes, and extends related boundary heat equation results to half-space scenarios.

Contribution

It provides new criteria for divergence of Feynman-Kac expectations with singular potentials for Brownian and stable processes, and relates these to boundary heat equation problems.

Findings

Identifies divergence conditions for expectations with singular potentials.

Extends boundary heat equation analysis to half-space for Brownian motion.

Connects divergence phenomena with boundary behavior in PDEs.

Abstract

Motivated by the work of Baras-Goldstein (1984), we discuss when expectations of the Feynman-Kac type with singular potentials are divergent. Underlying processes are Brownian motion and $\alpha$-stable process. In connection with the work of Ishige-Ishiwata (2012) concerned with the heat equation in the half-space with a singular potential on the boundary, we also discuss the same problem in the half-space for the case of Brownian motion.