Random potentials for Markov processes

TL;DR

This paper investigates integral functionals of Markov processes in higher dimensions, representing them via vector-valued random measures, with applications to processes like Brownian motion and compound Poisson processes.

Contribution

It introduces a novel representation of integral functionals of Markov processes as integrals against vector-valued random measures, expanding understanding of such functionals in higher dimensions.

Findings

Representation of integral functionals via vector-valued random measures

Application to Brownian motion and diffusions

Extension to compound Poisson processes

Abstract

The paper is devoted to the integral functionals $\int_0^\infty f(X_t)\,{\mathrm{d}t}$ of Markov processes in $\X$ in the case $d\ge 3$. It is established that such functionals can be presented as the integrals $\int_{\X} f(y) \G(x, \mathrm{d}y, \omega)$ with vector valued random measure $\G(x, \mathrm{d}y, \omega)$. Some examples such as compound Poisson processes, Brownian motion and diffusions are considered.