The exit-time problem for a Markov jump process

TL;DR

This paper investigates the exit-time problem for finite-range Markov jump processes, modeling anomalous transport with nonlocal diffusion equations and employing nonlocal calculus for analysis.

Contribution

It introduces a variational formulation and applies nonlocal vector calculus to analyze the exit-time problem for bounded jump processes.

Findings

Formulation of volume-constrained nonlocal diffusion equations

Derivation of nonlocal Kolmogorov equations for moments

Establishment of well-posedness for the nonlocal model

Abstract

The purpose of this paper is to consider the exit-time problem for a finite-range Markov jump process, i.e, the distance the particle can jump is bounded independent of its location. Such jump diffusions are expedient models for anomalous transport exhibiting super-diffusion or nonstandard normal diffusion. We refer to the associated deterministic equation as a volume-constrained nonlocal diffusion equation. The volume constraint is the nonlocal analogue of a boundary condition necessary to demonstrate that the nonlocal diffusion equation is well-posed and is consistent with the jump process. A critical aspect of the analysis is a variational formulation and a recently developed nonlocal vector calculus. This calculus allows us to pose nonlocal backward and forward Kolmogorov equations, the former equation granting the various moments of the exit-time distribution.