Eur. Phys. J. Special Topics 223, 3257–3271 (2014)
https://doi.org/10.1140/epjst/e2014-02331-7

Regular Article

The exit-time problem for a Markov jump process

¹ Department of Mathematics, Gonzaga University, Spokane, WA 99258, USA
² Sandia National Laboratories, Albuquerque, NM 87185-{1321, 1320}, USA

^a e-mail: burchn@gonzaga.edu
^b e-mail: mdelia@sandia.gov
^c e-mail: rblehou@sandia.gov

Received: 17 October 2014
Revised: 5 November 2014
Published online: 15 December 2014

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.