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
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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.
Sandia is a multi-program laboratory managed and operated by Sandia Corporation, a wholly subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.
© EDP Sciences, Springer-Verlag, 2014
