The exit-time problem for a Markov jump process

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 di...

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Bibliographic Details
Published in:The European physical journal. ST, Special topics Special topics, 2014-12, Vol.223 (14), p.3257-3271
Main Authors: Burch, N., D’Elia, M., Lehoucq, R. B.
Format: Article
Language:English
Subjects:
Atomic
Brownian Motion
Brownian Motion in Confined Geometries. Guest Editors: S.M. Bezrukov
Classical and Continuum Physics
CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
Condensed Matter Physics
European Physical Journal Special Topic
Jump Process
L. Schimansky-Geier and G. Schmid (Eds.)
Materials Science
MATHEMATICS AND COMPUTING
Measurement Science and Instrumentation
Molecular
Optical and Plasma Physics
Physics
Physics and Astronomy
Regular Article
Sandia National Laboratory
Volume Constraint
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Summary: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.
ISSN:1951-6355
1951-6401
DOI:10.1140/epjst/e2014-02331-7