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.
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Burch, N., D’Elia, M. & Lehoucq, R.B. The exit-time problem for a Markov jump process. Eur. Phys. J. Spec. Top. 223, 3257–3271 (2014). https://doi.org/10.1140/epjst/e2014-02331-7
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DOI: https://doi.org/10.1140/epjst/e2014-02331-7