45 research outputs found
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
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.Comment: 15 pages, 7 figure
The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator
We analyze a nonlocal diffusion operator having as special cases the
fractional Laplacian and fractional differential operators that arise in
several applications. In our analysis, a nonlocal vector calculus is exploited
to define a weak formulation of the nonlocal problem. We demonstrate that, when
sufficient conditions on certain kernel functions hold, the solution of the
nonlocal equation converges to the solution of the fractional Laplacian
equation on bounded domains as the nonlocal interactions become infinite. We
also introduce a continuous Galerkin finite element discretization of the
nonlocal weak formulation and we derive a priori error estimates. Through
several numerical examples we illustrate the theoretical results and we show
that by solving the nonlocal problem it is possible to obtain accurate
approximations of the solutions of fractional differential equations
circumventing the problem of treating infinite-volume constraints.Comment: 27 pages, 5 figure