Computing the exit-time for a finite-range symmetric jump process

Abstract

This paper investigates the exit-time for a broad class of symmetric finite-range jump processes via the corresponding master equation, a nonlocal diffusion equation suitably constrained. In direct analogy to the classical diffusion equation with a homogeneous Dirichlet boundary condition, the nonlocal diffusion equation is augmented with a homogeneous volume-constraint. The volume-constrained master equation provides an efficient alternative over Monte Carlo simulation for computing an important statistic of the process. Several numerical examples are given.

A Galerkin Radial Basis Function Method for Nonlocal Diffusion

“…Comparing (2) and (3), we see at least two differences. The first difference is that the nonlocal problem replaces a differential operator with an integral operator.…”

“…Imposing the volume constraint guarantees that the nonlocal problem is well-posed [5]. When g Á 0, then the solution u (2) is the probability density for the exit-time problem; see [2,4,6] for further details and information. In this note, we propose a Galerkin radial basis function (RBF) method to numerically solve (2).…”

“…When g Á 0, then the solution u (2) is the probability density for the exit-time problem; see [2,4,6] for further details and information. In this note, we propose a Galerkin radial basis function (RBF) method to numerically solve (2). We also exploit a recently developed Lagrange function quadrature scheme [7] for the necessary integrations.…”

“…In contrast, the primary advantage of the Galerkin RBF method is that entries in the stiffness matrix only require a pointwise evaluation of the kernel and multiplication by quadrature weightscomplications arising from overlapping partial element volumes are irrelevant. A disadvantage of the use of radial basis functions relative to discontinuous piecewise polynomials includes a "Gibbs phenomenon" at any discontinuities of the solution u for (2).…”

Identification of the Diffusion Parameter in Nonlocal Steady Diffusion Problems

“…where the equality constraint (the nonlocal counterpart of a Dirichlet boundary condition for PDEs) acts on an interaction volume Ω I that is disjoint from Ω. Nonlocal diffusion problems such as (2) have been analyzed in the recent works [19,20,21,22], and techniques for an accurate numerical solution have been developed and applied to diverse applications [1,23,24,25,26,27,28]. However, these mathematical models are not exact; parameters such as volume constraint data, diffusivity coefficients, and source terms are often unknown or subject to uncertainty.…”