Computation of Local Time of Reflecting Brownian Motion and Probabilistic Representation of the Neumann Problem

TL;DR

This paper develops numerical methods to compute the local time of reflecting Brownian motion in three dimensions and uses these methods to probabilistically solve the Neumann boundary problem for Laplace's equation.

Contribution

It introduces walk-on-spheres and lattice-based algorithms for approximating RBM local time and demonstrates their convergence through numerical experiments.

Findings

Numerical methods accurately approximate RBM local time.

Convergence observed with increased path length and sampling.

Applicable to various domain shapes like cube, sphere, and ellipsoid.

Abstract

In this paper, we propose numerical methods for computing the boundary local time of reflecting Brownian motion (RBM) in R3 and its use in the probabilistic representation of the solution of the Laplace equation with the Neumann boundary condition. Approximations of the RBM based on a walk-on-spheres (WOS) and random walk on lattices are discussed and tested for sampling the RBM paths and their applicability in finding accurate approximation of the local time and discretization of the probabilistic formula. Numerical tests for several types of domains (cube, sphere, and ellipsoid) have shown the convergence of the numerical methods as the length of the RBM path and number of paths sampled increase.