Escape-from-a-layer approach for simulating the boundary local time in Euclidean domains

TL;DR

This paper introduces an efficient numerical method combining walk-on-spheres and escape problem solutions to simulate boundary local time of reflected Brownian motion, improving accuracy and efficiency in complex domains.

Contribution

It presents a novel escape-from-a-layer approach that simplifies boundary reflections, enabling more efficient simulations of reflected Brownian motion in various Euclidean domains.

Findings

Validated against exact solutions for simple domains

Compared with finite-element method results for complex domains

Demonstrated effectiveness in multi-scale environments

Abstract

We propose an efficient numerical approach to simulate the boundary local time of reflected Brownian motion, as well as the time and position of the associated reaction event on a smooth boundary of a Euclidean domain. This approach combines the standard walk-on-spheres algorithm in the bulk with the approximate solution of the escape problem in a boundary layer. In this way, the most time-consuming simulation of multiple reflections on the boundary is replaced by an equivalent escape event. We validate the proposed escape-from-a-layer approach by comparing simulated statistics of the boundary local time with exact results known for simple domains (a disk, a circular annulus, a sphere, a spherical shell) and with the numerical results obtained by a finite-element method in more sophisticated domains. This approach offers a powerful tool for simulating reflected Brownian motion in multi-scale confinements such as porous media or biological environments, and for solving the related partial differential equations. Its applications in the context of diffusion-controlled reactions in chemical physics are discussed.