The rate of convergence of the Walk on Spheres Algorithm

TL;DR

This paper analyzes how quickly the Walk on Spheres algorithm converges when estimating Brownian motion exit probabilities, linking convergence rates to the domain's local geometry.

Contribution

It provides a complete characterization of the convergence rate of WoS based on the local geometry of the domain, advancing understanding of its efficiency.

Findings

Convergence rate depends on local geometric properties.

Complete characterization of WoS convergence in terms of domain geometry.

Insights into optimizing WoS for different domain shapes.

Abstract

In this paper we examine the rate of convergence of one of the standard algorithms for emulating exit probabilities of Brownian motion, the Walk on Spheres (WoS) algorithm. We obtain the complete characterization of the rate of convergence of WoS in terms of the local geomnetry of a domain.