Random walks and Brownian motion: A method of computation for first-passage times and related quantities in confined geometries

TL;DR

This paper develops a computational method for calculating mean first-passage times and related quantities for random walks and Brownian motion in confined geometries, including multiple targets and distribution details.

Contribution

It extends previous work by providing a comprehensive approach to compute first-passage times, splitting probabilities, and distribution characteristics in bounded domains for both discrete and continuous processes.

Findings

Computed mean first-passage times for random walks and Brownian motion.

Analyzed splitting probabilities and conditional mean first-passage times.

Explored higher-order moments and full distribution of first-passage times.

Abstract

In this paper we present a computation of the mean first-passage times both for a random walk in a discrete bounded lattice, between a starting site and a target site, and for a Brownian motion in a bounded domain, where the target is a sphere. In both cases, we also discuss the case of two targets, including splitting probabilities, and conditional mean first-passage times. In addition, we study the higher-order moments and the full distribution of the first-passage time. These results significantly extend our earlier contribution [Phys. Rev. Lett. 95, 260601].