Joint distribution of multiple boundary local times and related first-passage time problems with multiple targets

TL;DR

This paper develops a mathematical framework to analyze the joint distribution of boundary local times for a diffusing particle encountering multiple boundary subsets, providing explicit solutions for simple geometries and insights into related first-passage times.

Contribution

It extends existing methods to compute joint boundary local time distributions for multiple boundary subsets in various geometries, including explicit solutions for simple cases.

Findings

Explicit solutions for joint boundary local times in interval, circular annulus, and spherical shell.

Derivation of distributions of related first-passage times.

Probabilistic interpretation of the joint local time statistics.

Abstract

We investigate the statistics of encounters of a diffusing particle with different subsets of the boundary of a confining domain. The encounters with each subset are characterized by the boundary local time on that subset. We extend a recently proposed approach to express the joint probability density of the particle position and of its multiple boundary local times via a multi-dimensional Laplace transform of the conventional propagator satisfying the diffusion equation with mixed Robin boundary conditions. In the particular cases of an interval, a circular annulus and a spherical shell, this representation can be explicitly inverted to access the statistics of two boundary local times. We provide the exact solutions and their probabilistic interpretation for the case of an interval and sketch their derivation for two other cases. We also obtain the distributions of various associated first-passage times and discuss their applications.