The narrow capture problem on general Riemannian surfaces

TL;DR

This paper investigates the narrow capture problem on Riemannian surfaces by deriving the mean first passage time for a surface-bound Brownian particle using advanced mathematical techniques.

Contribution

It introduces a novel analytical approach combining layer potential methods and microlocal analysis to derive precise asymptotics of the mean first passage time on Riemannian 2-manifolds.

Findings

Derived the leading order singularity of the mean first passage time.

Calculated the O(1) term and spatial average of the mean first passage time.

Provided a mathematical framework applicable to surface-bound stochastic processes.

Abstract

In this article, we study the narrow capture problem on a Riemannian 2-manifold. This involves the derivation of the mean first passage (sojourn) time of a surface-bound ion modelled as a Brownian particle. We use a layer potential argument in conjunction with microlocal analysis in order to derive the leading order singularity as well as the O(1) term of the mean first passage time and the associated spatial average.