Narrow escape problem in the presence of the force field

TL;DR

This paper analyzes the narrow escape problem for Brownian particles in a 3D Riemannian manifold with force fields, deriving asymptotic mean sojourn times and exploring the Green's function structure.

Contribution

It provides the first asymptotic expansion of mean sojourn time for particles under force fields in curved spaces, with new insights into the Green's function.

Findings

Asymptotic expansion of mean sojourn time derived

Singular structure of the restricted Neumann Green's function characterized

Results applicable to complex geometries and force fields

Abstract

This paper considers the narrow escape problem of a Brownian particle within a three-dimensional Riemannian manifold under the influence of the force field. We compute an asymptotic expansion of mean sojourn time for Brownian particles. As an auxiliary result, we obtain the singular structure for the restricted Neumann Green's function which may be of independent interest.