Uniform asymptotic approximation of diffusion to a small target

TL;DR

This paper develops uniform asymptotic expansions for the first passage time of a diffusing molecule to a small spherical target in a large domain, combining short- and long-time methods for accurate approximation.

Contribution

It introduces a novel uniform asymptotic approach that combines pseudo-potential and eigenfunction expansions to accurately approximate diffusion to a small target.

Findings

Accurately approximates first passage time density for small targets.

Validates the method in a spherically symmetric domain.

Provides a unified framework for short- and long-time asymptotics.

Abstract

The problem of the time required for a diffusing molecule, within a large bounded domain, to first locate a small target is prevalent in biological modeling. Here we study this problem for a small spherical target. We develop uniform in time asymptotic expansions in the target radius of the solution to the corresponding diffusion equation. Our approach is based on combining short-time expansions using pseudo-potential approximations with long-time expansions based on first eigenvalue and eigenfunction approximations. These expansions allow the calculation of corresponding expansions of the first passage time density for the diffusing molecule to find the target. We demonstrate the accuracy of our method in approximating the first passage time density and related statistics for the spherically symmetric problem where the domain is a large concentric sphere about a small target centered at the origin.