Residence Time Near an Absorbing Set

TL;DR

This paper analyzes the residence time of diffusing particles near an absorbing boundary, providing explicit formulas for various scenarios and extending results to biased diffusion, finite intervals, and higher dimensions.

Contribution

It introduces a unified approach to calculate residence times for diffusing particles and random walks, including biased and confined cases, with new analytical results.

Findings

Explicit residence time formulas in 1D for unbiased diffusion.

Extension of methods to biased diffusion and finite intervals.

Distribution of first revisit times for a random walk.

Abstract

We determine how long a diffusing particle spends in a given spatial range before it dies at an absorbing boundary. In one dimension, for a particle that starts at $x_0$ and is absorbed at $x=0$, the average residence time in the range $[x,x+dx]$ is $T(x)=\frac{x}{D}\,dx$ for $x<x_0$ and $\frac{x_0}{D}\,dx$ for $x>x_0$, where $D$ is the diffusion coefficient. We extend our approach to biased diffusion, to a particle confined to a finite interval, and to general spatial dimensions. We use the generating function technique to derive parallel results for the average residence time of the one-dimensional symmetric nearest-neighbor random walk that starts at $x_0=1$ and is absorbed at $x=0$. We also determine the distribution of times at which the random walk first revisits $x=1$ before being absorbed.