Residence time near an absorbing set

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 x0 and is absorbed at x = 0, the average residence time of the particle in the range is for x < x0 and for x > x0, 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 then use the generating function technique to derive parallel results for the average number of times that a one-dimensional symmetric nearest-neighbor random walk visits site x when the walk starts at x0 = 1 and is absorbed at x = 0. We also determine the distribution of times when the random walk first revisits x = 1 before being absorbed.