First Passage in the Presence of Jump Rate Fluctuations: A Transition from Infinite to Finite Mean

Abstract

A simple symmetric random walk on a one-dimensional lattice is guaranteed to reach any target site, yet its mean first-passage time diverges. Here, we show that stochastic fluctuations in the jump rate can fundamentally alter this classic result. We find a phase transition in the mean first-passage time: sufficiently broad rate fluctuations render it finite, whereas weaker fluctuations leave it infinite. Our results demonstrate that temporal fluctuations in the jump rate---and hence in the diffusivity---can overcome spatial wandering and regularize otherwise divergent first-passage times.