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

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.