A Note on Limiting Behaviour of Disastrous Environment Exponents

Abstract

We consider a random walk on the $d$-dimensional lattice and investigate the asymptotic probability of the walk avoiding a "disaster" (points put down according to a regular Poisson process on space-time). We show that, given the Poisson process points, almost surely, the chance of surviving to time $t$ is like $e^{-\alpha \log (\frac1k) t } $, as $t$ tends to infinity if $k$, the jump rate of the random walk, is small.