Annealed Survival Probability of Random Walk in an Inhomogeneous Poisson Environment of Mobile Traps

Abstract

We study the annealed survival probability of a random walk in an inhomogeneous Poisson environment of mobile traps on $\mathbb{Z}^d$. In dimensions $d=1,2$, we determine the asymptotics for the annealed survival probability of the walker under suitable assumptions on the average trap intensity and study how inhomogeneity in the initial trap configuration affects these asymptotics. Our results extend the asymptotics proved for the homogeneous setting in \cite{DGRS2012}. We also establish a law of large numbers, central limit theorem and large deviation principle for the inhomogeneous Poisson trap environment in $d\ge 1$, generalising the corresponding results proved for the homogeneous case considered in \cite{CG1984}. We present a class of examples of inhomogeneous trap environments where the decay rate of survival probability can be identified and also instances where it does not decay with time.