Random walk in a birth-and-death dynamical environment

TL;DR

This paper studies a particle performing a random walk in a dynamic environment modeled by birth-and-death processes, establishing laws of large numbers and central limit theorems under ergodic conditions.

Contribution

It introduces new probabilistic techniques to analyze a random walk in a birth-and-death environment without requiring a uniform jump rate lower bound.

Findings

Proves LLN and CLT for the particle position in ergodic environments.

Develops stochastic domination and subadditive methods for non-uniform jump rates.

Analyzes the environment's asymptotics as seen by the particle.

Abstract

We consider a particle moving in continuous time as a Markov jump process; its discrete chain is given by an ordinary random walk on ${\mathbb Z}^d$ , and its jump rate at $({\mathbf x},t)$ is given by a fixed function $\varphi$ of the state of a birth-and-death (BD) process at $\\mathbf x$ on time $t$; BD processes at different sites are independent and identically distributed, and $\varphi$ is assumed non increasing and vanishing at infinity. We derive a LLN and a CLT for the particle position when the environment is 'strongly ergodic'. In the absence of a viable uniform lower bound for the jump rate, we resort instead to stochastic domination, as well as to a subadditive argument to control the time spent by the particle to give $n$ jumps; and we also impose conditions on the initial (product) environmental initial distribution. We also present results on the asymptotics of the environment seen by the particle (under different conditions on $\varphi$).