Random walk in a birth-and-death dynamical environment

TL;DR

The paper analyzes a time-inhomogeneous random walk on ${\mathbb Z}^d$ whose jump rates are given by a monotone function of an evolving birth–death environment, with BD processes at different sites being independent and ergodic. The authors establish a Law of Large Numbers and a Central Limit Theorem for the particle position under a strongly ergodic regime, deriving a key LLN for the jump times via subadditive ergodic theory and stochastic domination arguments. They extend the limit theorems to broader product initial environmental distributions, using coupling and regeneration techniques that depend on dimension and the jump-distribution support. Additionally, they study the environment seen from the particle, proving convergence to a limiting distribution (absolutely continuous with respect to the BD product measure in several cases) and providing tail bounds that underpin this convergence. Together, the results illuminate diffusion properties and aging-like phenomena in dynamical trap environments without ellipticity, offering a rigorous framework for space-time random environments driven by birth–death dynamics.

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 $\varphi$).

Figures (3)

• Figure 1: ${\mathcal{E}}_1,{\mathcal{E}}_2,\ldots$ are iid standard exponentials; $x$(resp., $y$)-axis indicates constancy intervals of $\omega_\mathbf 0$ (resp., $\check\omega_\mathbf 0$) in a realization where with $\omega_{\mathbf 0}(0)=\check\omega_{\mathbf 0}(0)= k\in{\mathbb N}$.
• Figure 2: Illustration of occurrences of random variables introduced in (\ref{['vt1']},\ref{['vt2']}). In (b) and (c), edges in red represent single jumps. In (c) we have $\vec{\vartheta}_{\ell+1}=\reflectbox{\vec{\reflectbox{\vartheta}}}_{\ell}$, but not in (b).
• Figure 3: Schematic depiction of a stretch of the (backward) trajectory of $\mathsf z\in\{\Upsilon_{{\mathcal{L}}_m}\leq m^{1+b}\}$.

Theorems & Definitions (44)

• Remark 2.1
• Lemma 2.2
• proof
• Theorem 3.1: Law of Large Numbers for $X$
• Theorem 3.2: Central Limit Theorem for $X$
• Lemma 3.3
• Corollary 3.4
• Lemma 3.5
• Corollary 3.6
• Lemma 3.7