The parabolic Anderson model in a dynamic random environment: space-time ergodicity for the quenched Lyapunov exponent

TL;DR

This paper studies the parabolic Anderson model with a dynamic random environment, showing that the quenched Lyapunov exponent converges to the environment's mean as the diffusion rate increases, indicating space-time ergodicity in high diffusivity regimes.

Contribution

It proves that the quenched Lyapunov exponent approaches the environment mean under space-time mixing conditions as diffusivity tends to infinity, revealing space-time ergodicity for the model.

Findings

The quenched Lyapunov exponent converges to the environment mean as diffusivity increases.

Under certain mixing conditions, the model exhibits space-time ergodicity in the large diffusivity limit.

The result applies even when the annealed Lyapunov exponent is infinite, indicating strong catalytic behavior.

Abstract

We continue our study of the parabolic Anderson equation $\partial u(x,t)/\partial t = \kappa\Delta u(x,t) + \xi(x,t)u(x,t)$, $x\in\Z^d$, $t\geq 0$, where $\kappa \in [0,\infty)$ is the diffusion constant, $\Delta$ is the discrete Laplacian, and $\xi$ plays the role of a \emph{dynamic random environment} that drives the equation. The initial condition $u(x,0)=u_0(x)$, $x\in\Z^d$, is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate $2d\kappa$, split into two at rate $\xi \vee 0$, and die at rate $(-\xi) \vee 0$. We assume that $\xi$ is stationary and ergodic under translations in space and time, is not constant and satisfies $\E(|\xi(0,0)|)<\infty$, where $\E$ denotes expectation w.r.t.\ $\xi$. Our main object of interest is the quenched Lyapunov exponent $\lambda_0 (\kappa) = \lim_{t\to\infty} \frac{1}{t}\log u(0,t)$. In earlier work we showed that under certain mild space-time mixing assumptions the limit exists $\xi$-a.s., is finite and continuous on $[0,\infty)$, is globally Lipschitz on $(0,\infty)$, is not Lipschitz at 0, and satisfies $\lambda_0(0) = \E(\xi(0,0))$ and $\lambda_0(\kappa) > \E(\xi(0,0))$ for $\kappa \in (0,\infty)$.In the present paper we show that $\lim_{\kappa\to\infty} \lambda_0(\kappa) =\E(\xi(0,0))$ under an additional space-time mixing condition on $\xi$. This result shows that the parabolic Anderson model exhibits space-time ergodicity in the limit of large diffusivity. This fact is interesting because there are choices of $\xi$ that fulfill our assumption for which the annealed Lyapunov exponent $\lambda_1(\kappa) = \lim_{t\to\infty} \frac{1}{t}\log \E(u(0,t))$ is infinite on $[0,\infty)$, a situation that is referred to as strongly catalytic behavior.