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

Abstract

We continue our study of the parabolic Anderson equation $\partial u(x,t)/\partial t=\kappa \Delta u(x,t)+\xi (x,t)u(x,t)$, $x\in {\mathbb{Z}}^d$, $t\ge 0$, 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 $u(x,0)=u_0(x)$, $x\in {\mathbb{Z}}^d$, 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 $\E(|ξ(0,0)|)<\infty$, 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 $λ_0(0) = \E(ξ(0,0))$ and $λ_0(κ) > \E(ξ(0,0))$ for $\kappa \in (0,\infty )$.In the present paper we show that $\lim_{κ\to\infty} λ_0(κ) =\E(ξ(0,0))$ 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 $λ_1(κ) = \lim_{t\to\infty} \frac{1}{t}\log \E(u(0,t))$ is infinite on $[0,\infty )$, a situation that is referred to as strongly catalytic behavior.

35 pages, 4 figures, the main result in this version is stronger than in the previous version

References in corpus (1)

• The parabolic Anderson model in a dynamic random environment: basic properties of the quenched Lyapunov exponent