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

We continue our study of the parabolic Anderson equation $\partial u(x,t)/\partial t = κΔu(x,t) + ξ(x,t)u(x,t)$, $x\in\Z^d$, $t\geq 0$, where $κ\in [0,\infty)$ is the diffusion constant, $Δ$ is the discrete Laplacian, and $ξ$ 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 $2dκ$, split into two at rate $ξ\vee 0$, and die at rate $(-ξ) \vee 0$. We assume that $ξ$ 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.\ $ξ$. Our main object of interest is the quenched Lyapunov exponent $λ_0 (κ) = \lim_{t\to\infty} \frac{1}{t}\log u(0,t)$. In earlier work we showed that under certain mild space-time mixing assumptions the limit exists $ξ$-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 $κ\in (0,\infty)$.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 $ξ$. 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 $ξ$ 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.

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