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

Abstract

35 pages, 4 figures.International audienceWe 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

The Parabolic Anderson Model

“…The results of [ErhHolMai14] (see Remark 8.10), interesting as they are on their own, are only the preparation for a much deeper result that is derived in the follow-up paper [ErhHolMai15a]. Here a stronger mixing property (called Gärtner-hyper-mixing) is imposed, under which it is shown that (at least for the localised initial condition u 0 D ı 0 ) lim Ä!1 0 .Ä/ D h .0; 0/i.…”

Anderson polymer in a fractional Brownian environment: asymptotic behavior of the partition function

“…The parabolic Anderson model (PAM) has been extensively studied both when the potential ξ(t, x) is a real-valued field (see e.g. [7,8,13] and the references therein) and when it is a white Gaussian noise which is a distributional-valued field (see e.g. [2,1] and their cited references).…”