Quenched Lyapunov exponent for the parabolic Anderson model in a dynamic random environment

Abstract

We continue our study of the parabolic Anderson equation $\partial u/\partial t = \kappa\Delta u + \gamma\xi u$ for the space-time field $u\colon\,\Z^d\times [0,\infty)\to\R$, where $\kappa \in [0,\infty)$ is the diffusion constant, $\Delta$ is the discrete Laplacian, $\gamma\in (0,\infty)$ is the coupling constant, and $\xi\colon\,\Z^d\times [0,\infty)\to\R$ is a space-time random environment that drives the equation. The solution of this equation describes the evolution of a "reactant" $u$ under the influence of a "catalyst" $\xi$, both living on $\Z^d$. In earlier work we considered three choices for $\xi$: independent simple random walks, the symmetric exclusion process, and the symmetric voter model, all in equilibrium at a given density. We analyzed the \emph{annealed} Lyapunov exponents, i.e., the exponential growth rates of the successive moments of $u$ w.r.t.\ $\xi$, and showed that these exponents display an interesting dependence on the diffusion constant $\kappa$, with qualitatively different behavior in different dimensions $d$. In the present paper we focus on the \emph{quenched} Lyapunov exponent, i.e., the exponential growth rate of $u$ conditional on $\xi$. We first prove existence and derive some qualitative properties of the quenched Lyapunov exponent for a general $\xi$ that is stationary and ergodic w.r.t.\ translations in $\Z^d$ and satisfies certain noisiness conditions. After that we focus on the three particular choices for $\xi$ mentioned above and derive some more detailed properties. We close by formulating a number of open problems.