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

In this paper we study 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 the u-field and the \xi-field are \R-valued, \kappa \in [0,\infty) is the diffusion constant, and $\Delta$ is the discrete Laplacian. 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. Our goal is to prove a number of basic properties of the solution u under assumptions on $\xi$ that are as weak as possible. Throughout the paper 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. Under a mild assumption on the tails of the distribution of \xi, we show that the solution to the parabolic Anderson equation exists and is unique for all \kappa\in [0,\infty). Our main object of interest is the quenched Lyapunov exponent \lambda_0(\kappa)=\lim_{t\to\infty}\frac{1}{t}\log u(0,t). Under certain weak space-time mixing conditions on \xi, we show the following properties: (1)\lambda_0(\kappa) does not depend on the initial condition u_0; (2)\lambda_0(\kappa)<\infty for all \kappa\in [0,\infty); (3)\kappa \mapsto \lambda_0(\kappa) is continuous on [0,\infty) but not Lipschitz at 0. We further conjecture: (4)\lim_{\kappa\to\infty}[\lambda_p(\kappa)-\lambda_0(\kappa)]=0 for all p\in\N, where \lambda_p (\kappa)=\lim_{t\to\infty}\frac{1}{pt}\log\E([u(0,t)]^p) is the p-th annealed Lyapunov exponent. Finally, we prove that our weak space-time mixing conditions on \xi are satisfied for several classes of interacting particle systems.