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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=κΔu(x,t)+ξ(x,t)u(x,t), x\in\Z^d, t\geq 0, where the u-field and the ξ-field are \R-valued, κ\in [0,\infty) is the diffusion constant, and $Δ$ 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 2dκ, split into two at rate ξ\vee 0, and die at rate (-ξ)\vee 0. Our goal is to prove a number of basic properties of the solution u under assumptions on $ξ$ that are as weak as possible. Throughout the paper 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. Under a mild assumption on the tails of the distribution of ξ, we show that the solution to the parabolic Anderson equation exists and is unique for all κ\in [0,\infty). Our main object of interest is the quenched Lyapunov exponent λ_0(κ)=\lim_{t\to\infty}\frac{1}{t}\log u(0,t). Under certain weak space-time mixing conditions on ξ, we show the following properties: (1)λ_0(κ) does not depend on the initial condition u_0; (2)λ_0(κ)<\infty for all κ\in [0,\infty); (3)κ\mapsto λ_0(κ) is continuous on [0,\infty) but not Lipschitz at 0. We further conjecture: (4)\lim_{κ\to\infty}[λ_p(κ)-λ_0(κ)]=0 for all p\in\N, where λ_p (κ)=\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 ξare satisfied for several classes of interacting particle systems.