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

TL;DR

This paper investigates the fundamental properties of the quenched Lyapunov exponent in the parabolic Anderson model with a dynamic random environment, establishing existence, uniqueness, and continuity under weak assumptions.

Contribution

It proves basic properties of the quenched Lyapunov exponent for the parabolic Anderson model with minimal assumptions on the environment.

Findings

Existence and uniqueness of solutions for all diffusion constants.

The quenched Lyapunov exponent is independent of initial conditions.

Continuity of the Lyapunov exponent with respect to the diffusion constant.

Abstract

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.