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

Abstract

We continue our study of the parabolic Anderson equation @u(x;t)=@t = u(x;t) + (x;t)u(x;t), x2 Z d , t 0, where 2 [0;1) is the diusion constant, is the discrete Laplacian, and plays the role of a dynamic random environment that drives the equation. The initial condition u(x; 0) = u0(x), x2 Z d , is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a eld of particles performing independent simple random walks with binary branching: particles jump at rate 2d , split into two at rate _ 0, and die at rate ( )_ 0. We assume that is stationary and ergodic under translations in space and time, is not constant and satises E(j (0; 0)j) E( (0; 0)) for 2 (0;1). In the present paper we show that lim !1 0( ) = E( (0; 0)) under an additional @ ;