Parabolic Anderson Model in a Dynamic Random Environment: Random Conductances

Abstract

The parabolic Anderson model is defined as the partial differential 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 \kappa \in [0,\infty) is the diffusion constant, \Delta is the discrete Laplacian, and \xi is a dynamic random environment that drives the equation. 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 focus is on the Lyapunov exponents \lambda_p(\kappa) = \lim_{t\to\infty} \frac{1}{t} \log \E([u(0,t)]^p)^{1/p}, p \in \N, and \lambda_0(\kappa) = \lim_{t\to\infty} \frac{1}{t}\log u(0,t). We investigate what happens when \kappa\Delta is replaced by \Delta^\cK, where \cK = \{\mathcal{K}(x,y)\colon\,x,y\in\Z^d,\,x \sim y\} is a collection of random conductances between neighbouring sites replacing the constant conductances \kappa in the homogeneous model. We show that the associated annealed Lyapunov exponents are given by the formula \lambda_p(\cK) = \sup\{\lambda_p(\kappa) \colon\,\kappa\in\Supp(\cK)\}, where \Supp(\cK) is the set of values taken by the \cK-field. We also show that for the associated quenched Lyapunov exponent this formula only provides a lower bound. Our proof is valid for three classes of reversible \xi, and for all \cK satisfying a certain clustering property, namely, there are arbitrarily large balls where \cK is almost constant and close to any value in \Supp(\cK). What our result says is that the Lyapunov exponents are controlled by those pockets of \cK where the conductances are close to the value that maximises the growth in the homogeneous setting.