Quenched Lyapunov Exponent for the Parabolic Anderson Model in a Dynamic Random Environment

Abstract

We continue our study of the parabolic Anderson equation ∂u ∕ ∂t = κΔu + γξu for the space–time field $u: \,{\mathbb{Z}}^{d} \times [0,\infty ) \rightarrow \mathbb{R}$, where κ ∈ [0, ∞) is the diffusion constant, Δ is the discrete Laplacian, γ ∈ (0, ∞) is the coupling constant, and $\xi : \,{\mathbb{Z}}^{d} \times [0,\infty ) \rightarrow \mathbb{R}$ is a space–time random environment that drives the equation. The solution of this equation describes the evolution of a “reactant” u under the influence of a “catalyst” ξ, both living on ${\mathbb{Z}}^{d}$.

In earlier work we considered three choices for ξ: independent simple random walks, the symmetric exclusion process, and the symmetric voter model, all in equilibrium at a given density. We analyzed the annealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of u w.r.t. ξ, and showed that these exponents display an interesting dependence on the diffusion constant κ, with qualitatively different behavior in different dimensions d. In the present paper we focus on the quenched Lyapunov exponent, i.e., the exponential growth rate of u conditional on ξ.

We first prove existence and derive qualitative properties of the quenched Lyapunov exponent for a general ξ that is stationary and ergodic under translations in space and time and satisfies certain noisiness conditions. After that we focus on the three particular choices for ξ mentioned above and derive some further properties. We close by formulating open problems.