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

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In this paper we study the parabolic Anderson equation ∂ u(x,t)/∂ t=κΔu(x,t)+ξ(x,t)u(x,t), x∈ᵈ, t≥ 0, where the u-field and the ξ-field are -valued, κ∈ [0,∞) is the diffusion constant, and Δ is the discrete Laplacian. The initial condition u(x,0)=u₀(x), x∈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 0, and die at rate (-ξ) 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 (|ξ(0,0)|)<∞, where 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∈ [0,∞). Our main object of interest is the quenched Lyapunov exponent λ₀(κ)=ₜ→∞1/t u(0,t). Under certain weak space-time mixing conditions on ξ, we show the following properties: (1)λ₀(κ) does not depend on the initial condition u₀; (2)λ₀(κ)<∞ for all κ∈ [0,∞); (3)κ λ₀(κ) is continuous on [0,∞) but not Lipschitz at 0. We further conjecture: (4)κ→∞[λₚ(κ)-λ₀(κ)]=0 for all p∈, where λₚ (κ)=ₜ→∞1/pt([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.