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

2012/08/01 by Erhard, Dirk, Hollander, Frank den, Maillard, Grégory
#35B40 (Secondary) #60F10 #60H25 #82C44 (Primary) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1208.0330

Abstract

In this paper we study the parabolic Anderson equation ∂ u(x,t)/∂ t=κΔu(x,t)+ξ(x,t)u(x,t), x∈\Z^d, t≥ 0, where the u-field and the ξ-field are \R-valued, κ∈ [0,∞) is the diffusion constant, and Δ is the discrete Laplacian. The initial condition u(x,0)=u₀(x), x∈\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 2dκ, split into two at rate ξ\vee 0, and die at rate (-ξ)\vee 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 \E(|ξ(0,0)|)

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