2013/04/08 by Dirk Erhard, Frank den Hollander, Erhard, Dirk +3
Mathematics · #60F10 #60H25 #82C44 #FOS: Mathematics #Primary 60K35 #Probability (math.PR) #Secondary 35B40 #math.PR #msc:35B40 #msc:60F10 #msc:60H25 #msc:60K35 #msc:82C44
paper · pdf · doi:10.48550/arxiv.1304.2274
35 pages, 4 figures, the main result in this version is stronger than in the previous version
arxiv created 2013/07/11 · arxiv updated 2013/07/15
We continue our study of the parabolic Anderson equation ∂ u(x,t)/∂ t = κΔu(x,t) + ξ(x,t)u(x,t), x∈\Zd, t≥ 0, where κ∈ [0,∞) is the diffusion constant, Δ is the discrete Laplacian, and ξ plays the role of a dynamic random environment that drives the equation. The initial condition u(x,0)=u0(x), x∈\Zd, 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. We assume that ξ is stationary and ergodic under translations in space and time, is not constant and satisfies \E(|ξ(0,0)|)<∞, where \E denotes expectation w.r.t. ξ. Our main object of interest is the quenched Lyapunov exponent λ0 (κ) = limt→∞ (1)/(t)log u(0,t). In earlier work we showed that under certain mild space-time mixing assumptions the limit exists ξ-a.s., is finite and continuous on [0,∞), is globally Lipschitz on (0,∞), is not Lipschitz at 0, and satisfies λ0(0) = \E(ξ(0,0)) and λ0(κ) > \E(ξ(0,0)) for κ∈ (0,∞).In the present paper we show that limκ→∞ λ0(κ) =\E(ξ(0,0)) under an additional space-time mixing condition on ξ. This result shows that the parabolic Anderson model exhibits space-time ergodicity in the limit of large diffusivity. This fact is interesting because there are choices of ξ that fulfill our assumption for which the annealed Lyapunov exponent λ1(κ) = limt→∞ (1)/(t)log \E(u(0,t)) is infinite on [0,∞), a situation that is referred to as strongly catalytic behavior.