The parabolic Anderson model in a dynamic random environment: space-time ergodicity for the quenched Lyapunov exponent

• 成果类型: Article
• 署名作者: Erhard, D.; den Hollander, F.; Maillard, G.
• 署名单位: Leiden University; Leiden University - Excl LUMC; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Aix-Marseille Universite
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-014-0564-x
• 发表日期: 2015
• 页码: 1-46
• 关键词: intermittency; catalysts

摘要

We continue our study of the parabolic Anderson equation , where is the diffusion constant, is the discrete Laplacian, and plays the role of a dynamic random environment that drives the equation. The initial condition , 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 , split into two at rate , and die at rate . We assume that is stationary and ergodic under translations in space and time, is not constant and satisfies , where denotes expectation w.r.t. . Our main object of interest is the quenched Lyapunov exponent . In earlier work (Erhard et al. Ann Inst Henri Poincar, Probab Stat, to appear; Gartner et al., Probability in Complex Physical Systems, 2012), we established a number of basic properties of under certain mild space-time mixing and noisiness assumptions on . In particular, we showed that the limit exists -a.s., is finite and continuous on , is globally Lipschitz on , is not Lipschitz at , and satisfies and for . In the present paper we show that under an additional space-time mixing condition on we call Gartner-hyper-mixing. This result, which completes our study of the quenched Lyapunov exponent for general , 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 that are Gartner-hyper-mixing for which the annealed Lyapunov exponent is infinite on , a situation that is referred to as strongly catalytic behavior. Our proof is based on a multiscale analysis of , in combination with discrete rearrangement inequalities for local times of simple random walk and spectral bounds for discrete Schrodinger operators.