Intermittency in a catalytic random medium

• 成果类型: Article
• 署名作者: Gaertner, J.; den Hollander, F.
• 署名单位: Technical University of Berlin; Leiden University; Leiden University - Excl LUMC
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117906000000467
• 发表日期: 2006
• 页码: 2219-2287
• 关键词: parabolic anderson model; asymptotics; tails

摘要

In this paper, we study intermittency for the parabolic Anderson equation partial derivative u/partial derivative t = K Delta u + xi u, where u: Z(d) x [0, infinity) -> R, K is the diffusion constant, Delta is the discrete Laplacian and xi : Z(d) X [0, infinity) -> R is a space-time random medium. We focus on the case where xi is gamma times the random medium that is obtained by running independent simple random walks with diffusion constant rho starting from a Poisson random field with intensity v. Throughout the paper, we assume that K, gamma, rho, v is an element of (0, infinity). The solution of the equation describes the evolution of a reactant u under the influence of a catalyst. We consider the annealed Lyapunov exponents, that is, the exponential growth rates of the successive moments of u, and show that they display an interesting dependence on the dimension d and on the parameters K, gamma, rho, v, with qualitatively different intermittency behavior in d = 1, 2, in d = 3 and in d >= 4. Special attention is given to the asymptotics of these Lyapunov exponents for K down arrow 0 and K -> infinity.