Geometric characterization of intermittency in the parabolic Anderson model

• 成果类型: Article
• 署名作者: Gaertner, Juergen; Koenig, Wolfgang; Molchanov, Stanislav
• 署名单位: Technical University of Berlin; University of North Carolina; University of North Carolina Charlotte; Leipzig University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117906000000764
• 发表日期: 2007
• 页码: 439-499
• 关键词: Asymptotics

摘要

We consider the parabolic Anderson problem partial derivative(t)u = Delta u + xi(x)u on R+ x Z(d) with localized initial condition u (0, x) = delta(0)(x) and random i.i.d. potential xi. Under the assumption that the distribution of xi(0) has a double-exponential, or slightly heavier, tail, we prove the following geometric characterization of intermittency: with probability one, as t -> infinity, the over-whelming contribution to the total mass Sigma x u(t, x) comes from a slowly increasing number of islands which are located far from each other. These islands are local regions of those high exceedances of the field xi in a box of side length 2t log(2) t for which the (local) principal Dirichlet eigenvalue of the random operator Delta + xi is close to the top of the spectrum in the box. We also prove that the shape of xi in these regions is nonrandom and that u(t,center dot) is close to the corresponding positive eigenfunction. This is the geometric picture suggested by localization theory for the Anderson Hamiltonian.