WEAK AND ALMOST SURE LIMITS FOR THE PARABOLIC ANDERSON MODEL WITH HEAVY TAILED POTENTIALS

• 成果类型: Article
• 署名作者: van der Hofstad, Remco; Morters, Peter; Sidorova, Nadia
• 署名单位: Eindhoven University of Technology; University of Bath; University of London; University College London
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/08-AAP526
• 发表日期: 2008
• 页码: 2450-2494
• 关键词: random-walk; intermittency; asymptotics; obstacles

摘要

We study the parabolic Anderson problem, that is, the heat equation partial derivative(t)u = Delta u + xi u on (0,infinity) x Z(d) with independent identically distributed random potential {xi(Z): Z is an element of Z(d)) and localized initial condition u(0, x) = 1(0)(x). Our interest is in the long-term behavior of the random total mass U(t) = Sigma(z) u (t, z) of the unique nonnegative solution in the case that the distribution of xi(0) is heavy tailed. For this, we study two paradigm cases of distributions with infinite moment generating functions: the case of polynomial or Pareto tails, and the case of stretched exponential or Weibull tails. In both cases we find asymptotic expansions for the logarithm of the total mass up to the first random term, which we describe in terms of weak limit theorems. In the case of polynomial tails, already the leading term in the expansion is random. For stretched exponential tails, we observe random fluctuations in the almost sure asymptotics of the second term of the expansion, but in the weak sense the fourth term is the first random term of the expansion. The main tool in our proofs is extreme value theory.