A TWO CITIES THEOREM FOR THE PARABOLIC ANDERSON MODEL

• 成果类型: Article
• 署名作者: Koenig, Wolfgang; Lacoin, Hubert; Moerters, Peter; Sidorova, Nadia
• 署名单位: Leipzig University; Universite Paris Cite; University of Bath; University of London; University College London
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/08-AOP405
• 发表日期: 2009
• 页码: 347-392
• 关键词: intermittency

摘要

The parabolic Anderson problem is the Cauchy problem for the heat equation partial derivative(t)u(t, z) = Delta u(t,z) + xi(z)u(t,z) on (0,infinity) x Z(d) with random potential (xi(z): z is an element of Z(d)). We consider independent and identically distributed potentials, such that the distribution function of (z) converges polynomially at infinity. If u is initially localized in the origin, that is, if u(0, z) = 1(0)(z), we show that, as time goes to infinity, the solution is completely localized in two points almost surely and in one point with high probability. We also identify the asymptotic behavior of the concentration sites in terms of a weak limit theorem.