LOCALISATION AND AGEING IN THE PARABOLIC ANDERSON MODEL WITH WEIBULL POTENTIAL

• 成果类型: Article
• 署名作者: Sidorova, Nadia; Twarowski, Aleksander
• 署名单位: University of London; University College London
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/13-AOP882
• 发表日期: 2014
• 页码: 1666-1698
• 关键词: intermittency; diffusion

摘要

The parabolic Anderson model is the Cauchy problem for the heat equation on the integer lattice with a random potential xi. We consider the case when {xi(z) : z is an element of Z(d)} is a collection of independent identically distributed random variables with Weibull distribution with parameter 0 < gamma < 2, and we assume that the solution is initially localised in the origin. We prove that, as time goes to infinity, the solution completely localises at just one point with high probability, and we identify the asymptotic behaviour of the localisation site. We also show that the intervals between the times when the solution re-localises from one site to another increase linearly over time, a phenomenon known as ageing.