On the existence and position of the farthest peaks of a family of stochastic heat and wave equations

• 成果类型: Article
• 署名作者: Conus, Daniel; Khoshnevisan, Davar
• 署名单位: Utah System of Higher Education; University of Utah
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-010-0333-4
• 发表日期: 2012
• 页码: 681-701
• 关键词: intermittency

摘要

We study the stochastic heat equation partial derivative(t)u = Lu + sigma(u)(W) over dot in (1 + 1) dimensions, where (W) over dot is space-time white noice sigma : R -> R is Lipschitz continuous, and L is the generator of a symmetric L,vy process that has finite exponential moments, and u (0) has exponential decay at +/- a. We prove that under natural conditions on sigma : (i) The nu th absolute moment of the solution to our stochastic heat equation grows exponentially with time; and (ii) The distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. Very little else seems to be known about the location of the high peaks of the solution to the stochastic heat equation under the present setting (see, however, Gartner et al. in Probab Theory Relat Fields 111:17-55, 1998; Gartner et al. in Ann Probab 35:439-499, 2007 for the analysis of the location of the peaks in a different model). Finally, we show that these results extend to the stochastic wave equation driven by Laplacian.