ON THE CHAOTIC CHARACTER OF THE STOCHASTIC HEAT EQUATION, BEFORE THE ONSET OF INTERMITTTENCY

• 成果类型: Article
• 署名作者: Conus, Daniel; Joseph, Mathew; Khoshnevisan, Davar
• 署名单位: Lehigh University; Utah System of Higher Education; University of Utah
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/11-AOP717
• 发表日期: 2013
• 页码: 2225-2260
• 关键词: partial-differential-equations; martingales; INTEGRALS

摘要

We consider a nonlinear stochastic heat equation at partial derivative(t)u = 1/2 partial derivative(xx)u + sigma(u)partial derivative W-xt, where partial derivative(xt) W denotes space time white noise and sigma : R -> R is Lipschitz continuous. We establish that, at every fixed time t > 0, the global behavior of the solution depends in a critical manner on the structure of the initial function u(0): under suitable conditions on u(0) and sigma, sup(x is an element of R)u(t)(x) is a.s. finite when u(0) has compact support, whereas with probability one, lim sup(vertical bar x vertical bar ->infinity) u(t)(x)/(log vertical bar x vertical bar)(1/6) > 0 when u(0) is bounded uniformly away from zero. This sensitivity to the initial data of the stochastic heat equation is a way to state that the solution to the stochastic heat equation is chaotic at fixed times, well before the onset of intermittency.