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

被引:43
作者
Conus, Daniel [1 ]
Joseph, Mathew [2 ]
Khoshnevisan, Davar [2 ]
机构
[1] Lehigh Univ, Dept Math, Bethlehem, PA 18015 USA
[2] Univ Utah, Dept Math, Salt Lake City, UT 84112 USA
基金
瑞士国家科学基金会;
关键词
Stochastic heat equation; chaos; intermittency; PARTIAL-DIFFERENTIAL-EQUATIONS; MARTINGALES; INTEGRALS;
D O I
10.1214/11-AOP717
中图分类号
O21 [概率论与数理统计]; C8 [统计学];
学科分类号
020208 ; 070103 ; 0714 ;
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.
引用
收藏
页码:2225 / 2260
页数:36
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