On the chaotic character of the stochastic heat equation, before the onset of intermitttency

• Daniel Conus ^[1] ; Mathew Joseph ^[2] ; Davar Khoshnevisan ^[2]
  1. [1] Lehigh University

     Lehigh University

     City of Bethlehem, Estados Unidos

     
  2. [2] University of Utah

     University of Utah

     Estados Unidos

     
• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 41, Nº. 3, 2, 2013, págs. 2225-2260
• Idioma: inglés
• DOI: 10.1214/11-AOP717
• Texto completo no disponible (Saber más ...)
• Resumen

  • We consider a nonlinear stochastic heat equation ∂tu=12∂xxu+σ(u)∂xtW, where ∂xtW denotes space–time white noise and σ: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 u0: under suitable conditions on u0 and σ, supx∈Rut(x) is a.s. finite when u0 has compact support, whereas with probability one, lim sup|x|→∞ut(x)/(log|x|)1/6>0 when u0 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.