Nonlinear noise excitation of intermittent stochastic PDEs and the topology of LCA groups

• Khoshnevisan, Davar ^[1] ; Kim, Kunwoo ^[1]
  1. [1] 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. 43, Nº. 4, 2015, págs. 1944-1991
• Idioma: inglés
• DOI: 10.1214/14-AOP925
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• Resumen

  • Consider the stochastic heat equation ∂tu=Lu+λσ(u)ξ, where L denotes the generator of a Lévy process on a locally compact Hausdorff Abelian group G, σ:R→R is Lipschitz continuous, λ≫1 is a large parameter, and ξ denotes space–time white noise on R+×G.

    The main result of this paper contains a near-dichotomy for the (expected squared) energy E(∥ut∥2L2(G)) of the solution. Roughly speaking, that dichotomy says that, in all known cases where u is intermittent, the energy of the solution behaves generically as exp{const⋅λ2} when G is discrete and ≥exp{const⋅λ4} when G is connected.