Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions
-
Chen, Le [1] ; Dalang, Robert C. [1]
-
[1] École Polytechnique Féderale de Lausanne
-
- Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 43, Nº. 6, 2015, págs. 3006-3051
- Idioma: inglés
- Enlaces
-
Resumen
We study the nonlinear stochastic heat equation in the spatial domain R, driven by space–time white noise. A central special case is the parabolic Anderson model. The initial condition is taken to be a measure on R, such as the Dirac delta function, but this measure may also have noncompact support and even be nontempered (e.g., with exponentially growing tails). Existence and uniqueness of a random field solution is proved without appealing to Gronwall’s lemma, by keeping tight control over moments in the Picard iteration scheme. Upper bounds on all pth moments (p≥2) are obtained as well as a lower bound on second moments. These bounds become equalities for the parabolic Anderson model when p=2. We determine the growth indices introduced by Conus and Khoshnevisan [Probab. Theory Related Fields 152 (2012) 681–701].
