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November 2015 Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions
Le Chen, Robert C. Dalang
Ann. Probab. 43(6): 3006-3051 (November 2015). DOI: 10.1214/14-AOP954


We study the nonlinear stochastic heat equation in the spatial domain $\mathbb{R}$, 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 $\mathbb{R}$, 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 $p$th moments $(p\ge2)$ 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].


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Le Chen. Robert C. Dalang. "Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions." Ann. Probab. 43 (6) 3006 - 3051, November 2015.


Received: 1 June 2013; Revised: 1 February 2014; Published: November 2015
First available in Project Euclid: 11 December 2015

zbMATH: 1338.60155
MathSciNet: MR3433576
Digital Object Identifier: 10.1214/14-AOP954

Primary: 60H15
Secondary: 35R60 , 60G60

Keywords: growth indices , Nonlinear stochastic heat equation , Parabolic Anderson model , rough initial data

Rights: Copyright © 2015 Institute of Mathematical Statistics

Vol.43 • No. 6 • November 2015
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