## The Annals of Probability

### Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions

#### Abstract

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].

#### Article information

Source
Ann. Probab., Volume 43, Number 6 (2015), 3006-3051.

Dates
Revised: February 2014
First available in Project Euclid: 11 December 2015

https://projecteuclid.org/euclid.aop/1449843625

Digital Object Identifier
doi:10.1214/14-AOP954

Mathematical Reviews number (MathSciNet)
MR3433576

Zentralblatt MATH identifier
1338.60155

#### Citation

Chen, Le; Dalang, Robert C. Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions. Ann. Probab. 43 (2015), no. 6, 3006--3051. doi:10.1214/14-AOP954. https://projecteuclid.org/euclid.aop/1449843625

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