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

TL;DR

This paper analyzes the nonlinear stochastic heat equation with rough initial conditions, establishing existence, uniqueness, and moment bounds, and determining growth indices for solutions driven by space-time white noise.

Contribution

It provides new existence and uniqueness results for solutions with measure-valued initial conditions, and characterizes growth indices for the model.

Findings

Established existence and uniqueness without Gronwall's lemma

Derived upper bounds on all moments and a lower bound on second moments

Determined growth indices for the solutions

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].