Moment estimates for some renormalized parabolic Anderson models

Abstract

The theory of regularity structures enables the definition of the following parabolic Anderson model in a very rough environment: ${\partial }_{\mathit{t}}{\mathit{u}}_{\mathit{t}}(\mathit{x})=\frac{1}{2}\mathrm{\Delta }{\mathit{u}}_{\mathit{t}}(\mathit{x})+{\mathit{u}}_{\mathit{t}}(\mathit{x}){\dot{\mathit{W}}}_{\mathit{t}}(\mathit{x})$, for $\mathit{t}\in {\mathbb{R}}_{+}$ and $\mathit{x}\in {\mathbb{R}}^{\mathit{d}}$, where ${\dot{\mathit{W}}}_{\mathit{t}}(\mathit{x})$ is a Gaussian noise whose space time covariance function is singular. In this rough context we shall give some information about the moments of ${\mathit{u}}_{\mathit{t}}(\mathit{x})$ when the stochastic heat equation is interpreted in the Skorohod as well as the Stratonovich sense. Of special interest is the critical case, for which one observes a blowup of moments for large times.