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: ${\partial }_t{u}_t(x)=\frac{1}{2}\Delta u_t(x)+u_t(x){\dot{W}}_t(x)$, for $t\in {\mathbb{R}}_{+}$ and $x\in {\mathbb{R}}^d$, where ${\dot{W}}_t(x)$ is a Gaussian noise whose space time covariance function is singular. In this rough context, we shall give some information about the moments of $u_t(x)$ 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.

The paper has been split. The construction of Stratonovich solution has been removed from this version

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