  The theory of regularity structures enables the definition of the following
parabolic Anderson model in a very rough environment: $\partial_{t} u_{t}(x) =
\frac12 \Delta u_{t}(x) + u_{t}(x) \, \dot W_{t}(x)$, 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.
