Moment estimates for some renormalized parabolic Anderson models

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.