Moment Estimates for some Renormalized Parabolic Anderson Models

In this talk I will first introduce and recall some basic facts about the parabolic Anderson model. This model can simply be seen as a linear heat equation in a random environment. I will spend some time describing a couple of manifestations of what is usually called localization, which is the main physical phenomenon observed in this context. Then I will talk about some recent developments concerning parabolic Anderson models in a very rough environment (namely a noise with very singular space-time covariance function). The theory of regularity structures enables the definition parabolic Anderson models in rough contexts. If we call \(u(t,x)\) the solution to our renormalized stochastic heat equation, I will give some information about the moments of \(u(t,x)\) when the stochastic heat equation is interpreted in the Skorohod as well as in the Stratonovich sense. Of special interest is the critical case, for which one observes a blowup of moments for large times.