Lecture notes on stationary critical and super-critical SPDEs

The goal of these lecture notes is to present recent results regarding the large-scale behaviour of critical and super-critical non-linear stochastic PDEs, that fall outside the realm of the theory of Regularity Structures. These include the two-dimensional Anisotropic KPZ equation, the stochastic Burgers equation in dimension $d\ge 2$ and the stochastic Navier-Stokes equation with divergence-free noise in dimension $d=2$. Rather than providing complete proofs, we try to emphasise the main ideas, and some crucial aspects of our approach: the role of the generator equation and of the Fluctuation-Dissipation Theorem to identify the limit process; Wiener chaos decomposition with respect to the stationary measure and its truncation; and the so-called Replacement Lemma, which controls the weak coupling limit of the equations in the critical dimension and identifies the limiting diffusivity. For pedagogical reasons, we will focus exclusively on the stochastic Burgers equation. The notes are based on works in collaboration with Dirk Erhard and Massimiliano Gubinelli.