标题： 关于平稳临界和超临界SPDE的讲义 显示英文标题

标题： Lecture notes on stationary critical and super-critical SPDEs

摘要： 这些讲义的目标是介绍关于临界和超临界非线性随机偏微分方程的大尺度行为的最新结果，这些结果超出了正则性结构理论的范畴。 这些包括二维各向异性KPZ方程、维度$d\ge 2$的随机Burgers方程以及维度$d=2$中具有散度自由噪声的随机Navier-Stokes方程。 我们并没有提供完整的证明，而是试图强调主要思想以及我们方法中的某些关键方面：生成器方程和涨落-耗散定理在识别极限过程中的作用；相对于平稳测度的维纳混沌分解及其截断；以及所谓的替换引理，它控制临界维数中方程的弱耦合极限并确定极限扩散率。 出于教学原因，我们将仅专注于随机Burgers方程。 这些讲义基于与Dirk Erhard和Massimiliano Gubinelli的合作工作。 显示英文摘要

摘要： 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.