Lecture notes on stationary critical and super-critical SPDEs

Abstract

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.

Figures (4)

• Figure 1: Left drawing: a schematic representation of term (I) in $(T^\eps_+ T^\eps_+e_k)(k_{1:3})$. The drawing should be read from right to left: each branching corresponds to the application of $\mathcal{A}^{\eps}_+$ (or equivalently $T^\eps_+$), the labels next to the lines denote the Fourier variable (momentum) and at each branching, the sum of outgoing momenta equals the incoming momentum. Center and right drawings: a schematic representation of the direct term (I)$^2$ and of the cross term (I)$\times$(II).
• Figure 2: A schematic representation of the terms (I'), (II'), (III'). When $T^\eps_+$ acts on $e_k\otimes e_k$, the first momentum $k$ is split into two momenta $k_i,k_j$ with three possible choices for $1\le i<j\le 3$. By conservation of momentum, $k_i+k_j=k$, and the third outgoing momentum also equals $k$.
• Figure 3: A schematic, graphical, representation of the three contributions that are obtained by applying $T^\eps_-$ (with the choice $j=1$ in the definition \ref{['e:gen']} of $\mathcal{A}^{\eps}_-$) to the three terms (I'), (II'), (III') defining $T^\eps_+[e_k\otimes e_k]$. The left-most drawing corresponds to a diagonal diagram (a similar one is obtained with the choice $j=2$) while the second and third are two of the four off-diagonal diagrams (again, the remaining two are obtained with the choice $j=2$). As the drawing indicates, in diagonal diagrams the action of $T^\eps_-$ merges exactly the two momenta that have been generated by the application of $T^\eps_+$, which splits one incoming momentum $k$. In off-diagonal diagrams, instead, one of the two merged momenta has not been modified by the action of $T^\eps_+$. Note that because of momentum conservation, in the two off-diagonal diagrams, $\ell$ is forced to be equal to $k_1-k=k-k_2$ (recall that $k_1+k_2=2k$ also because of momentum conservation), while in the diagonal one $\ell$ free and will be summed over.
• Figure 4: A schematic representation of three diagrams arising in the expression of $\mathcal{T}_p^\eps[e_k\otimes e_k]$ for paths belonging to $\Pi^{(n)}_{j+2,1},j=2$. The left-most one satisfies properties (i)-(ii) and it gives a non-vanishing contribution as $\eps\to0$. The two others do not, either because they are not disconnected, or because both connected components involve branching and merging.

Theorems & Definitions (39)

• theorem 1
• theorem 2
• theorem 3
• remark 1
• definition 1
• theorem 4
• remark 2
• remark 3
• lemma 1
• proof