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Weak coupling limit of the Anisotropic KPZ equation

Giuseppe Cannizzaro, Dirk Erhard, Fabio Toninelli

Abstract

In the present work, we study the two-dimensional anisotropic KPZ equation (AKPZ), which is formally given by \begin{equation*} \partial_t h=\tfrac12 Δh + λ((\partial_1 h)^2)-(\partial_2 h)^2) +ξ\,, \end{equation*} where $ξ$ denotes a space-time white noise and $λ>0$ is the so-called coupling constant. The AKPZ equation is a {\it critical} SPDE, meaning that not only it is analytically ill-posed but also the breakthrough path-wise techniques for singular SPDEs [M. Hairer, Ann. Math. 2014] and [M. Gubinelli, P. Imkeller and N. Perkowski, Forum of Math., Pi, 2015] are not applicable. As shown in [G. Cannizzaro, D. Erhard, F. Toninelli, arXiv, 2020], the equation regularised at scale $N$ has a diffusion coefficient that diverges logarithmically as the regularisation is removed in the limit $N\to\infty$. Here, we study the \emph{weak coupling limit} where $λ=λ_N=\hatλ/\sqrt{\log N}$: this is the correct scaling that guarantees that the nonlinearity has a still non-trivial but non-divergent effect. In fact, as $N\to\infty$ the sequence of equations converges to the linear stochastic heat equation \begin{equation*} \partial_t h =\tfrac{ν_{\rm eff}}{2} Δh + \sqrt{ν_{\rm eff}}ξ\,, \end{equation*} where $ν_{\rm eff} >1$ is explicit and depends non-trivially on $\hatλ$. This is the first full renormalization-type result for a critical, singular SPDE which cannot be linearised via Cole-Hopf or any other transformation.

Weak coupling limit of the Anisotropic KPZ equation

Abstract

In the present work, we study the two-dimensional anisotropic KPZ equation (AKPZ), which is formally given by \begin{equation*} \partial_t h=\tfrac12 Δh + λ((\partial_1 h)^2)-(\partial_2 h)^2) +ξ\,, \end{equation*} where denotes a space-time white noise and is the so-called coupling constant. The AKPZ equation is a {\it critical} SPDE, meaning that not only it is analytically ill-posed but also the breakthrough path-wise techniques for singular SPDEs [M. Hairer, Ann. Math. 2014] and [M. Gubinelli, P. Imkeller and N. Perkowski, Forum of Math., Pi, 2015] are not applicable. As shown in [G. Cannizzaro, D. Erhard, F. Toninelli, arXiv, 2020], the equation regularised at scale has a diffusion coefficient that diverges logarithmically as the regularisation is removed in the limit . Here, we study the \emph{weak coupling limit} where : this is the correct scaling that guarantees that the nonlinearity has a still non-trivial but non-divergent effect. In fact, as the sequence of equations converges to the linear stochastic heat equation \begin{equation*} \partial_t h =\tfrac{ν_{\rm eff}}{2} Δh + \sqrt{ν_{\rm eff}}ξ\,, \end{equation*} where is explicit and depends non-trivially on . This is the first full renormalization-type result for a critical, singular SPDE which cannot be linearised via Cole-Hopf or any other transformation.

Paper Structure

This paper contains 25 sections, 37 theorems, 30 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The equation and the main result
  3. Preliminaries
  4. Elements of Wiener space analysis
  5. Stochastic Burgers equation and its Generator
  6. The action of Burgers' generator on $L^2(\eta)$
  7. The effective diffusion coefficient
  8. The replacement argument
  9. The diffusivity
  10. The truncated generator equation
  11. The martingale problem
  12. Theorem \ref{['thm:Conv']}: strategy of proof
  13. Recursive estimates
  14. Proof of Theorem \ref{['thm:ControlGen']} and Proposition \ref{['p:H1Norm']}
  15. The convergence
  16. ...and 10 more sections

Key Result

theorem 1

For $N\in\mathbb{N}$ and $\hat{\lambda}>0$, let $h^N$ be the unique almost-stationary solution (see Definition def:QSsol) to e:AKPZ with coupling constant $\lambda_N$ given by e:const. For any $\hat{\lambda}>0$ the bulk diffusion coefficient $\mathrm{D}_\mathrm{bulk}^N$ of $h^N$, defined as in e:Dbu where $\nu_{\rm eff}$ is the effective diffusion coefficient for any $\hat{\lambda}>0$.

Figures (7)

  • Figure 1: A graphical representation of formula \ref{['e:trem3']} for $j_1=6,j_2=4$. Each of the four horizontal strings of dots represent the vertices in $V_1,\dots,V_4$ and to each line we associate the corresponding factor ${\mathfrak{t}}^{u}$. An edge connecting two dots means that the variables labelled by the two endpoints coincide modulo a sign change. Thick edges correspond to edges in $G,\tilde{G}$ (note that they are repeated identically in the first and second pair of rows) while dashed edges correspond to edges in $P$. There is necessarily at least one dashed line, as $V[G]$ is non-empty. In fact, the collection of endpoints of dashed lines in the first two rows is exactly $V[G]$. The two dotted edges are present in every Feynman diagram $\gamma$ and encode the conditions $p_{(1,0)}= -p_{(2,0)}$ and $p_{(3,0)}=-p_{(4,0)}$. Finally, the positions of the rectangles depend on the permutation $\sigma$ and correspond to the variables which are added up in $\mathfrak{t}^{(i)}$ (see \ref{['eq:deft']}) - for instance, in the first line they represent $p_{(1,0)}+p_{(1,1)}$ in $\mathfrak{t}^{(1)}(p_{(1)})={\mathfrak{t}}_{a_1,j_1,\sigma^{(1)}}(p_{(1)})$. Rectangles can either contain one of the variables $p_{(u,0)},\, u=1,\dots,4$ (as in row 1) or not (as in row 2). Note that the two vertices contained in a rectangle are in the same row but need not be adjacent, although for clarity of the pictures we will always draw cases where they are.
  • Figure 2: Left: A Feynman diagram in the case $j_1=a_1=1$ and $j_2=4$. In this example, the rectangles do not contain the vertices $(2,0),(4,0)$, that is, $\kappa=0$. Right: a diagram in the case $j_1=a_1=2$ and $j_2=3$. The horizontal dashed lines in the first/third rows indicate that, because of the form of the kernel of $\mathfrak{f}^{N,n}_{2,2}$ (see \ref{['e:kernh2']}), the two variables at the endpoints take opposite values.
  • Figure 3: Two examples of graphs in $\Gamma_0[j_1,j_2], j_1=7,j_2=5$: the two special edges (dotted) have no endpoints in the rectangles. Non-special edges not interesecting the rectangles are undirected edges $U$, while those intersecting the rectangles are the directed edges $D$. Connected components of directed edges can be either closed loops (left drawing) or open paths (right drawing). For a graph $\gamma$, it can happen that both loops and open paths exist.
  • Figure 4: The two possible types of connectivities of special edges in Feynman diagrams occurring when $\kappa=2$, as described in the text. For clarity, only directed and special edges $(s_1,s_2)$ are drawn. In the left figure there are two connected components $C_1,C_2$ that we orient towards $s_1,s_2$. In the right drawing there is a single component, that we orient from $s_1$ to $s_2$. Special edges are not oriented. In some Feynman diagrams, also closed loops of oriented edges (and therefore purple/orange edges) can be present.
  • Figure 5: The two different possible situations in the case $\kappa=4$ and $D\cup S$ has a single connected components. The thick dashed edge is the edge $e=((\bar{u},i^1_{\bar{u}}),(\bar{v},i^1_{\bar{v}}))$ that we treat separately. We don't need to assign it a direction and a color.
  • ...and 2 more figures

Theorems & Definitions (76)

  • definition 1
  • remark 1
  • theorem 1
  • theorem 2
  • remark 2: Extensions
  • remark 3
  • lemma 1
  • theorem 3
  • remark 4
  • lemma 2
  • ...and 66 more