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Mesoscopic averaging of the two-dimensional KPZ equation

Ran Tao

TL;DR

The result shows that a proper spatial averaging of the KPZ equation converges in distribution to the sum of the solution to a deterministic KPZ equations and a Gaussian random variable that depends solely on the scale of averaging.

Abstract

We study the limit of a local average of the KPZ equation in dimension $d=2$ with general initial data in the subcritical regime. Our result shows that a proper spatial averaging of the KPZ equation converges in distribution to the sum of the solution to a deterministic KPZ equation and a Gaussian random variable that depends solely on the scale of averaging. This shows a unique mesoscopic averaging phenomenon that is only present in dimension two. Our work is inspired by the recent findings by Chatterjee \cite{chatterjee2021weak}.

Mesoscopic averaging of the two-dimensional KPZ equation

TL;DR

The result shows that a proper spatial averaging of the KPZ equation converges in distribution to the sum of the solution to a deterministic KPZ equations and a Gaussian random variable that depends solely on the scale of averaging.

Abstract

We study the limit of a local average of the KPZ equation in dimension with general initial data in the subcritical regime. Our result shows that a proper spatial averaging of the KPZ equation converges in distribution to the sum of the solution to a deterministic KPZ equation and a Gaussian random variable that depends solely on the scale of averaging. This shows a unique mesoscopic averaging phenomenon that is only present in dimension two. Our work is inspired by the recent findings by Chatterjee \cite{chatterjee2021weak}.
Paper Structure (15 sections, 13 theorems, 113 equations)

This paper contains 15 sections, 13 theorems, 113 equations.

Table of Contents

  1. Introduction and Main result
  2. Main Result
  3. Context
  4. Sketch of proof
  5. Moment bounds
  6. Second moments
  7. Higher moments
  8. Negative moments
  9. Proof of Theorem \ref{['thm:main']}
  10. Proof of Proposition \ref{['pr:SHEdetermin']}
  11. Proof of Proposition \ref{['pr:KPZdetermin']}
  12. Proof of Proposition \ref{['pr:localave']}
  13. Proof of Corollary \ref{['co:localfield']}
  14. Proof of Lemma \ref{['le:condmean']}
  15. Proof of Lemma \ref{['le:condvar']}

Key Result

Theorem 1.1

Let $h^{\varepsilon} (t,x)$ be the solution to the mollified 2-dimensional KPZ equation eq:kpz2m with the initial condition $h_0 : \mathbb{R}^2 \to \mathbb{R}$ being bounded and Lipschitz continuous. Let $0< \beta < \sqrt{2\pi}$ and $r_{\varepsilon} = \varepsilon^{1-\gamma}$ for some $0\leq \gamma \ as $\varepsilon \to 0$, where $\bar{h}(t,x)$ is the solution to the 2-dimensional deterministic KPZ

Theorems & Definitions (16)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Remark 1.4
  • Theorem 1.5: caravenna2017universality, Theorem 2.15, Remark 2.16
  • Remark 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3: caravenna2020two, (5.13)
  • Corollary 2.4
  • ...and 6 more