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Gaussian fluctuations of a nonlinear stochastic heat equation in dimension two

Ran Tao

Abstract

We study the Gaussian fluctuations of a nonlinear stochastic heat equation in spatial dimension two. The equation is driven by a Gaussian multiplicative noise. The noise is white in time, smoothed in space at scale $\varepsilon$, and tuned logarithmically by a factor $\frac{1}{\sqrt{\log \varepsilon^{-1}}}$ in its strength. We prove that, after centering and rescaling, the solution random field converges in distribution to an Edwards-Wilkinson limit as $\varepsilon \downarrow 0$. The tool we used here is the Malliavin-Stein's method. We also give a functional version of this result.

Gaussian fluctuations of a nonlinear stochastic heat equation in dimension two

Abstract

We study the Gaussian fluctuations of a nonlinear stochastic heat equation in spatial dimension two. The equation is driven by a Gaussian multiplicative noise. The noise is white in time, smoothed in space at scale , and tuned logarithmically by a factor in its strength. We prove that, after centering and rescaling, the solution random field converges in distribution to an Edwards-Wilkinson limit as . The tool we used here is the Malliavin-Stein's method. We also give a functional version of this result.
Paper Structure (14 sections, 14 theorems, 116 equations)

This paper contains 14 sections, 14 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Context
  4. $\beta$-region
  5. General initial condition
  6. Organization of the paper
  7. Preliminaries
  8. Scaling
  9. The Malliavin calculus
  10. Moment bounds
  11. Stein's method
  12. Local statistics of the limit
  13. Proof of Theorem \ref{['thm:mainthm']}
  14. Proof of Theorem \ref{['thm:functionalCLT']}

Key Result

Theorem 1.1

There exists some $\beta_0 \in (0, \frac{\sqrt{2\pi}}{\sigma_{\mathrm{Lip}}})$ such that if $\beta <\beta_0$, for any fixed $T>0$ and any fixed Schwartz function $g\in C_c^\infty(\mathbb{R}^2)$, the random variable converges in law to a Gaussian distribution as $\varepsilon \to 0$. Here $U$ is the random distribution that solves the Edwards-Wilkinson equation in dimension two: $\Xi_{1,2}(\cdot)

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Proposition 2.2: Clark-Ocone Formula
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Remark 2.6
  • Proposition 2.7
  • Remark 2.8
  • ...and 7 more