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Edwards-Wilkinson fluctuations in subcritical 2D stochastic heat equations

Alexander Dunlap, Cole Graham

TL;DR

The paper analyzes a 2D vector-valued stochastic heat equation with logarithmically attenuated white noise in the subcritical regime and proves that macroscopic fluctuations converge to the Edwards–Wilkinson equation as the microscopic scale parameter $\rho$ vanishes. Using a renormalization-flow framework, approximate mild solutions, and a martingale central limit theorem, it reveals that the limiting fluctuation field $\mathcal{U}$ decomposes into a noise-measurable part and an independent part, both driven by Gaussian noises, and satisfies $d\mathcal{U}_{t}(x)=\frac{1}{2}\Delta\mathcal{U}_{t}(x)\,dt+ (\overline{J}_{1}\circ \overline{u}_{t})(x)\,dW_{t}(x) + (\widetilde{J}_{1}\circ \overline{u}_{t})(x)\,d\widetilde{W}_{t}(x)$. The analysis introduces an “approximate mild solution” notion with three quantitative conditions, proves concentration of macroscopic nonlinearities, and establishes universality: the macroscopic Edwards–Wilkinson statistics are robust to fine-scale model details. By decomposing the noise contributions and matching quadratic variations, the work extends Ran Tao’s results by relaxing Lipschitz constraints on $\sigma$ and clarifying the noise structure in the limit, with potential implications for broader classes of low-dimensional SPDEs exhibiting Gaussian fluctuations at large scales.

Abstract

We study 2D nonlinear stochastic heat equations under a logarithmically attenuated white-noise limit with subcritical coupling. We show that solutions asymptotically exhibit Edwards-Wilkinson fluctuations. This extends work of Ran Tao, which required a stricter condition on the coupling. Part of the limiting fluctuation is measurable with respect to the original noise and the remainder is independent. We also show that these statistics are universal in the sense that they are independent of the fine details of the model.

Edwards-Wilkinson fluctuations in subcritical 2D stochastic heat equations

TL;DR

The paper analyzes a 2D vector-valued stochastic heat equation with logarithmically attenuated white noise in the subcritical regime and proves that macroscopic fluctuations converge to the Edwards–Wilkinson equation as the microscopic scale parameter vanishes. Using a renormalization-flow framework, approximate mild solutions, and a martingale central limit theorem, it reveals that the limiting fluctuation field decomposes into a noise-measurable part and an independent part, both driven by Gaussian noises, and satisfies . The analysis introduces an “approximate mild solution” notion with three quantitative conditions, proves concentration of macroscopic nonlinearities, and establishes universality: the macroscopic Edwards–Wilkinson statistics are robust to fine-scale model details. By decomposing the noise contributions and matching quadratic variations, the work extends Ran Tao’s results by relaxing Lipschitz constraints on and clarifying the noise structure in the limit, with potential implications for broader classes of low-dimensional SPDEs exhibiting Gaussian fluctuations at large scales.

Abstract

We study 2D nonlinear stochastic heat equations under a logarithmically attenuated white-noise limit with subcritical coupling. We show that solutions asymptotically exhibit Edwards-Wilkinson fluctuations. This extends work of Ran Tao, which required a stricter condition on the coupling. Part of the limiting fluctuation is measurable with respect to the original noise and the remainder is independent. We also show that these statistics are universal in the sense that they are independent of the fine details of the model.
Paper Structure (3 sections, 7 theorems, 60 equations)

This paper contains 3 sections, 7 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Concentration
  3. The martingale argument

Key Result

Theorem 1.2

Let $\sigma$ be $L^2$-subcritical and ${u}_{0}\in L^{\infty}(\mathds{R}^{2})$ be deterministic. Then the pair of random distributions $(\gamma_{\rho}^{-1}({u^\rho}-\overline{{u}}),\overline{\mathcal{U}})$ converges jointly in law, in the topology of distributions on $\mathds{R}_{+}\times\mathds{R}^{

Theorems & Definitions (14)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 4 more