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Fluctuations of stochastic PDEs with long-range correlations

Luca Gerolla, Martin Hairer, Xue-Mei Li

TL;DR

This paper analyzes the large-scale fluctuations of a nonlinear stochastic heat equation in $d\ge 3$ subjected to long-range, non-integrable spatial noise with covariance decaying as $|x|^{-\kappa}$, $\kappa\in(2,d)$. By combining Malliavin calculus with Stein’s method and a Feynman-diagram framework, the authors show that diffusively scaled fluctuations persist in the limit and converge to an additive stochastic heat equation driven by a spatially colored noise with Riesz kernel $|x-y|^{-\kappa}$, with an effective variance $\nu_{\text{eff}}^2=|\mathbb{E}[\sigma(Z(0))]|^2$. They establish both finite-dimensional distribution convergence and tightness in negative Hölder spaces, and provide detailed bounds via diagrammatic estimates to control residual terms. The results extend the Edwards–Wilkinson-type limits to long-range correlated noise and quantify how correlation decay controls the scaling and effective variance, highlighting persistent spatial structure in the limit. This contributes to a deeper understanding of SPDE fluctuations under nonlocal spatial correlations and the robustness of Gaussian fluctuation limits beyond compactly supported noise.

Abstract

We study the large-scale dynamics of the solution to a nonlinear stochastic heat equation (SHE) in dimensions $d \geq 3$ with long-range dependence. This equation is driven by multiplicative Gaussian noise, which is white in time and coloured in space with non-integrable spatial covariance that decays at the rate of $|x|^{-κ}$ at infinity, where $κ\in (2, d)$. Inspired by recent studies on SHE and KPZ equations driven by noise with compactly supported spatial correlation, we demonstrate that the correlations persist in the large-scale limit. The fluctuations of the diffusively scaled solution converge to the solution of a stochastic heat equation with additive noise whose correlation is the Riesz kernel of degree $-κ$. Moreover, the fluctuations converge as a distribution-valued process in the optimal Hölder topologies.

Fluctuations of stochastic PDEs with long-range correlations

TL;DR

This paper analyzes the large-scale fluctuations of a nonlinear stochastic heat equation in subjected to long-range, non-integrable spatial noise with covariance decaying as , . By combining Malliavin calculus with Stein’s method and a Feynman-diagram framework, the authors show that diffusively scaled fluctuations persist in the limit and converge to an additive stochastic heat equation driven by a spatially colored noise with Riesz kernel , with an effective variance . They establish both finite-dimensional distribution convergence and tightness in negative Hölder spaces, and provide detailed bounds via diagrammatic estimates to control residual terms. The results extend the Edwards–Wilkinson-type limits to long-range correlated noise and quantify how correlation decay controls the scaling and effective variance, highlighting persistent spatial structure in the limit. This contributes to a deeper understanding of SPDE fluctuations under nonlocal spatial correlations and the robustness of Gaussian fluctuation limits beyond compactly supported noise.

Abstract

We study the large-scale dynamics of the solution to a nonlinear stochastic heat equation (SHE) in dimensions with long-range dependence. This equation is driven by multiplicative Gaussian noise, which is white in time and coloured in space with non-integrable spatial covariance that decays at the rate of at infinity, where . Inspired by recent studies on SHE and KPZ equations driven by noise with compactly supported spatial correlation, we demonstrate that the correlations persist in the large-scale limit. The fluctuations of the diffusively scaled solution converge to the solution of a stochastic heat equation with additive noise whose correlation is the Riesz kernel of degree . Moreover, the fluctuations converge as a distribution-valued process in the optimal Hölder topologies.
Paper Structure (17 sections, 29 theorems, 32 equations)

This paper contains 17 sections, 29 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Proof and organisation of the paper
  4. Preliminary estimates
  5. Elementary Gaussian and covariance estimates
  6. Feynman diagrams and integration lemma
  7. Moments and large time asymptotics of SHE
  8. Proof of Theorem \ref{['thm:basic-convergence']}
  9. Convergence of covariances
  10. Convergence of residual terms
  11. Elementary estimates
  12. Main Estimates
  13. Weak convergence of X eps in negative Hölder topology
  14. Proof of Theorem \ref{['thm:weak_conv_1']}
  15. Preliminaries
  16. ...and 2 more sections

Key Result

Theorem 1.3

Let $d\geqslant 3$, $\sigma$ be Lipschitz continuous, and suppose that $\xi$ satisfies Assumption assump-noise. Then, there exists a $\beta_0>0$ such that for all $\beta < \beta_0$, and for any time indices $0 < t_1 \leqslant \dots \leqslant t_n$, and for any smooth functions with compact support $\ Here, $\mathcal{U}$ denotes the solution to the additive stochastic heat equation with effective v

Theorems & Definitions (72)

  • Example 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Lemma 2.1
  • proof
  • ...and 62 more