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Central limit theorems for nonlinear stochastic wave equations in dimension three

Masahisa Ebina

TL;DR

This paper addresses Gaussian fluctuations for spatial averages of the 3D nonlinear stochastic wave equation driven by Gaussian noise with spatial covariance. Using the Malliavin–Stein framework, it proves that the properly normalized spatial average $F_R(t)$ converges to a Gaussian limit as the observation radius $R$ grows, under both a short-range Dalang condition ($\gamma\in L^1$ with $\gamma>0$) and a long-range Riesz kernel regime ($\gamma(x)=|x|^{-\beta}$, $0<\beta<2$). It further establishes functional central limit theorems for the processes $t\mapsto F_R(t)$, identifying the limiting Gaussian processes and their covariance structures, and providing nondegeneracy and convergence-rate insights. The results rely on a Picard-approximation approach tailored to the 3D wave kernel, together with Malliavin derivatives and variance bounds that bridge to Gaussian limits, and they extend CLTs for spatial averages to the physically relevant dimension three with both short- and long-range spatial correlations.

Abstract

In this paper, we consider three-dimensional nonlinear stochastic wave equations driven by the Gaussian noise which is white in time and has some spatial correlations. Using the Malliavin-Stein's method, we prove the Gaussian fluctuation for the spatial average of the solution under the Wasserstein distance in the cases where the spatial correlation is given by an integrable function and by the Riesz kernel. In both cases we also establish functional central limit theorems.

Central limit theorems for nonlinear stochastic wave equations in dimension three

TL;DR

This paper addresses Gaussian fluctuations for spatial averages of the 3D nonlinear stochastic wave equation driven by Gaussian noise with spatial covariance. Using the Malliavin–Stein framework, it proves that the properly normalized spatial average converges to a Gaussian limit as the observation radius grows, under both a short-range Dalang condition ( with ) and a long-range Riesz kernel regime (, ). It further establishes functional central limit theorems for the processes , identifying the limiting Gaussian processes and their covariance structures, and providing nondegeneracy and convergence-rate insights. The results rely on a Picard-approximation approach tailored to the 3D wave kernel, together with Malliavin derivatives and variance bounds that bridge to Gaussian limits, and they extend CLTs for spatial averages to the physically relevant dimension three with both short- and long-range spatial correlations.

Abstract

In this paper, we consider three-dimensional nonlinear stochastic wave equations driven by the Gaussian noise which is white in time and has some spatial correlations. Using the Malliavin-Stein's method, we prove the Gaussian fluctuation for the spatial average of the solution under the Wasserstein distance in the cases where the spatial correlation is given by an integrable function and by the Riesz kernel. In both cases we also establish functional central limit theorems.
Paper Structure (16 sections, 34 theorems, 260 equations)

This paper contains 16 sections, 34 theorems, 260 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stochastic integrals
  4. The Malliavan calculus and Malliavin-Stein bound
  5. Some other basic results
  6. The Picard Approximation and its Malliavin derivative
  7. The Picard approximation
  8. The Malliavin derivative of the Picard approximation
  9. Central Limit Theorems
  10. Nondegeneracy of the variance
  11. Proof of Theorem \ref{['main result1']}
  12. Functional Central Limit Theorems
  13. Limits of the covariance functions
  14. Convergence of finite-dimensional distributions
  15. Tightness
  16. ...and 1 more sections

Key Result

Theorem 1.3

Assume that the spatial correlation function $\gamma$ satisfies one of the two conditions below: Then, for any fixed $t \in (0,T]$, we have $\sigma_R(t) > 0$ for every $R >0$ and

Theorems & Definitions (59)

  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Proposition 2.2
  • Lemma 2.3: Burkholder-Davis-Gundy inequality
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • ...and 49 more