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Almost sure central limit theorems for parabolic/hyperbolic Anderson models with Gaussian colored noises

Panqiu Xia, Guangqu Zheng

Abstract

This short note is devoted to establishing the almost sure central limit theorem for the parabolic/hyperbolic Anderson models driven by colored-in-time Gaussian noises, completing recent results on quantitative central limit theorems for stochastic partial differential equations. We combine the second-order Gaussian Poincaré inequality with Ibragimov and Lifshits' method of characteristic functions, effectively overcoming the challenge from the lack of Itô tools in this colored-in-time setting, and achieving results that are inaccessible with previous methods.

Almost sure central limit theorems for parabolic/hyperbolic Anderson models with Gaussian colored noises

Abstract

This short note is devoted to establishing the almost sure central limit theorem for the parabolic/hyperbolic Anderson models driven by colored-in-time Gaussian noises, completing recent results on quantitative central limit theorems for stochastic partial differential equations. We combine the second-order Gaussian Poincaré inequality with Ibragimov and Lifshits' method of characteristic functions, effectively overcoming the challenge from the lack of Itô tools in this colored-in-time setting, and achieving results that are inaccessible with previous methods.
Paper Structure (9 sections, 6 theorems, 83 equations)

This paper contains 9 sections, 6 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Framework
  3. Main result
  4. Preliminaries
  5. Basic Malliavin calculus.
  6. Proof of Proposition \ref{['prop1818']}
  7. Proof of Theorem \ref{['thm_main']}
  8. Proof of (\ref{['COV_b1']})
  9. Proof of (\ref{['ob2']})

Key Result

Theorem 1.3

Let the above hypotheses (H1) and (H2) hold and we make further assumptions on the temporal/spatial correlation kernels: Let $u$ denote the solution CP1 to pam for any spatial dimension $d$ or HAM with $d\leq 2$. Fix any $t_0\in(0,\infty)$ throughout this note. We define for any $R > 0$ that with $\sigma_R \coloneqq \sqrt{ \textup{Var}(F_R) } > 0$ for each $R>0$. Then, the following quantitati

Theorems & Definitions (12)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Definition 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Proposition 1.8
  • Proposition 1.9
  • Proposition 2.1
  • ...and 2 more