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A frequency-independent bound on trigonometric polynomials of Gaussians and applications

Fanhao Kong, Wenhao Zhao

TL;DR

This work proves a frequency-independent bound on moments for trig-polynomial functionals of a class of singular Gaussian fields, enabling control of nonlinearities in weak universality problems for KPZ and dynamical $\Phi^4_3$. The authors extend clustering and localization techniques to a two-frequency setting, producing uniform bounds on the $2n$-th moments of derivatives of a two-frequency trig chaos after Wick renormalization and kernel integration. By combining pointwise correlation bounds with a careful domain decomposition and a localization argument, they achieve estimates that are uniform in the mollification parameter $\varepsilon$ and the scale $\lambda$, even when one frequency dominates the other. The results reduce the required regularity for nonlinearities to $F\in\mathcal{C}^{2+}$ in KPZ and $G\in\mathcal{C}^{3+}$ in dynamical $\Phi^4_3$, thereby strengthening the connection between analytic PDE structure and probabilistic universality, and providing a framework potentially applicable to other singular SPDE universality problems.

Abstract

We prove a frequency-independent bound on trigonometric functions of a class of singular Gaussian random fields, which arise naturally from weak universality problems for singular stochastic PDEs. This enables us to reduce the regularity assumption on the nonlinearity of the microscopic models in KPZ and dynamical $Φ^4_3$ in [HX19] and [FG19] to that required by PDE structures.

A frequency-independent bound on trigonometric polynomials of Gaussians and applications

TL;DR

This work proves a frequency-independent bound on moments for trig-polynomial functionals of a class of singular Gaussian fields, enabling control of nonlinearities in weak universality problems for KPZ and dynamical . The authors extend clustering and localization techniques to a two-frequency setting, producing uniform bounds on the -th moments of derivatives of a two-frequency trig chaos after Wick renormalization and kernel integration. By combining pointwise correlation bounds with a careful domain decomposition and a localization argument, they achieve estimates that are uniform in the mollification parameter and the scale , even when one frequency dominates the other. The results reduce the required regularity for nonlinearities to in KPZ and in dynamical , thereby strengthening the connection between analytic PDE structure and probabilistic universality, and providing a framework potentially applicable to other singular SPDE universality problems.

Abstract

We prove a frequency-independent bound on trigonometric functions of a class of singular Gaussian random fields, which arise naturally from weak universality problems for singular stochastic PDEs. This enables us to reduce the regularity assumption on the nonlinearity of the microscopic models in KPZ and dynamical in [HX19] and [FG19] to that required by PDE structures.
Paper Structure (15 sections, 28 theorems, 153 equations)

This paper contains 15 sections, 28 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Statement of the main bound
  4. Applications to weak universality problems
  5. Proof of Theorem \ref{['thm:main_result']}
  6. Some pointwise correlation bounds
  7. Proof of Theorem \ref{['thm:main_result']}
  8. Application to weak universality problems
  9. Preliminaries
  10. The KPZ equation -- proof of Theorem \ref{['thm:kpz']}
  11. The case
  12. The case
  13. The case
  14. Dynamical F -- proof of Theorem \ref{['thm:phi43']}
  15. Proof of the correlation bounds in Section \ref{['sec:correlation_bounds']}

Key Result

Theorem 1.5

Suppose the Gaussian field $\Psi_{\varepsilon}(x)$ and the constants used to define $\mathcal{A}_{\varepsilon,\lambda}$ satisfy the assumptions above. Then for every small $\eta > 0$, every $n\in \mathbf{N}$ and every $\mathbf{r} \in \mathbf{N}^2$, we have uniformly over $\varepsilon, \lambda \in (0,1)$, $\boldsymbol\theta \in \mathbf{R}^2$ and smooth test functions $\varphi$ supported in the uni

Theorems & Definitions (62)

  • Remark 1.2
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Definition 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Remark 1.10
  • Remark 1.11
  • Proposition 2.1
  • ...and 52 more