Sharp convergence of nonlinear functionals of a class of Gaussian random fields

TL;DR

This paper provides improved bounds on multi-point correlations of Gaussian random fields and proves their convergence when composed with general functions, aiding the study of singular stochastic PDEs.

Contribution

It offers a self-contained proof with improved estimates for Gaussian fields, extending convergence results for more general functions compared to prior work.

Findings

Established uniform bounds on correlations of Gaussian fields.

Proved convergence of Gaussian fields composed with broader classes of functions.

Facilitated analysis of weak universality in singular stochastic PDEs.

Abstract

We present a self-contained proof of a uniform bound on multi-point correlations of trigonometric functions of a class of Gaussian random fields. It corresponds to a special case of the general situation considered in [Hairer-Xu], but with improved estimates. As a consequence, we establish convergence of a class of Gaussian fields composite with more general functions. These bounds and convergences are useful ingredients to establish weak universalities of several singular stochastic PDEs.