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 and Xu (large-scale limit of interface fluctuation models. ArXiv e-prints arXiv:1802.08192, 2018), 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.
Highlights
1.1 Motivation from Weak UniversalitiesThe study of singular stochastic PDEs has received much attention recently, and powerful theories are being developed to enhance the general understanding of this area
One of the motivations to study singular SPDEs is that many of them are expected to be universal objects in crossover regimes of their respective universality classes, a phenomenon known as weak universality
We shall note that the convergence results in this article are not sufficient to establish weak universality in general situations
Summary
The study of singular stochastic PDEs has received much attention recently, and powerful theories are being developed to enhance the general understanding of this area. Such convergences need to be established in the optimal regularity space, which requires one to pth obtain moment bounds of these stochastic objects for At least arbitrarily large pth. In [13], the authors expanded F( εΨε) in terms of Fourier transform, developed a procedure in obtaining pointwise correlation bounds on trigonometric functions of Gaussians, and deduced the desired convergence from those bounds. The weak universality of Φ34 equation for a large class of symmetric phase coexistence models with polynomial potential was established in [13] for Gaussian noise and extended in [17] to non-Gaussian noise. We obtain a better bound in this special case, yielding convergence results for functions F with lower regularity
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