Sharp bounds for non-trace class noise and applications to SPDEs

TL;DR

This paper derives sharp conditions for the convergence of Gaussian series in Sobolev spaces, crucial for understanding stochastic PDEs with complex noise structures, and applies these to the stochastic heat equation.

Contribution

It provides necessary and sufficient conditions for Gaussian series convergence in Bessel potential spaces, advancing the analysis of SPDEs with non-trace class noise.

Findings

Established convergence criteria for Gaussian series in Sobolev spaces.

Linked the noise's spectral properties to the heat equation's scaling.

Provided sharp estimates for stochastic PDEs with colored noise.

Abstract

In the study of stochastic PDEs with colored, non-trace class space-time noise, one frequently encounters Gaussian series of the form $$ \sum_{n\geq 1} \gamma_n \mu_n f_n, $$ where $(\gamma_n)_{n}$ is a sequence of standard independent Gaussian variables, $g$ is an $L^\eta(\mathcal{O})$ function, $(\mu_n)_{n}$ is a sequence of scalars, and $(f_n)_n$ is an orthonormal system in $L^2(\mathcal{O})$ where $\mathcal{O} \subseteq \mathbb{R}^d$ is an open set. In this manuscript, we establish necessary and sufficient conditions for the above sum to converge in Bessel potential spaces $H^{-s,q}(\mathcal{O})$. The latter can be interpreted as a Sobolev embedding for Gaussian series. Our main theorem is formulated using weighted sequence spaces that encode the $L^\infty$-growth of the orthonormal system $(f_n)_{n}$, a feature that is crucial for obtaining sharp estimates. We apply our results to the stochastic heat equation with additive non-trace class noise. In this case, our conditions capture the scaling relationship between the heat operator and the coloring of the noise.