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

Antonio Agresti, Fabian Germ, Mark Veraar

TL;DR

The paper addresses convergence and regularity of Gaussian series modeling non-trace class space-time noise in SPDEs, introducing weighted sequence spaces $\ell^\zeta(\mathcal{S}_{f})$ to quantify $L^\infty$-growth of the ONB. It develops sharp, necessary-and-sufficient Sobolev embeddings in negative Bessel potential spaces $H^{-s,q}(\mathcal{O})$ via $\gamma$-radonifying operator theory and multilinear interpolation, identifying precise scaling conditions on $(s,q,\eta,\zeta)$ for the bounds to hold and proving endpoint sharpness. The main results deliver tight bounds for $M_gR_{\mu}$ and their homogeneous/periodic variants, with a robust interpolation framework that extends to random fields and SPDEs, including the stochastic heat equation with non-trace class noise. Applications to Matérn-type random fields and to SPDEs yield optimal space-time regularity and stochastic maximal regularity estimates, thereby enabling nonlinear analyses in critical spaces. The work advances the theory by decoupling from operator eigenfunctions, allowing unbounded domains, and capturing scaling through sharp, flexible parameter regimes, substantially enriching the toolbox for SPDEs 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 $$ \sum_{n\geq 1} γ_n μ_n f_n, $$ where $(γ_n)_{n}$ is a sequence of standard independent Gaussian variables, $g$ is an $L^η(\mathcal{O})$ function, $(μ_n)_{n}$ 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.

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

TL;DR

The paper addresses convergence and regularity of Gaussian series modeling non-trace class space-time noise in SPDEs, introducing weighted sequence spaces to quantify -growth of the ONB. It develops sharp, necessary-and-sufficient Sobolev embeddings in negative Bessel potential spaces via -radonifying operator theory and multilinear interpolation, identifying precise scaling conditions on for the bounds to hold and proving endpoint sharpness. The main results deliver tight bounds for and their homogeneous/periodic variants, with a robust interpolation framework that extends to random fields and SPDEs, including the stochastic heat equation with non-trace class noise. Applications to Matérn-type random fields and to SPDEs yield optimal space-time regularity and stochastic maximal regularity estimates, thereby enabling nonlinear analyses in critical spaces. The work advances the theory by decoupling from operator eigenfunctions, allowing unbounded domains, and capturing scaling through sharp, flexible parameter regimes, substantially enriching the toolbox for SPDEs 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 where is a sequence of standard independent Gaussian variables, is an function, is a sequence of scalars, and is an orthonormal system in where is an open set. In this manuscript, we establish necessary and sufficient conditions for the above sum to converge in Bessel potential spaces . The latter can be interpreted as a Sobolev embedding for Gaussian series. Our main theorem is formulated using weighted sequence spaces that encode the -growth of the orthonormal system , 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.
Paper Structure (26 sections, 26 theorems, 181 equations)

This paper contains 26 sections, 26 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Scaling for the stochastic heat equation with colored noise
  3. Comparison with the literature
  4. Proofs strategy and structure of the paper
  5. Preliminaries
  6. Function spaces
  7. $\gamma$-radonifying operators
  8. The $\gamma$-Young inequality and consequences
  9. Proof of Theorem \ref{['t:gammaYoung']} and endpoints
  10. Proof of Propositions \ref{['prop:M_g_Sobolev']} and \ref{['prop:M_g_Sobolev_hom']}
  11. Main results -- Sharp Sobolev embeddings for Gaussian series
  12. Two endpoints of Theorem \ref{['thm:MgTmu delta']}
  13. Proofs of Theorems \ref{['thm:MgTmu delta']} and \ref{['thm:MgTmu delta_hom']} and Proposition \ref{['p:necessityLinfty_hom']}
  14. Bird's eye view on the interpolation argument and applications to random fields
  15. The periodic setting
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $\mathcal{O}\subseteq {\mathbb R}^d$ be an open set. Let $\mathcal{S}_{f} = (f_n)_{n\geq 1}$ be an orthonormal system in $L^2(\mathcal{O})$, and $\mu\in \ell^\zeta(\mathcal{S}_{f})$. Assume that $q\in (1, \infty)$, $\eta\in (1, q)$, $\zeta\in [2, \infty]$, and $s\in(0,d)$ satisfy Then, for all $g\in L^{\eta}(\mathcal{O})$ the series eq:BM converges in $L^2(\Omega;H^{-s,q}(\mathcal{O}))$ and

Theorems & Definitions (51)

  • Theorem 1.1: Sobolev embedding for Gaussian series
  • Theorem 1.2: Optimal space regularity estimates for the stochastic heat equation
  • Theorem 3.1: $\gamma$-Young inequality
  • Proposition 3.2: $\gamma$-bounds for multiplication operators in Sobolev spaces
  • Proposition 3.3: $\gamma$-bounds for multiplication operators in homogeneous Sobolev spaces
  • Remark 3.4
  • proof : Proof of Theorem \ref{['t:gammaYoung']}
  • Proposition 3.5: Endpoint case $\eta=2$
  • proof
  • Proposition 3.6: Endpoint case $\eta=q$
  • ...and 41 more