Sharp bounds for non-trace class noise and applications to SPDEs
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866911709715759104 |
|---|---|
| author | Agresti, Antonio Germ, Fabian Veraar, Mark |
| author_facet | Agresti, Antonio Germ, Fabian Veraar, Mark |
| contents | In the study of stochastic PDEs with colored, non-trace class space-time noise, one frequently encounters Gaussian series of the form $$g \sum_{n\geq 1} γ_n μ_n f_n, $$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_19639 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sharp bounds for non-trace class noise and applications to SPDEs Agresti, Antonio Germ, Fabian Veraar, Mark Probability Analysis of PDEs Functional Analysis Primary: 60B12, 60H15, Secondary: 46E35, 47B10, 60B11, 60G15 In the study of stochastic PDEs with colored, non-trace class space-time noise, one frequently encounters Gaussian series of the form $$g \sum_{n\geq 1} γ_n μ_n f_n, $$ 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. |
| title | Sharp bounds for non-trace class noise and applications to SPDEs |
| topic | Probability Analysis of PDEs Functional Analysis Primary: 60B12, 60H15, Secondary: 46E35, 47B10, 60B11, 60G15 |
| url | https://arxiv.org/abs/2601.19639 |