Characterization of the (fractional) Malliavin-Watanabe-Sobolev spaces $\mathcal{D}^{α,2}$ via the Bargmann-Segal norm
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866915835327545344 |
|---|---|
| author | Bock, Wolfgang Grothaus, Martin |
| author_facet | Bock, Wolfgang Grothaus, Martin |
| contents | Motivated by an open question going back to P.Malliavin and P.-A.Meyer (and closely related to the foundational work of S.Watanabe) on whether Malliavin-Watanabe-Sobolev regularity admits a characterization in terms of a holomorphic Laplace image similar as for Hida distributions, we establish a characterization of the spaces $\mathcal{D}^{α,2}$ for all $α\in\mathbb{R}$ via the Bargmann-Segal norm of the $S$-transform. More precisely, we express $\mathcal{D}^{α,2}$-regularity, $α> 0$, of $F\in L^{2}(μ)$, as well as dual regularity of distributions, in terms of integrability, differentiability and growth properties of the function \[ (0,1) \ni λ\longmapsto \int_{\mathcal{S}'_{\mathbb{C}}} |SF(λu)|^{2}\,dν(u) \] involving integer-order derivatives in $λ$ for $α\in\mathbb{N}$ and Riemann-Liouville fractional derivatives/integrals for non-integer $α$. Here $ν$ is the Gaussian Bargmann-Segal measure. This yields practical criteria for both positive and negative (including fractional) orders of Malliavin regularity and thereby bridges Malliavin calculus and Bargmann-Segal techniques from white noise analysis. Applications are worked out for Donsker's delta, self-intersection local times of Gaussian processes, and Gauss kernels. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_04594 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Characterization of the (fractional) Malliavin-Watanabe-Sobolev spaces $\mathcal{D}^{α,2}$ via the Bargmann-Segal norm Bock, Wolfgang Grothaus, Martin Probability Complex Variables Functional Analysis 60H07, 60H40, 46E35, 46C15 Motivated by an open question going back to P.Malliavin and P.-A.Meyer (and closely related to the foundational work of S.Watanabe) on whether Malliavin-Watanabe-Sobolev regularity admits a characterization in terms of a holomorphic Laplace image similar as for Hida distributions, we establish a characterization of the spaces $\mathcal{D}^{α,2}$ for all $α\in\mathbb{R}$ via the Bargmann-Segal norm of the $S$-transform. More precisely, we express $\mathcal{D}^{α,2}$-regularity, $α> 0$, of $F\in L^{2}(μ)$, as well as dual regularity of distributions, in terms of integrability, differentiability and growth properties of the function \[ (0,1) \ni λ\longmapsto \int_{\mathcal{S}'_{\mathbb{C}}} |SF(λu)|^{2}\,dν(u) \] involving integer-order derivatives in $λ$ for $α\in\mathbb{N}$ and Riemann-Liouville fractional derivatives/integrals for non-integer $α$. Here $ν$ is the Gaussian Bargmann-Segal measure. This yields practical criteria for both positive and negative (including fractional) orders of Malliavin regularity and thereby bridges Malliavin calculus and Bargmann-Segal techniques from white noise analysis. Applications are worked out for Donsker's delta, self-intersection local times of Gaussian processes, and Gauss kernels. |
| title | Characterization of the (fractional) Malliavin-Watanabe-Sobolev spaces $\mathcal{D}^{α,2}$ via the Bargmann-Segal norm |
| topic | Probability Complex Variables Functional Analysis 60H07, 60H40, 46E35, 46C15 |
| url | https://arxiv.org/abs/2603.04594 |