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On the domain of elliptic operators defined in subsets of Wiener spaces
- Source :
- Infinite Dimensional Analysis, Quantum Probability and Related Topics Vol. 23, No. 1 (2020) 2050004 (42 pages)
- Publication Year :
- 2017
-
Abstract
- Let $X$ be a separable Banach space endowed with a non-degenerate centered Gaussian measure $\mu$. The associated Cameron-Martin space is denoted by $H$. Consider two sufficiently regular convex functions $U:X\rightarrow\mathbb{R}$ and $G:X\rightarrow \mathbb{R}$. We let $\nu=e^{-U}\mu$ and $\Omega=G^{-1}(-\infty,0]$. In this paper we are interested in the domain of the the self-adjoint operator associated with the quadratic form \begin{gather} (\psi,\varphi)\mapsto \int_\Omega\langle\nabla_H\psi,\nabla_H\varphi\rangle_Hd\nu\qquad\psi,\varphi\in W^{1,2}(\Omega,\nu).\qquad\qquad (\star) \end{gather} In particular we obtain a complete characterization of the Ornstein-Uhlenbeck operator on half-spaces, namely if $U\equiv 0$ and $G$ is an affine function, then the domain of the operator defined via $(\star)$ is the space \[\{u\in W^{2,2}(\Omega,\mu)\,|\, \langle\nabla_H u(x),\nabla_H G(x)\rangle_H=0\text{ for }\rho\text{-a.e. }x\in G^{-1}(0)\},\] where $\rho$ is the Feyel-de La Pradelle Hausdorff-Gauss surface measure.<br />Comment: arXiv admin note: text overlap with arXiv:1609.07337
Details
- Database :
- arXiv
- Journal :
- Infinite Dimensional Analysis, Quantum Probability and Related Topics Vol. 23, No. 1 (2020) 2050004 (42 pages)
- Publication Type :
- Report
- Accession number :
- edsarx.1706.05260
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1142/S0219025720500046