Generalized parabolic functions on white noise space
type: Item
instance of: scholarly article
title: Generalized parabolic functions on white noise space
zbMATH Open document ID: 0890.60038
DOI: 10.1016/S0304-4149(97)00130-5
author: Zhongmin Qian
published in: Stochastic Processes and their Applications
publication date: 1998-03-29T00:00:00Z
review text: Let \(({\mathcal S}'(\mathbb R), {\mathcal B}({\mathcal S}'(\mathbb R)),\mu )\) be the standard white noise space and denote by \(({\mathcal S})\subset ({\mathcal L}^{2})\subset ({\mathcal S})^{*}\) the Gelfand triple of white noise functionals. The Ornstein-Uhlenbeck semigroup \((P_{t})_{t\geq 0}\) on the white noise space is defined by \[ P_{t}\varphi (x) = \int _{{\mathcal S}'(\mathbb R)} \varphi \left (e^{-t}x + \sqrt {1-e^{-2t}}y\right )d\mu (y), \quad \varphi \in ({\mathcal S}). \] A family \(u=(u(t,\cdot ))_{t\in \mathbb R}\) of positive functionals in \(({\mathcal S})^{*}\) is called a generalized positive parabolic function, if \(\nu _{s}P_{t}= \nu _{s+t}\) for any \(s\in \mathbb R\), \(t\geq 0\), where \(\nu _s\) stands for the unique finite Borel measure on \({\mathcal S}'(\mathbb R)\) representing the positive white noise functional \(u(s,\cdot )\) according to \textit{Y. Yokoi}'s theorem [Hiroshima Math. J. 20, No. 1, 137-157 (1990)]. Integral representations of generalized positive parabolic functions in terms of generalized Cameron-Martin densities are investigated. Moreover, it is shown that any generalized parabolic function \(u\) solves the heat equation \(du(t,\cdot )/dt = -{\mathcal N}u(t,\cdot )\) in \(({\mathcal S})^{*}\), \(\mathcal N\) denoting the number operator on \(({\mathcal S})^{*}\). Recently, the parabolic Martin boundary of the Ornstein-Uhlenbeck semigroup on an abstract Wiener space has been studied by rather different methods by \textit{M. Röckner} [Ann. Probab. 20, No. 2, 1063-1085 (1992; Zbl 0761.60067)].
reviewed by: Jan Seidler
Mathematics Subject Classification ID: 60G20
zbMATH DE Number: 1134819
zbMATH Keywords: white noise space
zbMATH Keywords: heat equation
zbMATH Keywords: Hida functionals
zbMATH Keywords: positive parabolic functions
MaRDI profile type: Publication
cites work: Q3992729
cites work: Q3870044
cites work: Q4202615
cites work: Q4021661
cites work: Q4026019
cites work: A characterization of Hida distributions
cites work: On the parabolic Martin boundary of the Ornstein-Uhlenbeck operator on Wiener space
cites work: Q3979056
cites work: Positive generalized white noise functionals
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description: scientific article; zbMATH DE number 1134819