Generalized parabolic functions on white noise space
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)].
scientific article; zbMATH DE number 1134819
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