Positive generalized white noise functionals
Let \(E^*\) be the space of tempered distributions on \(R\) and \(\mu\) be the standard Gaussian random measure on \(E^*\). Consider a triplet (\({\mathcal S})\subset L^ 2(E^*,\mu)\subset ({\mathcal S}')\) of functions on \(E^*\). Following T. Hida an element of (\({\mathcal S})\) is called a generalized Brownian functional. \textit{I. Kubo} and the author proved [Nagoya Math. J. 115, 139-149 (1989; Zbl 0661.60097)] that every element \(\phi\) of (\({\mathcal S})\) has a unique continuous version \({\tilde \phi}\) as a random variable on \((E^*,\mu)\). Using this fact, the author obtained that for any positive generalized Brownian functional \(\psi\) there exists a positive finite measure \(\nu_{\psi}\) on \(E^*\) such that \(<\psi,\phi >=\int_{E^*}{\tilde \phi}(x) d\nu_{\psi}(x)\). For example generalized functionals \(:\exp[\lambda\dot B(t)]:\), \(\lambda\) and \(t\) are fixed, \(:\exp[c\int\dot B(u)^ 2du]:\), \(c<1/2\), and \(\delta_ x\), \(x\in E^*\), are positive and the corresponding positive measures are \(N(\lambda \xi (t),\| \xi \|^ 2)\), the Gaussian white noise with variance \((1-2c)^{-1}\), and the delta measure supported on a single point \(x\). The author also mentioned the differences of his results and Sugita's results obtained in the framework of Malliavin calculus.
scientific article; zbMATH DE number 4176143