Fourier-Mehler transforms of generalized Brownian functionals
Let \((L^ 2)^-\) be the space of generalized Brownian functionals in the sense of T. Hida. In his previous paper [J. Multivariate Anal. 12, 415-431 (1982; Zbl 0491.60039)] the author defined a generalized notion of Fourier transform on \((L^ 2)^-\). In this paper he considers the fractional power \(\{ {\mathcal G}_{\tau}\); \(\tau \in R\}\) of the Fourier transform above. For \(\phi \in(L^ 2)^- ({\mathcal G}_{\tau}\phi)(\dot b)\equiv \int K_{\tau}(\dot b,\dot B)\phi(\dot B)d\mu(\dot B),\) where b and B are the Brownian motions considered as variables of Brownian functionals, \(\mu\) is the Wiener measure w.r.t. B and \[ K_{\tau}(\dot b,\dot B)\equiv:\exp [(i/\sin \tau)\int \dot b(u)\dot B(u)du-(i/2 \tan \tau)\int \dot b(u)^ 2du]:_{\dot b}. \] The operation : \(:_{\dot b}\) means the renormalization w.r.t. variable b in the sense of T. Hida. For details see relevant articles of T. Hida and the author.
scientific article; zbMATH DE number 3868345