Fourier-Mehler transforms of generalized Brownian functionals
type: Item
instance of: scholarly article
title: Fourier-Mehler transforms of generalized Brownian functionals
zbMATH Open document ID: 0545.60045
author: Hui-Hsiung Kuo
published in: Proceedings of the Japan Academy. Series A
publication date: 1983-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60G20
Mathematics Subject Classification ID: 60H99
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 42B15
zbMATH DE Number: 3868345
zbMATH Keywords: causal analysis
zbMATH Keywords: generalized Brownian functionals
zbMATH Keywords: generalized notion of Fourier transform
zbMATH Keywords: renormalization
MaRDI profile type: Publication
full work available at URL: pjaa.59.312
OpenAlex ID: W1977419061
cites work: Q5471003
cites work: Generalized multiple Wiener integrals
cites work: Q3870044
cites work: On Fourier transform of generalized Brownian functionals
cites work: Q5806170
DOI: 10.3792/PJAA.59.312
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description: scientific article; zbMATH DE number 3868345