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H. Ouerdiane, Hafedh Rguigui (2012)
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(2020)
Two generalizations ofMehler’s formula inwhite noise analysis
In this paper, we introduce a space of θ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\theta $$\end{document}-admissible distributions denoted by Aθ∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {A}}_\theta ^*$$\end{document} as well as the notion of θ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\theta $$\end{document}-admissible operators. We study the regularity properties of the classical conditional expectation acting on Aθ∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {A}}_\theta ^*$$\end{document} and acting on L(Aθ,Aθ∗)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {L}}({\mathcal {A}}_\theta ,{\mathcal {A}}_\theta ^*)$$\end{document} which is the space of linear continuous operators from Aθ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {A}}_\theta $$\end{document} into Aθ∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {A}}_\theta ^*$$\end{document}. An integral representation with respect to the coordinate system of the quantum white noise (QWN) derivatives and their adjoints {Dt±,Dt±∗,t∈R}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\{D_t^{\pm }, D_t^{\pm *},\,t\in {\mathbb {R}}\}$$\end{document} of such conditional expectation is given. Then, we give a quantum white noise counterpart of the Clark formula. Finally, we introduce the QWN Hitsuda–Skorokhod integrals. Such integrals are shown to be QWN martingales using a new notion of QWN conditional expectation.
Bulletin of the Malaysian Mathematical Sciences Society – Springer Journals
Published: Jul 8, 2020
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