Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-28T14:20:24.795Z Has data issue: false hasContentIssue false
  • English
  • Français

On Meyer’s equivalence

Published online by Cambridge University Press:  22 January 2016

Jürgen Potthoff
Jürgen Potthoff*
Affiliation:
Department of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464 Japan, Japan, Department of Mathematics, Technical University Berlin, Strasse d. 17. Juni 135, D-1000, Berlin, 12
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In the recent years there has been a considerable effort to construct and analyze spaces of test and generalized functionals in infinite dimensional situations, cf. [3, 5, 12, 14] and literature quoted there. In particular Meyer [4, 5] has introduced a certain space of “smooth” functionals on the Wiener space, which was used by Watanabe [14] for an elegant formulation of “Malliavin’s calculus” (i.e. he proved a criterion for the existence and regularity of densities of Wiener functionals). This functional space is countably normed and one of its important properties is its algebraic structure. The proof of this property follows from an equivalence of the norms defining the space with a system of norms of Sobolev type [4, 5] (cf. also (1.5), (1.6)).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 111 , September 1988 , pp. 99 - 109
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1988

References

[1] Gel’fand, I. M. and Shilov, G. E., Generalized functions II, New York and London, Academic Press 1968.Google Scholar
[2] Gel’fand, I. M. and Vilenkin, N. Y., Generalized functions IV, New York and London, Academic Press 1964.Google Scholar
[3] Hida, T., Brownian motion; Berlin, Heidelberg, New York, Springer 1980.Google Scholar
[4] Meyer, P. A., Note sur les processus d’Ornstein-Uhlenbeck, Seminaire de probabilitiés XVI, ed. by Azema, J. and Yor, M., Berlin, Heidelberg, New York, Springer 1980.Google Scholar
[5] Meyer, P. A., Quelques resultats analytiques sur le semigroupe d’Ornstein-Uhlenbeck en dimension infinie, Theory and Application of Random Fields, ed. by Kallianpur, G., Berlin, Heidelberg, New York, Springer 1983.Google Scholar
[6] Nelson, E., Probability theory and Euclidean quantum field theory, Constructive quantum field theory, ed. by Velo, G. and Wightman, A., Berlin, Heidelberg, New York, Springer 1973.Google Scholar
[7] Potthoff, J., On positive generalized functionals, J. Funct. Anal., 74 (1987), 81–95.CrossRefGoogle Scholar
[8] Potthoff, J., Littlewood-Paley theory on Gaussian spaces, Nagoya Math. J., 109 (1988), 47–61.Google Scholar
[9] Reed, M. and Simon, B., Methods in Mathematical Physics I, II, New York, London, Academic Press 1972 and 1975.Google Scholar
[10] Simon, B., The P(Φ) 2 Euclidean field theory, Princeton, Princeton University Press 1970.Google Scholar
[11] Stein, E. M., Singular integrals and differentiability properties of functions, Princeton, Princeton University Press 1970.Google Scholar
[12] Sugita, H., Sobolev spaces of Wiener functionals and Malliavin’s calculus, J. Math. Kyoto Univ., 25 (1985), 31–48.Google Scholar
[13] Velo, G. and Wightman, A. (ed.s), Constructive quantum field theory, Berlin, Heidelberg, New York, Springer 1973.Google Scholar
[14] Watanabe, S., Malliavin’s calculus in terms of generalized Wiener functionals, Theory and Applications of Random Fields, ed. by Kallianpur, G., Berlin, Heidelberg, New York, Springer 1983.Google Scholar