Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-03T08:49:45.695Z Has data issue: false hasContentIssue false
  • English
  • Français

A characterization of the Lévy Laplacian in terms of infinite dimensional rotation groups

Published online by Cambridge University Press:  22 January 2016

Nobuaki Obata
Nobuaki Obata*
Affiliation:
Department of Mathematics, School of Science Nagoya University, Nagoya, 464-01, Japan
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.

P. Lévy introduced, in his celebrated books [21] and [22], an infinite dimensional Laplacian called the Lévy Laplacian in connection with a number of interesting topics in variational calculus. One of the most significant features of the Lévy Laplacian is observed when it acts on the singular part of the second functional derivatives. For this reason the Lévy Laplacian has become important also in white noise analysis initiated by T. Hida [12]. On the other hand, as was pointed out by H. Yoshizawa [29], infinite dimensional rotation groups are profoundly concerned with the structure of white noise, and therefore, play essential roles in certain problems of stochastic calculus. Motivated by these works, we aim at developing harmonic analysis on infinite dimensional spaces by means of the Lévy Laplacian and infinite dimensional rotation groups.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 118 , June 1990 , pp. 111 - 132
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1990

References

[1] Balakrishnan, A. V., A white noise version of the Girsanov formula, in: Itô, K. (ed.) Proceedings of the international symposium on stochastic differential equations, Kyoto, 1976, pp. 1–19. Kinokuniya, Tokyo, 1978.Google Scholar
[2] Courant, R., Hilbert, D., Methods of mathematical physics, Vol.1 (1st Eng. ed.), Interscience, New York, 1953.Google Scholar
[3] Dixmier, J., Von Neumann algebras, North-Holland, Amsterdam-New York-Oxford, 1981.Google Scholar
[4] Dixmier, J., C*-Algebras, North-Holland, Amsterdam-New York-Oxford, 1982.Google Scholar
[5] Dorfman, I. Ya., On means and the Laplacian of functions on Hilbert space, Math. USSR Sbornik, 10 (1970), 181–196.Google Scholar
[6] Dunford, N., Schwartz, J. T., Linear operators, Part I: General theory, Wiley, New York, 1988.Google Scholar
[7] Feller, M. N., Infinite dimensional elliptic equations and operators of Lévy type, Russian Math. Surveys, 41–4 (1986), 119–170.Google Scholar
[8] Gross, L., Potential theory on Hilbert space, J. Funct. Anal., 1 (1967), 123–181.Google Scholar
[9] Halmos, P. R., von Neumann, J., Operator methods in classical mechanics, II, Ann. of Math., 43 (1942), 332–350.Google Scholar
[10] Hasegawa, Y., Lévy’s functional analysis in terms of an infinite dimensional Brownian motion I-III, Osaka J. Math., 19 (1982), 405–428; 549–570; Nagoya Math. J., 90 (1983), 155–173.Google Scholar
[11] Hewitt, E., Ross, K. A., Abstract harmonic analysis II, Springer-Verlag, Berlin-Heidelberg-New York, 1970.Google Scholar
[12] Hida, T., Analysis of Brownian functionals, Carleton Math. Lect. Notes Vol. 13, 1975. 2nd ed., 1978.Google Scholar
[13] Hida, T., Brownian motion, Applications of Mathematics Vol. 11, Springer-Verlag, New York-Heidelberg-Berlin, 1980.Google Scholar
[14] Hida, T., Brownian motion and its functionals, Ricerche Mat., 34 (1985), 183–222.Google Scholar
[15] Hida, T., Analysis of Brownian functionals, Lecture Notes, IMA, University of Minnesota, 1986.Google Scholar
[16] Hida, T., Saitô, K., White noise analysis and the Levy Laplacian, in: Albeverio, S., Blanchard, Ph., Hazewinkel, M., Streit, L. (eds.) Stochastic processes in physics and engineering, pp. 177–184. D. Reidel Pub. Co., Dordrecht-Boston-Lancaster-Tokyo, 1988.Google Scholar
[17] Johnson, B. E., Parrott, S. K., Operators commuting with a von Neumann algebra modulo the set of compact operators, J. Funct. Anal., 11 (1972), 39–61.Google Scholar
[18] Kubo, I., Takenaka, S., Calculus on Gaussian white noise II, Proc. Japan Acad., 56A (1980), 411–416.Google Scholar
[19] Kuo, H.-H., On Laplacian operators of generalized Brownian functionals, in: Itô, K., Hida, T. (eds.) Stochastic processes and their applications. Proceedings, Nagoya, 1985 (Lect. Notes Math., vol. 1203), pp. 119–128. Springer-Verlag, Berlin-Heidelberg-New York, 1986.Google Scholar
[20] Kuo, H.-H., Obata, N., Saitô, K., Lévy Laplacian of generalized functions on a nuclear space, to appear in J. Funct. Anal. (1990).Google Scholar
[21] Lévy, P., Leçons d’analyse fonctionnelle, Gauthier-Villars, Paris, 1922.Google Scholar
[22] Lévy, P., Problèmes concrets d’analyse fonctionnelle, Gauthier-Villars, Paris, 1951.Google Scholar
[23] Obata, N., A note on certain permutation groups in the infinite dimensional rotation group, Nagoya Math. J., 109 (1988), 91–107.Google Scholar
[24] Obata, N., Analysis of the Levy Laplacian, Soochow J. Math., 14 (1988), 105–109.Google Scholar
[25] Obata, N., Density of natural numbers and the Levy group, J. Number Theory, 30 (1988), 288–297.Google Scholar
[26] Obata, N., The Levy Laplacian and mean value theorem, in: Heyer, H. (ed.) Probability measures on groups IX. Proceedings, Oberwolfach, 1988 (Lect. Notes Math., vol. 1379), pp. 242–253. Springer-Verlag, Berlin-Heidelberg-New York, 1989.Google Scholar
[27] Shilov, G. E., On some questions of analysis in Hilbert space, I—III. Funct. Anal. Appl., 1 (1967), 158–165; Amer. Math. Soc. Transl. (2), 90 (1970), 1–16; Math. USSR Sbornik, 3 (1967), 153–158.Google Scholar
[28] von Neumann, J., Einige Sätze über messbare Abbildungen, Ann. of Math., 33 (1932), 574–586.CrossRefGoogle Scholar
[29] Yoshizawa, H., Rotation group of Hilbert space and its application to Brownian motion, in: Proceedings of the international conference on functional analysis and related topics, Tokyo, 1969, pp. 414–423. Univ. of Tokyo Press, 1970.Google Scholar