Hostname: page-component-7bb8b95d7b-l4ctd Total loading time: 0 Render date: 2024-09-18T10:41:07.043Z Has data issue: false hasContentIssue false
  • English
  • Français

Transformations on white noise functions associated with second order differential operators of diagonal type

Published online by Cambridge University Press:  22 January 2016

Dong Myung Chung ,
Un Cig Ji  and
Nobuaki Obata
Dong Myung Chung
Affiliation:
Department of Mathematics, Sogang University, Seoul, 121-742, Korea, dmchung@ccs.sogang.ac.kr
Un Cig Ji
Affiliation:
Department of Mathematics, Sogang University, Seoul, 121-742, Korea, ucji@nuri.net
Nobuaki Obata
Affiliation:
Graduate School of Polymathematics, Nagoya University, Nagoya, 464-8602, Japan, obata@math.nagoya-u.ac.jp
Rights & Permissions [Opens in a new window]

Abstract.

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.

A generalized number operator and a generalized Gross Laplacian are introduced on the basis of white noise distribution theory. The equicontinuity is examined and associated one-parameter transformation groups are constructed. An infinite dimensional analogue of ax + b group and Cauchy problems on white noise space are discussed.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 149 , March 1998 , pp. 173 - 192
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1998

References

[1] Accardi, L., Yang-Mills equations and Levy Laplacians, in “Dirichlet Forms and Stochastic Processes (Z. M. Ma et al. eds.),” pp. 1–24, Walter de Gruyter, Berlin, 1995.Google Scholar
[2] Arai, A. and Mitoma, I., De Rham-Hodge-Kodaira decomposition in oo-dimensions, Math. Ann., 291 (1991), 51–73.CrossRefGoogle Scholar
[3] Chung, D. M. and Ji, U. C., Cauchy problems for a partial differential equation in white noise analysis, J. Korean Math. Soc, 33 (1996), 309–318.Google Scholar
[4] Chung, D. M. and Ji, U. C., Transformation groups on white noise functionals and their applications, to appear in Appl. Math. Optim.Google Scholar
[5] Chung, D. M. and Ji, U. C., Some Cauchy problems in white noise analysis and associated semigroups of operators, to appear in Stoch. Anal. Appl.Google Scholar
[6] Chung, D. M. and Ji, U. C., Transforms on white noise functionals with their applications to Cauchy problems, Nagoya Math. J., 147 (1997), 1–23.Google Scholar
[7] Dôku, I., On the Laplacian on a space of white noise functionals, Tsukuba J. Math., 19 (1995), 93–119.CrossRefGoogle Scholar
[8] Dôku, I., Kuo, H.-H. and Lee, Y.-J., Fourier transform and heat equation in white noise analysis, in “Stochastic Analysis on Infinite Dimensional Spaces (H. Kunita and H.-H. Kuo, eds.),” pp. 60–74, Longman, 1994.Google Scholar
[9] Gross, L., Potential theory on Hilbert space, J. Funct. Anal., 1 (1967), 123–181.Google Scholar
[10] Hida, T., “Analysis of Brownian Functionals,” Carleton Math. Lect. Notes no. 13, Carleton University, Ottawa, 1975.Google Scholar
[11] Hida, T., Kuo, H.-H. and Obata, N., Transformations on white noise functionals, J. Funct. Anal., 111 (1993), 259–277.Google Scholar
[12] Hida, T., Obata, N. and Saitô, K., Infinite dimensional rotations and Laplacians in terms of white noise calculus, Nagoya Math. J., 128 (1992), 65–93.Google Scholar
[13] Kuo, H.-H., On Laplacian operators of generalized Brownian functionals, in “Stochastic Processes and Applications (K. Itô and T. Hida, eds.),” pp. 119–128, Lect. Notes in Math. Vol. 1203, Springer-Verlag, 1986.Google Scholar
[14] Kuo, H.-H., Fourier-Mehler transforms in white noise analysis, in “Gaussian Random Fields (K. Itô and T. Hida eds.),” pp. 257–271, World Scientific, 1991.Google Scholar
[15] Kuo, H.-H., Lectures on white noise analysis, Soochow J. Math., 18 (1992), 229–300.Google Scholar
[16] Kuo, H.-H., “White Noise Distribution Theory,” CRC Press, 1996.Google Scholar
[17] Levy, P., “Problèmes Concrets d’Analyse Fonctionelle,” Gauthier-Viliars, Paris, 1951.Google Scholar
[18] Obata, N., Rotation-invariant operators on white noise functionals, Math. Z., 210 (1992), 69–89.Google Scholar
[19] Obata, N., “White Noise Calculus and Fock Space,” Lect. Notes in Math. Vol. 1577, Springer-Verlag, 1994.Google Scholar
[20] Obata, N., Lie algebras containing infinite dimensional Laplacians, in “Probability Measures on Groups and Related Structures (H. Heyer ed.),” pp. 260–273, World Scientific, 1995.Google Scholar
[21] Obata, N., Constructing one-parameter transformations on white noise functions in terms of equicontinuous generators, Monatsh. Math., 124 (1997), 317–335.Google Scholar
[22] Piech, M. A., A fundamental solution of the parabolic equation on Hilbert space, J. Funct. Anal, 3 (1969), 85–114.Google Scholar
[23] Piech, M. A., Parabolic equations associated with the number operator, Trans. Amer. Math. Soc, 194 (1974), 213–222.CrossRefGoogle Scholar
[24] Umemura, Y., On the infinite dimensional Laplacian operator, J. Math. Kyoto Univ., 4 (1965), 477–492.Google Scholar
[25] Umemura, Y. and Kôno, N., Infinite dimensional Laplacian and spherical harmonics, Publ. RIMS, 1 (1966), 163–186.Google Scholar
[26] Yosida, K., “Functional Analysis (6th ed.),” Springer-Verlag, 1980.Google Scholar