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The characterizations of Laplacians in white noise analysis*

Published online by Cambridge University Press:  22 January 2016

Sheng-Wu He ,
Jia-Gang Wang  and
Rong-Qin Yao
Sheng-Wu He
Affiliation:
Department of Statistics, East China Normal University, 200062 Shanghai, China
Jia-Gang Wang
Affiliation:
Department of Statistics, East China Normal University, 200062 Shanghai, China
Rong-Qin Yao
Affiliation:
Institute of Applied Mathematics, East China University of Science and Technology, 200237 Shanghai, China
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The Laplacians form a class of the most important differential operators in white noise analysis. The goal of this paper is to give their characterizations. Our main tools are the Fock expansions of operators in terms of integral kernel operators and rotation-invariance. In Section 1, the fundamental setting of white noise analysis is introduced briefly. In Section 2, integral kernel operators and the Fock expansions of operators are given. The characterization theorems for number operator, Gross-Laplacian and Euler operator are given in Sections 3, 4 and 5 respectively.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 143 , September 1996 , pp. 93 - 109
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1996

Footnotes

*

The projects supported by National Natural Science Foundation of China

References

[ 1 ] Haag, R., On quantum field theories, Dan. Mat. Fys. Medd., 29 (1995), No. 12, 1–37.Google Scholar
[ 2 ] Hida, T., Kuo, H. H., Potthoff, J. and Streit, L., White Noise—An Infinite Dimensional Calculus, Kluwer Academic Publ. 1993.Google Scholar
[ 3 ] Hida, T. Obata, N. and Saito, K. Infinite dimensional rotations and Laplacians in terms of white noise calculus, Nagoya Math. J., 128 (1992), 65–93.Google Scholar
[ 4 ] Huang, Z. Y., Quantum white noises—White noise approach to quantum stochastic calculus, Nagoya Math. J., 129 (1993), 23–42.Google Scholar
[ 5 ] Krée, P., La théorie des distributions en dimension quelconque et l’intégration stochastique, Lect. Notes in Math., 1316, 170–233, Springer. 1988.Google Scholar
[ 6 ] Kubo, I. and Takenaka, S., Calculus on Gaussian white noise I - IV, Proc. Japan Acad., 56A (1980), 376–380; 56A (1980), 411–416; 57A (1981), 433–437; 58A (1982), 186–189.Google Scholar
[ 7 ] Liu, K. and Yan, J. A., Euler operator and homogeneous Hida distributions. 1992. Preprint.Google Scholar
[ 8 ] Luo, S. L., Wick algebra of generalized operators involving quantum white noise, 1994. Preprint.Google Scholar
[ 9 ] Obata, N., Rotation-invariant operators on white noise functionals, Math, z., 210 (1992), 69–89.CrossRefGoogle Scholar
[10] Obata, N., White Noise Calculus and Fock Space, Springer. 1994.Google Scholar
[11] Obata, N., Lie algebras containing infinite dimensional Laplacians, 1995. Preprint.Google Scholar
[12] Yan, J. A., Some recent developments in white noise analysis, in Probability and statistics, Badrikian, A. et al. (eds.), 221–247. World Scientific. 1993.Google Scholar