Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-12T13:35:29.919Z Has data issue: false hasContentIssue false
  • English
  • Français

QUANTUM TANAKA FORMULA IN TERMS OF QUANTUM BROWNIAN MOTION

Part of: General quantum mechanics and problems of quantization

Published online by Cambridge University Press:  01 April 2011

YULAN ZHOU  and
CAISHI WANG
YULAN ZHOU*
Affiliation:
School of Mathematics and Information Science, Northwest Normal University, Lanzhou 730070, PR China (email: zhouylw123@163.com)
CAISHI WANG
Affiliation:
School of Mathematics and Information Science, Northwest Normal University, Lanzhou 730070, PR China (email: wangcs@nwnu.edu.cn)
*
✉For correspondence; e-mail: zhouylw123@163.com
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 quantum local time, which is a generalized operator-valued process, is defined for quantum Brownian motion, and a quantum analogue of the classical Tanaka formula is then established.

Keywords

white noise analysisquantum local timequantum Tanaka formulasymbol of generalized operator

MSC classification

Secondary: 81S25: Quantum stochastic calculus
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 83 , Issue 3 , June 2011 , pp. 401 - 412
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

The research was supported by the Natural Science Foundation of China (11061032).

References

[1]Chung, D. M., Chung, T. S. and Ji, U. C., ‘A characterization theorem for operators on white noise functionals’, J. Math. Soc. Japan 51 (1999), 437–447.Google Scholar
[2]Chung, K. L. and Williams, R. J., Introduction to Stochastic Integration (Birkhäuser, Boston, 1983).CrossRefGoogle Scholar
[3]Hida, T., Analysis of Brownian Functionals, Carleton Mathematical Lecture Notes, 13 (Carleton Univesity, Ottawa, 1975).Google Scholar
[4]Hida, T., Kuo, H.-H., Potthoff, J. and Streit, L., White Noise. An Infinite Dimensional Calculus, Mathematics and its Applications, 253 (Kluwer, Dordrecht, 1993).Google Scholar
[5]Huang, Z. Y., ‘Quantum white noises—white noise approach to quantum stochastic calculus’, Nagoya Math. J. 129 (1993), 23–42.CrossRefGoogle Scholar
[6]Huang, Z. Y., Wang, C. S. and Wang, X. J., ‘Quantum cable equations in terms of generalized operators. Recent developments in infinite-dimensional analysis and quantum probability’, Acta Appl. Math. 63 (2000), 151–164.Google Scholar
[7]Huang, Z. Y. and Yan, J. A., Introduction to Infinite Dimensional Stochastic Analysis (Kluwer, Dordrecht, 2000).Google Scholar
[8]Hudson, R. L. and Parthasarathy, K. R., ‘Quantum Itô’s formula and stochastic evolutions’, Comm. Math. Phys. 93 (1984), 301–323.Google Scholar
[9]Obata, N., White Noise Calculus and Fock Space, Lecture Notes in Mathematics, 1577 (Springer, Berlin, 1994).CrossRefGoogle Scholar
[10]Parthasarathy, K. R., An Introduction to Quantum Stochastic Calculus, Monographs in Mathematics, 85 (Birkhäuser, Basel, 1992).Google Scholar
[11]Potthoff, J. and Streit, L., ‘A characterization of Hida distributions’, J. Funct. Anal. 101 (1991), 212–229.Google Scholar
[12]Wang, C. S., ‘A new idea to define the δ-function of an observable in the context of white noise analysis’, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8(4) (2005), 659–668.Google Scholar
[13]Wang, C. S. and Huang, Z. Y., ‘A filtration of Wick algebra and its application to quantum SDEs’, Acta Math. Sin. (Engl. Ser.) 20 (2004), 999–1008.Google Scholar
[14]Wang, C. S., Huang, Z. Y. and Wang, X. J., ‘δ-function of an operator: a white noise approach’, Proc. Amer. Math. Soc. 133 (2005), 891–898.Google Scholar
[15]Wang, C. S., Huang, Z. Y. and Wang, X. J., ‘Analytic characterization for Hilbert–Schmidt operators on Foch space’, Acta Math. Sin. (Engl. Ser.) 21 (2005), 787–796.CrossRefGoogle Scholar
[16]Wang, C. S., Qu, M. Sh. and Chen, J. S., ‘A white noise approach to infinitely divisible distributions on Gel’fand triple’, J. Math. Anal. Appl. 315 (2006), 425–435.Google Scholar