Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-19T21:56:30.824Z Has data issue: false hasContentIssue false
  • English
  • Français

COHERENT STATES IN BERNOULLI NOISE FUNCTIONALS

Part of: Stochastic analysis

Published online by Cambridge University Press:  01 April 2011

CAISHI WANG  and
QI HAN
CAISHI WANG*
Affiliation:
School of Mathematics and Information Science, Northwest Normal University, Lanzhou, Gansu 730070, PR China (email: wangcs@nwnu.edu.cn, cswangnwnu@163.com)
QI HAN
Affiliation:
School of Mathematics and Information Science, Northwest Normal University, Lanzhou, Gansu 730070, PR China (email: hanqi1978@nwnu.edu.cn)
*
✉For correspondence; e-mail: cswangnwnu@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.

Let (Ω,ℱ,ℙ) be a probability space and Z=(ZK)k∈ℕ a Bernoulli noise on (Ω,ℱ,ℙ) which has the chaotic representation property. In this paper, we investigate a special family of functionals of Z, which we call the coherent states. First, with the help of Z, we construct a mapping ϕ from l2(ℕ) to ℒ2(Ω,ℱ,ℙ) which is called the coherent mapping. We prove that ϕ has the continuity property and other properties of operation. We then define functionals of the form ϕ(f) with f∈l2 (ℕ) as the coherent states and prove that all the coherent states are total in ℒ2 (Ω,ℱ,ℙ) . We also show that ϕ can be used to factorize ℒ2 (Ω,ℱ,ℙ) . Finally we give an application of the coherent states to calculus of quantum Bernoulli noise.

Keywords

Bernoulli noisecoherent statefactorization of functional space

MSC classification

Secondary: 60H40: White noise theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 84 , Issue 1 , August 2011 , pp. 116 - 126
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

The authors are supported by National Natural Science Foundation of China (Grant No. 11061032) and Natural Science Foundation of Gansu Province (Grant No. 0710RJZA106). The first author is also partly supported by a grant from Northwest Normal University (Grant No. NWNU-KJCXGC-03-61).

References

[1]Émery, M., ‘A discrete approach to the chaotic representation property’, in: Séminaire de Probabilités, XXXV, Lecture Notes in Mathematics, 1755 (Springer, Berlin, 2001), pp. 123–138.CrossRefGoogle Scholar
[2]Parthasarathy, K. R., An Introduction to Quantum Stochastic Calculus (Birkhäuser, Basel, 1992).Google Scholar
[3]Privault, N., ‘Stochastic analysis of Bernoulli processes’, Probab. Surv. 5 (2008), 435–483.CrossRefGoogle Scholar
[4]Wang, C. S., Chai, H. F. and Lu, Y. C., ‘Discrete-time quantum Bernoulli noises’, J. Math. Phys. 51 (2010), 053528.CrossRefGoogle Scholar
[5]Wang, C. S., Lu, Y. C. and Chai, H. F., ‘An alternative approach to Privault’s discrete-time chaotic calculus’, J. Math. Anal. Appl. 373 (2011), 643–654.CrossRefGoogle Scholar