Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-30T22:06:55.555Z Has data issue: false hasContentIssue false
  • English
  • Français

Exponential formulae in quantum stochastic calculus

Published online by Cambridge University Press:  14 November 2011

A. S. Holevo
A. S. Holevo
Affiliation:
Steklov Mathematical Institute, Vavilova 42, 117966 Moscow, Russia e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Extract

The rigorous definition of time-ordered exponentials, solving quantum linear stochastic differential equations, is extended to Boson and Fermion stochastic calculi with infinitely many degrees of freedom. The relation to the classicalmultiplicative stochastic integral, solving the Doleans exponential equation, is discussed.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 2 , 1996 , pp. 375 - 389
Copyright
Copyright © Royal Society of Edinburgh 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Applebaum, D. B.. Spectral families of quantum stochastic integrals (Preprint, 1992).Google Scholar
2Applebaum, D. B. and Hudson, R. L., Fermion Itô's formula and stochastic evolutions. Comm. Math. Phys. 96 (1984), 473–96.CrossRefGoogle Scholar
3Applebaum, D. B. and Kunita, H.. Lévy flows on manifolds and Lévy processeson Lie groups. J. Math. Kyoto Univ. 33 (1993), 1103–23.Google Scholar
4Emery, M.. Stabilité des solutions des équations differentielles stochastiquc application aux integrates multiplicatives stochastiques. Z. Wahr. verw. Gebiete 41 (1978), 241–62.CrossRefGoogle Scholar
5Holevo, A. S.. Stochastic representations of quantum dynamical semigroups. Proc. Steklov Inst. Math. 191 (1989). 130–9 (in Russian; English translation: 191 Issue 2 (1992), 145–54Google Scholar
6Holevo, A. S.. An analog of the Itô decomposition for multiplicative processes with values in a Lie group. Sankhyā Ser. A 53 (1991), 158–61.Google Scholar
7Holevo, A. S.. Time-ordered exponentials in quantum stochastic calculus. In Quantum Probability and Related Topics VIII, 175–202 (Singapore: World Scientific).Google Scholar
8Hudson, R. L. and Parthasarathy, K. R.. Quantum Itô formula and stochastic evolutions. Comm. Math. Phys. 93 (1984), 301–23.CrossRefGoogle Scholar
9Hudson, R. L. and Parthasarathy, K. R.. Unification of Fermion and Boson stochastic calculus. Comm. Math. Phys. 104 (1986). 457–70.CrossRefGoogle Scholar
10Meyer, P.-A.. Quantum Probability for Probahilists, Lecture Notes in Mathematics 1538(Berlin: Springer, 1993).CrossRefGoogle Scholar
11Mohari, A. and Sinha, K. B.. Quantum stochastic flows with infinite degrees of freedom and countable Markov processes. Sankhyā Ser. A 52 (1990), 43–57.Google Scholar
12Parthasarathy, K. R.. An Introduction to Quantum Stochastic Calculus (Basel: Birkhauser, 1992).Google Scholar