Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T07:45:17.113Z Has data issue: false hasContentIssue false
  • English
  • Français

Quantum stochastic integrals as belated integrals

Published online by Cambridge University Press:  18 May 2009

Chris Barnett ,
J. M. Lindsay  and
Ivan F. Wilde
Chris Barnett
Affiliation:
Department of Mathematics, Imperial College of Science, Technology and Medicine, London SW7 2BZ
J. M. Lindsay
Affiliation:
Department of Mathematics, University of Nottingham, University Park, Nottingham NG7 2RD
Ivan F. Wilde
Affiliation:
Department of Mathematics, Kings College, Strand, London WC2R 2LS
Rights & Permissions [Opens in a new window]

Extract

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.

Quantum stochastic integrals have been constructed in various contexts [2, 3, 4, 5, 9] by adapting the construction of the classical L2-Itô-integral with respect to Brownian motion. Thus, the integral is first defined for simple integrands as a finite sum, then one establishes certain isometry relations or suitable bounds to allow the extension, by continuity, to more general integrands. The integrator is typically operator-valued, the integrand is vector-valued or operator-valued and the quantum stochastic integral is then given as a vector in a Hilbert space, or as an operator on the Hilbert space determined by its action on suitable vectors.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 34 , Issue 2 , May 1992 , pp. 165 - 173
Copyright
Copyright © Glasgow Mathematical Journal Trust 1992

References

REFERENCES

1.Appelbaum, D., The strong Markov property for fermion brownian motion, J. Functional Analysis 65 (1986), 273–291.CrossRefGoogle Scholar
2.Appelbaum, D. and Hudson, R. L., Fermion Itô's formula and stochastic evolutions, Commun. Math. Phys. 96 (1984), 473–496.Google Scholar
3.Barnett, C., Streater, R. F. and Wilde, I. F., The Itô-Clifford integral, J. Functional Analysis 48 (1982), 172–212.CrossRefGoogle Scholar
4.Barnett, C., Streater, R. F. and Wilde, I. F., Quasi-free quantum stochastic integrals for the CAR and CCR, J. Functional Analysis 52 (1983), 19–47.CrossRefGoogle Scholar
5.Barnett, C., Streater, R. F. and Wilde, I. F., Quantum stochastic integrals under standing hypotheses, J. Math. Analysis and Applications 127 (1987), 181–192.Google Scholar
6.Barnett, C. and Wilde, I. F., Belated integrals, J. Functional Analysis 66 (1986), 283–307.CrossRefGoogle Scholar
7.Bartle, R. G., A general bilinear vector integral, Studia Math. 15 (1956), 337–352.CrossRefGoogle Scholar
8.Dell'Antonio, G. F., Structure of the algebras of some free systems, Commun. Math. Phys. 9 (1968), 81–117.CrossRefGoogle Scholar
9.Hudson, R. L. and Lindsay, J. M., A non-commutative martingale representation theorem for non-Fock quantum Brownian motion, J. Functional Analysis 61 (1985), 202–221.CrossRefGoogle Scholar
10.Hudson, R. L. and Parthasarathy, K. R., Quantum Itô's formula and stochastic evolutions, Commun. Math. Phys. 93 (1984), 901–923.CrossRefGoogle Scholar
11.Hudson, R. L. and Parthasarathy, K. R., Unification of fermion and boson stochastic calculus, Commun. Math. Phys. 104 (1986), 457–470.CrossRefGoogle Scholar
12.Lindsay, J. M., A quantum stochastic calculus, Ph.D. thesis (University of Nottingham 1985).Google Scholar
13.Lindsay, J. M. and Wilde, I. F., On non-Fock boson stochastic integrals, J. Functional Analysis 65 (1986), 76–82.CrossRefGoogle Scholar