Abstract
We consider series of iterated non-commutative stochastic integrals of scalar operators on the boson Fock space. We give a sufficient condition for these series to converge and to define a reasonable operator. An application of this criterion gives a condition for the convergence of some formal series of generalized integrator processes such as considered in [CEH].
Work supported by EU HCM Contract CHRX-CT93-0094
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Attal S.: “Non-commutative chaotic expansion of Hilbert-Schmidt operators on Fock spâce”, Comm. Math. Phys.175, (1996), p. 43–62.
Attal S.: “Semimartingales non commutatives et applications aux endomorphismes browniens”, Thèse de Doctorat, Univ. L. Pasteur, Strasbourg (1993).
Attal S.: “Problèmes d'unicité dans les représentations d'opérateurs sur l'espace de Fock”, Séminaire de probabilités XXVI, Springer Verlag L.N.M. 1526 (1992), p. 619–632.
Belavkin V.P. & Lindsay J.M.: “The kernel of a Fock space operator II”, Quantum Prob. & Rel. Topics IX, World Scientific p. 87–94.
Cohen P.B., Eyre T.W.M. & Hudson R.L.: “Higher order Ito product formula, and generators of evolutions and flows”, Int. Journ. Theor. Phys., 34 (1995), p. 1481–1486.
Dermoune A.: “Formule de composition pour une classe d'opérateurs”, Séminaire de probabilités XXIV, Springer Verlag L.N.M. 1426 (1990), p 397–401.
Hudson R.L. & Parthasarathy K.R.: “Quantum Itô's formula and stochastic evolutions”, Comm. Math. Phys. 93 (1984), p 301–323.
Hudson R.L. & Parthasarathy K.R.: “The Casimir chaos map for U(N)”, Tatra Mountains Mathematicals Proceedings 3 (1993), p 81–88.
Lindsay J.M. & Parthasarathy K.R.: “Cohomology of power sets with applications in quantum probability”, Comm. in Math. Phys. 124 (1989), p. 337–364.
Maassen H.: “Quantum Markov processes on Fock space described by integral kernels”, Quantum Prob. & Appl. II, Springer Verlag L.N.M. 1136 1985, p 361–374.
Meyer P.A.: “Quantum probability for probabilists”, second edition, Springer Verlag L.N.M. 1538, (1995).
Meyer P.A.: “Eléments de probabilités quantiques”, Séminaire de Probabilités XX, Springer Verlag L.N.M. 1204 (1986), p 186–312.
Parthasarathy K.R.: “An introduction to quantum stochastic calculus”, Monographs in Mathematics, Birkhäuser, 1992.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2000 Springer-Verlag
About this chapter
Cite this chapter
Attal, S., Hudson, R.L. (2000). Series of iterated quantum stochastic integrals. In: Azéma, J., Ledoux, M., Émery, M., Yor, M. (eds) Séminaire de Probabilités XXXIV. Lecture Notes in Mathematics, vol 1729. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0103801
Download citation
DOI: https://doi.org/10.1007/BFb0103801
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-67314-9
Online ISBN: 978-3-540-46413-6
eBook Packages: Springer Book Archive