Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-09T14:44:09.476Z Has data issue: false hasContentIssue false
  • English
  • Français

Quantum stochastic operator cocycles via associated semigroups

Published online by Cambridge University Press:  01 May 2007

J. MARTIN LINDSAY  and
STEPHEN J. WILLS
J. MARTIN LINDSAY
Affiliation:
Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF. e-mail: [email protected]
STEPHEN J. WILLS
Affiliation:
School of Mathematical Sciences, University College Cork, Cork, Ireland. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A recent characterisation of Fock-adapted contraction operator stochastic cocycles on a Hilbert space, in terms of their associated semigroups, yields a general principle for the construction of such cocycles by approximation of their stochastic generators. This leads to new existence results for quantum stochastic differential equations. We also give necessary and sufficient conditions for a cocycle to satisfy such an equation.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 142 , Issue 3 , May 2007 , pp. 535 - 556
Copyright
Copyright © Cambridge Philosophical Society 2007

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

REFERENCES

[AJL]Accardi, L., Journé, J.-L. and Lindsay, J. M.. On multidimensional Markovian cocycles. In Quantum Probability and Applications IV, eds. Accardi, L. & von Waldenfels, W.. Lecture Notes in Math. 1396 (Springer, 1989), pp. 59–67.Google Scholar
[Bou]Bourbaki, N.. Elements of Mathematics – Topological Vector Spaces (Springer-Verlag, 1987).CrossRefGoogle Scholar
[Dav]Davies, E. B.. One-parameter Semigroups (Academic Press, 1980).Google Scholar
[EfR]Effros, E. G. and Ruan, Z.-J.. Operator Spaces (Oxford University Press, 2000).Google Scholar
[Fa1]Fagnola, F.. Pure birth and death processes as quantum flows in Fock space. Sankhyā Ser. A 53 (1991), 288–297.Google Scholar
[Fa2]Fagnola, F.. Characterization of isometric and unitary weakly differentiable cocycles in Fock space. In Quantum Probability and Related Topics VIII, ed. Accardi, L.. (World Scientific, 1993), pp. 143–164.CrossRefGoogle Scholar
[FaW]Fagnola, F. and Wills, S. J.. Solving quantum stochastic differential equations with unbounded coefficients. J. Funct. Anal. 198 (2003), 279–310.CrossRefGoogle Scholar
[GS1]Goswami, D. and Sinha, K. B.. Hilbert modules and stochastic dilation of a quantum dynamical semigroup on a von Neumann algebra. Comm. Math. Phys. 205 (1999), 377–405.CrossRefGoogle Scholar
[GS2]Goswami, D. and Sinha, K. B.. Stochastic dilation of symmetric completely positive semigroups arXiv:math-ph/0201005. (See also Quantum Stochastic Processes and Noncommutative Geometry, Cambridge Trach in Mathematics 169 (Cambridge University Press, 2007).)Google Scholar
[HuL]Hudson, R. L. and Lindsay, J. M.. On characterizing quantum stochastic evolutions. Math. Proc. Camb. Phil. Soc. 102 (1987), 363–369.CrossRefGoogle Scholar
[HuP]Hudson, R. L. and Parthasarathy, K. R.. Quantum Itô's formula and stochastic evolutions. Comm. Math. Phys. 93 (1984), 301–323.CrossRefGoogle Scholar
[Jou]Journé, J.-L.. Structure des cocycles markoviens sur l'espace de Fock. Probab. Theory Related Fields 75 (1987), 291–316.CrossRefGoogle Scholar
[Lie]Liebscher, V.. How to generate Markovian cocycles on boson Fock space. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4 (2001), 215–219.CrossRefGoogle Scholar
[L]Lindsay, J. M.. Quantum stochastic analysis – an introduction. In Quantum Independent Increment Processes I: From Classical Probability to Quantum Stochastic Calculus, eds. Franz, U. & Schürmann, M.. Lecture Notes in Math. 1865 (Springer, 2005), pp. 181–271.CrossRefGoogle Scholar
[LW1]Lindsay, J. M. and Wills, S. J.. Existence, positivity, and contractivity for quantum stochastic flows with infinite dimensional noise. Probab. Theory Related Fields 116 (2000), 505–543.CrossRefGoogle Scholar
[LW2]Lindsay, J. M. and Wills, S. J.. Markovian cocycles on operator algebras, adapted to a Fock filtration. J. Funct. Anal. 178 (2000), 269–305.CrossRefGoogle Scholar
[LW3]Lindsay, J. M. and Wills, S. J.. Quantum stochastic cocycles and completely bounded semigroups on operator spaces I, in preparation.Google Scholar
[LW4]Lindsay, J. M. and Wills, S. J.. Construction of some quantum stochastic operator cocycles by the semigroup method. Proc. Indian Acad. Sci. Math. Sci. 116 (2006), 519–529. arXiv:math.FA/0606545.CrossRefGoogle Scholar
[Mey]Meyer, P.-A.. Quantum Probability for Probabilists. 2nd Edition. Lecture Notes in Math. 1538 (Springer, 1993).CrossRefGoogle Scholar
[Moh]Mohari, A.. Quantum stochastic differential equations with unbounded coefficients and dilations of Feller's minimal solution. Sankhyā Ser. A 53 (1991), 255–287.Google Scholar
[MoP]Mohari, A. and Parthasarathy, K. R.. A quantum probabilistic analogue of Feller's condition for the existence of unitary Markovian cocycles in Fock spaces. In Statistics and Probability: A Raghu Raj Bahadur Festschrift, eds. Ghosh, J. K., Mitra, S. K., Parthasarathy, K. R. & Rao, B. L. S. Prakasa. (Wiley Eastern, 1993), pp. 475–497.Google Scholar
[Pis]Pisier, G.. Introduction to Operator Space Theory. London Mathematical Society Lecture Note Series 294 (Cambridge University Press, 2003).CrossRefGoogle Scholar
[ReS]Reed, M. and Simon, B.. Methods of Modern Mathematical Physics II: Fourier Analysis, Self-adjointness (Academic Press, 1975).Google Scholar
[Wal]Waldenfels, W. von. Continuous Maassen kernels and the inverse oscillator. In Séminaire de Probabilités XXX, eds. Azéma, J., Emery, M. & Yor, M.. Lecture Notes in Math. 1626 (Springer, 1996), pp. 117–161.CrossRefGoogle Scholar