Hostname: page-component-848d4c4894-75dct Total loading time: 0 Render date: 2024-04-30T15:53:37.223Z Has data issue: false hasContentIssue false
  • English
  • Français

Quantum stochastic differential equations on *-bialgebras

Published online by Cambridge University Press:  24 October 2008

Peter Glockner
Peter Glockner
Affiliation:
Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 294, D-6900 Heidelberg, Germany
Get access Rights & Permissions [Opens in a new window]

Extract

Many examples of quantum independent stationary increment processes are solutions of quantum stochastic differential equations. We give a common characterization of these examples by a quantum stochastic differential equation on an abstract *-bialgebra. Specializing this abstract *-bialgebra and the coefficients of the equation, we obtain the equations for the Unitary Noncommutative Stochastic processes of [12], the Quantum Wiener Process [2], the Azéma martingales [11] and for other examples. The existence and uniqueness of a solution of the general equation is shown. Assuming the boundedness of this solution, we prove that it is a continuous and stationary independent increment process.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 109 , Issue 3 , May 1991 , pp. 571 - 595
Copyright
Copyright © Cambridge Philosophical Society 1991

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

[1]Abe, E.. Hopf Algebras. Cambridge University Press (1980).Google Scholar
[2]Accardi, L., Schürmann, M. and von Waldenfels, W.. Quantum independent increment processes on superalgebras. Math. Z. 198 (1988), 451–477.CrossRefGoogle Scholar
[3]Evans, M. P.. Existence of quantum diffusions. Prob. Theory and Rel. Fields 81 (1989), 473–483.CrossRefGoogle Scholar
[4]Evans, M. P. and Hudson, R. L.. Multidimensional quantum diffusions. Quantum Probability and Applications III. Proceedings, Oberwolfach (1987). Lecture Notes Math. 1303. Springer, 69–88.Google Scholar
[5]Glockner, P.. *-Bialgebren in der Quantenstochastik. Dissertation, Heidelberg (1989).Google Scholar
[6]Glockner, P. and von Waldenfels, W.. The relations of the non-commutative coefficient algebra of the unitary group. Quantum Probability and Applications IV. Proceedings, Rom (1987). Lecture Notes Math. 1396. Springer.Google Scholar
[7]Guichardet, A.. Symmetric Hilbert spaces and related topics. Lecture Notes Math. 261. Springer (1972).CrossRefGoogle Scholar
[8]Heyer, H.. Probability Measures on Locally Compact Groups. Springer (1977).Google Scholar
[9]Hudson, R. L.. Quantum diffusions and cohomology of algebras. Proc. 1st World Congress Bernoulli Soc., Tashkent (1986), vol. 1, 479–485.Google Scholar
[10]Hudson, R. L. and Parthasarathy, K. R.. Quantum Ito's formula and stochastic evolutions. Comm. Math. Phys. 93 (1984), 301–323.CrossRefGoogle Scholar
[11]Parthasarathy, K. R.. Azéma martingales and quantum stochastic calculus. Preprint (1989).Google Scholar
[12]Schürmann, M.. Über *-Bialgebren und quantenstochastische Zuwachsprozesse. Dissertation, Heidelberg (1985).Google Scholar
[13]Schürmann, M.. Non-commutative stochastic processes with independent and stationary increments satisfy quantum stochastic differential equations. Prob. Th. Rel. Fields 84 (1990), 473–490.Google Scholar
[14]Schürmann, M.. Noncommutative stochastic processes with independent and stationary additive increments. Quantum Probability and Applications IV. Proceedings, Rom (1987). Lecture Notes Math. 1396. Springer.Google Scholar
15]Schürmann, M.. The Azéma Martingales as components of quantum independent increment processes. Preprint (1990).CrossRefGoogle Scholar