Stationary Quantum Markov processes as solutions of stochastic differential equations

Jürgen Hellmich; Claus Köstler; Burkhard Kümmerer

Banach Center Publications (1998)

  • Volume: 43, Issue: 1, page 217-229
  • ISSN: 0137-6934

Abstract

top
From the operator algebraic approach to stationary (quantum) Markov processes there has emerged an axiomatic definition of quantum white noise. The role of Brownian motion is played by an additive cocycle with respect to its time evolution. In this report we describe some recent work, showing that this general structure already allows a rich theory of stochastic integration and stochastic differential equations. In particular, if a quantum Markov process is represented by a unitary cocycle, we can reconstruct an additive cocycle ('quantum Brownian motion') and the unitary cocycle ('quantum Markov process') appears as the solution of a certain stochastic differential equation. This establishes a one-to-one correspondence between multiplicative and additive adapted cocycles. As an application of this result we construct stationary Markov processes, driven by squeezed white noise and q-white noise.

How to cite

top

Hellmich, Jürgen, Köstler, Claus, and Kümmerer, Burkhard. "Stationary Quantum Markov processes as solutions of stochastic differential equations." Banach Center Publications 43.1 (1998): 217-229. <http://eudml.org/doc/208842>.

@article{Hellmich1998,
abstract = {From the operator algebraic approach to stationary (quantum) Markov processes there has emerged an axiomatic definition of quantum white noise. The role of Brownian motion is played by an additive cocycle with respect to its time evolution. In this report we describe some recent work, showing that this general structure already allows a rich theory of stochastic integration and stochastic differential equations. In particular, if a quantum Markov process is represented by a unitary cocycle, we can reconstruct an additive cocycle ('quantum Brownian motion') and the unitary cocycle ('quantum Markov process') appears as the solution of a certain stochastic differential equation. This establishes a one-to-one correspondence between multiplicative and additive adapted cocycles. As an application of this result we construct stationary Markov processes, driven by squeezed white noise and q-white noise.},
author = {Hellmich, Jürgen, Köstler, Claus, Kümmerer, Burkhard},
journal = {Banach Center Publications},
keywords = {quantum white noise; Brownian motion; stochastic integration; stochastic differential equations; quantum Markov process; stationary Markov processes; squeezed white noise; -white noise},
language = {eng},
number = {1},
pages = {217-229},
title = {Stationary Quantum Markov processes as solutions of stochastic differential equations},
url = {http://eudml.org/doc/208842},
volume = {43},
year = {1998},
}

TY - JOUR
AU - Hellmich, Jürgen
AU - Köstler, Claus
AU - Kümmerer, Burkhard
TI - Stationary Quantum Markov processes as solutions of stochastic differential equations
JO - Banach Center Publications
PY - 1998
VL - 43
IS - 1
SP - 217
EP - 229
AB - From the operator algebraic approach to stationary (quantum) Markov processes there has emerged an axiomatic definition of quantum white noise. The role of Brownian motion is played by an additive cocycle with respect to its time evolution. In this report we describe some recent work, showing that this general structure already allows a rich theory of stochastic integration and stochastic differential equations. In particular, if a quantum Markov process is represented by a unitary cocycle, we can reconstruct an additive cocycle ('quantum Brownian motion') and the unitary cocycle ('quantum Markov process') appears as the solution of a certain stochastic differential equation. This establishes a one-to-one correspondence between multiplicative and additive adapted cocycles. As an application of this result we construct stationary Markov processes, driven by squeezed white noise and q-white noise.
LA - eng
KW - quantum white noise; Brownian motion; stochastic integration; stochastic differential equations; quantum Markov process; stationary Markov processes; squeezed white noise; -white noise
UR - http://eudml.org/doc/208842
ER -

References

top
  1. [AcLu] L. Accardi and Y.G. Lu, The Wigner semi-circle law in quantum electro dynamics, Comm. Math. Phys. 180, 605-632, 1996. Zbl0856.46047
  2. [AFL] L. Accardi, A. Frigerio and J. T. Lewis, Quantum stochastic processes, Publ. RIMS Kyoto 18, 97-133, 1982. Zbl0498.60099
  3. [AFQ] L. Accardi, F. Fagnola and J. Quaegebeur, A representation free quantum stochastic calculus, J. Funct. Anal. 104, 149-197, 1992. Zbl0759.60068
  4. [ApFr] D. Applebaum and A. Frigerio, Stationary dilations of W*-dynamical systems constructed via quantum stochastic differential equations, in: From local times to global geometry, control and physics (Coventry, 1984/85), Pitman Res. Notes Math. Ser., 150, 1-38, Longman Sci. Tech., Harlow 1986. 
  5. [Ba] A. Barchielli, Applications of quantum stochastic calculus to quantum optics, Quantum probability & related topics, QP-PQ, VI, 111-125, 1991. Zbl0953.60529
  6. [Be] V. P. Belavkin, A new form and a *-algebraic structure of integrals in Fock space, Rend. Sem. Mat. Fis. Milano 58, 177-193, 1988. Zbl0708.60048
  7. [BiSp] P. Biane and R. Speicher, Stochastic calculus with respect to free Brownian motion and analysis on Wigner space, preprint, Paris, 1997. 
  8. [BKS] M. Bożejko, B. Kümmerer and R. Speicher, q-Gaussian processes: non-commutative and classical aspects, Comm. Math. Phys. 185, 129-154, 1997. Zbl0873.60087
  9. [BoSp] M. Bożejko and R. Speicher, An example of generalized Brownian motion, Comm. Math. Phys. 137, 519-531, 1991. Zbl0722.60033
  10. [BSW] C. Barnett, R. F. Streater and I. F. Wilde, Quasi-free quantum stochastic integrals for the CAR and CCR, J. Funct. Anal. 52, 19-47, 1983. Zbl0513.60063
  11. [Fr] M. Frank, Self-duality and C*-reflexivity of Hilbert C*-moduli, Zeitschr. Anal. Anw. 9, 165-176, 1990. 
  12. [GHJ] F. M. Goodman, P. de la Harpe and V. F. R. Jones, Coxeter Graphs and Towers of Algebras, Springer Verlag, New York, 1989. Zbl0698.46050
  13. [HKR] J. Hellmich, R. Honegger, C. Köstler, B. Kümmerer, A. Rieckers and C. Rupp, The quantum stochastic calculus of classical and non-classical squeezed white noise, preprint, Tübingen, 1997. 
  14. [Hi] T. Hida, Brownian Motion, Springer Verlag, New York, 1980. 
  15. [HuLi] R. L. Hudson and J. M. Lindsay, A noncommutative martingale representation theorem for non-Fock quantum Brownian motion, J. Funct. Anal. 61, 202-221, 1985. Zbl0577.60055
  16. [KFGV] A. Kossakowski, A. Frigerio, V. Gorini and M. Verri, Quantum detailed balance and KMS condition, Comm. Math. Phys. 57, 97-110, 1977. Zbl0374.46060
  17. [Kü1] B. Kümmerer, Markov dilations on W*-algebras, J. Funct. Anal. 63, 139-177, 1985. Zbl0601.46062
  18. [Kü2] B. Kümmerer, Stochastic processes with values in M n as couplings to free evolutions, preprint, 1993. 
  19. [KüMa] B. Kümmerer and H. Maassen, Elements of quantum probability, to appear in Quantum Probability Communications, X. 
  20. [KüSp] B. Kümmerer and R. Speicher, Stochastic integration on the Cuntz algebra O ∞ , J. Funct. Anal. 103, 372-408, 1992. Zbl0787.46052
  21. [La] E. C. Lance, Hilbert C*-modules, London Mathematical Society Lecture Notes Series 210, 1995. 
  22. [Me] P. A. Meyer, Quantum Probability for Probabilists, Springer Verlag, Berlin, Heidelberg, 1993. 
  23. [P] K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus, Birkhäuser, Basel, 1992. Zbl0751.60046
  24. [Pa] W. L. Paschke, Inner product modules over B*-algebras, Trans. Amer. Math. Soc. 182, 443-468, 1973. Zbl0239.46062
  25. [Pr] J. Prin, Verallgemeinertes weißes Rauschen und nichtkommutative stochastische Integration, Diplomarbeit, Tübingen, 1989. 
  26. [Sc] J. Schweizer, Interplay between noncommutative topology and operators on C*-algebras, thesis, Tübingen, 1996. 
  27. [Sk] M. Skeide, Hilbert modules in quantum electro dynamics and quantum probability, to appear in Comm. Math. Phys. Zbl0928.46063

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.