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1998 | 43 | 1 | 217-229
Tytuł artykułu

Stationary Quantum Markov processes as solutions of stochastic differential equations

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Rocznik
Tom
43
Numer
1
Strony
217-229
Opis fizyczny
Daty
wydano
1998
Twórcy
  • Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72 076 Tübingen, Germany
  • Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72 076 Tübingen, Germany
  • Mathematisches Institut A, Universität Stuttgart, Pfaffenwaldring 57, D-70 569 Stuttgart, Germany
Bibliografia
  • [AcLu] L. Accardi and Y.G. Lu, The Wigner semi-circle law in quantum electro dynamics, Comm. Math. Phys. 180, 605-632, 1996.
  • [AFL] L. Accardi, A. Frigerio and J. T. Lewis, Quantum stochastic processes, Publ. RIMS Kyoto 18, 97-133, 1982.
  • [AFQ] L. Accardi, F. Fagnola and J. Quaegebeur, A representation free quantum stochastic calculus, J. Funct. Anal. 104, 149-197, 1992.
  • [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.
  • [Ba] A. Barchielli, Applications of quantum stochastic calculus to quantum optics, Quantum probability & related topics, QP-PQ, VI, 111-125, 1991.
  • [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.
  • [BiSp] P. Biane and R. Speicher, Stochastic calculus with respect to free Brownian motion and analysis on Wigner space, preprint, Paris, 1997.
  • [BKS] M. Bożejko, B. Kümmerer and R. Speicher, q-Gaussian processes: non-commutative and classical aspects, Comm. Math. Phys. 185, 129-154, 1997.
  • [BoSp] M. Bożejko and R. Speicher, An example of generalized Brownian motion, Comm. Math. Phys. 137, 519-531, 1991.
  • [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.
  • [Fr] M. Frank, Self-duality and C*-reflexivity of Hilbert C*-moduli, Zeitschr. Anal. Anw. 9, 165-176, 1990.
  • [GHJ] F. M. Goodman, P. de la Harpe and V. F. R. Jones, Coxeter Graphs and Towers of Algebras, Springer Verlag, New York, 1989.
  • [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.
  • [Hi] T. Hida, Brownian Motion, Springer Verlag, New York, 1980.
  • [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.
  • [KFGV] A. Kossakowski, A. Frigerio, V. Gorini and M. Verri, Quantum detailed balance and KMS condition, Comm. Math. Phys. 57, 97-110, 1977.
  • [Kü1] B. Kümmerer, Markov dilations on W*-algebras, J. Funct. Anal. 63, 139-177, 1985.
  • [Kü2] B. Kümmerer, Stochastic processes with values in $M_n$ as couplings to free evolutions, preprint, 1993.
  • [KüMa] B. Kümmerer and H. Maassen, Elements of quantum probability, to appear in Quantum Probability Communications, X.
  • [KüSp] B. Kümmerer and R. Speicher, Stochastic integration on the Cuntz algebra $O_∞$, J. Funct. Anal. 103, 372-408, 1992.
  • [La] E. C. Lance, Hilbert C*-modules, London Mathematical Society Lecture Notes Series 210, 1995.
  • [Me] P. A. Meyer, Quantum Probability for Probabilists, Springer Verlag, Berlin, Heidelberg, 1993.
  • [P] K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus, Birkhäuser, Basel, 1992.
  • [Pa] W. L. Paschke, Inner product modules over B*-algebras, Trans. Amer. Math. Soc. 182, 443-468, 1973.
  • [Pr] J. Prin, Verallgemeinertes weißes Rauschen und nichtkommutative stochastische Integration, Diplomarbeit, Tübingen, 1989.
  • [Sc] J. Schweizer, Interplay between noncommutative topology and operators on C*-algebras, thesis, Tübingen, 1996.
  • [Sk] M. Skeide, Hilbert modules in quantum electro dynamics and quantum probability, to appear in Comm. Math. Phys.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-bcpv43i1p217bwm
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