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A Particular Case of Evans-Hudson Diffusion

  • Cristina SerbănescuEmail author
Chapter
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Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 343)

Abstract

We know that the Markov processes are the solutions of certain stochastic equations. In this article we will construct a noncommutative Markov process by noncommutative stochastic calculus. We will also show that these are particular cases of Evans-Hudson diffusions. At the end we will present two examples starting from the classical theory of probabilities (the Brownian motion and the Poisson process) which lead to particular cases of the noncommutative Markov processes.

Keywords

Noncommutative Markov process C*-algebra Brownian motion Poisson process Stochastic equation 

References

  1. 1.
    Lindblad, G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys. 48, 119–130 (1976)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Alicky, R.: Quantum Dynamic Semigroups and Applications. Lecture Notes in Physics 286, Part 1. Springer, Berlin (1987)Google Scholar
  3. 3.
    Gorini, V., Kossakowsky, A., Sudarshan, E.C.G.: Completely positive dynamical semigroups of n-level systems. J. Math. Phys. 17, 821–825 (1976)CrossRefGoogle Scholar
  4. 4.
    Stinespring, W.F.: Positive functions on C*-algebras. Proc. Am. Math. Soc. 6, 211–216 (1955)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Evans, M.P.: Existence of quantum diffusions. Probab. Theory Related Fields 81, 473–483 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Parthasarathy, K.R., Bhat, R.B.V.: Markov dilations of nonconservative dynamical semigroups and a quantum boundary theory. Annales de l’Institut Henri Poincaré, Probabilités et Statistique 30, 601–652 (1995)MathSciNetGoogle Scholar
  7. 7.
    Cuculescu, I., Oprea, A.: Noncommutative Probabilities. Kluwer, Boston (1994)CrossRefGoogle Scholar
  8. 8.
    Barnett, C., Streater, R.F., Wilde, I.F.: The Itô-Clifford integral II, stochastic differential equations. J. London Math. Soc. 27, 373–384 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Serbanescu, C.: An Ito product formula for Fermion stochastic integrals. Scientific Bulletin Politehnica Univ. Bucharest 60, 71–81 (1998)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Hudson, R.L., Parthasarathy, K.R.: Quantum Itô’s formula and stochastic evolutions. Commun. Math. Phys. 93, 301–323 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Applebaum, D.B., Hudson, R.L.: Fermion Itô’s formula and stochastic evolutions. Commun. Math. Phys. 96, 473–496 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Hudson, R.L., Parthasarathy, K.R.: Stochastic dilations of uniformly continuous completely positive semigroups. Acta Appl. Math. 2, 353–378 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Serbanescu C.: Fermoin Stochastic Integrals of Continuous Processes. Analele Universităţii, Bucureşti, pp. 277–288Google Scholar
  14. 14.
    Applebaum, D.B., Hudson, R.L.: Fermion diffusions. J. Math. Phys. 25, 858–861 (1984)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Applebaum, D.: Fermion Itô’s formula II the gauge process in fermion Fock space. Publications Res. Inst. Math. Sci., Kyoto Univ. 23, 17–56 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, 2nd edn. North Holland Mathematical Library, Amsterdam (1989)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity “Politehnica” BucharestBucharestRomania

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