Communications in Mathematical Physics

, Volume 93, Issue 3, pp 301–323

Quantum Ito's formula and stochastic evolutions

  • R. L. Hudson
  • K. R. Parthasarathy
Article

Abstract

Using only the Boson canonical commutation relations and the Riemann-Lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operator processes. This leads to a noncommutative Ito product formula, a realisation of the classical Poisson process in Fock space which gives a noncommutative central limit theorem, the construction of solutions of certain noncommutative stochastic differential equations, and finally to the integration of certain irreversible equations of motion governed by semigroups of completely positive maps. The classical Ito product formula for stochastic differentials with respect to Brownian motion and the Poisson process is a special case.

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References

  1. 1.
    Accardi, L.: On the quantum Feynman-Kac formula. Rend. Sem. Mat. Fis. Milano48, 135–180 (1980)Google Scholar
  2. 2.
    Applebaum, D., Hudson, R.L.: Fermion diffusions. J. Math. Phys. (to appear)Google Scholar
  3. 3.
    Barnett, C., Steater, R.F., Wilde, I.: The Ito-Clifford integral. J. Funct. Anal.48, 172–212 (1982)Google Scholar
  4. 4.
    Gorini, V., Kossakowski, A., Sudarshan, E.C.G.: Completely positive dynamical semigroups ofn-level systems. J. Math. Phys.17, 821–5 (1976)Google Scholar
  5. 5.
    Hudson, R.L., Ion, P.D.F., Parthasarathy, K.R.: Time-orthogonal unitary dilations and noncommutative Feynman-Kac formulae. Commun. Math. Phys.83, 261–80 (1982)Google Scholar
  6. 6.
    Hudson, R.L., Karandikar, R.L., Parthasarathy, K.R.: Towards a theory of noncommutative semimartingales adapted to Brownian motion and a quantum Ito's formula; and Hudson, R.L., Parthasarathy, K.R.: Quantum diffusions. In: Theory and applications of random fields, Kallianpur, (ed.). Lecture Notes in Control Theory and Information Sciences 49, Berlin, Heidelberg, New York, Tokyo: Springer 1983Google Scholar
  7. 7.
    Hudson, R.L., Parthasarathy, K.R.: Construction of quantum diffusions. In: Quantum probability and applications to the quantum theory of irreversible processes, Accardi (ed.) (to appear)Google Scholar
  8. 8.
    Hudson, R.L., Parthasarathy, K.R.: Stochastic dilations of uniformly continuous completely positive semigroups. Acta Math. Applicandae (to appear)Google Scholar
  9. 9.
    Ikeda, N., Watenabe, S.: Stochastic differential equations and diffusion processes. Amsterdam: North-Holland 1981Google Scholar
  10. 10.
    Lindblad, G.: On the generators of quantum dynamical semigroups. Commun. Math. Phys.48, 119–30 (1976)Google Scholar
  11. 11.
    Liptser, R.S., Shiraev, A.N.: Statistics of random processes. I. General theory. Berlin, Heidelberg, New York: Springer 1977Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • R. L. Hudson
    • 1
  • K. R. Parthasarathy
    • 2
  1. 1.Mathematics DepartmentUniversity of NottinghamNottinghamEngland
  2. 2.Indian Statistical InstituteNew DelhiIndia

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