Markov dilations of nonconservative dynamical semigroups and a quantum boundary theory

B. V. Rajarama Bhat; K. R. Parthasarathy

Annales de l'I.H.P. Probabilités et statistiques (1995)

  • Volume: 31, Issue: 4, page 601-651
  • ISSN: 0246-0203

How to cite

top

Rajarama Bhat, B. V., and Parthasarathy, K. R.. "Markov dilations of nonconservative dynamical semigroups and a quantum boundary theory." Annales de l'I.H.P. Probabilités et statistiques 31.4 (1995): 601-651. <http://eudml.org/doc/77522>.

@article{RajaramaBhat1995,
author = {Rajarama Bhat, B. V., Parthasarathy, K. R.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {weak Markov flow; exit and entrance cocycles; exit time; boson Fock space; boundary theory for quantum Markov processes; nonconservative one parameter semigroups of completely positive linear contractions on a von Neumann algebra},
language = {eng},
number = {4},
pages = {601-651},
publisher = {Gauthier-Villars},
title = {Markov dilations of nonconservative dynamical semigroups and a quantum boundary theory},
url = {http://eudml.org/doc/77522},
volume = {31},
year = {1995},
}

TY - JOUR
AU - Rajarama Bhat, B. V.
AU - Parthasarathy, K. R.
TI - Markov dilations of nonconservative dynamical semigroups and a quantum boundary theory
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 1995
PB - Gauthier-Villars
VL - 31
IS - 4
SP - 601
EP - 651
LA - eng
KW - weak Markov flow; exit and entrance cocycles; exit time; boson Fock space; boundary theory for quantum Markov processes; nonconservative one parameter semigroups of completely positive linear contractions on a von Neumann algebra
UR - http://eudml.org/doc/77522
ER -

References

top
  1. [AFL] L. Accardi, A. Frigerio and J.T. Lewis, Quantum stochastic processes, Proc. Res. Inst. Math. Sci., Kyoto, Vol. 18, 1982, pp. 97-133. Zbl0498.60099MR660823
  2. [B] B.V.R. Bhat, Markov Dilations of Nonconservative Quantum Dynamical Semigroups and a Quantum Boundary Theory, Ph. D. Thesis, Indian Statistical Institute, Delhi Centre, 1993. 
  3. [BP] B.V.R. Bhat and K.R. Parthasarathy, Kolmogorov's existence theorem for Markov processes in C* algebras, Proc. Ind. Acad. Sci. (Mathematical Sciences), Vol. 104, 1994, pp. 253-262. Zbl0796.60092MR1280073
  4. [C1] K.L. Chung, Markov Chains with Stationary Transition Probabilities, Springer, Berlin, Gottingen. Heidelberg, 1960. Zbl0092.34304MR116388
  5. [C2] K.L. Chung, Lectures on the Boundary Theory for Markov Chains, Princeton University Press, 1970. Zbl0204.51003MR267644
  6. [CW] K.L. Chung and R.J. Williams, Introduction to Stochastic Integration, Birkhäuser, Boston. Basil. Stuttgart, 1983. Zbl0527.60058MR711774
  7. [Da] E.B. Davies, Quantum dynamical semigroups and the neutron diffusion equation, Rep. Math. Phys., Vol. 11, 1977, pp. 169-189. Zbl0372.47020MR503331
  8. [Dy] E.B. Dynkin, Markov Processes, Vol. I and II, Springer-Verlag, Berlin. Gottingen. Heidelberg, 1965. Zbl0132.37901MR193671
  9. [EL] D.E. Evans and J.T. Lewis, Dilations of Irreducible Evolutions in Algebraic Quantum Theory, Communications of the Dublin Institute for Advanced Studies, Series A (Theoretical Physics), No. 24, Dublin, 1977. Zbl0365.46059
  10. [Em] G.G. Emch, Minimal dilations of CP flows, C* algebras and applications to physics, Springer LNM 650, 1978, pp. 156-159. Zbl0385.46035MR504761
  11. [Fa] F. Fagnola, Unitarity of solutions to quantum stochastic differential equations and conservativity of the associated semigroups, in Quantum Probability and Related Topics, Vol. VII, 1992, pp. 139-148 (World Scientific, River Edge NJ). Zbl0789.47027MR1186660
  12. [Fe1] W. Feller, Boundaries induced by non-negative matrices, Trans. of the Amer. Math. Soc., Vol. 83, No. 1, 1956, pp. 19-54. Zbl0071.34901MR90927
  13. [Fe2] W. Feller, On boundaries and lateral conditions for the Kolmogorov differential equations, Annals of Mathematics, Vol. 65, No. 3, 1957, pp. 527-70. Zbl0084.35503MR90928
  14. [Fe3] W. Feller, Notes to my paper "On boundaries and lateral conditions for the Kolmogorov differential equations", Annals of Mathematics, Vol. 68, No. 3, 1958, pp. 735-736. Zbl0084.35503MR106508
  15. [H] R.L. Hudson, The strong Markov property for canonical Wiener processes, J. Func. Anal., Vol. 34, 1979, pp. 266-281. Zbl0424.46048MR552705
  16. [HIP] R.L. Hudson, P.D.F. Ion and K.R. Parthasarathy, Time-orthogonal unitary dilations and non-commutative Feynman-Kac formulae, Commun. Math. Phys., Vol. 83, 1982, pp. 261-280. Zbl0485.46038MR649162
  17. [Me] P.A. Meyer, Quantum Probability for Probabilists, LNM 1538, Springer Verlag, 1993. Zbl0773.60098MR1222649
  18. [Mo] A. Mohari, Quantum Stochastic Calculus with Infinite Degrees of Freedom and Its Applications, Ph. D. Thesis, Indian Statistical Institute, Delhi Centre, New Delhi, 1991. 
  19. [P1] K.R. Parthasarathy, An Introduction to Quantum Stochastic Calculus, Monographs in Mathematics, Birkhauser Verlag, Basil, 1991. Zbl0751.60046
  20. [P2] K.R. Parthasarathy, A continuous time version of Stinespring's theorem on completely positive maps, in Quantum Probability and Applications V, L. Accardi, W.Von Walden-fels (Eds.), Lecture Notes in Mathematics, 1442, Springer-Verlag, Heidelberg, 1990, pp. 296-300. Zbl0743.46060
  21. [PS] K.R. Parthasarathy and K.B. Sinha, Stoptimes in Fock space stochastic calculus, Probab. Th. Rel. Fields, Vol. 75, 1987, pp. 317-349. Zbl0599.60044MR890283
  22. [Sa] J. Sauvageot, Markov quantum semigroups admit covariant Markov C* dilations, Commun. Math. Phys., Vol. 106, 1986, pp. 91-103. Zbl0606.60079MR853979
  23. [St] W.F. Stinespring, Positive functions on C* algebras, Proc. Amer. Math. Soc., Vol. 6, 1955, pp. 211-16. Zbl0064.36703MR69403
  24. [Vi-S] G.F. Vincent-Smith, Dilations of a dissipative quantum dynamical system to a quantum Markov process, Proc. London Math. Soc. (3), Vol. 49, 1984, pp. 58-72. Zbl0572.46055MR743370
  25. [Y] K. Yosida, Functional Analysis, Springer Verlag, Berlin, 1965. Zbl0126.11504

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.