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Calcul stochastique non-commutatif

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Références

  1. L. Accardi, A. Frigerio, J.T. Lewis, Quantum Stochastic processes, Publ. R.I.M.S. Kyoto, 18, 1982, 94–133.

    MathSciNet  MATH  Google Scholar 

  2. L. Accardi, W. von Waldenfels Eds., Quantum Probability and applications, I–V Lect. Notes in Maths. Springer 1055, 1136, 1303, 1325, 1442, VI-World Scientific, Singapore.

    Google Scholar 

  3. D. Applebaum, Unitary evolutions and horizontal lifts in quantum stochastic calculus, Comm. Math. Phys., 140, 1991, 63–80.

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Applebaum, An operator theoretic approach to stochastic flows on manifolds, Séminaire de Probabilités XXVI, Lect. Notes in Maths. Springer 1526, 1992, 514–532.

    Article  MathSciNet  MATH  Google Scholar 

  5. C. Barnett, R.F. Streater, I.F. Wilde, The Ito-Clifford integral I, Jour. Funct. An., 48, 1982, 172–212.

    Article  MathSciNet  MATH  Google Scholar 

  6. V.P. Belavkin, A quantum non-adapted Ito formula and stochastic analysis in Fock scale, Jour. Funct. An., 102, 1991, 414–447.

    Article  MathSciNet  MATH  Google Scholar 

  7. V.P. Belavkin, Chaotic states and stochastic integration in quantum systems, Russian Math. Surv. 14:1 1992, 53–116.

    Article  MathSciNet  MATH  Google Scholar 

  8. V.P. Belavkin, The unified ito formula has the pseudo-Poisson structur,, preprint n o 98 Centro Vito Volterra, Universita degli studi di Roma II, 1992.

    Google Scholar 

  9. B.V.R. Bhat, K.R. Parthasarathy, Markov dilations of non-conservative dynamical semi-groups and a quantum boundary theory, preprint, 1992.

    Google Scholar 

  10. P. Biane, Chaotic representation for finite Markov chains, Stochastics, 30, 1990, 61–68.

    MathSciNet  MATH  Google Scholar 

  11. P. Biane, Marches de Bernoulli quantiques, Séminaire de Probabilités XXIV, Lect. Notes in Maths. Springer 1426, 1990, 329–344.

    MathSciNet  MATH  Google Scholar 

  12. P. Biane, Quantum random walks on the dual of SU(n), Prob. Th. and Rel. Fields, 89, 1991, 117–129.

    Article  MathSciNet  MATH  Google Scholar 

  13. P. Biane, Minuscule weights and random walks on lattices, Quantum probability and applications, VII World Scientific, Singapore, 1992, 51–65.

    MATH  Google Scholar 

  14. N. Bourbaki, Groupes et algèbres de Lie, Chapitre 9, Hermann, Paris, 1969.

    MATH  Google Scholar 

  15. M. Bozejko, R Speicher, An example of generalized brownian motion. Comm. Math. Phys., 137, 1991, 519–531.

    Article  MathSciNet  MATH  Google Scholar 

  16. T. Chihara, An introduction to orthgonal polynomials, Gordon and Breach, New York, 1978.

    MATH  Google Scholar 

  17. K.L. Chung, R.J. Williams, Introduction to stochastic integration, Progress in Mathematics, Birkhaüser, 1983

    Google Scholar 

  18. E.B. Davies, Quantum theory of open systems, Academic Press, 1976.

    Google Scholar 

  19. E.B. Davies, J.M. Lindsay, Non-communitative Markov semi-group, preprint, 1990.

    Google Scholar 

  20. C. Dellacherie, P.A. Meyer, Probabilités et potentiel, Chapitre I à IV, Hermann, Paris, 1975.

    MATH  Google Scholar 

  21. C. Dellacherie, P.A. Meyer, Probabilités et potentiel, Chapitre V à VIII, Hermann, Paris, 1980.

    MATH  Google Scholar 

  22. C. Dellacherie, P.A. Meyer, Probabilités et potential, Chapitre IX à XI, Hermann, Paris, 1983.

    MATH  Google Scholar 

  23. C. Dellacherie, P.A. Meyer, B. Maisonneuve, Probabilités et potential, Chapitre XVII à XXIV, Hermann, Paris, 1993.

    Google Scholar 

  24. J. Dixmier, Les algèbres d'opérateurs dans l'espace hilbertien, (Algèbres de von Neumann), Gauthier-Villars, Paris, 1957.

    MATH  Google Scholar 

  25. J. Dixmier, Les C*-algèbres et leurs représentations, Gauthier-Villars, Paris, 1964.

    MATH  Google Scholar 

  26. M. Emery, On the Azéma martingales, Séminaire de Probabilités XXIII, Lect. Notes in Maths. Springer 1372, 1990, 66–87.

    Google Scholar 

  27. M. Emery, Quelques cas de représentation chaotique, Séminaire de Probabilités XXIV, Lect. Notes in Maths. Springer 1426, 1991, 10–23.

    Google Scholar 

  28. M. Emery, On the chaotic representation property for martingales, preprint 1993.

    Google Scholar 

  29. D.E. Evans, J.T. Lewis, Dilations of dynamical semi-groups, Comm. Math. Phys., 50, 1976, 219–228.

    Article  MathSciNet  MATH  Google Scholar 

  30. M. Evans, Existence of quantum diffusions, Prob. Th. and Rel. Fields, 81, 1989, 473–483.

    Article  MathSciNet  MATH  Google Scholar 

  31. M. Evans, R.L. Hudson, Multidimensionnal quantum diffusions, Quantum probability and applications, III, Lect. Notes in Maths. Springer 1303, 1988, 69–88.

    Google Scholar 

  32. P. Eymard, B. Roynette, Marches aléatoires sur le dual de, SU(2), Marches aléatoires sur les groupes, Lect. Notes in Maths. Springer 624, 1977.

    Google Scholar 

  33. F. Fagnola, K.B. Sinha, Quantum flows with unbounded structure maps and finite degrees of freedom, à paraître dans Jour. Lond. Math. Soc.

    Google Scholar 

  34. W. Fellen, An introduction, to probability theory and its applications, Vol I, John Wiley & Sons, New York 1977.

    Google Scholar 

  35. R.P. Feynman, R.B. Leighton, M. Sands, The Feynman Lectures on physics, Vol III, Addison Wesley, Reading Mass. 1965.

    MATH  Google Scholar 

  36. B. Gaveau, P. Trauber, L'intégrale stochastique comme opérateur de divergence dans l'espace fonctionnel, Jour. Funct. An. 46, 1982, 230–238.

    Article  MathSciNet  MATH  Google Scholar 

  37. V. Guillemin, S. Sternberg, Symplectic techniques in Physics, Cambridge University Press, 1984.

    Google Scholar 

  38. A. Guichardet, Symmetric Hilbert space and related topics, Lect. Notes in Maths. Springer 261, 1970.

    Google Scholar 

  39. R. Halmos, Introduction to Hilbert space, Chelsea, New York, 1951.

    MATH  Google Scholar 

  40. R.L. Hudson, K.R. Parthasarathy, Quantum Ito's formula and stochastic evolutions, Comm. Math. Phys., 93, 1984, 301–323.

    Article  MathSciNet  MATH  Google Scholar 

  41. R.L. Hudson, K.R. Parthasarathy, Unification of Fermion and Boson stochastic calculus, Comm. Math. Phys., 115, 1988, 47–53

    MathSciNet  MATH  Google Scholar 

  42. R.L. Hudson, K.R. Parthasarathy, Casimir chaos in Boson Fock space, preprint, 1993.

    Google Scholar 

  43. N. Ikeda, S. Watanabe, Stochastic differential equations and diffusion processes, North-Holland, Kodansha, 1981.

    MATH  Google Scholar 

  44. K. Itô, Multiple Wiener integrals, Jour. Math. Soc. Japan. 3, 1951, 157–169.

    Article  MATH  Google Scholar 

  45. M. Jolis, M. Sanz, Integrator properties of the Skorokhod integral, Stochastics, 41, 1992, 163–176.

    MATH  Google Scholar 

  46. J.L. Journé, P.A. Meyer, Une martingale d'opérateurs bornés non représentable en intégrale stochastique, Séminaire de Probabilités XX, Lect. Notes in Maths. Springer 1204, 1986, 313–316.

    MATH  Google Scholar 

  47. M. Lindsay, Quantum and non-causal stochastic calculus, à paraître dans Prob. Theory and rel. Fields.

    Google Scholar 

  48. G. Lion, M. Vergne, The Weil representation, Maslov index, and theta series, Progress in Mathematics, Vol 6, Birkhaüser, 1980. *** DIRECT SUPPORT *** A00I6B63 00002

    Google Scholar 

  49. H. Maassen, Quantum Markov processes on Fock space described by integral kernels, Quantum probability and applications, II, Lect. Notes in Maths. Springer 1136, 1985, 361–374.

    Google Scholar 

  50. P.A. Meyer, Eléments de Probabilités quantiques, Séminaire de Probabilités XX, Lect. Notes in Maths. Springer 1204, 1986, 186–312.

    Google Scholar 

  51. P.A. Meyer, Quantum Probability for Probabilists, Lect. Notes in Maths. Springer 1538, 1993.

    Google Scholar 

  52. P.A. Meyer, Approximation de l'oscillatcur harmonique (d'après L.Accardi et A. Bach), Séminaire de Probabilités XXIII, Lect. Notes in Maths. Springer 1372, 1990, 175–182.

    Google Scholar 

  53. P.A. Meyer, Progrès récent en calcul stochastique quantique, Séminaire Bourbaki, exposé 761, 1992.

    Google Scholar 

  54. A. Mohari, K.R. Parthasarathy, A quantum probabilistic analogue of Feller's condition for the existence of unitary markovian cocycles in Fock space, I.S.I. preprint, 1992.

    Google Scholar 

  55. E. Nelson, Analytic vectors, Ann. Math. 70, 1959, 572–615.

    Article  MathSciNet  MATH  Google Scholar 

  56. J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1951.

    Google Scholar 

  57. J. Neveu, Processus Aléatoires Gaussiens, Presses de l'Université de Montréal, 1968.

    Google Scholar 

  58. J. Neveu, Processus Ponctuels, Ecole d'élé de Probabilités de Saint-Flour VI, Lect. Notes in Maths. Springer 598, 1976, 250–445.

    Google Scholar 

  59. D. Nualart, Non-causal stochastic integrals and calculus, Stochastic analysis and related topics, Lect. Notes in Maths. Springer 1316, 1986, 80–129.

    Google Scholar 

  60. K.R. Parthasarathy, An introduction to quantum stochastic calculus, Monographs in mathematics, Vol 85, Birkhäuser, 1992.

    Google Scholar 

  61. K.R. Parthasarathy, Azéma martingales and quantum stochastic calculus, Proc. R.C. Bose Symposium, Wiley Eastern, 1990, 551–569.

    Google Scholar 

  62. K.R. Parthasarathy, A generalized Biane's process, Séminaire de Probabilités XXIV, Lect. Notes in Maths. Springer 1426, 1990, 345–248.

    Google Scholar 

  63. K.R. Parthasarathy, K.B. Sinha, Representation of bounded martingales in Fock space, Jour. Funct. An., 67, 1986, 126–151

    Article  MathSciNet  MATH  Google Scholar 

  64. K.R. Parthasarathy, K.B. Sinha, Representation of a class of bounded martingales, II, Quantum probability and applications, III, Lect. Notes in Maths. Springer 1303, 1988, 232–250.

    Google Scholar 

  65. K.R. Parthasarathy, K.B. Sinha, Unification of quantum noise processes in Fock spaces, Quantum probability and applications, VI, World Scientific, Singapore, 1991.

    MATH  Google Scholar 

  66. K.R. Parthasarathy, K.B. Sinha, Markov chains as Evans-Hudson diffusions in Fock space, Séminaire de Probabilités XXIV, Lect. Notes in Maths. Springer 1426, 1990, 362–369.

    Google Scholar 

  67. D. Petz, Conditionnal expectations in quantum probability, Quantum probability and applications, III, Lect. Notes in Maths. Springer 1303, 1988, 251–260.

    Google Scholar 

  68. J. Pitman, One dimensionnal brownian motion and the three dimensionnal Bessel process, Adv. Appl. Prob., 7, 1975, 511–526.

    Article  MathSciNet  MATH  Google Scholar 

  69. P. Protter, Stochastic integration and differential equations, a new approach, Springer, 1990.

    Google Scholar 

  70. M. Reed, B. Simon, Methods of modern mathematical physics, II, Fourier analysis and self-adjointness, Academic press, 1970.

    Google Scholar 

  71. S. Reynaud, Introduction à la réduction du bruit quantique, Ann. Phys. Fr. 15, 1990, 63–162.

    Article  Google Scholar 

  72. W. Rudin, Functionnal analysis, Mac Graw Hill, 1973.

    Google Scholar 

  73. J. de Sam Lazaro, P.A. Meyer, Méthodes de martingales et théorie des flots, Zeit. f. Wahr. 18, 1971, 116–140.

    Article  MATH  Google Scholar 

  74. J.L. Sauvageot, Markov quantum semi-groups admit covariant Markov C*-dilations, Comm. Math. Phys. 106, 1986, 91–103.

    Article  MathSciNet  MATH  Google Scholar 

  75. J.L. Sauvageot, Quantum Dirichlet forms, differential calculus and semi-groups, Quantum probability and applications, V, Lect. Notes in Maths. Springer 1442, 1990, 334–346.

    Google Scholar 

  76. J.L. Sauvageot, Semi-groupe de la chaleur transverse sur la C*-algèbre d'un feuilletage riemannien, C.R.A.S. 210, Série I, 1990, 531–536.

    MathSciNet  MATH  Google Scholar 

  77. M. Schürmann, The Azéma martingales as components of quantum independent increment processes, Séminaire de Probabilités XXV, Lect. Notes in Maths. Springer 1485, 1991, 24–30.

    Google Scholar 

  78. J.P. Serre, Algèbres de Lie semi-simples complexes, Benjamin, New York, 1966.

    MATH  Google Scholar 

  79. D. Shale, Linear symmetries of free boson fields, Trans. Am. Math. Soc. 103, 1962, 149–167.

    Article  MathSciNet  MATH  Google Scholar 

  80. A.V. Skorokhod, On a generalisation of a stochastic integral, Teor. Ver. 20, 1975, 219–233.

    MATH  Google Scholar 

  81. R. Speicher, A new example of “independance” and “white noise”, Prob. Th. and Rel. Fields, 84, 1990, 141–159.

    Article  MathSciNet  MATH  Google Scholar 

  82. R. Speicher, Stochastic integration on the full Fock space with the help of a fernel calculus, Publ. R.I.M.S. Kyoto, 27, 1991, 149–184.

    Article  MathSciNet  MATH  Google Scholar 

  83. N.J. Vilenkin, Special functions and the theory of group representations, Translations of the A.M.S., 22, 1968.

    Google Scholar 

  84. G.F. Vincent-Smith, Dilation of a dissipative quantum dynamical system into a quantum Markov process, Proc. Lond. Math. Soc. 49, 1984, 58–72.

    Article  MathSciNet  MATH  Google Scholar 

  85. D. Voiculescu, Free non-commutative random variables, random matrices and the II1 factors of free groups, Quantum probability and applications, VI, World Scientific, Singapore, 1991, 473–487.

    MATH  Google Scholar 

  86. W. von Waldenfels, The Markov process of total spin, Séminaire de Probabilités XXIV, Lect. Notes in Maths. Springer 1426, 1990, 357–361.

    Google Scholar 

  87. S. Watanabe, Stochastic differential equations and Malliavin calculus, Lectures on Mathematics and Physics, 73, Tat Institute of Fundamental Research, 1984.

    Google Scholar 

  88. N. Wiener, The homogeneous chaos, Amer. Jour. Math. 55, 1938, 897–936.

    Article  MathSciNet  MATH  Google Scholar 

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Biane, P. (1995). Calcul stochastique non-commutatif. In: Bernard, P. (eds) Lectures on Probability Theory. Lecture Notes in Mathematics, vol 1608. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0095746

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