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Introduction to Random Walks on Noncommutative Spaces

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Quantum Potential Theory

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1954))

Abstract

We introduce several examples of random walks on noncommutative spaces and study some of their probabilistic properties. We emphasize connections between classical potential theory and group representations.

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References

  • [B1] P. Biane, Some properties of quantum Bernoulli random walks. Quantum probability & related topics, 193–203, QP-PQ, VI, World Sci. Publ., River Edge, NJ, 1991.

    Google Scholar 

  • [B2] P. Biane, Quantum random walk on the dual of SU (n). Probab. Theory Related Fields 89 (1991), no. 1, 117–129.

    Article  MATH  MathSciNet  Google Scholar 

  • [B3] P. Biane, Minuscule weights and random walks on lattices. Quantum probability & related topics, 51–65, QP-PQ, VII, World Sci. Publ., River Edge, NJ, 1992.

    Google Scholar 

  • [B4] P. Biane, Équation de Choquet-Deny sur le dual ďun groupe compact. Probab. Theory Related Fields 94 (1992), no. 1, 39–51.

    Article  MATH  MathSciNet  Google Scholar 

  • [B5] P. Biane, Théorème de Ney-Spitzer sur le dual de SU(2). Trans. Amer. Math. Soc. 345 (1994), no. 1, 179–194.

    Article  MATH  MathSciNet  Google Scholar 

  • [B6] P. Biane, Quelques propriétés du mouvement brownien non-commutatif. Hommage à P. A. Meyer et J. Neveu. Astérisque No. 236 (1996), 73–101.

    MathSciNet  Google Scholar 

  • [B7] P. Biane, Le théorème de Pitman, le groupe quantique SUq(2), et une question de P. A. Meyer. In memoriam Paul-André Meyer: Séminaire de Probabilités XXXIX, 61–75, Lecture Notes in Math., 1874, Springer, Berlin, 2006.

    Chapter  Google Scholar 

  • [BPO] P. Biane, P. Bougerol, N. O’Connell, Littelmann paths and Brownian paths. Duke Math. J. 130 (2005), no. 1, 127–167.

    Article  MATH  MathSciNet  Google Scholar 

  • [BtD] T. Bröcker, T. tom Dieck, Representations of compact Lie groups. Graduate Texts in Mathematics, 98. Springer-Verlag, New York, 1985.

    Google Scholar 

  • [Col] B. Collins, Martin boundary theory of some quantum random walks. Ann. Inst. H. Poincaré Probab. Statist. 40 (2004), no. 3, 367–384.

    MATH  Google Scholar 

  • [C] A. Connes, Noncommutative geometry. Academic Press, 1995.

    Google Scholar 

  • [CD] G. Choquet, J. Deny, Sur ľéquation de convolution μ=μ*σ. C. R. Acad. Sci. Paris 250 1960 799–801.

    MATH  MathSciNet  Google Scholar 

  • [D1] J. Dixmier Les C *-algèbres et leurs représentations. Gauthier-Villars. Paris, 1969.

    Google Scholar 

  • [D2] J. Dixmier Les algèbres ďopérateurs dans ľespace hilbertien (algèbres de von Neumann). Gauthier-Villars, Paris, 1969.

    Google Scholar 

  • [GW] R. Goodman, N.R. Wallach, Representations and invariants of the classical groups. Encyclopedia of Mathematics and its Applications, 68. Cambridge University Press, Cambridge, 1998.

    Google Scholar 

  • [H] U. Haagerup, An example of a nonnuclear C *-algebra, which has the metric approximation property. Invent. Math. 50 (1978/79), no. 3, 279–293.

    Article  MathSciNet  Google Scholar 

  • [I] M. Izumi, Non-commutative Poisson boundaries and compact quantum group actions. Adv. Math. 169 (2002), no. 1, 1–57.

    Article  MATH  MathSciNet  Google Scholar 

  • [INT] M. Izumi, S. Neshveyev, L. Tuset, Poisson boundary of the dual of SUq(n). Comm. Math. Phys. 262 (2006), no. 2, 505–531.

    Article  MATH  MathSciNet  Google Scholar 

  • [K] M. Kashiwara, Bases cristallines des groupes quantiques. Rédigé par C. Cochet. Cours Spcialisés, 9. Société Mathmatique de France, Paris (2002).

    Google Scholar 

  • [Ke] H. Kesten, Symmetric random walks on groups. Trans. Amer. Math. Soc. 92 1959 336–354.

    Article  MATH  MathSciNet  Google Scholar 

  • [KKS] J.G. Kemeny, J.L. Snell, A.W. Knapp, Denumerable Markov chains. Second edition. With a chapter on Markov random fields, by David Griffeath. Graduate Texts in Mathematics, No. 40. Springer-Verlag, New York-Heidelberg-Berlin, 1976.

    Google Scholar 

  • [KV] A.U. Klimyk, N.Ja. Vilenkin, Representation of Lie groups and special functions. Vol. 2. Class I representations, special functions and integral transforms. Mathematics and its Applications (Soviet Series), 74. Kluwer Academic Publishers Group, Dordrecht (1993).

    Google Scholar 

  • [NS] P. Ney, F. Spitzer, The Martin boundary for random walk. Trans. Amer. Math. Soc. 121 1966 116–132.

    Article  MATH  MathSciNet  Google Scholar 

  • [Pa] K.R. Parthasarathy, An introduction to quantum stochastic calculus. Monographs in Mathematics, 85. Birkhuser Verlag, Basel, 1992.

    Google Scholar 

  • [Pi] J.W. Pitman, One-dimensional Brownian motion and the three-dimensional Bessel process. Advances in Appl. Probability 7 (1975), no. 3, 511–526.

    Article  MATH  MathSciNet  Google Scholar 

  • [PS] K.R. Parthasarathy, K. Schmidt, Positive definite kernels, continuous tensor products, and central limit theorems of probability theory. Lecture Notes in Mathematics, Vol. 272. Springer-Verlag, Berlin-New York, 1972.

    Chapter  Google Scholar 

  • [R] D. Revuz, Markov chains. Second edition. North-Holland Mathematical Library, 11. North-Holland Publishing Co., Amsterdam, 1984.

    MATH  Google Scholar 

  • [Rie] M.A. Rieffel, Gromov-Hausdorff distance for quantum metric spaces. Matrix algebras converge to the sphere for quantum Gromov-Hausdorff distance. Mem. Amer. Math. Soc. 168 (2004), no. 796. American Mathematical Society, Providence, RI, 2004.

    Google Scholar 

  • [S] J.L. Sauvageot, Markov quantum semigroups admit covariant Markov C *-dilations. Comm. Math. Phys. 106 (1986), no. 1, 91–103.

    Article  MATH  MathSciNet  Google Scholar 

  • [Sc] M. Schürmann, White noise on bialgebras. Lecture Notes in Mathematics, 1544. Springer-Verlag, Berlin, 1993.

    Google Scholar 

  • [T] M. Takesaki, Theory of operator algebras. I, II, III. Encyclopaedia of Mathematical Sciences, 124, 125, 127 Operator Algebras and Non-commutative Geometry, 5, 6, 8. Springer-Verlag, Berlin, 2002/2003.

    Google Scholar 

  • [Z] D.P. Želobenko, Compact Lie groups and their representations. Translated from the Russian by Israel Program for Scientific Translations. Translations of Mathematical Monographs, Vol. 40. American Mathematical Society, Providence, R.I., 1973.

    Google Scholar 

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Biane, P. (2008). Introduction to Random Walks on Noncommutative Spaces. In: Franz, U., Schürmann, M. (eds) Quantum Potential Theory. Lecture Notes in Mathematics, vol 1954. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69365-9_3

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