Abstract
We begin with a review and analytical construction of quantum Gaussian process (and quantum Brownian motions) in the sense of Franz (The Theory of Quantum Levy Processes, http://arxiv.org/abs/math/0407488v1 [math.PR], 2009), Schürmann (White noise on bioalgebras. Volume 1544 of Lecture Notes in Mathematics. Berlin: Springer-Verlag, 1993) and others, and then formulate and study in details (with a number of interesting examples) a definition of quantum Brownian motions on those non-commutative manifolds (a la Connes) which are quantum homogeneous spaces of their quantum isometry groups in the sense of Goswami (Commun Math Phys 285(1):141–160, 2009). We prove that bi-invariant quantum Brownian motion can be ‘deformed’ in a suitable sense. Moreover, we propose a non-commutative analogue of the well-known asymptotics of the exit time of classical Brownian motion. We explicitly analyze such asymptotics for a specific example on non-commutative two-torus \({\mathcal{A}_\theta}\) , which seems to behave like a one-dimensional manifold, perhaps reminiscent of the fact that \({\mathcal{A}_\theta}\) is a non-commutative model of the (locally one-dimensional) ‘leaf-space’ of the Kronecker foliation.
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References
Attal, S., Sinha, K. B.: Stopping semimartingales on Fock space. In: Quantum probability communications, QP-PQ, X, River Edge, NJ: World Sci. Publ., 1998, pp. 171–185
Banica T., Goswami D.: Quantum isometries and noncommutative spheres. Commun. Math. Phys. 298(2), 343–356 (2009)
Barnett, C., Wilde, I.F.: Quantum stopping-times. In: Quantum probability & related topics, QP-PQ, VI, River Edge, NJ: World Sci. Publ., 1991, pp. 127–135
Bhowmick, J.: Quantum isometry groups. Phd Thesis. http://arxiv.org/abs/0907.0618v1 [math.OA], 2009
Bhowmick J., Goswami D.: Quantum isometry groups: examples and computations. Commun. Math. Phys. 285(2), 421–444 (2009)
Boca, F.P.: Ergodic actions of compact matrix pseudogroups on C*-algebras. In: Recent Advances in Operator Algebras (Orleans, 1992). Asterisque, No. 232, 93–109 (1995)
Bhowmick J., Goswami D.: Quantum isometry groups: examples and computations. Commun. Math. Phys. 285(2), 421–444 (2009)
Connes, A.: Noncommutative geometry. San Diego, CA: Academic Press Inc., 1994
Connes A., Dubois-Violette M.: Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples. Commun. Math. Phys. 230(3), 539–579 (2002)
Davidson, K.R.: C*-algebras by example. Volume 6 of Fields Institute Monographs. Providence, RI: Amer. Math. Soc., 1996
Franz, U.: The Theory of Quantum Levy Processes. Habilitation thesis EMAU Greifswald. http://arxiv.org/abs/math/0407488v1 [math.PR], 2009
Goswami D.: Quantum group of isometries in classical and noncommutative geometry. Commun. Math. Phys. 285(1), 141–160 (2009)
Gray A.: The volume of a small geodesic ball of a Riemannian manifold. Michigan Math. J. 20, 329–344 (1974)
Itô K.: Brownian motions in a Lie group. Proc. Japan Acad. 26(8), 4–10 (1950)
Liao M., Zheng W.A.: Radial part of Brownian motion on a Riemannian manifold. Ann. Probab. 23(1), 173–177 (1995)
Liao, M.: Lévy processes in Lie groups. Volume 162 of Cambridge Tracts in Mathematics. Cambridge: Cambridge University Press, 2004
Maes A., Van Daele A.: Notes on compact quantum groups. Nieuw Arch. Wisk. (4) 16(1-2), 73–112 (1998)
Goswami D., Sahu L.: Invariants for Normal Completely Positive Maps on the Hyperfinite I I 1 Factor. Proc. Ind. Acad. Sci. (Math. Sci.) 116(4), 411–422 (2006)
Mohari A., Sinha K.B.: Quantum stochastic flows with infinite degrees of freedom and countable state Markov processes. Sankhyā Ser. A 52(1), 43–57 (1990)
Parthasarathy, K.R.: An introduction to quantum stochastic calculus. Volume 85 of Monographs in Mathematics. Basel: Birkhäuser Verlag, 1992
Parthasarathy, K.R., Sinha, K.B.: Stop times in Fock space stochastic calculus. In: Proceedings of the 1st World Congress of the Bernoulli Society, Vol. 1 (Tashkent, 1986), Utrecht: VNU Sci. Press, 1987, pp. 495–498
Parthasarathy, K.R., Sunder, V.S.: Exponentials of indicator functions are total in the boson Fock space Γ(L 2[0, 1]). In: Quantum probability communications, QP-PQ, X, River Edge, NJ: World Sci. Publ., 1998, pp. 281–284
Pinsky M.A.: Mean exit time from a bumpy sphere. Proc. Amer. Math. Soc. 122(3), 881–883 (1994)
Podleś P.: Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU(2) and SO(3) groups. Commun. Math. Phys 170(1), 1–20 (1995)
Rieffel, M.A.: Deformation quantization for actions of R d. Mem. Amer. Math. Soc. 106(506) (1993)
Schürmann, M.: White noise on bialgebras. Volume 1544 of Lecture Notes in Mathematics. Berlin: Springer-Verlag, 1993
Sinha, K.B., Goswami, D.: Quantum stochastic processes and noncommutative geometry. Volume 169 of Cambridge Tracts in Mathematics. Cambridge: Cambridge University Press, 2007
Skeide, M.: Indicator functions of intervals are totalizing in the symmetric Fock space. In: Accardi, L., Kuo, H.-H., Obata, N., Saito, K., Si, S., Streit, L., eds. Trends in Contemporary Infinite Dimensional Analysis and Quantum Probability. Volume in honour of Takeyuki Hida, Istituto Italiano di Cultura (ISEAS), Kyoto 2000 (Rome, Volterra-Preprint 1999/0395), 1999
Stroock, D.W., Varadhan, S.R.S.: Multidimensional diffusion processes. Classics in Mathematics. Berlin: Springer-Verlag, 2006, Reprint of the 1997 edition
Wang S.: Deformations of compact quantum groups via Rieffel’s quantization. Commun. Math. Phys. 178(3), 747–764 (1996)
Yosida K.: A characterization of the second order elliptic differential operators. Proc. Japan Acad. 31, 406–409 (1955)
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Communicated by A. Connes
Research partially supported by Indian National Science Academy and Dept. of Science and Technology, Govt. of India (Swarnajayanti Fellowship).
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Das, B., Goswami, D. Quantum Brownian Motion on Non-Commutative Manifolds: Construction, Deformation and Exit Times. Commun. Math. Phys. 309, 193–228 (2012). https://doi.org/10.1007/s00220-011-1368-9
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DOI: https://doi.org/10.1007/s00220-011-1368-9