Das, Biswarup ; Goswami, Debashish
(2012)
*Quantum Brownian motion on non-commutative manifolds: construction, deformation and exit times*
Communications in Mathematical Physics, 309
(1).
pp. 193-228.
ISSN 0010-3616

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Official URL: http://link.springer.com/article/10.1007/s00220-01...

Related URL: http://dx.doi.org/10.1007/s00220-011-1368-9

## 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 A_{θ}, which seems to behave like a one-dimensional manifold, perhaps reminiscent of the fact that A_{θ} is a non-commutative model of the (locally one-dimensional) ‘leaf-space’ of the Kronecker foliation.

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ID Code: | 102165 |

Deposited On: | 01 Feb 2018 04:02 |

Last Modified: | 01 Feb 2018 04:02 |

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