Quantum Brownian Motion on noncommutative manifolds: construction, deformation and exit times

Details Quantum Brownian Motion on noncommutative manifolds: construction, deformation and exit times Quantum Brownian Motion on noncommutative manifolds: construction, deformation and exit times

2011-01-06

arxiv.org/abs/1101.1200v2

Mathematics

Operator Algebras

Scientific paper

We begin with a review and analytical construction of quantum Gaussian process (and quantum Brownian motions) in the sense of [25],[10] and others, and then formulate and study in details (with a number of interesting examples) a definition of quantum Brownian motions on those noncommutative manifolds (a la Connes) which are quantum homogeneous spaces of their quantum isometry groups in the sense of [11]. We prove that bi-invariant quantum Brownian motion can be 'deformed' in a suitable sense. Moreover, we propose a noncommutative analogue of the well-known asymptotics of the exit time of classical Brownian motion. We explicitly analyze such asymptotics for a specific example on noncommutative two-torus A{\theta}, which seems to behave like a one-dimensional manifold, perhaps reminiscent of the fact that A{\theta} is a noncommutative model of the (locally one-dimensional) 'leaf-space' of the Kronecker foliation.

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