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

TL;DR

This paper develops a framework for quantum Brownian motions on noncommutative manifolds, demonstrating deformation properties and analyzing exit time asymptotics, with explicit examples on the noncommutative two-torus.

Contribution

It introduces a definition of quantum Brownian motion on noncommutative manifolds and shows how bi-invariant processes can be deformed, with applications to noncommutative geometry.

Findings

Quantum Brownian motions can be constructed on noncommutative manifolds.

Bi-invariant quantum Brownian motion admits deformation.

Exit time asymptotics on noncommutative two-torus resemble one-dimensional behavior.

Abstract

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.