Title: Quantum Brownian motion and exit time asymptotics
Abstract:
In the recent years, there has been a number of non-commutative
generalization of probability theory. One such generalization is called
Quantum probability which stems from the observation that a large number
of classical stochastic processes can be realized as family of operators
(possibly unbounded) on symmetric (sometimes called Boson) Fock space,
consisting of linear combinations of four fundamental operators :
creation, annihilation, number and time.
Noncommutative geometry can be thought of as a non-commutative
generalization of classical Riemannian geometry. Amongst the available
approach to the subject, one approach is the `spectral triple
formulation' which was introduced by Allain Connes. Roughly speaking, a
spectral triple is a set of objects which encodes geometric data in an
analytic way.
Within the framework of classical Riemannian geometry and probability
theory, there exists a remarkable connection between the two: stochastic
geometry. Many geometric invariants on a compact Riemannian manifold can
be read from the asymptotic expansion of the expected exit time of a
Brownian motion from a small metric ball in the manifold.
The aim of this talk will be an attempt to do similar things with
Noncommutative geometry and Quantum probability. We will give a
definition of quantum Brownian motion on spectral triples and then we
will give a possible definition of exit time of the quantum Brownian
motion and do computations similar to the classical set-up.