On the Structure of Classical and Quantum Flows

Abstract

In the framework of quantum probability, stochastic flows on manifolds and the interaction representation of quantum physics become unified under the notion ofMarkov cocycle. We prove a structure theorem forσ-weakly continuous Markov cocycles which shows that they are solutions of quantum stochastic differential equations on the largest *-subalgebra, contained in the domain of the generator of the Markov semigroup, canonically associated to the cocycle. The result is applied to prove that any Markov cocycle on the Clifford bundle of a compact Riemannian manifold, whose structure maps preserve the smooth sections and satisfy some natural compatibility conditions, uniquely determines a family of smooth vector fields and a connection, with the property that the cocycle itself is induced by the stochastic flow along the paths of the classical diffusion on the manifold, defined by these vector fields and by the Ito stochastic parallel transport associated to the connection. We use the language and techniques of quantum probability but, even when restricted to the classical case, our results seem to be new.