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 of Markov cocycle. We prove a structure theorem for _-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. AcademicPress, Inc.

Markovian Cocycles on Operator Algebras Adapted to a Fock Filtration

“…where s, t 0, extending and refining the work in [HL2,Jou,AJL,Fa2,Bra,AcM]. A solution of (0.1) consists of a family of operators on h F + .…”

Boson fock representations of stochastic processes

“…This fact allows not only to generalize known results of classical probability theory, but also to obtain new results by looking at classical objects from a nonclassical point of view. Several papers in the last years have shown that this program can be substantiated in numerous ways in the case of classical diffusions on R n [2][3][4], or on manifolds [5], and for general birth and death processes [4]. In the present note we initiate a similar program for stationary processes.…”

Quantum stochastic convolution cocycles I

“…Thus, for example, a (1) ⊗ a (2) ⊗ a (3) becomes a neutral notation for the effect of (1.1) on an element a. The Fundamental Theorem on Coalgebras states that the coalgebra generated by a finite subset of a coalgebra is necessarily finite dimensional.…”

“…In turn such cocycles may often be shown to have stochastic generators in the sense that they satisfy an exponential Itô type stochastic differential equation. In the context of quantum stochastic calculus [19,32,31,24] this has been studied in [18,20,2,8,10,3,27,29], and in an abstract context of noncommutative white noise in [16]. Quantum Lévy processes [5,34,35,13,36,31,12,11] are natural noncommutative analogues of stochastic semigroups [38].…”