A classical stochastic process which is Markovian for its past filtration is also Markovian for its future filtration. We show with a counterexample based on quantum liftings of a finite state classical Markov chain that this property cannot hold in the category of expected Markov processes. Using a duality theory for von Neumann algebras with weights, developed by Petz on the basis of previous results by Groh and Kümmerer, we show that a quantum version of this symmetry can be established in the category of weak Markov processes in the sense of Bhat and Parthasarathy. Here time reversal is implemented by an anti-unitary operator and a weak Markov process is time reversal invariant if and only if the associated semigroup coincides with its Petz dual. This construction allows one to extend to the quantum case, both for backward and forward processes, the Misra–Prigogine–Courbage internal time operator and to show that the two operators are intertwined by the time reversal anti-automorphism.

We study asymptotic behavior of a Markov semigroup on a von-Neumann algebra by exploring a maximal von-Neumann subalgebra where the Markov semigroup is an automorphism. This enables us to prove that strong mixing is equivalent to ergodic property for continuous time Markov semigroup on a type-I von-Neumann algebra with center completely atomic. For discrete time dynamics we prove that an aperiodic ergodic Markov semigroup on a type-I von-Neumann algebra with center completely atomic is strong mixing. There exists a tower of isomorphic von-Neumann algebras generated by the weak Markov process and a unique up to isomorphism minimal dilated quantum dynamics of endomorphisms associated with the Markov semigroup. The dilated endomorphism is pure in the sense of Powers if and only if the adjoint Markov semigroup satisfies Kolmogorov property. As an application of our general results we find a necessary and sufficient condition for a translation invariant state on a quantum spin chain to be pure. We also find a tower of type-II 1 factors canonically associated with the canonical conditional expectation on a sub-factor of a type-II 1 factor. This tower of factors unlike Jones's tower do not preserve index. This gives a sequence of Jones's numbers as an invariance for the inclusion of a finite sub-factor of a type-II 1 factor.(H, S i , P, V i , Ω) is a Popescu system [BJKW], where P is the support projection of the state ψ Ω (X) =< Ω, XΩ > on the von-Neumann algebra π(O d ) ′′ , S i = π(s i ) and V i = P S i P . Let Q be the support projection of the state ψ Ω on the von-Neumann algebra {S I S * J : |I| = |J| < ∞} ′′ and A 0 = QS I S * J Q : |I| = |J| < ∞}. We also set l k = QS k Q for all 1 ≤ k ≤ d and define Markov semigroup τ on A 0 by τ (x) = i l i xl * i . The normal state ψ 0 , defined

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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.

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