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Operator Markovian Cocycles via Associated Semigroups

J. Martin Lindsay, Stephen J. Wills

Published Dec 16, 2005• 1 citation
OpenAlex

Abstract

Abstract. A recent characterisation of Fock-adapted contraction operator stochastic cocycles on a Hilbert space, in terms of their associated semigroups, yields a general principle for the construction of such cocycles by approximation of their stochastic generators. This leads to new existence results for quantum stochastic differential equations. We also give necessary and sufficient conditions for a cocycle to satisfy such an equation. In this paper we study the functional equation V0 = I, Vr+t = Vrσr(Vt) for all r, t ≥ 0. for a family of contractions on h ⊗ F adapted to the Fock operator filtration. Here F is the symmetric Fock space over L 2 (R+; k), h and k are fixed but arbitrary

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Quantum flows

Franco Fagnola

We develop a rather general framework for constructing quantum Markov processes through Markov operator cocycles (see Chapter 2, Section 3) that satisfy a quantum stochastic differential equation.In order to achieve this goal we first recall the basic facts of Boson Fock quantum stochastic calculus and then give the fundamental results in the theory of quantum stochastic differential equations concerning existence, uniqueness, time reversal, isometricity and coisometricity of solutions. Next we construct the quantum flow associated with a Markov operator cocycle, and give a condition that guarantees that the restriction to a commutative subalgebra is a commutative flow.

Proyecciones (Antofagasta)•Apr 4, 2018•1 citation
Abstract. A recent characterisation of Fock-adapted contraction operator stochastic cocycles on a Hilbert space, in terms of their associated semigroups, yields a general principle for the construction of such cocycles by approximation of their stochastic generators. This leads to new existence results for quantum stochastic differential equations. We also give necessary and sufficient conditions for a cocycle to satisfy such an equation. In this paper we study the functional equation V0 = I, Vr+t = Vrσr(Vt) for all r, t ≥ 0. for a family of contractions on h ⊗ F adapted to the Fock operator filtration. Here F is the symmetric Fock space over L 2 (R+; k), h and k are fixed but arbitrary
An operator space analysis of quantum stochastic cocycles is undertaken. These are cocycles with respect to an ampliated CCR flow, adapted to the associated filtration of subspaces, or subalgebras. They form a noncommutative analog of stochastic semigroups in the sense of Skorohod. One-to-one correspondences are established between classes of cocycle of interest and corresponding classes of one-parameter semigroups on associated matrix spaces. Each of these “global” semigroups may be viewed as the expectation semigroup of an associated quantum stochastic cocycle on the corresponding matrix space. Proof of the two key characterizations, namely that of completely positive contraction cocycles on a C*-algebra, and contraction cocycles on a Hilbert space, involves all of the analysis undertaken here. As indicated by Accardi and Kozyrev, the Schur-action matrix semigroup viewpoint circumvents technical (domain) limitations inherent in the theory of quantum stochastic differential equations.
Abstract. We consider normal Markovian cocycles on a von Neumann al-gebra which are adapted to a Fock filtration. Every such cocycle k which is Markov-regular and consists of completely positive contractions is realised as a conditioned ∗-homomorphic cocycle. This amounts to a stochastic generalisa-tion of a recent dilation result for norm-continuous normal completely positive contraction semigroups. To achieve this stochastic dilation we use the fact that k is governed by a quantum stochastic differential equation whose coef-ficient matrix has a specific structure, and extend a technique for obtaining stochastic flow generators from Markov semigroup generators, to the context of cocycles. Number/exchange-free dilatability is seen to be related to locality in the case where the cocycle is a Markovian semigroup. In the same spirit unitary dilations of Markov-regular contraction cocycles on a Hilbert space are also described. The paper ends with a discussion of connections with measure-valued diffusion.
Every Markov-regular quantum Levy process on a multiplier C*-bialgebra is shown to be equivalent to one governed by a quantum stochastic differential equation, and the generating functionals of norm-continuous convolution semigroups on a multiplier C*-bialgebra are then completely characterised. These results are achieved by extending the theory of quantum Levy processes on a compact quantum group, and more generally quantum stochastic convolution cocycles on a C*-bialgebra, to locally compact quantum groups and multiplier C*-bialgebras. Strict extension results obtained by Kustermans, together with automatic strictness properties developed here, are exploited to obtain existence and uniqueness for coalgebraic quantum stochastic differential equations in this setting. Then, working in the universal enveloping von Neumann bialgebra, we characterise the stochastic generators of Markov-regular, *-homomorphic (respectively completely positive and contractive), quantum stochastic convolution cocycles.
We develop a rather general framework for constructing quantum Markov processes through Markov operator cocycles (see Chapter 2, Section 3) that satisfy a quantum stochastic differential equation.In order to achieve this goal we first recall the basic facts of Boson Fock quantum stochastic calculus and then give the fundamental results in the theory of quantum stochastic differential equations concerning existence, uniqueness, time reversal, isometricity and coisometricity of solutions. Next we construct the quantum flow associated with a Markov operator cocycle, and give a condition that guarantees that the restriction to a commutative subalgebra is a commutative flow.