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.
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.
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
Finding dilations of Markov semigroups to cocycle perturbations of noises is a fundamental problem of quantum probablity and quantum dynamics. A noise is hereby a reversible quantum dynamics with a filtration that is independent in some notion of quantum independence. The noise is a dilation of the trivial Markov semigroup that is turned into a dilation of a nontrivial Moarkov semigroup by perturbation with a unitary cocycle. In 2004, we pointed out that, in a sense, every noise has a filtration that is conditional (or amalgamated, or operator-valued)) monotone independent; it is a monotone noise. But there may be filtration that are independent on other notions of independences, for instance, in conditional (or amalgamated) free independence. In 2009, we proved that a Markov semigroup admits a dilation to a cocycle perturbation of a noise if (and, obviously, only if) it is spatial. A Markov semigroup is spatial if it "dominates ̨ an ̨elementary ́CP-semigroup. Spatiality of as Markov semigroup is reflected by its GNS product system and, in fact, in the construction of the dilation the GNS-system and the classification of E0-semigroups by their product systems up to ́stable ́cocycle conjugacy plays a crucial role. But this construction is "gereral abstract nonsense". In particular, it is unclear if the unitrary cocycle is adapted to a suitable independent filtration or if it fulfills a quantum stochastic differential equation. After recalling some details about these facts, we introduce free product systems an illustrate how to contruct a conditional (or amalgamated) free noise from every free productr system. We show how every spatial (tensor!) product system generates a free product system. For instance, as we pointed out in 2001, the time ordered product systems (that is, the spatial type I systems) generate the free product system of full Fock modules with the same index. For uniformly continuous continuous Markov semigroups, we constructed suitable unitary cocycles with our Quantum stochastic calculus on the full Fock module from 2000. Here, we ask if it is possible to do the same, purely algebraically without any caclulus, starting from the GNS-systsem of an arbitrary spatial Markov semigroup.
A characterisation of the generators of quantum stochastic cocycles of completely positive (CP) maps is given in terms of the complete dissipativity (CD) of its form-generator. The pseudo-Hilbert dilation of the stochastic form-generator and the pre-Hilbert dilation of the corresponding dissipator is found. The general form of the linear continuous structural maps for the algebra of all bounded operators is derived and the quantum stochastic flow for the corresponding cocycle is outlined. It is proved that any w*-analytical bounded CD form-generator give rise to a quantum stochastic CP cocycle over a von Neumann algebra.
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.
This is a continuation of the study of the theory of quantum stochastic dilation of completely positive semigroups on a von Neumann or C∗ algebra, here with unbounded generators. The additional assumption of symmetry with respect to a semifinite trace allows the use of the Hilbert space techniques, while the covariance gives rise to better handle on domains. An Evans-Hudson flow is obtained, dilating the given semigroup.