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arXiv:1008.0774 · Communications on Stochastic Analysis

E-semigroups subordinate to CCR flows

Stephen J. Wills

Published Dec 1, 2010• 0 citations
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Abstract

The subordinate E-semigroups of a fixed E-semigroup α are in one-to-one correspondence with local projection-valued cocycles of α.For the CCR flow we characterise these cocycles in terms of their stochastic generators, that is, in terms of the coefficient driving the quantum stochastic differential equation of Hudson-Parthasarathy type that such cocycles necessarily satisfy.In addition various equivalence relations and order-type relations on E-semigroups are considered, and shown to work especially well in the case of those semigroups subordinate to the CCR flows by exploiting our characterisation.

DOI
10.31390/cosa.4.4.08

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The subordinate E-semigroups of a fixed E-semigroup α are in one-to-one correspondence with local projection-valued cocycles of α.For the CCR flow we characterise these cocycles in terms of their stochastic generators, that is, in terms of the coefficient driving the quantum stochastic differential equation of Hudson-Parthasarathy type that such cocycles necessarily satisfy.In addition various equivalence relations and order-type relations on E-semigroups are considered, and shown to work especially well in the case of those semigroups subordinate to the CCR flows by exploiting our characterisation.
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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
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 present, for the first time, the result (from 2008) that (normal, strongly continuous) Markov semigroups on B(G) (G a separable Hilbert space) admit a Hudson-Parthasarathy dilation (that is, a dilation to a cocycle perturbation of a noise) if and only if the Markov semigroup is spatial (that is, if it dominates an elementary CP-semigroup).The proof is by general abstract nonsense (taken from Arveson's classification of E 0 -semigroups on B(H) by Arveson systems up to cocycle conjugacy) and not, as usual, by constructing the cocycle as a solution of a quantum stochastic differential equation in the sense of Hudson and Parthasarathy.All other results that make similar statements (especially, [Mem.Amer.Math.Soc.240 (2016), vi+126 pages, arXiv:0901.1798]) for more general C * -algebras) have been proved later by suitable adaptations of the methods exposed here.(They use Hilbert module techniques, which we carefully avoid here in order to make the result available without any appeal to Hilbert modules.)
We consider the GNS Hilbert space H of a uniformly hyper-finite C * -algebra and study a class of unbounded Lindbladian arises from commutators.Exploring the local structure of UHF algebra, we have shown that the associated Hudson-Parthasarathy type quantum stochastic differential equation admits a unitary solution.The vacuum expectation of homomorphic cocycle, implemented by the Hudson-Parthasarathy flow, is conservative and gives the minimal semi-group associated with the formal Lindbladian.We also associate conservative minimal semi-groups to another class of Lindbladian by solving the corresponding Evan-Hudson equation.