On quantum stochastic differential equations

Lindsay, J. Martin; Skalski, Adam

doi:10.1016/j.jmaa.2006.07.105

Cited by 8 publications

(9 citation statements)

References 20 publications

(37 reference statements)

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“…A detailed summary of the relevant results from QS analysis [18][19][20][21]15] is given in [16]. We shall therefore be brief here.…”

Section: Quantum Stochastic Processes Differential Equations and Cocmentioning

confidence: 99%

“…Recall the notation for the solution of a QS differential (ii) The first identity expresses the general relation between k φ,η and k φ [15]. By where (σ s ) s≥0 is the injective *-homomorphic semigroup of right shifts on B(F) and the identification…”

Section: Coalgebraic Quantum Stochastic Differential Equationsmentioning

confidence: 99%

“…Our essential strategy for analysing QS convolution cocycles is to work in the universal enveloping von Neumann bialgebra B and, by transferring between convolution and standard QS cocycles using the maps R σ and E σ , to apply the theory developed in [18][19][20][21], and [15].…”

Section: Coalgebraic Quantum Stochastic Differential Equationsmentioning

confidence: 99%

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Quantum stochastic convolution cocycles III

Lindsay

Skalski

2011

Math. Ann.

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Every Markov-regular quantum Lévy 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 Lévy 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. Mathematics Subject Classification (2000)Primary 46L53 · 81S25; Secondary 22A30 · 47L25 · 16W30 IntroductionLet G be a locally compact quantum semigroup. The main results of this paper are as follows: a concrete realisation of each abstract quantum Lévy process on G

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“…A detailed summary of the relevant results from QS analysis [18][19][20][21]15] is given in [16]. We shall therefore be brief here.…”

Section: Quantum Stochastic Processes Differential Equations and Cocmentioning

confidence: 99%

Section: Coalgebraic Quantum Stochastic Differential Equationsmentioning

confidence: 99%

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Quantum stochastic convolution cocycles III

Lindsay

Skalski

2011

Math. Ann.

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“…The above facts are essentially contained in [35], supplemented by [32]; the minor modifications needed for the present generality are explained in [28].…”

Section: Hölder Continuous Cocyclesmentioning

confidence: 99%

How to differentiate a quantum stochastic cocycle

Lindsay¹

2010

COSA

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Abstract. Two new approaches to the infinitesimal characterisation of quantum stochastic cocycles are reviewed. The first concerns mapping cocycles on an operator space and demonstrates the role of Hölder continuity; the second concerns contraction operator cocycles on a Hilbert space and shows how holomorphic assumptions yield cocycles enjoying an infinitesimal characterisation which goes beyond the scope of quantum stochastic differential equations.

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“…for all t 0, ζ, ζ ∈ k and v ∈ V, then k is called a strong solution. The equation (1•2) has a unique weakly regular weak solution, which is also a strong solution ( [16], [18]). We denote it by k θ,φ , or simply k φ if V = W and θ = id W .…”

Section: Qs Differential Equations and Standard Qs Cocyclesmentioning

confidence: 99%

Completely positive quantum stochastic convolution cocycles and their dilations

Skalski¹

2007

Math. Proc. Camb. Phil. Soc.

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Abstract. Stochastic generators of completely positive and contractive quantum stochastic convolution cocycles on a C * -hyperbialgebra are characterised. The characterisation is used to obtain dilations and stochastic forms of Stinespring decomposition for completely positive convolution cocycles on a C * -bialgebra.Stochastic (or Markovian) cocycles on operator algebras are basic objects of interest in quantum probability ([Acc]) and have been extensively investigated using quantum stochastic analysis (see [Lin]). There is also a well-developed theory of quantum Lévy processes, that is stationary, independent-increment, *-homomorphic processes on a *-bialgebra (see [Sch], [Fra] and references therein). Close examination of these two directions has naturally led to the notion of quantum stochastic convolution cocycle on a quantum group (or, more generally, on a coalgebra), as introduced and investigated in [ . Compact quantum hypergroups differ from compact quantum groups in that their coproduct need not be multiplicative. However, it remains completely positive, which makes compact quantum hypergroups, or more generally C * -hyperbialgebras, an appropriate category for the consideration of completely positive quantum stochastic convolution cocycles in a topological context (for the purely algebraic case see [FrS]). These cocycles may be viewed as natural counterparts of stationary, independent-increment processes on hypergroups. In [LS 4 ] it is shown that, under certain regularity conditions, they satisfy coalgebraic quantum stochastic differential equations.The aim of this paper is to prove dilation theorems for quantum stochastic convolution cocycles on a C * -bialgebra. To this end it is first necessary to establish the detailed structure of the stochastic generators of completely positive and contractive convolution cocycles. We give a direct derivation of this exploiting ideas used in the analysis of standard quantum stochastic cocycles with finite-dimensional noise space ([LiP]). Once the structure of generators is known, one may consider question of dilating completely positive convolution cocycles to * -homomorphic ones. In the context of standard quantum stochastic cocycles this problem was treated in [GLSW] and [GLW] (see also [Bel]). In the first of these papers it was shown

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On quantum stochastic differential equations

Cited by 8 publications

References 20 publications

Quantum stochastic convolution cocycles III

Quantum stochastic convolution cocycles III

How to differentiate a quantum stochastic cocycle

Completely positive quantum stochastic convolution cocycles and their dilations

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