Markovian Cocycles on Operator Algebras Adapted to a Fock Filtration

Lindsay, J. Martin; Wills, Stephen J.

doi:10.1006/jfan.2000.3658

Cited by 39 publications

(40 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(i) When Z is an E (S,S) -weak solution of (7.1), Z is also an E (S,S) -weak solution of a time reversed version of the differential equation (7.1). This is better described by means of two time parameters (see [LW2]). …”

Section: H-p Processesmentioning

confidence: 99%

See 1 more Smart Citation

Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise

Lindsay¹,

Wills²

2000

Probab Theory Relat Fields

Self Cite

View full text Add to dashboard Cite

Abstract. Quantum stochastic differential equations of the form dkt = kt • θ α β dΛ β α (t) govern stochastic flows on a C * -algebra A. We analyse this class of equation in which the matrix of fundamental quantum stochastic integrators Λ is infinite dimensional, and the coefficient matrix θ consists of bounded linear operators on A. Weak and strong forms of solution are distinguished, and a range of regularity conditions on the mapping matrix θ are considered, for investigating existence and uniqueness of solutions. Necessary and sufficient conditions on θ are determined, for any sufficiently regular weak solution k to be completely positive. The further conditions on θ for k to also be a contraction process are found; and when A is a von Neumann algebra and the components of θ are normal, these in turn imply sufficient regularity for the equation to have a strong solution. Weakly multiplicative and * -homomorphic solutions and their generators are also investigated. We then consider the right and left Hudson-Parthasarathy equations:, in which F is a matrix of bounded Hilbert space operators. Their solutions are interchanged by a time reversal operation on processes. The analysis of quantum stochastic flows is applied to obtain characterisations of the generators F of contraction, isometry and coisometry processes. In particular weak solutions that are contraction processes are shown to have bounded generators, and to be necessarily strong solutions. IntroductionThe quantum stochastic differential equation (QSDE)

show abstract

Section: H-p Processesmentioning

confidence: 99%

“…A converse question to the main concern of the present paper is the algebraic characterisation of the collection of processes on an algebra which satisfy a QSDE of the form (0.1). This is analysed in a sister paper [LW2], where the semigroup representation again plays an important role.…”

Section: Introductionmentioning

confidence: 99%

Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise

Lindsay¹,

Wills²

2000

Probab Theory Relat Fields

Self Cite

View full text Add to dashboard Cite

show abstract

“…We will identify bounded densely defined operators with their continuous extension, and consequently write P b (h, k), P b (A, k) and P cb (A, k) for the space of bounded processes on h, bounded processes on A and completely bounded processes on A respectively. For a von Neumann algebra M acting on h, if k ∈ P cb (M, k) is normal then it is a Markovian cocycle if it satisfies LW2]). Let k ∈ P b (A, k) be a completely positive contraction process on a unital C * -algebra A acting on h. Then the following are equivalent:…”

Section: [Lw1]) However If a Is A Von Neumann Algebra And θ Is Compmentioning

confidence: 99%

“…In this paper we consider completely positive Markovian cocycles on a von Neumann algebra which are adapted to a Fock filtration ([LW2]). We show that every such cocycle k which is sufficiently regular satisfies k t (a) = W * t j t (a)W t (in which k is a certain extension of k) where j is a * -homomorphic Markovian cocycle, and W is a process affiliated to the algebra and satisfying a quantum stochastic differential equation of the form…”

Section: Introductionmentioning

confidence: 99%

A stochastic Stinespring Theorem

2001

Self Cite

View full text Add to dashboard Cite

Abstract. Completely positive Markovian cocycles on a von Neumann algebra, adapted to a Fock filtration, are realised as conjugations of * -homomorphic Markovian cocycles. The conjugating processes are affiliated to the algebra, and are governed by quantum stochastic differential equations whose coefficients evolve according to the * -homomorphic process. Some perturbation theory for quantum stochastic flows is developed in order to achieve the above Stinespring decomposition. To appear in Mathematische Annalen IntroductionThe classical Stinespring Theorem ([Sti]) states that every bounded completely positive linear map T from a unital C * -algebra A into the algebra of bounded operators on a Hilbert space H may be decomposed asfor a unital representation (π, K) of A and a bounded operator V from H to K. A decomposition of the same form holds for linear maps T : A → L(D; H), in which D is a subspace of H which is not necessarily closed. In this case V ∈ L(D; K) and is unbounded if T is. Complete positivity for such a map means that for any finite collection {(a i , ξ i )} from A × D,In this paper we consider completely positive Markovian cocycles on a von Neumann algebra which are adapted to a Fock filtration ([LW2]). We show that every such cocycle k which is sufficiently regular satisfies k t (a) = W * t j t (a)W t (in which k is a certain extension of k) where j is a * -homomorphic Markovian cocycle, and W is a process affiliated to the algebra and satisfying a quantum stochastic differential equation of the form). To achieve this continuous time stochastic Stinespring decomposition it may be necessary to enlarge the dimension of the quantum noise driving the given process k. Interest in completely positive stochastic flows and their structure comes from several sources. They arise naturally in the quantum theory of filtering ([Be1]); they will be required for any quantum theory of measure-valued diffusions (

show abstract

“…For situations where the expectation semigroup is not norm continuous, the characterization problem is discussed in [5,1]. In [10,11], by extended semigroup methods, Lindsay and Wills have studied such problems for Fock adapted contractive operator cocycles and completely positive cocyles.…”

Section: §1 Introductionmentioning

confidence: 99%

Unitary Processes with Independent Increments and Representations of Hilbert Tensor Algebras

Sahu¹,

Schürmann²,

Sinha³

2009

Publ. Res. Inst. Math. Sci.

View full text Add to dashboard Cite

The aim of this article is to characterize unitary increment process by a quantum stochastic integral representation on symmetric Fock space. Under certain assumptions we have proved its unitary equivalence to a Hudson-Parthasarathy flow. §1. IntroductionIn the framework of the theory of quantum stochastic calculus developed by pioneering work of Hudson and Parthasarathy [6], quantum stochastic differential equations (qsde) of the form

show abstract

Markovian Cocycles on Operator Algebras Adapted to a Fock Filtration

Cited by 39 publications

References 24 publications

Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise

Existence, positivity and contractivity for quantum stochastic flows with infinite dimensional noise

A stochastic Stinespring Theorem

Unitary Processes with Independent Increments and Representations of Hilbert Tensor Algebras

Contact Info

Product

Resources

About