Quantum stochastic convolution cocycles III

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.     

On dissipative stochastic equations in a Hilbert space

Summary A general existence and uniqueness theorem for solutions of linear dissipative stochastic differential equation in a Hilbert space is proved. The dual equation is introduced and the duality relation is established. Proofs take inspirations from quantum stochastic calculus, however without using it. Solutions of both equations provide classical stochastic representation for a quantum dynamical semigroup, describing quantum Markovian evolution. The problem of the mean-square norm conservation, closely related to the unitality (non-explosion) of the quantum dynamical semigroup, is considered and a hyperdissipativity condition, ensuring such conservation, is discussed. Comments are given on the existence of solutions of a nonlinear stochastic differential equation, introduced and discussed recently in physical literature in connection with continuous quantum measurement processes.     

Dilations of Contraction Cocycles and Cocycle Perturbations of the Translation Group of the Line

The class of contraction cocycles which can be dilated to unitary Markovian cocycles of a translation group S on the straight line is introduced. The class of cocycle perturbations of S by unitary Markovian cocycles W with the property W _t ? I ∈ S ₂ (the Hilbert—Schmidt class) is investigated. The results are applied to perturbations of Kolmogorov flows on hyperfinite factors generated by the algebra of canonical anticommutation relations.     

On quantum stochastic differential equations

Existence and uniqueness theorems for quantum stochastic differential equations with nontrivial initial conditions are proved for coefficients with completely bounded columns. Applications are given for the case of finite-dimensional initial space or, more generally, for coefficients satisfying a finite localisability condition. Necessary and sufficient conditions are obtained for a conjugate pair of quantum stochastic cocycles on a finite-dimensional operator space to strongly satisfy such a quantum stochastic differential equation. This gives an alternative approach to quantum stochastic convolution cocycles on a coalgebra.     

Minimal isometric dilations of operator cocycles

We obtain existence, uniqueness results for minimal isometric dilations of contractive cocycles of semigroups of unital *-endomorphisms ofB(H. This generalizes the result of Sz. Nagy on minimal isometric dilations of semigroups of contractive operators on a Hilbert space. In a similar fashion we explore results analogus to Sarason's characterization that subspaces to which compressions of semigroups are again semigroups are semi-invariant subspaces, in the context of cocycles and quantum dynamical semigroups.This research is supported by the Indian National Science Academy under Young Scientist Project.     

Construction of some quantum stochastic operator cocycles by the semigroup method

A new method for the construction of Fock-adapted quantum stochastic operator cocycles is outlined, and its use is illustrated by application to a number of examples arising in physics and probability. The construction uses the Trotter-Kato theorem and a recent characterisation of such cocycles in terms of an associated family of contraction semigroups. In celebration of Kalyan Sinha’s sixtieth birthday     

Sesquilinear quantum stochastic analysis in Banach space

A theory of quantum stochastic processes in Banach space is initiated. The processes considered here consist of Banach space valued sesquilinear maps. We establish an existence and uniqueness theorem for quantum stochastic differential equations in Banach modules, show that solutions in unital Banach algebras yield stochastic cocycles, give sufficient conditions for a stochastic cocycle to satisfy such an equation, and prove a stochastic Lie–Trotter product formula. The theory is used to extend, unify and refine standard quantum stochastic analysis through different choices of Banach space, of which there are three paradigm classes: spaces of bounded Hilbert space operators, operator mapping spaces and duals of operator space coalgebras. Our results provide the basis for a general theory of quantum stochastic processes in operator spaces, of which Lévy processes on compact quantum groups is a special case.     

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

Quantum stochastic differential equations of the form

Existence of Feller Cocycles on a C*-Algebra

The quantum stochastic differential equation is considered on a unital C^*-algebra, with separablenoise dimension space. Necessary conditions on the matrix ofbounded linear maps for the existence of a completely positivecontractive solution are shown to be sufficient. It is knownthat for completely positive contraction processes, k satisfiessuch an equation if and only if k is a regular Markovian cocycle.‘Feller’ refers to an invariance condition analogousto probabilistic terminology if the algebra is thought of asa non-commutative topological space. 2000 Mathematics SubjectClassification 81S25, 46L07, 46L53, 47D06.     

Stabilizability of a class of stochastic bilinear hybrid systems

The problem of the stabilizability of stochastic nonlinear hybrid systems with a Markovian or any switching rule is considered. Using the Lyapunov technique sufficient conditions for the asymptotic stabilizability in probability by a smooth controller in every structure are found. In particular, the asymptotic stabilizability in probability problem of stochastic bilinear hybrid systems with a Markovian or any switching rule is discussed and a closed-loop controller is found. Also the sufficient conditions for the exponential mean-square stabilizability for bilinear hybrid systems with any switching based on the Lie algebra approach are formulated and an open-loop controller is designed. The obtained results are illustrated by examples and simulations.     

Evolution Equations for Markov Cocycles Obtained by Second Quantization in the Symplectic Fock Space

We derive a quantum stochastic differential equation satisfied by the unitary Markov cocycles obtained for a model situation during second quantization in the symmetric Fock space. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 1, pp. 186–192, January, 2006.     

On the relation between the singular and the weak coupling limits

After having recalled some definitions concerning quantum stochastic processes and, in particular, quantum Brownian motions, a general scheme is introduced which allows a unified approach to the weak coupling and singular coupling limits. The analogies and differences between the two are discussed. The main difference consists of the fact that, in the singular coupling limit, the use of a Hamiltonian unbounded below seems to be unavoidable, while this is not the case for the weak coupling limit.     

Structure des cocycles markoviens sur l'espace de Fock

Summary We show that strongly continuous unitary Markov cocycles on Fock space are solutions of a quantum stochastic Schrödinger equation and give their explicit form through a decomposition of Fock space on the eigenspaces of the number operator. We also give necessary and sufficient conditions for a generalized Hamiltonian to be the generator of such a cocycle. This generalizes the work of Hudson and Parthasarathy in the norm-continuous case.     

Categorified central extensions, étale Lie 2-groups and Lie?s Third Theorem for locally exponential Lie algebras

Lie?s Third Theorem, asserting that each finite-dimensional Lie algebra is the Lie algebra of a Lie group, fails in infinite dimensions. The modern account on this phenomenon is the integration problem for central extensions of infinite-dimensional Lie algebras, which in turn is phrased in terms of an integration procedure for Lie algebra cocycles.This paper remedies the obstructions for integrating cocycles and central extensions from Lie algebras to Lie groups by generalising the integrating objects. Those objects obey the maximal coherence that one can expect. Moreover, we show that they are the universal ones for the integration problem.The main application of this result is that a Mackey-complete locally exponential Lie algebra (e.g., a Banach–Lie algebra) integrates to a Lie 2-group in the sense that there is a natural Lie functor from certain Lie 2-groups to Lie algebras, sending the integrating Lie 2-group to an isomorphic Lie algebra.     

Quantum and non-causal stochastic calculus

Summary The quantum stochastic calculus initiated by Hudson and Parthasarathy, and the non-causal stochastic calculus originating with the papers of Hitsuda and Skorohod, are two potent extensions of the Itô calculus, currently enjoying intensive development. The former provides a quantum probabilistic extension of Schrödinger's equation, enabling the construction of a Markov process for a quantum dynamical semigroup. The latter allows the treatment of stochastic differential equations which involve terms which anticipate the future. In this paper the close relationship between these theories is displayed, and a noncausal quantum stochastic calculus, already in demand from physics, is described.     

Renormalization as a functor on bialgebras

The Hopf algebra of renormalization in quantum field theory is described at a general level. The products of fields at a point are assumed to form a bialgebra B and renormalization endows T(T(B)⁺), the double tensor algebra of B, with the structure of a noncommutative bialgebra. When the bialgebra B is commutative, renormalization turns S(S(B)⁺), the double symmetric algebra of B, into a commutative bialgebra. The usual Hopf algebra of renormalization is recovered when the elements of S¹(B) are not renormalized, i.e., when Feynman diagrams containing one single vertex are not renormalized. When B is the Hopf algebra of a commutative group, a homomorphism is established between the bialgebra S(S(B)⁺) and the Faà di Bruno bialgebra of composition of series. The relation with the Connes-Moscovici Hopf algebra is given. Finally, the bialgebra S(S(B)⁺) is shown to give the same results as the standard renormalization procedure for the scalar field.     

Lie algebras and recurrence relations IV: Representations of the Euclidean group and Bessel functions

Representations of a complex form of the Lie algebra of the Euclidean group of the plane are found. Bases for the corresponding Hilbert spaces are given in terms of Lommel polynomials and Bessel functions. An element of the Lie algebra is interpreted as a quantum random variable leading to the Lommel distributions. The tails of these distributions and limit theorems are studied. Applications to electronic music are noted, as well as connection with quantum mechanics on a one-dimensional lattice.     

Correspondences of ribbon categories

Much of algebra and representation theory can be formulated in the general framework of tensor categories. The aim of this paper is to further develop this theory for braided tensor categories. Several results are established that do not have a substantial counterpart for symmetric tensor categories. In particular, we exhibit various equivalences involving categories of modules over algebras in ribbon categories. Finally, we establish a correspondence of ribbon categories that can be applied to, and is in fact motivated by, the coset construction in conformal quantum field theory.     

The lagrangian approach to stochastic variational principles on curved manifolds

The classical definition of the action functional, for a dynamical system on curved manifolds, can be extended to the case of diffusion processes. For the stochastic action functional so obtained, we introduce variational principles of the type proposed by Morato. In order to generalize the class of process variations, from the flat case originally given by Morato to general curved manifolds, we introduce the notion of stochastic differential systems. These give a synthetic characterization of the process and its variations as a generalized controlled stochastic process on the tangent bundle of the manifold. The resulting programming equations are equivalent to the quantum Schrödinger equation, where the wave function is coupled to an additional vector potential, satisfying a plasma-like equation with a peculiar dissipative behavior.