J. Martin Lindsay

Quantum Stochastic Analysis – an Introduction

By quantum stochastic analysis is meant the analysis arising from the natural operator filtration of a symmetric Fock space over a Hilbert space of squareintegrable vector-valued functions on the positive half-line. Current texts on quantum stochastics are the monograph [Par], the lecture notes [Mey], the St. Flour lectures [Bia], and the Grenoble lectures [Hud]. Excellent background together with a wealth of examples may be found in these, each of which has its own emphasis. The point of view of these notes is closest to [Bia], as far as the basic construction of quantum stochastic integrals goes. Beyond that, particular emphasis is given to Markovian cocycles.

Quantum and non-causal stochastic calculus

SummaryThe 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 Schrödingers 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.

Existence of Feller Cocycles on a C*-Algebra

The quantum stochastic differential equation dk t = k t ∘ θ α β d Λ β α ( t ) is considered on a unital C *-algebra, with separable noise dimension space. Necessary conditions on the matrix of bounded linear maps θ for the existence of a completely positive contractive solution are shown to be sufficient. It is known that for completely positive contraction processes, k satisfies such an equation if and only if k is a regular Markovian cocycle. ‘Feller’ refers to an invariance condition analogous to probabilistic terminology if the algebra is thought of as a non-commutative topological space.

Quantum Stochastic Convolution Cocycles II

Schürmann’s theory of quantum Lévy processes, and more generally the theory of quantum stochastic convolution cocycles, is extended to the topological context of compact quantum groups and operator space coalgebras. Quantum stochastic convolution cocycles on a C*-hyperbialgebra, which are Markov-regular, completely positive and contractive, are shown to satisfy coalgebraic quantum stochastic differential equations with completely bounded coefficients, and the structure of their stochastic generators is obtained. Automatic complete boundedness of a class of derivations is established, leading to a characterisation of the stochastic generators of *-homomorphic convolution cocycles on a C*-bialgebra. Two tentative definitions of quantum Lévy process on a compact quantum group are given and, with respect to both of these, it is shown that an equivalent process on Fock space may be reconstructed from the generator of the quantum Lévy process. In the examples presented, connection to the algebraic theory is emphasised by a focus on full compact quantum groups.

Superderivations and symmetric Markov semigroups

Unbounded superderivations are used to construct non-commutative elliptic operators on semi-finite von Neumann algebras. The method exploits the interplay between dynamical semigroups and Dirichlet forms. The elliptic operators may be viewed as generators of irreversible dynamics for fermion systems with infinite degrees of freedom.

Quantum stochastic operator cocycles via associated semigroups

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.

Homomorphic Feller Cocycles on a C*-Algebra

When a Fock-adapted Feller cocycle on a C*-algebra is regular, completely positive and contractive, it possesses a stochastic generator that is necessarily completely bounded. Necessary and sufficient conditions are given, in the form of a sequence of identities, for a completely bounded map to generate a weakly multiplicative cocycle. These are derived from a product formula for iterated quantum stochastic integrals. Under two alternative assumptions, one of which covers all previously considered cases, the first identity in the sequence is shown to imply the rest.

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.

A stochastic Stinespring Theorem

Abstract. Completely positive Markovian cocycles on a von Neumann algebra, adapted to a Fock filtration, are realised as conjugations of

Quantum Feynman–Kac perturbations

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