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The quantum stochastic Schroedinger equation or Hudson-Parthasareathy (HP) equation is a powerful tool to construct unitary dilations of quantum dynamical semigroups and to develop the theory of measurements in continuous time via the construction of output fields. An important feature of such an equation is that it allows to treat not only absorption and emission of quanta, but also scattering processes, which however had very few applications in physical modelling. Moreover, recent developments have shown that also some non-Markovian dynamics can be generated by suitable choices of the state of the quantum noises involved in the HP-equation. This paper is devoted to an application involving these two features, non-Markovianity and scattering process. We consider a micro-mirror mounted on a vibrating structure and reflecting a laser beam, a process giving rise to a radiation-pressure force on the mirror. We show that this process needs the scattering part of the HP-equation to be described. On the other side, non-Markovianity is introduced by the dissipation due to the interaction with some thermal environment which we represent by a phonon field, with a nearly arbitrary excitation spectrum, and by the introduction of phase noise in the laser beam. Finally, we study the full power spectrum of the reflected light and we show how the laser beam can be used as a temperature probe., Comment: 17 pages

We study the continuity property of multiple Q-adapted quantum stochastic integrals with respect to noncommuting integrands given by the non-adapted multiple integral kernels in Fock scale. The noncommutative algebra of relatively (exponentially) bounded nonadapted quantum stochastic processes is studied in the kernel form as introduced by Belavkin in 1991. The differential Q-adapted formula generalizing Ito product formula for adapted integrals is presented in both strong and weak sense as a particular case of the quantum stochastic nonadapted Ito formula., Comment: Due to appear in communications on stochastic analysis journal (KRP volume). 21 pages

In this paper we first introduce the Fock-Guichardet formalism for the quantum stochastic integration, then the four fundamental processes of the dynamics are introduced in the canonical basis as the operator-valued measures of the QS integration over a space-time. Then rigorous analysis of the QS integrals is carried out, and continuity of the QS derivative is proved. Finally, Q-adapted dynamics is discussed, including Bosonic Q=1, Fermionic Q=-1, and monotone Q=0 quantum dynamics. These may be of particular interest to quantum field theory, quantum open systems, and quantum theory of stochastic processes., Comment: A similar version is due to appear in the proceedings of the 13th workshop: non-commutative harmonic analysis. 18 pages

We study infinite weighted graphs with view to \textquotedblleft limits at infinity,\textquotedblright or boundaries at infinity. Examples of such weighted graphs arise in infinite (in practice, that means \textquotedblleft very\textquotedblright large) networks of resistors, or in statistical mechanics models for classical or quantum systems. But more generally our analysis includes reproducing kernel Hilbert spaces and associated operators on them. If $X$ is some infinite set of vertices or nodes, in applications the essential ingredient going into the definition is a reproducing kernel Hilbert space; it measures the differences of functions on $X$ evaluated on pairs of points in $X$. And the Hilbert norm-squared in $\mathcal{H}(X)$ will represent a suitable measure of energy. Associated unbounded operators will define a notion or dissipation, it can be a graph Laplacian, or a more abstract unbounded Hermitian operator defined from the reproducing kernel Hilbert space under study. We prove that there are two closed subspaces in reproducing kernel Hilbert space $\mathcal{H}(X)$ which measure quantitative notions of limits at infinity in $X$, one generalizes finite-energy harmonic functions in $\mathcal{H}(X)$, and the other a deficiency index of a natural operator in $\mathcal{H}(X)$ associated directly with the diffusion. We establish these results in the abstract, and we offer examples and applications. Our results are related to, but different from, potential theoretic notions of \textquotedblleft boundaries\textquotedblright in more standard random walk models. Comparisons are made., Comment: 38 pages, 4 tables, 3 figures

We derive the Hamiltonian associated to a quantum stochastic flow by extending the Albeverio-Kurasov construction of self-adjoint extensions to finite rank perturbations of nonsemibounded operators to Fock space., Comment: 13 pages

We study the Classical Probability analogue of the dilations of a quantum dynamical semigroup defined in Quantum Probability via quantum stochastic differential equations. Given a homogeneous Markov chain in continuous time in a finite state space E, we introduce a second system, an environment, and a deterministic invertible time-homogeneous global evolution of the system E with this environment such that the original Markov evolution of E can be realized by a proper choice of the initial random state of the environment. We also compare this dilations with the dilations of a quantum dynamical semigroup in Quantum Probability: given a classical Markov semigroup, we extend it to a proper quantum dynamical semigroup for which we can find a Hudson-Parthasarathy dilation which is itself an extension of our classical dilation.

A concept of quantum stochastic convolution cocycle is introduced and studied in two different contexts -- purely algebraic and operator space theoretic. A quantum stochastic convolution cocycle is a quantum stochastic process on a coalgebra satisfying the convolution cocycle relation and the initial condition given by the counit. The notion generalises that of quantum Levy process, which in turn is a noncommutative probability counterpart of classical Levy process on a group. Convolution cocycles arise as solutions of quantum stochastic differential equations. In turn every sufficiently regular cocycle satisfies an equation of that type. This is proved along with the corresponding existence and uniqueness of solutions for coalgebraic quantum stochastic differential equations. The stochastic generators of unital *-homomorphic cocycles are characterised in terms of structure maps on a *-bialgebra. This yields a simple proof of the Schurmann Reconstruction Theorem for a quantum Levy process; it also yields a topological version for a quantum Levy process on a C*-bialgebra. Precise characterisation of the stochastic generators of completely positive and contractive quantum stochastic convolution cocycles in the C*-algebraic context is given, leading to some dilation results. A few examples are presented and some interpretations offered for quantum stochastic convolution cocycles and their stochastic generators on different types of *-bialgebra., Comment: The article is a revised version of the PhD thesis of the author, submitted to the University of Nottingham in December 2005 and defended on the 28 July 2006

A new method for the construction of Fock-adapted operator Markovian 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., Comment: 9 pages; to appear in special volume of Proc. Indian Acad. Sci. Math. Sci. dedicated to Kalyan Sinha

This paper provides an introduction to quantum filtering theory. An introduction to quantum probability theory is given, focusing on the spectral theorem and the conditional expectation as a least squares estimate, and culminating in the construction of Wiener and Poisson processes on the Fock space. We describe the quantum It\^o calculus and its use in the modelling of physical systems. We use both reference probability and innovations methods to obtain quantum filtering equations for system-probe models from quantum optics., Comment: 41 pages, 1 figure

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., Comment: 20 pages; to appear in Math Proc Cambridge Philos Soc. Minor edits carried out

Given a finite state space E, we build a universal dilation for all possible discrete time Markov chains on E, homogeneous or not: we introduce a second system (an ``environment'') and a deterministic invertible time-homogeneous global evolution of the system E with this environment such that any Markov evolution of E can be realized by a proper choice of the initial (random) state of the environment, which therefore determines the transition probabilities of the system. We also compare this dilation with the quantum dilations of a Quantum Dynamical Semigroup: given a Classical Markov Semigroup, we show that it can be extended to a Quantum Dynamical Semigroup for which we can find a quantum dilation to a group of *-automorphisms admitting an invariant abelian subalgebra where this quantum dilation gives just our classical dilation.

The stochastic generators of Markov-regular operator cocycles on symmetric Fock space are studied in a variety of cases: positive cocycles, projection cocycles, and partially isometric cocycles. Moreover a class of transformations of positive contraction cocycles is exhibited which leads to a polar decomposition result., Comment: 15 pages; accepted in special volume of Markov Processes and Related Fields for the John Lewis Memorial Conference, DIT Dublin 2005. An error rectified; illustrative examples added

We define tensor product decompositions of $E_0$-semigroups with a structure analogous to a classical theorem of Beurling. Such decompositions can be characterized by adaptedness and exactness of unitary cocycles. For CCR-flows we show that such cocycles are convergent., Comment: minor corrections, 10 pages

Various recent results on quantum L\'evy processes are presented. The first part provides an introduction to the theory of L\'evy processes on involutive bialgebras. The notion of independence used for these processes is tensor independence, which generalizes the notion of independence used in classical probability and corresponds to independent observables in quantum physics. In quantum probability there exist other notions of independence and L\'evy processes can also be defined for the five so-called universal independences. This is the topic of the second part. In particular, it is shown that boolean, monotone, and anti-monotone independence can be reduced to tensor independence. Finally, in the third part, several classes of quantum L\'evy processes of special interest are considered, e.g., L\'evy processes on real Lie algebras or Brownian motions on braided spaces. Several applications of these processes are also presented., Comment: Habilitation thesis EMAU Greifswald, 204 pages

We consider simply generated trees and study multiplicative functions on rooted plane trees. We show that the associated generating functions satisfy differential equations or difference equations. Our approach considers B-series from Butcher's theory, the generating functions are seen as generalized Runge-Kutta methods, Comment: 19 pages, 1 figure

We review some of our results from the theory of product systems of Hilbert modules. We explain that the product systems obtained from a CP-semigroup in a paper by Bhat and Skeide and in a paper by Muhly and Solel are commutants of each other. Then we use this new commutant technique to construct product systems from E_0-semigroups on B^a(E) where E is a strongly full von Neumann module. (This improves the construction from a paper by Skeide for Hilbert modules where existence of a unit vector is required.) Finally, we point out that the Arveson system of a CP-semigroup constructed by Powers from two spatial E_0-semigroups is the product of the corresponding spatial Arveson systems as defined (for Hilbert modules) in a paper by Skeide. It need not coincide with the tensor product of Arveson systems., Comment: To appear in Proceedings of ``Advances in Quantum Dynamics'', Mount Holyoke, 2002

We discuss the meaning of renormalization used for deriving quadratic bosonic commutation relations introduced by Accardi and find a representation of these relations on an interacting Fock space. Also, we investigate classical stochastic processes which can be constructed from noncommutative quadratic white noise. We postulate quadratic free white noise commutation relations and find their representation on an interacting Fock space., Comment: 17 pages

We give a short introduction to the theory of L\'evy processes on dual groups. As examples we consider L\'evy processes with additive increments and L\'evy processes on the dual affine group., Comment: 12 pages, Extended abstract to be published in Mini-proceedings: Second MaPhySto Conference on ``L\'evy Processes - Theory and Applications,'' January 2002, Aarhus, Denmark

In a noncommutative torus, effect of perturbation by inner derivation on the associated quantum stochastic process and geometric parameters like volume and scalar curvature have been studied. Cohomological calculations show that the above perturbation produces new spectral triples. Also for the Weyl C^*-algebra, the Laplacian associated with a natural stochastic process is obtained and associated volume form is calculated. Keywords: noncommutative torus, Laplacian, Dixmier trace, quantum stochastic process., Comment: To appear in Journal of Operator Theory

In this paper we investigate a quantum stochastic calculus build of creation, annihilation and number of particles operators which fulfill some deformed commutation relations. Namely, we introduce a deformation of a number of particles operator which has simple commutation relations with well known q-deformed creation and annihilation operators. Since all operators considered in this theory are bounded we do not deal with some difficulties of a non deformed theory of Hudson and Parthasarathy. We define stochastic integrals and estimate them in the operator norm. We prove Ito's formula as well., Comment: 18 pages

A derivation operator and a divergence operator are defined on the algebra of bounded operators on the symmetric Fock space over the complexification of a real Hilbert space $\eufrak{h}$ and it is shown that they satisfy similar properties as the derivation and divergence operator on the Wiener space over $\eufrak{h}$. The derivation operator is then used to give sufficient conditions for the existence of smooth Wigner densities for pairs of operators satisfying the canonical commutation relations. For $\eufrak{h}=L^2(\mathbb{R}_+)$, the divergence operator is shown to coincide with the Hudson-Parthasarathy quantum stochastic integral for adapted integrable processes and with the non-causal quantum stochastic integrals defined by Lindsay and Belavkin for integrable processes., Comment: 28 pages, amsart style