Lévy processes through time shift on oscillator Weyl algebra (original) (raw)

Classical Versions of Quantum Stochastic Processes Associated with the Oscillator Algebra

It has been known for a long time that any infinitely divisible distribution (I.D.D) can be realized on a symmetric Fock space with an appropriate noise space. This realization led to a kind of correspondence between Lie algebras and I.D.D. Namely, each I.D.D (or equivalently Lévy process) leads to a such Lie algebra commutation relations. In this context, it was shown (see [1]) that the the quantum stochastic processes corresponding to the bounded form of the oscillator algebra can not cover a large classes of Lévy processes, in particular the non standard Meixner classes. For this reason, we consider the unbounded form of the oscillator algebra called the adapted oscillator algebra. Then, we prove that its Fock representation can give rise to the infinitely divisible processes such as the Gamma, Pascal and the Meixner-Pollaczek.

Quantum stochastic process associated with quantum Lévy Laplacian

Communications on Stochastic Analysis, 2007

A noncommutative extension of the Lévy Laplacian, called the quantum Lévy Laplacian, is introduced and its relevant properties are studied, in particular, a relation between the classical and quantum Lévy Laplacians is studied. We construct a semigroup and a quantum stochastic process generated by the quantum Lévy Laplacian.

Quantum Poisson processes and dilations of dynamical semigroups

Probability Theory and Related Fields, 1989

The notion of a quantum Poisson process over a quantum measure space is introduced. This process is used to construct new quantum Markov processes on the matrix algebra M, with stationary faithful state qS. If (Jd, #) is the quantum measure space in question (Jt a yon Neumann algebra and # a faithful normal weight), then the semigroup e tL of transition operators on (M,, ~b) has generator L: M,~M,: a~i[h,a]+(id|174174 where u is an arbitrary unitary element of the centraliser of (M, | Jg, ~b | #).

White noise quantum time shifts

Random Operators and Stochastic Equations, 2000

In the present paper we extend the notion of quantum time shift, and the related results obtained in , from representations of current algebras of the Heisenberg Lie algebra to representations of current algebras of the Oscillator Lie algebra. This produces quantum extensions of a class of classical Lévy processes much wider than the usual Brownian motion. In particular this class processes includes the Meixner processes and, by an approximation procedure, we construct quantum extensions of all classical Lévy processes with a Lévy measure with finite variance. Finally we compute the explicit form of the action, on the Weyl operators of the initial space, of the generators of the quantum Markov processes canonically associated to the above class of Lévy processes. The emergence of the Meixner classes in connection with the renormalized second order white noise, is now well known. The fact that they also emerge from first order noise in a simple and canonical way, comes somehow as a surprise.

Quantum stochastic Weyl operators

2003 IEEE International Workshop on Workload Characterization (IEEE Cat. No.03EX775), 2003

. The second author wishes to express his gratitude to Professor Luigi Accardi for his support and guidance over the years, as well as for the hospitality of the Centro Vito Volterra of the Universita di Roma TorVergata on many occasions.

Non-abelian Weyl commutation relations and the series product of quantum stochastic evolutions

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2012

We show that the series product, which serves as an algebraic rule for connecting state-based input–output systems, is intimately related to the Heisenberg group and the canonical commutation relations. The series product for quantum stochastic models then corresponds to a non-abelian generalization of the Weyl commutation relation. We show that the series product gives the general rule for combining the generators of quantum stochastic evolutions using a Lie–Trotter product formula.

Lévy processes and Fourier analysis on compact Lie groups

The Annals of Probability, 2004

We study the Fourier expansion of the distribution density of a Lévy process in a compact Lie group based on the Peter-Weyl theorem. 1. Introduction. Let G be a Lie group with identity element e and of dimension d. A stochastic process g t in G, with right continuous paths having left limits, is called a Lévy process if it has independent and stationary increments in the sense that for any s < t, g −1 s g t is independent of F 0 s , the σ-algebra generated by g u for u ∈ [0, s], and has a distribution depending only on t − s. In this paper a Lévy process g t will always be assumed to start at e, that is, g 0 = e, unless when explicitly stated otherwise. Lévy processes in noncompact semi-simple Lie groups possess interesting limiting properties. These were studied in Liao (1998, 2002) motivated by the results on Brownian motion in symmetric spaces by Dynkin and Malliavia-Malliavin, and Lie group valued random walks by Furstenberg-Kesten and Guivarc'h-Raugi. See the references in the above cited papers and also Applebaum (2000a) for some of the other related results. Lévy processes in a compact Lie group possess completely different properties. Instead of exhibiting any sample path convergence, the process is ergodic and one would expect that its distribution converges to the normalized Haar measure as time t → ∞. The purpose of this paper is to study the Fourier expansion of the distribution density of a Lévy process in a compact Lie group based on the Peter-Weyl theorem and from which to obtain the exponential convergence of the distribution of the process to the normalized Haar measure. Fourier transformation of bounded measures on locally compact groups was studied in Heyer (1968) and a related central limit theorem was established in Siebert (1981). Fourier method has been proved useful for studying random walks on finite groups and, in some special cases, on Lie groups. See, for example, Diaconis (1988), Rosenthal (1994) and Klyachko (2000). This paper may be regarded as a first attempt at applying this useful method to study a general Lévy process in a compact Lie group. We will now describe a little more precisely the content of this paper. In the next section after a discussion of the generator of a Lévy process, we first establish

Weyl calculus in QED I. The unitary group

Journal of Mathematical Physics

In this work, we consider fixed 1/2 spin particles interacting with the quantized radiation field in the context of quantum electrodynamics (QED). We investigate the time evolution operator in studying the reduced propagator (interaction picture). We first prove that this propagator belongs to the class of infinite dimensional Weyl pseudodifferential operators recently introduced in [3] on Wiener spaces. We give a semiclassical expansion of the symbol of the reduced propagator up to any order with estimates on the remainder terms. Next, taking into account analyticity properties for the Weyl symbol of the reduced propagator, we derive estimates concerning transition probabilities between coherent states.