Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

CRC Press eBooks

The Semi-Martingale Property of the Square of White Noise Integrators

Luigi Accardi, Andreas Boukas

Published Apr 5, 2002• 10 citations
Publisher

Abstract

The abstract commutation relations of the algebra of the square of white noise of Accardi, Lu, and Volovich are shown to be realized by operator processes acting on the Fock space of Accardi and Skeide which is very closely related to the Finite Difference Fock space of Boukas and Feinsilver. The processes are shown to satisfy the necessary conditions for inclusion in the framework of the representation free quantum stochastic calculus of Accardi, Fagnola, and Quaegebeur. The connection between the Finite-Difference operators and the creation, annihilation, and conservation operators on usual symmetric Boson Fock space is further studied.

DOI
10.1201/9780203910177-1

Related papers

ON THE RELATION OF THE SQUARE OF WHITE NOISE AND THE FINITE DIFFERENCE ALGEBRA

Luigi Accardi, Michael Skeide

The algebra of square of white noise1 contains a subalgebra generated by elements fulfilling the relations of Feinsilver's finite difference algebra.6 Moreover, Boukas' representation space3 is the same as the representation space of the algebra of square of white noise discovered in Ref. 2. In other words, Boukas' representation extends to a representation of the algebra of square of white noise.

Infinite Dimensional Analysis Quantum Probability and Related Topics•Mar 1, 2000•21 citations

Application of Quantum Stochastic Calculus to Feedback Control

Andreas Boukas

The basic aspects of the Hudson-Parthasarathy quantum stochastic calculus and of the Accardi-Fagnola-Quaegebeur representation free stochastic calculus are presented. The basic features of the stochastic calculus for the square of white noise recently developed by Accardi-Boukas are described. The feedback control problem for stochastic processes driven by quantum noise is solved

arXiv:1308.1948•Aug 7, 2013•4 citations

Quantum stochastic integral representations of Fock space operators

Un Cig Ji, Nobuaki Obata

An (unbounded) operator Ξ on Boson Fock space over L 2(R +) is called regular if it is an admissible white noise operator such that the conditional expectations give rise to a regular quantum martingale. We prove that an admissible white noise operator is regular if and only if it admits a quantum stochastic integral representation.

Stochastics•Jun 1, 2009•18 citations

Powers of quantum white noise derivatives

Aymen Ettaieb, Habib Ouerdiane, Hafedh M. S. Rguigui

We introduce a new operator obtained from the quantum white noise (QWN) derivatives which satisfies new important commutation relations generalizing those of the renormalized power white noise Lie algebra introduced by Accardi, Boukas and Franz (see Ref. 5) without using renormalization conditions.

Infinite Dimensional Analysis Quantum Probability and Related Topics•Jul 30, 2014•15 citations

Quantum stochastic calculus on interacting Fock spaces: semimartingale estimates and stochastic integral

Vitonofrio Crismale

A quantum stochastic integration theory on interacting Fock spaces (IFS) is developed.We present the semi-martingale inequalities either in standard general IFS or in 1-mode type IFS, which allow us to introduce the definitions of integrable processes and construct stochastic integrals satisfying some useful properties which will be presented in [13].

Communications on Stochastic Analysis•Aug 1, 2007•6 citations

MALLIAVIN CALCULUS AND SKOROHOD INTEGRATION FOR QUANTUM STOCHASTIC PROCESSES

Uwe Franz, Rémi Léandre, René Schott

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 [Formula: see text] and it is shown that they satisfy similar properties as the derivation and divergence operator on the Wiener space over [Formula: see text]. 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 [Formula: see text], the divergence operator is shown to coincide with the Hudson–Parthasarathy quantum stochastic integral for adapted integrable processes and with the noncausal quantum stochastic integrals defined by Lindsay and Belavkin for integrable processes.

arXiv:math/0004088•Infinite Dimensional Analysis Quantum Probability and Related Topics•Mar 1, 2001•5 citations
The abstract commutation relations of the algebra of the square of white noise of Accardi, Lu, and Volovich are shown to be realized by operator processes acting on the Fock space of Accardi and Skeide which is very closely related to the Finite Difference Fock space of Boukas and Feinsilver. The processes are shown to satisfy the necessary conditions for inclusion in the framework of the representation free quantum stochastic calculus of Accardi, Fagnola, and Quaegebeur. The connection between the Finite-Difference operators and the creation, annihilation, and conservation operators on usual symmetric Boson Fock space is further studied.
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 [Formula: see text] and it is shown that they satisfy similar properties as the derivation and divergence operator on the Wiener space over [Formula: see text]. 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 [Formula: see text], the divergence operator is shown to coincide with the Hudson–Parthasarathy quantum stochastic integral for adapted integrable processes and with the noncausal quantum stochastic integrals defined by Lindsay and Belavkin for integrable processes.