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

Stochastic evolutions driven by non--linear quantum noise II

Luigi Accardi, Andreas Boukas

Published Jan 1, 2001• 2 citations
OpenAlex

Abstract

We consider the renormalized Ito table for higher powers of white noise and, assuming that there exists a Hilbert space where these powers have an operator realization, we prove existence, uniqueness, and unitarity for stochastic equations driven by these powers.

Related papers

ON THE UNITARITY OF STOCHASTIC EVOLUTIONS DRIVEN BY THE SQUARE OF WHITE NOISE

Luigi Accardi, Andreas Boukas, HUI-HSUNG KUO

Using the closed Itô's table for the renormalized square of white noise, recently obtained by Accardi, Hida, and Kuo in Ref. 4, we consider the problem of providing necessary and sufficient conditions for the unitarity of the solutions of a certain type of quantum stochastic differential equations.

Infinite Dimensional Analysis Quantum Probability and Related Topics•Dec 1, 2001•10 citations

QUANTUM STOCHASTIC ANALYSIS VIA WHITE NOISE OPERATORS IN WEIGHTED FOCK SPACE

Dong Myung Chung, Un Cig Ji, Nobuaki Obata

White noise theory allows to formulate quantum white noises explicitly as elemental quantum stochastic processes. A traditional quantum stochastic differential equation of Itô type is brought into a normal-ordered white noise differential equation driven by lower powers of quantum white noises. The class of normal-ordered white noise differential equations covers quantum stochastic differential equations with highly singular noises such as higher powers or higher order derivatives of quantum white noises, which are far beyond the traditional Itô theory. For a general normal-ordered white noise differential equation unique existence of a solution is proved in the sense of white noise distribution. Its regularity properties are investigated by means of weighted Fock spaces interpolating spaces of white noise distributions and associated characterization theorems for S-transform and for operator symbols.

Reviews in Mathematical Physics•Mar 1, 2002•37 citations

Unitarity conditions for stochastic differential equations driven by nonlinear quantum noise

Luigi Accardi, Andreas Boukas

We prove the stochastic independence of the basic integrators of the renormalized square of white noise (SWN). We use this result to deduce the unitarity conditions for stochastic differential equations driven by the SWN.

Random Operators and Stochastic Equations•Jan 1, 2002•11 citations

Wick product of white noise operators and quantum stochastic differential equations

Nobuaki Obata

The Wick product of operators on Fock space is introduced on the basis of the analytic characterization theorem for operator symbols established within the framework of white noise distribution theory. Existence and uniqueness of solutions are proved for a certain class of ordinary differential equations for Fock space operators. Quantum stochastic differential equations of Itô type and their generalizations involving higher powers of quantum white noises enter into our consideration.

Journal of the Mathematical Society of Japan•Jul 1, 1999•29 citations

Higher Powers of Quantum White Noises in Terms of Integral Kernel Operators

Dong Myung Chung, Un Cig Ji, Nobuaki Obata

A rigorous mathematical formulation of higher powers of quantum white noises is given on the basis of the most recent theory of white noise distributions due to Cochran, Kuo and Sengupta. The renormalized quantum Itô formula due to Accardi, Lu and Volovich is derived from the renormalized product formula based on integral kernel operators on white noise functions. During the discussion, the analytic characterization of operator symbols and the expansion theorem for a white noise operator in terms of integral kernel operators are established.

Infinite Dimensional Analysis Quantum Probability and Related Topics•Oct 1, 1998•22 citations

Module Free White Noise Flows

Wided Ayed

The main result of this paper is to extend to Hilbert module level the proof of the inclusion of (non-Hamiltonian) stochastic differential equations based on free noise into the class of Hamiltonian equations driven by free white noise. To achieve this goal, free white noise calculus is extended to a trivial Hilbert module. The white noise formulation of the Ito table is radically different from the usual Itô tables, both classical and quantum and, combined with the Accardi–Boukas approach to Ito algebra, allows to drastically simplify calculations. Infinitesimal generators of Hilbert module free flows are characterized in terms of stochastic derivations from an initial algebra into a white noise Itô algebra. We prove that any such derivation is the difference of a ⋆-homomorphism and a trivial embedding.

Open Systems & Information Dynamics•Dec 1, 2018•0 citations
White noise theory allows to formulate quantum white noises explicitly as elemental quantum stochastic processes. A traditional quantum stochastic differential equation of Itô type is brought into a normal-ordered white noise differential equation driven by lower powers of quantum white noises. The class of normal-ordered white noise differential equations covers quantum stochastic differential equations with highly singular noises such as higher powers or higher order derivatives of quantum white noises, which are far beyond the traditional Itô theory. For a general normal-ordered white noise differential equation unique existence of a solution is proved in the sense of white noise distribution. Its regularity properties are investigated by means of weighted Fock spaces interpolating spaces of white noise distributions and associated characterization theorems for S-transform and for operator symbols.
A rigorous mathematical formulation of higher powers of quantum white noises is given on the basis of the most recent theory of white noise distributions due to Cochran, Kuo and Sengupta. The renormalized quantum Itô formula due to Accardi, Lu and Volovich is derived from the renormalized product formula based on integral kernel operators on white noise functions. During the discussion, the analytic characterization of operator symbols and the expansion theorem for a white noise operator in terms of integral kernel operators are established.
The main result of this paper is to extend to Hilbert module level the proof of the inclusion of (non-Hamiltonian) stochastic differential equations based on free noise into the class of Hamiltonian equations driven by free white noise. To achieve this goal, free white noise calculus is extended to a trivial Hilbert module. The white noise formulation of the Ito table is radically different from the usual Itô tables, both classical and quantum and, combined with the Accardi–Boukas approach to Ito algebra, allows to drastically simplify calculations. Infinitesimal generators of Hilbert module free flows are characterized in terms of stochastic derivations from an initial algebra into a white noise Itô algebra. We prove that any such derivation is the difference of a ⋆-homomorphism and a trivial embedding.