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

Journal of Physics A Mathematical and General

On the existence and uniqueness of non-commutative stochastic process

G. O. S. Ekhaguere

Published Nov 1, 1982• 1 citation
Publisher

Abstract

Introduces a general notion of non-commutative stochastic processes over complete locally convex *-algebras and proves a theorem concerning the existence and uniqueness of such processes. The author shows that quantum fields are stochastic processes in the sense of this paper. Moreover, the author applies the theory to infer the existence and uniqueness of quantum fields.

DOI
10.1088/0305-4470/15/11/022

Related papers

Quantum Stochastic Processes

Luigi Accardi, Alberto Frigerio, John T. Lewis

A class of non-commutative stochastic processes is defined.These processes are defined up to equivalence by their multi-time correlation kernels.A reconstruction theorem, generalizing the Kolmogorov theorem for classical processes, is proved.Markov processes and their associated semigroups are studied, and some non-quasi free examples are constructed on the Clifford algebra, with the use of a perturbation theory of Markov processes.The connection with the Hepp-Lieb models is discussed.

Publications of the Research Institute for Mathematical Sciences•Apr 30, 1982•265 citations

Quantum stochastic integration and quantum stochastic differential equations

Chris Barnett, Stanisław Goldstein, Ivan F. Wilde

Abstract Quantum stochastic integrals are constructed using the non-commutative L p -space theory of Haagerup. The existence and uniqueness of the solution to quantum stochastic differential equations driven by quasi-Wiener noises, or noises satisfying generalized standing hypotheses, is established as is the Markov behaviour of the solution. Various examples of the theory are discussed, and quantum Ornstein-Uhlenbeck processes are obtained as explicit solutions.

Mathematical Proceedings of the Cambridge Philosophical Society•Nov 1, 1994•6 citations

Quantum Lévy–Itô algebras and non-commutative stochastic analysis

V. P. Belavkin

We review the basic concepts of quantum probability and revise the classical and quantum stochastic (QS) calculus using the universal Itô B*-algebra approach. A non-commutative generalization of Lévy process is defined in Fock–Guichardet space in terms of the underlying B*-Itô modular algebra. The main notions and results of classical and QS analysis are reformulated in terms of the non-commutative but associative stochastic covariation of quantum integrals defined as sesquilinear forms adapted to a quantum Lévy process in Fock space. The general semi-Markov QS dynamics is defined in terms of a hemigroup of semi-morphic transformations as adjoints to a semiunitary hemigroup resolving a QS Schrödinger equation in Fock space. The corresponding semi-Markov hemigroup of completely positive contractions resolving a quantum master equation with generalized Lindblad generator as a non-commutative analogue of Feller–Kolmogorov equation is derived by taking vacuum conditional expectation in Fock space.

Stochastics•Nov 21, 2011•0 citations

Quantum stochastic convolution cocycles

Adam Skalski

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.

arXiv:math/0608756•Aug 30, 2006•2 citations

Existence, uniqueness and approximation for stochastic Schrodinger equation: the Poisson case

Clément Pellegrini

In quantum physics, recent investigations deal with the so-called "quantum trajectory" theory. Heuristic rules are usually used to give rise to "stochastic Schrodinger equations" which are stochastic differential equations of non-usual type describing the physical models. These equations pose tedious problems in terms of mathematical justification: notion of solution, existence, uniqueness, justification... In this article, we concentrate on a particular case: the Poisson case. Random measure theory is used in order to give rigorous sense to such equations. We prove existence and uniqueness of a solution for the associated stochastic equation. Furthermore, the stochastic model is physically justified by proving that the solution can be obtained as a limit of a concrete discrete time physical model.

arXiv:0709.3713•Sep 24, 2007•5 citations

Hidden quantum Markov processes

Luigi Accardi, El Gheteb Soueidi, Yun Gang Lu +1 more

Starting from the notion of algebraic hidden processes introduced in Ref. 9 , we propose a definition of quantum hidden Markov process which extends the notion of quantum Markov chain (QMC) in the same sense in which classical hidden Markov processes (HMPs) extend classical Markov chains. We give several structure theorems for this new family of quantum states which shares with the QMCs the unique property that the finite dimensional joint expectations are explicitly given without these states being linearly equivalent to product states (as the Gaussian ones). The special feature of this new class of quantum states is best exemplified by looking at their restrictions to diagonal sub-algebras: we prove that they lead to a new family of classical stochastic processes that generalize in a non-trivial way the classical HMPs.

Infinite Dimensional Analysis Quantum Probability and Related Topics•May 9, 2024•9 citations
Introduces a general notion of non-commutative stochastic processes over complete locally convex *-algebras and proves a theorem concerning the existence and uniqueness of such processes. The author shows that quantum fields are stochastic processes in the sense of this paper. Moreover, the author applies the theory to infer the existence and uniqueness of quantum fields.
We review the basic concepts of quantum probability and revise the classical and quantum stochastic (QS) calculus using the universal Itô B*-algebra approach. A non-commutative generalization of Lévy process is defined in Fock–Guichardet space in terms of the underlying B*-Itô modular algebra. The main notions and results of classical and QS analysis are reformulated in terms of the non-commutative but associative stochastic covariation of quantum integrals defined as sesquilinear forms adapted to a quantum Lévy process in Fock space. The general semi-Markov QS dynamics is defined in terms of a hemigroup of semi-morphic transformations as adjoints to a semiunitary hemigroup resolving a QS Schrödinger equation in Fock space. The corresponding semi-Markov hemigroup of completely positive contractions resolving a quantum master equation with generalized Lindblad generator as a non-commutative analogue of Feller–Kolmogorov equation is derived by taking vacuum conditional expectation in Fock space.
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.
In quantum physics, recent investigations deal with the so-called "quantum trajectory" theory. Heuristic rules are usually used to give rise to "stochastic Schrodinger equations" which are stochastic differential equations of non-usual type describing the physical models. These equations pose tedious problems in terms of mathematical justification: notion of solution, existence, uniqueness, justification... In this article, we concentrate on a particular case: the Poisson case. Random measure theory is used in order to give rigorous sense to such equations. We prove existence and uniqueness of a solution for the associated stochastic equation. Furthermore, the stochastic model is physically justified by proving that the solution can be obtained as a limit of a concrete discrete time physical model.
Starting from the notion of algebraic hidden processes introduced in Ref. 9 , we propose a definition of quantum hidden Markov process which extends the notion of quantum Markov chain (QMC) in the same sense in which classical hidden Markov processes (HMPs) extend classical Markov chains. We give several structure theorems for this new family of quantum states which shares with the QMCs the unique property that the finite dimensional joint expectations are explicitly given without these states being linearly equivalent to product states (as the Gaussian ones). The special feature of this new class of quantum states is best exemplified by looking at their restrictions to diagonal sub-algebras: we prove that they lead to a new family of classical stochastic processes that generalize in a non-trivial way the classical HMPs.