Wick Calculus of Generalized Operators and its Applications to Quantum Stochastic Calculus

1998 ◽  
Vol 01 (03) ◽  
pp. 455-466 ◽  
Author(s):  
Zhiyuan Huang ◽  
Shunlong Luo
Keyword(s):  
Differential Equations ◽  
White Noise ◽  
Stochastic Analysis ◽  
Stochastic Calculus ◽  
Stochastic Integrals ◽  
Algebra Structure ◽  
Quantum Stochastic Calculus ◽  
Fundamental Properties ◽  
White Noise Calculus ◽  
Wick Calculus

A nonlinear and stochastic analysis of free Bose field is established in the framework of white noise calculus. Wick algebra structure is introduced in the space of generalized operators generated by quantum white noise, some fundamental properties of the calculus based on the Wick algebra are investigated. As applications, quantum stochastic integrals and quantum stochastic differential equations are treated from the viewpoint of Wick calculus.

Exponential formulae in quantum stochastic calculus

1996 ◽  
Vol 126 (2) ◽  
pp. 375-389 ◽  
Author(s):  
A. S. Holevo
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Degrees Of Freedom ◽  
Stochastic Integral ◽  
Stochastic Calculus ◽  
Exponential Equation ◽  
Quantum Stochastic Calculus ◽  
Rigorous Definition ◽  
Definition Of

The rigorous definition of time-ordered exponentials, solving quantum linear stochastic differential equations, is extended to Boson and Fermion stochastic calculi with infinitely many degrees of freedom. The relation to the classicalmultiplicative stochastic integral, solving the Doleans exponential equation, is discussed.

Some New Examples of Lie Super-Algebra Representations Arising from Quantum Stochastic Calculus

1998 ◽  
Vol 01 (01) ◽  
pp. 33-42
Author(s):  
K. R. Parthasarathy
Keyword(s):  
Stochastic Calculus ◽  
Stochastic Integrals ◽  
Quantum Stochastic Calculus

Following the methods of Refs. 1 and 2 we construct here a family of Lie super-algebra representations in terms of quantum stochastic integrals.

scholarly journals Wick calculus on spaces of regular generalized functions of Levy white noise analysis

2018 ◽  
Vol 10 (1) ◽  
pp. 82-104 ◽  
Author(s):  
M.M. Frei
Keyword(s):  
White Noise ◽  
Noise Analysis ◽  
Gaussian White Noise ◽  
Holomorphic Functions ◽  
Generalized Functions ◽  
Leibniz Rule ◽  
Wick Product ◽  
Stochastic Integrals ◽  
White Noise Analysis ◽  
Wick Calculus

Many objects of the Gaussian white noise analysis (spaces of test and generalized functions, stochastic integrals and derivatives, etc.) can be constructed and studied in terms of so-called chaotic decompositions, based on a chaotic representation property (CRP): roughly speaking, any square integrable with respect to the Gaussian measure random variable can be decomposed in a series of Ito's stochastic integrals from nonrandom functions. In the Levy analysis there is no the CRP (except the Gaussian and Poissonian particular cases). Nevertheless, there are different generalizations of this property. Using these generalizations, one can construct different spaces of test and generalized functions. And in any case it is necessary to introduce a natural product on spaces of generalized functions, and to study related topics. This product is called a Wick product, as in the Gaussian analysis. The construction of the Wick product in the Levy analysis depends, in particular, on the selected generalization of the CRP. In this paper we deal with Lytvynov's generalization of the CRP and with the corresponding spaces of regular generalized functions. The goal of the paper is to introduce and to study the Wick product on these spaces, and to consider some related topics (Wick versions of holomorphic functions, interconnection of the Wick calculus with operators of stochastic differentiation). Main results of the paper consist in study of properties of the Wick product and of the Wick versions of holomorphic functions. In particular, we proved that an operator of stochastic differentiation is a differentiation (satisfies the Leibniz rule) with respect to the Wick multiplication.

Module Free White Noise Flows

2018 ◽  
Vol 25 (04) ◽  
pp. 1850018
Author(s):  
Wided Ayed
Keyword(s):  
Differential Equations ◽  
White Noise ◽  
Stochastic Differential Equations ◽  
Hilbert Module ◽  
Hamiltonian Equations ◽  
Infinitesimal Generators ◽  
Initial Algebra ◽  
The Difference ◽  
White Noise Calculus

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.

scholarly journals Quantum white noises—White noise approach to quantum stochastic calculus

1993 ◽  
Vol 129 ◽  
pp. 23-42 ◽  
Author(s):  
Zhiyuan Huang
Keyword(s):  
Hilbert Space ◽  
White Noise ◽  
Fock Space ◽  
Stochastic Calculus ◽  
Quantum Stochastic Calculus ◽  
Integrable Functions ◽  
Boson Fock Space ◽  
Square Integrable Functions ◽  
Complex Valued ◽  
White Noises

Let H = L2 (R) be the Hilbert space of all complex-valued square integrable functions defined on R, Ф = Γ(H) be the Boson Fock space over H. For each h ∈ H, denote by ε(h) the corresponding exponential vector:in particular ε(0) is the Fock vacuum.

Quantum Lévy white noise derivatives with their applications to implementation problem

2014 ◽  
Vol 22 (4) ◽  
Author(s):  
Anis Riahi
Keyword(s):  
Differential Equations ◽  
White Noise ◽  
Fock Space ◽  
Wick Product ◽  
System Of Differential Equations ◽  
Fundamental Properties ◽  
Implementation Problem ◽  
Lévy White Noise ◽  
Derivatives Of

AbstractOn the basis of the quantum Lévy white noise theory we introduce the notion of creation and annihilation derivatives of Fock space operators and examine their fundamental properties, in particular, with respect to the Wick product. Next, we solve the implementation problem witch is reduced to a system of differential equations containing the quantum Lévy white noise derivatives.

Sign in / Sign up

Export Citation Format

Share Document