scholarly journals The q-Gamma White Noise

2016 ◽  
Vol 66 (1) ◽  
pp. 81-90
Author(s):  
Hakeem A. Othman
Keyword(s):  
White Noise ◽  
Fock Space ◽  
Infinite Dimensional ◽  
Chaos Decomposition ◽  
Noise Measure ◽  
Interacting Fock Space

Abstract For 0 < q < 1 and 0 < α < 1, we construct the infinite dimensional q-Gamma white noise measure γα,q by using the Bochner-Minlos theorem. Then we give the chaos decomposition of an L2 space with respect to the measure γα,q via an isomorphism with the 1-mode type interacting Fock space associated to the q-Gamma measure.

scholarly journals Generalized free Gaussian white noise

2016 ◽  
Vol 4 (1) ◽  
pp. 18
Author(s):  
Hakeem Othman
Keyword(s):  
White Noise ◽  
Gaussian White Noise ◽  
Infinite Dimensional ◽  
Chaos Decomposition ◽  
Noise Measure

<p>Based on an adequate new Gel'fand triple,  we construct the infinite dimensional free Gaussian white noise measure \(\mu\) using the Bochner-Minlos theorem. Next, we give the chaos decomposition of an \(L^{2}\) space with respect to the measure \(\mu\).</p>

scholarly journals White noise delta functions and continuous version theorem

1993 ◽  
Vol 129 ◽  
pp. 1-22
Author(s):  
Nobuaki Obata
Keyword(s):  
White Noise ◽  
Harmonic Analysis ◽  
Quantum Physics ◽  
Fock Space ◽  
Continuous Version ◽  
Infinite Dimensional ◽  
Schwartz Distribution ◽  
Delta Functions ◽  
Gelfand Triple ◽  
Gaussian Space

The recently developed Hida calculus of white noise [5] is an infinite dimensional analogue of Schwartz’ distribution theory besed on the Gelfand triple (E) ⊂ (L2) = L2 (E*, μ) ⊂ (E)*, where (E*, μ) is Gaussian space and (L2) is (a realization of) Fock space. It has been so far discussed aiming at an application to quantum physics, for instance [1], [3], and infinite dimensional harmonic analysis [7], [8], [13], [14], [15].

scholarly journals OPERATORS OF GAMMA WHITE NOISE CALCULUS

2000 ◽  
Vol 03 (03) ◽  
pp. 303-335 ◽  
Author(s):  
YURI G. KONDRATIEV ◽  
EUGENE W. LYTVYNOV
Keyword(s):  
White Noise ◽  
Fock Space ◽  
Noise Analysis ◽  
Gamma Field ◽  
Chaos Decomposition ◽  
The Creation ◽  
The Fourier Transform ◽  
White Noise Calculus ◽  
Adjoint Operators ◽  
Natural Way

The paper is devoted to the study of Gamma white noise analysis. We define an extended Fock space ℱ ext (ℋ) over ℋ= L2(ℝd, dσ) and show how to include the usual Fock space ℱ(ℋ) in it as a subspace. We introduce in ℱ ext (ℋ) operators a(ξ)=∫ℝddxξ(x)a(x), ξ∈ S, with [Formula: see text], where [Formula: see text] and ∂x are the creation and annihilation operators at x. We show that (a(ξ))ξ∈S is a family of commuting self-adjoint operators in ℱ ext (ℋ) and construct the Fourier transform in generalized joint eigenvectors of this family. This transform is a unitary I between ℱ ext (ℋ) and the L2-space L2(S', dμ G ), where μ G is the measure of Gamma white noise with intensity σ. The image of a(ξ) under I is the operator of multiplication by <·,ξ>, so that a(ξ)'s are Gamma field operators. The Fock structure of the Gamma space determined by I coincides with that discovered in Ref. 22. We note that I extends in a natural way the multiple stochastic integral (chaos) decomposition of the "chaotic" subspace of the Gamma space. Next, we introduce and study spaces of test and generalized functions of Gamma white noise and derive explicit formulas for the action of the creation, neutral, and Gamma annihilation operators on these spaces.

WHITE NOISE LÉVY–MEIXNER PROCESSES THROUGH A TRANSFER PRINCIPAL FROM ONE-MODE TO ONE-MODE TYPE INTERACTING FOCK SPACES

2010 ◽  
Vol 13 (03) ◽  
pp. 435-460 ◽  
Author(s):  
LUIGI ACCARDI ◽  
ABDESSATAR BARHOUMI ◽  
ANIS RIAHI
Keyword(s):  
White Noise ◽  
Fock Space ◽  
Delta Function ◽  
Fock Spaces ◽  
Jacobi Fields ◽  
Derivative Formula ◽  
Scalar Products ◽  
The Fourier Transform ◽  
The One ◽  
Interacting Fock Space

Consider the Lévy–Meixner one-mode interacting Fock space {ΓLM, 〈 ⋅, ⋅ 〉LM}. Inspired by a derivative formula appearing in 〈 ⋅, ⋅ 〉LM, we define scalar products 〈 ⋅, ⋅ 〉LM , nin symmetric n-particle spaces. Then, we introduce a class of one-mode type interacting Fock spaces [Formula: see text] naturally associated to the one-dimensional infinitely divisible distributions with Lévy–Meixner type {μr; r > 0}. The Fourier transform in generalized joint eigenvectors of a family [Formula: see text] of Lévy–Meixner Jacobi fields provides a way to explicit a unitary isomorphism 𝔘LMbetween [Formula: see text] and the so-called Lévy–Meixner white noise space [Formula: see text]. We derive a chaotic decomposition property of the quadratic integrable functionals of the Lévy–Meixner white noise processes in terms of an appropriate Wick tensor product. For their stochastic regularity, we give explicit form and sharp estimate of the associated Donsker's delta function.

ON THE QUADRATIC HEISENBERG GROUP

2010 ◽  
Vol 13 (04) ◽  
pp. 551-587 ◽  
Author(s):  
LUIGI ACCARDI ◽  
HABIB OUERDIANE ◽  
HABIB REBEÏ
Keyword(s):  
White Noise ◽  
Heisenberg Group ◽  
Lie Group ◽  
Unitary Operator ◽  
Fock Space ◽  
Local Chart ◽  
Rational Expression ◽  
The One ◽  
Interacting Fock Space ◽  
Group Law

In this paper we introduce the quadratic Weyl operators canonically associated to the one mode renormalized square of white noise (RSWN) algebra as unitary operator acting on the one mode interacting Fock space {Γ, {ωn, n ∈ ℕ}, Φ} where {ωn, n ∈ ℕ} is the principal Jacobi sequence of the nonstandard (i.e. neither Gaussian nor Poisson) Meixner classes. We deduce the quadratic Weyl relations and construct the quadratic analogue of the Heisenberg Lie group with one degree of freedom. In particular, we determine the manifold structure of the group and introduce a local chart containing the identity on which the group law has a simple rational expression in the chart coordinates (see Theorem 6.3).

scholarly journals An Analytic Characterization of p , q -White Noise Functionals

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Anis Riahi ◽  
Amine Ettaieb ◽  
Wathek Chammam ◽  
Ziyad Ali Alhussain
Keyword(s):  
White Noise ◽  
Entire Functions ◽  
Fock Space ◽  
Gaussian White Noise ◽  
Characterization Theorem ◽  
Quantum Oscillator ◽  
Exponential Map ◽  
Infinite Dimensional ◽  
White Noise Functionals ◽  
Young Functions

In this paper, a characterization theorem for the S -transform of infinite dimensional distributions of noncommutative white noise corresponding to the p , q -deformed quantum oscillator algebra is investigated. We derive a unitary operator U between the noncommutative L 2 -space and the p , q -Fock space which serves to give the construction of a white noise Gel’fand triple. Next, a general characterization theorem is proven for the space of p , q -Gaussian white noise distributions in terms of new spaces of p , q -entire functions with certain growth rates determined by Young functions and a suitable p , q -exponential map.

ANALYTIC CHARACTERIZATION OF GENERALIZED FOCK SPACE OPERATORS AS TWO-VARIABLE ENTIRE FUNCTIONS WITH GROWTH CONDITION

2002 ◽  
Vol 05 (03) ◽  
pp. 395-407 ◽  
Author(s):  
UN CIG JI ◽  
NOBUAKI OBATA ◽  
HABIB OUERDIANE
Keyword(s):  
Differential Equation ◽  
White Noise ◽  
Growth Condition ◽  
Growth Rates ◽  
Entire Functions ◽  
Fock Space ◽  
Infinite Dimensional ◽  
Singular Coefficients ◽  
Young Functions

Duality is established for new spaces of entire functions in two infinite dimensional variables with certain growth rates determined by Young functions. These entire functions characterize the symbols of generalized Fock space operators. As an application, a proper space is found for a solution to a normal-ordered white noise differential equation having highly singular coefficients.

scholarly journals Infinite dimensional rotations and Laplacians in terms of white noise calculus

1992 ◽  
Vol 128 ◽  
pp. 65-93 ◽  
Author(s):  
Takeyuki Hida ◽  
Nobuaki Obata ◽  
Kimiaki Saitô
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Quantum Physics ◽  
Variational Calculus ◽  
Rotation Group ◽  
Infinite Dimensional ◽  
Lévy Laplacian ◽  
White Noise Functionals ◽  
White Noise Calculus

The theory of generalized white noise functionals (white noise calculus) initiated in [2] has been considerably developed in recent years, in particular, toward applications to quantum physics, see e.g. [5], [7] and references cited therein. On the other hand, since H. Yoshizawa [4], [23] discussed an infinite dimensional rotation group to broaden the scope of an investigation of Brownian motion, there have been some attempts to introduce an idea of group theory into the white noise calculus. For example, conformal invariance of Brownian motion with multidimensional parameter space [6], variational calculus of white noise functionals [14], characterization of the Levy Laplacian [17] and so on.

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