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 ANALYSIS: SOME APPLICATIONS IN COMPLEX SYSTEMS, BIOPHYSICS AND QUANTUM MECHANICS

2012 ◽  
Vol 26 (29) ◽  
pp. 1230014 ◽  
Author(s):  
CHRISTOPHER C. BERNIDO ◽  
M. VICTORIA CARPIO-BERNIDO
Keyword(s):  
Quantum Mechanics ◽  
Probability Density Function ◽  
Complex Systems ◽  
White Noise ◽  
Noise Analysis ◽  
Social Systems ◽  
White Noise Analysis ◽  
The Past ◽  
White Noise Calculus ◽  
With Memory

The white noise calculus originated by T. Hida is presented as a powerful tool in investigating physical and social systems. Combined with Feynman's sum-over-all histories approach, we parameterize paths with memory of the past, and evaluate the corresponding probability density function. We discuss applications of this approach to problems in complex systems and biophysics. Examples in quantum mechanics with boundaries are also given where Markovian paths are considered.

scholarly journals ROLES OF LOG-CONCAVITY, LOG-CONVEXITY, AND GROWTH ORDER IN WHITE NOISE ANALYSIS

2001 ◽  
Vol 04 (01) ◽  
pp. 59-84 ◽  
Author(s):  
NOBUHIRO ASAI ◽  
IZUMI KUBO ◽  
HUI-HSIUNG KUO
Keyword(s):  
White Noise ◽  
Noise Analysis ◽  
Holomorphic Functions ◽  
Legendre Transform ◽  
Legendre Functions ◽  
Growth Order ◽  
White Noise Analysis ◽  
Systematic Method ◽  
Growth Functions ◽  
Natural Way

In this paper we will develop a systematic method to answer the questions (Q1) (Q2) (Q3) (Q4) (stated in Sec. 1) with complete generality. As a result, we can solve the difficulties (D1) (D2) (discussed in Sec. 1) without uncertainty. For these purposes we will introduce certain classes of growth functions u and apply the Legendre transform to obtain a sequence which leads to the weight sequence {α(n)} first studied by Cochran et al.6 The notion of (nearly) equivalent functions, (nearly) equivalent sequences and dual Legendre functions will be defined in a very natural way. An application to the growth order of holomorphic functions on ℰc will also be discussed.

COMPLEX WHITE NOISE ANALYSIS

2001 ◽  
Vol 04 (03) ◽  
pp. 347-375
Author(s):  
MYLAN REDFERN
Keyword(s):  
Stochastic Processes ◽  
White Noise ◽  
Fock Space ◽  
Noise Analysis ◽  
White Noise Analysis ◽  
Space Decomposition ◽  
New Space ◽  
Class Of Functions

This paper describes a new space, [Formula: see text], of complex Wiener distributions for the analysis of multi-parameter generalized stochastic processes [Formula: see text]. For a certain class of functions [Formula: see text] and complex Wiener integrals Φ1, …, Φm, F(Φ1, …, Φm) is defined as an element of [Formula: see text] and its Fock space decomposition determined.

A NOTE ON CONVOLUTION OPERATORS IN WHITE NOISE CALCULUS

2011 ◽  
Vol 14 (04) ◽  
pp. 661-674 ◽  
Author(s):  
NOBUAKI OBATA ◽  
HABIB OUERDIANE
Keyword(s):  
Laplace Transform ◽  
White Noise ◽  
Convolution Operator ◽  
Fock Space ◽  
Wick Product ◽  
Convolution Product ◽  
Convolution Operators ◽  
S Transform ◽  
The Laplace Transform ◽  
White Noise Calculus

We derive some characteristic properties of the convolution operator acting on white noise functions and prove that the convolution product of white noise distributions coincides with their Wick product. Moreover, we show that the S-transform and the Laplace transform coincide on Fock space.

A white noise analysis of Volterra processes

2015 ◽  
Vol 36 ◽  
pp. 1560005 ◽  
Author(s):  
H. P. Suryawan
Keyword(s):  
White Noise ◽  
Explicit Expression ◽  
Noise Analysis ◽  
Analysis Framework ◽  
White Noise Analysis ◽  
Delta Functions ◽  
Chaos Decomposition ◽  
Tensor Powers

In this paper we present a realization of Volterra processes within the white noise analysis framework. We show that Donsker's delta functions of Volterra processes are elements from the space of Hida distributions. An explicit expression for the corresponding chaos decomposition in terms of Wick tensor powers of white noise is also given.

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.

Pascal white noise harmonic analysis on configuration spaces and applications

2018 ◽  
Vol 21 (04) ◽  
pp. 1850024
Author(s):  
Anis Riahi ◽  
Habib Rebei
Keyword(s):  
White Noise ◽  
Harmonic Analysis ◽  
Configuration Space ◽  
Fock Space ◽  
Noise Analysis ◽  
Random Measure ◽  
Configuration Spaces ◽  
Wick Product ◽  
White Noise Analysis ◽  
Natural Realization

In this paper, we unify techniques of Pascal white noise analysis and harmonic analysis on configuration spaces establishing relations between the main structures of both ones. Fix a Random measure [Formula: see text] on a Riemannian manifold [Formula: see text], we construct on the space of finite compound configuration space [Formula: see text] the so-called Lebesgue–Pascal measure [Formula: see text] and as a consequence we obtain the Pascal measure [Formula: see text] on the compound configuration space [Formula: see text]. Next, the natural realization of the symmetric Fock space over [Formula: see text] as the space [Formula: see text] leads to the unitary isomorphism [Formula: see text] between the space [Formula: see text] and [Formula: see text]. Finally, in the first application we study some algebraic products, namely, the Borchers product on the Fock space, the Wick product on the Pascal space, and the ⋆-convolution on the Lebesgue–Pascal space and we prove that the Pascal white noise analysis and harmonic analysis are related through an equality of operators involving [Formula: see text]. The second application is devoted to solve the implementation problem.

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.

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