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.

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.

scholarly journals Interconnection between Wick multiplication and integration on spaces of nonregular generalized functions in the Lévy white noise analysis

2019 ◽  
Vol 11 (1) ◽  
pp. 70-88
Author(s):  
N.A. Kachanovsky ◽  
T.O. Kachanovska
Keyword(s):  
White Noise ◽  
Stochastic Integral ◽  
Noise Analysis ◽  
Stochastic Equations ◽  
Generalized Functions ◽  
Wick Product ◽  
Pettis Integral ◽  
White Noise Analysis ◽  
Lévy White Noise

We deal with spaces of nonregular generalized functions in the Lévy white noise analysis, which are constructed using Lytvynov's generalization of a chaotic representation property. Our aim is to describe a relationship between Wick multiplication and integration on these spaces. More exactly, we show that when employing the Wick multiplication, it is possible to take a time-independent multiplier out of the sign of an extended stochastic integral; establish an analog of this result for a Pettis integral (a weak integral); and prove a theorem about a representation of the extended stochastic integral via the Pettis integral from the Wick product of the original integrand by a Lévy white noise. As examples of an application of our results, we consider some stochastic equations with Wick type nonlinearities.

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

2018 ◽  
Vol 10 (1) ◽  
pp. 114-132
Author(s):  
N.A. Kachanovsky
Keyword(s):  
White Noise ◽  
Noise Analysis ◽  
Holomorphic Functions ◽  
Generalized Functions ◽  
Leibniz Rule ◽  
Wick Product ◽  
Stochastic Integrals ◽  
White Noise Analysis ◽  
Actual Problem ◽  
Lévy White Noise

Development of a theory of test and generalized functions depending on infinitely many variables is an important and actual problem, which is stipulated by requirements of physics and mathematics. One of  successful approaches to building of such a theory consists in introduction of spaces of the above-mentioned functions in such a way that the dual pairing between test and generalized functions is generated by integration with respect to some probability measure. First it was the Gaussian measure, then it were realized numerous generalizations. In particular, important results can be obtained if one uses the Levy white noise measure, the corresponding theory is called the Levy white noise analysis. In the Gaussian case one can construct spaces of test and generalized functions and introduce some important operators (e.g., stochastic integrals and derivatives) on these spaces by means of a so-called chaotic representation property (CRP): roughly speaking, any square integrable random variable can be decomposed in a series of repeated Itos stochastic integrals from nonrandom functions. In the Levy analysis there is no the CRP, but there are different generalizations of this property. In this paper we deal with one of the most useful and challenging generalizations of the CRP in the Levy analysis, which is proposed by E.W. Lytvynov, and with corresponding spaces of nonregular generalized functions. The goal of the paper is to introduce a natural product (a Wick product) on these spaces, and to study some related topics. Main results are theorems about properties of the Wick product and of Wick versions of holomorphic functions. In particular, we prove that an operator of stochastic differentiation satisfies the Leibniz rule with respect to the Wick multiplication. In addition we show that the Wick products and the Wick versions of holomorphic functions, defined on the spaces of regular and nonregular generalized functions, constructed by means of Lytvynov's generalization of the CRP, coincide on intersections of these spaces. Our research is a contribution in a further development of the Levy white noise analysis.

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.

scholarly journals NO ZERO DIVISOR FOR WICK PRODUCT IN (S)*

2008 ◽  
Vol 11 (02) ◽  
pp. 307-311 ◽  
Author(s):  
TAKAHIRO HASEBE ◽  
IZUMI OJIMA ◽  
HAYATO SAIGO
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Zero Divisor ◽  
Noise Analysis ◽  
Operational Calculus ◽  
Generalized Functions ◽  
Wick Product ◽  
White Noise Analysis ◽  
Remarkable Property ◽  
Zero Divisors

In White Noise Analysis (WNA), various random quantities are analyzed as elements of (S)*, the space of Hida distributions.1 Hida distributions are generalized functions of white noise, which is to be naturally viewed as the derivative of the Brownian motion. On (S)*, the Wick product is defined in terms of the [Formula: see text]-transform. We have found such a remarkable property that the Wick product has no zero divisors among Hida distributions. This result is a WNA version of Titchmarsh's theorem and is expected to play fundamental roles in developing the "operational calculus" in WNA along the line of Mikusiński's version for solving differential equations.

White-noise analysis in retinal physiology

1986 ◽  
Vol 4 ◽  
pp. S141-S152
Author(s):  
Masanori Sakuranaga ◽  
Yu-Ichiro Ando ◽  
Ken-Ichi Naka
Keyword(s):  
White Noise ◽  
Noise Analysis ◽  
White Noise Analysis

A Poisson Structure in White-Noise Analysis

2009 ◽  
Author(s):  
R. Léandre ◽  
Piotr Kielanowski ◽  
S. Twareque Ali ◽  
Anatol Odzijewicz ◽  
Martin Schlichenmaier ◽  
...  
Keyword(s):  
White Noise ◽  
Poisson Structure ◽  
Noise Analysis ◽  
White Noise Analysis

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.

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