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.

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.

White-noise analysis in visual neuroscience

1988 ◽  
Vol 1 (3) ◽  
pp. 287-296 ◽  
Author(s):  
Hiroko M. Sakai ◽  
Naka Ken-Ichi ◽  
Michael J. Korenberg
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Light Stimulus ◽  
Noise Analysis ◽  
Gaussian White Noise ◽  
White Noise Analysis ◽  
Response Dynamics ◽  
Visual Neuroscience ◽  
Final Output ◽  
Mean Luminance

AbstractIn 1827, plant biologist Robert Brown discovered what is known as Brownian motion, a class of chaos. Formal derivative of Brownian motion is Gaussian white-noise. In 1938, Norbert Wiener proposed to use the Gaussian white-noise as an input probe to identify a system by a series of orthogonal functionals known as the Wiener G-functionals.White-noise analysis is uniquely suited for studying the response dynamics of retinal neurons because (1) white-noise light stimulus is a modulation around a mean luminance, as are the natural photic inputs, and it is a highly efficient input; and (2) the analysis defines the response dynamics and can be extended to spike trains, the final output of the retina. Demonstrated here are typical examples and results from applications of white-noise analysis to a visual system.

scholarly journals Local times of self-intersection for multidimensional Brownian motion

1995 ◽  
Vol 138 ◽  
pp. 51-64 ◽  
Author(s):  
Sheng-Wu He ◽  
Wen-Qiang Yang ◽  
Rong-Qin Yao ◽  
Jia-Gang Wang
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Main Part ◽  
Noise Analysis ◽  
Local Times ◽  
White Noise Analysis ◽  
Deep Insight ◽  
Insight Into

We will define local times of self-intersection for multidimensional Brownian motion as generalized Wiener functionals under the framework of white noise analysis as in H. Watanabe ([6]). By making use of the chaotic representation of -function and precise computation we get a deep insight into the problem. In the section 1 multiple Wiener integrals with respect to multidimensional Brownian motion and chaotic representations for square-integrable Wiener functionals are given. They are indispensable, but seem not to be formulated clearly and correctly before. The useful concepts and results of white noise analysis are illustrated in the section 2. Section 3 is the main part of the paper. The applications to local times are introduced in the section 4 briefly.

scholarly journals On the Self-Intersection Local Time of Subfractional Brownian Motion

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Junfeng Liu ◽  
Zhihang Peng ◽  
Donglei Tang ◽  
Yuquan Cang
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Local Time ◽  
Noise Analysis ◽  
The Self ◽  
White Noise Analysis ◽  
Intersection Local Time ◽  
Subfractional Brownian Motion

We study the problem of self-intersection local time ofd-dimensional subfractional Brownian motion based on the property of chaotic representation and the white noise analysis.

scholarly journals Weighted Local Times of a Sub-fractional Brownian Motion as Hida Distributions

2019 ◽  
Vol 15 (2) ◽  
pp. 81 ◽  
Author(s):  
Herry Pribawanto Suryawan
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Fractional Brownian Motion ◽  
Short Range ◽  
Probability Space ◽  
Noise Analysis ◽  
Local Times ◽  
White Noise Analysis ◽  
Self Similarity ◽  
Sample Paths

The sub-fractional Brownian motion is a Gaussian extension of the Brownian motion. It has the properties of self-similarity, continuity of the sample paths, and short-range dependence, among others. The increments of sub-fractional Brownian motion is neither independent nor stationary. In this paper we study the sub-fractional Brownian motion using a white noise analysis approach. We recall the represention of sub-fractional Brownian motion on the white noise probability space and show that Donsker's delta functional of a sub-fractional Brownian motion is a Hida distribution. As a main result, we prove the existence of the weighted local times of a $d$-dimensional sub-fractional Brownian motion as Hida distributions.

scholarly journals White noise analysis and Tanaka formula for intersections of planar Brownian motion

1991 ◽  
Vol 122 ◽  
pp. 1-17 ◽  
Author(s):  
Narn-Rueih Shieh
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Stochastic Integral ◽  
Integral Formula ◽  
Noise Analysis ◽  
Local Times ◽  
White Noise Analysis ◽  
Planar Brownian Motion ◽  
Tanaka Formula ◽  
Intersection Local Times

In this paper, we shall use Hida’s [5, 7, 9] theory of generalized Brownian functionals (or named white noise analysis) to establish a stochastic integral formula concerning the multiple intersection local times of planar Brownian motion B(t).

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.

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