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 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 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).

scholarly journals The 𝒮-Transform of Sub-fBm and an Application to a Class of Linear Subfractional BSDEs

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Zhi Wang ◽  
Litan Yan
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
White Noise ◽  
Stochastic Differential Equations ◽  
Stochastic Integral ◽  
Noise Analysis ◽  
Explicit Solutions ◽  
Pettis Integral ◽  
White Noise Analysis ◽  
Subfractional Brownian Motion

Let SH be a subfractional Brownian motion with index 0<H<1. Based on the 𝒮-transform in white noise analysis we study the stochastic integral with respect to SH, and we also prove a Girsanov theorem and derive an Itô formula. As an application we study the solutions of backward stochastic differential equations driven by SH of the form -dYt=f(t,Yt,Zt)dt-ZtdStH, t∈[0,T],YT=ξ, where the stochastic integral used in the above equation is Pettis integral. We obtain the explicit solutions of this class of equations under suitable assumptions.

scholarly journals Operators of stochastic differentiation on spaces of nonregular generalized functions of Levy white noise analysis

2016 ◽  
Vol 8 (1) ◽  
pp. 83-106
Author(s):  
N.A. Kachanovsky
Keyword(s):  
White Noise ◽  
Stochastic Integral ◽  
Noise Analysis ◽  
Gaussian White Noise ◽  
Generalized Functions ◽  
Test Functions ◽  
White Noise Analysis ◽  
Lévy White Noise ◽  
Stochastic Derivative ◽  
Further Development

The operators of stochastic differentiation, which are closely related with the extended Skorohod stochastic integral and with the Hida stochastic derivative, play an important role in the classical (Gaussian) white noise analysis. In particular, these operators can be used in order to study some properties of the extended stochastic integral and of solutions of stochastic equations with Wick-type nonlinearities. During recent years the operators of stochastic differentiation were introduced and studied, in particular, in the framework of the Meixner white noise analysis, in the same way as on spaces of regular test and generalized functions and on spaces of nonregular test functions of the Levy white noise analysis. In the present paper we make the next natural step: introduce and study operators of stochastic differentiation on spaces of nonregular generalized functions of the Levy white noise analysis (i.e., on spaces of generalized functions that belong to the so-called nonregular rigging of the space of square integrable with respect to the measure of a Levy white noise functions). In so doing, we use Lytvynov's generalization of the chaotic representation property. The researches of the present paper can be considered as a contribution in a further development of the Levy white noise analysis. 

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

2020 ◽  
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 On operators of stochastic differentiation on spaces of regular test and generalized functions of Lévy white noise analysis

2014 ◽  
Vol 6 (2) ◽  
pp. 212-229 ◽  
Author(s):  
M.M. Dyriv ◽  
N.A. Kachanovsky
Keyword(s):  
White Noise ◽  
Stochastic Integral ◽  
Noise Analysis ◽  
Gaussian White Noise ◽  
Stochastic Equations ◽  
Generalized Functions ◽  
White Noise Analysis ◽  
Unbounded Operators ◽  
Lévy White Noise ◽  
Stochastic Derivative

The operators of stochastic differentiation, which are closely related with the extended Skorohod stochastic integral and with the Hida stochastic derivative, play an important role in the classical (Gaussian) white noise analysis. In particular, these operators can be used in order to study properties of the extended stochastic integral and of solutions of stochastic equations with Wick-type nonlinearities. In this paper we introduce and study bounded and unbounded operators of stochastic differentiation in the Levy white noise analysis. More exactly, we consider these operators on spaces from parametrized regular rigging of the space of square integrable with respect to the measure of a Levy white noise functions, using the Lytvynov's generalization of the chaotic representation property. This gives a possibility to extend to the Levy white noise analysis and to deepen the corresponding results of the classical white noise analysis.

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