Gaussian white noise analysis and its application to Feynman path integral

2016 ◽  
Author(s):  
Herry Pribawanto Suryawan
Keyword(s):  
White Noise ◽  
Path Integral ◽  
Noise Analysis ◽  
Gaussian White Noise ◽  
Feynman Path Integral ◽  
White Noise Analysis ◽  
Feynman Path

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 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.

A RIGOROUS CONSTRUCTION OF ABELIAN CHERN-SIMONS PATH INTEGRALS USING WHITE NOISE ANALYSIS

1996 ◽  
Vol 08 (03) ◽  
pp. 445-456 ◽  
Author(s):  
PETER LEUKERT ◽  
JÖRG SCHÄFER
Keyword(s):  
White Noise ◽  
Noise Analysis ◽  
Path Integrals ◽  
Simons Theory ◽  
White Noise Analysis ◽  
Feynman Path ◽  
Feynman Path Integrals ◽  
Linking Numbers ◽  
Chern Simons ◽  
Chern Simons Theory

Using recent results in white noise analysis we rigorously construct Feynman path integrals for abelian Chern-Simons theory. We define Wilson loop variables and show that their expectation is a topological invariant, namely computable in terms of pairwise linking numbers as conjectured by E. Witten.

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 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.

PATH INTEGRAL APPROACH TO A SINGLE POLYMER CHAIN WITH RANDOM MEDIA

2004 ◽  
Vol 18 (10n11) ◽  
pp. 1465-1478 ◽  
Author(s):  
CH. KUNSOMBAT ◽  
V. SA-YAKANIT
Keyword(s):  
Polymer Chain ◽  
Path Integral ◽  
Random Media ◽  
Feynman Path Integral ◽  
Feynman Path ◽  
Short Range Correlation ◽  
Range Correlation ◽  
End Distance ◽  
Non Local ◽  
The Mean

In this paper we consider the problem of a polymer chain in random media with finite correlation. We show that the mean square end-to-end distance of a polymer chain can be obtained using the Feynman path integral developed by Feynman for treating the polaron problem and successfuly applied to the theory of heavily doped semiconductor. We show that for short-range correlation or the white Gaussian model we derive the results obtained by Edwards and Muthukumar using the replica method and for long-range correlation we obtain the result of Yohannes Shiferaw and Yadin Y. Goldschimidt. The main idea of this paper is to generalize the model proposed by Edwards and Muthukumar for short-range correlation to finite correlation. Instead of using a replica method, we employ the Feynman path integral by modeling the polymer Hamiltonian as a model of non-local quadratic trial Hamiltonian. This non-local trial Hamiltonian is essential as it will reflect the translation invariant of the original Hamiltonian. The calculation is proceeded by considering the differences between the polymer propagator and the trial propagator as the first cumulant approximation. The variational principle is used to find the optimal values of the variational parameters and the mean square end-to-end distance is obtained. Several asymptotic limits are considered and a comparison between this approaches and replica approach will be discussed.

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
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