scholarly journals Calculus on Gaussian and Poisson white noises

1988 ◽  
Vol 111 ◽  
pp. 41-84 ◽  
Author(s):  
Yoshifusa Ito ◽  
Izumi Kubo
Keyword(s):  
White Noise ◽  
Gaussian White Noise ◽  
Exceptional Point ◽  
Stochastic Integrals ◽  
Generalized Poisson ◽  
Gaussian Case ◽  
White Noises

Recently one of the authors has introduced the concept of generalized Poisson functionals and discussed the differentiation, renormalization, stochastic integrals etc. ([8], [9]), analogously to the works of T. Hida ([3], [4], [5]). Here we introduce a transformation for Poisson fnnctionals with the idea as in the case of Gaussian white noise (cf. [10], [11], [12], [13]). Then we can discuss the differentiation, renormalization, multiple Wiener integrals etc. in a way completely parallel with the Gaussian case. The only one exceptional point, which is most significant, is that the multiplications are described by for the Gaussian case, for the Poisson case,as will be stated in Section 5. Conversely, those formulae characterize the types of white noises.

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.

Generalized Fokker-Planck equation for a class of stochastic dynamical systems driven by additive Gaussian and Poissonian fractional white noises of order α

Open Physics ◽  
2008 ◽  
Vol 6 (3) ◽  
Author(s):  
Guy Jumarie
Keyword(s):  
White Noise ◽  
Fractional Order ◽  
Gaussian White Noise ◽  
Linear Partial Differential Equation ◽  
Planck Equation ◽  
Stochastic Dynamical Systems ◽  
Poissonian Noise ◽  
Liouville Fractional Derivative ◽  
White Noises ◽  
Taylor’S Series

AbstractIn a first stage, the paper deals with the derivation and the solution of the equation of the probability density function of a stochastic system driven simultaneously by a fractional Gaussian white noise and a fractional Poissonian white noise both of the same order. The key is the Taylor’s series of fractional order f(x + h) = E α(hαDx α)f(x) where E α() denotes the Mittag-Leffler function, and D x α is the so-called modified Riemann-Liouville fractional derivative which removes the effects of the non-zero initial value of the function under consideration. The corresponding fractional linear partial differential equation is solved by using a suitable extension of the Lagrange’s technique involving an auxiliary set of fractional differential equations. As an example, one considers a half-oscillator of fractional order driven by a fractional Poissonian noise.

HIDA DISTRIBUTION CONSTRUCTION OF NON-GAUSSIAN REFLECTION POSITIVE GENERALIZED RANDOM FIELDS

2009 ◽  
Vol 12 (01) ◽  
pp. 21-49 ◽  
Author(s):  
SERGIO ALBEVERIO ◽  
MINORU W. YOSHIDA
Keyword(s):  
White Noise ◽  
Random Fields ◽  
Gaussian White Noise ◽  
Random Variables ◽  
Rotation Invariance ◽  
Stochastic Integrals ◽  
Multiple Stochastic Integrals ◽  
Space Rotation ◽  
Non Gaussian ◽  
Euclidean Invariance

Multiple stochastic integrals with respect to the Gaussian white noise indexed by Rd are used for the construction of reflection positive non-Gaussian [Formula: see text] valued random variables (random fields). This construction is then extended to a class of Hida distribution to consider several interesting examples. In particular, some new fields (without full Euclidean invariance but satisfying all other axioms, including space rotation invariance) are constructed through this probabilistic (and purely Euclidean) procedure.

scholarly journals On stochastic differential equations driven by the renormalized square of the Gaussian white noise

2015 ◽  
Vol 18 (04) ◽  
pp. 1550025
Author(s):  
Bilel Kacem Ben Ammou ◽  
Alberto Lanconelli
Keyword(s):  
Differential Equations ◽  
White Noise ◽  
Stochastic Differential Equations ◽  
Gaussian White Noise ◽  
Wick Product ◽  
Nonlinear Functions ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Definition Of ◽  
White Noises

We investigate the properties of the Wick square of Gaussian white noises through a new method to perform nonlinear operations on Hida distributions. This method lays in between the Wick product interpretation and the usual definition of nonlinear functions. We prove an Itô-type formula and solve stochastic differential equations driven by the renormalized square of the Gaussian white noise. Our approach works with standard assumptions on the coefficients of the equations, global Lipschitz continuity, and produces existence and uniqueness results in the space where the noise lives. The linear case is studied in details and positivity of the solution is proved.

A Wavelet Spatial Correlation Algorithm to Partial Discharge Denoising Based on MDL Criterion

2013 ◽  
Vol 340 ◽  
pp. 642-646
Author(s):  
Li Song Tian ◽  
Wei Xuan Chen
Keyword(s):  
White Noise ◽  
Spatial Correlation ◽  
Partial Discharge ◽  
Parameters Estimation ◽  
Optimization Strategy ◽  
Signal Parameter ◽  
Detection Systems ◽  
Correlation Algorithm ◽  
White Noises ◽  
General Method

The partial discharge (PD) detection systems are often vulnerable to strong external interferences, and sometimes the PD signals are submerged in noises (white noise for example) completely. So the signals acquired must be preprocessed to obtain the reliable PD information. While there are many methods for white noise denoising, mostly are not very suitable for partial discharge. The wavelet transform (WT) coefficient of PD and white noises have different spread characteristics in different WT scales. Based on the Information Theory, The Minimum Information Description Length (MDL) criterion is a optimization strategy, a small amount of signal parameter is requried to the PD signals representation, the paper proposes a wavelet spatial correlation algorithm to partial discharge denoising based on MDL criterion: optimal wavelet function is selected based on MDL, then have the white noise reduced in WT, the algorithm has wonderful virtues such as free from any parameters estimation about noise, free from presetting threshhold and threshold chooseing behavior, so the algorithm is highly adaptive. Large amount of experimental results illustrate that the method presented in this paper are efficient and feasible and outperforms other general method of PD noise reduction.

scholarly journals Stochastic P-bifurcation in a bistable Van der Pol oscillator with fractional time-delay feedback under Gaussian white noise excitation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yajie Li ◽  
Zhiqiang Wu ◽  
Guoqi Zhang ◽  
Feng Wang ◽  
Yuancen Wang
Keyword(s):  
Time Delay ◽  
White Noise ◽  
Gaussian White Noise ◽  
Singularity Theory ◽  
Van Der Pol Oscillator ◽  
Order System ◽  
Damping Force ◽  
Van Der Pol ◽  
Restoring Force ◽  
White Noise Excitation

Abstract The stochastic P-bifurcation behavior of a bistable Van der Pol system with fractional time-delay feedback under Gaussian white noise excitation is studied. Firstly, based on the minimal mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of damping force and restoring force, and the original system is further simplified to an equivalent integer order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and the critical parametric conditions for stochastic P-bifurcation of system amplitude are determined according to the singularity theory. Finally, the types of stationary PDF curves of system amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical solutions and Monte Carlo simulation results verifies the theoretical analysis in this paper.

Exact Stationary Response Solution for Second Order Nonlinear Systems Under Parametric and External White Noise Excitations: Part II

1988 ◽  
Vol 55 (3) ◽  
pp. 702-705 ◽  
Author(s):  
Y. K. Lin ◽  
Guoqiang Cai
Keyword(s):  
Nonlinear System ◽  
Nonlinear Systems ◽  
Probability Distribution ◽  
White Noise ◽  
Probability Density ◽  
Detailed Balance ◽  
Stationary Probability ◽  
Probability Flow ◽  
Parametric And External Excitations ◽  
White Noises

A systematic procedure is developed to obtain the stationary probability density for the response of a nonlinear system under parametric and external excitations of Gaussian white noises. The procedure is devised by separating the circulatory portion of the probability flow from the noncirculatory flow, thus obtaining two sets of equations that must be satisfied by the probability potential. It is shown that these equations are identical to two of the conditions established previously under the assumption of detailed balance; therefore, one remaining condition for detailed balance is superfluous. Three examples are given for illustration, one of which is capable of exhibiting limit cycle and bifurcation behaviors, while another is selected to show that two different systems under two differents sets of excitations may result in the same probability distribution for their responses.

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