A New Look at the Fractional Brownian Motion Definition

2007 ◽  
Author(s):  
Manuel Duarte Ortigueira ◽  
Arnaldo Guimara˜es Batista
Keyword(s):  
Brownian Motion ◽  
Fractional Derivative ◽  
White Noise ◽  
Fractional Brownian Motion ◽  
Fractional Derivatives ◽  
Autocorrelation Functions ◽  
Fractional Noise ◽  
Simulation Procedure ◽  
Definition Of ◽  
Fractional Noises

A reinterpretation of the classic definition of fractional Brownian motion leads to a new definition involving a fractional noise obtained as a fractional derivative of white noise. To obtain this fractional noise, two sets of fractional derivatives are considered: a) the forward and backward and b) the central derivatives. For these derivatives the autocorrelation functions of the corresponding fractional noises have the same representations. The obtained results are used to define and propose a new simulation procedure.

White noise delta functions and infinite-dimensional Laplacians

2020 ◽  
pp. 2050028
Author(s):  
Luigi Accardi ◽  
Ai Hasegawa ◽  
Un Cig Ji ◽  
Kimiaki Saitô
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Time Derivative ◽  
Delta Function ◽  
Itô Formula ◽  
Ito Formula ◽  
Infinite Dimensional ◽  
Delta Functions ◽  
Definition Of ◽  
White Noise Distribution

In this paper, we introduce a new white noise delta function based on the Kubo–Yokoi delta function and an infinite-dimensional Brownian motion. We also give a white noise differential equation induced by the delta function through the Itô formula introducing a differential operator directed by the time derivative of the infinite-dimensional Brownian motion and an extension of the definition of the Volterra Laplacian. Moreover, we give an extension of the Itô formula for the white noise distribution of the infinite-dimensional Brownian motion.

scholarly journals FRACTIONAL DERIVATIVE AS FRACTIONAL POWER OF DERIVATIVE

2007 ◽  
Vol 18 (03) ◽  
pp. 281-299 ◽  
Author(s):  
VASILY E. TARASOV
Keyword(s):  
Fourier Series ◽  
Fractional Derivative ◽  
Taylor Series ◽  
Fractional Derivatives ◽  
Fractional Power ◽  
Weyl Quantization ◽  
Fractional Powers ◽  
Derivative Operator ◽  
Quantization Procedure ◽  
Definition Of

Definitions of fractional derivatives as fractional powers of derivative operators are suggested. The Taylor series and Fourier series are used to define fractional power of selfadjoint derivative operator. The Fourier integrals and Weyl quantization procedure are applied to derive the definition of fractional derivative operator. Fractional generalization of concept of stability is considered.

scholarly journals On the relation between the fractional Brownian motion and the fractional derivatives

2008 ◽  
Vol 372 (7) ◽  
pp. 958-968 ◽  
Author(s):  
Manuel Duarte Ortigueira ◽  
Arnaldo Guimarães Batista
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Fractional Derivatives

scholarly journals A New Conformable Fractional Derivative and Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Ahmed Kajouni ◽  
Ahmed Chafiki ◽  
Khalid Hilal ◽  
Mohamed Oukessou
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Conformable Fractional Derivative ◽  
First Order ◽  
Rolle’S Theorem ◽  
Power Rule ◽  
The Mean ◽  
Definition Of

This paper is motivated by some papers treating the fractional derivatives. We introduce a new definition of fractional derivative which obeys classical properties including linearity, product rule, quotient rule, power rule, chain rule, Rolle’s theorem, and the mean value theorem. The definition D α f t = lim h ⟶ 0 f t + h e α − 1 t − f t / h , for all t > 0 , and α ∈ 0,1 . If α = 0 , this definition coincides to the classical definition of the first order of the function f .

scholarly journals New Approach for the Analysis of Damped Vibrations of Fractional Oscillators

2009 ◽  
Vol 16 (4) ◽  
pp. 365-387 ◽  
Author(s):  
Yuriy A. Rossikhin ◽  
Marina V. Shitikova
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Time Derivative ◽  
Fractional Power ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Derivative Term ◽  
Mechanical Oscillators ◽  
Definition Of ◽  
Nonlinear Elastic

The dynamic behavior of linear and nonlinear mechanical oscillators with constitutive equations involving fractional derivatives defined as a fractional power of the operator of conventional time-derivative is considered. Such a definition of the fractional derivative enables one to analyse approximately vibratory regimes of the oscillator without considering the drift of its position of equilibrium. The assumption of small fractional derivative terms allows one to use the method of multiple time scales whereby a comparative analysis of the solutions obtained for different orders of low-level fractional derivatives and nonlinear elastic terms is possible to be carried out. The interrelationship of the fractional parameter (order of the fractional operator) and nonlinearity manifests itself in full measure when orders of the small fractional derivative term and of the cubic nonlinearity entering in the oscillator's constitutive equation coincide.

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