Fractional Brownian motion and motion governed by the fractional Langevin equation in confined geometries

2010 ◽  
Vol 81 (2) ◽  
Author(s):  
Jae-Hyung Jeon ◽  
Ralf Metzler
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Langevin Equation ◽  
Confined Geometries ◽  
Fractional Langevin Equation

Numerics for the fractional Langevin equation driven by the fractional Brownian motion

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Peng Guo ◽  
Caibin Zeng ◽  
Changpin Li ◽  
YangQuan Chen
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Fractional Order ◽  
Langevin Equation ◽  
Mean Square Displacement ◽  
Hurst Parameter ◽  
Mean Square ◽  
Fractional Langevin Equation ◽  
Normal Diffusion ◽  
The Mean

AbstractWe study analytically and numerically the fractional Langevin equation driven by the fractional Brownian motion. The fractional derivative is in Caputo’s sense and the fractional order in this paper is α = 2 − 2H, where H ∈ ($\tfrac{1} {2} $, 1) is the Hurst parameter (or, index). We give numerical schemes for the fractional Langevin equation with or without an external force. From the figures we can find that the mean square displacement of the fractional Langevin equation has the property of the anomalous diffusion. When the fractional order tends to an integer, the diffusion reduces to the normal diffusion.

scholarly journals A Wavelet-Based Almost-Sure Uniform Approximation of Fractional Brownian Motion with a Parallel Algorithm

2014 ◽  
Vol 51 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Dawei Hong ◽  
Shushuang Man ◽  
Jean-Camille Birget ◽  
Desmond S. Lun
Keyword(s):  
Brownian Motion ◽  
Convergence Rate ◽  
Fractional Brownian Motion ◽  
Parallel Algorithm ◽  
Uniform Approximation ◽  
Haar Wavelets ◽  
Hurst Index ◽  
Sample Paths ◽  
Vanishing Moment ◽  
Uniform Expansion

We construct a wavelet-based almost-sure uniform approximation of fractional Brownian motion (FBM) (Bt(H))_t∈[0,1] of Hurst index H ∈ (0, 1). Our results show that, by Haar wavelets which merely have one vanishing moment, an almost-sure uniform expansion of FBM for H ∈ (0, 1) can be established. The convergence rate of our approximation is derived. We also describe a parallel algorithm that generates sample paths of an FBM efficiently.

Global attracting sets of neutral stochastic functional integro-differential equations driven by a fractional Brownian motion

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
A. Bakka ◽  
S. Hajji ◽  
D. Kiouach
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Sufficient Conditions ◽  
Integrodifferential Equations ◽  
Hurst Parameter ◽  
Fixed Point Principle ◽  
Stochastic Functional

Abstract By means of the Banach fixed point principle, we establish some sufficient conditions ensuring the existence of the global attracting sets of neutral stochastic functional integrodifferential equations with finite delay driven by a fractional Brownian motion (fBm) with Hurst parameter H ∈ ( 1 2 , 1 ) {H\in(\frac{1}{2},1)} in a Hilbert space.

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