scholarly journals Generalized Fractional Master Equation for Self-Similar Stochastic Processes Modelling Anomalous Diffusion

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Gianni Pagnini ◽  
Antonio Mura ◽  
Francesco Mainardi
Keyword(s):  
Brownian Motion ◽  
Stochastic Processes ◽  
Master Equation ◽  
Fractional Brownian Motion ◽  
Anomalous Diffusion ◽  
Natural Generalization ◽  
Fractional Diffusion ◽  
Gaussian Density ◽  
Master Equation Approach ◽  
Self Similar

The Master Equation approach to model anomalous diffusion is considered. Anomalous diffusion in complex media can be described as the result of a superposition mechanism reflecting inhomogeneity and nonstationarity properties of the medium. For instance, when this superposition is applied to the time-fractional diffusion process, the resulting Master Equation emerges to be the governing equation of the Erdélyi-Kober fractional diffusion, that describes the evolution of the marginal distribution of the so-called generalized grey Brownian motion. This motion is a parametric class of stochastic processes that provides models for both fast and slow anomalous diffusion: it is made up of self-similar processes with stationary increments and depends on two real parameters. The class includes the fractional Brownian motion, the time-fractional diffusion stochastic processes, and the standard Brownian motion. In this framework, the M-Wright function (known also as Mainardi function) emerges as a natural generalization of the Gaussian distribution, recovering the same key role of the Gaussian density for the standard and the fractional Brownian motion.

scholarly journals Erdélyi-Kober fractional diffusion

2012 ◽  
Vol 15 (1) ◽  
Author(s):  
Gianni Pagnini
Keyword(s):  
Brownian Motion ◽  
Stochastic Processes ◽  
Fractional Brownian Motion ◽  
Anomalous Diffusion ◽  
Short Note ◽  
Natural Generalization ◽  
Fractional Diffusion ◽  
Gaussian Density ◽  
Fractional Integral Equation ◽  
Mainardi Function

AbstractThe aim of this Short Note is to highlight that the generalized grey Brownian motion (ggBm) is an anomalous diffusion process driven by a fractional integral equation in the sense of Erdélyi-Kober, and for this reason here it is proposed to call such family of diffusive processes as Erdélyi-Kober fractional diffusion. The ggBm is a parametric class of stochastic processes that provides models for both fast and slow anomalous diffusion. This class is made up of self-similar processes with stationary increments and it depends on two real parameters: 0 < α ≤ 2 and 0 < β ≤ 1. It includes the fractional Brownian motion when 0 < α ≤ 2 and β = 1, the time-fractional diffusion stochastic processes when 0 < α = β < 1, and the standard Brownian motion when α = β = 1. In the ggBm framework, the Mainardi function emerges as a natural generalization of the Gaussian distribution recovering the same key role of the Gaussian density for the standard and the fractional Brownian motion.

FRACTIONAL DIFFUSION EQUATION, QUANTUM SUBDYNAMICS AND EINSTEIN'S THEORY OF BROWNIAN MOTION

2007 ◽  
Vol 21 (23n24) ◽  
pp. 3993-3999
Author(s):  
SUMIYOSHI ABE
Keyword(s):  
Brownian Motion ◽  
Quantum Theory ◽  
Fractional Calculus ◽  
Diffusion Equation ◽  
Master Equation ◽  
Anomalous Diffusion ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Quantum Master Equation ◽  
Objective System

The fractional diffusion equation for describing the anomalous diffusion phenomenon is derived in the spirit of Einstein's 1905 theory of Brownian motion. It is shown how naturally fractional calculus appears in the theory. Then, Einstein's theory is examined in view of quantum theory. An isolated quantum system composed of the objective system and the environment is considered, and then subdynamics of the objective system is formulated. The resulting quantum master equation is found to be of the Lindblad type.

AN INTRODUCTION TO THE THEORY OF SELF-SIMILAR STOCHASTIC PROCESSES

2000 ◽  
Vol 14 (12n13) ◽  
pp. 1399-1420 ◽  
Author(s):  
PAUL EMBRECHTS ◽  
MAKOTO MAEJIMA
Keyword(s):  
Probability Theory ◽  
Brownian Motion ◽  
Stochastic Processes ◽  
Fractional Brownian Motion ◽  
High Energy Physics ◽  
High Energy ◽  
Point Of View ◽  
Long Range Dependence ◽  
Self Similar ◽  
Energy Physics

Self-similar processes such as fractional Brownian motion are stochastic processes that are invariant in distribution under suitable scaling of time and space. These processes can typically be used to model random phenomena with long-range dependence. Naturally, these processes are closely related to the notion of renormalization in statistical and high energy physics. They are also increasingly important in many other fields of application, as there are economics and finance. This paper starts with some basic aspects on self-similar processes and discusses several topics from the point of view of probability theory.

scholarly journals Dimensions of Fractional Brownian Images

2021 ◽  
Author(s):  
Stuart A. Burrell
Keyword(s):  
Brownian Motion ◽  
Hausdorff Dimension ◽  
Stochastic Processes ◽  
Fractional Brownian Motion ◽  
Box Dimension ◽  
General Strategy ◽  
Borel Sets ◽  
Exceptional Sets ◽  
Explicit Condition ◽  
Box Dimensions

AbstractThis paper concerns the intermediate dimensions, a spectrum of dimensions that interpolate between the Hausdorff and box dimensions. Potential-theoretic methods are used to produce dimension bounds for images of sets under Hölder maps and certain stochastic processes. We apply this to compute the almost-sure value of the dimension of Borel sets under index-$$\alpha $$ α fractional Brownian motion in terms of dimension profiles defined using capacities. As a corollary, this establishes continuity of the profiles for Borel sets and allows us to obtain an explicit condition showing how the Hausdorff dimension of a set may influence the typical box dimension of Hölder images such as projections. The methods used propose a general strategy for related problems; dimensional information about a set may be learned from analysing particular fractional Brownian images of that set. To conclude, we obtain bounds on the Hausdorff dimension of exceptional sets, with respect to intermediate dimensions, in the setting of projections.

scholarly journals Optimal Sequential Change Detection for Fractional Diffusion-Type Processes

2013 ◽  
Vol 50 (1) ◽  
pp. 29-41
Author(s):  
Alexandra Chronopoulou ◽  
Georgios Fellouris
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Abrupt Change ◽  
Fractional Diffusion ◽  
Hurst Index ◽  
Continuous Path ◽  
Cusum Test ◽  
Random Drift ◽  
Diffusion Type ◽  
Drift Term

The problem of detecting an abrupt change in the distribution of an arbitrary, sequentially observed, continuous-path stochastic process is considered and the optimality of the CUSUM test is established with respect to a modified version of Lorden's criterion. We apply this result to the case that a random drift emerges in a fractional Brownian motion and we show that the CUSUM test optimizes Lorden's original criterion when a fractional Brownian motion with Hurst index H adopts a polynomial drift term with exponent H+1/2.

scholarly journals Entropic Approach to the Detection of Crucial Events

Entropy ◽  
2019 ◽  
Vol 21 (2) ◽  
pp. 178 ◽  
Author(s):  
Garland Culbreth ◽  
Bruce West ◽  
Paolo Grigolini
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Anomalous Diffusion ◽  
Correlation Functions ◽  
Joint Action ◽  
Clear Distinction ◽  
Aging Effects ◽  
Recent Advances ◽  
The Brain

In this paper, we establish a clear distinction between two processes yielding anomalous diffusion and 1 / f noise. The first process is called Stationary Fractional Brownian Motion (SFBM) and is characterized by the use of stationary correlation functions. The second process rests on the action of crucial events generating ergodicity breakdown and aging effects. We refer to the latter as Aging Fractional Brownian Motion (AFBM). To settle the confusion between these different forms of Fractional Brownian Motion (FBM) we use an entropic approach properly updated to incorporate the recent advances of biology and psychology sciences on cognition. We show that although the joint action of crucial and non-crucial events may have the effect of making the crucial events virtually invisible, the entropic approach allows us to detect their action. The results of this paper lead us to the conclusion that the communication between the heart and the brain is accomplished by AFBM processes.

A multivariate Weierstrass–Mandelbrot function

1985 ◽  
Vol 400 (1819) ◽  
pp. 331-350 ◽  
Keyword(s):  
Brownian Motion ◽  
Stochastic Processes ◽  
Fractional Brownian Motion ◽  
Continuum Limit ◽  
Small Length ◽  
Stochastic Description ◽  
Fractal Surfaces ◽  
Higher Dimensional ◽  
The Many ◽  
The Continuum

The univariate Weierstrass–Mandelbrot function is generalized to many variables to model higher dimensional stochastic processes such as undersea topography. Because this topography is difficult to measure at small length scales over the many large regions that affect long-ranged acoustic propagation in the ocean, one needs a stochastic description that can be extrapolated to both large and small features. Fractal surfaces are a convenient framework for such a description. Computer-generated plots for the two-variable case are presented. It is shown that in the continuum limit the multivariate function is equivalent to the multivariate fractional Brownian motion.

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