One Dimensional Random Motion on Segment with Reflecting Edges and Dependent Increments

Gurami Tsitsiashvili

doi:10.4236/jamp.2018.63045

Abstract

In previous papers, the author considered the model of anomalous diffusion, defined by stable random process on an interval with reflecting edges. Estimates of the rate convergence of this process distribution to a uniform distribution are constructed. However, recent physical studies require consideration of models of diffusion, defined not only by stable random process with independent increments but multivariate fractional Brownian motion with dependent increments. This task requires the development of special mathematical techniques evaluation of the rate of convergence of the distribution of multivariate Brownian motion in a segment with reflecting boundaries to the limit. In the present work, this technology is developed and a power estimate of the rate of convergence to the limiting uniform distribution is built.

Highlights

In recent years, fractional Brownian motion has experienced significant growth in the applied problems of physics [1] [2] [3] [4] in connection with the necessity of modelling chaotic behaviour of the diffusing impurity in a variety of environments and alloys
The author considered the model of anomalous diffusion, defined by stable random process on an interval with reflecting edges
There is a need to analyse the speed of mixing of impurities in areas with reflecting boundaries, which cannot be obtained by the method of Fourier series

Summary

Introduction

Fractional Brownian motion has experienced significant growth in the applied problems of physics [1] [2] [3] [4] in connection with the necessity of modelling chaotic behaviour of the diffusing impurity in a variety of environments and alloys. G. Tsitsiashvili diffusion [6], simulating stable random process with independent increments. Tsitsiashvili diffusion [6], simulating stable random process with independent increments It is based on reflection formula for the density of the anomalous diffusion process. In the present work the algorithm of the corresponding estimates for the fractional Brownian motion on interval with reflecting edges is constructed. This algorithm is based on a calculation of a derivative of series which is describing density of fractional Brownian motion distribution [7]

Preliminaries

Examples of Random Processes Anomalous Diffusion

Conclusions