Lévy Walk:Stochastic Analysis And Two-dimensional Nonlinear Modeling

The continuous time random walk model is one of the most popular and useful mod-els in statistical physics and mathematics,and the Levy walk model is a special type of the continuous time random walk model.This paper is based on the classical one-dimensional linear coupled Levy walk(i.e.|x|= v0t).Firstly,we give the random analysis of Levy walk:the definition of the Levy walk random integral and self-similarity of the limiting process of Levy walk.Next,we consider the one-dimensional nonlinear coupled Levy walk(i.e.|x|= v0t?,?>0),discuss in detail the master equation of one-dimensional nonlinear coupled Levy walk and its corresponding mean squared displacement,when it takes different probability density index of step length and waiting time.The most im-portant thing is that we extend one-dimensional nonlinear coupled Levy walk to the two-dimensional nonlinear coupled situation,calculate the corresponding probability density function,draw the probability density function image and present the numerical simula-tion.Finally,we sum up the results and point out the defect of the article.