Series Representations for the Characteristic Function of the Multidimensional Markov Random Flight

Abstract

We consider the symmetric Markov random flight, also called the persistent random walk, performed by a particle that moves at constant finite speed in the Euclidean space $\mathbb {R}^m, \; m\ge 2,$ and changes its direction at Poisson-distributed time instants by taking it at random according to the uniform distribution on the surface of the unit $(m-1)$-dimensional sphere. Such stochastic motion has become a very popular object of modern statistical physics because it can serve as an appropriate model for describing the isotropic finite-velocity transport in multidimensional Euclidean spaces. In recent decade this approach was also developed in the framework of the run-and-tumble theory. In this article we study one of the most important characteristics of the multidimensional symmetric Markov random flight, namely, its characteristic function. We derive two series representations of the characteristic function of the process with respect to Bessel functions with variable indices and with respect to the powers of time variable. The coefficients of these series are given by recurrent relations, as well as in the form of special determinants. As an application of these results, an asymptotic formula for the second moment function $\mu _{(2,2,2)}(t), \; t>0,$ of the three-dimensional Markov random flight, is presented. The moment function $\mu _{(2,0,0)}(t), \; t>0,$ is obtained in an explicit form.