On Random Flights with Non-uniformly Distributed Directions

Abstract

This paper deals with a new class of random flights in a"e (d) , d >= 2, characterized by non-uniform probability distributions on the multidimensional sphere. These random motions differ from similar models appeared in literature where the directions are taken according to the uniform law. The family of angular probability distributions introduced in this paper depends on a parameter nu >= 0, which gives the anisotropy of the motion. Furthermore, we assume that the number of changes of direction performed by the random flight is fixed. The time lengths between two consecutive changes of orientation have joint probability distribution given by a Dirichlet density function. The analysis of the position (X) under bar (d) t > 0, obtained as projection onto the lower space R-m , m < d, of the original random motion in R-d . In its general framework, the analysis of (X) under bar (d) t > 0, is very complicated; nevertheless for some values of nu, we provide some explicit results about the process. Indeed, for nu=1 we get the characteristic function of the random flight moving in a"e (d) . By inverting the obtained characteristic function, we derive the analytic form (up to some constants) of the probability distribution of (X) under bar (d), t > 0. It is worth to mention that the stochastic processes considered in this paper belong to the class of the non-isotropic random walks, which has several applications in the mechanical statistics

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