A family of random walks with generalized Dirichlet steps

We analyze a class of continuous time random walks in R-d, d >= 2, with uniformly distributed directions. The steps performed by these processes are distributed according to a generalized Dirichlet law. Given the number of changes of orientation, we provide the analytic form of the probability density function of the position {(X) under bar (d) (t), t > 0} reached, at time t > 0, by the random motion. In particular, we analyze the case of random walks with two steps. In general, it is a hard task to obtain the explicit probability distributions for the process {(X) under bar (d) (t), t > 0}. Nevertheless, for suitable values for the basic parameters of the generalized Dirichlet probability distribution, we are able to derive the explicit conditional density functions of {(X) under bar (d) (t), t > 0}. Furthermore, in some cases, by exploiting the fractional Poisson process, the unconditional probability distributions of the random walk are obtained. This paper extends in a more general setting, the random walks with Dirichlet displacements introduced in some previous papers. (C) 2014 AIP Publishing LLC.