Dottorato in SCUOLA DI SCIENZE STATISTICHE

Titolo della tesi: Multidimensional finite-velocity stochastic motions and their fractional versions

We study multidimensional motions with finite velocities, generalizations of the well-known telegraph process. We begin by presenting orthogonal motions moving in $\mathbb{R}^2$ and $\mathbb{R}^3$ and governed by a non-homogeneous Poisson process. One of the main result is that the planar process can be represented as the linear trasformations of two independent one-dimensional telegraph processes. We then consider processes in higher dimensions, studying in depth the minimal motions, obtaining the form of the distribution inside the support and on its border. We also consider more general motions, focusing also on the connections between motions developing in different dimensions. Finally, we study Dzherbasyhan-Caputo fractional Cauchy problems of higher order, obtaining also some interesting results involving the inverses of stable subordinators. The study of fractional problems permits us to show fractional versions of muldimensional random motions.