Random Motions with Space-Varying Velocities

Abstract

Random motions on the line and on the plane with space-varying velocities are considered and analyzed in this paper. On the line we investigate symmetric and asymmetric telegraph processes with space-dependent velocities and we are able to present the explicit distribution of the position \(\mathcal {T}(t)\), \(t>0\), of the moving particle. Also the case of a nonhomogeneous Poisson process (with rate \(\lambda = \lambda (t)\)) governing the changes of direction is analyzed in three specific cases. For the special case \(\lambda (t)= \alpha /t\), we obtain a random motion related to the Euler–Poisson–Darboux (EPD) equation which generalizes the well-known case treated, e.g., in (Foong, S.K., Van Kolck, U.: Poisson random walk for solving wave equations. Prog. Theor. Phys. 87(2), 285–292, 1992, [6], Garra, R., Orsingher, E.: Random flights related to the Euler-Poisson-Darboux equation. Markov Process. Relat. Fields 22, 87–110, 2016, [8], Rosencrans, S.I.: Diffusion transforms. J. Differ. Equ. 13, 457–467, 1973, [16]). A EPD-type fractional equation is also considered and a parabolic solution (which in dimension \(d=1\) has the structure of a probability density) is obtained. Planar random motions with space-varying velocities and infinite directions are finally analyzed in Sect. 5. We are able to present their explicit distributions, and for polynomial-type velocity structures we obtain the hyper- and hypoelliptic form of their support (of which we provide a picture).