Isotropic Random Motion at Finite Speed with K-Erlang Distributed Direction Alternations

Abstract

We study uniformly distributed direction of motion at finite speed where the direction alternations occur according to the renewal epochs of a K-Erlang pdf. At first sight, our generalizations of previous Markovian results appears to be a small step, however, it must be seen as an important non-Markovian case where we have found closed-form expressions for the pdf and the conditional characteristic function of this semi-Markov transport process. We present detailed calculations of a three-dimensional example for the 2-Erlang case, which is important not only from physical applications point of view but also to understand more general models. For instance, in principle the example of the 2-Erlang case can be extended to a K-Erlang case (K = 3, 4, . . .) but some of the mathematical expressions may be cumbersome.

Random Motion with Uniformly Distributed Directions and Random Velocity

“…Thus, we are extending some of the results found in [1,2,7]. Furthermore, we consider this random motion in one-, two-, three-, and four-dimensions, and we show that the transition densities in some of the cases have an explosive growth near the sphere of their singularity.…”

“…One-dimensional non-Markovian random evolutions generalizes these results by changing the underlying Poisson process, where motion is driven by an alternating semi-Markov process with Erlang distributed interrenewal times [3][4][5][6]. Random flights in R n with K−Erlang distributed displacements and uniformly distributed directions have been analyzed in [7]. A planar random motion performed by a particle that changes direction at even-valued Poisson epochs is studied in [8].…”

The distribution of random motion at non-constant velocity in semi-Markov media

“…De Gregorio and A. Kolesnik in (1), (2). In this work, we consider multidimensional random motions with uniformly distributed directions with general distributed steps and non-constant velocity, and we have extended some results of (4), (6), (7). We show some interesting solvable cases no reported before.…”

“…In papers (3)- (6) it is considered a non-Markov generalization of one-dimensional random evolutions of the telegrapher's random process where motion is driven by an alternating semi-Markov process with Erlang distributed interrenewal times. Random flights in R n with KErlang distributed displacements and uniformly distributed directions have been studied in (7). A planar random motion performed by a particle, which, at even-valued Poisson events, changes direction, is studied in (8).…”

Random motion with gamma steps in higher dimensions

“…One-dimensional non-Markovian generalizations of the telegrapher's random process were obtained in Di Crescenzo (2001) and Pogorui and Rodriguez-Dagnino (2005) with velocities alternating at k-Erlang-distributed sojourn times, see also Pogorui and Rodriguez-Dagnino (2011) and . Isotropic random motions with motion driven by a homogeneous Poisson process in higher spaces or dimensions has been studied by Orsingher and De Gregorio in higher dimensions (Orsingher and De Gregorio, 2007), see also De Gregorio and Orsingher (2012) and references therein.…”