Cyclic Random Motions with Orthogonal Directions

#### R. Garra, E. Orsingher, A.I . Zeifman

2020, v.26, Issue 3, 381-402

ABSTRACT

A cyclic random motion at finite velocity with orthogonal directions is considered in the plane and in $\mathbb{R}^3$.
We obtain in both cases the explicit conditional distributions of the position of the moving particle
when the number of switches of directions is fixed. The explicit unconditional distributions
are also obtained and are expressed in terms of Bessel functions.
The governing equations are derived and given as products of D'Alembert operators.
The limiting form of the equations is provided in the Euclidean space $\mathbb{R}^d$ and
takes the form of a heat equation with variance parameter $1/d$.

Keywords: cyclic random motions, Bessel functions, random flights, Klein\tire Gordon equations