Random Flights Related to the Euler-Poisson- Darboux Equation}

#### R. Garra, E. Orsingher

2016, v.22, Issue 1, 87-110

ABSTRACT

This paper is devoted to the analysis of random motions on the line and in the space $\mathbb{R}^d$
($d>1$) performed at finite velocity and governed by a non-homogeneous
Poisson process with rate $\lambda(t)$. The explicit distributions $p(\mathbf{x},t)$ of the position of
the randomly moving particles are obtained by solving initial-value problems for the Euler-Poisson-Darboux
equation when $\lambda(t)= \alpha/t$, $\alpha >0$. We consider also the case where $\lambda(t)= \lambda \coth \lambda t$
and $\lambda(t)= \lambda \tanh\lambda t$ where some Riccati differential equations emerge and the explicit distributions
are obtained for $d=1$. We also examine planar random motions with random velocities obtained by projecting random flights in $\mathbb{R}^d$
onto the plane. Finally the case of planar motions with four orthogonal directions is considered and the corresponding
higher-order equations with time-varying coefficients obtained.

Keywords: Euler-Poisson-Darboux equation, inhomogeneous Poisson process, finite velocity random motions, telegraph process