Asymptotic relation for the transition density of the three-dimensional Markov random flight on small time intervals

TL;DR

This paper derives an asymptotic relation for the transition density of a three-dimensional Markov random flight process with constant speed and random direction changes, valid for small time intervals, including error estimates.

Contribution

It provides the first asymptotic formula for the transition density of the 3D Markov random flight, including error bounds and applicability for small times.

Findings

Asymptotic formula for the transition density as t→0

Error order of o(t^3) in the approximation

Effective small-time interval approximation with error estimates

Abstract

We consider the Markov random flight $\bold X(t), \; t>0,$ in the three-dimensional Euclidean space $\Bbb R^3$ with constant finite speed $c>0$ and the uniform choice of the initial and each new direction at random time instants that form a homogeneous Poisson flow of rate $\lambda>0$. Series representations for the conditional characteristic functions of $\bold X(t)$ corresponding to two and three changes of direction, are obtained. Based on these results, an asymptotic formula, as $t\to 0$, for the unconditional characteristic function of $\bold X(t)$ is derived. By inverting it, we obtain an asymptotic relation for the transition density of the process. We show that the error in this formula has the order $o(t^3)$ and, therefore, it gives a good approximation on small time intervals whose lengths depend on $\lambda$. Estimate of the accuracy of the approximation is analysed.