文摘
We consider the Markov random flight \(\varvec{X}(t), \; t>0,\) in the three-dimensional Euclidean space \({\mathbb {R}}^{3}\) 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 \(\varvec{X}(t)\) corresponding to two and three changes of direction, are obtained. Based on these results, an asymptotic formula, as \(t\rightarrow 0\), for the unconditional characteristic function of \(\varvec{X}(t)\) 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 \(\lambda \). An asymptotic formula, as \(t\rightarrow 0\), for the probability of being in a three-dimensional ball of radius \(r<ct\), is also derived. Estimate of the accuracy of the approximation is analysed.
