Integral equation for the transition density of the multidimensional Markov random flight

TL;DR

This paper derives an integral equation for the transition density of a multidimensional Markov random flight, providing a series solution and analyzing special cases like uniform and Gaussian distributions on spheres.

Contribution

It introduces a new integral equation for the transition density of multidimensional Markov random flights with arbitrary directional distributions.

Findings

Derived convolution-type relations for densities

Established an integral equation with series solution

Analyzed special cases of directional distributions

Abstract

We consider the Markov random flight $\bold X(t)$ in the Euclidean space $\Bbb R^m, \; m\ge 2,$ starting from the origin $\bold 0\in\Bbb R^m$ that, at Poisson-paced times, changes its direction at random according to arbitrary distribution on the unit $(m-1)$-dimensional sphere $S^m(\bold 0,1)$ having absolutely continuous density. For any time instant $t>0$, the convolution-type recurrent relations for the joint and conditional densities of process $\bold X(t)$ and of the number of changes of direction, are obtained. Using these relations, we derive an integral equation for the transition density of $\bold X(t)$ whose solution is given in the form of a uniformly converging series composed of the multiple double convolutions of the singular component of the density with itself. Two important particular cases of the uniform distribution on $S^m(\bold 0,1)$ and of the Gaussian distributions on the unit circumference $S^2(\bold 0,1)$ are separately considered.