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

We consider the Markov random flight $\mathbf{X}(t)$ in the Euclidean space $\Bbb R^m, \; m\ge 2,$ starting from the origin $0\in\Bbb R^m$ that, at Poisson-paced times, changes its direction at random according to arbitrary distribution on the unit $(m-1)$-dimensional sphere $S^m(0,1)$ having absolutely continuous density. For any time instant $t>0$, the convolution-type recurrent relations for the joint and conditional densities of the process $\mathbf{X}(t)$ and of the number of changes of direction, are obtained. Using these relations, we derive an integral equation for the transition density of $\mathbf{X}(t)$ whose solution is given in the form of a uniformly convergent 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 $S^m( 0,1)$ and of the circular Gaussian law on the unit circle $S^2(0,1)$ are considered separately.