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

Alexander D. Kolesnik
Theory of Stochastic Processes
Vol.20 (36), no.2, 2015, pp.42-53
We consider the Markov random flight X(t) in the Euclidean space R^m, m ≥ 2, starting from the origin 0 ∈ 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 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 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 S²(0,1) are considered separately.