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

zbMATH Open1363.60124arXiv1601.06998MaRDI QIDQ2818853

Alexander D. Kolesnik

Publication date: 27 September 2016

Published in: Theory of Stochastic Processes (Search for Journal in Brave)

Abstract: We consider the Markov random flight $$ in the Euclidean space $Bbb{R}^m,;mge2,$ starting from the origin $$ that, at Poisson-paced times, changes its direction at random according to arbitrary distribution on the unit $(m-1)$-dimensional sphere $$ having absolutely continuous density. For any time instant $t>0$, the convolution-type recurrent relations for the joint and conditional densities of process $$ and of the number of changes of direction, are obtained. Using these relations, we derive an integral equation for the transition density of $$ whose solution is given in the form of a uniformly converging series composed of the multiple double convolutions of the singular component of the density with itself. Two important particular cases of the uniform distribution on $$ and of the Gaussian distributions on the unit circumference $$ are separately considered.

Full work available at URL: https://arxiv.org/abs/1601.06998

zbMATH Keywords

Fourier transformconvolutioncharacteristic functiontransition densityconditional densityjoint densitycontinuous-time random walkrandom flightuniform distribution on spherecircular Gaussian law

Mathematics Subject Classification ID

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