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Cyclic random motions in $\mathbb{R}^d$ -space withn directions

Published online by Cambridge University Press:  08 September 2006

Aimé Lachal
Aimé Lachal*
Affiliation:
Institut National des Sciences Appliquées de Lyon, Bâtiment Léonard de Vinci, 20 avenue Albert Einstein, 69621 Villeurbanne Cedex, France; aime.lachal@insa-lyon.fr; http://maths.univ-lyon1.Fr/~lachal
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Abstract

We study the probability distribution of the location of a particleperforming a cyclic random motion in $\mathbb{R}^d$ . The particle can taken possible directions with different velocities and the changes ofdirection occur at random times. The speed-vectors as well as thesupport of the distribution form a polyhedron (the first one havingconstant sides and the other expanding with time t). Thedistribution of the location of the particle is made up of twocomponents: a singular component (corresponding to the beginning ofthe travel of the particle) and an absolutely continuous component.
We completely describe the singular component and exhibit anintegral representation for the absolutely continuous one. Thedistribution is obtained by using a suitable expression of thelocation of the particle as well as some probability calculustogether with some linear algebra. The particular case of theminimal cyclic motion (n=d+1) with Erlangian switching times isalso investigated and the related distribution can be expressed interms of hyper-Bessel functions with several arguments.

Keywords

Cyclic random motionslinear image of arandom vectorsingular and absolutely continuous measuresconvexityhyper-Bessel functions with several arguments.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 10 , February 2006 , pp. 277 - 316
Copyright
© EDP Sciences, SMAI, 2006

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