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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; [email protected]; 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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References

Cooper, R.B., Niu, S.C. and Srinivasan, M.M., Setups in polling models: does it make sense to set up if no work is waiting? J. Appl. Prob. 36 (1999) 585–592. CrossRef
Di Crescenzo, A., On random motions with velocities alternating at Erlang-distributed random times. Adv. Appl. Prob. 33 (2001) 690–701. CrossRef
Di Crescenzo, A., Exact transient analysis of a planar random motion with three directions. Stoch. Stoch. Rep. 72 (2002) 175–189. CrossRef
V.A. Fok, Works of the State Optical Institute, 4, Leningrad Opt. Inst. 34 (1926) (in Russian).
Goldstein, S., On diffusion by discontinuous movements, and on the telegraph equation. Quart. J. Mech. Appl. Math. 4 (1951) 129–156. CrossRef
Griego, R. and Hersh, R., Theory of random evolutions with applications to partial differential equations. Trans. Amer. Math. Soc. 156 (1971) 405–418. CrossRef
Kac, M., A stochastic model related to the telegrapher's equation. Rocky Mountain J. Math. 4 (1974) 497–509. CrossRef
Kolesnik, A.D. and Orsingher, E., Analysis of a finite-velocity planar random motion with reflection. Theory Prob. Appl. 46 (2002) 132–140. CrossRef
A. Lachal, S. Leorato and E. Orsingher, Random motions in $\mathbb{R}^n$ -space with (n + 1) directions, to appear in Ann. Inst. Henri Poincaré Sect. B.
S. Leorato and E. Orsingher, Bose-Einstein-type statistics, order statistics and planar random motions with three directions. Adv. Appl. Probab. 36(3) (2004) 937–970.
S. Leorato, E. Orsingher and M. Scavino, An alternating motion with stops and the related planar, cyclic motion with four directions. Adv. Appl. Probab. 35(4) (2003) 1153–1168.
Orsingher, E., Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff's laws. Stoch. Proc. Appl. 34 (1990) 49–66. CrossRef
E. Orsingher, Exact joint distribution in a model of planar random motion. Stoch. Stoch. Rep. 69 (2000) 1–10.
Orsingher, E., Bessel functions of third order and the distribution of cyclic planar motions with three directions. Stoch. Stoch. Rep. 74 (2002) 617–631. CrossRef
Orsingher, E. and Kolesnik, A.D., Exact distribution for a planar random motion model, controlled by a fourth-order hyperbolic equation. Theory Prob. Appl. 41 (1996) 379–387.
Orsingher, E. and Ratanov, N., Planar random motions with drift. J. Appl. Math. Stochastic Anal. 15 (2002) 205–221. CrossRef
E. Orsingher and N. Ratanov, Exact distributions of random motions in inhomogeneous media, submitted.
E. Orsingher and A. San Martini, Planar random evolution with three directions, in Exploring stochastic laws, A.V. Skorokhod and Yu.V. Borovskikh, Eds., VSP, Utrecht (1995) 357–366.
E. Orsingher and A.M. Sommella, A cyclic random motion in $\mathbb{R}^3$ with four directions and finite velocity. Stoch. Stoch. Rep. 76(2) (2004) 113–133.
M.A. Pinsky, Lectures on random evolution. World Scientific, River Edge (1991).
Samoilenko, I.V., Markovian random evolutions in $\mathbb{R}^n$ . Random Oper. Stochastic Equ. 9 (2001) 139–160.
I.V. Samoilenko, Analytical theory of Markov random evolutions in $\mathbb{R}^n$ . Doctoral thesis, University of Kiev (in Russian) (2001).