Geodesic
The distribution of random motion in semi-Markov media
Teoriâ slučajnyh processov, Tome 18 (2012) no. 1, pp. 111-118.

Voir la notice de l'article provenant de la source Math-Net.Ru

This paper deals with the random motion with finite speed along uniformly distributed directions, where the direction alternations occur according to renewal epochs of a general distribution. We derive a renewal equation for the characteristic function of a transition density of multidimensional motion. By using the renewal equation, we study the behavior of the transition density near the sphere of its singularity in two- and three-dimensional cases. For $\left(n-1\right)$-Erlang distributed steps of the motion in an $n$-dimensional space ($n\geq 2$), we have obtained the characteristic function as a solution of the renewal equation. As an example, we have derived the distribution for the three-dimensional random motion.
Keywords: Random motion, characteristic function, Dirac delta-function.
Mots-clés : convolution, Fourier transform, Laplace transform 
@article{THSP_2012_18_1_a6,
     author = {A. Pogorui},
     title = {The distribution of random motion in {semi-Markov} media},
     journal = {Teori\^a slu\v{c}ajnyh processov},
     pages = {111--118},
     publisher = {mathdoc},
     volume = {18},
     number = {1},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/THSP_2012_18_1_a6/}
}
TY  - JOUR
AU  - A. Pogorui
TI  - The distribution of random motion in semi-Markov media
JO  - Teoriâ slučajnyh processov
PY  - 2012
SP  - 111
EP  - 118
VL  - 18
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/THSP_2012_18_1_a6/
LA  - en
ID  - THSP_2012_18_1_a6
ER  - 
%0 Journal Article
%A A. Pogorui
%T The distribution of random motion in semi-Markov media
%J Teoriâ slučajnyh processov
%D 2012
%P 111-118
%V 18
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/THSP_2012_18_1_a6/
%G en
%F THSP_2012_18_1_a6
A. Pogorui. The distribution of random motion in semi-Markov media. Teoriâ slučajnyh processov, Tome 18 (2012) no. 1, pp. 111-118. http://geodesic.mathdoc.fr/item/THSP_2012_18_1_a6/

[1] E. Orsingher, A. De Gregorio, “Random flights in higher spaces”, J. Theor. Probab., 20 (2007), 769–806 | DOI | MR | Zbl

[2] A. D. Kolesnik, “Random motions at finite speed in higher dimensions”, J. Stat. Phys., 131 (2008), 1039–1065 | DOI | MR | Zbl

[3] A. Di Crescenzo, “On random motions with velocities alternating at Erlang-distributed random times”, Adv. Appl. Probab., 61 (2001), 690–701 | MR

[4] A. A. Pogorui, R. M. Rodriguez-Dagnino, “One-dimensional semi-Markov evolutions with general Erlang sojourn times”, Random Oper. Stoch. Equ., 13 (2005), 1720–1724 | DOI | MR

[5] A. A. Pogorui, “Fading evolution in multidimensional spaces”, Ukr. Math. J., 62:11 (2010), 1828–1834 | MR

[6] A. A. Pogorui, “The distribution of random evolution in Erlang semi-Markov media”, Theory of Stoch. Processes, 17:1 (2011), 90–99 | MR | Zbl

[7] A. A. Pogorui, R. M. Rodriguez-Dagnino, “Isotropic random motion at finite speed with $K$-Erlang distributed direction alternations”, J. Stat. Phys., 145 (2011), 102–114 | DOI | MR

[8] L. Beghin, E. Orsingher, “Moving randomly amid scattered obstacles”, Stochastics, 82 (2010), 201–229 | MR | Zbl

[9] G. Le Caer, “A Pearson-Dirichlet random walk”, J. Stat. Phys., 140 (2010), 728–751 | DOI | MR | Zbl

[10] G. Le Caer, “A new family of solvable Pearson-Dirichlet random walks”, J. Stat. Phys., 144 (2011), 23–45 | DOI | MR | Zbl

[11] S. Bochner, K. Chandrasekhar, Fourier Transforms, Annals of Math. Studies, 19, Princeton University Press, 1949 | MR | Zbl

[12] N. W. McLachlan, Bessel Functions for Engineers, Clarendon Press, Oxford, 1995

[13] I. S. Gradshtein, I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products, Academic Press, New York, 1980