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Random motions in inhomogeneous media

Authors: E. Orsingher and N. E. Ratanov
Translated by: The authors
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 76 (2007).
Journal: Theor. Probability and Math. Statist. 76 (2008), 141-153
MSC (2000): Primary 60K99; Secondary 62G30, 35L25, 60C05
Published electronically: July 16, 2008
MathSciNet review: 2368746
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Abstract | References | Similar Articles | Additional Information

Abstract: Space inhomogeneous random motions of particles on the line and in the plane are considered in the paper. The changes of the movement direction are driven by a Poisson process. The particles are assumed to move according to a finite velocity field that depends on a spatial argument.

The explicit distribution of particles is obtained in the paper for the case of dimension 1 in terms of characteristics of the governing equations. In the case of dimension 2, the distribution is obtained if a rectifying diffeomorphism exists.

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  • 1. V. I. Arnol'd, Ordinary Differential Equations, ``Nauka'', Moscow, 1971; English transl., MIT Press, Cambridge, MA-London, 1978. MR 0508209 (58:22707)
  • 2. D. G. Aronson, N. V. Mantzaris, and H. G. Othmer, Wave propagation and blocking in inhomogeneous media, Discrete and Continuous Dynamical Systems 13 (2005), 843-876. MR 2166708 (2006k:35287)
  • 3. S. Goldstein, On diffusion by discontinuous movements and the telegraph equation, Quart. J. Mech. Appl. Math. 4 (1951), 129-156. MR 0047963 (13:960b)
  • 4. K. P. Hadeler, Reaction transport systems in biological modelling, Mathematics Inspired by Biology, 95-150, CIME Lectures 1997, Florence (V. Capasso and O. Diekmann, eds.), Lecture Notes in Mathematics 1714, Springer-Verlag, 1999. MR 1737306
  • 5. D. D. Joseph and L. Preziosi, Addendum to the paper: ``Heat waves'' [Rev. Mod. Phys. 61 (1989), no. 1, 41-73]; Rev. Mod. Phys. 62 (1990), no. 2, 375-391. MR 1056235 (91e:80003)
  • 6. M. Kac, Probability and Related Topics in Physical Sciences, Interscience, London, 1959. MR 0102849 (21:1635)
  • 7. M. Kac, A stochastic model related to the telegrapher's equation, Rocky Mountain J. Math. 4 (1974), 497-509. MR 0510166 (58:23185)
  • 8. A. Kolesnik and E. Orsingher, Analysis of a finite-velocity planar random motion with reflection, Theory of Probability and its Applications 46 (2001), no. 1, 138-147. MR 1968710 (2004g:60109)
  • 9. H. G. Othmer, S. R. Dunbar, and W. Alt, Models of dispersal in biological systems, J. Math. Biol. 26 (1988), 263-298. MR 949094 (90a:92064)
  • 10. T. Hillen and H. G. Othmer, The diffusion limit of transport equations derived from velocity-jump processes, SIAM J. Appl. Math. 61 (2000), no. 3, 751-775. MR 1788017 (2001m:35302)
  • 11. H. G. Othmer and T. Hillen, The diffusion limit of transport equations. II. Chemotaxis equations, SIAM J. Appl. Math. 62 (2002), no. 4, 1222-1250. MR 1898520 (2003c:35154)
  • 12. E. Orsingher, Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff's law, Stoch. Processes Appl. 34 (1990), 49-66. MR 1039562 (91g:60086)
  • 13. E. Orsingher, Exact joint distribution in a model of planar random motion, Stochastics and Stochastics Reports 69 (2000), no. 1-2, 1-10. MR 1751715 (2000m:60117)
  • 14. E. Orsingher and N. Ratanov, Planar random motions with drift, J. Appl. Math. Stoch. Anal. 15 (2002), no. 3, 205-221. MR 1930947 (2003i:60113)
  • 15. E. Orsingher, Bessel functions of third order and the distribution of cyclic planar motions with three directions, Stochastics and Stochastics Reports 74 (2002), no. 3-4, 617-631. MR 1943582 (2003j:60144)
  • 16. H. G. Othmer, On significance of finite propagation speeds in multicomponent reacting systems, J. Chem. Phys. 64 (1976), 460-470. MR 0449372 (56:7676)
  • 17. M. A. Pinsky, Lectures on Random Evolution, World Scientific, Singapore-New Jersey-London-Hong Kong, 1992. MR 1143780 (93b:60160)
  • 18. N. E. Ratanov, Random walks of a particle in a one-dimensional inhomogeneous environment with reflection and absorption, Teor. Matem. Fiz. 112 (1997), no. 1, 81-91. (Russian) MR 1478901 (98k:82079)
  • 19. N. Ratanov, Reaction-advection random motions in inhomogeneous media, Physica D 189 (2004), 130-140. MR 2044720 (2004m:35167)
  • 20. N. Ratanov, Branching random motions, nonlinear hyperbolic systems and travelling waves, European Series in Applied and Industrial Mathematics (ESAIM:PS) 10 (2006), 236-257. MR 2219342 (2007g:60101)
  • 21. D. W. Stroock, Some stochastic processes which arise from a model of the motion of a bacterium, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 28 (1973/74), 303-315. MR 0386038 (52:6897)
  • 22. G. H. Weiss, Some applications of persistent random walks and the telegrapher's equation, Physica A 311 (2002), 381-410. MR 1943373 (2004h:82096)

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Additional Information

E. Orsingher
Affiliation: Dipartimento di Statistica, Probabilitá e Statistiche Applicate, Universitá degli Studi di Roma “La Sapienza”, 00185 Rome, Italy

N. E. Ratanov
Affiliation: Universidad del Rosario, Bogotá, Colombia

Keywords: Bessel functions, Poisson process, rectifying diffeomorphism, hyperbolic equations, telegraph process
Received by editor(s): May 16, 2006
Published electronically: July 16, 2008
Article copyright: © Copyright 2008 American Mathematical Society

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