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Random Motions with Space-Varying Velocities

  • Roberto Garra
  • Enzo OrsingherEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 208)

Abstract

Random motions on the line and on the plane with space-varying velocities are considered and analyzed in this paper. On the line we investigate symmetric and asymmetric telegraph processes with space-dependent velocities and we are able to present the explicit distribution of the position \(\mathcal {T}(t)\), \(t>0\), of the moving particle. Also the case of a nonhomogeneous Poisson process (with rate \(\lambda = \lambda (t)\)) governing the changes of direction is analyzed in three specific cases. For the special case \(\lambda (t)= \alpha /t\), we obtain a random motion related to the Euler–Poisson–Darboux (EPD) equation which generalizes the well-known case treated, e.g., in (Foong, S.K., Van Kolck, U.: Poisson random walk for solving wave equations. Prog. Theor. Phys. 87(2), 285–292, 1992, [6], Garra, R., Orsingher, E.: Random flights related to the Euler-Poisson-Darboux equation. Markov Process. Relat. Fields 22, 87–110, 2016, [8], Rosencrans, S.I.: Diffusion transforms. J. Differ. Equ. 13, 457–467, 1973, [16]). A EPD-type fractional equation is also considered and a parabolic solution (which in dimension \(d=1\) has the structure of a probability density) is obtained. Planar random motions with space-varying velocities and infinite directions are finally analyzed in Sect. 5. We are able to present their explicit distributions, and for polynomial-type velocity structures we obtain the hyper- and hypoelliptic form of their support (of which we provide a picture).

Keywords

Planar random motions Damped wave equations Euler–Poisson–Darboux fractional equation 

MSC 2010

60G60 35R11 

Notes

Acknowledgements

We are very greatful to both referees for their suggestions and comments.

References

  1. 1.
    Beghin, L., Nieddu, L., Orsingher, E.: Probabilistic analysis of the telegrapher’s process with drift by means of relativistic transformations. Int. J. Stoch. Anal. 14(1), 11–25 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Cattaneo, C.R.: Sur une forme de l’ équation de la chaleur éliminant le paradoxe d’une propagation instantanée. Comptes Rendus 247(4), 431–433 (1958)zbMATHGoogle Scholar
  3. 3.
    De Gregorio, A.: Transport processes with random jump rate. Stat. Probab. Lett. 118, 127–134 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    De Gregorio, A., Orsingher, E.: Flying randomly in \(\mathbb{R}^d\) with dirichlet displacements. Stoch. Process. Appl. 122(2), 676–713 (2012)Google Scholar
  5. 5.
    D’Ovidio, M., Orsingher, E., Toaldo, B.: Time-changed processes governed by space-time fractional telegraph equations. Stoch. Anal. Appl. 32(6), 1009–1045 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Foong, S.K., Van Kolck, U.: Poisson random walk for solving wave equations. Prog. Theor. Phys. 87(2), 285–292 (1992)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Garra, R., Orsingher, E., Polito, F.: Fractional Klein-Gordon equations and related stochastic processes. J. Stat. Phys. 155(4), 777–809 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Garra, R., Orsingher, E.: Random flights related to the Euler-Poisson-Darboux equation. Markov Process. Relat. Fields 22, 87–110 (2016)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Iacus, S.M.: Statistical analysis of the inhomogeneous telegrapher’s process. Stat. Probab. Lett. 55, 83–88 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kaplan, S.: Differential equations in which the poisson process plays a role. Bull. Am. Math. Soc. 70(2), 264–268 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, vol. 204. Elsevier Science Limited, Amsterdam (2006)Google Scholar
  12. 12.
    Kolesnik, A.D., Orsingher, E.: A planar random motion with an infinite number of directions controlled by the damped wave equation. J. Appl. Probab. 42(4), 1168–1182 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Masoliver, J., Weiss, G.H.: Telegraphers equations with variable propagation speeds. Phys. Rev. E. 49(5), 3852–3854 (1994)CrossRefGoogle Scholar
  14. 14.
    Orsingher, E.: Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff’s laws. Stoch. Process. Appl. 34(1), 49–66 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Pogorui, A.A., Rodríguez-Dagnino, R.M.: Random motion with uniformly distributed directions and random velocity. J. Stat. Phys. 147(6), 1216–1225 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Rosencrans, S.I.: Diffusion transforms. J. Differ. Equ. 13, 457–467 (1973)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.“Sapienza” Università di RomaRomeItaly

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