Stochastic motions driven by wave equations

  • Enzo Orsingher


In this paper we present some random motions whose probability law is a solution of the telegraph equation. We give the explicit form of the flow function and of the probability law whose convergence to the Gaussian distribution is also discussed.

A planar version of motion governed by the equation of damped vibrations of membranes is also presented. Finally the connection of this motions with Maxwell equations of electrodynamics is analysed.


Brownian Motion Wave Equation Random Motion Telegraph Equation Homogeneous Poisson Process 
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In questo articolo sono presentati dei moti aleatori la cui legge di probabilità è soluzione dell’equazione del telegrafo. Sono presentate le espressioni esplicite sia della funzione di flusso che della legge di probabilità di cui è discussa la convergenza alla legge normale.

E’ infine data una versione bidimensionale del moto guidato dall’equazione delle vibrazioni smorzate delle membrane di cui è esaminata la connessione con le equazioni di Maxwell.


  1. [1]
    Bartlett M. S. (1957), Some problems associated with random velocity.Pubbl. Inst. Stat. Univ. Paris 6, 261–270.MATHMathSciNetGoogle Scholar
  2. [2]
    Bartlett M. S. (1978), A note on random walks at constant speed.Adv. Appl. Probab. 10, 704–707.MATHCrossRefGoogle Scholar
  3. [3]
    Cane V. R. (1967), Random walks and physical processes.Bull. Internat. Stat. 42, 622–640.Google Scholar
  4. [4]
    Cane V. R. (1975), Diffusion models with relativity effects, in Perspectives in Probability and Statistics, ed. Gani, distributed by Academic Press for the Applied Probability Trust, Sheffield, 263–273.Google Scholar
  5. [5]
    Gillis J. (1955), Correlated random walks.Proc. Camb. Phil. Soc. 51, 639–651.MATHMathSciNetGoogle Scholar
  6. [6]
    Henderson R., Renshaw E. andFord D. (1984), A correlated random walk model for two-dimensional diffusion.J. Appl. Probab. 21, 233–246.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    Kac M. (1959), Probability and related topics in physical sciences. Interscience, New York.MATHGoogle Scholar
  8. [8]
    Kac M. (1974), A stochastic model related to the telegrapher’s equation.Rocky Mountain Journal of Math. 4, 497–509.MATHCrossRefGoogle Scholar
  9. [9]
    Orsingher E. (1985), Hyperbolic equations arising in random models,Stoch. Proc. Appl. 21, 93–106.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    Orsingher E. (1986), A planar random motion governed by the two-dimensional telegraph equation.J. Appl. Probab. 23, 2, 385–397.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Orsingher E. (1987), Probabilistic models connected with wave equations.Boll. U.M.I. (7) 1.B., 423–437.MathSciNetGoogle Scholar
  12. [12]
    Orsingher E. (1987), Stochastic motions on the 3-sphere governed by wawe and heat equations.J. Appl. Probab., 24, 2, 315–327.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser-Verlag 1987

Authors and Affiliations

  • Enzo Orsingher
    • 1
  1. 1.Dipartimento di Informatica e Applicazionidell’Università di SalernoSalernoItalia

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