Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T08:24:11.450Z Has data issue: false hasContentIssue false
  • English
  • Français

The Equations of Markovian Random Evolution on the Line

Part of: Markov processes Special processes

Published online by Cambridge University Press:  14 July 2016

Alexander Kolesnik
Alexander Kolesnik*
Affiliation:
Institute of Mathematics, Kishinev
*
∗Postal address: Institute of Mathematics, Academy Street 5, Kishinev 277028, Moldova. e-mail address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a general model of one-dimensional random evolution with n velocities and rates of a switching Poisson process (n ≥ 2). A governing nth-order hyperbolic equation in a determinant form is given. For two important particular cases it is written in an explicit form. Some known hyperbolic equations are obtained as particular cases of the general model

Keywords

Random evolutionsrandom motionshigh-order hyperbolic equations

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60J60: Diffusion processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 1 , March 1998 , pp. 27 - 35
Copyright
Copyright © Applied Probability Trust 1998 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bartlett, M. (1957). Some problems associated with random velocity. Publ. Inst. Stat. Univ. Paris 6, 261–270.Google Scholar
Bartlett, M. (1978). A note on random walks at constant speed. Adv. Appl. Prob. 10, 704–707.Google Scholar
Cane, V. (1967). Random walks and physical processes Bull. Int. Statist. Inst. 42, 622–640.Google Scholar
Cane, V. (1975). Diffusion models with relativity effects. In Perspectives in Probability and Statistics. ed. Gani, J. Applied Probability Trust, Sheffield. pp. 263–273.Google Scholar
Faddeev, D. K., and Sominsky, I. S. (1972). Problems in Algebra. Nauka, Moscow. (In Russian.)Google Scholar
Goldstein, S. (1951). On diffusion by discontinuous movements and the telegraph equation. Quart. J. Mech. Appl. Math. 4, 129–156.CrossRefGoogle Scholar
Kac, M. (1974). A stochastic model related to the telegrapher's equation Rocky Mountain J. Math. 4, 497–509.Google Scholar
Kaplan, S. (1964). Differential equations in which the Poisson process plays a role Bull. AMS 70, 264–268.Google Scholar
Kolesnik, A. D. (1990). One-dimensional models of the Markovian random evolutions Preprint 90.57. Kiev. (In Russian.)Google Scholar
Kolesnik, A. D., and Turbin, A. F. (1991). Infinitesimal hyperbolic operator of the Markovian random evolutions in ℝ n . Dokl. Acad. Nauk Ukrain. 1, 11–14. (In Russian.)Google Scholar
Lang, S. (1965). Algebra. Addision-Wesley, Reading, MA.Google Scholar
Mizohata, S. (1977). Theory of the Partial Differential Equations. Nauka, Moscow. (In Russian.)Google Scholar
Orsingher, E. (1985). Hyperbolic equations arising in random models. Stoc. Proc. Appl. 21, 93–106.CrossRefGoogle Scholar
Orsingher, E. (1987). Stochastic motions driven by wave equations. Rend. Sem. Mat. Fis. di Milano LVII, 365–380.Google Scholar
Orsingher, E. (1990). Random motions governed by third-order equations. Adv. Appl. Prob. 22, 915–928.Google Scholar
Orsingher, E., and Bassan, B. (1992). On a 2n-valued telegraph signal and the related integrated process. Stoch. Stoch. Rep. 38, 159–173.Google Scholar