The equation of symmetric Markovian random evolution in a plane

Stochastic Processes and their Applications

Volumn 75, Issue 1, 1998, Pages 67-87

  Kolesnik, Alexander D ^a   Turbin, Anatoly F ^b  

^a INSTITUTE OF MATHEMATICS AND COMPUTER SCIENCE   (Moldova)

^b INSTITUTE OF MATHEMATICS   (Ukraine)

Author keywords

Diffusion with finite speed; High order hyperbolic equation; Random evolution; Random motion

Indexed keywords

EID: 0010038267     PISSN: 03044149     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0304-4149(98)00003-9     Document Type: Article

References (34)

1. Bartlett M. Some problems associated with random velocity. Publ. Inst. Statist. Paris. 6:1957;261-270.
2. Bartlett M. A note on random walks at constant speed. Adv. Appl. Probab. 10:1978;704-707.
3. Bers, L., John, F., Schechter, M., 1966. Partial Differential Equations. Mir, Moscow (Russian translation).
4. Cane V. Random walks and physical processes. Bull. Int. Statist. Inst. 42:1967;622-640.
5. Cane, V., 1975. Diffusion models with relativity effects. In: Gani, J. (Ed.), Perspectives in Probability and Statistics. Applied Probability Trust, Sheffield, pp. 263-273.
6. Foong, S.K., 1992. First-passage time, maximum displacement, and Kac's solution of the telegraph equation. Phys. Rev. 46 (2), 707-710.
7. Foong S.K., Kanno S. Properties of the telegrapher's random process with or without a trap. Stochastic Process. Appl. 53:1994;147-173.
8. Goldstein S. On diffusion by discontinuous movements and the telegraph equation. Quart. J. Mech. Appl. Math. 4:1951;129-156.
9. Gorostiza L. An invariance principle for a class of. d -dimensional polygonal random functions Trans. Amer. Math. Soc. 177:1973;413-445.
10. Griego R., Hersh R. Theory of random evolutions with applications to partial differential equations. Trans. Amer. Math. Soc. 156:1971;405-418.
11. Hall, M., 1962. The Theory of Groups. Inostrannaya Literatura, Moscow (Russian Translation).
12. Hersh, R., 1974. Random evolutions: a survey of results and problems. Rocky Mount. J. Math. 4, 443-477.
13. Hersh R., Pinsky M. Random evolutions are asymptotically Gaussian. Comm. Pure Appl. Math. 25:1972;33-44.
14. Janssen A. The distance between the Kac process and the Wiener process with applications to generalized telegraph equations. J. Theor. Probab. 3:1990;349-360.
15. Kabanov, Yu.M., 1992. On the probabilistic representation of a solution of the telegraph equation. Theor. Probab. Appl. 37, 425-426 (in Russian).
16. Kac, M., 1967. Some probabilistic problems of physics and mathematics. Nauka, Moscow (Russian Translation).
17. Kac M. A stochastic model related to the telegrapher's equation. Rocky Mount. J. Math. 4:1974;497-509.
18. Kaplan S. Differential equations in which the Poisson process plays a role. Bull. Amer. Math. Soc. 70:1964;264-268.
19. Kolesnik, A.D., 1989. On a model of Markovian random evolution in a plane. Analytical Methods of Investigation of Evolution of Stochastic Systems. Kiev, Inst. Math. Ukrain., pp. 55-61 (in Russian).
20. Kolesnik, A.D., 1991. The equations of Markovian random evolutions. Doctoral Dissertation, Inst. Math. Ukrain. Acad. Sci., Kiev, p. 130 (in Russian).
21. Kolesnik, A.D., 1998. The equations of Markovian random evolutions in a line. J. Appl. Probab. 35 (1), to appear.
22. 2 . Dokl. Acad. Nauk Ukrain 2, 10-11 (in Russian).
23. n . Dokl. Acad. Nauk Ukrain 1, 11-14 (in Russian).
24. Korolyuk, V.S., Swishchuk, A.V., 1992. Semi-Markov Random Evolutions. Kiev, Naukova Dumka (in Russian).
25. Mizohata, S., 1997. Theory of the Partial Differential Equations. Moscow, Mir (Russian translation).
26. Orsingher E. Hyperbolic equations arising in random models. Stochastic Process. Appl. 21:1985;93-106.
27. Orsingher E. A planar random motion governed by the two-dimensional telegraph equation. J. Appl. Probab. 23:1986;385-397.
28. Orsingher E. Stochastic motions on the 3-sphere governed by wave and heat equations. J. Appl. Probab. 24:1987;315-327.
29. Orsingher E. Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff's laws. Stochastic. Process. Appl. 34:1990;49-66.
30. Orsingher E. Random motions governed by third-order equations. Adv. Appl. Probab. 22:1990;915-928.
31. Orsingher E., Bassan B. On a. 2n -valued telegraph signal and the related integrated process Stochastics and Stochastics Rep. 38:1992;159-173.
32. Orsingher, E., Kolesnik, A.D., 1996. The explicit probability law of a planar random motion governed by a fouth order hyperbolic equation. Theor. Probab. Appl. 41, 451-459 (in Russian).
33. Pinsky M. Differential equations with a small parameter and the central limit theorem for functions defined on a finite Markov chain. Zeit. Wahr. 9:1968;101-111.
34. Pinsky, M., 1991. Lectures on Random Evolution. World Scientific, Singapore.