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A four-dimensional random motion at finite speed

Published online by Cambridge University Press:  14 July 2016

Alexander D. Kolesnik*
Affiliation:
Academy of Sciences of Moldova
*
Postal address: Institute of Mathematics and Computer Science, Academy of Sciences of Moldova, Academy Street 5, Kishinev, MD-2028, Moldova. Email address: [email protected]
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Abstract

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We consider the random motion of a particle that moves with constant finite speed in the space 4 and, at Poisson-distributed times, changes its direction with uniform law on the unit four-sphere. For the particle's position, X(t) = (X1(t), X2(t), X3(t), X4(t)), t > 0, we obtain the explicit forms of the conditional characteristic functions and conditional distributions when the number of changes of directions is fixed. From this we derive the explicit probability law, f(x, t), x4, t ≥ 0, of X(t). We also show that, under the Kac condition on the speed of the motion and the intensity of the switching Poisson process, the density, p(x,t), of the absolutely continuous component of f(x,t) tends to the transition density of the four-dimensional Brownian motion with zero drift and infinitesimal variance σ2 = ½.

Type
Research Papers
Copyright
© Applied Probability Trust 2006 

References

Fichtenholtz, G. M. (1970). Course of Differential and Integral Calculus, Vol. 3. Nauka, Moscow.Google Scholar
Goldstein, S. (1951). On diffusion by discontinuous movements and on the telegraph equation. Quart. J. Mech. Appl. Math. 4, 129156.Google Scholar
Gradshteyn, I. S. and Ryzhik, I. M. (1980). Tables of Integrals, Series and Products. Academic Press, New York.Google Scholar
Kac, M. (1974). A stochastic model related to the telegrapher's equation. Rocky Mountain J. Math. 4, 497509.Google Scholar
Kolesnik, A. D. (2006). Exact probability distribution of a random motion at finite speed in {R}4 . In Proc. 31st Internat. Conf. Stoch. Process. Appl. (Paris, July 2006), pp. 5556.Google Scholar
Kolesnik, A. D. and Orsingher, E. (2005). A planar random motion with an infinite number of directions controlled by the damped wave equation. J. Appl. Prob. 42, 11681182.CrossRefGoogle Scholar
Kolesnik, A. D. and Turbin, A. F. (1998). The equation of symmetric Markovian random evolution in a plane. Stoch. Process. Appl. 75, 6787.Google Scholar
Korolyuk, V. S. and Swishchuk, A. V. (1994). Semi-Markov Random Evolutions. Kluwer, Amsterdam.Google Scholar
Masoliver, J., Porrá, J. M. and Weiss, G. H. (1993). Some two and three-dimensional persistent random walks. Physica A 193, 469482.Google Scholar
Orsingher, E. (1990). Probability law, flow function, maximum distribution of wave-governed random motions and their connections with Kirchoff's laws. Stoch. Process. Appl. 34, 4966.Google Scholar
Pinsky, M. (1976). Isotropic transport process on a Riemannian manifold. Trans. Amer. Math. Soc. 218, 353360.Google Scholar
Pinsky, M. (1991). Lectures on Random Evolution. World Scientific, River Edge, NJ.Google Scholar
Stadje, W. (1987). The exact probability distribution of a two-dimensional random walk. J. Statist. Phys. 46, 207216.Google Scholar
Stadje, W. (1989). Exact probability distributions for non-correlated random walk models. J. Statist. Phys. 56, 415435.Google Scholar
Tolubinsky, E. V. (1969). The Theory of Transfer Processes. Naukova Dumka, Kiev.Google Scholar