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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 

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