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A four-dimensional random motion at finite speed

Part of: Special processes Nonparametric inference Stochastic analysis Markov processes

Published online by Cambridge University Press:  14 July 2016

Alexander D. Kolesnik
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: kolesnik@math.md
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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 ℝ4 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), x ∈ ℝ4, 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.

Keywords

Random motionfinite speedrandom evolutioncharacteristic functionuniformly distributed directionsBessel functionmultidimensional Brownian motion

MSC classification

Primary: 60K99: None of the above, but in this section
Secondary: 62G30: Order statistics; empirical distribution functions 60K35: Interacting random processes; statistical mechanics type models; percolation theory 60J60: Diffusion processes 60H30: Applications of stochastic analysis (to PDE, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 43 , Issue 4 , December 2006 , pp. 1107 - 1118
Copyright
© Applied Probability Trust 2006 

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