A four-dimensional random motion at finite speed

Abstract

We consider the random motion of a particle that moves with constant finite speed in the space ℝ 4 and, at Poisson-distributed times, changes its direction with uniform law on the unit four-sphere. For the particle's position, X (t) = (X 1(t), X 2(t), X 3(t), X 4(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 ∈ ℝ 4, 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.