Exact formula for the 2-marginal second moment function of the multidimensional symmetric Markov random flight

Abstract

We consider the symmetric Markov random flight $\bold X(t), \; t>0,$ in the Euclidean space $\Bbb R^m, \; m\ge 3$, performed by a particle that moves in $\Bbb R^m$ with constant finite speed and changes its directions at Poisson-distributed random time instants by choosing the initial and each new direction at random according to the uniform distribution on the unit $(m-1)$-dimensional sphere. The 2-marginal second moment function $\mu_{(2,2,0,\dots,0)}(t), \; t>0,$ of $\bold X(t)$, corresponding to the multi-index $(2,2,0,\dots,0)$, is examined. An explicit formula for function $\mu_{(2,2,0,\dots,0)}(t)$ is obtained. This formula is also valid for all other 2-marginal second moment functions corresponding to any multi-indices of the form $(0,\dots,0,2,0,\dots,0,2,0,\dots,0)$. It is also shown that this moment function, under the standard Kac scaling condition, turns into the product of the variances of two coordinates of the $m$-dimensional homogeneous Brownian motion.