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

Abstract

We consider the symmetric Markov random flight $\mathbf{X}(t),t>0,$ in the Euclidean space ${\mathbb{R}}^m,m\ge 3$, performed by a particle that moves in ${\mathbb{R}}^m$ 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 $\hat{I}{¼}_{(2,2,0,\dots ,0)}(t),t>0,$ of $\mathbf{X}(t)$, corresponding to the multi-index $(2,2,0,\dots ,0)$, is examined. An explicit formula for function $\hat{I}{¼}_{(2,2,0,\dots ,0)}(t)$ 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.

18 pages, 1 figure