Occupation time of a randomly accelerated particle on the positive half axis: Results for the first five moments

Abstract

In the random acceleration process a point particle moving in one dimension is accelerated by Gaussian white noise with zero mean. Although several fundamental statistical properties of the motion have been analyzed in detail, the statistics of occupation times is still not well understood. We consider the occupation or residence time $T_+$ on the positive $x$ axis of a particle which is randomly accelerated on the unbounded $x$ axis for a time $t$. The first two moments of $T_+$ were recently derived by Ouandji Boutcheng et al. \cite{OB}. With an alternate approach utilizing basis functions which have proved useful in other studies of randomly accelerated motion, results for the first five moments are obtained in this paper.