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

TL;DR

This paper investigates the statistical properties of the occupation time of a randomly accelerated particle on the positive axis, deriving the first five moments using a novel basis function approach.

Contribution

It provides new analytical results for the first five moments of occupation time, extending previous work limited to the first two moments.

Findings

First five moments of occupation time derived

New basis function method applied successfully

Extends understanding of occupation time statistics

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.