Occupation time statistics of the random acceleration model

TL;DR

This paper investigates the occupation time of the one-dimensional random acceleration model, providing analytical moments, comparing it with the maximum time distribution, and using simulations to understand their relationship.

Contribution

It offers the first analytical calculation of occupation time moments for the random acceleration model and compares its distribution with the maximum time distribution.

Findings

First two moments of occupation time calculated analytically.

Distributions of occupation time and maximum time are similar but not identical.

Monte Carlo simulations support the analytical findings.

Abstract

The random acceleration model is one of the simplest non-Markovian stochastic systems and has been widely studied in connection with applications in physics and mathematics. However, the occupation time and related properties are non-trivial and not yet completely understood. In this paper we consider the occupation time $T_+$ of the one-dimensional random acceleration model on the positive half-axis. We calculate the first two moments of $T_+$ analytically and also study the statistics of $T_+$ with Monte Carlo simulations. One goal of our work was to ascertain whether the occupation time $T_+$ and the time $T_m$ at which the maximum of the process is attained are statistically equivalent. For regular Brownian motion the distributions of $T_+$ and $T_m$ coincide and are given by L\'evy's arcsine law. We show that for randomly accelerated motion the distributions of $T_+$ and $T_m$ are quite similar but not identical. This conclusion follows from the exact results for the moments of the distributions and is also consistent with our Monte Carlo simulations.