Occupation time statistics of the random acceleration model

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 Lévy'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.

10 pages, 4 figures