Occupation time statistics of the random acceleration model

Hermann Joel Ouandji Boutcheng 1, 2 Thomas Bouetou Bouetou 3, 1 Theodore W. Burkhardt 4 Alberto Rosso 5 Andrea Zoia 6 Kofane Timoleon Crepin 2, 1

Journal of Statistical Mechanics: Theory and Experiment, IOP Science, 2016, pp.053213

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.

  • 1. CETIC asbl - Centre d’Excellence en Technologies de l’Information et de la Communication
  • 2. University of Yaoundé [Cameroun]
  • 3. ENSP - Ecole Nationale Supérieure Polytechnique [Yaoundé]
  • 4. Temple University [Philadelphia]
  • 5. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
  • 6. CEA-DEN - CEA-Direction de l'Energie Nucléaire