Occupation time statistics of the random acceleration model

Hermann Joel Ouandji Boutcheng; Thomas Bouetou Bouetou; Theodore W. Burkhardt; Alberto Rosso; Andrea Zoia; Kofane Timoleon Crepin

arXiv · 1603.06883

Occupation time statistics of the random acceleration model

Hermann Joel Ouandji Boutcheng·Thomas Bouetou Bouetou·Theodore W. Burkhardt·Alberto Rosso·Andrea Zoia·Kofane Timoleon Crepin

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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.

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Hermann Joel Ouandji Boutcheng, Thomas Bouetou Bouetou, Theodore W. Burkhardt, Alberto Rosso, Andrea Zoia, Kofane Timoleon Crepin. 2016-03-22. Occupation time statistics of the random acceleration model. https://arxiv.org/abs/1603.06883

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