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Occupation time of a randomly accelerated particle on the positive half axis: results for the first five moments. (English) Zbl 1383.82048

Summary: In the random acceleration process a point particle 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 H. J. Ouandji Boutcheng et al. [“Occupation time statistics of the random acceleration model”, J. Stat. Mech., 053213, 1–10 (2016)]. 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.

MSC:

82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
35Q53 KdV equations (Korteweg-de Vries equations)
44A10 Laplace transform
45B05 Fredholm integral equations
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