Distribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time

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dc.creatorRandon-Furling, Julien
dc.creatorMajumdar, Satya N.
dc.date2007-08-15
dc.date2008-02-25
dc.date.accessioned2026-07-07T09:22:42Z
dc.date.available2026-07-07T09:22:42Z
dc.descriptionWe calculate analytically the probability density $P(t_m)$ of the time $t_m$ at which a continuous-time Brownian motion (with and without drift) attains its maximum before passing through the origin for the first time. We also compute the joint probability density $P(M,t_m)$ of the maximum $M$ and $t_m$. In the driftless case, we find that $P(t_m)$ has power-law tails: $P(t_m)\sim t_m^{-3/2}$ for large $t_m$ and $P(t_m)\sim t_m^{-1/2}$ for small $t_m$. In presence of a drift towards the origin, $P(t_m)$ decays exponentially for large $t_m$. The results from numerical simulations are in excellent agreement with our analytical predictions.
dc.description13 pages, 5 figures. Published in Journal of Statistical Mechanics: Theory and Experiment (J. Stat. Mech. (2007) P10008, doi:10.1088/1742-5468/2007/10/P10008)
dc.identifierhttps://arxiv.org/abs/0708.2101
dc.identifierhttp://arxiv.org/abs/0708.2101
dc.identifierJournal of Statistical Mechanics: Theory and Experiment (2007) P10008
dc.identifierdoi:10.1088/1742-5468/2007/10/P10008
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/155468
dc.subjectStatistical Mechanics
dc.subjectProbability
dc.titleDistribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time
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