Article Dans Une Revue Journal of Statistical Mechanics: Theory and Experiment Année : 2007

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

Résumé

We 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.

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Dates et versions

hal-00167112 , version 1 (14-08-2007)
hal-00167112 , version 2 (19-09-2007)
hal-00167112 , version 3 (09-10-2007)
hal-00167112 , version 4 (25-02-2008)

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Julien Randon-Furling, Satya N. Majumdar. Distribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time. Journal of Statistical Mechanics: Theory and Experiment, 2007, pp.P10008. ⟨10.1088/1742-5468/2007/10/P10008⟩. ⟨hal-00167112v4⟩
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