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hal-00167112, version 1

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

Julien Randon-Furling () 1, Satya N. Majumdar 1

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.

  • 1 :  Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS)
  • CNRS : UMR8626 – Université Paris XI - Paris Sud
  • Domaine : Physique/Matière Condensée/Mécanique statistique
    Mathématiques/Probabilités
  • Mots-clés : Brownian motion – first-passage problems – extreme value problems
  • Commentaire : Submitted to Journal of Statistical Mechanics: Theory and Experiment
  • Versions disponibles :  v1 (15-08-2007) v2 (19-09-2007) v3 (09-10-2007) v4 (25-02-2008)
 
  • hal-00167112, version 1
  • http://hal.archives-ouvertes.fr/hal-00167112
  • oai:hal.archives-ouvertes.fr:hal-00167112
  • Contributeur : Julien Randon-Furling <>
  • Soumis le : Mardi 14 Août 2007, 18:08:48
  • Dernière modification le : Jeudi 16 Août 2007, 10:29:55