# Distribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time - Condensed Matter > Statistical Mechanics

Distribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time - Condensed Matter > Statistical Mechanics - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online.

Abstract: We calculate analytically the probability density \$Pt m\$ of the time \$t m\$at which a continuous-time Brownian motion with and without drift attains itsmaximum before passing through the origin for the first time. We also computethe joint probability density \$PM,t m\$ of the maximum \$M\$ and \$t m\$. In thedriftless case, we find that \$Pt m\$ has power-law tails: \$Pt m\simt m^{-3-2}\$ for large \$t m\$ and \$Pt m\sim t m^{-1-2}\$ for small \$t m\$. Inpresence of a drift towards the origin, \$Pt m\$ decays exponentially for large\$t m\$. The results from numerical simulations are in excellent agreement withour analytical predictions.

Autor: Julien Randon-Furling LPTMS, Satya N. Majumdar LPTMS

Fuente: https://arxiv.org/