Physics – Condensed Matter – Statistical Mechanics
Scientific paper
2007-08-15
Journal of Statistical Mechanics: Theory and Experiment (2007) P10008
Physics
Condensed Matter
Statistical Mechanics
13 pages, 5 figures. Published in Journal of Statistical Mechanics: Theory and Experiment (J. Stat. Mech. (2007) P10008, doi
Scientific paper
10.1088/1742-5468/2007/10/P10008
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.
Majumdar Satya N.
Randon-Furling Julien
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