2007/08/31 by Julien Randon-Furling, Satya N. Majumdar, Satya N Majumdar · 3 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Brownian motion #Diffusion and Search Dynamics #Distribution (mathematics) #Fractional Brownian motion #Large deviations theory #Lévy distribution #Motion (physics) #Probability distribution #Stochastic process #cond-mat.stat-mech #math.PR #stochastic dynamics and bifurcation
paper · pdf · doi:10.1088/1742-5468/2007/10/p10008
published as Journal of Statistical Mechanics: Theory and Experiment (2007) P10008 · 13 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)
openalex publication_date 2007/10/12 · arxiv created 2008/02/25 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 )∼ t m −3/2 for large t m and P ( t m )∼ t m −1/2 for small t m . In the 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.