NearExtreme Statistics of Brownian Motion
Abstract
We study the statistics of nearextreme events of Brownian motion (BM) on the time interval [0,t]. We focus on the density of states (DOS) near the maximum \rho(r,t) which is the amount of time spent by the process at a distance r from the maximum. We develop a path integral approach to study functionals of the maximum of BM, which allows us to study the full probability density function (PDF) of \rho(r,t) and obtain an explicit expression for the moments, \langle [\rho(r,t)]^k \rangle, for arbitrary integer k. We also study nearextremes of constrained BM, like the Brownian bridge. Finally we also present numerical simulations to check our analytical results.
 Publication:

Physical Review Letters
 Pub Date:
 December 2013
 DOI:
 10.1103/PhysRevLett.111.240601
 arXiv:
 arXiv:1305.6490
 Bibcode:
 2013PhRvL.111x0601P
 Keywords:

 05.40.Fb;
 02.50.Cw;
 05.40.Jc;
 Random walks and Levy flights;
 Probability theory;
 Brownian motion;
 Condensed Matter  Statistical Mechanics;
 Condensed Matter  Disordered Systems and Neural Networks;
 Mathematics  Probability
 EPrint:
 5 pages, 2 figures