Temporal Correlations of the Running Maximum of a Brownian Trajectory
Résumé
We study the correlations between the maxima m and M of a Brownian motion (BM) on the time intervals [0,t1] and [0,t2], with t2>t1. We determine the exact forms of the distribution functions P(m,M) and P(G=M−m), and calculate the moments E{(M−m)k} and the cross-moments E{mlMk} with arbitrary integers l and k. We show that correlations between m and M decay as t1/t2−−−−√ when t2/t1→∞, revealing strong memory effects in the statistics of the BM maxima. We also compute the Pearson correlation coefficient ρ(m,M) and the power spectrum of Mt, and we discuss a possibility of extracting the ensemble-averaged diffusion coefficient in single-trajectory experiments using a single realization of the maximum process.