title: Temporal Correlations of the Running Maximum of a Brownian Trajectory.
abstract: We study the correlations between the maxima m and M of a Brownian motion (BM) on the time intervals [0,t_{1}] and [0,t_{2}], with t_{2}>t_{1}. 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{m^{l}M^{k}} with arbitrary integers l and k. We show that correlations between m and M decay as sqrt[t_{1}/t_{2}] when t_{2}/t_{1}→∞, 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 M_{t}, and we discuss a possibility of extracting the ensemble-averaged diffusion coefficient in single-trajectory experiments using a single realization of the maximum process.
authors: Laboratoire de Physique Théorique de la Matière Condensée, UPMC, CNRS UMR 7600, Sorbonne Universités, 4 Place Jussieu, 75252 Paris Cedex 05, France.
authors: Olivier
authors: O
authors: Bénichou
authors: Department of Physics, Boston University, Boston, Massachusetts 02215, USA.
authors: P L
authors: PL
authors: Krapivsky
authors: Laboratory of Physical Properties, Technical University of Madrid, Avenida Complutense s/n, 28040 Madrid, Spain.
authors: Carlos
authors: C
authors: Mejía-Monasterio
authors: Laboratoire de Physique Théorique de la Matière Condensée, UPMC, CNRS UMR 7600, Sorbonne Universités, 4 Place Jussieu, 75252 Paris Cedex 05, France.
authors: Gleb
authors: G
authors: Oshanin