Temporal correlations of the running maximum of a Brownian trajectory

O. Benichou, P. L. Krapivsky, C. Mejia-Monasterio, G. Oshanin

Published 2016-02-22Version 1

We study the correlations between the maxima $m$ and $M$ of a Brownian motion on the time intervals $[0,t_1]$ and $[0,t_2]$, with $t_2>t_1$. We determine exact forms of the distribution functions $P(m,M)$ and $P(G = M - m)$, and calculate the moments $\mathbb{E}\{\left(M - m\right)^k\}$ and the cross-moments $\mathbb{E}\{m^l M^k\}$ with arbitrary integers $l$ and $k$. We compute the Pearson correlation coefficient $\rho(m,M)$ and show that $\rho(m,M) \sim \sqrt{t_1/t_2}$ when $t_2 \to \infty$ with $t_1$ kept fixed, revealing strong memory effects in the statistics of the maxima of a Brownian motion. As an application, we discuss a possibility of extracting the ensemble-average diffusion coefficient in single-trajectory experiments using a single realization of the maximum process of a Brownian motion.