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 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 show that correlations between $m$ and $M$ decay as $\sqrt{t_1/t_2}$ when $t_2/t_1 \to \infty$, revealing strong memory effects in the statistics of the BM maxima. We also compute the Pearson correlation coefficient $\rho(m,M)$, 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.