Temporal correlations of the running maximum of a Brownian trajectory

TL;DR

This paper analyzes the statistical correlations between the maxima of a Brownian motion over different time intervals, providing exact distributions, moments, and insights into memory effects and potential experimental applications.

Contribution

It derives exact distribution functions and moments for the maxima of Brownian motion and explores their correlations and implications for single-trajectory experiments.

Findings

Correlations decay as t_1/t_2 when t_2/t_1    

Exact forms of distribution functions P(m,M) and P(G) are obtained

Insights into extracting diffusion coefficients from maximum process in experiments

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.