Temporal Correlations of the Running Maximum of a Brownian Trajectory

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 $\mathbb{E}\{{(M-m)}^k\}$ 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 $\rho (m,M)$ 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.

Figure 1

A realization of a BM (blue) and the corresponding maximum process $M_t$ (red). $M$ and $m$ are the maxima of BM on $[0,t_2]$ and $[0,t_1]$, respectively.

Figure 2

Comparison of the power spectra of the maximum process [solid line, (15)] and $S_{\nu }^{(BM)}(T)$ for the BM (dashed line). Symbols denote the results of Monte Carlo (MC) simulations.

Figure 3

The variance of $f_k$ as a function of $k$ for $D=0.25$, $D=0.01$, and $D=0.005$ (from top to bottom). Solid lines are Eq. (19) with $k$ considered as a continuous variable, while the symbols are the results of the MC simulations.

Figure 4

$\text{Var}(D_{\text{es}}=f_k^{2/k})/D^2$ using a nonlinearly transformed estimator in Eq. (18) as a function of $k$ for $D=0.25$, $D=0.01$, and $D=0.005$ (from top to bottom). The solid line is $1/k^2$. The inset shows the dependence of the deviation ${\delta }^2(D_{\text{es}})$ (see the text) on the number $N$ of recorded points of a discretized trajectory. Different colors correspond to $k=10$ (blue), $k=50$ (green), and $k=10^2$ (red).

Figure 5

The variance of the discretized time-averaged functional in Eq. (16), divided by $D^2$, versus the number $N$ of recorded positions of the trajectory $B_t$. The solid line is the theoretical result, $\text{Var}({\mathrm{D}}_{\text{msd}})/D^2\sim 3/\mathrm{N}$, while the symbols (the same as in Figs. 3 and 4) are results of MC simulations for $D=0.25$, $D=0.01$, and $D=0.005$. The horizontal dashed lines give $\text{Var}(D_{\text{es}})/D^2$ for $k=10^2$, $5\times 10^2$, and $10^3$.