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Temporal correlations of the running maximum of a Brownian trajectory | | O. Benichou
; P. L. Krapivsky
; C. Mejia-Monasterio
; G. Oshanin
; | | Date: |
22 Feb 2016 | | Abstract: | 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 $mathbb{E}{left(M - m
ight)^k}$ and the
cross-moments $mathbb{E}{m^l M^k}$ with arbitrary integers $l$ and $k$. We
compute the Pearson correlation coefficient $
ho(m,M)$ and show that
$
ho(m,M) sim sqrt{t_1/t_2}$ when $t_2 o infty$ 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. | | Source: | arXiv, 1602.6770 | | Services: | Forum | Review | PDF | Favorites | |
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