Longest excursion of fractional Brownian motion: Numerical evidence of non-Markovian effects

We study, using exact numerical simulations, the statistics of the longest excursion $l_{\text{max}}\left(t\right)$ up to time $t$ for the fractional Brownian motion with Hurst exponent $0<H<1$. We show that in the large $t$ limit, $⟨l_{\text{max}}\left(t\right)⟩\propto Q_{\infty }t$, where $Q_{\infty }\equiv Q_{\infty }\left(H\right)$ depends continuously on $H$. These results are compared with exact analytical results for a renewal process with an associated persistence exponent $\theta =1-H$. This comparison shows that $Q_{\infty }\left(H\right)$ carries the clear signature of non-Markovian effects for $H\ne 1/2$. The preasymptotic behavior of $⟨l_{\text{max}}\left(t\right)⟩$ is also discussed.

Figure 1

Intervals between zero crossings (excursions) for the fBm in the particular case $H=0.75$, which was generated numerically using the Levinson algorithm. The longest excursion $l_{\text{max}}\left(t\right)$, studied in this Rapid Communication, is defined in Eq. (2).

Figure 2

(Color online) $Q\left(t\right)$ as a function of $t$ for $H=0.1$, 0.5, and 0.9. The straight lines correspond to the value of $Q_{\infty }^R\left(1-H\right)$ for a renewal process given in Eq. (5). This clearly illustrates that the fBm is not a renewal process.

Figure 3

(Color online) Plot of $Q\left(t\right)$ as a function of $t$ on a linear-linear scale for $H=0.9$. The solid line indicates the fit as in Eq. (13) with $a\left(0.9\right)\simeq -0.1$ and $b\left(0.9\right)\simeq 0.3$. Inset: plot of $Q_{\infty }\left(H=0.9\right)-Q\left(t\right)$ (same data as in the main figure) as a function of $t$ in a log-log plot. The solid line corresponds to $-a\left(0.9\right)t^{-b\left(H\right)}$: this suggests a good quality of the fitting procedure in Eq. (13).

Figure 4

(Color online) The symbols indicate the numerical estimate of $Q_{\infty }\left(\theta =1-H\right)$, extracted from the fitting procedure in Eq. (13). For comparison, we have also plotted $Q_{\infty }^R\left(\theta =1-H\right)$ for a renewal process (5). This plot clearly shows that, except for $H=1/2$, the fBm is not a renewal process.

Figure 5

(Color online) Plot of $a\left(H\right)$ and $b\left(H\right)$ as functions of $H=1-\theta$.