The longest excursion of fractional Brownian motion : numerical evidence of non-Markovian effects

TL;DR

This paper investigates the statistics of the longest excursion in fractional Brownian motion, revealing non-Markovian effects through numerical simulations and comparison with renewal process models.

Contribution

It provides numerical evidence showing how the longest excursion statistics depend on the Hurst exponent, highlighting non-Markovian effects in fractional Brownian motion.

Findings

< l_{max}(t) > scales linearly with t for large t

Q_ infty(H) varies continuously with H and indicates non-Markovian behavior

Comparison with renewal process models highlights non-Markovian effects for H ≠ 1/2

Abstract

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