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arXiv · 0911.1897

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

Reinaldo Garcia-Garcia·Alberto Rosso·Gregory Schehr
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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 \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 θ= 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.

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Reinaldo Garcia-Garcia, Alberto Rosso, Gregory Schehr. 2009-11-10. The longest excursion of fractional Brownian motion : numerical evidence of non-Markovian effects. https://doi.org/10.1103/physreve.81.010102

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