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Article Dans Une Revue Physical Review Letters Année : 2015

Maximum of a Fractional Brownian Motion: Analytic Results from Perturbation Theory

Résumé

Fractional Brownian motion is a non-Markovian Gaussian process $X_t$, indexed by the Hurst exponent $H$. It generalises standard Brownian motion (corresponding to $H=1/2$). We study the probability distribution of the maximum $m$ of the process and the time $t_{\rm max}$ at which the maximum is reached. They are encoded in a path integral, which we evaluate perturbatively around a Brownian, setting $H=1/2 + \varepsilon$. This allows us to derive analytic results beyond the scaling exponents. Extensive numerical simulations for different values of $H$ test these analytical predictions and show excellent agreement, even for large $\varepsilon$.

Dates et versions

hal-02281084 , version 1 (07-09-2019)

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Mathieu Delorme, Kay Joerg Wiese. Maximum of a Fractional Brownian Motion: Analytic Results from Perturbation Theory. Physical Review Letters, 2015, 115 (21), ⟨10.1103/PhysRevLett.115.210601⟩. ⟨hal-02281084⟩
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