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

Abstract

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 $t_{\max }$ at which the maximum is reached. They are encoded in a path integral, which we evaluate perturbatively around a Brownian, setting $H=1/2+\epsilon$. 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 $\epsilon$.

5 pages, 7 figures

References in corpus (3)

• Asymptotic behavior of self-affine processes in semi-infinite domains
• Fractional Brownian motion approach to polymer translocation: the governing equation of motion
• Probabilistic Phase Space Trajectory Description for Anomalous Polymer Dynamics

Cited by in corpus (22)

• Generalized arcsine laws for fractional Brownian motion
• Extreme-Value Statistics of Fractional Brownian Motion Bridges
• Perturbative Expansion for the Maximum of Fractional Brownian Motion
• Survival probability of stochastic processes beyond persistence exponents
• First passage in an interval for fractional Brownian motion
• Pickands' constant at first order in an expansion around Brownian motion
• Everlasting impact of initial perturbations on first-passage times of non-Markovian random walks
• Time-changed fractional Ornstein-Uhlenbeck process
• An integrate-and-fire model to generate spike trains with long-range dependence
• Path integrals for fractional Brownian motion and fractional Gaussian noise
• Long-term memory induced correction to Arrhenius law
• Sampling first-passage times of fractional Brownian Motion using adaptive bisections
• Probability density of fractional Brownian motion and the fractional Langevin equation with absorbing walls
• Evidence and quantification of memory effects in competitive first passage events
• Log-correlated Random Energy Models with extensive free energy fluctuations: pathologies caused by rare events as signatures of phase transitions
• Theory and Experiments for Disordered Elastic Manifolds, Depinning, Avalanches, and Sandpiles
• Simulation paradoxes related to a fractional Brownian motion with small Hurst index
• Functionals of fractional Brownian motion and the three arcsine laws
• On the inverse problem of fractional Brownian motion and the inverse of infinite Toeplitz matrices
• Tempered fractional Brownian motion on finite intervals
• Noise sensitivity of functionals of fractional Brownian motion driven stochastic differential equations: Results and perspectives
• Location of the Path Supremum for Self-similar Processes with Stationary Increments