Generalized Arcsine Laws for Fractional Brownian Motion

Area of Science:

• Stochastic processes
• Extreme value statistics
• Gaussian processes

Background:

• Brownian motion exhibits three arcsine laws for key time-based observables.
• These laws describe the probability distributions of the time spent positive, last visit to the origin, and time to reach maximum/minimum.
• Fractional Brownian motion (fBm) generalizes Brownian motion with tunable persistence.

Purpose of the Study:

• To investigate how arcsine laws are modified for fractional Brownian motion.
• To analyze the impact of the Hurst exponent (H) on these extreme-value statistics.
• To compare the behavior of fBm with standard Brownian motion (H=1/2).

Main Methods:

• Utilizing a perturbative expansion around H=1/2 (ϵ = H - 1/2).
• Deriving the modified probability distributions for the three observables.
• Conducting high-precision numerical simulations to validate theoretical predictions.

Main Results:

• The three arcsine laws for fBm yield distinct probability distributions.
• Differences in these distributions emerge at the second order of the perturbative expansion (ϵ^2).
• Theoretical predictions show excellent agreement with simulation results.

Conclusions:

• Arcsine laws for Brownian motion are modified in the non-Markovian regime of fractional Brownian motion.
• The Hurst exponent H dictates the specific deviations from standard Brownian motion.
• This study provides a theoretical and numerical framework for understanding extreme-value statistics in generalized Brownian processes.