Near-extreme statistics of Brownian motion

Area of Science:

• Stochastic processes
• Statistical mechanics
• Probability theory

Background:

• Brownian motion (BM) is a fundamental stochastic process.
• Understanding extreme events in BM is crucial for various scientific fields.
• Previous studies often focused on the maximum value itself, not the time spent near it.

Purpose of the Study:

• To investigate the statistical properties of near-extreme events in Brownian motion.
• To develop a theoretical framework for analyzing the time spent by BM near its maximum.
• To extend these findings to constrained Brownian motion, such as the Brownian bridge.

Main Methods:

• Development of a path integral approach to study functionals of the maximum of BM.
• Derivation of an explicit expression for the moments <[ρ(r,t)]k> of the time density near the maximum.
• Analysis of near extremes for constrained Brownian motion.

Main Results:

• The study provides the full probability density function of ρ(r,t), the time spent near the maximum.
• Explicit formulas for arbitrary integer moments of ρ(r,t) were obtained.
• Analytical results for constrained BM, including the Brownian bridge, were derived.

Conclusions:

• The path integral method offers a powerful tool for analyzing near-extreme statistics of Brownian motion.
• The findings provide a deeper understanding of the temporal behavior of BM near its extremes.
• Numerical simulations confirm the validity of the derived analytical results.