Aging arcsine law in Brownian motion and its generalization

Area of Science:

• Probability theory
• Stochastic processes
• Mathematical physics

Background:

• The classical arcsine law describes the limiting distribution of occupation time in Brownian motion.
• This law typically assumes a system starts at time zero and is observed over a long duration.
• Deviations from this standard setup can alter the expected statistical behavior.

Purpose of the Study:

• To investigate the impact of system preparation and aging on the arcsine law.
• To derive a generalized distributional theorem for occupation time statistics in aged Brownian motion.
• To extend these findings to renewal processes.

Main Methods:

• Derivation of an aging distributional theorem for occupation time statistics.
• Analysis of Brownian motion with a delayed start or observation window.
• Generalization of the derived theorem to renewal processes.

Main Results:

• The fraction of occupation time in Brownian motion exhibits an "aging" effect, deviating from the classical arcsine distribution.
• The ratio of the start time of measurement to the total measurement time significantly influences the resulting distribution's shape.
• The aging phenomenon and its distributional characteristics are successfully generalized to renewal processes.

Conclusions:

• System preparation and aging fundamentally alter the arcsine law in Brownian motion.
• The timing of measurements is a critical factor in characterizing aged occupation time distributions.
• The developed aging distributional limit theorem provides a broader framework applicable to various stochastic processes, including renewal processes.