Time to reach the maximum for a stationary stochastic process.

Area of Science:

• Statistical Physics
• Time Series Analysis
• Nonlinear Dynamics

Background:

• Understanding the behavior of stationary time series is crucial in various scientific fields.
• The time at which a process reaches its global maximum provides valuable insights into its dynamics.
• Distinguishing between equilibrium and nonequilibrium processes is a fundamental challenge.

Purpose of the Study:

• To compute the probability density function P(t_m|T) for the time t_m of the global maximum in stationary time series.
• To investigate the properties of P(t_m|T) for both equilibrium and nonequilibrium processes.
• To explore the universality of P(t_m|T) for diffusive particles in confining potentials.

Main Methods:

• Utilized a path-decomposition technique to derive the probability density function P(t_m|T).
• Analyzed equilibrium processes like the Ornstein-Uhlenbeck process.
• Studied nonequilibrium processes such as Brownian motion with stochastic resetting.

Main Results:

• For equilibrium processes, P(t_m|T) is symmetric around T/2 due to time-reversal symmetry.
• This symmetry serves as a marker for detecting nonequilibrium fluctuations in stationary time series.
• For diffusive particles, the scaled P(t_m|T) becomes universal at late times, uniform in the bulk and with specific shapes at edge regimes.

Conclusions:

• The symmetry of the maximum time distribution is a key indicator of equilibrium in stationary processes.
• The universality of the maximum time distribution for diffusive particles offers a model-independent characterization.
• These findings provide new tools for analyzing complex time series data in physics and beyond.