Role of conditional probability in multiscale stationary markovian processes

Area of Science:

• Stochastic processes
• Statistical physics
• Time series analysis

Background:

• Stochastic stationary Markovian processes are fundamental in modeling complex systems.
• Understanding the influence of multiple time scales is crucial for accurately describing process dynamics.
• Persistence and mean first passage time are key metrics for characterizing stochastic behavior.

Purpose of the Study:

• To investigate how incorporating numerous time scales affects the conditional probability of stochastic stationary Markovian processes.
• To analyze the impact of different time scale structures (bounded vs. unbounded) on process properties.
• To explore the relationship between time scales, persistence, and the speed of reaching new states.

Main Methods:

• Analysis of two Gaussian processes: short-range correlated (bounded time scales) and power-law correlated (unbounded time scales).
• Investigation of equal position conditional probability P(x,t|x,0) as a measure of persistence.
• Calculation of mean first passage time Tx(Λ) to quantify process speed.

Main Results:

• Increased time scales lead to greater persistence in stochastic processes.
• Power-law correlated processes demonstrate a slow power-law decay in P(x,t|x,0), unlike the exponential cutoff of short-range processes.
• An infinite, unbounded set of time scales is necessary but not sufficient for slow power-law decay, which is linked to algebraic autocorrelation decay.
• Larger time scales increase mean first passage time for distant states (Λ) but decrease it for nearby states.

Conclusions:

• The structure of time scales significantly influences the persistence and dynamics of stochastic processes.
• Power-law correlated processes exhibit enhanced persistence due to their unbounded time scale structure.
• The interplay between time scales and distance (Λ) dictates whether a process becomes faster or slower in reaching new positions.