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Persistence in a stationary time series.

S N Majumdar¹, D Dhar

¹Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-400005, India.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|November 3, 2001

Summary

We investigate persistence in continuous stochastic processes. Under specific conditions, their persistence simplifies to that of a discrete sequence, offering insights into non-Markovian processes and diffusion on hierarchical lattices.

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Area of Science:

Statistical Physics
Stochastic Processes
Complex Systems

Background:

Persistence, the probability of a system remaining in its initial state, is crucial in various physical phenomena.
Continuous stochastic processes stationary under integer time shifts present unique analytical challenges.

Purpose of the Study:

To analyze the persistence of continuous stochastic processes exhibiting stationarity only under integer time shifts.
To establish a connection between the persistence of continuous processes and their discrete counterparts.

Main Methods:

Developing theoretical conditions for reducing continuous process persistence to discrete sequence persistence.
Constructing a specific non-Markovian discrete sequence for persistence computation.
Investigating the relationship to diffusion processes on hierarchical lattices.

Main Results:

Demonstrated that persistence in certain continuous stochastic processes can be accurately represented by a discrete sequence derived from integer-time measurements.
Successfully computed the persistence of a constructed non-Markovian sequence.
Established this computation as a limiting case for persistence in diffusion on hierarchical lattices.

Conclusions:

The study provides a method to simplify persistence calculations for a class of continuous stochastic processes.
The findings offer a new perspective on non-Markovian dynamics and their relation to lattice-based diffusion models.

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