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Persistence of a Continuous Stochastic Process with Discrete-Time Sampling

Satya N. Majumdar·Alan J. Bray·George C. M. A. Ehrhardt
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Abstract

We introduce the concept of `discrete-time persistence', which deals with zero-crossings of a continuous stochastic process, X(T), measured at discrete times, T = n ΔT. For a Gaussian Markov process with relaxation rate μ, we show that the persistence (no crossing) probability decays as ρ(a)^n for large n, where a = \exp(-μΔT), and we compute ρ(a) to high precision. We also define the concept of `alternating persistence', which corresponds to a<0. For a>1, corresponding to motion in an unstable potential (μ<0), there is a nonzero probability of having no zero-crossings in infinite time, and we show how to calculate it.

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Satya N. Majumdar, Alan J. Bray, George C. M. A. Ehrhardt. 2001-09-19. Persistence of a Continuous Stochastic Process with Discrete-Time Sampling. https://doi.org/10.1103/physreve.64.015101

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