Persistence of a Continuous Stochastic Process with Discrete-Time Sampling

Abstract

We introduce the concept of `discrete-time persistence&#39;, 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&#39;, 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.