Persistence of a continuous stochastic process with discrete-time sampling

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=nΔT. 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.