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=n DeltaT. For a Gaussian Markov process with relaxation rate mu, we show that the persistence (no crossing) probability decays as [rho (a)](n) for large n, where a = exp(-mu DeltaT), and we compute rho (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 (mu <0), there is a nonzero probability of having no zero-crossings in infinite time; and we show how to calculate it.