Persistence of a continuous stochastic process with discrete-time sampling- art. no. 015101

We introduce the concept of ''discrete-time persistence,'' which deals with zero-crossings of a continuous stochastic process, X(T), measured at discr ete times, T=n DeltaT. For a Gaussian Markov process with relaxation rate m u, 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 hig h precision. We also define the concept of "alternating persistence,'' whic h corresponds to a<0. For a>1, corresponding to motion in an unstable poten tial (mu <0), there is a nonzero probability of having no zero-crossings in infinite time; and we show how to calculate it.