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We consider the persistence probability, the occupation-time distribution andthe distribution of the number of zero crossings for discrete or (equivalently)discretely sampled Gaussian Stationary Processes (GSPs) of zero mean. We firstconsider the Ornstein-Uhlenbeck process, finding expressions for the mean andvariance of the number of crossings and the `partial survival' probability. Wethen elaborate on the correlator expansion developed in an earlier paper [G. C.M. A. Ehrhardt and A. J. Bray, Phys. Rev. Lett. 88, 070602 (2001)] to calculatediscretely sampled persistence exponents of GSPs of known correlator by meansof a series expansion in the correlator. We apply this method to the processesd^n x/dt^n=\eta(t) with n > 2, incorporating an extrapolation of the series tothe limit of continuous sampling. We extend the correlator method to calculatethe occupation-time and crossing-number distributions, as well as theirpartial-survival distributions and the means and variances of the occupationtime and number of crossings. We apply these general methods to the d^nx/dt^n=\eta(t) processes for n=1 (random walk), n=2 (random acceleration) andlarger n, and to diffusion from random initial conditions in 1-3 dimensions.The results for discrete sampling are extrapolated to the continuum limit wherepossible.Comment: 35 pages, 31 figure

Persistence exponents and the statistics of crossings and occupation times for Gaussian stationary processes