Persistence of a continuous stochastic process with discrete-time sampling: Non-Markov processes

We consider the problem of “discrete-time persistence,” which deals with the zero crossings of a continuous stochastic process X(T) measured at discrete times T=nΔT. For a Gaussian stationary process the persistence (no crossing) probability decays as exp(-θ_DT)=[ρ(a)]ⁿ for large n, where a=exp(-ΔT/2) and the discrete persistence exponent θ_D is given by θ_D=(ln ρ)/(2 ln a). Using the “independent interval approximation,” we show how θ_D varies with ΔT for small ΔT and conclude that experimental measurements of persistence for smooth processes, such as diffusion, are less sensitive to the effects of discrete sampling than measurements of a randomly accelerated particle or random walker. We extend the matrix method developed by us previously [Phys. Rev. E 64, 015101(R) (2001)] to determine ρ(a) for a two-dimensional random walk and the one-dimensional random-acceleration problem. We also consider “alternating persistence,” which corresponds to a<0, and calculate ρ(a) for this case.