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 = n(\Delta T). For a Gaussian Stationary Process the persistence (no crossing) probability decays as exp(-\theta_D T) = [\rho(a)]^n for large n, where a = \exp[-(\Delta T)/2], and the discrete persistence exponent, \theta_D, is given by \theta_D = \ln(\rho)/2\ln(a). Using the Independent Interval Approximation’, we show how \theta_D varies with (\Delta T) for small (\Delta T) 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, 015151® (2001)] to determine \rho(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 \rho(a) for this case.