Persistence with Discrete-Time Sampling
Satya N. Majumdar, Alan J. Bray, George C. M. A. Ehrhardt · arXiv (Cornell University) · 2000
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 \Delta T. 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 \Delta T), and we compute \rho(a) to high precision. We also define the concept of `alternating persistence', which corresponds to 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.