Persistence of a Continuous Stochastic Process with Discrete-Time Sampling

Details Persistence of a Continuous Stochastic Process with Discrete-Time Sampling Persistence of a Continuous Stochastic Process with Discrete-Time Sampling

2000-11-16

arxiv.org/abs/cond-mat/0011290v2

Phys. Rev. E 64, 015101(R) (2001)

Physics

Condensed Matter

Statistical Mechanics

5 pages, some minor changes

Scientific paper

10.1103/PhysRevE.64.015101

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<0. For 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.

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