Crossing intervals of non-Markovian Gaussian processes

Abstract

We review the properties of time intervals between the crossings at a level M of a smooth stationary Gaussian temporal signal. The distribution of these intervals and the persistence are derived within the Independent Interval Approximation (IIA). These results grant access to the distribution of extrema of a general Gaussian process. Exact results are obtained for the persistence exponents and the crossing interval distributions, in the limit of large |M|. In addition, the small time behavior of the interval distributions and the persistence is calculated analytically, for any M. The IIA is found to reproduce most of these exact results and its accuracy is also illustrated by extensive numerical simulations applied to non-Markovian Gaussian processes appearing in various physical contexts.

Final version: Minor typos corrected, section IV extended, references added

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Cited by in corpus (6)

• Pickands' constant at first order in an expansion around Brownian motion
• Distribution of zeros in the rough geometry of fluctuating interfaces
• Dipole diffusion in a random electrical potential
• Effective computations of joint excursion times for stationary Gaussian processes
• Persistence of Non-Markovian Gaussian Stationary Processes in Discrete Time
• A simple method to calculate first-passage time densities of non-smooth processes