Persistence Exponents and the Statistics of Crossings and Occupation Times for Gaussian Stationary Processes

TL;DR

This paper develops a series expansion method to analyze persistence, crossing, and occupation-time statistics of Gaussian stationary processes, including various derivatives of Brownian motion and diffusion, with results extrapolated to continuous sampling.

Contribution

It introduces an extended correlator expansion technique for calculating persistence exponents and crossing statistics of discretely sampled Gaussian processes, applicable to higher derivatives and diffusion.

Findings

Derived expressions for mean and variance of crossings in Ornstein-Uhlenbeck process.

Extended correlator method to compute occupation-time and crossing distributions.

Applied methods to processes like random walk, random acceleration, and diffusion, with continuum extrapolation.

Abstract

We consider the persistence probability, the occupation-time distribution and the distribution of the number of zero crossings for discrete or (equivalently) discretely sampled Gaussian Stationary Processes (GSPs) of zero mean. We first consider the Ornstein-Uhlenbeck process, finding expressions for the mean and variance of the number of crossings and the `partial survival' probability. We then elaborate on the correlator expansion developed in an earlier paper [G. C. M. A. Ehrhardt and A. J. Bray, Phys. Rev. Lett. 88, 070602 (2001)] to calculate discretely sampled persistence exponents of GSPs of known correlator by means of a series expansion in the correlator. We apply this method to the processes d^n x/dt^n=\eta(t) with n > 2, incorporating an extrapolation of the series to the limit of continuous sampling. We extend the correlator method to calculate the occupation-time and crossing-number distributions, as well as their partial-survival distributions and the means and variances of the occupation time and number of crossings. We apply these general methods to the d^n x/dt^n=\eta(t) processes for n=1 (random walk), n=2 (random acceleration) and larger n, and to diffusion from random initial conditions in 1-3 dimensions. The results for discrete sampling are extrapolated to the continuum limit where possible.