Effective persistency evaluation via exact excursion distributions for random processes and fields

Abstract

Finding the probability that a stochastic system stays in a certain region of its state space over a specified time – a long-standing problem both in computational physics and in applied and theoretical mathematics – is approached through the extended and multivariate Rice formula. In principle, it applies to any smooth process multivariate both in argument and in value given that efficient numerical implementations of the high-dimensional integration are available. The method offers an exact integral representation yielding remarkably accurate results and provides an alternative method of computing persistency probability and exponent for a physical system. The computational methodology can be viewed as an implementation of path integration for a smooth Gaussian process with an arbitrary covariance. The high accuracy of the proposed method is due to efficient computation of expectations with respect to high-dimensional nearly singular Gaussian distributions. For Gaussian processes, the computations are effective and more precise than those based on the Rice series expansions and the independent interval approximation. For the benchmark diffusion process, it produces the persistency exponent that is essentially the same as the recently obtained analytical value and surpasses in the accuracy, interpretability as well as control of the error, previous methods including the independent or Markovian excursion approximation. The method solves the two-step excursion dependence for a general stationary differentiable Gaussian process, in both theoretical and practical numerical sense. The solution is based on exact expressions for the probability density for one and two successive excursion lengths. The numerical routine RIND computes the densities using recent advances in scientific computing and is easily accessible for a general covariance function, via a simple numerical interface. The work offers also some analytical results that explain the effectiveness of the implemented methodology and elaborates how it can be utilized for non-Gaussian processes.

Characteristics of the switch process and geometric divisibility

“…Approximating exceedance time distributions is important in statistical physics because the explicit analytical solution is a well-known difficult problem in probability theory. There are several possible approaches and among them are numerical based on the generalized Rice formula, see for example Lindgren et al (2022), and quasi-analytical such as the independent interval approximation (IIA) framework, see for example Sire (2007). Our derived relation between expected value, covariance and the switching time distribution contributes to this framework.…”