Counting of level crossings for inertial random processes: Generalization of the Rice formula

TL;DR

This paper generalizes Rice's formula to count level crossings in all Gaussian inertial processes, providing exact crossing intensities for models like Brownian motion and noisy oscillators, with applications and simulations.

Contribution

It extends Rice's classical formula to a broader class of Gaussian processes, enabling precise crossing counts for inertial stochastic models.

Findings

Exact crossing intensities derived for various inertial processes

Long- and short-time behavior of crossing rates analyzed

Numerical simulations confirm theoretical results

Abstract

We address the counting of level crossings for inertial stochastic processes. We review Rice's approach to the problem and generalize the classical Rice formula to include all Gaussian processes in their most general form. We apply the results to some second-order (i.e., inertial) processes of physical interest, such as Brownian motion, random acceleration and noisy harmonic oscillators. For all models we obtain the exact crossing intensities and discuss their long- and short-time dependence. We illustrate these results with numerical simulations.