Crossing intervals of non-Markovian Gaussian processes

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.

💡 Research Summary

The paper investigates the statistical properties of time intervals between successive crossings of a smooth stationary Gaussian process X(t) at a prescribed level M. The authors focus on two complementary sets of quantities: (i) the persistence probabilities P⁽>⁾(t) and P⁽<⁾(t), which are the probabilities that a trajectory that starts above (or below) the threshold M never crosses it up to time t, and (ii) the interval‑length distributions P₊(t) and P₋(t), which give the probability density that a so‑called “+” (above‑M) or “–” (below‑M) interval lasts exactly t. By assuming that successive intervals are statistically independent – the Independent Interval Approximation (IIA) – the authors derive exact relations in Laplace space that connect the persistence functions to the interval distributions. In the time domain these relations become simple integral equations (Eqs. 16‑19) and, after double differentiation, the compact forms P₊(t)=τ₊ P⁽>⁾″(t) and P₋(t)=τ₋ P⁽<⁾″(t) (Eqs. 18‑19).

The analysis begins with a precise definition of “smooth” processes: the velocity X′(t) is continuous, which translates into a two‑time correlation function f(t)=⟨X(t₀)X(t₀+t)⟩ that is at least twice differentiable at t=0. “Very smooth” processes have a fourth derivative at the origin. For such processes the mean crossing rate τ⁻¹ can be expressed in terms of the second derivative of the correlator, a²=−f″(0), and the level M, yielding τ=π √a² e^{M²/2} (Eq. 13). The mean lengths of + and – intervals, τ₊ and τ₋, are related to τ by simple algebraic relations (Eqs. 14‑15).

Two asymptotic regimes are treated analytically. In the limit of large |M| the crossing events become rare. The authors obtain exact expressions for the persistence exponents θ₊(M) and θ₋(M) that govern the exponential decay P⁽>⁾(t)∼e^{−θ₊t} and P⁽<⁾(t)∼e^{−θ₋t} as t→∞. They show that for M≫0 the +‑interval distribution approaches a Wigner‑type form while the –‑interval distribution becomes Poissonian. The IIA reproduces these results almost perfectly, except for a small discrepancy in the large‑M asymptotics of θ₊.

For short times (t→0) the authors expand the interval distributions analytically. For very smooth processes the leading term is linear in t, P₊(t)≈(π/2) a t e^{−M²/2}, with higher‑order O(t³) corrections. For merely smooth processes additional cubic terms appear. Remarkably, the IIA yields exactly the same short‑time expansions, confirming its consistency in this regime.

To validate the theory, the paper presents extensive numerical simulations for several physically relevant Gaussian processes: (a) the Ornstein‑Uhlenbeck process (exponential correlator, non‑smooth velocity), (b) a diffusion‑based process with Gaussian correlator f(t)=e^{−t²/2}, and (c) the height field of a Kardar‑Parisi‑Zhang interface (algebraic decay of correlations). For each case the persistence exponents and interval distributions obtained from direct Monte‑Carlo simulations are compared with the IIA predictions. The agreement is excellent for the smooth and very smooth examples, with relative errors typically below 5 % for moderate values of M. In the non‑smooth Ornstein‑Uhlenbeck case the IIA fails, illustrating its dependence on the smoothness assumption.

The authors also emphasize the connection between persistence and extreme‑value statistics. Since P⁽<⁾(t)=Pr