Persistence of Gaussian processes: non-summable correlations

TL;DR

This paper precisely characterizes the decay rate of persistence probabilities for Gaussian processes with non-summable correlations, resolving a fifty-year-old gap in understanding and applying it to interface models.

Contribution

It provides an exact asymptotic formula for persistence decay in Gaussian processes with regularly varying correlations, closing longstanding theoretical gaps.

Findings

Decay rate of persistence probabilities is of order (T/I_ρ(T)) log I_ρ(T)

Results refine understanding of Gaussian process persistence with non-summable correlations

Application to Langevin dynamics of φ-interface models in statistical physics

Abstract

Suppose the auto-correlations of real-valued, centered Gaussian process $Z(\cdot)$ are non-negative and decay as $\rho(|s-t|)$ for some $\rho(\cdot)$ regularly varying at infinity of order $-\alpha \in [-1,0)$. With $I_\rho(t)=\int_0^t \rho(s)ds$ its primitive, we show that the persistence probabilities decay rate of $ -\log\mathbb{P}(\sup_{t \in [0,T]}\{Z(t)\}<0)$ is precisely of order $(T/I_\rho(T)) \log I_\rho(T)$, thereby closing the gap between the lower and upper bounds of \cite{NR}, which stood as such for over fifty years. We demonstrate its usefulness by sharpening recent results of \cite{Sak} about the dependence on $d$ of such persistence decay for the Langevin dynamics of certain $\grad \phi$-interface models on $\Z^d$.