Persistence of Gaussian processes: non-summable correlations

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 $\grad ϕ$-interface models on ${\mathbb{Z}}^d$.

Minor typos corrected. To appear in Probability Theory and Related Fields