Persistence of Gaussian processes: non-summable correlations

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 $\Z^d$.