Persistence of Gaussian processes: non-summable correlations

Suppose the auto-correlations of real-valued, centered Gaussian process Z(·) are non-negative and decay as ρ(|s − t |) for some ρ(·) regularly varying at infinity of order −α ∈ [−1, 0). With Iρ(t) = ∫ t 0 ρ(s)ds its primitive, we show that the persistence probabilities decay rate of − logP(supt∈[0,T ]{Z(t)} < 0) is precisely of order (T/Iρ(T )) log Iρ(T ), thereby closing the gap between the lower and upper bounds of Newell andRosenblatt (Ann.Math. Stat. 33:1306–1313, 1962), which stood as such for over fifty years. We demonstrate its usefulness by sharpening recent results of Sakagawa (Adv. Appl. Probab. 47:146–163, 2015) about the dependence on d of such persistence decay for the Langevin dynamics of certain ∇φ-interface models on Z d .