Almost Sure Central Limit Theorems for Parabolic/Hyperbolic Anderson Models with Gaussian Colored Noises

Abstract This short note is devoted to establishing the almost sure central limit theorem for the parabolic/hyperbolic Anderson models driven by colored-in-time Gaussian noises, completing recent results on quantitative central limit theorems for stochastic partial differential equations. We combine the second-order Gaussian Poincaré inequality with the method of characteristic functions of Ibragimov and Lifshits, effectively overcoming the challenge from the lack of Itô tools in this colored-in-time setting, and achieving results that are inaccessible with previous methods.