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 Poincar\'e inequality with Ibragimov and Lifshits' method of characteristic functions, effectively overcoming the challenge from the lack of It^o tools in this colored-in-time setting, and achieving results that are inaccessible with previous methods.