In this talk, I will first give an overview of stochastic homogenization for nondivergence form elliptic equations (from the PDE perspective) and quenched invariance principles for nonreversible diffusion processes (from the probability perspective). I will then present new quantitative homogenization results for the parabolic Green Function and for the unique mutually absolutely continuous invariant measure. I will discuss the implications of these homogenization results, such as heat kernel bounds on the parabolic Green function and quantitative ergodicity for the environment process. This talk is based on joint work with Scott Armstrong and Benjamin Fehrman.