Homogenization for space-time-dependent KPP reaction–diffusion equations and G-equations

We prove stochastic homogenization for reaction–advection–diffusion equations with random space-time-dependent KPP reactions with temporal correlations that are decaying in an appropriate sense. We show that the limiting homogenized dynamic has the simple form of spreading with some deterministic direction-dependent speeds from the support of the initial datum. We obtain analogous results for G-equations with random flame speeds and incompressible background advections. Important ingredients in our proofs are a non-autonomous subadditive theorem and the principle of virtual linearity for KPP reactions from the companion papers (Zhang and Zlatoš in Electron J Probab 28:1–23, 2023; Zlatoš in Virtual linearity for KPP reaction–diffusion equations).