  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 [30, 35].
