  We study the homogenization of nonlinear, first-order equations with highly
oscillatory mixing spatio-temporal dependence. It is shown in a variety of
settings that the homogenized equations are stochastic Hamilton-Jacobi
equations with deterministic, spatially homogenous Hamiltonians driven by white
noise in time. The paper also contains proofs of some general regularity and
path stability results for stochastic Hamilton-Jacobi equations, which are
needed to prove some of the homogenization results and are of independent
interest.
