Scaling limits and homogenization for mixing Hamilton-Jacobi equations

I will present results about scaling limits of Hamilton-Jacobi equations with highly oscillatory, mixing spatio-temporal dependence. When the Hamiltonian is centered, then the long time, large space behavior of the solutions is governed by a spatially homogenous, stochastic Hamilton-Jacobi equation. The result relies on explicit convergence rates for the stochastic homogenization of Hamilton-Jacobi equations and new stability estimates for stochastic viscosity solutions. I will also discuss a class of problems for which the hyperbolic scaling limit leads to spatio-temporal stochastic homogenization.