Homogenization of Hamilton-Jacobi equations with rough time dependence

We consider viscosity solutions of Hamilton-Jacobi equations with oscillatory spatial dependence and rough time dependence. The time dependence is in the form of the derivative of a continuous path that converges to a possibly nowhere-differentiable path, for example a Brownian motion. In the case where the path is one-dimensional, we prove that the solutions converge locally uniformly to the solution of a spatially homogenous, stochastic Hamilton-Jacobi equation in the sense of Lions and Souganidis. We also provide examples of equations in which the path is multi-dimensional, and show that many different behaviors are possible, including diverging to infinity or converging in law to a Brownian motion.