  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.
