The study of random Hamilton-Jacobi PDE is motivated by mathematical physics, and in particular, the study of random Burgers equations. We will show, for a family of randomly kicked equations on the torus, the unique stationary solution is locally smooth. Moreover, the solution to the Cauchy problem converges to the stationary solution exponentially fast. This generalizes the one-dimensional result of E, Khanin, Mazel and Sinai to arbitrary dimensions. The proof uses a nice connection between Hamilton-Jacobi PDE, weak KAM theory, and non-uniform hyperbolic dynamics. Based on joint works with K. Khanin and R. Iturriaga.