Hyperbolicity of minimizers and regularity of viscosity solutions for random Hamilton-Jacobi equations

We show that for a family of randomly kicked Hamilton-Jacobi equations, the unique global minimizer is hyperbolic, almost surely. Furthermore, we prove the unique forward and backward viscosity solutions, though in general only Lipshitz, are smooth in a neighbourhood of the global minimizer. Our result generalizes the result of E, Khanin, Mazel and Sinai (\cite{EKMS00}) to dimension $d\ge 2$, and extends the result of Iturriaga and Khanin in \cite{IK03}.