Hyperbolicity of Minimizers and Regularity of Viscosity Solutions for a Random Hamilton-Jacobi Equation

We show that for a large class of randomly kicked Hamilton-Jacobi equations, the unique global minimizer is almost surely hyperbolic. Furthermore, we prove that the unique forward and backward viscosity solutions, though in general only Lipshitz, are smooth in a neighborhood of the global minimizer. Related results in the one-dimensional case were obtained by E, Khanin et al. (Ann Math (2) 151(3):877-960, 2000). However, the methods in the above paper are purely one-dimensional and cannot be extended to the case of higher dimensions. Here we develop a completely different approach.