Georgia Institute of Technology College of Sciences

The study of random Hamilton-Jacobi PDE is motivated by mathematical physics, and in particular, the study of random Burgers equations. We will show that, almost surely, there is a unique stationary solution, which also has better regularity than expected. The solution to any initial value problem converges to the stationary solution exponentially fast. These properties are closely related to the hyperbolicity of global minimizer for the underlying Lagrangian system. Our result generalizes the one-dimensional result of E, Khanin, Mazel and Sinai to arbitrary dimensions. Based on joint works with K. Khanin and R. Iturriaga.