Scaling Limits of Deterministic Surface Growth Models and Fully Nonlinear Parabolic PDE with Discontinuities

I will discuss the scaling limits of a class of deterministic surface growth models on lattices, which includes “Edwards-Wilkinson growth” (the height of the surface evolves according to the discrete heat equation) and “Solid-on-Solid growth” (the height of the surface is the value function of a two-player, tug-of-war game). The novelty from a PDE point-of-view is, with the exception of the Edwards-Wilkinson case, the limiting evolution is determined by a fully nonlinear parabolic PDE with discontinuous gradient dependence. I will try to explain how the discontinuities arise as a result of the geometry of the underlying lattice. This insight can then be exploited to develop a theory of viscosity solutions for such strange PDE, which I will also briefly describe. Finally, I will conclude by describing some other applications, future directions, and open problems.