Nonlinear Evolution Equations Driven by Nonlocal Infinitesimal Generators and Their Probabilistic Interpretations

ABSTRACT: One of the motivations of our program was to develop an understanding of the interplay between the nonlinear and nonlocal components in evolution equations driven by the infinitesimal generators of stochastic processes with jumps, such as Levy processes and flights. In particular, we also studied probabilistic approximations (propagation of chaos) for several extensions of the classical quasilinear and strongly linear PDEs, including the conservation laws, porous medium and Hamilton-Jacobi equations, and reaction-diffusion type equations for Darwinian evolutionary population models where the hydrodynamic limits may still preserve some "background" random noise.