A Schauder regularity theory for nonlocal viscous Hamilton-Jacobi equations

Viscous Hamilton–Jacobi equations arise as dynamic programming equations in optimal control and differential games, where the unknown function represents the value function of an optimally controlled process. We use a heat-semigroup approach to study the regularity of such equations with nonlocal and mixed local–nonlocal diffusions. Our framework covers a broad class of Lévy-type operators, including fractional Laplacians as well as nonsymmetric and strongly anisotropic operators. Using the Duhamel formulation together with smoothing estimates in Hölder spaces, we obtain optimal spatial Schauder estimates and explicit blow-up rates for higher-order Hölder norms as time approaches zero. We focus on two canonical regimes: Lipschitz initial data, leading to solutions with uniformly bounded gradients, and lower-regularity Hölder initial data, where gradients may blow up and their behavior is linked to the growth of the Hamiltonian. Under additional assumptions on the diffusion, we also obtain optimal regularity in time and joint space-time Schauder estimates consistent with the natural fractional parabolic scaling. We further discuss short- and long-time existence, uniqueness, and classical solutions. Our motivation comes from mean field games (MFGs), with the goal of applying this regularity theory to nonlocal MFGs with low-regularity data. The talk is based on our paper "A Schauder regularity theory for nonlocal and mixed local--nonlocal viscous Hamilton--Jacobi equations" (joint work with Espen R. Jakobsen and Artur Rutkowski).