Towards a Schauder theory for fractional viscous Hamilton--Jacobi equations

We survey some results on Lipschitz and Schauder regularity estimates for viscous Hamilton–Jacobi equations with subcritical Lévy diffusions. The Schauder estimates, along with existence of smooth solutions, are obtained with the help of a Duhamel formula and $L^1$ bounds on the spatial derivatives of the heat kernel. Our results cover very general nonlocal and mixed local-nonlocal diffusions, including strongly anisotropic, nonsymmetric, mixed order, and spectrally one-sided models.