Non-local gradient dependent operators

In this paper we study a general class of “quasilinear non-local equations” depending on the gradient which arises from tug-of-war games. We establish a Cα/C1,α/C2,αCα/C1,α/C2,α regularity theory for these equations (the kind of regularity depending on the assumptions on the kernel), and we construct different non-local approximations of the pp-Laplacian.