Comparison Principle for General Nonlocal Hamilton–Jacobi Equations with Superlinear Gradient

Abstract. We obtain the comparison principle for discontinuous viscosity sub- and supersolutions of nonlocal Hamilton–Jacobi equations, with superlinear and coercive gradient terms. The nonlocal terms are integrodifferential operators in Lévy form, with general measures: [Formula: see text]-dependent, possibly degenerate, and without any restriction on the order. The measures must satisfy a combined Wasserstein/total variation-continuity assumption, which is one of the weakest conditions used in the context of viscosity approach for this type of integrodifferential PDEs. The proof relies on a regularizing effect due to the gradient growth. We present several examples of applications to PDEs with different types of nonlocal operators (measures with density, operators of variable order, Lévy–Itô operators).