Regularity results for fully nonlinear integro-differential operators with nonsymmetric positive kernels : Subcritical Case

Abstract

We introduce a new class of fully nonlinear integro-differential operators with possible nonsymmetric kernels, which includes the ones that arise from stochastic control problems with purely jump Lèvy processes. If the index of the operator $\sigma$ is in $(1,2)$ (subcritical case), then we obtain a comparison principle, a nonlocal version of the Alexandroff-Backelman-Pucci estimate, a Harnack inequality, a Hölder regularity, and an interior ${\mathrm{C}}^{1,\alpha }$-regularity for fully nonlinear integro-differential equations associated with such a class. Moreover, our estimates remain uniform as the index $\sigma$ of the operator is getting close to two, so that they can be regarded as a natural extension of regularity results for elliptic partial differential equations.

References in corpus (1)

• Second-Order Elliptic Integro-Differential Equations: Viscosity Solutions' Theory Revisited

Cited by in corpus (1)

• Stochastic Homogenization for Some Nonlinear Integro-Differential Equations