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Fully nonlinear integro-differential equations with deforming kernels. (English) Zbl 1452.35055
In this interesting paper, the authors develop a regularity theory for integro-differential equations with kernels deforming in space like sections of a convex solution of a Monge-Ampère equation. Such a kind of equations appear in stochastic control problems, for example if, in a competitive stochastic game, two players are allowed to choose from different strategies at every step in order to maximize the expected value at the first exit point of a domain. The authors prove a non-local version of the Aleksandrov-Bakelman-Pucci estimate and a Harnack inequality, and derive Hölder and $$C^{1,\alpha}$$ regularity results for solutions.

##### MSC:
 35B65 Smoothness and regularity of solutions to PDEs 35R09 Integro-partial differential equations 35J60 Nonlinear elliptic equations 35J96 Monge-Ampère equations 47G20 Integro-differential operators 35D40 Viscosity solutions to PDEs
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