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Regularity for solutions of nonlocal, nonsymmetric equations. (English) Zbl 1317.35278
The authors consider integro-differential equations that arise when one studies discontinuous stochastic processes and uses the celebrated Lévy-Khintchine formula. More precisely, the authors study the regularity of solutions of fully nonlinear integro-differential equations with respect to nonsymmetric kernels. They assume that the operator is elliptic with respect to a family of integro-differential linear operators, where the symmetric parts of the kernels have a fixed homogeneity \(\sigma\) and the skew-symmetric parts have strictly smaller homogeneity \(\tau\). The authors prove a weak Alexandrov-Bellman-Pucci estimate and \(C^{1,\alpha}\) regularity. Note that the estimates are uniform as \(\sigma \rightarrow2\) and \(\tau \rightarrow 1\), so this result extends the regularity theory for elliptic differential equations with dependence on the gradient.

MSC:
35R09 Integral partial differential equations
35R60 PDEs with randomness, stochastic partial differential equations
35B65 Smoothness and regularity of solutions to PDEs
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