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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.

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