Analyticity for Solution of Integro-Differential Operators

Abstract: We prove that for a certain class of kernels $K(y)$ that viscosity solutions of the integro-differential equation $$\int_{\mathbb R^n} (u(x+y) - 2 u(x) + u(x-y)) K(y) dy = f(x,u(x))$$ are locally analytic if $f$ is an analytic function. This extends the result of Albanese, Fiscella, Valdinoci that such solutions belong to certain Gevrey classes.