In this talk, I will present some recent results on the asymptotic behavior of periodic viscosity solutions of parabolic integro-differential equations (where nonlocal terms are associated with Levy-It\^o operators). In particular, we address the problem to mixed integro-differential equations, e.g. where fractional diffusion occurs in one direction and classical diffusion in the complementary one. The typical results states that, under some suitable assumptions, the solution of the initial value problem behaves like $ct + v(x) + o(1)$ as $t\rightarrow\infty$, where $v$ is a solution of the stationary ergodic problem corresponding to the ergodic constant $c$. In order to obtain this asymptotic behavior, two additional results are shown: Lipschitz regularity of solutions for mixed integro-differential equations and the strong maximum principle. This is a joint work with Guy Barles, Emmanuel Chasseigne and Cyril Imbert.