Homogenization and Orowan's law for anisotropic fractional operators of any order

We consider an anisotropic L\'evy operator $\mathcal{I}_s$ of any order $s\in(0,1)$ and we consider the homogenization properties of an evolution equation. The scaling properties and the effective Hamiltonian that we obtain is different according to the cases $s<1/2$ and $s>1/2$. In the isotropic onedimensional case, we also prove a statement related to the so-called Orowan's law, that is an appropriate scaling of the effective Hamiltonian presents a linear behavior.