Linear Integro-differential Equations

In this chapter, we study general elliptic operators of order 2s, which are linear and translation invariant. We start with their probabilistic motivation and then move on to establishing their basic properties and different notions of solution (strong, weak, and distributional). After that, we develop the interior regularity theory for these operators and then extend it to the more general setting of x-dependent operators, both in divergence and in non-divergence form. Finally, we study the Hölder regularity of solutions up to the boundary and conclude the chapter by presenting some further results and open problems.