On the Cauchy problem for integro-differential operators in Sobolev   classes and the martingale problem
  The existence and uniqueness in Sobolev spaces of solutions of the Cauchy problem to parabolic integro-differential equation of the order {\alpha}\in(0,2) is investigated. The principal part of the operator has kernel m(t,x,y)/|y|^{d+{\alpha}} with a bounded nondegenerate m, H\"older in x and measurable in y. The lower order part has bounded and measurable coefficients. The result is applied to prove the existence and uniqueness of the corresponding martingale problem. 
R. Mikulevicius and H. Pragarauskas
math.AP math.PR