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 with variable coefficients of the order α∈(0,2)α∈(0,2) is investigated. The principal part of the operator has kernel m(t,x,y)/|y|d+αm(t,x,y)/|y|d+α with a bounded nondegenerate m, Hölder 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.