THE CAUCHY PROBLEM AND THE MARTINGALE PROBLEM FOR INTEGRO-DIFFERENTIAL OPERATORS WITH NON-SMOOTH KERNELS

We consider the linear integro-differential operator L defined by Lu(x) = integral(Rn) (u(x + y) - u(x) - 1([1,2])(alpha)1([vertical bar y vertical bar <= 2])(y)y. del u(x))k(x, y)dy. Here the kernel k(x, y) behaves like vertical bar y vertical bar(-n-alpha), alpha is an element of (0, 2), for small y and is Holdercontinuous in the first variable, precise definitions are given below. We study the unique solvability of the Cauchy problem corresponding to L. As an application we obtain well-posedness of the martingale problem for L. Our strategy follows the classical path of Stroock-Varadhan. The assumptions allow for cases that have not been dealt with so far.