The Cauchy problem and the martingale problem for integro-differential operators with non-smooth kernels

Abstract

We consider the linear integro-differential operator $L$ defined by \begin{equation*} Lu(x) =\int_{\mathbb{R}^{n}}(u(x+y)-u(x) -\mathbbm{1}_{[1,2]}(\alpha)\mathbbm{1}_{\{|y|\leq 2\}}(y)y \cdot \nabla u(x))k(x,y)\, dy. \end{equation*} Here the kernel $k(x,y)$ behaves like $|y|^{-n-\alpha}$, $\alpha \in (0,2)$, for small $y$ and is Hölder-continuous 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.