The Cauchy problem and the martingale problem for integro-differential operators with non-smooth kernels
Helmut Abels, Moritz Kaßmann · Osaka City University (Osaka City University) · 2009
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 \ abla 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.