The martingale problem for geometric stable-like processes

Abstract

We prove that the martingale problem is well posed for pure-jump Lévy-type operators of the form $$(\mathcal{L}f)(x)={\int }_{{\mathbb{R}}^d\setminus \{0\}}\left(f(x+h)-f(x)-(\nabla f(x)\cdot h)1_{\Vert h\Vert <1}\right)K(x,h)dh,$$ where $K(x,\cdot )$ is a jump kernel of the form $K(x,h)\sim \frac{l(\Vert h\Vert )}{\Vert h{\Vert }^d}$ for each $x\in {\mathbb{R}}^d,\Vert h\Vert <1$, and $l$ is a positive function that is slowly varying at $0$, under suitable assumptions on $K$. This includes jump kernels such as those of $\hat{I}\pm$-geometric stable processes, $\hat{I}\pm \in (0,2]$.

29 pages, to be submitted to Stochastic Processes and Applications