Consider the following Levy type operator: $$ L_t\varphi(x)=\int_{R^d}\big[\varphi(x+z)-\varphi(x)-1_{|z|\leq 1}z\cdot\nabla\varphi(x)\big]\kappa(x,z)\nu(dz),\quad\forall \varphi\in C_c^2(R^d), $$ where $\nu$ is an symmetric Levy measure, $\kappa(\cdot,z)\in C^\infty(R^d)$ (smooth in $x$, but maybe degenerate and unbounded), and for each $x$, $$ \int_{R^d}(|z|^2\wedge1)\kappa(x,z)\nu(dz)<\infty. $$ Then, is the martingale problem for $L$ has a unique solution? I think the conclusion should be true, since the coefficient is smooth, but I can not find a reference. Thanks for the help.
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