Uniqueness of stable-like processes

Abstract

In this work we consider the following $\alpha$-stable-like operator (a class of pseudo-differential operator) $$ {\mathscr L} f(x):=\int_{\mathbb R^d}[f(x+\sigma_x y)-f(x)-1_{\alpha\in[1,2)}1_{|y|\leq 1}\sigma_x y\cdot\nabla f(x)]\nu_x(d y), $$ where the L\'evy measure $\nu_x(d y)$ is comparable with a non-degenerate $\alpha$-stable-type L\'evy measure (possibly singular), and $\sigma_x$ is a bounded and nondegenerate matrix-valued function. Under H\"older assumption on $x\mapsto\nu_x(d y)$ and uniformly continuity assumption on $x\mapsto\sigma_x$, we show the well-posedness of martingale problem associated with the operator $\mathscr L$. Moreover, we also obtain the existence-uniqueness of strong solutions for the associated SDE when $\sigma$ belongs to the first order Sobolev space $\mathbb W^{1,p}(\mathbb R^d)$ provided $p>d(1+\alpha\vee 1)$ and $\nu_x=\nu$ is a non-degenerate $\alpha$-stable-type L\'evy measure.