Uniqueness of stable-like processes

Abstract

In this work we consider the following $α$-stable-like operator (a class of pseudo-differential operator) $$ {\mathscr L} f(x):=\int_{\mathbb R^d}[f(x+σ_x y)-f(x)-1_{α\in[1,2)}1_{|y|\leq 1}σ_x y\cdot\nabla f(x)]ν_x(d y), $$ where the Lévy measure $ν_x(d y)$ is comparable with a non-degenerate $α$-stable-type Lévy measure (possibly singular), and $σ_x$ is a bounded and nondegenerate matrix-valued function. Under Hölder assumption on $x\mapstoν_x(d y)$ and uniformly continuity assumption on $x\mapstoσ_x$, 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 $σ$ belongs to the first order Sobolev space $\mathbb W^{1,p}(\mathbb R^d)$ provided $p>d(1+α\vee 1)$ and $ν_x=ν$ is a non-degenerate $α$-stable-type Lévy measure.