Parametrix method and the weak solution to an SDE driven by an $α$-stable noise

Let $L:= a(x) (-\Delta)^{-\alpha/2}+ (b(x), \nabla)$, where $\alpha\in (0,2)$, and $a:\mathbb{R}^d\to \mathbb{R}$, $b: \mathbb{R}^d\to \mathbb{R}^d$ are H\"older continuous. We show that the $C_\infty(\mathbb{R}^d)$-closure of $(L, C_\infty^2(\mathbb{R}^d))$ is the generator of a Feller Markov process $X$, which possesses a transition probability density $p_t(x,y)$. Complete description of this process is given both in terms of a martingale problem and as a weak solution to an SDE driven by an $\alpha$-stable noise. To construct the transition probability density and to obtain the two-sided estimates for it, we develop a new version of the parametrix method, which allows one to handle the case where $0<\alpha\leq 1$ and $b\neq 0$; that is, in our approach the gradient part of the generator is not required to be dominated by the jump part.