Drift reduction method for SDEs driven by inhomogeneous singular L{é}vy noise

We study SDE $$d{X}_t=b(X_t)dt+A(X_{t-})d{Z}_t,X_0=x\in {\mathbb{R}}^d,t\ge 0$$ where $Z=(Z^1,\dots ,Z^d{)}^T$, with $Z^i,i=1,\dots ,d$ being independent one-dimensional symmetric jump Lévy processes, not necessarily identically distributed. In particular, we cover the case when each $Z^i$ is one-dimensional symmetric ${\alpha }_i$-stable process (${\alpha }_i\in (0,2)$ and they are not necessarily equal). Under certain assumptions on $b$, $A$ and $Z$ we show that the weak solution to the SDE is uniquely defined and Markov, we provide a representation of the transition probability density and we establish H{ö}lder regularity of the corresponding transition semigroup. The method we propose is based on a reduction of an SDE with a drift term to another SDE without such a term but with coefficients depending on time variable. Such a method have the same spirit with the classic characteristic method and seems to be of independent interest.