  We analyse analytic properties of nonlocal transition semigroups associated
with a class of stochastic differential equations (SDEs) in $\mathbb{R}^d$
driven by pure jump--type L\'evy processes. First, we will show under which
conditions the semigroup will be analytic on the Besov space $B_{p,q}^
m(\mathbb{R}^d)$ with $1\le p, q<\infty$ and $m\in\mathbb{R}$. Secondly, we
present some applications by proving the strong Feller property and give weak
error estimates for approximating schemes of the SDEs over the Besov space
$B_{\infty,\infty}^ m(\mathbb{R}^d)$.
