  A stable-like process is a Feller process $(X_t)_{t\geq 0}$ taking values in
$\mathbb{R}^d$ and whose generator behaves, locally, like an $\alpha$-stable
L\'evy process, but the index $\alpha$ and all other characteristics may depend
on the state space. More precisely, the jump measure need not to be symmetric
and it strongly depends on the current state of the process; moreover, we do
not require the gradient term to be dominated by the pure jump part. Our
approach is to understand the above phenomena as suitable microstructural
perturbations.
  We show that the corresponding martingale problem is well-posed, and its
solution is a strong Feller process which admits a transition density. For the
transition density we obtain a representation as a sum of an explicitly given
principal term -- this is essentially the density of an $\alpha$-stable random
variable whose parameters depend on the current state $x$ -- and a residual
term; the $L^\infty\otimes L^1$-norm of the residual term is negligible and so
is, under an additional structural assumption, the $L^\infty\otimes
L^\infty$-norm. Concrete examples illustrate the relation between the
assumptions and possible transition density estimates.
