  We show existence of an infinitesimally invariant measure $m$ for a large
class of divergence and non-divergence form elliptic second order partial
differential operators with locally Sobolev regular diffusion coefficient and
drift of some local integrability order. Subsequently, we derive regularity
properties of the corresponding semigroup which is defined in
$L^s(\mathbb{R}^d,m)$, $s\in [1,\infty]$, including the classical strong Feller
property and classical irreducibility. This leads to a transition function of a
Hunt process that is explicitly identified as a solution to an SDE. Further
properties of this Hunt process, like non-explosion, moment inequalities,
recurrence and transience, as well as ergodicity, including invariance and
uniqueness of $m$, and uniqueness in law, can then be studied using the derived
analytical tools and tools from generalized Dirichlet form theory.
