  We present conservativeness criteria for sub-Markovian semigroups generated
by divergence type operators with specified infinitesimally invariant measures.
The conservativeness criteria in this article are derived by $L^1$-uniqueness
and imply that a given infinitesimally invariant measure becomes an invariant
measure. We explore further conditions on the coefficients of the partial
differential operators that ensure the uniqueness of the invariant measure
beyond the case where the corresponding semigroups are recurrent. A main
observation is that for conservativeness and uniqueness of invariant measures
in this article, no growth conditions are required for the partial derivatives
related to the anti-symmetric matrix of functions $C=(c_{ij})_{1 \leq i,j \leq
d}$ that determine a part of the drift coefficient. As stochastic counterparts,
our results can be applied to show not only the existence of a pathwise unique
and strong solution up to infinity to a corresponding It\^{o}-SDE, but also the
existence and uniqueness of invariant measures for the family of strong
solutions.

Probability