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We present conservativeness criteria for sub-Markovian semigroups generatedby divergence type operators with specified infinitesimally invariant measures.The conservativeness criteria in this article are derived by $L^1$-uniquenessand imply that a given infinitesimally invariant measure becomes an invariantmeasure. We explore further conditions on the coefficients of the partialdifferential operators that ensure the uniqueness of the invariant measurebeyond the case where the corresponding semigroups are recurrent. A mainobservation is that for conservativeness and uniqueness of invariant measuresin this article, no growth conditions are required for the partial derivativesrelated to the anti-symmetric matrix of functions $C=(c_{ij})_{1 \leq i,j \leqd}$ 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 uniqueand strong solution up to infinity to a corresponding It\^{o}-SDE, but also theexistence and uniqueness of invariant measures for the family of strongsolutions.

Conservativeness and uniqueness of invariant measures related to non-symmetric divergence type operators