In the first part of this paper, the authors extends the sufficient conditions for the existence of an invariant measure for Markov semigroups from T. Komorowski, S. Peszat and T. Szarek [Ann. Probab. 38, No. 4, 1401–1443 (2010; Zbl 1214.60035)]. They also formulate criteria for stability and prove results concerning the sweeping property of a Markov semigroup. In the second part, applications are given to general stochastic differential equations. In particular, the general results apply to the equation describing the passive tracer in a compressible random flow. See A. Fannjiang, T. Komorowski and S. Peszat [Stochastic Processes Appl. 97, No. 2, 171–198 (2002; Zbl 1058.60048)]. It is found that the velocity of the passive tracer converges weakly to some random vector.