We give a new criterion for the existence and statistical stability of an invariant probability measure of a Markov process taking values in a Polish phase space. The stability property in question can be formulated as follows: ergodic averages of the laws of the process starting with any initial distribution converge, in the sense of weak convergence of measures, to the invariant measure. The principal assumptions required of the process are: the lower bound on the ergodic averages of the transition of probability function and uniform continuity of (P_tψ), t≥0, where (P_tψ), t≥0 is transition semigroup of the process and ψ is a bounded Lipschitz function. In the second part of the talk we apply this result to address the question of existence and stability of an invariant probability measure for a stochastic partial differential equation with an additive noise. With this result we show the existence and weak* mean ergodicity of an invariant measure for the Lagrangian observation process appearing in the passive tracer model of transport in a random, compressible environment. The above results were obtained in joint papers with S. Peszat & T. Komorowski, and M. Ślęczka & M. Urbański.