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
(Ptψ), t≥0, where (Ptψ), 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.