Ergodicity of a measure-valued Markov chain induced by random transformations
Let \(Z_ 1,Z_ 2,..\). be a sequence of i.i.d. random transformations (possibly discontinuous) of a compact metric space M, and let E denote the space of normalized mass distributions on M. Given \(\mu\) in E, let \(\mu _ n\) denote the random measure \(\mu \circ (Z_ n\circ...\circ Z_ 1)^{-1}\) (when well-defined). We construct the transition probability P of the E-valued Markov chain \((\mu _ n)\), and give a necessary and sufficient condition for P to have a unique invariant measure concentrated on the degenerate mass distributions. Convergence to 'statistical equilibrium' of the associated discrete-time stochastic flow is investigated.
scientific article; zbMATH DE number 4007370
scientific article