Abstract
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 μ in E, let μ n denote the random measure μ°(Z n °...°Z 1)−1 (when well-defined). We construct the transition probability P of the E-valued Markov chain (μ 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.
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Research supported by NSF grants INT-8420360 and DMS-8502802. This work was done while the author was visiting the Universities of Warwick and Minnesota
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Darling, R.W.R. Ergodicity of a measure-valued Markov chain induced by random transformations. Probab. Th. Rel. Fields 77, 211–229 (1988). https://doi.org/10.1007/BF00334038
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DOI: https://doi.org/10.1007/BF00334038