Asymptotic properties of semigroups of Markov operators

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The author provides an overview of some main ideas and methods developed in the study of asymptotic behavior of Markov operators, and the following three properties of a Markov operator P are considered: (1) Asymptotic stability, (2) Asymptotic periodicity, (3) Sweeping with respect to a family A0⊂A: lim∫APnfdm=0 for all AA0 and fD. Applications of these results appear in the study of locally expanding mappings (on intervals, on the real line and on manifolds), integral operators (especially of Volterra types), dynamical systems with stochastic perturbations, and evolution equations. In an excellent book by the author and M. C. Mackey [Chaos, fractals, and noise, Second edition, Springer, New York, 1994; MR1244104], these and many more related results are presented in much more detail. The last section of the present paper concerns Markov operators acting on finite Borel measures on a Polish metric space. The author shows how this result yields insight into attractors of iterated function systems: they appear to be supports of invariant measures of corresponding Markov operators. The paper is the Wacław Sierpiński Lecture for the year 2002.