Asymptotic Periodicity of the Iterates of Markov Operators

Abstract

ABSTRACT. We say P: L1 -. L1 is a Markov operator if (i) Pf > 0 for / > 0 and (ii) ||P/|| = U/H if / > 0. It is shown that any Markov operator P has certain spectral decomposition if, for any / 6 ¿' with / > 0 and ||/|| = 1, Pnf -> 7 when n -> oo, where 7 is a strongly compact subset of L1. It follows from this decomposition that Pn f is asymptotically periodic for any / G Ll.Introduction.In the theory of stationary discrete time Markov processes, the sequence {Pn} of the iterates of a linear operator P: L1 -♦ L1 plays an important role. This operator is positive, bounded and generalizes the notion of the transition function [2,4]. Asymptotic behavior of {Pn} depends heavily on the spectral properties of P [12], but the construction of the spectral decomposition for P is not an easy problem. It can be solved under some additional hypothesis such as quasicompactness [14] of P or the uniform stability of {Pn} in mean [6]. In general, there is no technique available for studying the asymptotic behavior of {Pn} even if P is given by a simple explicit formula. A typical example where such a situation occurs is the statistical theory of deterministic systems. If 5 is a mapping of the unit interval into itself and / is a probability density, then the sequence {Pg }, with

Asymptotic periodicity of recurrent flows in infinite networks

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An extension of the Vu–Sine theorem and compact-supercyclicity

“…This theorem was proved in [4] for the Markov semigroups in L 1 . Its general case was proved by Vu [10] and Sine [11].…”

“…I am grateful to Eduard Emelyanov, who attracted my attention to papers [4,10,11] and formulated a hypothesis that led to Theorems 2 and 3.…”

“…The second part of the paper is devoted to the application of Theorem 1 to the abovementioned strengthening of the results of [4,10,11].…”

Asymptotic behavior of Markov semigroups on preduals of von Neumann algebras

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