Invariant Probabilities of Markov-Feller Operators and Their Supports

Zaharopol, Radu

doi:10.1007/b98076

Cited by 23 publications

(54 citation statements)

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“…Although we have studied the supports of the elementary measures in [52], actually, we did not find "formulas" for their supports. In this paper, in addition to extending the KBBY-decomposition, we will obtain such "formulas" (valid also in the general not necessarily Feller case) for elementary measures of a fairly general kind.…”

Section: Introductionmentioning

confidence: 63%

“…13 of Yosida's monograph [51]); for more recent works on the topic, see Hernández-Lerma and Lasserre [13] and Chap. 5 of their recent monograph [14], and our monograph [52]; for related results, see also Costa and Dufour [9,10], and [11].…”

Section: Introductionmentioning

confidence: 93%

“…While discussing the KBBY-decomposition in the Feller case in [52], we have singled out a type of measures that we have called elementary. Although we have studied the supports of the elementary measures in [52], actually, we did not find "formulas" for their supports.…”

Section: Introductionmentioning

confidence: 99%

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An Ergodic Decomposition Defined by Transition Probabilities

Zaharopol

2008

Acta Appl Math

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Our main goal in this paper is to prove that any transition probability P on a locally compact separable metric space (X, d) defines a Kryloff-Bogoliouboff-BeboutoffYosida (KBBY) ergodic decomposition of the state space (X, d). Our results extend and strengthen the results of Chap. 5 of Hernández-Lerma and Lasserre (Markov Chains and Invariant Probabilities, 2003) and extend our KBBY-decomposition for Markov-Feller operators that we have obtained in Chap. 2 of our monograph (Zaharopol in Invariant Probabilities of Markov-Feller Operators and Their Supports, 2005). In order to deal with the decomposition that we present in this paper, we had to overcome the fact that the LasotaYorke lemma (Theorem 1.2.4 in our book (op. cit.)) and two results of Lasota and Myjak (Proposition 1.1.7 and Corollary 1.1.8 of our work (op. cit.)) are no longer true in general in the non-Feller case.In the paper, we also obtain a "formula" for the supports of elementary measures of a fairly general type. The result is new even for Markov-Feller operators.We conclude the paper with an outline of the KBBY decomposition for a fairly large class of transition functions. The results for transition functions and transition probabilities seem to us surprisingly similar. However, as expected, the arguments needed to prove the results for transition functions are significantly more involved and are not presented here. We plan to discuss the KBBY decomposition for transition functions with full details in a small monograph that we are currently trying to write.

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Section: Introductionmentioning

confidence: 63%

Section: Introductionmentioning

confidence: 93%

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An Ergodic Decomposition Defined by Transition Probabilities

Zaharopol

2008

Acta Appl Math

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“…Thanks to these properties (see Section 1.1, [23]), for every n ∈ N and a sequence of constants (h i ) i∈N fixed, there exists a unique regular Markov operator P n h 1 ,...,hn , for which Π n h 1 ,...,hn is a transition probability function, and it is given by the formula…”

Section: Iterated Function Systemsmentioning

confidence: 99%

Limit theorems for some Markov chains

Hille

Horbacz

Szarek

et al. 2016

Journal of Mathematical Analysis and Applications

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“…Moreover, it has been proved, see [8], that this semigroup satisfies the assumptions of Theorem 3. Having this we were able to derive that the Markov family (Z ξ (t)) t≥0 is mean * ergodic (see [8,14]). Applying Theorem 3 we obtain, in turn, that the transition semigroup (P t ) t≥0 is asymptotically stable.…”

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confidence: 99%