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1
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Continuous subadditive processes and formulae for Lyapunov characteristic exponents

100%
Słomczyński W.
Annales Polonici Mathematici
|
1995
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tom 61
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nr 2
101-134
EN
Asymptotic properties of various semidynamical systems can be examined by means of continuous subadditive processes. To investigate such processes we consider different types of exponents: characteristic, central, singular and global exponents and we study their properties. We derive formulae for central and singular exponents and show that they provide upper bounds for characteristic exponents. The concept of conjugate processes introduced in this paper allows us to find lower bounds for characteristic exponents. We also give applications to continuous cocycles.
2
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On measure-preserving transformations and doubly stationary symmetric stable processes

100%
Gross A., Weron A.
Studia Mathematica
|
1995
|
tom 114
|
nr 3
275-287
EN
In a 1987 paper, Cambanis, Hardin and Weron defined doubly stationary stable processes as those stable processes which have a spectral representation which is itself stationary, and they gave an example of a stationary symmetric stable process which they claimed was not doubly stationary. Here we show that their process actually had a moving average representation, and hence was doubly stationary. We also characterize doubly stationary processes in terms of measure-preserving regular set isomorphisms and the existence of σ-finite invariant measures. One consequence of the characterization is that all harmonizable symmetric stable processes are doubly stationary. Another consequence is that there exist stationary symmetric stable processes which are not doubly stationary.
3
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Asymptotic Properties of Stochastic Semilinear Equations by the Method of Lower Measures

100%
Maslowski B., Simão I.
Colloquium Mathematicum
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1997
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tom 72
|
nr 1
147-171
4
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On invariant measures for power bounded positive operators

100%
Sato R.
Studia Mathematica
|
1996
|
tom 120
|
nr 2
183-189
EN
We give a counterexample showing that $\overline{(I-T*)L_{∞}} ∩ L^{+}_{∞} = {0}$ does not imply the existence of a strictly positive function u in $L_1$ with Tu = u, where T is a power bounded positive linear operator on $L_1$ of a σ-finite measure space. This settles a conjecture by Brunel, Horowitz, and Lin.
5
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Strong Unique Ergodicity of Random Dynamical Systems on Polish Spaces

100%
Płonka P.
Annales Mathematicae Silesianae
|
2016
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tom 30
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nr 1
129-142
EN
In this paper we want to show the existence of a form of asymptotic stability of random dynamical systems in the sense of L. Arnold using arguments analogous to those presented by T. Szarek in [6], that is showing it using conditions generalizing the notion of tightness of measures. In order to do that we use tightness theory for random measures as developed by H. Crauel in [2].
6
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Ergodic theory approach to chaos: Remarks and computational aspects

88%
Mitkowski P., Mitkowski W.
International Journal of Applied Mathematics and Computer Science
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2012
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tom 22
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nr 2
259-267
EN
We discuss basic notions of the ergodic theory approach to chaos. Based on simple examples we show some characteristic features of ergodic and mixing behaviour. Then we investigate an infinite dimensional model (delay differential equation) of erythropoiesis (red blood cell production process) formulated by Lasota. We show its computational analysis on the previously presented theory and examples. Our calculations suggest that the infinite dimensional model considered possesses an attractor of a nonsimple structure, supporting an invariant mixing measure. This observation verifies Lasota's conjecture concerning nontrivial ergodic properties of the model.
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