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1
Content available remote On lifting invariant probability measures
100%
Cieśla T.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2020
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tom Vol. 68, no. 1
69--74
EN
We study when an invariant probability measure lifts to an invariant measure. Consider a standard Borel space X, a Borel probability measure μ on X, a Borel map T : X → X preserving μ, a Polish space Y , a continuous map S : Y → Y , and a Borel surjection p: Y → X with p ◦ S = T ◦ p. We prove that if the fibers of p are compact then μ lifts to an S-invariant measure on Y.
2
Content available remote On measure-preserving transformations and doubly stationary symmetric stable processes
80%
Gross A. , Weron A.
Studia Mathematica
|
1995
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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
Content available remote Strong Unique Ergodicity of Random Dynamical Systems on Polish Spaces
80%
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].
4
Content available remote On invariant measures for power bounded positive operators
80%
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
Content available remote A Note on Existence of Global Solutions and Invariant Measures for Jump Sdes with Locally One-Sided Lipschitz Drift
80%
Majka M. B.
Probability and Mathematical Statistics
|
2020
|
tom Vol. 40, Fasc. 1
37--55
EN
We extend some methods developed by Albeverio, Brzeźniak and Wu and we show how to apply them in order to prove existence of global strong solutions of stochastic differential equations with jumps, under a local one-sided Lipschitz condition on the drift (also known as a monotonicity condition) and a local Lipschitz condition on the diffusion and jump coefficients, while an additional global one-sided linear growth assumption is satisfied. Then we use these methods to prove existence of invariant measures for a broad class of such equations.
6
Content available remote Continuous subadditive processes and formulae for Lyapunov characteristic exponents
80%
Słomczyński W.
Annales Polonici Mathematici
|
1995
|
tom 61
|
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.
7
Content available A note on invariant measures
80%
Niemiec P.
Opuscula Mathematica
|
2011
|
tom Vol. 31, no. 3
425-431
EN
The aim of the paper is to show that if F is a family of continuous transformations of a nonempty compact Hausdorff space Ω, then there is no F-invariant probabilistic regular Borel measures on Ω iff there are φ1..., φp ∈ F (for some p ≥ 2) and a continuous function u: Ω, p ? R such that Σ σ ∈Spu(x&sigma(1),...,x&sigmap = 0 and lim infn?∞1/n Σ n-1/k=0 (u o Φk)9x1,...xp) ≥ 1 for each x1,...xp ∈ Ω, where Φ: Ωp ∋ (x1,...xp) ? (φ1(x1,..., φp(xp)∈ Ωp and Φk is the k-th iterate of Φ. A modified version of this result in case the family F generates an equicontinuous semigroup is proved.
8
Content available remote Asymptotic Properties of Stochastic Semilinear Equations by the Method of Lower Measures
80%
Maslowski B. , Simão I.
Colloquium Mathematicum
|
1997
|
tom 72
|
nr 1
147-171
9
Content available remote Ergodic theory approach to chaos: Remarks and computational aspects
70%
Mitkowski P. , Mitkowski W.
|
|
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.
10
Content available remote Topological pressure for one-dimensional holomorphic dynamical systems
60%
Gelfert K. , Wolf C.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2007
|
tom Vol. 55, no 1
53-62
EN
For a class of one-dimensional holomorphic maps ƒ of the Riemann sphere we prove that for a wide class of potentials φ the topological pressure is entirely determined by the values of φ on the repelling periodic points of ƒ. This is a version of a classical result of Bowen for hyperbolic diffeomorphisms in the holomorphic non-uniformly hyperbolic setting.
11
Content available remote On a limit structure of the Galton-Watson branching processes with regularly varying generating functions
60%
Imomov A. A.
Probability and Mathematical Statistics
|
2019
|
tom Vol. 39, Fasc. 1
61--73
EN
We investigate limit properties of discrete time branching processes with application of the theory of regularly varying functions in the sense of Karamata. In the critical situation we suppose that the offspring probability generating function has an infinite second moment but its tail regularly varies. In the noncritical case, the finite moment of type E [x ln x] is required. The lemma on the asymptotic representation of the generating function of the process and its differential analogue will underlie our conclusions.
12
Content available remote Lower bound technique and its applications to function systems and stochastic partial differential equations
51%
Szarek T.
Mathematica Applicanda
|
2013
|
tom Vol. 41, No. 2
185--198
EN
We formulate some criteria for the existence of an invariant measure for Markov chains and Markov processes. We also show their application in the theory of function systems and stochastic differential equations
PL
W pracy formułujemy kryteria dla istnienia miary niezmienniczej dla łańcuchów i procesów Markowa. Następnie pokazujemy ich użyteczność w teorii iterowanych układów funkcyjnych i stochastycznych równań rózniczkowych.
13
Content available Ergodic theory approach to chaos: remarks and computational aspects
51%
Mitkowski P. J. , Mitkowski W.
International Journal of Applied Mathematics and Computer Science
|
2012
|
tom Vol. 22, no. 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.
14
Content available remote On a dimension of measures
41%
Lasota A. , Myjak J.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2002
|
tom Vol. 50, no 2
221-235
EN
Using the notion of the Levy concentration function we discuss a definition of the dimension for probability measures. This dimension is strongly connected with the correlation dimension of measures and with the Hausdorff dimension of sets. Moreover, we calculate some bounds of this dimension for measures generated by Iterated Function Systems and by a partial differential equation.
15
Content available remote Semifractals on Polish spaces
41%
Lasota A. , Myjak J.
Bulletin of the Polish Academy of Sciences. Mathematics
|
1998
|
tom Vol. 46, no 2
179--196
EN
We extend the theory of semifractals to arbitrary metric spaces. We also show kow to construct semifractals on Polish spaces by a use of Markov operators and Markov chain.
16
Content available remote Attractors of multifunetions
41%
Lasota A. , Myjak J.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2000
|
tom Vol. 48, no 3
319--334
EN
We introduce the notion of a semiattractor and attractor for multifunctions and we show that they have properties similar to semifractals and fractals. Further we show a relationship between the multifunctions and transition functions appearing in the theory of Markov operators.
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