Displaying similar documents to “Randomly connected dynamical systems - asymptotic stability”

Invariant measures for random dynamical systems

Katarzyna Horbacz

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We consider random dynamical systems with randomly chosen jumps on Polish spaces. They generalize Markov processes corresponding to iterated function systems, Poisson driven stochastic differential equations, and irreducible Markov systems. We formulate criteria for the existence of an invariant measure and asymptotic stability for these systems. Estimates of the lower pointwise and concentration dimension of invariant measures are also given.

A criterion of asymptotic stability for Markov-Feller e-chains on Polish spaces

Dawid Czapla (2012)

Annales Polonici Mathematici

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Stettner [Bull. Polish Acad. Sci. Math. 42 (1994)] considered the asymptotic stability of Markov-Feller chains, provided the sequence of transition probabilities of the chain converges to an invariant probability measure in the weak sense and converges uniformly with respect to the initial state variable on compact sets. We extend those results to the setting of Polish spaces and relax the original assumptions. Finally, we present a class of Markov-Feller chains with a linear state space...

Asymptotic stability of a linear Boltzmann-type equation

Roksana Brodnicka, Henryk Gacki (2014)

Applicationes Mathematicae

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We present a new necessary and sufficient condition for the asymptotic stability of Markov operators acting on the space of signed measures. The proof is based on some special properties of the total variation norm. Our method allows us to consider the Tjon-Wu equation in a linear form. More precisely a new proof of the asymptotic stability of a stationary solution of the Tjon-Wu equation is given.

Random Dynamical Systems with Jumps and with a Function Type Intensity

Joanna Kubieniec (2016)

Annales Mathematicae Silesianae

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In paper [4] there are considered random dynamical systems with randomly chosen jumps acting on Polish spaces. The intensity of this process is a constant λ. In this paper we formulate criteria for the existence of an invariant measure and asymptotic stability for these systems in the case when λ is not constant but a Lipschitz function.

Asymptotic stability in L¹ of a transport equation

M. Ślęczka (2004)

Annales Polonici Mathematici

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We study the asymptotic behaviour of solutions of a transport equation. We give some sufficient conditions for the complete mixing property of the Markov semigroup generated by this equation.

Markov operators on the space of vector measures; coloured fractals

Karol Baron, Andrzej Lasota (1998)

Annales Polonici Mathematici

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We consider the family 𝓜 of measures with values in a reflexive Banach space. In 𝓜 we introduce the notion of a Markov operator and using an extension of the Fortet-Mourier norm we show some criteria of the asymptotic stability. Asymptotically stable Markov operators can be used to construct coloured fractals.

Applications of the Kantorovich-Rubinstein maximum principle in the theory of Markov semigroups

Henryk Gacki

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We present new sufficient conditions for the asymptotic stability of Markov operators acting on the space of signed measures. Our results are based on two principles. The first one is the LaSalle invariance principle used in the theory of dynamical systems. The second is related to the Kantorovich-Rubinstein theorems concerning the properties of probability metrics. These criteria are applied to stochastically perturbed dynamical systems, a Poisson driven stochastic differential equation...

On a nonstandard approach to invariant measures for Markov operators

Andrzej Wiśnicki (2010)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We show the existence of invariant measures for Markov-Feller operators defined on completely regular topological spaces which satisfy the classical positivity condition.

Markov operators acting on Polish spaces

Tomasz Szarek (1997)

Annales Polonici Mathematici

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We prove a new sufficient condition for the asymptotic stability of Markov operators acting on measures. This criterion is applied to iterated function systems.

Strong and weak stability of some Markov operators

Ryszard Rudnicki (2000)

Colloquium Mathematicae

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An integral Markov operator P appearing in biomathematics is investigated. This operator acts on the space of probabilistic Borel measures. Let μ and ν be probabilistic Borel measures. Sufficient conditions for weak and strong convergence of the sequence ( P n μ - P n ν ) to 0 are given.

Kantorovich-Rubinstein Maximum Principle in the Stability Theory of Markov Semigroups

Henryk Gacki (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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A new sufficient condition for the asymptotic stability of a locally Lipschitzian Markov semigroup acting on the space of signed measures s i g is proved. This criterion is applied to the semigroup of Markov operators generated by a Poisson driven stochastic differential equation.

The uniqueness of invariant measures for Markov operators

Tomasz Szarek (2008)

Studia Mathematica

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It is shown that Markov operators with equicontinuous dual operators which overlap supports have at most one invariant measure. In this way we extend the well known result proved for Markov operators with the strong Feller property by R. Z. Khas'minski.