Displaying similar documents to “Strong and weak stability of some Markov operators”

On iterates of strong Feller operators on ordered phase spaces

Wojciech Bartoszek (2004)

Colloquium Mathematicae

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Let (X,d) be a metric space where all closed balls are compact, with a fixed σ-finite Borel measure μ. Assume further that X is endowed with a linear order ⪯. Given a Markov (regular) operator P: L¹(μ) → L¹(μ) we discuss the asymptotic behaviour of the iterates Pⁿ. The paper deals with operators P which are Feller and such that the μ-absolutely continuous parts of the transition probabilities P ( x , · ) x ∈ X are continuous with respect to x. Under some concentration assumptions on the asymptotic transition...

A note on Markov operators and transition systems

Bartosz Frej (2002)

Colloquium Mathematicae

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On a compact metric space X one defines a transition system to be a lower semicontinuous map X → 2 X . It is known that every Markov operator on C(X) induces a transition system on X and that commuting of Markov operators implies commuting of the induced transition systems. We show that even in finite spaces a pair of commuting transition systems may not be induced by commuting Markov operators. The existence of trajectories for a pair of transition systems or Markov operators is also investigated. ...

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.

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.

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.

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...

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...

Markov's property of the Cantor ternary set

Leokadia Białas, Alexander Volberg (1993)

Studia Mathematica

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We prove that the Cantor ternary set E satisfies the classical Markov inequality (see [Ma]): for each polynomial p of degree at most n (n = 0, 1, 2,...) (M) | p ' ( x ) | ≤ M n m s u p E | p | for x ∈ E, where M and m are positive constants depending only on E.

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.

Invariant measures related with randomly connected Poisson driven differential equations

Katarzyna Horbacz (2002)

Annales Polonici Mathematici

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We consider the stochastic differential equation (1) d u ( t ) = a ( u ( t ) , ξ ( t ) ) d t + ∫ Θ σ ( u ( t ) , θ ) p ( d t , d θ ) for t ≥ 0 with the initial condition u(0) = x₀. We give sufficient conditions for the existence of an invariant measure for the semigroup P t t ≥ 0 corresponding to (1). We show that the existence of an invariant measure for a Markov operator P corresponding to the change of measures from jump to jump implies the existence of an invariant measure for the semigroup P t t ≥ 0 describing the evolution of measures along trajectories and vice versa. ...

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.