Displaying similar documents to “Applications of the Kantorovich-Rubinstein maximum principle in the theory of Markov semigroups”

Asymptotic stability of a linear Boltzmann-type equation

Roksana Brodnicka, Henryk Gacki (2014)

Applicationes Mathematicae

Similarity:

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.

Markov operators acting on Polish spaces

Tomasz Szarek (1997)

Annales Polonici Mathematici

Similarity:

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

Piecewise-deterministic Markov processes

Jolanta Kazak (2013)

Annales Polonici Mathematici

Similarity:

Poisson driven stochastic differential equations on a separable Banach space are examined. Some sufficient conditions are given for the asymptotic stability of a Markov operator P corresponding to the change of distribution from jump to jump. We also give criteria for the continuous dependence of the invariant measure for P on the intensity of the Poisson process.

Asymptotic behaviour in the set of nonhomogeneous chains of stochastic operators

Małgorzata Pułka (2012)

Discussiones Mathematicae Probability and Statistics

Similarity:

We study different types of asymptotic behaviour in the set of (infinite dimensional) nonhomogeneous chains of stochastic operators acting on L¹(μ) spaces. In order to examine its structure we consider different norm and strong operator topologies. To describe the nature of the set of nonhomogeneous chains of Markov operators with a particular limit behaviour we use the category theorem of Baire. We show that the geometric structure of the set of those stochastic operators which have...

Markov operators on the space of vector measures; coloured fractals

Karol Baron, Andrzej Lasota (1998)

Annales Polonici Mathematici

Similarity:

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.

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

Dawid Czapla (2012)

Annales Polonici Mathematici

Similarity:

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...

Invariant measures for random dynamical systems

Katarzyna Horbacz

Similarity:

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.

Stochastic differential equation driven by a pure-birth process

Marta Tyran-Kamińska (2002)

Annales Polonici Mathematici

Similarity:

A generalization of the Poisson driven stochastic differential equation is considered. A sufficient condition for asymptotic stability of a discrete time-nonhomogeneous Markov process is proved.

Strong and weak stability of some Markov operators

Ryszard Rudnicki (2000)

Colloquium Mathematicae

Similarity:

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.

Dynamics and density evolution in piecewise deterministic growth processes

Michael C. Mackey, Marta Tyran-Kamińska (2008)

Annales Polonici Mathematici

Similarity:

A new sufficient condition is proved for the existence of stochastic semigroups generated by the sum of two unbounded operators. It is applied to one-dimensional piecewise deterministic Markov processes, where we also discuss the existence of a unique stationary density and give sufficient conditions for asymptotic stability.

Randomly connected dynamical systems - asymptotic stability

Katarzyna Horbacz (1998)

Annales Polonici Mathematici

Similarity:

We give sufficient conditions for asymptotic stability of a Markov operator governing the evolution of measures due to the action of randomly chosen dynamical systems. We show that the existence of an invariant measure for the transition operator implies the existence of an invariant measure for the semigroup generated by the system.

On the Kantorovich-Rubinstein maximum principle for the Fortet-Mourier norm

Henryk Gacki (2005)

Annales Polonici Mathematici

Similarity:

A new version of the maximum principle is presented. The classical Kantorovich-Rubinstein principle gives necessary conditions for the maxima of a linear functional acting on the space of Lipschitzian functions. The maximum value of this functional defines the Hutchinson metric on the space of probability measures. We show an analogous result for the Fortet-Mourier metric. This principle is then applied in the stability theory of Markov-Feller semigroups.

Asymptotic stability condition for stochastic Markovian systems of differential equations

Efraim Shmerling (2010)

Mathematica Bohemica

Similarity:

Asymptotic stability of the zero solution for stochastic jump parameter systems of differential equations given by d X ( t ) = A ( ξ ( t ) ) X ( t ) d t + H ( ξ ( t ) ) X ( t ) d w ( t ) , where ξ ( t ) is a finite-valued Markov process and w(t) is a standard Wiener process, is considered. It is proved that the existence of a unique positive solution of the system of coupled Lyapunov matrix equations derived in the paper is a necessary asymptotic stability condition.

Stochastic vortices in periodically reclassified populations

Gracinda Rita Guerreiro, João Tiago Mexia (2008)

Discussiones Mathematicae Probability and Statistics

Similarity:

Our paper considers open populations with arrivals and departures whose elements are subject to periodic reclassifications. These populations will be divided into a finite number of sub-populations. Assuming that: a) entries, reclassifications and departures occur at the beginning of the time units; b) elements are reallocated at equally spaced times; c) numbers of new elements entering at the beginning of the time units are...

On asymptotic cyclicity of doubly stochastic operators

Wojciech Bartoszek (1999)

Annales Polonici Mathematici

Similarity:

It is proved that a doubly stochastic operator P is weakly asymptotically cyclic if it almost overlaps supports. If moreover P is Frobenius-Perron or Harris then it is strongly asymptotically cyclic.

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

Henryk Gacki (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

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.

Random Dynamical Systems with Jumps and with a Function Type Intensity

Joanna Kubieniec (2016)

Annales Mathematicae Silesianae

Similarity:

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.