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

Henryk Gacki

Displaying similar documents to “Kantorovich-Rubinstein Maximum Principle in the Stability Theory of Markov Semigroups”

Invariant measures related with randomly connected Poisson driven differential equations

Katarzyna Horbacz (2002)

Annales Polonici Mathematici

Similarity:

We consider the stochastic differential equation (1) $d u (t) = a (u (t), ξ (t)) d t + \int_{Θ} σ {(u (t), θ)}_{p} (d t, d θ)$ for t ≥ 0 with the initial condition u(0) = x₀. We give sufficient conditions for the existence of an invariant measure for the semigroup ${P^{t}}_{t \geq 0}$ 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 ${P^{t}}_{t \geq 0}$ describing the evolution of measures along trajectories and vice versa. ...

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.

Strong mixing Markov semigroups on C₁ are meager

Wojciech Bartoszek, Beata Kuna (2006)

Colloquium Mathematicae

Similarity:

We show that the set of those Markov semigroups on the Schatten class ₁ such that in the strong operator topology $l i m_{t \to \infty} T (t) = Q$ , where Q is a one-dimensional projection, form a meager subset of all Markov semigroups.

Extending the Wong-Zakai theorem to reversible Markov processes

Richard F. Bass, B. Hambly, Terry Lyons (2002)

Journal of the European Mathematical Society

Similarity:

We show how to construct a canonical choice of stochastic area for paths of reversible Markov processes satisfying a weak Hölder condition, and hence demonstrate that the sample paths of such processes are rough paths in the sense of Lyons. We further prove that certain polygonal approximations to these paths and their areas converge in $p$ -variation norm. As a corollary of this result and standard properties of rough paths, we are able to provide a significant generalization of the classical...

Limit theorems for stochastic recursions with Markov dependent coefficients

Dariusz Buraczewski, Małgorzata Letachowicz (2012)

Colloquium Mathematicae

Similarity:

We consider the stochastic recursion $X ₙ = A ₙ X_{n - 1} + B ₙ$ for Markov dependent coefficients (Aₙ,Bₙ) ∈ ℝ⁺ × ℝ. We prove the central limit theorem, the local limit theorem and the renewal theorem for the partial sums Sₙ = X₁+ ⋯ + Xₙ.

Tangential Markov inequality in $L^{p}$ norms

Agnieszka Kowalska (2015)

Banach Center Publications

Similarity:

In 1889 A. Markov proved that for every polynomial p in one variable the inequality $| | p^{'} {| |}_{[- 1, 1]} {\leq (d e g p) ² | | p | |}_{[- 1, 1]}$ is true. Moreover, the exponent 2 in this inequality is the best possible one. A tangential Markov inequality is a generalization of the Markov inequality to tangential derivatives of certain sets in higher-dimensional Euclidean spaces. We give some motivational examples of sets that admit the tangential Markov inequality with the sharp exponent. The main theorems show that the results on certain arcs...

On iterates of strong Feller operators on ordered phase spaces

Wojciech Bartoszek (2004)

Colloquium Mathematicae

Similarity:

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 ${P (x, \cdot)}_{x \in X}$ are continuous with respect to x. Under some concentration assumptions on the asymptotic transition...

Compact subsets of $C^{n}$ preserving Markov’s inequality

W. Plesniak (1988)

Matematički Vesnik

Similarity:

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.

Markov's property of the Cantor ternary set

Leokadia Białas, Alexander Volberg (1993)

Studia Mathematica

Similarity:

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) $| p^{'} (x) | \leq M n^{m} s u p_{E} | p |$ for x ∈ E, where M and m are positive constants depending only on E.

Mean lower bounds for Markov operators

Eduard Emel&#039;yanov, Manfred Wolff (2004)

Annales Polonici Mathematici

Similarity:

Let T be a Markov operator on an L¹-space. We study conditions under which T is mean ergodic and satisfies dim Fix(T) < ∞. Among other things we prove that the sequence $(n^{- 1} \sum_{k = 0}^{n - 1} T^{k}) ₙ$ converges strongly to a rank-one projection if and only if there exists a function 0 ≠ h ∈ L¹₊ which satisfies $l i m_{n \to \infty} | | (h - n^{- 1} \sum_{k = 0}^{n - 1} T^{k} f) ₊ | | = 0$ for every density f. Analogous results for strongly continuous semigroups are given.

Semiflows and semigroups

Edoardo Vesentini (1996)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Similarity:

Given a compact Hausdorff space $K$ and a strongly continuous semigroup $T$ of linear isometries of the Banach space of all complex-valued, continuous functions on $K$ , the semiflow induced by $T$ on $K$ is investigated. In the particular case in which $K$ is a compact, connected, differentiable manifold, a class of semigroups $T$ preserving the differentiable structure of $K$ is characterized.

Eigenvalues of operators in $L_{p}$ -spaces in Markov chains with a general state space

Zbyněk Šidák (1967)

Czechoslovak Mathematical Journal

Similarity:

The Nagaev-Guivarc’h method via the Keller-Liverani theorem

Loïc Hervé, Françoise Pène (2010)

Bulletin de la Société Mathématique de France

Similarity:

The Nagaev-Guivarc’h method, via the perturbation operator theorem of Keller and Liverani, has been exploited in recent papers to establish limit theorems for unbounded functionals of strongly ergodic Markov chains. The main difficulty of this approach is to prove Taylor expansions for the dominating eigenvalue of the Fourier kernels. The paper outlines this method and extends it by stating a multidimensional local limit theorem, a one-dimensional Berry-Esseen theorem, a first-order...

Transition semigroups for stochastic semilinear equations on Hilbert spaces

Anna Chojnowska-Michalik

Similarity:

A large class of stochastic semilinear equations with measurable nonlinear term on a Hilbert space H is considered. Assuming the corresponding nonsymmetric Ornstein-Uhlenbeck process has an invariant measure μ, we prove in the $L^{p} (H, μ)$ spaces the existence of a transition semigroup $(P_{t})$ for the equations. Sufficient conditions are provided for hyperboundedness of $P_{t}$ and for the Log Sobolev Inequality to hold; and in the case of a bounded nonlinear term, sufficient and necessary conditions are obtained....

A Class of Contractions in Hilbert Space and Applications

Nick Dungey (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

We characterize the bounded linear operators T in Hilbert space which satisfy T = βI + (1-β)S where β ∈ (0,1) and S is a contraction. The characterizations include a quadratic form inequality, and a domination condition of the discrete semigroup ${(T ⁿ)}_{n = 1, 2, . . .}$ by the continuous semigroup ${(e^{- t (I - T)})}_{t \geq 0}$ . Moreover, we give a stronger quadratic form inequality which ensures that $s u p n ∥ T ⁿ - T^{n + 1} ∥ : n = 1, 2, . . . < \infty$ . The results apply to large classes of Markov operators on countable spaces or on locally compact groups.

Nash -equilibria for stochastic games with total reward functions: an approach through Markov decision processes

Francisco J. González-Padilla, Raúl Montes-de-Oca (2019)

Kybernetika

Similarity:

The main objective of this paper is to find structural conditions under which a stochastic game between two players with total reward functions has an $ϵ$ -equilibrium. To reach this goal, the results of Markov decision processes are used to find $ϵ$ -optimal strategies for each player and then the correspondence of a better answer as well as a more general version of Kakutani’s Fixed Point Theorem to obtain the $ϵ$ -equilibrium mentioned. Moreover, two examples to illustrate the theory developed...