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

Henryk Gacki

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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. ...

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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

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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...

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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.

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

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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

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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.

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

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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

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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)

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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.

An optimal strong equilibrium solution for cooperative multi-leader-follower Stackelberg Markov chains games

Kristal K. Trejo, Julio B. Clempner, Alexander S. Poznyak (2016)

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This paper presents a novel approach for computing the strong Stackelberg/Nash equilibrium for Markov chains games. For solving the cooperative $n$ -leaders and $m$ -followers Markov game we consider the minimization of the $L_{p} -$ norm that reduces the distance to the utopian point in the Euclidian space. Then, we reduce the optimization problem to find a Pareto optimal solution. We employ a bi-level programming method implemented by the extraproximal optimization approach for computing the strong...

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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.

Distortion inequality for the Frobenius-Perron operator and some of its consequences in ergodic theory of Markov maps in $ℝ^{d}$

Piotr Bugiel (1998)

Annales Polonici Mathematici

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Asymptotic properties of the sequences (a) ${P_{φ}^{j} g}_{j = 1}^{\infty}$ and (b) ${j^{- 1} \sum_{i = 0}^{j - 1} P ⁱ_{φ} g}_{j = 1}^{\infty}$ , where $P_{φ} : L ¹ \to L ¹$ is the Frobenius-Perron operator associated with a nonsingular Markov map defined on a σ-finite measure space, are studied for g ∈ G = f ∈ L¹: f ≥ 0 and ⃦f ⃦ = 1. An operator-theoretic analogue of Rényi’s Condition is introduced. It is proved that under some additional assumptions this condition implies the L¹-convergence of the sequences (a) and (b) to a unique g₀ ∈ G. The general result is applied to some smooth Markov...

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 the theory of remediability

Hassan Emamirad (2003)

Banach Center Publications

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Suppose ${G ₁ (t)}_{t \geq 0}$ and ${G ₂ (t)}_{t \geq 0}$ are two families of semigroups on a Banach space X (not necessarily of class C₀) such that for some initial datum u₀, G₁(t)u₀ tends towards an undesirable state u*. After remedying by means of an operator ρ we continue the evolution of the state by applying G₂(t) and after time 2t we retrieve a prosperous state u given by u = G₂(t)ρG₁(t)u₀. Here we are concerned with various properties of the semigroup (t): ρ → G₂(t)ρG₁(t). We define (X) to be the space of remedial operators...

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 $X \to 2^{X}$ . 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. ...