Any two irreducible Markov chains are finitarily orbit equivalent

Two invertible dynamical systems (X, gA, μ, T) and (Y, ℬ, ν, S), where X, Y are metrizable spaces and T, S are homeomorphisms on X and Y, are said to be finitarily orbit equivalent if there exists an invertible measure preserving mapping ϕ from a subset X ₀ of X of full measure to a subset Y ₀ of Y of full measure such that ϕ|x ₀ is continuous in the relative topology on X ₀, ϕ ⁻¹|Y ₀ is continuous in the relative topology on Y ₀ and ϕ(Orb _T (x)) = Orb _Sϕ (x) for μ-a.e. x ∈ X. In this article a finitary orbit equivalence mapping is shown to exist between any two irreducible Markov chains.     

Any two irrational rotations are nearly continuously Kakutani equivalent

Ergodic homeomorphisms T and S of Polish probability spaces X and Y are evenly Kakutani equivalent if there is an orbit equivalence ϕ: X ₀ → Y ₀ between full measure subsets of X and Y such that, for some A ⊂ X ₀ of positive measure, ϕ restricts to a measurable isomorphism of the induced systems T _A and S _ϕ(A). The study of even Kakutani equivalence dates back to the seventies, and it is well known that any two irrational rotations of the circle are evenly Kakutani equivalent. But even Kakutani equivalence is a purely measurable relation, while systems such as irrational rotations are both measurable and topological.     

Remarks on ergodicity of stationary irreducible transient Markov chains

Summary Consider a stationary process {X _n(), – < n < . If the measure of the process is finite (the measure of the whole sample space finite), it is well known that ergodicity of the process {X _n(), - < n < and of each of the subprocesses {X _n(), 0 n < , {X _n(), – < n 0 are equivalent (see [3]). We shall show that this is generally not true for stationary processes with a sigma-finite measure, specifically for stationary irreducible transient Markov chains. An example of a stationary irreducible transient Markov chain {X _n(), - < n <} with {itX_n(), 0 n < < ergodic but {X _n(), < n 0 nonergodic is given. That this can be the case has already been implicitly indicated in the literature [4]. Another example of a stationary irreducible transient Markov chain with both {X _n(), 0 n < and {itX _n(),-< < n 0} ergodic but {X _n(), - < n < nonergodic is presented. In fact, it is shown that all stationary irreducible transient Markov chains {X _n(), - < n < < are nonergodic.This research was supported in part by the Office of Naval Research.John Simon Guggenheim Memorial Fellow.     

Gradient flows of the entropy for finite Markov chains

Let K be an irreducible and reversible Markov kernel on a finite set X. We construct a metric W on the set of probability measures on X and show that with respect to this metric, the law of the continuous time Markov chain evolves as the gradient flow of the entropy. This result is a discrete counterpart of the Wasserstein gradient flow interpretation of the heat flow in Rn by Jordan, Kinderlehrer and Otto (1998). The metric W is similar to, but different from, the L²-Wasserstein metric, and is defined via a discrete variant of the Benamou-Brenier formula.     

Geodesic convexity of the relative entropy in reversible Markov chains

We consider finite-dimensional, time-continuous Markov chains satisfying the detailed balance condition as gradient systems with the relative entropy E as driving functional. The Riemannian metric is defined via its inverse matrix called the Onsager matrix K. We provide methods for establishing geodesic λ-convexity of the entropy and treat several examples including some discretizations of one-dimensional Fokker–Planck equations.     

Negative entropy,zero temperature and Markov chains on the interval

We consider ergodic optimization for the shift map on the modified Bernoulli space σ: [0, 1]^? → [0, 1]^?, where [0, 1] is the unit closed interval, and the potential A: [0, 1]^? → ? considered depends on the two first coordinates of [0, 1]^?. We are interested in finding stationary Markov probabilities µ_∞ on [0, 1]^? that maximize the value ∫ Adµ, among all stationary (i.e. σ-invariant) probabilities µ on [0, 1]^?. This problem correspond in Statistical Mechanics to the zero temperature case for the interaction described by the potential A. The main purpose of this paper is to show, under the hypothesis of uniqueness of the maximizing probability, a Large Deviation Principle for a family of absolutely continuous Markov probabilities µ _β which weakly converges to µ_∞. The probabilities µ _β are obtained via an information we get from a Perron operator and they satisfy a variational principle similar to the pressure in Thermodynamic Formalism. As the potential A depends only on the first two coordinates, instead of the probability µ on [0, 1]^?, we can consider its projection ν on [0, 1]². We look at the problem in both ways. If µ_∞ is the maximizing probability on [0, 1]^?, we also have that its projection ν _∞ is maximizing for A. The hypothesis about stationarity on the maximization problem can also be seen as a transhipment problem. Under the hypothesis of A being C ² and the twist condition, that is,

$\frac{{\partial ^2 A}}{{\partial x\partial y}}(x,y) \ne 0, for all (x,y) \in [0,1]^2 ,$

we show the graph property of the maximizing probability ν on [0, 1]². Moreover, the graph is monotonous. An important result we get is: the maximizing probability is unique generically in Mañé’s sense. Finally, we exhibit a separating sub-action for A.

     

Table algebras with exactly one irreducible character whose degree and multiplicity are not equal

Commutative standard table algebras with exactly one multiplicity not equal to 1 are characterized by the wreath product of some special table algebras in [1 Antonou, A. (2015). Commutative standard table algebras with at most one nontrivial multiplicity. Commun. Algebra 43:2516–2523.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]]. A natural and much more general question is the characterization of standard table algebras (not necessarily commutative) with exactly one irreducible character whose degree and multiplicity are not equal and the degree is 1. We will give a characterization of such table algebras, including the main result of [1 Antonou, A. (2015). Commutative standard table algebras with at most one nontrivial multiplicity. Commun. Algebra 43:2516–2523.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]] as a special case. Applications to association schemes are also discussed.     

Finitary isomorphisms of irreducible Markov shifts

It is shown that irreducible finite state, Markov shifts of the same entropy and period arefinitarily isomorphic.     

Approximation of Markov chains

Conditions are obtained, as well as quantitative estimates, of Markov chains with a common set of states that are uniformly continuous in time. Within the framework of the method under consideration, it is possible to show that such chains can be finitely approximated.Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei — Trudy Seminara, pp. 4–12, 1980.     

Boundary measures of Markov chains

It is known that each Markov chain has associated with it a polytope and a family of Markov measures indexed by the interior points of the polytope. Measure-preserving factor maps between Markov chains must preserve the associated families. In the present paper, we augment this structure by identifying measures corresponding to points on the boundary of the polytope. These measures are also preserved by factor maps. We examine the data they provide and give examples to illustrate the use of this data in ruling out the existence of factor maps between Markov chains. E. Cawley was partially supported by the Modern Analysis joint NSF grant in Berkeley. S. Tuncel was partially supported by NSF Grant DMS-9303240.     

Filtering of continuous-time Markov chains

This paper discusses finite-dimensional optimal filters for partially observed Markov chains. A model for a system containing a finite number of components where each component behaves like an independent finite state continuous-time Markov chain is considered. Using measure change techniques various estimators are derived.     

Usual and stochastic tail orders between hitting times for two Markov chains

We consider time‐homogeneous Markov chains with state space E_k≡{0,1,…,k} and initial distribution concentrated on the state 0. For pairs of such Markov chains, we study the Stochastic Tail Order and the stochastic order in the usual sense between the respective first passage times in the state k . On this purpose, we will develop a method based on a specific relation between two stochastic matrices on the state space E_k . Our method provides comparisons that are simpler and more refined than those obtained by the analysis based on the spectral gaps. Copyright © 2016 John Wiley & Sons, Ltd.