Ergodic decompositions associated with regular Markov operators on Polish spaces

2010 ◽  
Vol 31 (2) ◽  
pp. 571-597 ◽  
Author(s):  
DANIËL T. H. WORM ◽  
SANDER C. HILLE
Keyword(s):  
State Space ◽  
Metric Spaces ◽  
Transition Probabilities ◽  
Polish Space ◽  
Invariant Probability Measure ◽  
The State ◽  
Probability Measures ◽  
Ergodic Decomposition ◽  
Polish Spaces ◽  
Appl Math

AbstractFor any regular Markov operator on the space of finite Borel measures on a Polish space we give a Yosida-type decomposition of the state space, which yields a parametrization of the ergodic probability measures associated with this operator in terms of particular subsets of the state space. We use this parametrization to prove an integral decomposition of every invariant probability measure in terms of the ergodic probability measures and give an ergodic decomposition of the state space. This extends results by Yosida [Functional Analysis. Springer, Berlin, 1980, Ch. XIII.4], Hernández-Lerma and Lasserre [Ergodic theorems and ergodic decomposition for Markov chains. Acta Appl. Math.54 (1998), 99–119] and Zaharopol [An ergodic decomposition defined by transition probabilities. Acta Appl. Math.104 (2008), 47–81], who considered the setting of locally compact separable metric spaces. Our extension to Polish spaces solves an open problem posed by Zaharopol (loc. cit.) in a satisfactory manner.

scholarly journals The temporal logic of coalgebras via Galois algebras

2002 ◽  
Vol 12 (6) ◽  
pp. 875-903 ◽  
Author(s):  
BART JACOBS
Keyword(s):  
State Space ◽  
Temporal Logic ◽  
Metric Spaces ◽  
Linear Temporal Logic ◽  
The State ◽  
Temporal Logics ◽  
Computation Tree Logic ◽  
Computation Tree ◽  
Polynomial Functors

This paper introduces a temporal logic for coalgebras. Nexttime and lasttime operators are defined for a coalgebra, acting on predicates on the state space. They give rise to what is called a Galois algebra. Galois algebras form models of temporal logics like Linear Temporal Logic (LTL) and Computation Tree Logic (CTL). The mapping from coalgebras to Galois algebras turns out to be functorial, yielding indexed categorical structures. This construction gives many examples, for coalgebras of polynomial functors on sets. More generally, it will be shown how ‘fuzzy’ predicates on metric spaces, and predicates on presheaves, yield indexed Galois algebras, in basically the same coalgebraic manner.

On large deviations of empirical measures in the τ-topology

1994 ◽  
Vol 31 (A) ◽  
pp. 41-47 ◽  
Author(s):  
A. De Acosta
Keyword(s):  
Large Deviations ◽  
State Space ◽  
The State ◽  
Probability Measures ◽  
Empirical Measures

We prove a generalization of Sanov's theorem in which the state space S is arbitrary and the set of probability measures on S is endowed with the τ -topology.

Kolmogorov's differential equations for non-stationary, countable state Markov processes with uniformly continuous transition probabilities

1973 ◽  
Vol 73 (1) ◽  
pp. 119-138 ◽  
Author(s):  
Gerald S. Goodman ◽  
S. Johansen
Keyword(s):  
Markov Chain ◽  
Differential Equations ◽  
State Space ◽  
Markov Processes ◽  
Transition Probabilities ◽  
The State ◽  
Continuous Transition ◽  
Countable State Space ◽  
Countable State ◽  
Uniformly Continuous

1. SummaryWe shall consider a non-stationary Markov chain on a countable state space E. The transition probabilities {P(s, t), 0 ≤ s ≤ t <t0 ≤ ∞} are assumed to be continuous in (s, t) uniformly in the state i ε E.

On large deviations of empirical measures in the τ-topology

1994 ◽  
Vol 31 (A) ◽  
pp. 41-47 ◽  
Author(s):  
A. De Acosta
Keyword(s):  
Large Deviations ◽  
State Space ◽  
The State ◽  
Probability Measures ◽  
Empirical Measures

We prove a generalization of Sanov's theorem in which the state space S is arbitrary and the set of probability measures on S is endowed with the τ -topology.

On Levy's theorem concerning positiveness of transition probabilities of Markov processes: the circuit processes case

1993 ◽  
Vol 30 (01) ◽  
pp. 28-39
Author(s):  
S. Kalpazidou
Keyword(s):  
Probability Distribution ◽  
State Space ◽  
Markov Processes ◽  
Transition Probabilities ◽  
The State ◽  
Sample Paths ◽  
Lévy’S Theorem

We prove Lévy's theorem concerning positiveness of transition probabilities of Markov processes when the state space is countable and an invariant probability distribution exists. Our approach relies on the representation of transition probabilities in terms of the directed circuits that occur along the sample paths.

scholarly journals CONTINUOUS RAMSEY THEORY ON POLISH SPACES AND COVERING THE PLANE BY FUNCTIONS

2004 ◽  
Vol 04 (02) ◽  
pp. 109-145 ◽  
Author(s):  
STEFAN GESCHKE ◽  
MARTIN GOLDSTERN ◽  
MENACHEM KOJMAN
Keyword(s):  
Set Theory ◽  
Metric Spaces ◽  
Ramsey Theory ◽  
Polish Space ◽  
Lipschitz Functions ◽  
Cantor Space ◽  
Polish Spaces ◽  
Real Functions ◽  
Number Formula ◽  
Independence Results

We investigate the Ramsey theory of continuous graph-structures on complete, separable metric spaces and apply the results to the problem of covering a plane by functions. Let the homogeneity number[Formula: see text] of a pair-coloring c:[X]2→2 be the number of c-homogeneous subsets of X needed to cover X. We isolate two continuous pair-colorings on the Cantor space 2ω, c min and c max , which satisfy [Formula: see text] and prove: Theorem. (1) For every Polish space X and every continuous pair-coloringc:[X]2→2with[Formula: see text], [Formula: see text] (2) There is a model of set theory in which[Formula: see text]and[Formula: see text]. The consistency of [Formula: see text] and of [Formula: see text] follows from [20]. We prove that [Formula: see text] is equal to the covering number of (2ω)2 by graphs of Lipschitz functions and their reflections on the diagonal. An iteration of an optimal forcing notion associated to c min gives: Theorem. There is a model of set theory in which (1) ℝ2 is coverable byℵ1graphs and reflections of graphs of continuous real functions; (2) ℝ2 is not coverable byℵ1graphs and reflections of graphs of Lipschitz real functions. Figure 1.1 in the introduction summarizes the ZFC results in Part I of the paper. The independence results in Part II show that any two rows in Fig. 1.1 can be separated if one excludes [Formula: see text] from row (3).

scholarly journals First Hitting Place Probabilities for a Discrete Version of the Ornstein-Uhlenbeck Process

2009 ◽  
Vol 2009 ◽  
pp. 1-12 ◽  
Author(s):  
Mario Lefebvre ◽  
Jean-Luc Guilbault
Keyword(s):  
Markov Chain ◽  
State Space ◽  
Transition Probabilities ◽  
The State ◽  
Discrete Version ◽  
Uhlenbeck Process ◽  
Current State ◽  
Ornstein Uhlenbeck Process

A Markov chain with state space{0,…,N}and transition probabilities depending on the current state is studied. The chain can be considered as a discrete Ornstein-Uhlenbeck process. The probability that the process hitsNbefore 0 is computed explicitly. Similarly, the probability that the process hitsNbefore−Mis computed in the case when the state space is{−M,…,0,…,N}and the transition probabilitiespi,i+1are not necessarily the same wheniis positive andiis negative.

On Levy's theorem concerning positiveness of transition probabilities of Markov processes: the circuit processes case

1993 ◽  
Vol 30 (1) ◽  
pp. 28-39 ◽  
Author(s):  
S. Kalpazidou
Keyword(s):  
Probability Distribution ◽  
State Space ◽  
Markov Processes ◽  
Transition Probabilities ◽  
The State ◽  
Sample Paths ◽  
Lévy’S Theorem

We prove Lévy's theorem concerning positiveness of transition probabilities of Markov processes when the state space is countable and an invariant probability distribution exists. Our approach relies on the representation of transition probabilities in terms of the directed circuits that occur along the sample paths.

Modeling of CCLP in koryo extract production

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ji Chol ◽  
Ri Jun Il
Keyword(s):  
Process Control ◽  
Medical Care ◽  
State Space ◽  
State Space Model ◽  
The State ◽  
Space Model ◽  
Effective Components ◽  
Counter Current

Abstract The modeling of counter-current leaching plant (CCLP) in Koryo Extract Production is presented in this paper. Koryo medicine is a natural physic to be used for a diet and the medical care. The counter-current leaching method is mainly used for producing Koryo medicine. The purpose of the modeling in the previous works is to indicate the concentration distributions, and not to describe the model for the process control. In literature, there are no nearly the papers for modeling CCLP and especially not the presence of papers that have described the issue for extracting the effective components from the Koryo medicinal materials. First, this paper presents that CCLP can be shown like the equivalent process consisting of two tanks, where there is a shaking apparatus, respectively. It allows leachate to flow between two tanks. Then, this paper presents the principle model for CCLP and the state space model on based it. The accuracy of the model has been verified from experiments made at CCLP in the Koryo Extract Production at the Gang Gyi Koryo Manufacture Factory.

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