Krylov–Bogolyubov theorem

Krylov–Bogolyubov theorem In mathematics, the Krylov–Bogolyubov theorem (also known as the existence of invariant measures theorem) may refer to either of the two related fundamental theorems within the theory of dynamical systems. The theorems guarantee the existence of invariant measures for certain "nice" maps defined on "nice" spaces and were named after Russian-Ukrainian mathematicians and theoretical physicists Nikolay Krylov and Nikolay Bogolyubov who proved the theorems.[1] Contenuti 1 Formulation of the theorems 1.1 Invariant measures for a single map 1.2 Invariant measures for a Markov process 2 Guarda anche 3 Notes Formulation of the theorems Invariant measures for a single map Theorem (Krylov–Bogolyubov). Permettere (X, T) be a compact, metrizable topological space and F : X → X a continuous map. Then F admits an invariant Borel probability measure.

Questo è, if Borel(X) denotes the Borel σ-algebra generated by the collection T of open subsets of X, then there exists a probability measure μ : Borel(X) → [0, 1] such that for any subset A ∈ Borel(X), {displaystyle mu left(F^{-1}(UN)Giusto)=mu (UN).} In terms of the push forward, this states that {stile di visualizzazione F_{*}(in )=mu .} Invariant measures for a Markov process Let X be a Polish space and let {stile di visualizzazione P_{t},tgeq 0,} be the transition probabilities for a time-homogeneous Markov semigroup on X, cioè.

{stile di visualizzazione Pr[X_{t}in un|X_{0}=x]=P_{t}(X,UN).} Teorema (Krylov–Bogolyubov). If there exists a point {stile di visualizzazione xin X} for which the family of probability measures { Pt(X, ·) | t > 0 } is uniformly tight and the semigroup (Pt) satisfies the Feller property, then there exists at least one invariant measure for (Pt), cioè. a probability measure μ on X such that {stile di visualizzazione (P_{t})_{ast }(in )=mu {mbox{ per tutti }}t>0.} See also For the 1st theorem: Ya. G. Sinai (Ed.) (1997): Dynamical Systems II. Ergodic Theory with Applications to Dynamical Systems and Statistical Mechanics. Berlino, New York: Springer-Verlag. ISBN 3-540-17001-4. (Sezione 1). For the 2nd theorem: G. Da Prato and J. Zabczyk (1996): Ergodicity for Infinite Dimensional Systems. Cambridge Univ. Premere. ISBN 0-521-57900-7. (Sezione 3). Notes ^ N. N. Bogoliubov and N. M. Krylov (1937). "La theorie generalie de la mesure dans son application a l'etude de systemes dynamiques de la mecanique non-lineaire". Annali di matematica. Seconda serie (in francese). Annali di matematica. 38 (1): 65–113. doi:10.2307/1968511. JSTOR 1968511. Zbl. 16.86.

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Categorie: Ergodic theoryTheorems in dynamical systemsProbability theoremsRandom dynamical systemsTheorems in measure theory