# Krylov-Bogolyubov-Theorem

Krylov-Bogolyubov-Theorem In der Mathematik, das 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 "Hübsch" maps defined on "Hübsch" spaces and were named after Russian-Ukrainian mathematicians and theoretical physicists Nikolay Krylov and Nikolay Bogolyubov who proved the theorems.[1] Inhalt 1 Formulation of the theorems 1.1 Invariant measures for a single map 1.2 Invariant measures for a Markov process 2 Siehe auch 3 Notes Formulation of the theorems Invariant measures for a single map Theorem (Krylov–Bogolyubov). Lassen (X, T) be a compact, metrizable topological space and F : X → X a continuous map. Then F admits an invariant Borel probability measure.

Das ist, 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}(EIN)Rechts)= ein (EIN).} In terms of the push forward, this states that {Anzeigestil F_{*}(in )= ein .} Invariant measures for a Markov process Let X be a Polish space and let {Anzeigestil P_{t},tgeq 0,} be the transition probabilities for a time-homogeneous Markov semigroup on X, d.h.

{Anzeigestil Pr[X_{t}in einem|X_{0}=x]=P_{t}(x,EIN).} Satz (Krylov–Bogolyubov). If there exists a point {Anzeigestil 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), d.h. a probability measure μ on X such that {Anzeigestil (P_{t})_{Ast }(in )= ein {mbox{ für alle }}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. Berlin, New York: Springer-Verlag. ISBN 3-540-17001-4. (Abschnitt 1). For the 2nd theorem: G. Da Prato and J. Zabczyk (1996): Ergodicity for Infinite Dimensional Systems. Cambridge Univ. Drücken Sie. ISBN 0-521-57900-7. (Abschnitt 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". Annalen der Mathematik. Zweite Serie (auf Französisch). Annalen der Mathematik. 38 (1): 65–113. doi:10.2307/1968511. JSTOR 1968511. Zbl. 16.86.