View source for 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 [[theorem]]s within the theory of [[dynamical systems]]. The theorems guarantee the existence of [[invariant measure]]s for certain "nice" maps defined on "nice" spaces and were named after [[Russia]]n-[[Ukraine|Ukrainian]] [[mathematician]]s and [[theoretical physics|theoretical physicists]] [[Nikolay Mitrofanovich Krylov|Nikolay Krylov]] and [[Nikolay Bogolyubov]] who proved the theorems.<ref name="Bogoliubov1937">{{cite journal|author=N. N. Bogoliubov and N. M. Krylov|journal=Annals of Mathematics |series=Second Series|volume=38|pages=65–113|year=1937|jstor=1968511|title=La theorie generalie de la mesure dans son application a l'etude de systemes dynamiques de la mecanique non-lineaire|language=French|doi=10.2307/1968511|issue=1|publisher=Annals of Mathematics}} Zbl. 16.86.</ref>

==Formulation of the theorems==

===Invariant measures for a single map===
'''Theorem (Krylov–Bogolyubov)'''. Let (''X'',&nbsp;''T'') be a [[compact space|compact]], [[metrization theorem|metrizable]] [[topological space]] and ''F''&nbsp;:&nbsp;''X''&nbsp;→&nbsp;''X'' a [[continuous map]]. Then ''F'' admits an invariant [[Borel measure|Borel]] [[probability measure]].

That is, if Borel(''X'') denotes the [[Borel sigma algebra|Borel]] [[sigma algebra|&sigma;-algebra]] generated by the collection ''T'' of [[open set|open subsets]] of ''X'', then there exists a probability measure ''μ'' : Borel(''X'') → [0, 1] such that for any subset ''A'' ∈ Borel(''X''),
:<math>\mu \left( F^{-1} (A) \right) = \mu (A).</math>

In terms of the [[pushforward measure|push forward]], this states that
:<math>F_{*} (\mu) = \mu.</math>

===Invariant measures for a Markov process===
Let ''X'' be a [[Polish space]] and let <math>P_t, t\ge 0,</math> be the transition probabilities for a time-homogeneous [[Markov process|Markov]] [[semigroup]] on ''X'', i.e.
:<math>\Pr [ X_{t} \in A | X_{0} = x ] = P_{t} (x, A).</math>

'''Theorem (Krylov–Bogolyubov)'''. If there exists a point <math>x\in X</math> for which the family of probability measures {&nbsp;''P''<sub>''t''</sub>(''x'',&nbsp;·)&nbsp;|&nbsp;''t''&nbsp;&gt;&nbsp;0&nbsp;} is [[tightness of measures|uniformly tight]] and the semigroup (''P''<sub>''t''</sub>) satisfies the [[Feller-continuous process|Feller property]], then there exists at least one invariant measure for (''P''<sub>''t''</sub>), i.e. a probability measure ''μ'' on ''X'' such that
:<math>(P_{t})_{\ast} (\mu) = \mu \mbox{ for all } t > 0.</math>

==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}}. (Section 1).
* For the 2nd theorem: G. Da Prato and J. Zabczyk (1996): ''Ergodicity for Infinite Dimensional Systems''. Cambridge Univ. Press. {{isbn|0-521-57900-7}}. (Section 3).

==Notes==
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{{PlanetMath attribution|id=5525|title=Krylov-Bogolubov theorem}}

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[[Category:Ergodic theory]]
[[Category:Theorems in dynamical systems]]
[[Category:Probability theorems]]
[[Category:Random dynamical systems]]
[[Category:Measure theory]]