World Library  
Flag as Inappropriate
Email this Article

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]


  • Formulation of the theorems 1
    • Invariant measures for a single map 1.1
    • Invariant measures for a Markov process 1.2
  • See also 2
  • Notes 3

Formulation of the theorems

Invariant measures for a single map

Theorem (Krylov–Bogolyubov). Let (XT) be a compact, metrizable topological space and F : X → X a continuous map. Then F admits an invariant Borel probability measure.

That is, 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),

\mu \left( F^{-1} (A) \right) = \mu (A).

In terms of the push forward, this states that

F_{*} (\mu) = \mu.\

Invariant measures for a Markov process

Let X be a Polish space and let P_t, t\ge 0, be the transition probabilities for a time-homogeneous Markov semigroup on X, i.e.

\Pr [ X_{t} \in A | X_{0} = x ] = P_{t} (x, A).

Theorem (Krylov–Bogolyubov). If there exists a point x\in 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), i.e. a probability measure μ on X such that

(P_{t})_{\ast} (\mu) = \mu \mbox{ for all } 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. (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).


  1. ^ 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". Ann. Math. II (in French) (Annals of Mathematics) 38 (1): 65–113.   Zbl. 16.86.

This article incorporates material from Krylov-Bogolubov theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.