Krylov-Bogolyubov theorem

Krylov-Bogolyubov theorem

In mathematics, the Krylov-Bogolyubov theorem (also known as the existence of invariant measures theorem) may refer to one of two related fundamental theorems in the study of dynamical systems. It guarantees the existence of invariant measures for "nice" maps on "nice" spaces. It is named after the Russian-Ukrainian mathemathicians and theoretical physicists Nikolay Mitrofanovich Krylov and Nikolay Bogolyubov

tatement of the theorem

Invariant measures for a single map

Let ("X", "T") be a compact, metrizable topological space, and let "F" : "X" → "X" be 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 is a probability measure "μ" : Borel("X") → [0, 1] such that, for any subset "A" ∈ Borel("X"),

:mu left( F^{-1} (A) ight) = 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" 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).

The Krylov-Bogolyubov theorem states that if there exists a point "x" in "X" for which the family of probability measures { "P""t"("x", ·) | "t" > 0 } is uniformly tight and the semigroup ("P""t") has the Feller property, then there exists at least one invariant measure for ("P""t"), i.e. a probability measure "μ" on "X" such that

:(P_{t})_{ast} (mu) = mu mbox{ for all } t > 0.

----


Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Krylov — ( ru. Крылов) (masculine or Krylova (feminine) is a common Russian last name. Alternative spellings are Krilov, Kryloff, Kriloff (masculine) and Krilova (feminine; Крылова).This last name is shared by the following people:*Alexei Krylov, a… …   Wikipedia

  • Nikolay Bogolyubov — For the actor, see Nikolay Bogolyubov (actor). Nikolay Nikolaevich Bogolyubov Born 21 August 1909( …   Wikipedia

  • Nikolay Mitrofanovich Krylov — Nikolay Krylov Born 29 November [O.S. 17 November] 1879 St Petersburg, Russian Empire …   Wikipedia

  • List of Russian people — The Millennium of Russia monument in Veliky Novgorod, featuring the statues and reliefs of the most celebrated people in the first 1000 years of Russian history …   Wikipedia

  • List of Russian mathematicians — Andrey Kolmogorov, a preeminent 20th century mathematician. This list of Russian mathematicians includes the famous mathematicians from the Russian Empire, the Soviet Union and the Russian Federation. This list is incomplete; you can help by …   Wikipedia

  • Measure-preserving dynamical system — In mathematics, a measure preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Contents 1 Definition 2 Examples 3 Homomorphisms 4 …   Wikipedia

  • List of mathematics articles (K) — NOTOC K K approximation of k hitting set K ary tree K core K edge connected graph K equivalence K factor error K finite K function K homology K means algorithm K medoids K minimum spanning tree K Poincaré algebra K Poincaré group K set (geometry) …   Wikipedia

  • List of dynamical systems and differential equations topics — This is a list of dynamical system and differential equation topics, by Wikipedia page. See also list of partial differential equation topics, list of equations. Contents 1 Dynamical systems, in general 2 Abstract dynamical systems 3 …   Wikipedia

  • List of theorems — This is a list of theorems, by Wikipedia page. See also *list of fundamental theorems *list of lemmas *list of conjectures *list of inequalities *list of mathematical proofs *list of misnamed theorems *Existence theorem *Classification of finite… …   Wikipedia

  • Dynamical system (definition) — This article presents the many ways to define a dynamical system. See the main article, dynamical system, for an overview of the topic. The dynamical system concept is a mathematical formalization for any fixed rule which describes the time… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”