Krylov–Bogolyubov theorem

Source: Wikipedia, the free encyclopedia.

In

Nikolay Krylov and Nikolay Bogolyubov who proved the theorems.[1]

Formulation of the theorems

Invariant measures for a single map

Theorem (Krylov–Bogolyubov). Let X be a

continuous map. Then F admits an invariant Borel probability measure
.

That is, if Borel(X) denotes the

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),

In terms of the push forward, this states that

Invariant measures for a Markov process

Let X be a Polish space and let be the transition probabilities for a time-homogeneous

Markov semigroup
on X, i.e.

Theorem (Krylov–Bogolyubov). If there exists a point 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

See also

Notes

This article incorporates material from Krylov-Bogolubov theorem on

Creative Commons Attribution/Share-Alike License
.