Krylov–Bogolyubov theorem

Theorem (Krylov–Bogolyubov). Let (X, T) 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),

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

In terms of the push forward, this states that

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

Let X be a Polish space and let ${\displaystyle P_{t},t\geq 0,}$ be the transition probabilities for a time-homogeneous Markov semigroup on X, i.e.

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

Theorem (Krylov–Bogolyubov). If there exists a point ${\displaystyle x\in X}$ for which the family of probability measures { P_t(x, ·) | t > 0 } is uniformly tight and the semigroup (P_t) satisfies the Feller property, then there exists at least one invariant measure for (P_t), i.e. a probability measure μ on X such that

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