Krylov–Bogolyubov theorem

In

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

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

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

• For the 1st theorem:
  ISBN 3-540-17001-4
  . (Section 1).
• For the 2nd theorem:
  ISBN 0-521-57900-7
  . (Section 3).