Krylov–Bogolyubov theorem
Appearance
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
- For the 1st theorem: ISBN 3-540-17001-4. (Section 1).
- For the 2nd theorem: ISBN 0-521-57900-7. (Section 3).
Notes
- JSTOR 1968511. Zbl. 16.86.
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