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

:$(P\_\{t\})\_\{ast\}\; (mu)\; =\; mu\; mbox\{\; for\; all\; \}\; t\; >\; 0.$

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