Krylov-Bogolyubov theorem

The Krylov – Bogolyubov theorem states the existence of invariant measures for “good” mappings defined on “good” spaces. There are two variations of the theorem, for dynamical systems and for Markov processes

The theorem was proved by the mathematician N. M. Krylov and the theoretical physicist , mathematician N. N. Bogolyubov . ^[1] ^[2] (reprinted in ^[3] ).

Content

Dynamic wording

Let be $F$ ${\ displaystyle F}$ - continuous display of metric compact $X$ ${\ displaystyle X}$ in yourself. Then on $X$ ${\ displaystyle X}$ there is at least one $F$ ${\ displaystyle F}$ - invariant measure $\mu$ ${\ displaystyle \ mu}$ which can be chosen in such a way that it will be indecomposable, or ergodic ^[4] .

Remarks

Condition $F$ ${\ displaystyle F}$ -invariance, $F_{*}\mu =\mu$ ${\ displaystyle F _ {*} \ mu = \ mu}$ , means that the measure of the inverse image of any Borel set is equal to the measure of this set,
$\forall A\in {\mathcal {B}}(X)\quad \mu (F^{-1}(A))=\mu (A);$ ${\ displaystyle \ forall A \ in {\ mathcal {B}} (X) \ quad \ mu (F ^ {- 1} (A)) = \ mu (A);}$

in case of irreversible display

F

{\ displaystyle F}

measure

F(A)

{\ displaystyle F (A)}

not required to equal measure

A

{\ displaystyle A}

.

For example, the Lebesgue measure is invariant for doubling a circle. $x\mapsto 2x\mod 1$ ${\ displaystyle x \ mapsto 2x \ mod 1}$ however measure of arc $\left[0,{\frac {1}{3}}\right]$ ${\ displaystyle \ left [0, {\ frac {1} {3}} \ right]}$ not equal to the measure of her image, arc $\left[0,{\frac {2}{3}}\right]$ ${\ displaystyle \ left [0, {\ frac {2} {3}} \ right]}$ .

Proof

The proof of the theorem is based on the so-called Krylov-Bogolyubov procedure — the procedure for extracting a convergent subsequence from a sequence of time averages of an arbitrary initial measure.

Namely, an arbitrary initial measure is taken. $\mu _{0}$ ${\ displaystyle \ mu _ {0}}$ , and the sequence of its time averages is considered:

{\bar {\mu }}_{n}={\frac {1}{n}}\sum _{j=0}^{n-1}F_{*}^{j}(\mu _{0}).

{\ displaystyle {\ bar {\ mu}} _ {n} = {\ frac {1} {n}} \ sum _ {j = 0} ^ {n-1} F _ {*} ^ {j} (\ mu _ {0}).}

Temporary averages are more and more $F$ ${\ displaystyle F}$ -invariant:

F_{*}{\bar {\mu }}_{n}={\bar {\mu }}_{n}+{\frac {1}{n}}(F_{*}^{n}(\mu _{0})-\mu _{0}).

{\ displaystyle F _ {*} {\ bar {\ mu}} _ {n} = {\ bar {\ mu}} _ {n} + {\ frac {1} {n}} (F _ {*} ^ { n} (\ mu _ {0}) - \ mu _ {0}).}

Therefore, the limit of any convergent subsequence of a sequence of time means is an invariant measure for the map $F$ ${\ displaystyle F}$ . But the space of probability measures on a metric compact $F$ ${\ displaystyle F}$ compact (in the sense of a * -weak topology), therefore, at least one accumulation point of the sequence ${\bar {\mu }}_{n}$ ${\ displaystyle {\ bar {\ mu}} _ {n}}$ there is - which completes the proof. ■

Remarks

In case, as a measure $\mu _{0}$ ${\ displaystyle \ mu _ {0}}$ the Dirac measure (concentrated at a typical starting point) or the Lebesgue measure, the convergence of the sequence ${\bar {\mu }}_{n}$ ${\ displaystyle {\ bar {\ mu}} _ {n}}$ corresponds to the existence of the Sinai – Ruelle – Bowen measure .

Wording for Markov processes

Let X be a Polish space and let ( P _t ) be the family of probabilities of the transition of some homogeneous Markov semigroup to X , i.e.

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

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

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

(P_{t})_{\ast }(\mu )=\mu ,\quad \forall t>0.

{\ displaystyle (P_ {t}) _ {\ ast} (\ mu) = \ mu, \ quad \ forall t> 0.}

Variations and generalizations

Exactly the same reasoning, only related to averaging over the Fölner sequence , allows us to prove that for any continuous action of an amenable group on a metric compactum, there is a measure invariant with respect to this action.

Links

↑ Bogolyubov N.N., Krylov N.M. (1937): “General measure theory in nonlinear mechanics”. - Kiev.
↑ NN Bogoliubov and NM Krylov. La theorie generalie de la mesure dans son application a l'etude de systemes dynamiques de la mecanique non-lineaire (fr.) // Ann. Math. II. - 1937.- T. 38 . - S. 65-113 . Zbl. 16.86.
↑ "Nikolai Nikolaevich Bogolyubov. Collection of scientific papers in 12 volumes. RAS. Volume 1: Mathematics. " - M .: Nauka, 2005. ISBN 5-02-034463-X .
↑ Nonlinear Dynamics and Chaos, 2011 , p. 177.

Literature

Malinetskii G. G. , Potapov A. B. Non-linear dynamics and chaos: basic concepts. - M .: Librocom, 2011 .-- 240 p. - ISBN 978-5-397-01583-7 .

[1] Bogolyubov N.N., Krylov N.M. (1937): “General measure theory in nonlinear mechanics”. - Kiev.

[2] NN Bogoliubov and NM Krylov. La theorie generalie de la mesure dans son application a l'etude de systemes dynamiques de la mecanique non-lineaire (fr.) // Ann. Math. II. - 1937.- T. 38 . - S. 65-113 . Zbl. 16.86.

[3] "Nikolai Nikolaevich Bogolyubov. Collection of scientific papers in 12 volumes. RAS. Volume 1: Mathematics. " - M .: Nauka, 2005. ISBN 5-02-034463-X .

[_424afddc16ac7e3a-4] Nonlinear Dynamics and Chaos, 2011 , p. 177.

Krylov-Bogolyubov theorem

Content

Dynamic wording

Remarks

Proof

Remarks

Wording for Markov processes

Variations and generalizations

Links

Literature

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