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title="Theorem">theorems</a> within the theory of <a href="/wiki/Dynamical_systems" class="mw-redirect" title="Dynamical systems">dynamical systems</a>. The theorems guarantee the existence of <a href="/wiki/Invariant_measure" title="Invariant measure">invariant measures</a> for certain "nice" maps defined on "nice" spaces and were named after <a href="/wiki/Russia" title="Russia">Russian</a>-<a href="/wiki/Ukraine" title="Ukraine">Ukrainian</a> <a href="/wiki/Mathematician" title="Mathematician">mathematicians</a> and <a href="/wiki/Theoretical_physics" title="Theoretical physics">theoretical physicists</a> <a href="/wiki/Nikolay_Mitrofanovich_Krylov" class="mw-redirect" title="Nikolay Mitrofanovich Krylov">Nikolay Krylov</a> and <a href="/wiki/Nikolay_Bogolyubov" title="Nikolay Bogolyubov">Nikolay Bogolyubov</a> who proved the theorems.<sup id="cite_ref-Bogoliubov1937_1-0" class="reference"><a href="#cite_note-Bogoliubov1937-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Formulation_of_the_theorems">Formulation of the theorems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Krylov%E2%80%93Bogolyubov_theorem&action=edit&section=1" title="Edit section: Formulation of the theorems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Invariant_measures_for_a_single_map">Invariant measures for a single map</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Krylov%E2%80%93Bogolyubov_theorem&action=edit&section=2" title="Edit section: Invariant measures for a single map"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Theorem (Krylov–Bogolyubov)</b>. Let <i>X</i> be a <a href="/wiki/Compact_space" title="Compact space">compact</a>, <a href="/wiki/Metrizable_space" title="Metrizable space">metrizable</a> <a href="/wiki/Topological_space" title="Topological space">topological space</a> and <i>F</i> : <i>X</i> → <i>X</i> a <a href="/wiki/Continuous_map" class="mw-redirect" title="Continuous map">continuous map</a>. Then <i>F</i> admits an invariant <a href="/wiki/Borel_measure" title="Borel measure">Borel</a> <a href="/wiki/Probability_measure" title="Probability measure">probability measure</a>. </p><p>That is, if Borel(<i>X</i>) denotes the <a href="/wiki/Borel_sigma_algebra" class="mw-redirect" title="Borel sigma algebra">Borel</a> <a href="/wiki/Sigma_algebra" class="mw-redirect" title="Sigma algebra">σ-algebra</a> generated by the collection <i>T</i> of <a href="/wiki/Open_set" title="Open set">open subsets</a> of <i>X</i>, then there exists a probability measure <i>μ</i> : Borel(<i>X</i>) → [0, 1] such that for any subset <i>A</i> ∈ Borel(<i>X</i>), </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \left(F^{-1}(A)\right)=\mu (A).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu \left(F^{-1}(A)\right)=\mu (A).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3210088fe78b4794a48dfe215ac1f5897f0b81ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.318ex; height:3.343ex;" alt="{\displaystyle \mu \left(F^{-1}(A)\right)=\mu (A).}"></span></dd></dl> <p>In terms of the <a href="/wiki/Pushforward_measure" title="Pushforward measure">push forward</a>, this states that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{*}(\mu )=\mu .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>μ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{*}(\mu )=\mu .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6391180e2b1c27c5c82f635bff0b8dc4e5e929c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.907ex; height:2.843ex;" alt="{\displaystyle F_{*}(\mu )=\mu .}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Invariant_measures_for_a_Markov_process">Invariant measures for a Markov process</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Krylov%E2%80%93Bogolyubov_theorem&action=edit&section=3" title="Edit section: Invariant measures for a Markov process"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>X</i> be a <a href="/wiki/Polish_space" title="Polish space">Polish space</a> and let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{t},t\geq 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>,</mo> <mi>t</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{t},t\geq 0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c947c258f0a5652dd7c0737663c2f36137edef8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.1ex; height:2.509ex;" alt="{\displaystyle P_{t},t\geq 0,}"></span> be the transition probabilities for a time-homogeneous <a href="/wiki/Markov_process" class="mw-redirect" title="Markov process">Markov</a> <a href="/wiki/Semigroup" title="Semigroup">semigroup</a> on <i>X</i>, i.e. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pr[X_{t}\in A|X_{0}=x]=P_{t}(x,A).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">Pr</mo> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>∈</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Pr[X_{t}\in A|X_{0}=x]=P_{t}(x,A).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d46a3f3b2af9de6bb1eb916b8a707bfda97ceb0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.155ex; height:2.843ex;" alt="{\displaystyle \Pr[X_{t}\in A|X_{0}=x]=P_{t}(x,A).}"></span></dd></dl> <p><b>Theorem (Krylov–Bogolyubov)</b>. If there exists a point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}"></span> for which the family of probability measures { <i>P</i><sub><i>t</i></sub>(<i>x</i>, ·) | <i>t</i> > 0 } is <a href="/wiki/Tightness_of_measures" title="Tightness of measures">uniformly tight</a> and the semigroup (<i>P</i><sub><i>t</i></sub>) satisfies the <a href="/wiki/Feller-continuous_process" title="Feller-continuous process">Feller property</a>, then there exists at least one invariant measure for (<i>P</i><sub><i>t</i></sub>), i.e. a probability measure <i>μ</i> on <i>X</i> such that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (P_{t})_{\ast }(\mu )=\mu {\mbox{ for all }}t>0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> for all </mtext> </mstyle> </mrow> <mi>t</mi> <mo>></mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P_{t})_{\ast }(\mu )=\mu {\mbox{ for all }}t>0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d26563d482c3372076ca0ef4c004913a85d3a2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.624ex; height:2.843ex;" alt="{\displaystyle (P_{t})_{\ast }(\mu )=\mu {\mbox{ for all }}t>0.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Krylov%E2%80%93Bogolyubov_theorem&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>For the 1st theorem: <a href="/wiki/Ya._G._Sinai" class="mw-redirect" title="Ya. G. Sinai">Ya. G. Sinai</a> (Ed.) (1997): <i>Dynamical Systems II. Ergodic Theory with Applications to Dynamical Systems and Statistical Mechanics</i>. Berlin, New York: Springer-Verlag. <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-17001-4" title="Special:BookSources/3-540-17001-4">3-540-17001-4</a>. (Section 1).</li> <li>For the 2nd theorem: <a href="/wiki/Giuseppe_Da_Prato" title="Giuseppe Da Prato">G. Da Prato</a> and J. Zabczyk (1996): <i>Ergodicity for Infinite Dimensional Systems</i>. Cambridge Univ. Press. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-57900-7" title="Special:BookSources/0-521-57900-7">0-521-57900-7</a>. (Section 3).</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Krylov%E2%80%93Bogolyubov_theorem&action=edit&section=5" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Bogoliubov1937-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bogoliubov1937_1-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFN._N._Bogoliubov_and_N._M._Krylov1937" class="citation journal cs1 cs1-prop-foreign-lang-source">N. N. Bogoliubov and N. M. Krylov (1937). "La theorie generale de la mesure dans son application a l'etude de systemes dynamiques de la mecanique non-lineaire". <i>Annals of Mathematics</i>. Second Series (in French). <b>38</b> (1): 65–113. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1968511">10.2307/1968511</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1968511">1968511</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=La+theorie+generale+de+la+mesure+dans+son+application+a+l%27etude+de+systemes+dynamiques+de+la+mecanique+non-lineaire&rft.volume=38&rft.issue=1&rft.pages=65-113&rft.date=1937&rft_id=info%3Adoi%2F10.2307%2F1968511&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1968511%23id-name%3DJSTOR&rft.au=N.+N.+Bogoliubov+and+N.+M.+Krylov&rfr_id=info%3Asid%2Fen.wikipedia.org%3AKrylov%E2%80%93Bogolyubov+theorem" class="Z3988"></span> Zbl. 16.86.</span> </li> </ol></div></div> <p><i>This article incorporates material from <a rel="nofollow" class="external text" href="https://planetmath.org/krylovbogolubovtheorem">Krylov-Bogolubov theorem</a> on <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>, which is licensed under the <a href="/wiki/Wikipedia:CC-BY-SA" class="mw-redirect" title="Wikipedia:CC-BY-SA">Creative Commons Attribution/Share-Alike License</a>.</i> </p>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Krylov–Bogolyubov_theorem&oldid=1257143646">https://en.wikipedia.org/w/index.php?title=Krylov–Bogolyubov_theorem&oldid=1257143646</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Ergodic_theory" title="Category:Ergodic theory">Ergodic theory</a></li><li><a href="/wiki/Category:Theorems_in_dynamical_systems" title="Category:Theorems in dynamical systems">Theorems in dynamical systems</a></li><li><a href="/wiki/Category:Probability_theorems" title="Category:Probability theorems">Probability theorems</a></li><li><a href="/wiki/Category:Random_dynamical_systems" title="Category:Random dynamical systems">Random dynamical systems</a></li><li><a href="/wiki/Category:Theorems_in_measure_theory" title="Category:Theorems in measure theory">Theorems in measure theory</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_French-language_sources_(fr)" title="Category:CS1 French-language sources (fr)">CS1 French-language sources (fr)</a></li><li><a href="/wiki/Category:Wikipedia_articles_incorporating_text_from_PlanetMath" title="Category:Wikipedia articles incorporating text from PlanetMath">Wikipedia articles incorporating text from PlanetMath</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 13 November 2024, at 13:24<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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