On the ergodicity of nonlinear Fokker–Planck flows in $L^{1}(\mathbb R^d)$
UDC 517.9 We prove that a nonlinear semigroup $S(t)$ in $L^1(\mathbb{R}^d),$ $d\ge 3,$ associated with the nonlinear Fokker–Planck equation $u_t-\Delta\beta(u)+{\rm div }(Db(u)u)=0,$ $u(0)=u_0,$ in $(0,\infty)\times\mathbb{R}^d,$ with suitable conditions imposed on the coefficients $\beta\colon \mat...
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| Date: | 2026 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Institute of Mathematics, NAS of Ukraine
2026
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/8286 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860513017819561984 |
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| author | Barbu, Viorel Röckner, Michael Barbu, Viorel Röckner, Michael |
| author_facet | Barbu, Viorel Röckner, Michael Barbu, Viorel Röckner, Michael |
| author_sort | Barbu, Viorel |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2026-03-21T13:30:45Z |
| description | UDC 517.9
We prove that a nonlinear semigroup $S(t)$ in $L^1(\mathbb{R}^d),$ $d\ge 3,$ associated with the nonlinear Fokker–Planck equation $u_t-\Delta\beta(u)+{\rm div }(Db(u)u)=0,$ $u(0)=u_0,$ in $(0,\infty)\times\mathbb{R}^d,$ with suitable conditions imposed on the coefficients $\beta\colon \mathbb{R}\to\mathbb{R},$ $-D\colon \mathbb{R}^d\to\mathbb{R}^d,$ and $b\colon \mathbb{R}\to\mathbb{R},$ is mean ergodic. In particular, this implies the mean ergodicity of the time marginal laws for the solutions to the corresponding McKean–Vlasov stochastic differential equation. This completes the results established in [V. Barbu, M. Röckner, The invariance principle for nonlinear Fokker–Planck equations, J. Different. Equat., 315, 200–221 (2022)] on the nature of the corresponding omega-set $\omega(u_0)$ for $S(t)$ in the case where the flow $S(t)$ in $L^1(\mathbb{R}^d)$ does not have a fixed point and, hence, the corresponding stationary Fokker–Planck equation has no solutions. |
| doi_str_mv | 10.3842/umzh.v77i4.8286 |
| first_indexed | 2026-03-24T03:38:00Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-8286 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:38:00Z |
| publishDate | 2026 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | |
| spelling | umjimathkievua-article-82862026-03-21T13:30:45Z On the ergodicity of nonlinear Fokker–Planck flows in $L^{1}(\mathbb R^d)$ On the ergodicity of nonlinear Fokker–Planck flows in $L^{1}(\mathbb R^d)$ Barbu, Viorel Röckner, Michael Barbu, Viorel Röckner, Michael Fokker-Planck semigroup ergodic UDC 517.9 We prove that a nonlinear semigroup $S(t)$ in $L^1(\mathbb{R}^d),$ $d\ge 3,$ associated with the nonlinear Fokker–Planck equation $u_t-\Delta\beta(u)+{\rm div }(Db(u)u)=0,$ $u(0)=u_0,$ in $(0,\infty)\times\mathbb{R}^d,$ with suitable conditions imposed on the coefficients $\beta\colon \mathbb{R}\to\mathbb{R},$ $-D\colon \mathbb{R}^d\to\mathbb{R}^d,$ and $b\colon \mathbb{R}\to\mathbb{R},$ is mean ergodic. In particular, this implies the mean ergodicity of the time marginal laws for the solutions to the corresponding McKean–Vlasov stochastic differential equation. This completes the results established in [V. Barbu, M. Röckner, The invariance principle for nonlinear Fokker–Planck equations, J. Different. Equat., 315, 200–221 (2022)] on the nature of the corresponding omega-set $\omega(u_0)$ for $S(t)$ in the case where the flow $S(t)$ in $L^1(\mathbb{R}^d)$ does not have a fixed point and, hence, the corresponding stationary Fokker–Planck equation has no solutions. УДК 517.9 Ергодичність нелінійних потоків Фоккера–Планка в $L^{1}(\mathbb R^d)$ Доведено, що нелінійна напівгрупа $S(t)$ у $L^1(\mathbb{R}^d),$ $d\ge 3,$ пов'язана з нелінійним рівнянням Фоккера–Планка $u_t-\Delta\beta(u)+{\rm div }(Db(u)u)=0,$ $u(0)=u_0,$ у $(0,\infty)\times\mathbb{R}^d,$ за відповідних умов на коефіцієнти $\beta\colon \mathbb{R}\to\mathbb{R},$ $D\colon \mathbb{R}^d\to\mathbb{R}^d$ і $b\colon \mathbb{R}\to\mathbb{R}$ є середньою ергодичною. Зокрема, це означає середню ергодичність граничних за часом законів для розв'язків відповідного стохастичного диференціального рівняння Маккіна–Власова. Це доповнює результати, встановлені в роботі [V. Barbu, M. Röckner, The invariance principle for nonlinear Fokker–Planck equations, J. Different. Equat., 315, 200–221 (2022)], про природу відповідної омега-множини $\omega(u_0)$ для $S(t)$ у випадку, коли потік $S(t)$ у $L^1(\mathbb{R}^d)$ не має нерухомої точки, і тому відповідне стаціонарне рівняння Фоккера–Планка не має розв'язків. Institute of Mathematics, NAS of Ukraine 2026-03-21 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/8286 10.3842/umzh.v77i4.8286 Ukrains’kyi Matematychnyi Zhurnal; Vol. 77 No. 4 (2025); 279 Український математичний журнал; Том 77 № 4 (2025); 279 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/8286/10441 Copyright (c) 2025 Viorel Barbu, Michael Röckner |
| spellingShingle | Barbu, Viorel Röckner, Michael Barbu, Viorel Röckner, Michael On the ergodicity of nonlinear Fokker–Planck flows in $L^{1}(\mathbb R^d)$ |
| title | On the ergodicity of nonlinear Fokker–Planck flows in $L^{1}(\mathbb R^d)$ |
| title_alt | On the ergodicity of nonlinear Fokker–Planck flows in $L^{1}(\mathbb R^d)$ |
| title_full | On the ergodicity of nonlinear Fokker–Planck flows in $L^{1}(\mathbb R^d)$ |
| title_fullStr | On the ergodicity of nonlinear Fokker–Planck flows in $L^{1}(\mathbb R^d)$ |
| title_full_unstemmed | On the ergodicity of nonlinear Fokker–Planck flows in $L^{1}(\mathbb R^d)$ |
| title_short | On the ergodicity of nonlinear Fokker–Planck flows in $L^{1}(\mathbb R^d)$ |
| title_sort | on the ergodicity of nonlinear fokker–planck flows in $l^{1}(\mathbb r^d)$ |
| topic_facet | Fokker-Planck semigroup ergodic |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/8286 |
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