{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/284646"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/284646","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"Relaxation to equilibrium for kinetic Fokker-Planck equation","abstract":"We want to study long-time behaviour of solutions $f_t$ of kinetic Fokker-Planck equation in $\\mathbb{R}^d$, namely their convergence towards equilibrium $f_\\infty$ in the form \\[ \\textrm{d}(f_t,f_\\infty)\\leq C_1 e^{-C_2 t}\\textrm{d}(f_0,\\mu) \\] for appropriate distances $\\textit{d}$ and constants $C_1 \\geq 1$, $C_2>0$. In Section 1 we provide an introduction and motivation for the equation, together with the setting of {Villani, Hypocoercivity} which will be useful in Section 2. In Section 2 we will review the monograph {Villani, Hypocoercivity}, where such convergence is proved, for $h=f/\\mu$, in $H^1 (\\mu)$ and $H_\\mu +I_\\mu$, that is, the sum of relative entropy and Fisher information. Here results are stated in terms of general operators $\\partial_t +A^*A+B=0$, and commutation conditions on $A$ and $B$ are to be imposed. In Section 3 we shall take into consideration the work by Monmarch\\'{e} {Monmarche, Generalized Γ calculus} in which such convergence is established by rephrasing some concepts in term of $\\Gamma$-calculus: with respect to {Villani, Hypocoercivity} there is no need for regularization along the semigroup since the functional taken into account is a modified $H+I$ that at initial time only takes entropy into account, and the argument turns out to be shorter. Also, the convergence rate is $e^{-Ct(1-e^{-t})^2}$ instead of $C_1 e^{-C_2 t}$. However it turns out, as in {Villani, Hypocoercivity}, that for this case it is strictly needed to have a pointwise bound on $D^2 U$, where $U$ is the confinement potential. A drawback of this method with respect to {Villani, Hypocoercivity} is that, in a more general setting than kinetic Fokker-Planck equation, stronger commutation assumptions are required, which imply that the diffusion matrix is basically required to be constant. On this work a specific analysis was carried out, simplifying the proof for our Fokker-Planck case and finding explicit and improved expressions for convergence constants. The same author in {Monmarche, chaos kinetic particles}, which is the subject of Section 4, addresses a Vlasov-Fokker-Planck equation with a potential that generalizes $U$ and the related particle system. Chaos propagation in $W_2$, the $2$-Wasserstein distance, is proved, namely $W_2(f_t^{(1,N)},f_t)\\leq CN^{-\\epsilon}$. This leads to both Wasserstein and $L^1$ hypocoercivity, however dependence of the right hand side from the initial data is not linear as wished.","abstract_html":"We want to study long-time behaviour of solutions <span class=\"etd-inline-math\">f<sub>t</sub></span> of kinetic Fokker-Planck equation in <span class=\"etd-inline-math\">\\mathbb{R}<sup>d</sup></span>, namely their convergence towards equilibrium <span class=\"etd-inline-math\">f<sub>\\</sub>infty</span> in the form \\[ \\textrm{d}(f_t,f_\\infty)\\leq C_1 e^{-C_2 t}\\textrm{d}(f_0,\\mu) \\] for appropriate distances <span class=\"etd-inline-math\"><em>d</em></span> and constants <span class=\"etd-inline-math\">C<sub>1</sub> \\geq 1</span>, <span class=\"etd-inline-math\">C<sub>2</sub>&gt;0</span>. In Section 1 we provide an introduction and motivation for the equation, together with the setting of {Villani, Hypocoercivity} which will be useful in Section 2. In Section 2 we will review the monograph {Villani, Hypocoercivity}, where such convergence is proved, for <span class=\"etd-inline-math\">h=f/&mu;</span>, in <span class=\"etd-inline-math\">H<sup>1</sup> (&mu;)</span> and <span class=\"etd-inline-math\">H<sub>\\</sub>mu +I<sub>\\</sub>mu</span>, that is, the sum of relative entropy and Fisher information. Here results are stated in terms of general operators <span class=\"etd-inline-math\">\\partial<sub>t</sub> +A<sup>*</sup>A+B=0</span>, and commutation conditions on $A$ and $B$ are to be imposed. In Section 3 we shall take into consideration the work by Monmarch\\&#x27;{e} {Monmarche, Generalized Γ calculus} in which such convergence is established by rephrasing some concepts in term of $\\Gamma$-calculus: with respect to {Villani, Hypocoercivity} there is no need for regularization along the semigroup since the functional taken into account is a modified $H+I$ that at initial time only takes entropy into account, and the argument turns out to be shorter. Also, the convergence rate is <span class=\"etd-inline-math\">e<sup>-Ct(1-e<sup>-t</sup>)<sup>2</sup></sup></span> instead of <span class=\"etd-inline-math\">C<sub>1</sub> e<sup>-C<sub>2</sub> t</sup></span>. However it turns out, as in {Villani, Hypocoercivity}, that for this case it is strictly needed to have a pointwise bound on <span class=\"etd-inline-math\">D<sup>2</sup> U</span>, where $U$ is the confinement potential. A drawback of this method with respect to {Villani, Hypocoercivity} is that, in a more general setting than kinetic Fokker-Planck equation, stronger commutation assumptions are required, which imply that the diffusion matrix is basically required to be constant. On this work a specific analysis was carried out, simplifying the proof for our Fokker-Planck case and finding explicit and improved expressions for convergence constants. The same author in {Monmarche, chaos kinetic particles}, which is the subject of Section 4, addresses a Vlasov-Fokker-Planck equation with a potential that generalizes $U$ and the related particle system. Chaos propagation in <span class=\"etd-inline-math\">W<sub>2</sub></span>, the $2$-Wasserstein distance, is proved, namely <span class=\"etd-inline-math\">W<sub>2</sub>(f<sub>t</sub><sup>(1,N)</sup>,f<sub>t</sub>)\\leq CN<sup>-&epsilon;</sup></span>. This leads to both Wasserstein and <span class=\"etd-inline-math\">L<sup>1</sup></span> hypocoercivity, however dependence of the right hand side from the initial data is not linear as wished.","abstract_has_math":true,"creators":["Piazzoli, Davide"],"institution":"University of Cambridge","degree_name":"Master of Science (MSc)","degree_level":"Masters","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Mouhot, Clément"],"committee_chairs":[],"committee_members":[],"year":2019,"date_issued":"2019-02-01","date_published":"2019-02-01","updated_at":"2026-07-22T22:23:54Z","subjects":["Villani","hypocoercivity","kinetic","PDE","Wasserstein","entropy","Monmarche","Fokker","Planck","Fokker-Planck","Logarithmic Sobolev inequality","Gamma calculus","carré du champ","commutation","Mouhot","Bolley","particle system","Bakry-Emery"],"languages":["en"],"rights":[],"rights_urls":["https://www.repository.cam.ac.uk/bitstreams/4940a118-7755-4f79-aa8c-668b7fd30326/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.32020","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Mouhot, Clément"]},{"key":"dc:creator","label":"Author","values":["Piazzoli, Davide"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2019-02-01"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/284646"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Masters"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Master of Science (MSc)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Villani","hypocoercivity","kinetic","PDE","Wasserstein","entropy","Monmarche","Fokker","Planck","Fokker-Planck","Logarithmic Sobolev inequality","Gamma calculus","carré du champ","commutation","Mouhot","Bolley","particle system","Bakry-Emery"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["https://www.repository.cam.ac.uk/bitstreams/4940a118-7755-4f79-aa8c-668b7fd30326/download","https://www.rioxx.net/licenses/all-rights-reserved/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.17863/CAM.32020"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://www.repository.cam.ac.uk/bitstreams/85d8bc76-c8ed-4820-af22-fe59c16146c8/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We want to study long-time behaviour of solutions $f_t$ of kinetic Fokker-Planck equation in $\\mathbb{R}^d$, namely their convergence towards equilibrium $f_\\infty$ in the form \\[ \\textrm{d}(f_t,f_\\infty)\\leq C_1 e^{-C_2 t}\\textrm{d}(f_0,\\mu) \\] for appropriate distances $\\textit{d}$ and constants $C_1 \\geq 1$, $C_2>0$. In Section 1 we provide an introduction and motivation for the equation, together with the setting of {Villani, Hypocoercivity} which will be useful in Section 2. In Section 2 we will review the monograph {Villani, Hypocoercivity}, where such convergence is proved, for $h=f/\\mu$, in $H^1 (\\mu)$ and $H_\\mu +I_\\mu$, that is, the sum of relative entropy and Fisher information. Here results are stated in terms of general operators $\\partial_t +A^*A+B=0$, and commutation conditions on $A$ and $B$ are to be imposed. In Section 3 we shall take into consideration the work by Monmarch\\'{e} {Monmarche, Generalized Γ calculus} in which such convergence is established by rephrasing some concepts in term of $\\Gamma$-calculus: with respect to {Villani, Hypocoercivity} there is no need for regularization along the semigroup since the functional taken into account is a modified $H+I$ that at initial time only takes entropy into account, and the argument turns out to be shorter. Also, the convergence rate is $e^{-Ct(1-e^{-t})^2}$ instead of $C_1 e^{-C_2 t}$. However it turns out, as in {Villani, Hypocoercivity}, that for this case it is strictly needed to have a pointwise bound on $D^2 U$, where $U$ is the confinement potential. A drawback of this method with respect to {Villani, Hypocoercivity} is that, in a more general setting than kinetic Fokker-Planck equation, stronger commutation assumptions are required, which imply that the diffusion matrix is basically required to be constant. On this work a specific analysis was carried out, simplifying the proof for our Fokker-Planck case and finding explicit and improved expressions for convergence constants. The same author in {Monmarche, chaos kinetic particles}, which is the subject of Section 4, addresses a Vlasov-Fokker-Planck equation with a potential that generalizes $U$ and the related particle system. Chaos propagation in $W_2$, the $2$-Wasserstein distance, is proved, namely $W_2(f_t^{(1,N)},f_t)\\leq CN^{-\\epsilon}$. This leads to both Wasserstein and $L^1$ hypocoercivity, however dependence of the right hand side from the initial data is not linear as wished."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["186663108b77c6f25dacde19e0a228e9","87eda9de84448d1f82354d60eee3eb5f"]},{"key":"dc:title","label":"Title","values":["Relaxation to equilibrium for kinetic Fokker-Planck equation"]}]}],"canonical_facts":{"dc:contributor.advisor":["Mouhot, Clément"],"dc:creator":["Piazzoli, Davide"],"dc:date.issued":["2019-02-01"],"dc:description.abstract":["We want to study long-time behaviour of solutions $f_t$ of kinetic Fokker-Planck equation in $\\mathbb{R}^d$, namely their convergence towards equilibrium $f_\\infty$ in the form \\[ \\textrm{d}(f_t,f_\\infty)\\leq C_1 e^{-C_2 t}\\textrm{d}(f_0,\\mu) \\] for appropriate distances $\\textit{d}$ and constants $C_1 \\geq 1$, $C_2>0$. In Section 1 we provide an introduction and motivation for the equation, together with the setting of {Villani, Hypocoercivity} which will be useful in Section 2. In Section 2 we will review the monograph {Villani, Hypocoercivity}, where such convergence is proved, for $h=f/\\mu$, in $H^1 (\\mu)$ and $H_\\mu +I_\\mu$, that is, the sum of relative entropy and Fisher information. Here results are stated in terms of general operators $\\partial_t +A^*A+B=0$, and commutation conditions on $A$ and $B$ are to be imposed. In Section 3 we shall take into consideration the work by Monmarch\\'{e} {Monmarche, Generalized Γ calculus} in which such convergence is established by rephrasing some concepts in term of $\\Gamma$-calculus: with respect to {Villani, Hypocoercivity} there is no need for regularization along the semigroup since the functional taken into account is a modified $H+I$ that at initial time only takes entropy into account, and the argument turns out to be shorter. Also, the convergence rate is $e^{-Ct(1-e^{-t})^2}$ instead of $C_1 e^{-C_2 t}$. However it turns out, as in {Villani, Hypocoercivity}, that for this case it is strictly needed to have a pointwise bound on $D^2 U$, where $U$ is the confinement potential. A drawback of this method with respect to {Villani, Hypocoercivity} is that, in a more general setting than kinetic Fokker-Planck equation, stronger commutation assumptions are required, which imply that the diffusion matrix is basically required to be constant. On this work a specific analysis was carried out, simplifying the proof for our Fokker-Planck case and finding explicit and improved expressions for convergence constants. The same author in {Monmarche, chaos kinetic particles}, which is the subject of Section 4, addresses a Vlasov-Fokker-Planck equation with a potential that generalizes $U$ and the related particle system. Chaos propagation in $W_2$, the $2$-Wasserstein distance, is proved, namely $W_2(f_t^{(1,N)},f_t)\\leq CN^{-\\epsilon}$. This leads to both Wasserstein and $L^1$ hypocoercivity, however dependence of the right hand side from the initial data is not linear as wished."],"dc:format.checksum.md5":["186663108b77c6f25dacde19e0a228e9","87eda9de84448d1f82354d60eee3eb5f"],"dc:identifier.doi":["10.17863/CAM.32020"],"dc:identifier.uri":["https://www.repository.cam.ac.uk/bitstreams/85d8bc76-c8ed-4820-af22-fe59c16146c8/download"],"dc:language":["en"],"dc:publisher.institution":["University of Cambridge"],"dc:relation.isreferencedby.uri":["https://www.repository.cam.ac.uk/handle/1810/284646"],"dc:rights":["https://www.repository.cam.ac.uk/bitstreams/4940a118-7755-4f79-aa8c-668b7fd30326/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"dc:subject":["Villani","hypocoercivity","kinetic","PDE","Wasserstein","entropy","Monmarche","Fokker","Planck","Fokker-Planck","Logarithmic Sobolev inequality","Gamma calculus","carré du champ","commutation","Mouhot","Bolley","particle system","Bakry-Emery"],"dc:title":["Relaxation to equilibrium for kinetic Fokker-Planck equation"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Masters"],"dc:type.qualificationname":["Master of Science (MSc)"]},"updated_at":"2026-07-22T22:23:54Z"}