University of Cambridge
Relaxation to equilibrium for kinetic Fokker-Planck equation
Abstract
dc:description.abstractWe want to study long-time behaviour of solutions ft 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 d and constants C1 \geq 1, C2>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/μ, in H1 (μ) and H\mu +I\mu, that is, the sum of relative entropy and Fisher information. Here results are stated in terms of general operators \partialt +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 C1 e-C2 t. However it turns out, as in {Villani, Hypocoercivity}, that for this case it is strictly needed to have a pointwise bound on D2 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 W2, the $2$-Wasserstein distance, is proved, namely W2(ft(1,N),ft)\leq CN-ε. This leads to both Wasserstein and L1 hypocoercivity, however dependence of the right hand side from the initial data is not linear as wished.
Degree
thesis:*- Name dc:type.qualificationname
- Master of Science (MSc)
- Level dc:type.qualificationlevel
- Masters
- Grantor dc:publisher.institution
- University of Cambridge
- Year dc:date.issued
- 2019
Author and committee
dc:creator, dc:contributor.*- Author dc:creator
-
- Piazzoli, Davide
- Advisor dc:contributor.advisor
-
- Mouhot, Clément
Subjects
dc:subject × 18Rights
dc:rightsIdentifiers
dc:identifier.*- DOI dc:identifier.doi
- https://doi.org/10.17863/CAM.32020