## Relaxation to equilibrium for kinetic Fokker-Planck equation

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##### Authors

Piazzoli, Davide

##### Advisors

Mouhot, Clément

##### Date

2019-02-01##### Awarding Institution

University of Cambridge

##### Author Affiliation

Cambridge Centre for Analysis (CCA)

##### Qualification

Master of Science (MSc)

##### Language

English

##### Type

Thesis

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Show full item record##### Citation

Piazzoli, D. (2019). Relaxation to equilibrium for kinetic Fokker-Planck equation (Masters thesis). https://doi.org/10.17863/CAM.32020

##### 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.

##### Keywords

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

##### Identifiers

This record's DOI: https://doi.org/10.17863/CAM.32020

##### Rights

All rights reserved