We want to study long-time behaviour of solutions $ft$ of kinetic Fokker-Planck equation in $Rd$, 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 $C1\ge 1$, $C2>0$.

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 $h=f/\mu$, in $H1(\mu )$ and $H\mu +I\mu$, that is, the sum of relative entropy and Fisher information. Here results are stated in terms of general operators $\partial t+A\ast A+B=0$, 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 $C1e-C2t$. However it turns out, as in {Villani, Hypocoercivity}, that for this case it is strictly needed to have a pointwise bound on $D2U$, 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 $W2(ft(1,N),ft)\le CN-ϵ$. This leads to both Wasserstein and $L1$ hypocoercivity, however dependence of the right hand side from the initial data is not linear as wished.