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.