hyperrefbookmark level for unknown author 11institutetext: Emeric Bouin
22institutetext: Jean Dolbeault 33institutetext: Luca Ziviani 44institutetext:
CEREMADE (CNRS UMR n∘ 7534), PSL university, Université Paris-Dauphine, Place
de Lattre de Tassigny, 75775 Paris 16, France
44email<EMAIL_ADDRESS>(E. Bouin), 44email:
<EMAIL_ADDRESS>(J. Dolbeault), 44email:
<EMAIL_ADDRESS>(L. Ziviani)
https://www.ceremade.dauphine.fr/~bouin/ (E. Bouin),
https://www.ceremade.dauphine.fr/~dolbeaul/ (J. Dolbeault),
https://www.ceremade.dauphine.fr/fr/membres/detail-cv/profile/luca-
ziviani.html (L. Ziviani)

#  $\mathrm{L}^{2}$ Hypocoercivity methods for kinetic Fokker-Planck equations
with factorised Gibbs states

Emeric Bouin  Jean Dolbeault  Luca Ziviani

###### Abstract

This contribution deals with $\mathrm{L}^{2}$ hypocoercivity methods for
kinetic Fokker-Planck equations with integrable local equilibria and a
_factorisation_ property that relates the Fokker-Planck and the transport
operators. Rates of convergence in presence of a global equilibrium, or decay
rates otherwise, are estimated either by the corresponding rates in the
diffusion limit, or by the rates of convergence to local equilibria, under
moment conditions. On the basis of the underlying functional inequalities, we
establish a classification of decay and convergence rates for large times,
which includes for instance sub-exponential local equilibria and sub-
exponential potentials.
_MSC (2020):_ Primary: 82C40. Secondary: 35H10, 35K65, 35Q84, 76P05, 35P15.

###### Keywords:

Hypocoercivity, linear kinetic equations, entropy - entropy production
inequalities, Fokker–Planck operator, transport operator, diffusion limit,
Nash inequality, Caffarelli-Kohn-Nirenberg inequality, Hardy-Poincaré
inequality, weighted Poincaré inequality, Poincaré inequality

__

##  1 Introduction

_Hypocoercivity_ refers to the method developed by C. Villani in order to
capture large time asymptotics in kinetic equations, see Mouhot-Neumann ; Mem-
villani , which borrows ideas from Hörmander’s _hypoelliptic_ theory and from
the _carré du champ_ method introduced by D. Bakry and M. Emery in Bakry1985 .
For this reason, the Fisher information plays an important role and, to some
extent, we can consider it as an $\mathrm{H}^{1}$-theory. Here we shall focus
more on the notion of $\mathrm{L}^{2}$-hypocoercivity inspired by MR2215889
and introduced in Dolbeault2009511 ; DMS  in a simple case of a kinetic
Fokker-Planck equation, which puts the emphasis on the underlying diffusion
limit. The heuristic idea is simple: while the Fokker-Planck diffusion
operator controls the rate of convergence towards local equilibria in the
velocity space, the equilibration of the spatial density (its convergence to
the spatial density of a global equilibrium or its decay when no such
equilibrium exists) can be interpreted as a diffusion in the position space,
at least in a certain parabolic scaling, which results of the interplay of the
diffusion in the velocity direction and the transport and the mixing in the
phase space induced by the transport operator. The advantage of this approach
is that rates are fully determined by the functional inequalities associated
to the diffusion operator on the velocity space and to the diffusion limit in
the position space.

This paper is organized as follows. We start by recalling the abstract
$\mathrm{L}^{2}$ hypocoercivity method in Theorem 2.1 before applying it to
the framework of non-Maxwellian local equilibria and a compatible transport
operator in Corollary 1. Although an adaptation of the standard theory in
presence of microscopic and macroscopic coercivity associated respectively to
the Fokker-Planck operator and the diffusion limit, this result is new and
covers for instance the case of relativistic transport. This framework is also
well adapted to situations with weaker notions of coercivity corresponding to
either an external potential with slower growth at infinity or to local
equilibria with fatter tails than the Maxwellian. After reviewing various
families of interpolation inequalities which dictate the asymptotic behaviours
of the solutions to the Fokker-Planck equations, we extend the
$\mathrm{L}^{2}$ hypocoercivity method to the kinetic Fokker-Planck equations
and establish a classification in terms the (microscopic) local equilibria and
the (macroscopic) equilibria associated with the Fokker-Planck diffusion
limit.

Reviewing extensively the literature on the asymptotic behaviour of the
solutions of Fokker-Planck, degenerate Fokker-Planck and kinetic Fokker-Planck
equations goes beyond our scope. Let us simply quote some papers directly
related to our methods, with more details to be given later.

Concerning _Fokker-Planck_ equations, coercivity for the diffusion operator
means spectral gap and Poincaré inequality, and thus exponential decay of the
solutions. This is a basic example of application of the carré du champ
method: see MR3155209 . _Weak Poincaré_ inequalities are natural in the
absence of spectral gaps as explained in MR4056271  and have been quite
systematically explored: see MR1856277 ; MR2381160 ; MR4265692  and earlier
references therein. However, such methods require strong assumptions on the
initial data. This is the reason why we adopt an alternative approach based on
moments and weighted functional inequalities, where the extreme case
corresponds to Nash’s inequality in absence of an external potential. See
Table 1.

As for _kinetic Fokker-Planck_ equations, hypocoercivity primarily refers to
the method exposed in Mem-villani . We can also quote piazzoli2019relaxation
for a detailed presentation of the commutator method and of Bakry-Emery type
computations applied to estimates of the relaxation to equilibrium. More
results and further references can also be found in
doi:10.1142/S0218202518500574 . In MR3892316  S. Hu and X. Wang introduced a
weak hypocoercivity approach _à la_ Villani, using a weak Poincaré inequality,
and proved subexponential convergence to equilibrium. This was later extended
to a class of degenerate diffusion processes in MR4021241  by M. Grothaus and
F.-Y. Wang using weak Poincaré inequalities for the symmetric and
antisymmetric part of the generator, with non-exponential rates of
convergence. In the same vein, C. Cao proved quantitative convergence rates
for the kinetic Fokker-Planck equation with more general confinement forces in
MR4069622 ; MR4278431 .

Alternatively the method of DMS  was extended to cases without potentials in
BDMMS  while cases of weak or very weak potentials were considered
respectively in BDLS  and BDS-very-weak . Here the idea is to introduce
moments and weighted interpolation inequalities to prove non-exponential decay
or convergence rates. In these papers the effort has been mostly focused on
the role of the external potential and fat tail local equilibria were not
taken into consideration. However, it is known from mellet_fractional_2011
that fat-tail local equilibria can be responsible of a fractional diffusion
limit, which may govern decay rates in the case without external potential as
shown in Bouin_2022 , but this is not always the case. Non-Maxwellian local
equilibria have been less explored than the Maxwellian case, but one has to
mention Brigati2023KRM ; brigati2023construct  for such an extension of the
earlier works albritton2021variational ; cao2023explicit , with slightly
different methods based on weak norms, Lions’ lemma and time-averages.

##  2 From microscopic and macroscopic coercivity to hypocoercivity

Let us start by an expository section which collects some known results and
introduces our present purpose. Let us consider the general evolution equation

$\frac{dF}{dt}+\mathsf{T}F=\mathsf{L}F$ (1)

where $F$ is the density of a probability distribution defined on a real or
complex Hilbert space $\mathscr{H}$ with scalar product
$\langle\cdot,\cdot\rangle$ and norm $\|\cdot\|$. We assume that $\mathsf{T}$
and $\mathsf{L}$ are two linear operators, respectively anti-Hermitian and
Hermitian: $\mathsf{T}^{*}=-\,\mathsf{T}$ and
$\mathsf{\mathrm{missing}}L^{*}=\mathsf{L}$, where ∗ denotes the adjoint with
respect to $\langle\cdot,\cdot\rangle$. We are interested in the decay rate of
$F$ or in the convergence to a steady state $F_{\star}$. We assume that
$F_{\star}$ is unique, up to normalization. Since (1) is linear, we can always
replace $F$ by $F-F_{\star}$ and study the convergence to $0$ of an eventually
sign changing function $F$. We have in mind that $\mathsf{L}$ is an elliptic
degenerate operator. If $\Pi$ is the orthogonal projection onto the kernel of
$\mathsf{L}$, we assume that $\mathsf{L}$ has the _microscopic coercivity_
property in the sense that it is _coercive_ on $(1-\Pi)\,\mathscr{H}$, where
$1$ is here a shorthand notation for the identity that will make sense in the
functional setting of interest. In other words, we claim that

$\frac{1}{2}\,\frac{d}{dt}\,\|F\|^{2}=\langle\mathsf{L}F,F\rangle\leq-\,\lambda_{m}\,\|(1-\Pi)F\|^{2}$
(H1)

for some $\lambda_{m}>0$. This is not enough to conclude that $\|F(t,\cdot)\|$
decays exponentially as we have no decay rate on $\mathrm{Ker}(\mathsf{L})$,
but if the operators $\mathsf{L}$ and $\mathsf{T}$ do not commute, we can hope
that some of the decay properties on $(1-\Pi)\,\mathscr{H}$ are transferred on
$\Pi\,\mathscr{H}$. This points towards the computation of various commutators
and the whole machinery of Hörmander’s hypoellipticity theory. A micro/macro
approach offers a simpler framework, that has the advantage of clarifying the
role played by various functional inequalities in estimating decay rates of
$F$. The underlying ideas rely on the formal _macroscopic limit_ of the scaled
evolution equation

$\varepsilon\,\frac{dF}{dt}+\mathsf{T}F=\frac{1}{\varepsilon}\,\mathsf{L}F$

on the Hilbert space $\mathscr{H}$, which is a typical parabolic scaling when
$\varepsilon$ is a small parameter. Using a formal expansion of a solution
$F_{\varepsilon}=F_{0}+\varepsilon\,F_{1}+\varepsilon^{2}\,F_{2}+{\mathcal{O}(\varepsilon^{3})}$
as $\varepsilon\to 0_{+}$ and solving the equation order by order, we obtain

$\begin{array}[]{ll}\mbox{at
order}\,O\big{(}\varepsilon^{-1}\big{)}:&\quad\mathsf{L}F_{0}=0\,,\\\\[4.0pt]
\mbox{at
order}\,O\big{(}\varepsilon^{0}\big{)}:&\quad\mathsf{T}F_{0}=\mathsf{L}F_{1}\,,\\\\[4.0pt]
\mbox{at
order}\,O\big{(}\varepsilon^{1}\big{)}:&\quad\frac{dF_{0}}{dt}+\mathsf{T}F_{1}=\mathsf{L}F_{2}\,.\end{array}$

The first and second equation respectively read as $F_{0}=\Pi F_{0}$ and
$F_{1}=-\,(\mathsf{T}\Pi)\,F_{0}$. After projection on
$\mathrm{Ker}(\mathsf{L})$, the third equation becomes $\frac{d}{dt}\left(\Pi
F_{0}\right)-\,\Pi\mathsf{T}\,(\mathsf{T}\Pi)\,F_{0}=\Pi\mathsf{L}F_{2}=0$
that we can also write as

$\frac{\partial F_{0}}{\partial
t}+(\mathsf{T}\Pi)^{*}\,(\mathsf{T}\Pi)\,F_{0}=0\,.$ (2)

Assuming _macroscopic coercivity_ , _i.e._ , the property that the operator
$(\mathsf{T}\Pi)^{*}\,(\mathsf{T}\Pi)$ is coercive on $(1-\Pi)\,\mathscr{H}$,
we obtain

$\frac{1}{2}\,\frac{d}{dt}\,\|F_{0}\|^{2}=-\,\|(\mathsf{T}\Pi)\,F_{0}\|^{2}\leq-\,\lambda_{M}\,\|F_{0}\|^{2}$
(H2)

for some $\lambda_{M}>0$. In order to derive (2), we implicitly used the fact
that all terms are of order $\varepsilon$, which relies on the _parabolic
macroscopic dynamics_ condition

$\Pi\mathsf{T}\Pi\,F=0\,.$ (H3)

As in the _hypocoercivity_ method of DMS , let us consider the operator

$\mathsf{A}:=\big{(}1+(\mathsf{T}\Pi)^{*}(\mathsf{T}\Pi)\big{)}^{-1}(\mathsf{T}\Pi)^{*}$

where the $(\mathsf{T}\Pi)^{*}(\mathsf{T}\Pi)$ term is of course reminiscent
of (2), and, the Lyapunov functional, or _entropy_ ,

$\mathsf{H}[F]:=\frac{1}{2}\,\|F\|^{2}+\delta\,\mathrm{Re}\langle\mathsf{A}F,F\rangle\,.$
(3)

The parameter $\delta>0$, to be determined, as to be thought as a small
parameter so that $\mathsf{H}[F]$ is a perturbation of
$\frac{1}{2}\,\|F\|^{2}$. The following estimate is by now classical but
deserves some emphasis. Let us consider $G=\mathsf{A}F$, _i.e._ , the solution
of $(\mathsf{T}\Pi)^{*}F=G+(\mathsf{T}\Pi)^{*}\,\mathsf{T}\Pi\,G$. As in (DMS,
, Lemma 1), by a Cauchy-Schwarz inequality, we learn that

$\langle\mathsf{TA}F,F\rangle=\langle
G,(\mathsf{T}\Pi)^{*}\,F\rangle=\|G\|^{2}+\|\mathsf{T}\Pi
G\|^{2}=\|\mathsf{A}F\|^{2}+\|\mathsf{TA}F\|^{2}\\\
\leq\|\mathsf{TA}F\|\,\|(\mathrm{Id}-\Pi)F\|\leq\frac{1}{2\,\mu}\,\|\mathsf{TA}F\|^{2}+\frac{\mu}{2}\,\|(\mathrm{Id}-\Pi)F\|^{2}\,.$

Applied with either $\mu=1/2$ or $\mu=1$, this estimate proves that
$\|\mathsf{A}F\|\leq\frac{1}{2}\,\|(\mathrm{Id}-\Pi)F\|$ and
$\|\mathsf{TA}F\|\leq\|(\mathrm{Id}-\Pi)F\|$. Incidentally, this proves that

$\displaystyle\left|\langle\mathsf{TA}F,F\rangle\right|=\left|\langle\mathsf{TA}F,(\mathrm{Id}-\Pi)F\rangle\right|\leq\|(\mathrm{Id}-\Pi)F\|^{2}\,,$
(4a) $\displaystyle|\langle\mathsf{A}F,F\rangle|\leq\frac{1}{2}\,\|\Pi
F\|\,\|(\mathrm{Id}-\Pi)F\|\leq\frac{1}{4}\,\|F\|^{2}\,.$ (4b)

We read from (4b) that $\mathsf{H}[F]$ and $\|F\|^{2}$ are equivalent with

$\frac{2-\delta}{4}\,\|F\|^{2}\leq\mathsf{H}[F]\leq\frac{2+\delta}{4}\,\|F\|^{2}\,.$

However, the twist introduced in $\mathsf{H}[F]$ by
$\langle\mathsf{A}F,F\rangle$ makes it exponentially decaying in $t$ if $F$
solves (1). We can indeed compute

$-\,\frac{d}{dt}\mathsf{H}[F]=\mathsf{D}[F]$

where

$\mathsf{D}[F]:=-\,\langle\mathsf{L}F,F\rangle+\delta\,\langle\mathsf{AT}\Pi
F,F\rangle-\,\delta\,\Big{(}\mathrm{Re}\langle\mathsf{TA}F,F\rangle-\mathrm{Re}\langle\mathsf{AT}(1-\Pi)F,F\rangle+\mathrm{Re}\langle\mathsf{AL}F,F\rangle\Big{)}\,.$
(5)

By (H1), we know that
$-\,\langle\mathsf{L}F,F\rangle\geq\lambda_{m}\,\|(1-\Pi)F\|^{2}$. On the
other hand, (H2) amounts to

$\big{\langle}(\mathsf{T}\Pi)^{*}\,(\mathsf{T}\Pi)\,F,F\big{\rangle}\geq\lambda_{M}\,\|F\|^{2}\quad\mbox{if}\quad
F\in\mathrm{Ker}(\mathsf{L})$

and, by construction, the operator $\mathsf{A}$ is therefore such that

$\langle\mathsf{AT}\Pi F,F\rangle\geq\frac{\lambda_{M}}{1+\lambda_{M}}\,\|\Pi
F\|^{2}\,.$ (6)

The first two terms in the definition of $\mathsf{D}[F]$ can be combined to
prove that

$\mathsf{D}[F]\geq\lambda_{m}\,\|(1-\Pi)F\|^{2}+\frac{\delta\,\lambda_{M}}{1+\lambda_{M}}\,\|\Pi
F\|^{2}-\,\delta\,\Big{(}\mathrm{Re}\langle\mathsf{TA}F,F\rangle-\mathrm{Re}\langle\mathsf{AT}(1-\Pi)F,F\rangle+\mathrm{Re}\langle\mathsf{AL}F,F\rangle\Big{)}\,.$
(7)

Under the additional assumption that the last term in the above identity
involves only _bounded auxiliary operators_ in the sense that

$\|\mathsf{AT}(1-\Pi)F\|+\|\mathsf{AL}F\|\leq C_{M}\,\|(1-\Pi)F\|\,,$ (H4)

one obtains the _entropy – entropy production_ inequality

$\mathsf{D}[F]\geq\lambda\,\mathsf{H}[F]$

for some explicit constant $\lambda>0$. The precise statement goes as follows.
It has been established in DMS  in the case of a real Hilbert space
$\mathscr{H}$ and extended to complex Hilbert spaces in BDMMS .

######  Theorem 2.1 (BDMMS ; DMS )

Let $\mathsf{L}$ and $\mathsf{T}$ be closed linear operators in the complex
Hilbert space $\big{(}\mathcal{H},\langle\cdot,\cdot\rangle\big{)}$. We assume
that $\mathsf{L}$ is Hermitian and $\mathsf{T}$ is anti-Hermitian, and that
(H1)–(H4) hold for some positive constants $\lambda_{m}$, $\lambda_{M}$, and
$C_{M}$. Then there is some $\delta_{\star}\in(0,2)\cap(0,\lambda_{m})$ such
that, for any $\delta\in(0,\delta_{\star})$, there are explicit constants
$\lambda>0$ and $\mathscr{C}>1$ for which, if $F$ solves (1) with initial
datum $F_{0}\in\mathcal{H}$, then

$\mathsf{H}[F(t,\cdot)]\leq\mathsf{H}[F_{0}]\,e^{-\lambda\,t}\quad\mbox{and}\quad\|F(t,\cdot)\|^{2}\leq\mathscr{C}\,e^{-\lambda\,t}\,\|F_{0}\|^{2}\quad\forall\,t\geq
0\,.$ (8)

The estimates of $\lambda>0$ and $\mathscr{C}>1$ in (BDMMS, , Proposition 4)
have been improved in (Arnold2021, , Proposition 2) as follows. With
$X:=\|(\mathrm{Id}-\Pi)F\|$ and $Y:=\|\Pi F\|$. Using (4b), we read from (3)
that

$\mathsf{H}[F]\leq\frac{1}{2}\left(X^{2}+Y^{2}\right)+\frac{\delta}{2}\,X\,Y$

while it follows from (7), (4a) and (H4) that

$\mathsf{D}[F]-\lambda\,\mathsf{H}[F]\geq\left(\lambda_{m}-\,\delta-\frac{\lambda}{2}\right)X^{2}-\,\delta\left(C_{M}+\frac{\lambda}{2}\right)X\,Y+\left(\frac{\delta\,\lambda_{M}}{1+\lambda_{M}}-\frac{\lambda}{2}\right)Y^{2}\,.$

With $K_{M}:=\frac{\lambda_{M}}{1+\lambda_{M}}<1$ and
$\delta_{\star}:=\frac{4\,K_{M}\,\lambda_{m}}{4\,K_{M}+C_{M}^{2}}<\lambda_{m}$,
a simple discriminant condition shows that for any
$\delta\in(0,\delta_{\star})$, the right-hand side is nonnegative for the
largest (positive) solution of

$\delta^{2}\left(C_{M}+\frac{\lambda}{2}\right)^{2}-4\left(\lambda_{m}-\,\delta-\frac{\lambda}{2}\right)\left(\frac{\delta\,\lambda_{M}}{1+\lambda_{M}}-\frac{\lambda}{2}\right)=0\,.$

We refer to Arnold2021  for further details and to MR3522857  for more
considerations on the functional framework.

In the framework of kinetic equations, $\mathsf{T}$ and $\mathsf{L}$ are
respectively the _transport operator_ and the _collision operator_ acting on a
distribution function $f(t,x,v)$ where $t\geq 0$ is the time, $x$ is the
position and $v$ is the velocity. To fix ideas, we shall assume that $x$,
$v\in{\mathbb{R}}^{d}$ and consider

  * •

a transport operator defined by the Poisson bracket as

$\mathsf{T}f:=\nabla_{v}\mathscr{E}\cdot\nabla_{x}f-\nabla_{x}\mathscr{E}\cdot\nabla_{v}f$
(9)

corresponding to the Hamiltonian energy

$(x,v)\mapsto\mathscr{E}(x,v):=\frac{1}{\beta}\,\langle
v\rangle^{\beta}+\phi(x)\,,$

where $\phi$ denotes an external, given potential,

  * •

a collision operator of Fokker-Planck type given by

$\mathsf{L}f:=\nabla_{v}\cdot\left(\nabla_{v}f+v\,\langle
v\rangle^{\beta-2}\,f\right)\,.$ (10)

Here we use the notation

$\langle v\rangle:=\sqrt{1+|v|^{2}}\,.$

Unless $\beta=2$, our choice of the transport operator differs from the
transport operator corresponding to Newton’s equations, namely
$v\cdot\nabla_{x}-\nabla_{x}\phi\cdot\nabla_{v}$, which has been widely
studied in the literature. More general dependences of $\mathcal{E}$ and
$\mathsf{L}$ on $v$ than $\langle v\rangle^{\beta}$, with for instance a power
law asymptotic growth as $|v|\to+\infty$, could be considered under minor
changes. Our purpose is to study the asymptotic behaviour of the solution of

$\frac{\partial f}{\partial t}+\mathsf{T}f=\mathsf{L}f\,,\quad
f(t=0,\cdot,\cdot)=f_{0}$ (11)

with $\mathsf{T}$ and $\mathsf{L}$ given respectively by (9) and (10) as
$t\to+\infty$. With these choices and under the condition that $e^{-\phi}$ is
integrable, a remarkable property is that the _Gibbs state_

$f_{\star}(x,v):=\frac{1}{Z}\,e^{-\mathscr{E}(x,v)}\quad\mbox{where}\quad
Z=\int_{{\mathbb{R}}^{d}}{e^{-\phi}}\,dx\int_{{\mathbb{R}}^{d}}{e^{-\frac{1}{\beta}\,\langle
v\rangle^{\beta}}}\,dv$ (12)

is a stationary solution of mass
$\|f_{\star}\|_{\mathrm{L}^{1}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,dx\,dv)}=1$.
We consider $\mathsf{L}$ as an operator on
$\mathrm{L}^{2}\big{(}{\mathbb{R}}^{d},e^{\langle v\rangle^{\beta}}dv\big{)}$
acting on functions depending on the velocity variable $v$ and extend it to
the Hilbert space
$\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},d\mu)$ of functions
depending on $x$ and $v$ where

$d\mu:=\frac{dx\,dv}{f_{\star}(x,v)}\,.$

Since $f_{\star}$ is integrable, notice that
$\mathrm{L}^{1}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},dx\,dv)\subset\mathrm{L}^{2}\big{(}{\mathbb{R}}^{d}\times{\mathbb{R}}^{d},d\mu)$
by a Cauchy-Schwarz inequality. After these preliminaries, we observe that
$f_{\star}$ is local equilibrium, _i.e._ , belongs to
$\mathrm{Ker}(\mathsf{L})$ which is generated by functions of the type

$f_{\rho}(x,v):=\frac{\rho(x)\,e^{-\frac{1}{\beta}\,\langle
v\rangle^{\beta}}}{\int_{{\mathbb{R}}^{d}}{e^{-\frac{1}{\beta}\,\langle
v\rangle^{\beta}}}\,dv}$ (13)

where
$\rho\in\mathrm{L}^{2}\big{(}{\mathbb{R}}^{d},e^{-\phi}\,dx\big{)}\supset\mathrm{L}^{1}({\mathbb{R}}^{d},dx)$
is an arbitrary function. The property
$\mathsf{T}f_{\star}=\mathsf{L}f_{\star}=0$ sometimes appears in the physics
literature as a _factorization_ property. The orthogonal projector onto
$\mathrm{Ker}(\mathsf{L})$ is defined as the projection on local equilibria by

$\Pi
f=f_{\rho}(x,v)\quad\mbox{where}\quad\rho=\int_{{\mathbb{R}}^{d}}{f}\,dv\,.$

Notice that $f$ and $f_{\rho}$ have the same spatial density because
$\int_{{\mathbb{R}}^{d}}{f_{\rho}}\,dv=\rho$. Under the assumption that the
measure $e^{-\phi}\,dx$ admits a Poincaré inequality, that is, there is some
positive constant $\lambda_{\phi}$ for which

$\int_{{\mathbb{R}}^{d}}{|\nabla
u|^{2}\,e^{-\phi}}\,dx\geq\lambda_{\phi}\int_{{\mathbb{R}}^{d}}{|u|^{2}\,e^{-\phi}}\,dx\quad\forall\,u\in\mathscr{D}({\mathbb{R}}^{d})\quad\mbox{such
that}\quad\int_{{\mathbb{R}}^{d}}{u\,e^{-\phi}}\,dx=0\,,$ (14)

Theorem 2.1 applies as follows.

###### Corollary 1

Assume that $\phi$ is such that (14) holds for some $\lambda_{1}>0$ and
$\beta\geq 1$. If $f$ solves (11) for some nonnegative function
$f_{0}\in\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},d\mu)$ with
$\|f_{0}\|_{\mathrm{L}^{1}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,dx\,dv)}=1$,
then for some $\delta>0$, there exists $\lambda>0$ and $\mathscr{C}>1$ such
that (8) holds with
$\|\cdot\|:=\|\cdot\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}$.

The case $\beta=2$ is by now standard and covered in various papers: see
Dolbeault2009511 ; DMS  for an $\mathrm{L}^{2}$ hypocoercivity approach and
MR2034753 ; MR2130405 ; MR2215889  as well as references therein for earlier
results based on hypoelliptic methods. To our knowledge the case $\beta\neq 2$
has not been studied yet by $\mathrm{L}^{2}$-hypocoercivity methods, but
convergence results are known from cao2023explicit ; brigati2023construct
using other methods. Of particular interest in physics is the case $\beta=1$
where $\mathscr{E}$ is the standard energy for relativistic particles, up to
physical constants (mass and speed of light are taken equal to $1$), while the
corresponding $\mathsf{L}$ operator is not much more than a caricature of a
relativistic collision operator. Concerning $\mathsf{L}$ and from the point of
view of phenomenological models, it is however interesting to consider local
equilibria given by (13) and it makes sense to assume that stationary
solutions have the _factorization_ property. Throughout this paper, we will
make this simplifying assumption.

The strategy of the proof of Corollary 1 goes as follows. With $\beta\geq 1$,
we have the Poincaré inequality: there is some $\lambda_{m}>0$ such that, for
all $g\in\mathrm{L}^{2}({\mathbb{R}}^{d},e^{-\frac{1}{\beta}\,\langle
v\rangle^{\beta}}\,dv)$ such that
$\int_{{\mathbb{R}}^{d}}{g\,e^{-\frac{1}{\beta}\,\langle
v\rangle^{\beta}}}\,dv=0$, we have

$\int_{{\mathbb{R}}^{d}}{|\nabla g|^{2}\,e^{-\frac{1}{\beta}\,\langle
v\rangle^{\beta}}}\,dv\geq\lambda_{m}\int_{{\mathbb{R}}^{d}}{|g|^{2}\,e^{-\frac{1}{\beta}\,\langle
v\rangle^{\beta}}}\,dv\,.$ (15)

This is for instance a consequence of Persson’s lemma based on the observation
that $\psi(v):=\frac{1}{\beta}\,\langle v\rangle^{\beta}$ is such that

$\liminf_{|v|\to+\infty}\left(\frac{1}{4}\,|\nabla\psi(v)|^{2}-\frac{1}{2}\,\Delta\psi(v)\right)>0\,.$

See for instance (Bouin_2020, , Appendix A.1) for details. As a consequence,
(H1) holds. The macroscopic coercivity condition (H2) follows from (14). The
_parabolic macroscopic dynamics_ condition (H3) is a simple consequence of the
definitions of $\mathsf{T}$ and $\Pi$. Hence the only assumption that deserves
some attention is (H4), which is obtained by elliptic estimates. A detailed
proof is given in Section 4.6.

In the framework of (9) and (10), an elementary computation shows that (2)
written for $f_{\rho}$ defined by (13) reduces to the _Fokker-Planck_ equation

$\frac{\partial\rho}{\partial
t}=\sigma\,\big{(}\Delta\rho+\nabla\cdot(\rho\,\nabla\phi)\big{)}$ (16)

with diffusion coefficient $\sigma$ given in terms of $\beta$ by

$\sigma=\frac{1}{d}\int_{{\mathbb{R}}^{d}}{|v|^{2}\,\langle
v\rangle^{2\beta-4}\,e^{-\frac{1}{\beta}\,\langle v\rangle^{\beta}}}\,dv\,.$
(17)

In order to simplify the discussion, we shall assume that

$\phi(x)=\frac{1}{\alpha}\,\langle
x\rangle^{\alpha}\quad\forall\,x\in{\mathbb{R}}^{d}\,.$

Corollary 1 corresponds to $\alpha\geq 1$. Our purpose is to investigate the
decay rates of (11) in terms of $\beta>0$ and $\alpha>0$. Let us start by
studying the asymptotic behaviour of a solution $\rho$ of (16) depending on
the various cases for the potential $\phi$. For completeness, we will also
consider the limit case as $\alpha\to 0$ and distinguish several cases
depending on whether we take $\phi=0$ a.e., or (in the case of the Fokker-
Planck equation), depending on $\gamma>0$,

$\phi(x)=\gamma\,\log\langle x\rangle\quad\forall\,x\in{\mathbb{R}}^{d}\,.$

Up to minor technicalities, general potentials $\phi$ with asymptotic power
law or logarithmic growths as $|x|\to+\infty$ could also be covered.

##  3 Fokker-Planck equations with various external potentials, moments and
functional inequalities

We collect some results on the asymptotic behaviour of the solutions of (16)
as $t\to+\infty$ based on various functional inequalities. In this section we
omit the discussion of optimality cases and estimates on sharp constants in
the functional inequalities. By default, constants in the inequalities are
always taken to their optimal value. Table 1 collects the results in a
synthetic picture, although without all details on the assumptions.

###  3.1 Strong confinement case: Poincaré inequality

If $\phi(x)=\frac{1}{\alpha}\,\langle x\rangle^{\alpha}$ with $\alpha\geq 1$,
then (14) holds with $\lambda_{\phi}=\lambda_{M}>0$. We apply it to
$u=\rho/e^{-\phi}$. A solution $\rho$ of (16) with initial datum $\rho_{0}$ at
$t=0$ satisfies

$\frac{d}{dt}\|\rho(t,\cdot)\|_{\mathrm{L}^{2}({\mathbb{R}}^{d},\,e^{\phi}\,dx)}^{2}=-\,2\,\sigma\,\|\nabla\rho(t,\cdot)\|_{\mathrm{L}^{2}({\mathbb{R}}^{d},\,e^{\phi}\,dx)}^{2}\leq-\,2\,\lambda_{M}\,\sigma\,\|\rho(t,\cdot)\|_{\mathrm{L}^{2}({\mathbb{R}}^{d},\,e^{\phi}\,dx)}^{2}\,,$

which yields the estimate

$\|\rho(t,\cdot)\|_{\mathrm{L}^{2}({\mathbb{R}}^{d},\,e^{\phi}\,dx)}^{2}\leq\|\rho_{0}\|_{\mathrm{L}^{2}({\mathbb{R}}^{d},\,e^{\phi}\,dx)}^{2}\,e^{-2\,\lambda_{M}\,\sigma\,t}\quad\forall\,t\geq
0\,.$

###  3.2 Weak confinement case: weighted Poincaré inequality

The following results are taken from (BDLS, , Appendices A and B). We assume
here that $\alpha\in(0,1)$ and consider a solution of (16) with nonnegative
initial datum $\rho_{0}\in\mathrm{L}^{1}({\mathbb{R}}^{d},dx)$ such that
$\left\|{\rho_{0}}\right\|_{\mathrm{L}^{1}({\mathbb{R}}^{d})}=1$. The function
$u=\rho\,e^{\phi}$ is a solution of the _Ornstein-Uhlenbeck_ equation (also
known as the _backward Kolmogorov_ equation)

$\frac{\partial u}{\partial t}=\sigma\,\big{(}\Delta u-\nabla\phi\cdot\nabla
u\big{)}\,.$ (18)

With $k\geq 0$, let us compute

$\frac{d}{dt}\int_{{\mathbb{R}}^{d}}{|u(t,x)|^{2}\,\langle
x\rangle^{k}\,e^{-\phi}}\,dx+2\,\sigma\int_{{\mathbb{R}}^{d}}{|\nabla_{x}u(t,x)|^{2}\,\langle
x\rangle^{k}\,e^{-\phi}}\,dx\leq\int_{{\mathbb{R}}^{d}}{\big{(}a_{k}-b_{k}\,\langle
x\rangle^{\alpha-2}\big{)}\,|u(t,x)|^{2}\,\langle
x\rangle^{k}\,e^{-\phi}}\,dx$

for some $a_{k}\in{\mathbb{R}}$, $b_{k}\in(0,+\infty)$. As a consequence,
there exists a constant $\mathcal{K}(k)>0$ such that

$\int_{{\mathbb{R}}^{d}}{\langle
x\rangle^{k}\,|\rho(t,x)|^{2}\,e^{\phi}}\,dx\leq\mathcal{K}(k)\int_{{\mathbb{R}}^{d}}{\langle
x\rangle^{k}\,|\rho_{0}|^{2}\,e^{\phi}}\,dx\quad\forall\,t\geq 0\,.$

See (BDLS, , Proposition 4 and Appendix B.2) for details. With $k=0$, we
notice that $a_{0}=b_{0}=0$ and use the _weighted Poincaré inequality_

$\int_{{\mathbb{R}}^{d}}{|\nabla_{x}u(t,x)|^{2}\,e^{-\phi}}\,dx\geq\mathscr{C}_{\alpha}^{\mathrm{wP}}\int_{{\mathbb{R}}^{d}}{|u(t,x)-\bar{u}|^{2}\,\frac{e^{-\phi}}{\langle
x\rangle^{2\,(1-\alpha)}}}\,dx\quad\mbox{where}\quad\bar{u}=\frac{\int_{{\mathbb{R}}^{d}}{u\,e^{-\phi}}\,dx}{\int_{{\mathbb{R}}^{d}}{e^{-\phi}}\,dx}$
(19)

(notice that the average $\bar{u}$ is computed with respect to the measure of
the l.h.s.) to prove that

$\frac{d}{dt}\int_{{\mathbb{R}}^{d}}{|u(t,x)-\bar{u}|^{2}\,e^{-\phi}}\,dx=-\,2\,\sigma\int_{{\mathbb{R}}^{d}}{|\nabla_{x}u(t,x)|^{2}\,e^{-\phi}}\,dx\leq-\,2\,\sigma\,\mathscr{C}_{\alpha}^{\mathrm{wP}}\int_{{\mathbb{R}}^{d}}{|u(t,x)-\bar{u}|^{2}\,\frac{e^{-\phi}}{\langle
x\rangle^{2\,(1-\alpha)}}}\,dx\,.$

With $k\geq 2\,(1-\alpha)$ and $\theta=k/\big{(}k+2\,(1-\alpha)\big{)}$,
Hölder’s inequality

$\int_{{\mathbb{R}}^{d}}{|u-\bar{u}|^{2}\,e^{-\phi}}\,dx\leq\left(\int_{{\mathbb{R}}^{d}}{|u-\bar{u}|^{2}\,\frac{e^{-\phi}}{\langle
x\rangle^{2\,(1-\alpha)}}}\,dx\right)^{\theta}\left(\int_{{\mathbb{R}}^{d}}{|u-\bar{u}|^{2}\,\langle
x\rangle^{k}\,e^{-\phi}}\,dx\right)^{1-\theta}$

allows us to prove that

$\int_{{\mathbb{R}}^{d}}{|\rho(t,x)-\rho_{\star}(x)|^{2}\,e^{\phi}}\,dx\leq\left(\|\rho_{0}-\rho_{\star}\|_{\mathrm{L}^{2}({\mathbb{R}}^{d},\,e^{\phi}\,dx)}^{-\,4\,(1-\alpha)/k}+\frac{4\,(1-\alpha)\,\sigma\,\mathscr{C}_{\alpha}^{\mathrm{wP}}}{k\,\mathcal{K}_{*}^{\,2\,(1-\alpha)/k}}\,t\right)^{-\frac{k}{2\,(1-\alpha)}}\quad\forall\,t\geq
0$

with $\mathcal{K}_{*}:=\mathcal{K}(k)^{2}\,\int_{{\mathbb{R}}^{d}}{\langle
x\rangle^{k}\,|\rho_{0}|^{2}\,e^{\phi}}\,dx+\int_{{\mathbb{R}}^{d}}{\langle
x\rangle^{k}\,e^{-\phi}}\,dx\left\|{\rho_{0}}\right\|_{\mathrm{L}^{1}({\mathbb{R}}^{d})}^{2}$.

###  3.3 Weak confinement, a limit case: Hardy-Poincaré inequality

The results in this case are new. In the limit as $\alpha\to 0_{+}$, we can
assume that $\phi(x)=\gamma\,\log\langle x\rangle$ with $\gamma>d$ so that
$f_{\star}$ defined by (12) is integrable. Let $u=\rho\,e^{\phi}$ be a
solution of (18). With $k\geq 0$, let us compute

$\frac{d}{dt}\int_{{\mathbb{R}}^{d}}{|u(t,x)|^{2}\,\langle
x\rangle^{k}\,e^{-\phi}}\,dx+2\,\sigma\int_{{\mathbb{R}}^{d}}{|\nabla_{x}u(t,x)|^{2}\,\langle
x\rangle^{k}\,e^{-\phi}}\,dx\\\
=k\int_{{\mathbb{R}}^{d}}{|u(t,x)|^{2}\,\langle
x\rangle^{k-2}\left(d+(k-\gamma-2)\,\tfrac{|x|^{2}}{\langle
x\rangle^{2}}\right)e^{-\phi}}\,dx\\\ \leq
k\int_{{\mathbb{R}}^{d}}{|u(t,x)|^{2}\,\langle
x\rangle^{k-2}\left(d+k-\gamma-2-(k-\gamma-2)\,\langle
x\rangle^{-2}\right)e^{-\phi}}\,dx\,.$ (20)

By arguing as in (BDLS, , Proposition 4 and Appendix B.2), this is enough to
prove that there exists a constant $\mathcal{K}(k)>0$ such that

$\int_{{\mathbb{R}}^{d}}{\langle
x\rangle^{k}\,|u(t,x)|^{2}\,e^{-\phi}}\,dx\leq\mathcal{K}(k)\int_{{\mathbb{R}}^{d}}{\langle
x\rangle^{k}\,|\rho_{0}|^{2}\,e^{\phi}}\,dx\quad\forall\,t\geq 0$

if $k\in(\gamma-d,\gamma+2-d)$. Notice that a better range of $k$ can be
obtained as follows. Since $\langle x\rangle^{k}\,e^{-\phi}=\langle
x\rangle^{k-\gamma}$, we learn from MR2609957 ; MR2823993  that for some
positive constant $\mathscr{C}^{\mathrm{HP}}_{\gamma-k}$, we have the _Hardy-
Poincaré_ inequality

$\int_{{\mathbb{R}}^{d}}{|\nabla_{x}u|^{2}\,\langle
x\rangle^{k}\,e^{-\phi}}\,dx\geq\mathscr{C}^{\mathrm{HP}}_{\gamma-k}\int_{{\mathbb{R}}^{d}}{|u-\bar{u}|^{2}\,\langle
x\rangle^{k-2}\,e^{-\phi}}\,dx$ (21)

for an appropriate choice of $\bar{u}$ depending on $k-\gamma$. In any case,
Inequality (20) written with $k=0$, that is,

$\frac{d}{dt}\int_{{\mathbb{R}}^{d}}{|u(t,x)-\bar{u}|^{2}\,e^{-\phi}}\,dx\leq-\,2\,\sigma\int_{{\mathbb{R}}^{d}}{|\nabla_{x}u(t,x)|^{2}\,e^{-\phi}}\,dx$

and then (21) combined with Hölder’s inequality applied as in the case of the
weighted Poincaré inequality (with $\alpha=0$) show that

$\int_{{\mathbb{R}}^{d}}{|\rho(t,x)-\rho_{\star}(x)|^{2}\,e^{\phi}}\,dx\leq\int_{{\mathbb{R}}^{d}}{|\rho_{0}-\rho_{\star}|^{2}\,e^{\phi}}\,dx\,(1+c\,t)^{-\frac{k}{2}}\quad\forall\,t\geq
0$

for some constant $c$ which depends on $d$, $\gamma$, $\sigma$, $k$,
$\int_{{\mathbb{R}}^{d}}{|\rho_{0}|^{2}\,\langle x\rangle^{k-\gamma}}\,dx$ and
$\left\|{\rho_{0}}\right\|_{\mathrm{L}^{1}({\mathbb{R}}^{d})}^{2}$.

###  3.4 Very weak confinement case: Caffarelli-Kohn-Nirenberg inequality

According to (BDS-very-weak, , Theorem 1), if $1\leq\gamma<d$ and
$\phi(x)=\gamma\,\log\langle x\rangle$, a solution $\rho$ of (16) with
nonnegative initial datum $\rho_{0}\in\mathrm{L}^{1}({\mathbb{R}}^{d},\langle
x\rangle^{k}\,dx)\cap\mathrm{L}^{2}\big{(}{\mathbb{R}}^{d},e^{\phi}\,dx\big{)}$
with $k=\max\\{2,\gamma/2\\}$ satisfies the estimate

$M_{k}(t):=\int_{{\mathbb{R}}^{d}}{\langle x\rangle^{k}\,\rho(t,x)}\,dx\leq
2^{\frac{k-2}{2}}\left(M_{0}+\left(\big{(}M_{k}(0)-M_{0}\big{)}^{2/k}+2\,\sigma\,\big{(}d+k-2-\gamma\big{)}\,M_{0}^{2/k}t\right)^{k/2}\right)\,.$

With $e^{-\phi}=\langle x\rangle^{-\gamma}$ and $u=\rho\,\langle
x\rangle^{\gamma}$, a solution $u$ of (18) satisfies the estimate

$\frac{d}{dt}\int_{{\mathbb{R}}^{d}}{|u(t,x)|^{2}\,\langle
x\rangle^{-\gamma}}\,dx=-\,2\,\sigma\int_{{\mathbb{R}}^{d}}{|\nabla
u(t,x)|^{2}\,\langle x\rangle^{-\gamma}}\,dx$

Combined with the _inhomogeneous Caffarelli-Kohn-Nirenberg inequality_

$\int_{{\mathbb{R}}^{d}}{|u|^{2}\,\langle
x\rangle^{-\gamma}}\,dx\leq\mathscr{C}_{k,\gamma}^{\mathrm{CKN}}\left(\int_{{\mathbb{R}}^{d}}{|\nabla
u|^{2}\,\langle
x\rangle^{-\gamma}}\,dx\right)^{a}\left(\int_{{\mathbb{R}}^{d}}{u\,\langle
x\rangle^{k-\gamma}}\,dx\right)^{2(1-a)}\quad\mbox{with}\quad
a=\frac{d+2k-\gamma}{d+2+2k-\gamma}\,,$

this proves the decay estimate

$\|\rho(t,\cdot)\|_{\mathrm{L}^{2}({\mathbb{R}}^{d},\,e^{\phi}\,dx)}^{2}\leq\|\rho_{0}\|_{\mathrm{L}^{2}({\mathbb{R}}^{d},\,e^{\phi}\,dx)}^{2}\,(1+c\,t)^{-\frac{d-\gamma}{2}}\quad\forall\,t\geq
0$

where the constant $c$ depends on $d$, $\gamma$, $\sigma$,
$\|\rho_{0}\|_{\mathrm{L}^{2}({\mathbb{R}}^{d},\,e^{\phi}\,dx)}$,
$M_{0}=\left\|{u_{0}}\right\|_{1}$, and
$M_{k}(0)=\left\|{|x|^{k}\,\rho_{0}}\right\|_{1}$. For more details, as well
as a proof of the Caffarelli-Kohn-Nirenberg inequality, see (BDS-very-weak, ,
Appendix B).

###  3.5 No potential case: Nash’s inequality

We assume that $\phi=0$ so that (16) is the standard heat equation. By _Nash’s
inequality_

$\left\|{u}\right\|_{\mathrm{L}^{2}({\mathbb{R}}^{d})}\leq\mathscr{C}_{\mathrm{Nash}}\,\left\|{\nabla
u}\right\|_{\mathrm{L}^{2}({\mathbb{R}}^{d})}^{\frac{d}{d+2}}\,\left\|{u}\right\|_{\mathrm{L}^{1}({\mathbb{R}}^{d})}^{\frac{2}{d+2}}\quad\forall\,u\in\mathrm{H}^{1}({\mathbb{R}}^{d},dx)\,,$

a solution $\rho$ of (16) with initial datum $\rho_{0}$ at $t=0$ satisfies

$\frac{d}{dt}\left\|{\rho(t,\cdot)}\right\|_{\mathrm{L}^{2}({\mathbb{R}}^{d})}^{2}=-\,2\,\sigma\,\left\|{\nabla\rho(t,\cdot)}\right\|_{\mathrm{L}^{2}({\mathbb{R}}^{d})}^{2}\,.$

Hence
$y(t):=\left\|{\rho(t,\cdot)}\right\|_{\mathrm{L}^{2}({\mathbb{R}}^{d})}^{2}$
solves the differential inequality
$y^{\prime}\leq-\,2\,\sigma\,\mathscr{C}_{\mathrm{Nash}}^{-1}\,\left\|{\rho_{0}}\right\|_{\mathrm{L}^{1}({\mathbb{R}}^{d})}^{-\frac{4}{d}}\,y^{1+\frac{2}{d}}$
which, after integration, yields the estimate

$\left\|{\rho(t,\cdot)}\right\|_{\mathrm{L}^{2}({\mathbb{R}}^{d})}^{2}\leq\Big{(}\left\|{\rho_{0}}\right\|_{\mathrm{L}^{2}({\mathbb{R}}^{d})}^{-4/d}+\,\tfrac{4\,\sigma}{d\,\mathscr{C}_{\mathrm{Nash}}}\,\left\|{\rho_{0}}\right\|_{\mathrm{L}^{1}({\mathbb{R}}^{d})}^{-4/d}\,t\Big{)}^{-d/2}\quad\forall\,t\geq
0\,.$

Potential |  $\phi=0$ |  $\phi(x)=\gamma\,\log\langle x\rangle$ $\gamma<d$ |  $\phi(x)=\gamma\,\log\langle x\rangle$ $\gamma>d$ |  $\phi(x)=\frac{1}{\alpha}\,\langle x\rangle^{\alpha}$ $\alpha\in(0,1)$ |  $\phi(x)=\frac{1}{\alpha}\,\langle x\rangle^{\alpha}$ $\alpha\geq 1$
---|---|---|---|---|---
Inequality | Nash |  Caffarelli-Kohn- Nirenberg |  Hardy-Poincaré |  Weighted Poincaré | Poincaré
Asymptotic behavior |  $t^{-d/2}$ decay |  $t^{-(d-\gamma)/2}$ decay |  $t^{-k/2}$ convergence |  $t^{-\frac{k}{2\,(1-\alpha)}}$ convergence |  $e^{-\lambda\,t}$ convergence
References |  Nash58  |  BDS-very-weak  |  |  BDLS  |  $(*)$
Table 1:  Short summary of the behaviours as $t\to+\infty$ of the solution of
(16) depending on the choice of $\phi$, with some references. On the left side
($\phi=0$ or $\gamma<d$), there is no global stationary solution and we study
decay rates. On the right side, we investigate the convergence rates to a
global stationary solution. Under additional or different constraints on the
initial data, other behaviours can be obtained based for instance on _weak
Poincaré inequalities_ : see (MR1856277, , Theorem 2.1), (MR2381160, , Theorem
1.4) and MR4265692 .
$(*)$ The use of the Poincaré inequality in relation with the Fokker-Planck
equation has a long history, which we cannot cover entirely here: we can for
instance refer to Nash58 , and to (MR3155209, , Chapter 4) for an overview in
the context of Markov processes.

###  3.6 A short summary

In case of the Fokker-Planck equation (16), Table 1 summarizes what is known
on decay rates based on moment estimates and interpolation inequalities. Cases
in gray will be further considered in the case of kinetic equations.

##  4 Kinetic Fokker-Planck equations and hypocoercivity results

###  4.1 State of the art

Some known results are collected in Table 2. They are exclusively concerned
with the classical transport operator

$\mathsf{T}f:=v\cdot\nabla_{x}f-\nabla_{x}\phi\cdot\nabla_{v}f\,,$

_i.e._ , coincide with our framework if $\beta=2$ (at the level of the
transport operator).

Potential |  $\phi=0$ |  $\phi(x)=\frac{1}{\alpha}\,\langle x\rangle^{\alpha}$ $\alpha\in(0,1)$ |  $\phi(x)=\frac{1}{\alpha}\,\langle x\rangle^{\alpha}$ $\alpha\geq 1$, or $\mathbb{T}^{d}$ Macro Poincaré
---|---|---|---
$\psi(v)=\frac{1}{\beta}\,\langle v\rangle^{\beta}$ $\beta\geq 1$ Micro Poincaré |  $t^{-d/2}$ decay BDMMS  |  $e^{-t^{b}}$, $b<1$ $\beta=2$ convergence MR4069622  |  $e^{-\lambda\,t}$ convergence MR2215889 ; Mouhot-Neumann ; Dolbeault2009511 ; DMS ; MR3488535 ; albritton2021variational ; cao2023explicit ; Brigati2023KRM ; brigati2023construct
$\psi(v)=\frac{1}{\beta}\,\langle v\rangle^{\beta}$ $\beta\in(0,1)$ |  $t^{-\zeta}$ $\zeta=\min\\{\frac{d}{2},\frac{\ell}{2(1-\beta)}\\}$ decay, BDLS  |  $t^{-\zeta}$ convergence BDZ-2  |  $t^{-\zeta}$ convergence BDZ-2
Limit as $\beta\to 0_{+}$ $\psi(v)=-(d+\varepsilon)\,\log\langle v\rangle$ |  $\varepsilon\in(0,2)$ fractional dif- fusion limit, Bouin_2022  | BDZ-2  |  $t^{-\zeta}$ if $\varepsilon>2$ convergence BDZ-2
Table 2:  Rough classification of the asymptotic behaviour of the solutions of
$\partial_{t}f+v\cdot\nabla_{x}f-\nabla_{x}\phi\cdot\nabla_{v}f=f_{\star}\,\nabla_{v}\big{(}f_{\star}^{-1}\,\nabla_{v}f\big{)}$
as $t\to+\infty$ where
$f_{\star}(x,v)=Z^{-1}\,\exp\left(-\phi(x)-\psi(v)\right)$. Additional
assumptions on the initial datum $f_{0}=f(t=0,\cdot,\cdot)$ are needed: for
instance in the case $\alpha\geq 1$ and $\beta\in(0,1)$, the initial datum is
such that
$f_{0}\in\mathrm{L}^{2}\big{(}{\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\langle
v\rangle^{(1-\beta)\,\sigma}\,d\mu\big{)}$. In the case $\phi=0$ and
$\beta\in(0,1)$, we assume that
$f_{0}\in\mathrm{L}^{2}\big{(}{\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\langle
v\rangle^{\ell/2}\,d\mu\big{)}$. If $\alpha\in(0,1)$ and $\beta\in(0,1)$,
using the weak Poincaré inequality requires specific bounds. Further cases and
more detailed assumptions can be found in the references collected above.

In Table 2, if $\beta\geq 1$, _Micro_ Poincaré refers to a Poincaré inequality
written in the velocity variable $v$, which controls the convergence towards a
local equilibrium while, _Macro_ Poincaré refers to a Poincaré inequality
written in the position variable $x$, which controls the convergence of the
solution in the macroscopic or diffusion limit, towards a global equilibrium,
or to $0$ if there is no such equilibrium. Cases in gray will be further
considered in the case of the transport operator given by (9).

###  4.2 Notation and basic observations

From here on, we assume that $\psi(v)=\frac{1}{\beta}\,\langle
v\rangle^{\beta}$ and $\phi(x)=\frac{1}{\alpha}\,\langle x\rangle^{\alpha}$
for some $\beta>0$ and $\alpha>0$, use the notation

$\rho_{\star}:=\frac{e^{-\phi}}{\int_{{\mathbb{R}}^{d}}{e^{-\phi}}\,dx}\,,\quad\rho_{f}:=\int_{{\mathbb{R}}^{d}}{f}\,dv\quad\mbox{and}\quad
u_{f}:=\frac{\rho_{f}}{\rho_{\star}}\,,$

and consider the transport operator given by (9). We recall that $f_{\star}$
is defined by (12). The following observations can be omitted at first reading
and will be used only in Sections 4.4–4.6 for proving Theorem 4.1. We can
write

$\Pi
f=\rho_{f}\,\frac{e^{-\psi}}{\int_{{\mathbb{R}}^{d}}{e^{-\psi}}\,dv}=u_{f}\,f_{\star}\,,\quad\mathsf{T}\Pi
f=\langle
v\rangle^{\beta-2}\,(v\cdot\nabla_{x}u_{f})\,f_{\star}\quad\mbox{and}\quad\Pi\mathsf{T}f=\left(\nabla_{x}\cdot\int_{{\mathbb{R}}^{d}}{v\,\langle
v\rangle^{\beta-2}\,f}\,dv\right)\frac{f_{\star}}{\rho_{\star}}\,.$

If $f=u\,f_{\star}\in\mathrm{Ker}(\mathsf{L})$ and $\sigma$ is defined by
(17), then

$(\mathsf{T}\Pi)^{*}\,(\mathsf{T}\Pi)\,f=-\,\frac{\sigma}{\rho_{\star}}\,\nabla_{x}\cdot\big{(}\rho_{\star}\,\nabla_{x}u\big{)}\,f_{\star}=-\,\sigma\,\big{(}\Delta_{x}u-\nabla_{x}\phi\cdot\nabla_{x}u\big{)}\,f_{\star}\,,$

Solving $g=\big{(}1+(\mathsf{T}\Pi)^{*}(\mathsf{T}\Pi)\big{)}^{-1}f$ means
that $g=u\,f_{\star}$ where $u=u_{g}$ solves

$u-\sigma\,\big{(}\Delta_{x}u-\nabla_{x}\phi\cdot\nabla_{x}u\big{)}=u_{f}\,.$
(22)

In order to justify integrations by parts (see Section 4.4 below), one can
notice that $c\,f_{\star}$ with an arbitrary $c\in{\mathbb{R}}$ can be used as
a barrier function, so that we can assume that $u_{f}$ is bounded as
$|x|\to+\infty$. Standard elliptic estimates apply to the solution $u$ of (22)
and one can conclude using density arguments.

###  4.3 Main result

Our goal is to get a classification similar to the results summarized in Table
2 for $\beta\neq 2$ in the transport operator defined by (9), _i.e._ , with
$\mathsf{T}f:=\nabla_{v}\mathscr{E}\cdot\nabla_{x}f-\nabla_{x}\mathscr{E}\cdot\nabla_{v}f$.
As far as we know, this transport operator has not been studied yet in the
framework of hypocoercivity methods, except for some recent results in the
bounded domain case in albritton2021variational ; Brigati2023KRM  or when
$\alpha\geq 1$ in cao2023explicit ; brigati2023construct  which are based on
weak norms and Lions’ lemma.

###### Theorem 4.1

Let $f=f(t,x,v)$ be a solution of (11) with transport and collision operators
given respectively by (9) and (10) for some $\beta>0$ and $\alpha>0$. With
$f_{\star}$ defined by (12), we assume that the initial datum satisfies

$0\leq f_{0}\leq C\,f_{\star}$ (23)

for a suitable constant $C>0$. Depending on $\beta$ and $\alpha$, we have the
following convergence and decay estimates.

  1. 1.

Assume $\beta\geq 1$ and $\alpha\geq 1$. Then there exist constants
$\mathscr{C}>0$ and $\lambda>0$ such that any solution $f$ of (11) with
initial datum
$f_{0}\in\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},d\mu)$
satisfies

$\|f-f_{\star}\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2}\leq\mathscr{C}e^{-\lambda
t}\,\|f_{0}-f_{\star}\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2}\quad\forall\,t\geq
0\,.$

  2. 2.

Assume $\beta\in(0,1)$ and $\alpha\geq 1$. Then there exists a constant
$\mathscr{C}_{\ell}>0$ such that any solution $f$ of (11), with initial datum
$f_{0}\in\mathrm{L}^{2}\big{(}{\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\langle
v\rangle^{\ell}\,d\mu\big{)}$ for some $\ell>0$, satisfies

$\|f-f_{\star}\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2}\leq\mathscr{C}_{\ell}\,(1+t)^{-\frac{\ell}{2(1-\beta)}}\,\|f_{0}-f_{\star}\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2}\quad\forall\,t\geq
0\,.$

  3. 3.

Assume $\beta\geq 1$ and $\alpha\in(0,1)$. Then there exists a constant
$\mathscr{C}_{k}>0$ such that any solution $f$ of (11), with initial datum
$f_{0}\in\mathrm{L}^{2}\big{(}{\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\langle
x\rangle^{k}\,d\mu\big{)}$ for some $k>0$, satisfies

$\|f-f_{\star}\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2}\leq\mathscr{C}_{k}\,(1+t)^{-\frac{k}{2(1-\alpha)}}\,\|f_{0}-f_{\star}\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2}\quad\forall\,t\geq
0\,.$

  4. 4.

Assume $\beta\in(0,1)$ and $\alpha\in(0,1)$. Then there exist a constant
$\mathscr{C}_{k,\ell}>0$ such that any solution $f$ of (11), with initial
datum
$f_{0}\in\mathrm{L}^{2}\big{(}{\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\langle
x\rangle^{k}\,d\mu\big{)}\cap\mathrm{L}^{2}\big{(}{\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\langle
v\rangle^{\ell}\,d\mu\big{)}$ for some $k>0$ and $\ell>0$, satisfies

$\|f-f_{\star}\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2}\leq\mathscr{C}_{k,\ell}\,(1+t)^{-\zeta}\,\|f_{0}-f_{\star}\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2}\quad\forall\,t\geq
0\,.$

where
$\zeta=\min\big{\\{}\frac{k}{2(1-\alpha)},\frac{\ell}{2(1-\beta)}\big{\\}}$

  5. 5.

Assume $\beta\geq 1$ and $\phi=0$. Then there exist a constant $\mathscr{K}>0$
depending on
$\|f_{0}\|_{\mathrm{L}^{1}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,dx\,dv)}$
such that any solution $f$ of (11), with initial datum
$f_{0}\in\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},d\mu)$,
satisfies

$\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2}\leq\mathscr{K}\,(1+t)^{-\frac{d}{2}}\,\|f_{0}\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2}\quad\forall\,t\geq
0\,.$

  6. 6.

Assume $\beta\in(0,1)$ and $\phi=0$. Then there exist a constant
$\mathscr{K}_{\ell}>0$ such that any solution $f$ of (11), with initial datum
$f_{0}\in\mathrm{L}^{2}\big{(}{\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\langle
v\rangle^{\ell}\,d\mu\big{)}$ for some $\ell>0$, satisfies

$\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2}\leq\mathscr{K}_{\ell}\,(1+t)^{-\zeta}\,\|f_{0}\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2}\quad\forall\,t\geq
0\,,$

where $\zeta=\min\big{\\{}\frac{d}{2},\frac{\ell}{2(1-\beta)}\big{\\}}$.

In the statement of Theorem 4.1, even if it is not specified, the constants
may depend on norms of $f_{0}$. See Table 3 for a summary of the results.
Assumption (23) is a simplifying assumption which can be removed in various
cases: see for instance DMS ; MR3488535 ; MR4069622 ; BDLS . It allows an
immediate conservation of moments along the flow, see Lemma 3 below.

Potential |  $\phi=0$ |  $\phi(x)=\frac{1}{\alpha}\,\langle x\rangle^{\alpha}$ $\alpha\in(0,1)$ |  $\phi(x)=\frac{1}{\alpha}\,\langle x\rangle^{\alpha}$ $\alpha\geq 1$
---|---|---|---
$\psi(v)=\frac{1}{\beta}\,\langle v\rangle^{\beta}$ $\beta\geq 1$ Micro Poincaré |  $t^{-d/2}$ decay |  $t^{-\frac{k}{2(1-\alpha)}}$ convergence |  $e^{-\lambda t}$ convergence
$\psi(v)=\frac{1}{\beta}\,\langle v\rangle^{\beta}$ $\beta\in(0,1)$ |  $t^{-\min\big{\\{}\frac{d}{2},\frac{\ell}{2(1-\beta)}\big{\\}}}$ convergence |  $t^{-\min\big{\\{}\frac{k}{2(1-\alpha)},\frac{\ell}{2(1-\beta)}\big{\\}}}$ convergence |  $t^{-\frac{\ell}{2(1-\beta)}}$ convergence
Table 3:  Summary of the results of Theorem 4.1. See the statement for the
precise meaning of the rates and the assumptions.

_Remark._ The results of Theorem 4.1 can be extended to functions $\psi$ and
$\phi$ depending monotonously on $|v|$ and $|x|$ respectively, which behave
like $\langle v\rangle^{\beta}$ and $\langle x\rangle^{\alpha}$ as
$|v|\to+\infty$ and $|x|\to+\infty$. Typically, one has to assume that for any
$v\in{\mathbb{R}}^{d}$,

$C_{1}\,\langle v\rangle^{\beta}\leq\phi(v)\leq C_{2}\,\langle
v\rangle^{\beta}\,,\quad C_{3}\,|v|\,\langle v\rangle^{\beta-1}\leq
v\cdot\nabla_{v}\psi(v)\leq C_{4}\,|v|\,\langle
v\rangle^{\beta-1}\quad\mbox{and}\quad|\mathrm{Hess}(\psi)|(v)\leq
C_{5}\,\langle v\rangle^{\beta-2}$

for some positive constants $C_{i}$, with $i=1,\dots,5$, and similar estimates
for $\phi$.

###  4.4 An estimate of the entropy production

Let us introduce the weighted norm defined by

$\|f\|_{\beta}:=\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,\langle
v\rangle^{-2(1-\beta)_{+}}\,d\mu)}$

where $(1-\beta)_{+}$ denotes the positive part of $1-\beta$. As a consequence
$\|f\|_{2}$ denotes the standard norm with no weight and we keep using the
notation $\langle\cdot,\cdot\rangle$ for the associated scalar product. We can
rephrase the Poincaré inequality (15) corresponding to the case $\beta\geq 1$,
and the weighted Poincaré inequality (19) rewritten in the variable $v$ with
$\beta\in(0,1)$ instead of $\alpha$ and $\lambda_{m}=\mathscr{C}_{\beta}$, as

$-\,\langle\mathsf{L}f,f\rangle\geq\lambda_{m}\,\|(1-\Pi)f\|_{\beta}^{2}\,.$

In the language of Theorem 2.1, this inequality replaces (H1) while (H3) is
still satified. Next we use the notation of Section 2 for $\mathsf{A}$,
$\mathsf{H}$ and $\mathsf{D}$, with $\mathsf{T}$ and $\mathsf{L}$ given
respectively by (9) and (10).

###### Lemma 1

For any $\beta>0$, there is a positive constant $\kappa$ such that

$\mathsf{D}[f]\geq\kappa\left(\|(1-\Pi)f\|_{\beta}^{2}+\langle\mathsf{AT}\Pi
f,\Pi f\rangle\right)\,.$ (24)

###### Proof

We recall that by (5), $\mathsf{D}$ is defined as

$\mathsf{D}[f]:=-\,\langle\mathsf{L}f,f\rangle+\delta\,\langle\mathsf{AT}\Pi
f,f\rangle-\,\delta\,\Big{(}\mathrm{Re}\langle\mathsf{TA}f,f\rangle-\mathrm{Re}\langle\mathsf{AT}(1-\Pi)f,f\rangle+\mathrm{Re}\langle\mathsf{AL}f,f\rangle\Big{)}\,.$

In order to prove Lemma 1, we have to give estimates on the last three terms
using $\|(1-\Pi)f\|_{\beta}$ and $\langle\mathsf{AT}\Pi f,\Pi f\rangle$. We
obtain these estimates in four steps, as follows.

Step 1. Expressions of $\langle\mathsf{AT}\Pi f,\Pi f\rangle$. We consider the
function $u=u(x)$ implicitly defined by
$u\,f_{\star}=\big{(}1+(\mathsf{T}\Pi)^{\ast}(\mathsf{T}\Pi)\big{)}^{-1}\,\Pi
f$, that is, the solution of (22) that can be rewritten as

$u-\frac{\sigma}{\rho_{\star}}\,\nabla_{x}\cdot\big{(}\rho_{\star}\,\nabla_{x}u\big{)}=u_{f}=\frac{\rho_{f}}{\rho_{\star}}\,.$
(25)

We deduce from

$\mathsf{AT}\Pi
f=\big{(}1+(\mathsf{T}\Pi)^{\ast}(\mathsf{T}\Pi)\big{)}(\mathsf{T}\Pi)^{\ast}(\mathsf{T}\Pi)f=\Pi
f-\big{(}1+(\mathsf{T}\Pi)^{\ast}(\mathsf{T}\Pi)\big{)}^{-1}f=\Pi
f-u\,f_{\star}=\left(\frac{\rho_{f}}{\rho_{\star}}-u\right)f_{\star}$

and (25) that

$\langle\mathsf{AT}\Pi f,\Pi
f\rangle=-\,\sigma\int_{{\mathbb{R}}^{d}}{\big{(}\nabla_{x}\cdot\big{(}\rho_{\star}\,\nabla_{x}u\big{)}\big{)}\,\rho_{f}}\,dx=-\,\sigma\int_{{\mathbb{R}}^{d}}{\nabla_{x}\cdot\big{(}\rho_{\star}\,\nabla_{x}u\big{)}\left(u-\frac{\sigma}{\rho_{\star}}\,\nabla_{x}\cdot\big{(}\rho_{\star}\,\nabla_{x}u\big{)}\right)}\,dx\,,$

that is,

$\langle\mathsf{AT}\Pi f,\Pi
f\rangle=\sigma\int_{{\mathbb{R}}^{d}}{|\nabla_{x}u|^{2}\,\rho_{\star}}\,dx+\sigma^{2}\int_{{\mathbb{R}}^{d}}{\big{|}\nabla_{x}\cdot\big{(}\rho_{\star}\,\nabla_{x}u\big{)}\big{|}^{2}\,\rho_{\star}^{-1}}\,dx\,.$
(26)

Testing (25) with $u\,\rho_{\star}$, we learn after an integration by parts
that

$\int_{{\mathbb{R}}^{d}}{|u|^{2}\,\rho_{\star}}\,dx+\sigma\int_{{\mathbb{R}}^{d}}{|\nabla
u|^{2}\,\rho_{\star}}\,dx=\int_{{\mathbb{R}}^{d}}{u\,\rho_{f}}\,dx\leq\left(\int_{{\mathbb{R}}^{d}}{|u|^{2}\,\rho_{\star}}\,dx\right)^{1/2}\,\|\Pi
f\|$

where the inequality arises from a Cauchy-Schwarz estimate using $\|\Pi
f\|^{2}=\int_{{\mathbb{R}}^{d}}{|\rho_{f}|^{2}\,\rho_{\star}^{-1}}\,dx$. Hence

$\int_{{\mathbb{R}}^{d}}{|u|^{2}\,\rho_{\star}}\,dx\leq\|\Pi
f\|^{2}\quad\mbox{and}\quad\sigma\int_{{\mathbb{R}}^{d}}{|\nabla
u|^{2}\,\rho_{\star}}\,dx\leq\|\Pi f\|^{2}\,.$ (27)

Step 2. An estimate of $|\langle\mathsf{TA}f,f\rangle|$. We know from (4a)
that $|\langle\mathsf{TA}f,f\rangle|\leq\|(1-\Pi)f\|_{2}^{2}$ if $\beta\geq
1$. With $\sigma$ defined by (17), we claim that

$\big{|}\mathrm{Re}\langle\mathsf{TA}f,f\rangle\big{|}\leq\frac{1}{\sigma}\,\|(1-\Pi)f\|_{\beta}^{2}$
(28)

also holds if $\beta\in(0,1)$. In this later case, let us consider the
function $w=w(x)$ implicitly defined by $w\,f_{\star}=\mathsf{A}f$, that is,
the solution of

$w-\frac{\sigma}{\rho_{\star}}\,\nabla_{x}\cdot(\rho_{\star}\,\nabla_{x}w)=-\,\frac{1}{\rho_{\star}}\nabla_{x}\cdot\int_{{\mathbb{R}}^{d}}{v\,\langle
v\rangle^{\beta-2}f}\,dv\,.$

Testing with $w\,\rho_{\star}$ we obtain

$\sigma\int_{{\mathbb{R}}^{d}}{|\nabla_{x}w|^{2}\,\rho_{\star}}\,dx\leq\int_{{\mathbb{R}}^{d}}{|w|^{2}\,\rho_{\star}}\,dx+\sigma\int_{{\mathbb{R}}^{d}}{|{\nabla_{x}w}|^{2}\,\rho_{\star}}\,dx=\int_{{\mathbb{R}}^{d}}{\nabla_{x}w\cdot\left(\int_{{\mathbb{R}}^{d}}{v\,\langle
v\rangle^{\beta-2}f}\,dv\right)}\,dx\,.$

By applying the Cauchy-Schwarz inequality and after squaring, we obtain

$\sigma^{2}\int_{{\mathbb{R}}^{d}}{|\nabla_{x}w|^{2}\,\rho_{\star}}\,dx\leq\int_{{\mathbb{R}}^{d}}{\left(\int_{{\mathbb{R}}^{d}}{v\,\langle
v\rangle^{\beta-2}(1-\Pi)f}\,dv\right)^{2}\,\rho_{\star}^{-1}}\,dx\leq\|(1-\Pi)f\|_{\beta}^{2}$

using $|v|/\langle v\rangle\leq 1$ so that
$\int_{{\mathbb{R}}^{d}}{|v|^{2}\,\langle
v\rangle^{-2}\,e^{-\psi}}\,dv/\int_{{\mathbb{R}}^{d}}{e^{-\psi}}\,dv\leq 1$.
Altogether, we prove (28) with

$\|\mathsf{TA}f\,\langle
v\rangle^{(1-\beta)_{+}}\,\|_{2}^{2}=\|\mathsf{T}(w\,f_{\star})\,\langle
v\rangle^{(1-\beta)_{+}}\,\|_{2}^{2}=\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\big{|}\tfrac{v}{\langle
v\rangle}\cdot\nabla_{x}w\big{|}^{2}\,\langle
v\rangle^{2(\beta-1)_{+}}\,f_{\star}\,dx\,dv\leq\frac{1}{\sigma^{2}}\,\|(1-\Pi)f\|_{\beta}^{2}\,.$

Step 3. We claim that, for some explicit constant $C_{\beta}>0$,

$\big{|}\mathrm{Re}\langle\mathsf{AL}(1-\Pi)f,f\rangle\big{|}\leq\frac{C_{\beta}}{\sigma}\,\|(1-\Pi)f\|_{\beta}\,\langle\mathsf{AT}\Pi
f,\Pi f\rangle\,.$ (29)

This follows from a direct computation. Consider $u=u(x)$ defined by (25) and
observe that $\mathsf{A}^{\ast}\Pi f=\mathsf{T}(u\,f_{\star})$. Then
$\langle\mathsf{AL}(1-\Pi)f,\Pi
f\rangle=\big{\langle}(1-\Pi)f,\mathsf{L}\mathsf{T}(u\,f_{\star})\big{\rangle}$.
Since

$\mathsf{L}\mathsf{T}(u\,f_{\star})=\mathsf{L}\left(\langle
v\rangle^{\beta-2}\,v\cdot\nabla_{x}u\,f_{\star}\right)=\big{(}\xi(v)\cdot\nabla_{x}u\big{)}\,f_{\star}$

where
$\xi(v):=\nabla_{v}\cdot\mathrm{Hess}_{v}(\psi)-\nabla_{v}\psi\cdot\mathrm{Hess}_{v}(\psi)$
is a vector valued function of $v$, it follows that

$\|\mathsf{L}\mathsf{T}(u\,f_{\star})\langle
v\rangle^{(1-\beta)_{+}}\,\|^{2}_{2}\leq
C_{\beta}\int_{{\mathbb{R}}^{d}}{|\nabla_{x}u|^{2}\,\rho_{\star}}\,dx\quad\mbox{with}\quad
C_{\beta}=\int_{{\mathbb{R}}^{d}}{\frac{|\xi(v)|^{2}\,\langle
v\rangle^{2(1-\beta)_{+}}\,e^{-\psi}}{\int_{{\mathbb{R}}^{d}}{e^{-\psi}}\,dv}}\,dv\,.$

Then (29) follows from (26).

Step 4. We claim that, for some explicit constant $C>0$,

$\big{|}\langle\mathsf{AT}(1-\Pi)f,\Pi f\rangle\big{|}\leq
C\,\|(1-\Pi)f\|_{\beta}\,\langle\mathsf{AT}\Pi f,\Pi f\rangle^{1/2}.$ (30)

With $u=u(x)$ defined by (25), we have $(\mathsf{AT})^{\ast}\Pi
f=-\,\mathsf{T}^{2}(u\,f_{\star})$ and

$\langle\mathsf{AT}(1-\Pi)f,\Pi
f\rangle=\langle(1-\Pi)f,-\mathsf{T}^{2}(u\,f_{\star})\rangle\,.$

Using the expression (9) for $\mathsf{T}$, a computation yields

$\mathsf{T}^{2}(u\,f_{\star})=\Big{(}\nabla_{v}\psi\cdot\mathrm{Hess}_{x}(u)\cdot\nabla_{v}\psi-\nabla_{x}u\cdot\mathrm{Hess}_{v}(\psi)\cdot\nabla_{x}\phi\Big{)}\,f_{\star}\,.$

Adding and subtracting
$(\nabla_{v}\psi\cdot\nabla_{x}u)\,(\nabla_{v}\psi\cdot\nabla_{x}\phi)$, we
can rewrite

$\mathsf{T}^{2}(uf_{\star})=\nabla_{v}\psi\cdot\big{(}\mathrm{Hess}(u)-\nabla_{x}u\otimes\nabla_{x}\phi\big{)}\cdot\nabla_{v}\psi\,f_{\star}-\nabla_{x}u\cdot\big{(}\mathrm{Hess}(\psi)-\nabla_{v}\psi\otimes\nabla_{v}\psi\big{)}\cdot\nabla_{x}\phi\,f_{\star}$

where $\nabla_{x}u\otimes\nabla_{x}\phi$ and
$\nabla_{v}\psi\otimes\nabla_{v}\psi$ are respectively the matrices with
entries $\big{(}\partial_{x_{i}}u\,\partial_{x_{j}}\phi\big{)}_{i,j}$ and
$\big{(}\partial_{v_{i}}\psi\,\partial_{v_{j}}\psi\big{)}_{i,j}$ . We estimate
independently the two terms in the expression of $\mathsf{T}^{2}(uf_{\star})$.
(1) The second term is estimated by

$\displaystyle\big{\|}\nabla_{x}u\cdot\big{(}\mathrm{Hess}(\psi)-\nabla_{v}\psi\otimes\nabla_{v}\psi\big{)}$
$\displaystyle\cdot\nabla_{x}\phi\,f_{\star}\,\langle
v\rangle^{(1-\beta)_{+}}\big{\|}_{2}^{2}$
$\displaystyle\leq\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}|\nabla_{x}u|^{2}\,\big{|}\,\mathrm{Hess}(\psi)-\nabla_{v}\psi\otimes\nabla_{v}\psi\big{|}^{2}\,|\nabla_{x}\phi|^{2}\,\langle
v\rangle^{2(1-\beta)_{+}}\,f_{\star}\,dx\,dv$ $\displaystyle\leq
C_{\beta,2}\int_{{\mathbb{R}}^{d}}{|\nabla_{x}u|^{2}\,|\nabla_{x}\phi|^{2}\,\rho_{\star}}\,dx$

where
$C_{\beta,2}=\int_{{\mathbb{R}}^{d}}{\big{|}\,\mathrm{Hess}(\psi)-\nabla_{v}\psi\otimes\nabla_{v}\psi\big{|}^{2}\,\langle
v\rangle^{2(1-\beta)_{+}}\,e^{-\psi}}\,dv\big{/}\int_{{\mathbb{R}}^{d}}{e^{-\psi}}\,dv$.
Using the fact that $|\nabla_{x}\phi|^{2}$ is bounded for $\alpha\in(0,1)$ and
(DMS, , Lemma 8) if $\alpha\geq 1$, there is some constant $c_{\alpha}>0$ such
that the solution of (25) satisfies

$\int_{{\mathbb{R}}^{d}}{|\nabla_{x}u|^{2}\,|\nabla_{x}\phi|^{2}\,\rho_{\star}}\,dx\leq
c_{\alpha}\int_{{\mathbb{R}}^{d}}{|\nabla_{x}u|^{2}\,\rho_{\star}}\,dx\,,$
(31)

which, after using (26), is enough to obtain the bound

$\big{\|}\nabla_{x}u\cdot\big{(}\mathrm{Hess}(\psi)-\nabla_{v}\psi\otimes\nabla_{v}\psi\big{)}\cdot\nabla_{x}\phi\,f_{\star}\,\langle
v\rangle^{(1-\beta)_{+}}\big{\|}_{2}^{2}\leq\frac{c_{\alpha}\,C_{\beta,2}}{\sigma}\,\|(1-\Pi)f\|_{\beta}\,\langle\mathsf{AT}\Pi
f,\Pi f\rangle^{1/2}\,.$

(2) For the first term, we have

$\displaystyle\big{\|}\nabla_{v}\psi\cdot\big{(}\mathrm{Hess}(u)-\nabla_{x}u\otimes\nabla_{x}\phi\big{)}$
$\displaystyle\cdot\nabla_{v}\psi\,f_{\star}\,\langle
v\rangle^{(1-\beta)_{+}}\big{\|}^{2}_{2}$
$\displaystyle\leq\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}|\nabla_{v}\psi|^{4}\,\big{|}\,\mathrm{Hess}(u)-\nabla_{x}u\otimes\nabla_{x}\phi\big{|}^{2}\,\langle
v\rangle^{2(1-\beta)_{+}}\,f_{\star}\,dx\,dv$ $\displaystyle\leq
C_{\beta,3}\int_{{\mathbb{R}}^{d}}{\big{|}\,\mathrm{Hess}(u)-\nabla_{x}u\otimes\nabla_{x}\phi\big{|}^{2}\,\rho_{\star}}\,dx$

where $C_{\beta,3}=\int_{{\mathbb{R}}^{d}}{|\nabla_{v}\psi|^{4}\,\langle
v\rangle^{2(1-\beta)_{+}}\,e^{-\psi}}\,dv\big{/}\int_{{\mathbb{R}}^{d}}{e^{-\psi}}\,dv$.
Notice that $\mathrm{Hess}(u)-\nabla_{x}u\otimes\nabla_{x}\phi$ is the matrix
with entries
$\partial_{x_{i}x_{j}}u-\partial_{x_{i}}u\,\partial_{x_{j}}\phi=\partial_{x_{i}}\big{(}\rho_{\star}\,\partial_{x_{j}}u\big{)}\,\rho_{\star}^{-1}$
for $i,j=1,\dots,d$. Hence

$\int_{{\mathbb{R}}^{d}}\big{|}\,\mathrm{Hess}(u)-\nabla_{x}u\otimes\nabla_{x}\phi\big{|}^{2}\,\rho_{\star}\,dx=\sum_{i,j=1}^{d}\int_{{\mathbb{R}}^{d}}{\left(\partial_{x_{i}}\big{(}\rho_{\star}\,\partial_{x_{j}}u\big{)}\right)^{2}\,\rho_{\star}^{-1}}\,dx\\\
=\int_{{\mathbb{R}}^{d}}{\big{|}\nabla_{x}\cdot\big{(}\rho_{\star}\,\nabla_{x}u\big{)}\big{|}^{2}\,\rho_{\star}^{-1}}\,dx+\int_{{\mathbb{R}}^{d}}{|\nabla_{x}u|^{2}\,\Delta_{x}\phi\,\rho_{\star}}\,dx-\int_{{\mathbb{R}}^{d}}{\mathrm{Hess}(\phi):\nabla_{x}u\otimes\nabla_{x}u\,\rho_{\star}}\,dx\,.$
(32)

To prove (32), it is indeed enough to notice that

$\displaystyle\int_{{\mathbb{R}}^{d}}{\left(\partial_{x_{i}}(\rho_{\star}\,\partial_{x_{j}}u)\right)^{2}\,\rho_{\star}^{-1}}\,dx$
$\displaystyle=-\int_{{\mathbb{R}}^{d}}{\partial_{x_{i}}\phi\,\partial_{x_{j}}u\,\big{(}\partial_{x_{i}}(\rho_{\star}\,\partial_{x_{j}}u)\big{)}}\,dx+\int_{{\mathbb{R}}^{d}}{\partial_{x_{i}x_{j}}^{2}u\,\big{(}\partial_{x_{i}}(\rho_{\star}\,\partial_{x_{j}}u)\big{)}}\,dx\,.$

The observation on the solutions of (22) in Section 4.2 applies. By
integrating by parts, the two integrals are

$\displaystyle\int_{{\mathbb{R}}^{d}}{\partial_{x_{i}x_{j}}^{2}u\,\big{(}\partial_{x_{i}}(\rho_{\star}\,\partial_{x_{j}}u)\big{)}}\,dx=$
$\displaystyle\int_{{\mathbb{R}}^{d}}{\partial_{x_{i}x_{i}}^{2}u\,\big{(}\partial_{x_{j}}(\rho_{\star}\,\partial_{x_{j}}u)\big{)}}\,dx\,,$
$\displaystyle-\int_{{\mathbb{R}}^{d}}{\partial_{x_{i}}\phi\,\partial_{x_{j}}u\,\big{(}\partial_{x_{i}}(\rho_{\star}\,\partial_{x_{j}}u)\big{)}}\,dx=$
$\displaystyle\int_{{\mathbb{R}}^{d}}{\partial_{x_{i}x_{i}}^{2}\phi\,\partial_{x_{j}}u\,\big{(}\rho_{\star}\,\partial_{x_{j}}u\big{)}}\,dx+\int_{{\mathbb{R}}^{d}}{\partial_{x_{i}}\phi\,\partial_{x_{i}x_{j}}^{2}u\,\big{(}\rho_{\star}\,\partial_{x_{j}}u\big{)}}\,dx$
$\displaystyle=$
$\displaystyle\int_{{\mathbb{R}}^{d}}{\partial_{x_{i}x_{i}}^{2}\phi\,\partial_{x_{j}}u\,\big{(}\rho_{\star}\,\partial_{x_{j}}u\big{)}}\,dx-\int_{{\mathbb{R}}^{d}}{\partial_{x_{i}}\phi\,\partial_{x_{i}}u\partial_{x_{j}}\big{(}\rho_{\star}\,\partial_{x_{j}}u\big{)}}\,dx$
$\displaystyle\hskip
119.50148pt-\int_{{\mathbb{R}}^{d}}{\partial_{x_{j}x_{i}}^{2}\phi\,\partial_{x_{i}}u\,\big{(}\rho_{\star}\,\partial_{x_{j}}u\big{)}}\,dx\,.$

Putting everything together, we get

$\displaystyle\int_{{\mathbb{R}}^{d}}{\left(\partial_{x_{i}}(\rho_{\star}\,\partial_{x_{j}}u)\right)^{2}\,\rho_{\star}^{-1}}\,dx=\int_{{\mathbb{R}}^{d}}{\partial_{x_{i}x_{i}}^{2}\phi\,(\partial_{x_{j}}u)^{2}\,\rho_{\star}}\,dx$
$\displaystyle+\int_{{\mathbb{R}}^{d}}{\partial_{x_{i}}(\rho_{\star}\,\partial_{x_{i}}u)\,\partial_{x_{j}}\big{(}\rho_{\star}\,\partial_{x_{j}}u\big{)}}\,dx$
$\displaystyle-\int_{{\mathbb{R}}^{d}}{\partial_{x_{j}x_{i}}^{2}\phi\,\partial_{x_{i}}u\,\partial_{x_{j}}u\,\rho_{\star}}\,dx$

and (32) is obtained by summing over $i$ and $j$. Finally all integrals are
estimated by $\langle\mathsf{AT}\Pi f,\Pi f\rangle$ using (26) and the
improved Poincaré inequality (31), which completes the proof of (30).
In all cases, we conclude that $\|\mathsf{T}^{2}(u\,f_{\star})\langle
v\rangle^{(1-\beta)_{+}}\,\|_{2}^{2}\leq C\,\langle\mathsf{AT}\Pi f,\Pi
f\rangle$ using (26), for some explicit constant $C>0$. This completes the
proof of (30).

Conclusion. By (28), (29) and (30), we control
$|\langle\mathsf{TA}f,f\rangle|$,
$\big{|}\mathrm{Re}\langle\mathsf{AL}(1-\Pi)f,f\rangle\big{|}$ and
$\big{|}\langle\mathsf{AT}(1-\Pi)f,\Pi f\rangle\big{|}$. A discriminant
condition on $\delta$ completes the proof of Lemma 1 as in the proof of
Theorem 2.1. $\square$

###  4.5 Moment estimates

###### Lemma 2

Let $u=u(x)$ be defined in terms of $f$ as in (25) and assume For any $k\geq
0$, there exists a constant $C_{k}>0$ such that

$M_{k}:=\int_{{\mathbb{R}}^{d}}{|u|^{2}\,\langle
x\rangle^{k}\,\rho_{\star}}\,dx\leq
C_{k}\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}|\Pi f|^{2}\,\langle
x\rangle^{k}\,f_{\star}^{-1}\,dx\,dv\,.$ (33)

###### Proof

The case $k=0$ is true from (27). By squaring equation (25) and testing with
$\langle x\rangle^{k}\,\rho_{\star}$, we obtain

$\int_{{\mathbb{R}}^{d}}{u^{2}\,\langle
x\rangle^{k}\,\rho_{\star}}\,dx+2\,\sigma\int_{{\mathbb{R}}^{d}}{\nabla_{x}\big{(}u\,\langle
x\rangle^{k}\big{)}\cdot\nabla_{x}u\,\rho_{\star}}\,dx+\sigma^{2}\int_{{\mathbb{R}}^{d}}{|\nabla_{x}(\rho_{\star}\,\nabla_{x}u)|^{2}\,\frac{\langle
x\rangle^{k}}{\rho_{\star}}}\,dx=\int_{{\mathbb{R}}^{d}}{\rho_{f}^{2}\,\frac{\langle
x\rangle^{k}}{\rho_{\star}}}\,dx\,.$

Moreover, after an integration by parts, we have

$\displaystyle 2\,\sigma\int_{{\mathbb{R}}^{d}}{\nabla_{x}(u\,\langle
x\rangle^{k})\cdot\nabla_{x}u\,\rho_{\star}}\,dx$
$\displaystyle\,=2\,\sigma\int_{{\mathbb{R}}^{d}}{|\nabla_{x}u|^{2}\,\langle
x\rangle^{k}\,\rho_{\star}}\,dx-\sigma\int_{{\mathbb{R}}^{d}}{|u|^{2}(\Delta_{x}\langle
x\rangle^{k}-\nabla_{x}\langle
x\rangle^{k}\cdot\nabla_{x}\phi)\,\rho_{\star}}\,dx$
$\displaystyle\,=2\,\sigma\int_{{\mathbb{R}}^{d}}{|\nabla_{x}u|^{2}\,\langle
x\rangle^{k}\,\rho_{\star}}\,dx-\sigma\,k\,(k+d-2)\,M_{k-2}+\sigma\,k\,(k-2)\,M_{k-4}$
$\displaystyle\quad+\sigma\,k\,M_{k+\alpha-2}-\sigma\,k\,M_{k+\alpha-4}\,.$

After dropping the positive terms we get

$M_{k}\leq\int_{{\mathbb{R}}^{d}}{\frac{\rho_{f}^{2}}{\rho_{\star}}\,\langle
x\rangle^{k}}\,dx+\sigma\,k\,(k+d-2)\,M_{k-2}+\sigma\,k\,M_{k+\alpha-4}\,.$

Inequality (33) follows by induction and interpolation.

Under the simplifying assumption (23), we also obtain moment estimates
directly for the distribution function $f$. Here there is space for
improvements.

###### Lemma 3

Let $f=f(t,x,v)$ be a solution of (11) with transport and collision operators
given respectively by (9) and (10) for some $\beta>0$ and $\alpha>0$. Assume
that the initial datum $f_{0}$ satisfies the bound (23). Then for any $k>0$
and for any $\ell>0$ there exist positive constants $C_{k}$ and $C_{\ell}$
such that, for any $t\geq 0$

$\displaystyle
J_{k}(t):=\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}|f(t,x,v)|^{2}\,\langle
x\rangle^{k}\,f_{\star}^{-1}\,dx\,dv\leq C_{k}\,,$ (34) $\displaystyle
K_{\ell}(t):=\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}|f(t,x,v)|^{2}\,\langle
v\rangle^{\ell}\,f_{\star}^{-1}\,dx\,dv\leq C_{\ell}\,.$ (35)

###### Proof

Since $f_{\star}$ is a stationary solution, the maximum principle yields

$f(t,\cdot,\cdot)\leq C\,f_{\star}\quad\forall t\geq 0.$

Therefore (34) and (35) follow by taking

$C_{k}=C^{2}\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}f_{\star}\,\langle
x\rangle^{k}\,dx\,dv\quad\mbox{and}\quad
C_{\ell}=C^{2}\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}f_{\star}\,\langle
v\rangle^{\ell}\,dx\,dv\,.$

This completes the proof of Lemma 3.

Notice that, with the elementary estimates

$\displaystyle 2^{\frac{k}{2}-1}\,\big{(}1+r^{k}\big{)}\leq\langle
r\rangle^{k}\leq 1+r^{k}\quad$ $\displaystyle\mbox{if}\quad k\in(0,2)\,,$
$\displaystyle 1+r^{k}\leq\langle r\rangle^{k}\leq
2^{\frac{k}{2}-1}\big{(}1+r^{k}\big{)}\quad$ $\displaystyle\mbox{if}\quad
k\geq 2\,,$

we have the simple moment estimate

$M_{k,\eta}:=\int_{{\mathbb{R}}^{d}}{\langle
x\rangle^{k}\,e^{-\frac{1}{\eta}\,\langle
x\rangle^{\eta}}}\,dx\leq\max\left\\{1,\,2^{\frac{k}{2}-1}\right\\}\,\big{|}\mathbb{S}^{d-1}\big{|}\int_{0}^{\infty}r^{d-1}\,\big{(}1+r^{k}\big{)}\,e^{-\frac{r^{\eta}}{\eta}}\,dr\\\
=\max\left\\{1,\,2^{\frac{k}{2}-1}\right\\}\,\frac{2\,\pi^{\frac{d}{2}}}{\Gamma(\frac{d}{2})}\,\eta^{\frac{d-\eta}{\eta}}\left(\Gamma\big{(}\tfrac{d}{\eta}\big{)}+\eta^{\frac{k}{\eta}}\,\Gamma\big{(}\tfrac{d+k}{\eta}\big{)}\right)$

for any $k>0$ and $\Gamma$ is the Euler Gamma function. As a consequence,
$f_{\star}$ defined by (12) with $Z=M_{0,\beta}\,M_{0,\alpha}$ is such that

$\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\langle
x\rangle^{k}\,f_{\star}\,dx\,dv=\frac{M_{k,\alpha}}{M_{0,\alpha}}\quad\mbox{and}\quad\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}\langle
v\rangle^{\ell}\,f_{\star}\,dx\,dv=\frac{M_{k,\beta}}{M_{0,\beta}}\,.$

###  4.6 Proof of Theorem 4.1

In this section we will work in the framework of Theorem 4.1, i.e. we will
consider a solution $f=f(t,x,v)$ to the kinetic Fokker-Planck equation (11)
with initial datum $0\leq f_{0}\leq C\,f_{\star}$, for a certain $C>0$. As
sign plays no role, up to replacing $f$ with $f-f_{\star}$ when $f_{\star}$ is
integrable, we may assume that

$\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}f(t,x,v)\,dx\,dv=0\quad\forall\,t\geq
0\,.$ (36)

We distinguish various cases depending on the values of $\beta$ and $\alpha$.

####  $\bullet$ Case $\beta\geq 1$ and $\alpha\geq 1$

Thanks to Lemma 1, we have

$\mathsf{D}[f]\geq\kappa\left(\|(1-\Pi)f\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2}+\langle\mathsf{AT}\Pi
f,\Pi f\rangle\right)\,.$

for some $\kappa>0$. Because of the assumption $\alpha\geq 1$ the operator
$(\mathsf{T}\Pi)^{\ast}(\mathsf{T}\Pi)$ is coercive, that is (H2) hold. As a
consequence we also have (6), _i.e._ ,

$\langle\mathsf{AT}\Pi f,\Pi
f\rangle\geq\frac{\lambda_{M}}{1+\lambda_{M}}\,\|\Pi
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2}\,.$

Therefore

$\mathsf{D}[f]\geq\frac{\kappa\,\lambda_{M}}{1+\lambda_{M}}\,\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2}$

holds for some $\kappa>0$, which gives exponential convergence:

$\|f(t,\cdot,\cdot)\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2}\leq\frac{4}{2-\delta}\,\mathsf{H}[f(t,\cdot,\cdot)]\leq\frac{4}{2-\delta}\,\mathsf{H}[f_{0}]\,e^{-\lambda
t}\quad\mbox{where}\quad\lambda=\frac{\kappa\,\lambda_{M}}{1+\lambda_{M}}\,.$

Exactly the same proof applies in the case of Corollary 1, with $\lambda_{M}$
now given by the Poincaré inequality associated with the measure
$e^{-\phi}\,dx$, of which the case $\phi(x)=\frac{1}{\alpha}\,\langle
x\rangle^{\alpha}$ with $\alpha\geq 1$ is a special case.

####  $\bullet$ Case $\beta\in(0,1)$ and $\alpha\geq 1$

In this case, on the one hand we still have macroscopic coercivity (H2) due to
the fact that $\alpha\geq 1$, but on the other hand, a loss of weight now
appears for the microscopic component because of $\beta\in(0,1)$. Inequality
(24) now reads as

$\mathsf{D}[f]\geq\kappa\left(\|(1-\Pi)f\|^{2}_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\langle
v\rangle^{-2(1-\beta)}\,d\mu)}+\frac{\lambda_{M}}{1+\lambda_{M}}\,\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2}\right)\,.$

In order to recover the
$\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},d\mu)$ norm we need to
interpolate with the conservation of moments. Let $\ell>0$ and notice that
$f_{0}\in\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\langle
v\rangle^{\ell}d\mu)$ by our assumption on the initial datum (23). Setting
$a=\ell/\big{(}\ell+2(1-\beta)\big{)}$, by Hölder’s inequality and Lemma 3, we
have

$\displaystyle\|(1-\Pi)f\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2}$
$\displaystyle\leq\|(1-\Pi)f\|^{2a}_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\langle
v\rangle^{-2(1-\beta)}\,d\mu)}\,\|(1-\Pi)f\|^{2(1-a)}_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\langle
v\rangle^{\ell}\,d\mu)}$
$\displaystyle\leq\|(1-\Pi)f\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\langle
v\rangle^{-2(1-\beta)}\,d\mu)}^{2a}\,K_{\ell}(t)^{1-a}$ $\displaystyle\leq
C_{\ell}^{1-a}\,\|(1-\Pi)f\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\langle
v\rangle^{-2(1-\beta)}\,d\mu)}^{2a}\,.$

As a consequence, for a certain constant $C>0$ we have

$\mathsf{D}[f]\geq
C\left(\|(1-\Pi)f\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2\left(1+\frac{2(1-\beta)}{\ell}\right)}+\|\Pi
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2}\right)\geq
C\,\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2\left(1+\frac{2(1-\beta)}{\ell}\right)}\,.$

By the Bihari-LaSalle estimate, we finally have

$\mathsf{H}[f]\leq\mathsf{H}[f_{0}]\left(1+C\,\mathsf{H}[f_{0}]^{\frac{2(1-\beta)}{\ell}}\,t\right)^{-\frac{\ell}{2(1-\beta)}}\,.$

####  $\bullet$ Case $\beta\geq 1$ and $\alpha\in(0,1)$

In this case we have the symmetrical situation compared to the previous one.
The dissipation of entropy is

$\mathsf{D}[f]\geq\kappa\left(\|(1-\Pi)f\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2}+\langle\mathsf{AT}\Pi
f,\Pi f\rangle\right)$

where now $\langle\mathsf{AT}\Pi f,\Pi f\rangle$ does not produce macroscopic
coercivity, but is given in terms of $u$ by (26). Fix $k>0$ and assume that
$f_{0}\in\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\langle
x\rangle^{k}\,d\mu)$, then from Lemma 2 and 3, we have that the moments
$M_{k}(t)=\int_{{\mathbb{R}}^{d}}{|u|^{2}\,\langle
x\rangle^{k}\,\rho_{\star}}\,dx$ are uniformly bounded in time. With
$b=k/\big{(}k+2(1-\alpha)\big{)}\in(0,1)$, using Hölder’s inequality and the
weighted Poincaré inequality with non-classical average of (BDLS, , Cor. 10),
we obtain

$\displaystyle\|\Pi
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},d\mu)}^{2}$
$\displaystyle=\int_{{\mathbb{R}}^{d}}{|u|^{2}\,\rho_{\star}}\,dx+2\,\sigma\int_{{\mathbb{R}}^{d}}{|\nabla_{x}u|^{2}\,\rho_{\star}}\,dx+\sigma^{2}\int_{{\mathbb{R}}^{d}}{|\nabla_{x}\cdot(\rho_{\star}\,\nabla_{x}u)|^{2}\,\rho_{\star}}\,dx$
$\displaystyle\leq\left(\int_{{\mathbb{R}}^{d}}{|u|^{2}\,\langle
x\rangle^{-2(1-\alpha)}\,\rho_{\star}}\,dx\right)^{b}\left(\int_{{\mathbb{R}}^{d}}{|u|^{2}\,\langle
x\rangle^{k}\,\rho_{\star}}\,dx\right)^{1-b}+2\,\langle\mathsf{AT}\Pi f,\Pi
f\rangle$
$\displaystyle\leq\big{(}\mathscr{C}_{\alpha}^{\mathrm{wP}}\big{)}^{b}\left(\int_{{\mathbb{R}}^{d}}{|\nabla_{x}u|^{2}\,\rho_{\star}}\,dx\right)^{b}\left(\int_{{\mathbb{R}}^{d}}{|u|^{2}\,\langle
x\rangle^{k}\,\rho_{\star}}\,dx\right)^{1-b}+2\,\langle\mathsf{AT}\Pi f,\Pi
f\rangle$ $\displaystyle\leq C\,\langle\mathsf{AT}\Pi f,\Pi
f\rangle^{b}+2\,\langle\mathsf{AT}\Pi f,\Pi
f\rangle=:\Phi\big{(}\langle\mathsf{AT}\Pi f,\Pi f\rangle\big{)}$

where $C$ depends on $\mathscr{C}_{\alpha}^{\mathrm{wP}}$ and the bound on
$\|f_{0}\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\langle
x\rangle^{k}\,d\mu)}$. In the weighted Poincaré inequality we used (36), hence
$\int_{{\mathbb{R}}^{d}}{u\,\rho_{\star}}\,dx=\iint_{{\mathbb{R}}^{d}\times{\mathbb{R}}^{d}}f\,dx\,dv=0$.
Now we have

$\displaystyle\mathsf{H}[f]\leq\frac{2+\delta}{4}\,\|f\|^{2}_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},d\mu)}$
$\displaystyle\leq\frac{2+\delta}{4}\left(\|(1-\Pi)f\|^{2}_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},d\mu)}+\Phi\big{(}\langle\mathsf{AT}\Pi
f,\Pi f\rangle\big{)}\right)$
$\displaystyle\leq\frac{2+\delta}{4}\,\Phi\left(\|(1-\Pi)f\|^{2}_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},d\mu)}+\langle\mathsf{AT}\Pi
f,\Pi f\rangle\right)$
$\displaystyle\leq\frac{2+\delta}{4}\,\Phi\left(\kappa^{-1}\,\mathsf{D}[f]\right)\,.$

Therefore the decay of $\mathsf{H}[f]$ is estimated by the decay of the
solution $z(t)$ of

$z^{\prime}=\frac{dz}{dt}=-\,\kappa\,\Phi^{-1}\left(\frac{4\,z}{2+\delta}\right)\,,\quad
z(0)=\mathsf{H}[f_{0}]\,.$

In view of the expression of $\Phi$, we conclude that $z$ monotonically
converges to $0$ as $t\to+\infty$ and, as a consequence $z^{\prime}$ also
converges to $0$. This implies that after some time $t_{0}\geq 0$, we have

$\Phi\left(-\,\kappa^{-1}\,z^{\prime}\right)\leq
C\left(-\,\kappa^{-1}\,z^{\prime}\right)^{b}\,,$

Where $C$ denotes a positive constant that may change from line to line.
Altogether, we end up with the differential inequality

$z^{\prime}\leq\,-\,C\,z^{1/b}\,.$

Integrating and using $\frac{b}{1-b}=\frac{k}{2(1-\alpha)}$, we obtain that
$z(t)\leq C\left(1+t\right)^{-\frac{k}{2(1-\alpha)}}$.

####  $\bullet$ Case $\beta\in(0,1)$ and $\alpha\in(0,1)$

If $\beta\in(0,1)$ and $\alpha\in(0,1)$ we have neither microscopic coercivity
nor macroscopic coercivity. The dissipation of entropy is

$\mathsf{D}[f]\geq\kappa\left(\|(1-\Pi)f\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\langle
v\rangle^{-2(1-\beta)}\,d\mu)}^{2}+\langle\mathsf{AT}\Pi f,\Pi
f\rangle\right)$

and we have to interpolate with moments in both variables $x$ and $v$. As in
the previous cases, we have

$\|\Pi
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2}\leq
C\,\langle\mathsf{AT}\Pi f,\Pi f\rangle^{b}+2\,\langle\mathsf{AT}\Pi f,\Pi
f\rangle=\Phi\big{(}\langle\mathsf{AT}\Pi f,\Pi f\rangle\big{)}$

and

$\displaystyle\|(1-\Pi)f\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2}$
$\displaystyle\leq
C\,\|(1-\Pi)f\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\langle
v\rangle^{-2(1-\beta)}\,d\mu)}^{2a}=:\Psi\left(\|(1-\Pi)f\|^{2}_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\langle
v\rangle^{-2(1-\beta)}\,d\mu)}\right)$

with $a=\frac{\ell}{\ell+2(1-\beta)}$ and $b=\frac{k}{k+2(1-\alpha)}$. As
above we have

$\displaystyle\mathsf{H}[f]$
$\displaystyle\leq\frac{2+\delta}{4}\left(\Psi\left(\|(1-\Pi)f\|^{2}_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\langle
v\rangle^{-2(1-\beta)}\,d\mu)}\right)+\Phi\big{(}\langle\mathsf{AT}\Pi f,\Pi
f\rangle\big{)}\right)$
$\displaystyle\leq\frac{2+\delta}{4}\left(\Psi\big{(}\kappa^{-1}\,\mathsf{D}[f]\big{)}+\Phi\big{(}\kappa^{-1}\,\mathsf{D}[f]\big{)}\right)\,.$

Notice that the function $t\mapsto\Psi(t)+\Phi(t)$ is increasing, concave and
$\Psi(0)+\Phi(0)=0$. Moreover we have

$\frac{d}{dt}\mathsf{H}[f(t,\cdot,\cdot)]\leq-\,\kappa\,(\Psi+\Phi)^{-1}\left(\frac{4}{2+\delta}\,\mathsf{H}[f(t,\cdot,\cdot)]\right)\,.$

As a consequence, $\mathsf{H}[f(t,\cdot,\cdot)]$ can be estimated by the
solution $z$ of

$z^{\prime}=-\,\kappa\,(\Psi+\Phi)^{-1}\left(\frac{4\,z}{2+\delta}\right)\,.$

For the same reasons as before, $z^{\prime}$ converges to $0$ as
$t\to+\infty$. Using the explicit expressions of $\Phi$ and $\Psi$, we see
that there exists some $t_{0}\geq 0$ such that, for any $t\geq t_{0}$,

$(\Psi+\Phi)\left(-\,\kappa^{-1}\,z^{\prime}\right)\leq
C\left(-\,\kappa^{-1}\,z^{\prime}\right)^{\zeta}$

for some $C>0$, where $\zeta=\min\\{a,b\\}$. This inequality leads to
$z^{\prime}\leq-\,C\,z^{1/\zeta}$ and therefore to

$z(t)\leq
C\,(1+t)^{-\min\big{\\{}\frac{k}{2(1-\alpha)},\frac{\ell}{2(1-\beta)}\big{\\}}}\quad\forall\,t\geq
0$

by the Bihari-LaSalle estimate.

####  $\bullet$ Case $\beta\geq 1$ and $\phi=0$

In absence of a global equilibrium, we can still consider (24) written with
$\rho_{\star}=1$. Identity (26) now reads as

$\langle\mathsf{AT}\Pi f,\Pi
f\rangle=\sigma\int_{{\mathbb{R}}^{d}}{|\nabla_{x}u|^{2}}\,dx+\sigma^{2}\int_{{\mathbb{R}}^{d}}{|\Delta_{x}u|^{2}}\,dx\,.$

Because of Nash’s inequality and the conservation of mass, we have

$\displaystyle\|\Pi
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},d\mu)}^{2}$
$\displaystyle=\|u\|^{2}_{\mathrm{L}^{2}(dx)}+2\,\sigma\|\nabla_{x}u\|^{2}_{\mathrm{L}^{2}(dx)}+\sigma^{2}\|\Delta_{x}u\|^{2}_{\mathrm{L}^{2}(dx)}$
$\displaystyle\leq
C_{\mathrm{Nash}}\,\|u\|_{\mathrm{L}^{1}(dx)}^{\frac{4}{d+2}}\,\|\nabla_{x}u\|_{\mathrm{L}^{2}(dx)}^{\frac{2d}{d+2}}+2\,\langle\mathsf{AT}\Pi
f,\Pi f\rangle$ $\displaystyle\leq C\,\langle\mathsf{AT}\Pi f,\Pi
f\rangle^{\frac{d}{d+2}}+2\,\langle\mathsf{AT}\Pi f,\Pi f\rangle\,.$

In the asymptotic regime of small $\|\Pi
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2}$, we
have

$\langle\mathsf{AT}\Pi f,\Pi f\rangle\geq C\,\|\Pi
f\|^{2\left(1+\frac{2}{d}\right)}$

for some suitable constant $C>0$, and

$\mathsf{D}[f]\geq
C\left(\|(1-\Pi)f\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},d\mu)}^{2}+\|\Pi
f\|^{2\left(1+\frac{2}{d}\right)}_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},d\mu)}\right)\geq
C\,\|f\|^{2\left(1+\frac{2}{d}\right)}\,.$

By the Bihari-LaSalle estimate, we can finally conclude

$\mathsf{H}[f(t,\cdot,\cdot)]\leq\mathsf{H}[f_{0}]\left(1+C\,\mathsf{H}[f_{0}]^{\frac{2}{d}}\,t\right)^{-\frac{d}{2}}\forall\,t\geq
0\,.$

####  $\bullet$ Case $\beta\in(0,1)$ and $\phi=0$

We proceed as in the previous case. The macroscopic part obeys the same
estimate

$\langle\mathsf{AT}\Pi f,\Pi f\rangle\gtrsim C\,\|\Pi
f\|^{2\left(1+\frac{2}{d}\right)}\,.$

The microscopic component has to be interpolated with moments:

$\|(1-\Pi)f\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2}\leq
C\,\|(1-\Pi)f\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\langle
v\rangle^{-2(1-\beta)}\,d\mu)}^{2b}\,.$

We conclude that, as
$\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}\to 0$,

$\mathsf{D}[f]\geq
C\left(\|(1-\Pi)f\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2\left(1+\frac{2(1-\beta)}{k}\right)}+\|\Pi
f\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2\left(1+\frac{2}{d}\right)}\right)\geq
C\,\|f\|_{\mathrm{L}^{2}({\mathbb{R}}^{d}\times{\mathbb{R}}^{d},\,d\mu)}^{2\left(1+\frac{1}{\zeta}\right)}$

where $\zeta=\min\\{\frac{k}{2(1-\beta)},\frac{d}{2}\\}$. By the Bihari-
LaSalle estimate we conclude

$\mathsf{H}[f]\leq\mathsf{H}[f_{0}]\left(1+C\,\mathsf{H}[f_{0}]^{\frac{1}{\zeta}}\,t\right)^{-\zeta}\,.$

###### Acknowledgements.

L.Z. received funding from the European Union’s Horizon 2020 research and
innovation program under the Marie Skłodowska-Curie grant agreement No.
945322. J.D. thanks S. Menozzi, A. Pascucci and S. Polidoro for the
organization of the meeting on _Kolmogorov operators and their applications_
which has turned to be a great opportunity to write down and collect the
results of the present contribution. All authors thank an anonymous referee
for his/her valuable suggestions and remarks. © 2024 by the authors. This
paper may be reproduced, in its entirety, for non-commercial purposes. CC-BY
4.0

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