cases when invariant measures are not unique, whence our choice to focus on
the Kuramoto model with $\kappa>1$.

Throughout the section, we use freely the same general notations as in the
previous section. In particular, $m(\cdot\,;\mu)$ denotes the solution to
(1.4) with $\mu$ as initial condition and with $b$ as in (4.1). Here, it takes
the form

$\partial_{t}m(t\,;\mu)-\tfrac{1}{2}\partial^{2}_{xx}m(t\,;\mu)-\partial_{x}\Bigl{(}m(t\,;\mu)\bigl{(}J\star
m(t\,;\mu)\bigr{)}\Bigr{)}=0,$ (4.2)

where $\star$ denotes the standard convolution product and
$J(\cdot)=2\pi\kappa\sin(2\pi\cdot)$.

Thanks to the diffusive structure of (1.4), $m(\cdot\,;\mu)$ is absolutely
continuous in positive time, we therefore let
$p(t\,;\mu)=(\mathrm{d}/\mathrm{d}x)m(t;\mu):x\mapsto p(t\,;\mu)$ (which we
also write $p(t,x\,;\mu)$) be the density of $m(t\,;\mu)$, for $t>0$.
$p(t\,;\mu)(\cdot)$ is a smooth function of $x$, uniformly in
$t\in[t_{0},\infty)$ for any $t_{0}>0$. Similarly, we still denote by
$\mu^{N}$ the flow of empirical distributions defined in (1.1). Also, we
recall that each $p_{\infty,\psi}$ is (strictly) positive. Lastly, except when
it is initialised with another condition (in which case it is explicitly
stated), the (common) law of the I.I.D. initial positions
$\eta_{1},\cdots,\eta_{N}$ in (1.1) is denoted by $\mu_{\textrm{\rm init}}$.

###  4.1. Main result

As mentioned above, Kuramoto’s model exhibits two different kinds of
equilibrium point. While the invariant measures in ${\mathcal{I}}$ are stable,
the uniform measure is unstable. Below, we thus focus on initial conditions
that are sufficiently far away from the uniform measure, namely we let, for
any $\eta\in(0,1)$,

${\mathcal{Q}}_{\eta}=\bigl{\\{}\mu\in{\mathcal{P}}({\mathbb{T}}):|\mu^{1}|\geq\eta\bigr{\\}},\quad\mu^{1}:=\int_{{\mathbb{T}}}\exp(-{\mathrm{i}}2\pi\theta)\mu(\mathrm{d}\theta).$

The following result (whose proof is postponed to the end of the subsection
and is in fact mostly taken from [8] and [41]) shows that
${\mathcal{Q}}_{\eta}$ is attracted at an exponential rate by ${\mathcal{I}}$.

######  Proposition 4.1.

For any $\eta\in(0,1)$, and any integer $k\geq 1$, there exist an exponent
$\beta>0$ and a constant $C$, both depending on $K$, $\eta$ and $k$, such that

$\forall t\geq
1,\quad\sup_{\mu\in{\mathcal{Q}}_{\eta}}\inf_{\psi\in{\mathbb{T}}}\,\bigl{\|}p(t\,;\mu)-p_{\infty,\psi}\bigr{\|}_{k,\infty}\leq
C\exp(-\beta t).$

Proposition 4.1 plays a key role in our analysis. Notice that the constraint
$t\geq 1$ may be easily changed into $t\geq t_{0}$ for any $t_{0}>0$, in which
case the constant $C$ may depend on $t_{0}$ as well. This does not make any
conceptual difference, since the main interest of the result is about the long
time behaviour of $p(\cdot\,;\mu)$.

In order to state our main result precisely, we need the following additional
definition.

######  Definition 4.2.

We say that a function $\Phi:{\mathcal{P}}(\mathbb{T})\rightarrow{\mathbb{R}}$
is rotation invariant if, for any $\mu\in{\mathcal{P}}(\mathbb{T})$ and
$\psi\in\mathbb{T}$,

$\Phi\bigl{(}\mu\circ\tau_{\psi}^{-1}\bigr{)}=\Phi\bigl{(}\mu\bigr{)},$

where $\mu\circ\tau_{\psi}^{-1}$ is the image of $\mu$ by the translation
$\tau_{\psi}:\mathbb{T}\ni x\mapsto x+\psi$.

We now have all the ingredients to formulate the main theorem of this section.

######  Theorem 4.3.

Assume that $b$ satisfies _( Reg-$b$-(${4,2}$))_, _( Lip-$b$-(${3,2}$))_ and
_( Erg)_ and that $\Phi$ is rotation invariant and satisfies _(
Reg-$\Phi$-(${\alpha,2}$))_ for some $\alpha\in(0,1]$. Then, for any
$\eta\in(0,1)$, there exists a constant $C>0$ such that, for any
$\mu_{\textrm{\rm init}}\in{\mathcal{Q}}_{\eta}$ and any $N\geq 1$,

$\sup_{t\geq
0}\Big{|}\mathbb{E}\bigl{[}\Phi(\mu^{N}_{t})\bigr{]}-\Phi\bigl{(}m(t\,;{\mu_{\text{\emph{init}}}})\bigr{)}\Big{|}\leq\frac{C}{N}.$

As for examples of functions $\Phi$ that satisfy our assumption, the most
useful example is certainly the following one (very much in the spirit of
Proposition 2.4).

######  Proposition 4.4.

Let $p_{\infty,+}$ denote the unique element of ${\mathcal{I}}$ whose first
Fourier coefficient $p_{\infty,+}^{1}$ is a positive real and let
$\mu_{\infty,+}:=p_{\infty,+}\cdot\textrm{\rm Leb}_{\mathbb{T}}$. For
$\varepsilon\in(0,1]$ and for a smooth non-decreasing cut-off function
$\varphi:[0,1]\rightarrow[0,1]$ that is equal to $0$ on $[0,\delta/2]$ and $1$
on $[\delta,1]$, for some $\delta\in(0,1)$, let $\Phi$ be defined by

$\Phi(\mu):=\varphi\bigl{(}|\mu^{1}|\bigr{)}\Bigl{\|}\mu\circ\tau_{\frac{\mu^{1}}{|\mu^{1}|}}^{-1}-\mu_{\infty,+}\Bigr{\|}_{-(1+\varepsilon)/2,2}^{2}+1-\varphi\bigl{(}|\mu^{1}|\bigr{)},$

where $\|\cdot\|_{-(1+\varepsilon)/2,2}$ is defined as in Proposition 2.4 with
$d=1$ therein. Then $\Phi$ satisfies the assumption of Theorem 4.3.

Here, we need to say a few words about the meaning of $p_{\infty,+}$. Recall
that the elements of ${\mathcal{I}}$ are obtained by rotation. Therefore, the
collection of their first Fourier coefficients coincides with a circle, whose
radius is in fact non-zero (see for instance [8, Subsection 1.2]).
Consequently, we may indeed choose $p_{\infty,+}\in{\mathcal{I}}$ such that
$p_{\infty,+}^{1}>0$. Accordingly, we also notice that, in the notation
$\mu\circ\tau_{\mu^{1}/|\mu^{1}|}^{-1}$, we identify $\mu^{1}/|\mu^{1}|$ with
the unique element $\psi\in{\mathbb{T}}$ such that
$\mu^{1}=|\mu^{1}|\exp({\mathrm{i}}2\pi\psi)$. In particular, the first
Fourier coefficient $(\mu\circ\tau_{\mu^{1}/|\mu^{1}|}^{-1})^{1}$ of
$\mu\circ\tau_{\mu^{1}/|\mu^{1}|}^{-1}$, which is equal to
$\exp(-{\mathrm{i}}2\pi\psi)\mu^{1}=|\mu^{1}|$, is positive (at least when
$\mu^{1}\not=0$), which explains why we compare
$\mu\circ\tau_{\mu^{1}/|\mu^{1}|}^{-1}$ with $\mu_{\infty,+}$. We refer to
[49, Lemma 2.8] for another projection onto ${\mathcal{I}}$.

Moreover, we notice that this is precisely the role of the cut-off function
$\varphi$ in the definition of $\Phi$ to remove the measures $\mu$ for which
$\mu^{1}=0$. In fact, the cut-off function has no real consequence on our
result. Actually, what matters is that $\Phi(\mu)$ is small if and only if
$\mu$ is close enough to ${\mathcal{I}}$. Precisely, we can find a constant
$c>0$ such that

$\Phi(\mu)\geq
c\inf_{\psi\in\mathbb{T}}\bigl{\|}\mu-\mu_{\psi}\|_{-(1+\varepsilon)/2,2}^{2}.$
(4.3)

The proof of the above lower bound is quite easy. It suffices to prove it for
$|\mu^{1}|$ bounded away from zero. To do so, we may observe that
$\|\mu\circ\tau_{\mu^{1}/|\mu^{1}|}^{-1}-\mu_{0}\|_{-(1+\varepsilon)/2,2}^{2}$
is lower-bounded by
$\inf_{\psi\in\mathbb{T}}\bigl{\|}\mu-\mu_{\psi}\|_{-(1+\varepsilon)/2,2}^{2}$.
The following is a straightforward corollary.

######  Corollary 4.5.

For any $\eta\in(0,1)$ and $\varepsilon\in(0,1]$, there exist two (positive)
constants $c$ and $C$ such that, for $\mu_{\textrm{\rm init}}$ in
${\mathcal{Q}}_{\eta}$,

$\forall t\geq
0,\quad{\mathbb{E}}\Bigl{[}\inf_{\psi\in\mathbb{T}}\bigl{\|}\mu_{t}^{N}-\mu_{\psi}\|_{-(1+\varepsilon)/2,2}^{2}\Bigr{]}\leq\frac{C}{N}+C\exp(-ct).$

###### Proof of Corollary 4.5.

We take for granted the statements of Theorem 4.3 and Propositions 4.1 and
4.4. By (4.3) and with $\Phi$ as in Proposition 4.4, it suffices to prove that
$\Phi(m(t\,;\mu))$ decays exponentially fast for any
$\mu\in{\mathcal{Q}}_{\eta}$. In order to do so, we observe from Proposition
4.1 that, for some $\psi\in\mathbb{T}$,
$\|p(t\,;\mu)-p_{\infty,\psi}\|_{2}\leq C\exp(-ct)$, for $t\geq 1$. Hence, it
suffices to show that

$\biggl{\|}p_{\infty,\psi}-p_{\infty,+}\circ\tau_{\frac{(p(t,\mu))^{1}}{|(p(t,\mu))^{1}|}}\biggr{\|}_{2}\leq
C\exp(-ct),$

or equivalently that

$\biggl{\|}p_{\infty,+}\circ\tau_{\frac{(p_{\infty,\psi})^{1}}{|(p_{\infty,\psi})^{1}|}}-p_{\infty,+}\circ\tau_{\frac{(p(t,\mu))^{1}}{|(p(t,\mu))^{1}}|}\biggr{\|}_{2}\leq
C\exp(-ct),$

at least for $t$ sufficiently large. Since $p^{1}_{\infty,+}>0$, we know that
$|(p(t,\mu))^{1}|$ is lower-bounded by a positive constant, uniformly over all
$t$ greater than some $t_{0}>0$. In turn, we have

$\biggl{|}\frac{(p(t,\mu))^{1}}{|(p(t,\mu))^{1}|}-\frac{(p_{\infty,\psi})^{1}}{|(p_{\infty,\psi})^{1}|}\biggr{|}\leq
C\exp(-ct),$

for some possibly new value of $C$, which gives the expected result. ∎

#### Proofs of the auxiliary Propositions 4.1 and 4.4

We present the proofs of Propositions 4.1 and 4.4 here. The reader may skip
them ahead on a first reading.

###### Proof of Proposition 4.4..

We use a simplified notation $\tilde{\mu}$ for
$\mu\circ\tau_{{\mu^{1}}/{|\mu^{1}|}}^{-1}$. The tricky point is then to study
the smoothness of the mapping

$\tilde{\Phi}(\mu)=\bigl{\|}\tilde{\mu}-\nu_{0}\bigr{\|}_{-(1+\varepsilon)/2,2}^{2},$

at least when $\mu^{1}$ stays away from $0$, for a fixed $\nu_{0}$. We observe
that we have the following:

$\tilde{\mu}^{n}=\int_{\mathbb{T}}e^{-{\mathrm{i}}2\pi
nx}d\tilde{\mu}(x)=\Bigl{(}\frac{|\mu^{1}|}{\mu^{1}}\Bigr{)}^{n}\int_{\mathbb{T}}e^{-{\mathrm{i}}2\pi
nx}d\mu(x)=\Bigl{(}\frac{|\mu^{1}|}{\mu^{1}}\Bigr{)}^{n}\mu^{n}.$

Therefore, following (2.17), we get

$\Phi(\mu)=\sum_{n\in{\mathbb{N}}}\frac{1}{(1+n^{2})^{(1+\varepsilon)/2}}\biggl{[}\mu^{n}\bar{\mu}^{n}+\nu_{0}^{n}\bar{\nu}_{0}^{n}-\Bigl{(}\frac{|\mu^{1}|}{\mu^{1}}\Bigr{)}^{n}\mu^{n}\bar{\nu}_{0}^{n}-\Bigl{(}\frac{|\mu^{1}|}{\bar{\mu}^{1}}\Bigr{)}^{n}\nu_{0}^{n}\bar{\mu}^{n}\biggr{]},$

which we can then rewrite in the form

$\Phi(\mu)=\sum_{n\in{\mathbb{N}}}\frac{1}{(1+n^{2})^{(1+\varepsilon)/2}}\biggl{[}\mu^{n}\bar{\mu}^{n}+\nu_{0}^{n}\bar{\nu}_{0}^{n}-\bigl{(}\Psi(\mu)\bigr{)}^{n}\mu^{n}\bar{\nu}_{0}^{n}-\bigl{(}\overline{\Psi}(\mu)\bigr{)}^{n}\nu_{0}^{n}\bar{\mu}^{n}\biggr{]},$

with $\Psi(\mu)=|\mu^{1}|/\mu^{1}$. On the open subset $\\{\mu^{1}\not=0\\}$
(for the ${\mathcal{W}}^{1}$ topology), the function $\Psi$ is infinitely
differentiable with respect to $\mu$. The power $n$ creates additional factors
that are handled in the same way as in the proof of Proposition 2.4. As a
result, we get that, for the same values of $k$, $\Phi$ satisfies the same
properties as in Proposition 2.4, but on any domain where $\mu^{1}$ stays away
from $0$. ∎

We close this subsection with the following proof.

###### Proof of Proposition 4.1..

The proof is achieved in two steps, which are mostly adapted from [7] and
[41].

First step. The first step is to show that

$\lim_{t\rightarrow\infty}\sup_{\mu\in{\mathcal{Q}}_{\eta}}\inf_{\psi\in{\mathbb{T}}}\
\bigl{\|}p(t\,;\mu)-p_{\infty,\psi}\bigr{\|}_{2}=0.$

In order to do so, we follow the proof of Proposition 1.7 in [7]. We recall
indeed that the McKean-Vlasov equation (4.2) may be regarded as a gradient
flow, with potential

${\mathcal{F}}(p)=\frac{1}{2}\int_{{\mathbb{T}}}p(x)\ln\bigl{(}p(x)\bigr{)}\mathrm{d}x-\frac{\kappa}{2}\int_{\mathbb{T}}\int_{{\mathbb{T}}}p(x)\cos\bigl{(}2\pi(x-x^{\prime})\bigr{)}p(x^{\prime})\mathrm{d}x\mathrm{d}x^{\prime}.$

Recall that the solution of (4.2) has a smooth density in positive time. In
particular,

$\forall t>0,\quad{\mathcal{F}}\bigl{(}p(t\,;\mu)\bigr{)}<\infty.$

Moreover, the gradient flow structure says that, for any $0<t_{1}<t_{2}$,

${\mathcal{F}}\bigl{(}p(t_{2};\mu)\bigr{)}-{\mathcal{F}}\bigl{(}p(t_{1};\mu)\bigr{)}=-\int_{t_{1}}^{t_{2}}\int_{\mathbb{T}}\Bigl{|}2\pi\kappa\int_{\mathbb{T}}\sin\bigl{(}2\pi(x-x^{\prime})\bigr{)}m(r;\mu)(\mathrm{d}x^{\prime})+\frac{1}{2}\frac{p^{\prime}(r,x;\mu)}{p(r,x;\mu)}\Bigr{|}^{2}m(r;\mu)(\mathrm{d}x)\mathrm{d}r.$
(4.4)

Then, we know from Proposition 4.4 in [41] that, after some time
$t_{\eta,\epsilon}$ (independent of the choice of
$\mu\in{\mathcal{Q}}_{\eta}$), $p(t\,;\mu)\not\in
B_{L^{2}(\mathbb{T})}({\mathbbm{1}},\epsilon)$ for a given $\epsilon>0$ (here,
${\mathbbm{1}}$ is regarded as the constant function that is equal to 1 on
$\mathbb{T}$, and $B_{L^{2}(\mathbb{T})}({\mathbbm{1}},\epsilon)$ is the
$L^{2}(\mathbb{T})$-ball of center ${\mathbbm{1}}$ and radius $\epsilon$).
Also, after the same time $t_{\eta,\epsilon}$, we know that $p(t\,;\mu)$
belongs to a compact subset ${\mathcal{K}}$ of
${\mathcal{C}}^{1}({\mathbb{T}},(0,+\infty))$, which may be chosen
independently of $\mu$ (the lower bound on $p(t\,;\mu)$ is a mere consequence
of Harnack’s inequality, which can be found in, e.g., Proposition 7.37 of
[48]).

For $\delta>0$ as above, let
${\mathcal{K}}_{\delta}=\\{p\in{\mathcal{K}}:\|p-{\mathbbm{1}}\|_{{\mathcal{C}}^{1}}\geq\delta,\,\inf_{\psi\in\mathbb{T}}\|p-p_{\infty,\psi}\|_{{\mathcal{C}}^{1}}\geq\delta\\}$.
Clearly, ${\mathcal{K}}_{\delta}$ is a compact subset of ${\mathcal{C}}^{1}$.
More importantly, the quantity

$\int_{\mathbb{T}}\Bigl{|}2\pi\kappa\int_{\mathbb{T}}\sin\bigl{(}2\pi(x-x^{\prime})\bigr{)}p(x^{\prime})\mathrm{d}x^{\prime}+\frac{1}{2}\frac{p^{\prime}}{p}(x)\Bigr{|}^{2}p(x)\mathrm{d}x$

cannot vanish on ${\mathcal{K}}_{\delta}$, as otherwise there would exist
$p\in{\mathcal{K}}_{\delta}$ such that (recalling that
$p\in{\mathcal{K}}\Rightarrow\inf_{\mathbb{T}}p>0$)

$2\pi\kappa\int_{\mathbb{T}}\sin\bigl{(}2\pi(x-x^{\prime})\bigr{)}p(x^{\prime})\mathrm{d}x^{\prime}+\frac{1}{2}\frac{p^{\prime}}{p}(x)=0,\quad
x\in\mathbb{T}.$

However, by multiplying by $p(x)$ and then by taking the derivative with
respect to $x$, we would get

$\frac{\mathrm{d}}{\mathrm{d}x}\biggl{(}2\pi\kappa\,p(x)\int_{\mathbb{T}}\sin\bigl{(}2\pi(x-x^{\prime})\bigr{)}p(x^{\prime})\mathrm{d}x^{\prime}\biggr{)}+\frac{1}{2}p^{\prime\prime}(x)=0,\quad
x\in\mathbb{T},$

namely $p$ would be a stationary solution of (1.4), which is impossible as
there are only two types of stationary solutions, ${\mathbbm{1}}$ or
$(p_{\infty,\psi})_{\psi\in\mathbb{T}}$, and none of them belongs to
${\mathcal{K}}_{\delta}$.

By compactness of ${\mathcal{K}}_{\delta}$ and by a straightforward continuity
argument, we deduce that there exists a constant $c>0$ such that, for all
$p\in{\mathcal{K}}_{\delta}$,

$\int_{\mathbb{T}}\Bigl{|}2\pi\kappa\int_{\mathbb{T}}\sin\bigl{(}2\pi(x-x^{\prime})\bigr{)}p(x^{\prime})\mathrm{d}x^{\prime}+\frac{1}{2}\frac{p^{\prime}}{p}(x)\Bigr{|}^{2}p(x)\mathrm{d}x\geq
c.$

In particular, if we take $t_{1}=t_{\eta,\epsilon}$ and $t_{2}=\inf\\{t\geq
t_{\eta,\epsilon}:p(t,\mu)\not\in{\mathcal{K}}_{\delta}\\}$ in (4.4), we
obtain that

${\mathcal{F}}\bigl{(}p(t_{2},\mu)\bigr{)}-{\mathcal{F}}\bigl{(}p(t_{1},\mu)\bigr{)}\leq-c(t_{2}-t_{1}),$

for any $\mu\in{\mathcal{Q}}_{\eta}$. Since the density $p(t_{1},\mu)$ of
$m(t_{1},\mu)$ belongs to ${\mathcal{K}}$, we can find a constant $C$,
independent of $\mu$ and $t_{1}$, such that

${\mathcal{F}}\bigl{(}p(t_{2},\mu)\bigr{)}\leq C-c(t_{2}-t_{1}).$

By the same compactness argument, ${\mathcal{F}}$ must be lower-bounded on
${\mathcal{K}}$, from which we deduce that, for any
$\mu\in{\mathcal{Q}}_{\eta}$, $t_{2}$ is finite and that, most of all, there
exists $T<\infty$, independent of $\mu$, such that $t_{2}\leq T$.

Now, if we choose $\delta\leq\epsilon$ (with $\epsilon$ being introduced right
below (4.4)), then we cannot have
$\|p(t\,;\mu)-{\mathbbm{1}}\|_{{\mathcal{C}}^{1}}\leq\delta$ for $t\geq
t_{1}$, as otherwise $m(t\,;\mu)$ would belong to
$B_{L^{2}(\mathbb{T})}({\mathbbm{1}},\epsilon)$. Therefore, the condition that
$t_{2}<\infty$ implies
$\inf_{\psi\in\mathbb{T}}\|p(t_{2}\,;\mu)-p_{\infty,\psi}\|_{{\mathcal{C}}^{1}}<\delta$.
All in all, we deduce that, for any $\mu\in{\mathcal{Q}}_{\eta}$, there exists
$t_{2}$ (possibly depending on $\mu$), less than $T$ (which is independent of
$\mu$), such that
$\inf_{\psi\in\mathbb{T}}\|p(t_{2}\,;\mu)-p_{\infty,\psi}\|_{{\mathcal{C}}^{1}}\leq\delta$.

Second step. By choosing $\delta$ as small as we want, we deduce that, after
the same time $t_{2}\leq T$ as above, $p(t_{2}\,;\mu)$, regarded as the new
initial condition for the Fokker-Planck equation, satisfies the assumption of
Theorem 4.6 in [41].

It now remains to see that the exponential decay in [41, Theorem 4.6] is
uniform on $B_{L^{2}(\mathbb{T})}(p_{\infty,\psi},\delta)$, provided that
$\delta$ is small enough (independently of the value of $\psi$), which is
exactly Lemma 4.9. By combining with the smoothing estimate stated in [41,
Lemma 2.2], this completes the proof. ∎

###  4.2. Linearised operator

Following our agenda, we now address the linearisation of the nonlinear
Fokker-Planck equation at a (non-trivial) invariant measure (which hence has
the form $p_{\infty,\psi}$ for some phase $\psi\in\mathbb{T}$). Accordingly,
non-trivial invariant measures appear very often in the sequel. For that
reason, we prefer to write $p_{\psi}$ instead of $p_{\infty,\psi}$.

The linearised version (3.1) may be written in the form

$\partial_{t}q(t)-L_{m(t;\mu)}q(t)=0,$

with the convenient notation

$L_{m}q=\tfrac{1}{2}\partial^{2}_{xx}q+\partial_{x}\Bigl{(}q\bigl{(}J\star
m\bigr{)}+m\bigl{(}J\star q\bigr{)}\Bigr{)},$

for any two distributions $m$ and $q$ acting on smooth functions on the torus
(notice that $L_{m}q$ always makes sense as a distribution since $J\star m$
and $J\star q$ themselves should be smooth functions). When $m=p_{\psi}$ for
some element $\psi\in\mathbb{T}$, we merely write $L_{\psi}$ for $L_{m}$.
Notice in particular that, by choosing $m_{t}=p_{\psi}$ (which is hence
independent of $t$) in (4.2) and then by taking the derivative with respect to
$\psi$ (which coincides with the derivative in $x$),

$L_{\psi}p_{\psi}^{\prime}=0,$ (4.5)

where $p_{\psi}^{\prime}=(\mathrm{d}/\mathrm{d}x)p_{\psi}$. The above identity
was already used in [7] and in the subsequent works of the same authors. It
plays a key role here in our analysis as well.

We start with the long-run analysis of the linearised operator, which is the
most demanding step. In fact, the proof is made easier by all the existing
results on the Kuramoto model, but the reader must realise that this
preliminary step is the cornerstone of the whole analysis in this subsection.

######  Proposition 4.6.

For some smooth initial condition $q_{0}$ on $\mathbb{T}$, with
$\int_{\mathbb{T}}q_{0}(x)\mathrm{d}x=0$, let $q$ denote the solution of the
equation

$\partial_{t}q-L_{\psi}q=0,\quad\quad t\geq 0,$ (4.6)

and, for any $t\geq 0$, let $Q(t,\cdot)$ denote the (unique) periodic
primitive of $q(t,\cdot)$ satisfying $\int_{\mathbb{T}}Q(t,x)\mathrm{d}x=0$.

Then there exist two positive constants $\lambda$ and $C$, only depending on
$\kappa$, together with a constant $\bar{q}_{1/2}$, depending on $q(0,\cdot)$
and $\psi$, such that, for any $t\geq 0$,

$\displaystyle\int_{\mathbb{T}}|Q(t,x)|^{2}\mathrm{d}x\leq
C\int_{\mathbb{T}}|Q(0,x)|^{2}\mathrm{d}x,$ (4.7)
$\displaystyle\int_{\mathbb{T}}|Q(t,x)-\bar{q}_{1/2}(p_{\psi}(x)-1)|^{2}\mathrm{d}x\leq
Ce^{-\lambda t}\int_{\mathbb{T}}|Q(0,x)|^{2}\mathrm{d}x.$ (4.8)

######  Remark 4.7.

In fact, (4.7) follows from (4.8). Indeed, by expanding the square in the
right-hand side below and by using the fact that $Q(t,\cdot)$ is centred,
notice that

$\int_{\mathbb{T}}\frac{Q^{2}(t,x)}{p_{\psi}(x)}dx=\int_{\mathbb{T}}\frac{|Q(t,x)-\bar{q}_{1/2}p_{\psi}(x)|^{2}}{p_{\psi}(x)}dx+\bar{q}_{1/2}^{2}.$
(4.9)

from which we deduce that, by taking $t=0$ and by using the fact that
$p_{\psi}$ is lower-bounded, $\bar{q}_{1/2}^{2}\leq
C\int_{\mathbb{T}}|Q(0,x)|^{2}\mathrm{d}x$.

###### Proof.

Surely, the fact that Equation (4.6) has a unique solution is a mere
consequence of the analysis performed in the previous section (since the
Kuramoto model fits all the regularity conditions that are necessary here).
Since $\int_{\mathbb{T}}q(t,x)\mathrm{d}x=0$ for any $t\geq 0$, it makes sense
to define $Q(t,\cdot)$ as in the statement.

Thanks to Remark 4.7, it suffices to focus on the proof of (4.8). The proof
mostly relies on the work of [7]. Following the notation introduced in [7,
(1.23)], we indeed let

$\llangle
u,v\rrangle_{\psi}:=\int_{\mathbb{T}}\frac{\bar{U}(x)\bar{V}(x)}{p_{\psi}(x)}\mathrm{d}x,$
(4.10)

for any two distributions $u$ and $v$ on $\mathbb{T}$ and any two $\bar{U}$
and $\bar{V}$ in $L^{2}(\mathbb{T})$ such that $\bar{U}^{\prime}=u$,
$\bar{V}^{\prime}=v$ and
$\langle\bar{U},p_{\psi}^{-1}\rangle=\langle\bar{V},p_{\psi}^{-1}\rangle=0$.
Then we know from [7, (2.14), (2.16), (2.37)] that there exists a constant
$\lambda$, only depending on $\kappa$, such that

$\frac{\mathrm{d}}{\mathrm{d}t}\bigl{\llangle}q(t,\cdot),q(t,\cdot)\bigr{\rrangle}_{\psi}+\lambda\big{\llangle}q(t,\cdot)-\bar{q}_{1/2}(t)p_{\psi}^{\prime},q(t,\cdot)-\bar{q}_{1/2}(t)p_{\psi}^{\prime}\bigr{\rrangle}_{\psi}\leq
0,$ (4.11)

for any $t\geq 0$, with

$\bar{q}_{1/2}(t)=\frac{\big{\llangle}q(t,\cdot),p_{\psi}^{\prime}\big{\rrangle}_{\psi}}{\big{\llangle}p_{\psi}^{\prime},p_{\psi}^{\prime}\big{\rrangle}_{\psi}}.$

By [7, (2.14)] again, we observe that

$\frac{\mathrm{d}}{\mathrm{d}t}\big{\llangle}q(t,\cdot),p_{\psi}^{\prime}\big{\rrangle}_{\psi}=\big{\llangle}L_{\psi}q(t,\cdot),p_{\psi}^{\prime}\big{\rrangle}_{\psi}=\big{\llangle}q(t,\cdot),L_{\psi}p_{\psi}^{\prime}\big{\rrangle}_{\psi}=0,$

with the last equality following from (4.5). Hence, we can write
$\bar{q}_{1/2}(t)$ as $\bar{q}_{1/2}$. Moreover, applying (4.11) to
$q(t,\cdot)-\bar{q}_{1/2}p_{\psi}^{\prime}$, we deduce that

$\big{\llangle}q(t,\cdot)-\bar{q}_{1/2}p_{\psi}^{\prime},q(t,\cdot)-\bar{q}_{1/2}p_{\psi}^{\prime}\bigr{\rrangle}_{\psi}\leq\big{\llangle}q(0,\cdot)-\bar{q}_{1/2}p_{\psi}^{\prime},q(0,\cdot)-\bar{q}_{1/2}p_{\psi}^{\prime}\bigr{\rrangle}_{\psi}e^{-\lambda
t},\quad t\geq 0.$ (4.12)

By (4.10), the right-hand side in (4.12) reads

$\big{\llangle}q(0,\cdot)-\bar{q}_{1/2}p_{\psi}^{\prime},q(0,\cdot)-\bar{q}_{1/2}p_{\psi}^{\prime}\bigr{\rrangle}_{\psi}=\int_{\mathbb{T}}\frac{|Q(0,x)-\bar{q}_{1/2}p_{\psi}(x)-k(0)|^{2}}{p_{\psi}(x)}\mathrm{d}x,$

where $k(0)$ is a centring constant that forces the mean of
$(Q(0,\cdot)-\bar{q}_{1/2}p_{\psi}-k(0))/p_{\psi}$ to be zero. In particular,
the left-hand side is less than

$\begin{split}\big{\llangle}q(0,\cdot)-\bar{q}_{1/2}p_{\psi}^{\prime},q(0,\cdot)-\bar{q}_{1/2}p_{\psi}^{\prime}\bigr{\rrangle}_{\psi}\leq\int_{\mathbb{T}}\frac{|Q(0,x)-\bar{q}_{1/2}p_{\psi}(x)|^{2}}{p_{\psi}(x)}\mathrm{d}x&\leq\int_{\mathbb{T}}\frac{|Q(0,x)|^{2}}{p_{\psi}(x)}\mathrm{d}x\\\
&\leq C\int_{\mathbb{T}}|Q(0,x)|^{2}\mathrm{d}x,\end{split}$ (4.13)

where the penultimate inequality follows from (4.9) and the last one from the
fact that $p_{\psi}$ is lower-bounded, with the constant depending only on
$\kappa$.

Back to (4.12), we can handle the left-hand side in a similar manner, by
writing

$\big{\llangle}q(t,\cdot)-\bar{q}_{1/2}p_{\psi}^{\prime},q(t,\cdot)-\bar{q}_{1/2}p_{\psi}^{\prime}\bigr{\rrangle}_{\psi}=\int_{\mathbb{T}}\frac{|Q(t,x)-\bar{q}_{1/2}p_{\psi}(x)-k(t)|^{2}}{p_{\psi}(x)}\mathrm{d}x,$

for a new centring constant $k(t)$. Using now an upper bound for $p_{\psi}$
and assuming without any loss of generality that the constant $C$ right above
is large enough (as long as it only depends on $\kappa$), we obtain that

$\big{\llangle}q(t,\cdot)-\bar{q}_{1/2}p_{\psi}^{\prime},q(t,\cdot)-\bar{q}_{1/2}p_{\psi}^{\prime}\bigr{\rrangle}_{\psi}\geq
C^{-1}\int_{\mathbb{T}}|Q(t,x)-\bar{q}_{1/2}p_{\psi}(x)-k(t)|^{2}\mathrm{d}x.$

Using the fact that $Q(t,\cdot)-\bar{q}_{1/2}(p_{\psi}-1)$ has zero mean, we
deduce that

$\big{\llangle}q(t,\cdot)-\bar{q}_{1/2}p_{\psi}^{\prime},q(t,\cdot)-\bar{q}_{1/2}p_{\psi}^{\prime}\bigr{\rrangle}_{\psi}\geq
C^{-1}\int_{\mathbb{T}}|Q(t,x)-\bar{q}_{1/2}(p_{\psi}(x)-1)|^{2}\mathrm{d}x.$
(4.14)

Combining (4.12), (4.13) and (4.14), we get (4.8).

∎

The following result is the analogue of Theorem 3.11 in Section 3.3.

######  Proposition 4.8.

Let $t>0$, $\psi\in\mathbb{T}$ and $\xi\in W^{1,\infty}(\mathbb{T})$ and let
$L_{\psi}^{*}$ denote the adjoint if $L_{\psi}$ is in $L^{2}(\mathbb{T})$.
Then the Cauchy problem

$\begin{cases}\partial_{s}w+L_{\psi}^{*}w=0,\quad\quad s\in[0,t],\\\
w(t,x)=\xi(x),\end{cases}$ (4.15)

admits a unique classical solution $(w(s,\cdot))_{0\leq s\leq t}$. Moreover,
there exist constants $C,\lambda>0$ $($only depending on $\kappa$ and hence
independent of $t)$ such that

  * $(i)$

$\bigg{\|}w(s,\cdot)-\int_{\mathbb{T}}w(s,y)\mathrm{d}y\bigg{\|}_{\infty}\leq
C\Bigl{(}\|\xi\|_{\infty}e^{-\lambda(t-s)}+|\langle\xi,p_{\psi}^{\prime}\rangle|\Bigr{)},\quad\quad\forall
s\in[0,t];$

  * $(ii)$

for any $\alpha,\beta\in[0,1]$ $($allowing, in addition, the constants $C$ and
$\lambda$ to depend on $\alpha,\beta)$,

$\begin{split}&\big{\|}\partial_{x}w(s,\cdot)\big{\|}_{\beta,\infty}\leq{\frac{C}{1\wedge(t-s)^{(\alpha+\beta)/2}}}\Bigl{(}\|\xi\|_{1-\alpha,\infty}e^{-\lambda(t-s)}+|\langle\xi,p_{\psi}^{\prime}\rangle|\Bigr{)},\quad\quad\forall
s\in[0,t].\end{split}$

Now suppose that, for $k\in\\{2,3\\}$, $\xi\in W^{k,\infty}(\mathbb{T})$.
Then, for any $\alpha,\beta\in[0,1]$, there exist constants $C,\lambda>0$,
depending only $\alpha$, $\beta$ and $\kappa$, such that

  * $(iii)$

${\big{\|}\partial^{k}_{x}w(s,\cdot)\big{\|}_{\beta,\infty}\leq{\frac{C}{1\wedge(t-s)^{(\alpha+\beta)/2}}}\Bigl{(}\|\xi\|_{{k-\alpha,{\infty}}}e^{-\lambda(t-s)}}+|\langle\xi,p_{\psi}^{\prime}\rangle|\Bigr{)},\quad\quad\forall
s\in[0,t].\quad\quad$

###### Proof.

Existence and uniqueness of a classical solution to the Cauchy problem is
standard. The rest of the proof is similar to the proofs of Theorems 3.11 and
3.12, but with some differences that we clarify below.

First Step. The first step is to observe that all the three bounds hold when
$t-s\leq 1$. The proof follows from the same argument as the one used in the
proof of Theorem 3.17. By expanding the operator $L_{\psi}^{*}$, we indeed
observe that the equation satisfied by $w$ may be rewritten in the form

$\partial_{s}w(s,\cdot)+\tfrac{1}{2}\partial^{2}_{xx}w(s,\cdot)+V\partial_{x}w(s,\cdot)+\int_{\mathbb{T}}W(\cdot,y)\bar{w}(s,y)\mathrm{d}y=0,\quad
s\in[0,t],$ (4.16)

where $\bar{w}(s,x)=w(s,x)-\int_{\mathbb{T}}w(s,y)\mathrm{d}y$ and $V$ and $W$
are smooth functions on $\mathbb{T}$ and $\mathbb{T}^{2}$ respectively, whose
derivatives up to any order are bounded in terms of $\kappa$ only. By a
standard application of the maximum principle combined with Gronwall’s lemma,
it is easy to show that

$\sup_{\max(0,t-1)\leq s\leq t}\|w(s,\cdot)\|_{\infty}\leq C\|\xi\|_{\infty}.$
(4.17)

Since $W$ is smooth, this provides a bound for the derivatives of any order of
the third term in the left-hand side of (4.16). Following the proof of Theorem
3.17, we then split $w$ into $w=w_{1}+w_{2}$, with

$\begin{split}&\partial_{s}w_{1}(s,\cdot)+\tfrac{1}{2}\partial^{2}_{xx}w_{1}(s,\cdot)+V\partial_{x}w_{1}(s,\cdot)=0,\quad
s\in[0,t]\,;\quad w_{1}(t,\cdot)=\xi,\\\
&\partial_{s}w_{2}(s,\cdot)+\tfrac{1}{2}\partial^{2}_{xx}w_{2}(s,\cdot)+\int_{\mathbb{T}}W(\cdot,y)\bar{w}(s,y)\mathrm{d}y=0,\quad
s\in[0,t]\,;\quad w_{2}(t,\cdot)=0.\end{split}$ (4.18)

By Theorem 3.11, we get all the required bounds on $w_{1}(s,\cdot)$ and its
derivatives, at least for $t-s\leq 1$.

As for the equation satisfied by $w_{2}$, we know that the source term therein
(with $\bar{w}$ being frozen) is smooth. The solution $w_{2}(s,\cdot)$ thus
has bounded (spatial) derivatives of any order, for $t-s\leq 1$, with the
bounds being independent of $t$.

Second Step. The rest of the proof is hence dedicated to the case when $t\geq
1$. We start with the proof of $(i)$, using the same notations $q(0,\cdot)$,
$q$ and $\bar{q}_{1/2}$ as in the statement and the proof of Proposition 4.6.
By Proposition 4.6 and Remark 4.7, we deduce that there exist $\lambda$ and
$C$ as in the statement (but the values of which are allowed to vary from line
to line) such that

$\begin{split}\bigl{|}\bigl{\langle}\xi,q(t,\cdot)\bigr{\rangle}\bigr{|}&\leq\bigl{|}\bigl{\langle}\xi,\bigl{(}q(t,\cdot)-\bar{q}_{1/2}p_{\psi}^{\prime}\bigr{)}\bigr{\rangle}\bigr{|}+\bigl{|}\bar{q}_{1/2}\bigr{|}\,\bigl{|}\bigl{\langle}\xi,p_{\psi}^{\prime}\bigr{\rangle}\bigr{|}\\\
&\leq\bigl{|}\bigl{\langle}\xi^{\prime},\bigl{(}Q(t,\cdot)-\bar{q}_{1/2}[p_{\psi}-1]\bigr{)}\bigr{\rangle}\bigr{|}+C\|Q(0,\cdot)\|_{2}\bigl{|}\bigl{\langle}\xi,p_{\psi}^{\prime}\bigr{\rangle}\bigr{|}\\\
&\leq C\|Q(0,\cdot)\|_{2}\bigl{(}\|\xi\|_{1,\infty}e^{-\lambda
t}+\bigl{|}\bigl{\langle}\xi,p_{\psi}^{\prime}\bigr{\rangle}\bigr{|}\bigr{)}.\end{split}$

Next, we use the Sobolev bound

$\begin{split}\|Q(0,\cdot)\|_{2}&=\biggl{(}\int_{\mathbb{T}}\biggl{|}Q(0,x)-\int_{\mathbb{T}}Q(0,y)\mathrm{d}y\biggr{|}^{2}dx\biggr{)}^{1/2}\leq\biggl{(}\int_{\mathbb{T}}\int_{\mathbb{T}}\bigl{|}Q(0,x)-Q(0,y)\bigr{|}^{2}\mathrm{d}x\,\mathrm{d}y\biggr{)}^{1/2}\leq
C\|q(0,\cdot)\|_{1},\end{split}$

where we used the equality $\int_{\mathbb{T}}Q(0,x)\mathrm{d}x=0$ together
with the obvious identity $Q(0,x)-Q(0,y)=\int_{y}^{x}q(0,z)\mathrm{d}z$. By
the same duality argument as in the proof of Theorem 3.12, we observe that

$\begin{split}\frac{\mathrm{d}}{\mathrm{d}s}\langle
w(s,\cdot),q(s,\cdot)\rangle=0,\end{split}$ (4.19)

which implies that $\langle
w(0,\cdot),q(0,\cdot)\rangle=\langle\xi,q(t,\cdot)\rangle$. We therefore
deduce that

$\bigl{|}\langle w(0,\cdot),q(0,\cdot)\rangle\bigr{|}\leq
C\|q(0,\cdot)\|_{1}\bigl{(}\|\xi\|_{1,\infty}e^{-\lambda
t}+\bigl{|}\bigl{\langle}\xi,p_{\psi}^{\prime}\bigr{\rangle}\bigr{|}\bigr{)}.$

By choosing $q(0,x)=q^{n}(x)-1$, where $(q^{n})_{n\geq 1}$ is standard
mollifier of the Dirac mass at some point $x_{0}\in\mathbb{T}$, and letting
$n$ tend to $\infty$ and then taking the supremum over $x_{0}$, we obtain that

$\sup_{x\in\mathbb{T}}\biggl{|}w(0,x)-\int_{\mathbb{T}}w(0,y)\mathrm{d}y\biggr{|}\leq
C\bigl{(}\|\xi\|_{1,\infty}e^{-\lambda
t}+\bigl{|}\bigl{\langle}\xi,p_{\psi}^{\prime}\bigr{\rangle}\bigr{|}\bigr{)}.$
(4.20)

Obviously, the above bound does not exactly fit $(i)$. The first point to
recover $(i)$ is to replace $\|\xi\|_{1,\infty}$ by $\|\xi\|_{\infty}$. In
order to so, we apply (4.20), but on the interval $[0,t-1]$ and with
$\xi=w(t-1,\cdot)$ itself. For a new value of $C$ (that depends on the same
parameters as before),

$\begin{split}\sup_{x\in\mathbb{T}}\biggl{|}w(0,x)-\int_{\mathbb{T}}w(0,y)\mathrm{d}y\biggr{|}&\leq
C\Bigl{(}\|w(t-1,\infty)\|_{1,\infty}e^{-\lambda
t}+\bigl{|}\bigl{\langle}w(1,\infty),p_{\psi}^{\prime}\bigr{\rangle}\bigr{|}\Bigr{)}\leq
C\Bigl{(}\|\xi\|_{\infty}e^{-\lambda
t}+\bigl{|}\bigl{\langle}\xi,p_{\psi}^{\prime}\bigr{\rangle}\bigr{|}\Bigr{)},\end{split}$
(4.21)

where we used in the last line the fact that
$\langle\xi,p_{\psi}^{\prime}\rangle=\langle
w(1,\cdot),p_{\psi}^{\prime}\rangle$, which follows from (4.5) and then from
the same duality argument as in (4.19). In the above, we also used the bound
$\|w(t-1,\infty)\|_{1,\infty}\leq C\|\xi\|_{\infty}$, which follows from
$(ii)$ (in small time, as already studied in the first step).

Third Step. The end of the proof is similar to the end of the second step in
the proof of Theorem 3.17. Indeed, by Theorem 3.11, $w_{1}$ satisfies all the
required bounds. In particular, $w_{2}=w-w_{1}$ also satisfies (4.21).
Subsequently, we use the same expansion as in (3.29), by taking benefit of the
fact that the source term in the equation for $w_{2}$ is smooth and that all
its derivatives can be upper-bounded by (4.21). We then obtain $(ii)$ and
$(iii)$. ∎

The following proposition is one key step in our proof.

######  Proposition 4.9.

For fixed $\kappa>1$ and $\eta\in(0,1)$, the drift (4.1) satisfies (Erg) up to
the change that, in (Erg-($K$,$\gamma$,$\alpha$)), $\mu$ is taken in
${\mathcal{Q}}_{\eta}$ and that

  1. (1)

(3.17) holds when $t\leq 1$;

  2. (2)

(3.17) holds when $t\geq 1$, but with $q(t)$ replaced by
$q(t)-q_{\infty}p^{\prime}(t\,;\mu)$, where: $q_{\infty}$ is here a real
number depending on the input of the Cauchy problem (3.16), but independent of
$t$, and $q_{\infty}$ is bounded by
$C\max\\{1,\|q_{0}\|_{(2,\infty)^{\prime}}\\}$, for $C$ only depending on
$\eta$, $\kappa$, $K$ and $\gamma$.

###### Proof.

Throughout the proof, we merely write $m(t)$ for $m(t\,;\mu)$ and $p(t)$ for
$p(t\,;\mu)$ (recalling that $\mu$ is given in the definition of
(Erg-($K$,$\gamma$,$\alpha$))). Moreover, following the statement of
Proposition 4.1, we may call $\psi\in\mathbb{T}$ such that $p_{\psi}$ is the
limit of $p(t)$ as $t$ tends to $\infty$. The proof is then an adaptation of
the proof of Theorem 3.17, which is itself an adaptation of the proof of
Theorem 3.12.

First Step. The first step is completely similar to the first step in the
proof of Theorem 3.17. The point is to observe is that, for any $\tau>0$, the
analysis performed in the proof of Theorem 3.12 remains true on the time
interval $[0,\tau]$, whether (3.34) holds or not. This says that, in
(Erg-($K$,$\gamma$,$\alpha$)), (3.17) holds (under (3.15) and (3.16)) for
$t\in(0,\tau]$, with the constants being independent of the choice of $\mu$.

Second Step. As for the second step, we also follow the second step of the
proof of Theorem 3.17. We thus start from the same equation (3.2) (with the
same solution $q$), but instead of considering $w$ as the solution of (3.35),
we choose $w$ as the solution of (4.15). As in the proof of Theorem 3.17, this
leads to a new expansion in (3.36). In short, $T_{1}$ and $T_{2}$ are the
same, but $T_{3}$ takes the new form (very much in the spirit of (3.53)):

$\begin{split}T_{3}&=\int_{0}^{t}\Bigl{\langle}\Bigl{(}b\bigl{(}\cdot,m(s)\bigr{)}-b\bigl{(}\cdot,p_{\psi}\bigr{)}\Bigr{)}\partial_{x}w(s,\cdot),q(s)\Bigr{\rangle}\mathrm{d}s\\\
&\hskip 15.0pt+\int_{0}^{t}\int_{\mathbb{T}^{d}}\Bigl{\langle}\frac{\delta
b}{\delta
m}\big{(}x,m(s)\big{)}(\cdot)\,\partial_{x}w(s,x),q(s)\Bigr{\rangle}\,\Bigl{(}p(s,x)-p_{\psi}(x)\Bigr{)}\mathrm{d}x\,\mathrm{d}s\\\
&\hskip
15.0pt+\int_{0}^{t}\int_{\mathbb{T}^{d}}\Bigl{\langle}\Bigl{(}\frac{\delta
b}{\delta m}\big{(}x,m(s)\big{)}(\cdot)-\frac{\delta b}{\delta
m}\big{(}x,p_{\psi}\big{)}(\cdot)\Bigr{)}\partial_{x}w(s,x),q(s)\Bigr{\rangle}\,p_{\psi}(x)\mathrm{d}x\,\mathrm{d}s,\end{split}$
(4.22)

where, in the argument of $b$, we have identified the density $p_{\psi}$ with
the measure of density $p_{\psi}$.

Here comes now the subtlety. Unless we assume
$\langle\xi,p_{\psi}^{\prime}\rangle=0$, the conclusions $(i)$, $(ii)$ and
$(iii)$ in Proposition 4.8 just hold without the exponential decay. Although
this seems to be a serious drawback to follow the second step of the proof of
Theorem 3.17, this can be easily circumvented. Indeed, by the first step, we
can easily replace the initial point $0$ in $T_{3}$ by any time $\tau>0$ and
then assume accordingly that $t\geq\tau$. By Proposition 4.1, this permits to
render the distance between $p(s)$ (and $m(s)$) as small as needed by taking
$\tau$ large enough, uniformly in $\mu\in{\mathcal{Q}}_{\eta}$. Taking
$\tau\geq 1$ large enough, we get the following variant of (3.40):

$\|q(t)\|_{(k-\alpha,\infty)^{\prime}}\leq
C\|q_{0}\|_{(k,\infty)^{\prime}}+\frac{1}{2}\bigg{(}\sup_{\tau\leq s\leq
t}\|q(s)\|_{(k,\infty)^{\prime}}\bigg{)}+C(1+t)\bigg{(}\sup_{0\leq s\leq
t}\|r(s)\|_{(k,\infty)^{\prime}}\bigg{)},\quad t\geq 1,$ (4.23)

with the additional $t$ on the right-hand side coming from the fact that we do
not have any integrable decay for $w$ and its derivatives in Proposition 4.8.
As for the constant $C$, it only depends on $\kappa$, $\eta$, $k$ and
$\alpha$. Choosing first $\alpha=0$, restoring the value of $\alpha$ and
allowing $C$ to depend on $\kappa$, $\eta$, $\alpha$, $K$ and $\gamma$, we
deduce that, for $t\geq 1$,

$\|q(t)\|_{(k-\alpha,\infty)^{\prime}}\leq
C\max\bigl{\\{}1+t,\|q(0)\|_{(k,\infty)^{\prime}}\bigr{\\}},\quad t>0.$ (4.24)

Third Step. We now assume $\langle\xi,p_{\psi}^{\prime}\rangle=0$ in order to
benefit from the exponential decay in $(i)$, $(ii)$ and $(iii)$ in Proposition
4.8. As in the proof of Theorem 3.12, we refer back to (3.36), by initialising
the dynamics at some positive time. Here, we choose $t/2$ as initial time, for
$t\geq 1$. The equation (3.16), which is regarded on the sole $[t/2,\infty)$,
may be split into $q(s)=q_{1}(s)+q_{2}(s)$, for $s\geq t/2$, where $q_{1}$ is
the solution of (3.16) with $q_{1}(t/2)=q(t/2)$ as initial condition and with
a zero source term, and $q_{2}$ is the solution of (3.16) with a zero initial
condition and with $r$ as source term. Proceeding as in the second step,
$q_{1}$ and $q_{2}$ also satisfy (4.24).

The term $q_{2}$ can be easily addressed by returning to (3.36) (initialised
from $t/2$). The term $T_{1}$ therein is zero. Similar to (3.40) (but with
$s\geq t/2$ in the last supremum) and thanks to the exponential decay of $r$,
the term $T_{2}$ is less than $C\exp(-\lambda t/4)\|\xi\|_{k-\alpha,\infty}$.
Furthermore, by combining (4.24) (for $q_{2}$) and Proposition 4.1, the term
$T_{3}$ (as given by (4.22)) is also less than $C\exp(-\lambda
t/4)\|\xi\|_{k-\alpha,\infty}$. This is enough to prove that
$\|q_{2}(t)\|_{(k-\alpha,\infty)^{\prime}}\leq C\exp(-\lambda
t/4)\max\\{1,\|q_{0}\|_{(k,\infty)^{\prime}}\\}$.

In order to handle $q_{1}$, we must revisit (3.36). Since the source term in
the equation of $q_{2}$ is null, the corresponding $T_{2}$ in (3.36) is $0$.
Recalling the condition $\langle\xi,p_{\psi}^{\prime}\rangle=0$, we get the
following variant of (4.24):

$\Bigl{\|}q_{1}(t)-\frac{\langle q_{1}(t),p_{\psi}^{\prime}\rangle}{\langle
p_{\psi}^{\prime},p_{\psi}^{\prime}\rangle}p_{\psi}^{\prime}\Bigr{\|}_{(k-\alpha,\infty)^{\prime}}\leq
Ce^{-\lambda t/2}\max\bigl{\\{}1,\|q(t/2)\|_{(k,\infty)^{\prime}}\bigr{\\}}.$
(4.25)

Next, we prove that $\langle q_{1}(t),p_{\psi}^{\prime}\rangle/\langle
p_{\psi}^{\prime},p_{\psi}^{\prime}\rangle$ converges exponentially fast to a
constant $q_{\infty}$. We write $\langle
q_{1}(t),p_{\psi}^{\prime}\rangle=-\langle
Q_{1}(t,\cdot),p_{\psi}^{\prime\prime}\rangle$, with $Q_{1}$ being the
primitive of $q_{1}$ (in space) with a zero mean. Invoking Proposition 4.6
(with $q=q_{1}$), we know that $\langle
Q_{1}(t,\cdot),p_{\psi}^{\prime\prime}\rangle$ converges exponentially fast to
some constant. The rate of convergence, as given by (4.8), depends on

$\|Q_{1}(t/2+1,\cdot)\|_{2}\leq C\|q_{1}(t/2+1,\cdot)\|_{\infty}.$ (4.26)

We now provide an upper bound for the above right-hand side. Since the source
term in the equation for $q_{1}$ is zero, we can apply the second step twice.
We first apply it on the interval $[(t+1)/2,t/2+1]$ with $k=\alpha=1$, which
gives $\|q_{1}(t/2+1,\cdot)\|_{\infty}\leq
C\max\\{1,\|q_{1}((t+1)/2,\cdot)\|_{(1,\infty)^{\prime}}\\}$. We then reapply
it on the interval $[t/2,(t+1)/2]$ with $k=2$ and $\alpha=1$, which yields
$\|q_{1}((t+1)/2,\cdot)\|_{(1,\infty)^{\prime}}\leq
C\max\\{1,\|q(t/2,\cdot)\|_{(2,\infty)^{\prime}}\\}$. We end up with
$\|q_{1}((t+1)/2,\cdot)\|_{(1,\infty)^{\prime}}\leq
C\max\\{1+t,\|q(0,\cdot)\|_{(2,\infty)^{\prime}}\\}$. Inserting the latter in
(4.26), we deduce from (4.25) and from the second step again that, for
possibly new values of $C$ and $\lambda$,

$\Bigl{\|}q_{1}(t)-q_{\infty}p_{\psi}^{\prime}\Bigr{\|}_{(k-\alpha,\infty)^{\prime}}\leq
Ce^{-\lambda t/2}\max\bigl{\\{}1,\|q_{0}\|_{(k,\infty)^{\prime}}\bigr{\\}}.$

It then remains to recall from Proposition 4.1 that $p(t)$ converges
exponentially fast to $p_{\psi}$. We deduce that

$\Bigl{\|}q_{1}(t)-q_{\infty}p^{\prime}(t)\Bigr{\|}_{(k-\alpha,\infty)^{\prime}}\leq
Ce^{-\lambda
t/2}\max\bigl{\\{}1,\|q_{0}\|_{(k,\infty)^{\prime}}\bigr{\\}},\quad t\geq 1.$

Using the bound for $q_{2}$, we can easily replace $q_{1}$ by $q$. By the
first step, the constant $q_{\infty}$ is bounded by
$C\max\\{1,\|q_{0}\|_{(2,\infty)^{\prime}}\\}$. This completes the proof.∎

###  4.3. Estimates of the tangent processes

We now have all the ingredients in order to estimate the tangent processes
$m^{(1)}$, $m^{(2)}$, $d^{(1)}$ and $d^{(2)}$ in Theorems 3.2, 3.3 and 3.4
respectively. Notice that, since the model is one-dimensional, we may remove
the indices $i$ and $(i,j)$ in $d^{(1)}$ and $d^{(2)}$ respectively.

######  Proposition 4.10.

For any $\alpha\in[0,1]$, $\beta\in[0,2)$ and $\eta\in(0,1)$, there exist two
positive constants $\lambda$ and $C$, only depending on $\kappa$, $\alpha$,
$\beta$ and $\eta$, such that, for any $\mu$ in ${\mathcal{Q}}_{\eta}$, $\nu$,
$\nu_{1}$ and $\nu_{2}$ in ${\mathcal{P}}(\mathbb{T})$ and $z$, $z_{1}$ and
$z_{2}$ in ${\mathbb{T}}$, the following bound holds:

$\bigl{\|}m^{(1)}(t\,;\mu,\nu)\bigr{\|}_{(0,\infty)^{\prime}}+\bigl{\|}m^{(2)}(t\,;\mu,\nu_{1},\nu_{2})\bigr{\|}_{(0,\infty)^{\prime}}\leq
C,\qquad t\geq 0.$ (4.27)

Moreover, we can find two real numbers $q_{\infty}^{(1)}(\mu,z)$ and
$q_{\infty}^{(2)}(\mu,z_{1},z_{2})$ such that

$\begin{cases}&\min(1,t^{\alpha/2})\bigl{\|}d^{(1)}(t\,;\mu,z)\bigr{\|}_{(1-\alpha,\infty)^{\prime}}+\min(1,t^{\beta/2})\bigl{\|}d^{(2)}(t\,;\mu,z_{1},z_{2})\bigr{\|}_{(2-\beta,\infty)^{\prime}}\leq
C,\\\
&\min(1,t^{\alpha/2})\bigl{\|}\tilde{d}^{(1)}(t\,;\mu,z)\bigr{\|}_{(1-\alpha,\infty)^{\prime}}+\min(1,t^{\beta/2})\bigl{\|}\tilde{d}^{(2)}(t\,;\mu,z_{1},z_{2})\bigr{\|}_{(2-\beta,\infty)^{\prime}}\leq
Ce^{-\lambda t},\end{cases}$ (4.28)

for $t>0$, where

$\begin{split}&\tilde{d}^{(1)}(t\,;\mu,z)=d^{(1)}(t\,;\mu,z)-q_{\infty}^{(1)}(\mu,z)p^{\prime}(t\,;\mu),\\\
&\tilde{d}^{(2)}(t\,;\mu,z_{1},z_{2})=d^{(2)}(t\,;\mu,z_{1},z_{2})-q_{\infty}^{(1)}(\mu,z_{2})\partial_{x}d^{(1)}(t\,;\mu,z_{1})-q_{\infty}^{(2)}(\mu,z_{1},z_{2})p^{\prime}(t\,;\mu).\end{split}$
(4.29)

###### Proof.

First Step. We start with the proof of (4.28). Throughout the proof, we fix
$\eta\in(0,1)$. By Theorem 3.3, we know that, for any
$\mu\in{\mathcal{Q}}_{\eta}$ and $z\in\mathbb{T}$, $d^{(1)}(t\,;\mu,z)$ solves
the linearised equation

$\partial_{t}d^{(1)}(t\,;\mu,z)-L_{m(t;\mu)}d^{(1)}(t\,;\mu,z)=0,$ (4.30)

with $d^{(1)}(0\,;\mu,z)=D_{z}^{\prime}=-\partial_{x}(\delta_{z})$ and
$m(0\,;\mu)=\mu$. Then, by item (2) in Proposition 4.9, with $k=1$ in (3.17),
we get the two bounds for $d^{(1)}(t\,;\mu,z)$ and
$\tilde{d}^{(1)}(t\,;\mu,z)$, with $q_{\infty}^{(1)}(\mu,z)$ being given by
Proposition 4.9.

As for $d^{(2)}$, things are more complicated. By Theorem 3.4, we indeed know
that, for any $\mu\in{\mathcal{Q}}_{\eta}$ and any
$z_{1},z_{2}\in{\mathbb{T}}^{d}$, $d^{(2)}(t\,;\mu,z_{1},z_{2})$ solves the
equation

$\begin{split}&\partial_{t}d^{(2)}(t\,;\mu,z_{1},z_{2})-L_{m(t;\mu)}d^{(2)}(t\,;\mu,z_{1},z_{2})\\\
&\hskip 5.0pt-\partial_{x}\Bigl{(}d^{(1)}(t\,;\mu,z_{1})\bigl{(}J\star
d^{(1)}(t\,;\mu,z_{2})\bigr{)}+d^{(1)}(t\,;\mu,z_{2})\bigl{(}J\star
d^{(1)}(t\,;\mu,z_{1})\bigr{)}\Bigr{)}=0,\end{split}$ (4.31)

with $d^{(2)}(0\,;\mu,z_{1},z_{2})=0$. Next, if we want to apply item (2) in
Proposition 4.9, with $k=2$ in (3.17), we have to choose the following in
(3.16):

$r(t)=\partial_{x}\Bigl{(}d^{(1)}(t\,;\mu,z_{1})\bigl{(}J\star
d^{(1)}(t\,;\mu,z_{2})\bigr{)}+d^{(1)}(t\,;\mu,z_{2})\bigl{(}J\star
d^{(1)}(t\,;\mu,z_{1})\bigr{)}\Bigr{)}.$

However, we cannot prove that $\|r(t)\|_{(2,\infty)^{\prime}}$ decays
exponentially fast, since (4.28) just provides an exponential bound for
$\tilde{d}^{(1)}(t\,;\mu,z_{1})$ and $\tilde{d}^{(1)}(t\,;\mu,z_{2})$, and not
for $d^{(1)}(t\,;\mu,z_{1})$ and $d^{(1)}(t\,;\mu,z_{2})$. Instead, we focus
on

$\bar{d}^{(2)}(t\,;\mu,z_{1},z_{2}):=d^{(2)}(t\,;\mu,z_{1},z_{2})-q_{\infty}^{(1)}(\mu,z_{2})\partial_{x}d^{(1)}(t\,;\mu,z_{1}),$
(4.32)

for $q_{\infty}^{(1)}(\mu,z)$ as in (4.29). We easily see that
$(\partial_{x}d^{(1)}(t\,;\mu,z))_{t\geq 0}$ solves (in a weak sense) the
equation

$\partial_{t}\partial_{x}d^{(1)}(t\,;\mu,z)-L_{m(t\,;\cdot)}\partial_{x}d^{(1)}(t\,;\mu,z)-\partial_{x}\Bigl{(}d^{(1)}(t\,;\mu,z)\bigl{(}J\star
m^{\prime}(t\,;\mu)\bigr{)}+m^{\prime}(t\,;\mu)\bigl{(}J\star
d^{(1)}(t\,;\mu,z)\bigr{)}\Bigr{)}=0,$ (4.33)

in the space
$\cap_{T>0}L^{\infty}([0,T],(W^{2,\infty}(\mathbb{T}^{d}))^{\prime})\cap_{T>1}L^{\infty}([1/T,T],(W^{2-2\alpha,\infty}(\mathbb{T}^{d}))^{\prime})$,
with $-\partial^{2}_{x}(\delta_{z})$ as initial condition. Then, choosing
$z=z_{1}$ in (4.33), multiplying by $q_{\infty}^{(1)}(\mu,z_{2})$ and then
subtracting to (4.31), we obtain

$\begin{split}&\partial_{t}\bar{d}^{(2)}(t\,;\mu,z_{1},z_{2})-L_{m(t;\mu)}\bar{d}^{(2)}(t\,;\mu,z_{1},z_{2})-r(t)=0,\end{split}$

with

$r(t)=\partial_{x}\Bigl{(}d^{(1)}(t\,;\mu,z_{1})\bigl{(}J\star\tilde{d}^{(1)}(t\,;\mu,z_{2})\bigr{)}+\tilde{d}^{(1)}(t\,;\mu,z_{2})\bigl{(}J\star
d^{(1)}(t\,;\mu,z_{1})\bigr{)}\Bigr{)}.$ (4.34)

We first evaluate the $(W^{2,\infty}(\mathbb{T}))^{\prime}$ norm of $r(t)$.
Using the fact that the convolution kernel $J$ is odd, we get, for any $\xi\in
W^{2,\infty}(\mathbb{T})$,

$\begin{split}\langle
r(t),\xi\rangle&=-\Bigl{\langle}d^{(1)}(t\,;\mu,z_{1})\bigl{(}J\star\tilde{d}^{(1)}(t\,;\mu,z_{2})\bigr{)}+\tilde{d}^{(1)}(t\,;\mu,z_{2})\bigl{(}J\star
d^{(1)}(t\,;\mu,z_{1})\bigr{)},\xi^{\prime}\Bigr{\rangle}\\\
&=\Bigl{\langle}\tilde{d}^{(1)}(t\,;\mu,z_{2}),J\star\bigl{(}d^{(1)}(t\,;\mu,z_{2})\xi^{\prime}\bigr{)}\Bigr{\rangle}-\Bigl{\langle}\tilde{d}^{(1)}(t\,;\mu,z_{2}),\bigl{(}J\star
d^{(1)}(t\,;\mu,z_{1})\bigr{)}\xi^{\prime}\Bigr{\rangle}.\end{split}$

Invoking (4.28) with $\alpha=0$ (using in addition the smoothness of $J$ and
noticing that the bound obviously holds at time $t=0$), we then have

$\|r(t)\|_{(2,\infty)^{\prime}}\leq Ce^{-\lambda t},\qquad t\geq 0,$

where the values of $C$ and $\lambda$ are allowed to vary as long as they only
depend on $\kappa$, $\alpha$, $\beta$ and $\eta$. By item (2) in Proposition
4.9, with $k=2$ in (3.17), we deduce that there exists a constant
$q_{\infty}^{(2)}(\mu,z_{1},z_{2})$ such that

$\min(1,t^{\beta/2})\bigl{\|}\tilde{d}^{(2)}(t\,;\mu,z_{1},z_{2})\bigr{\|}_{(2-\beta,\infty)^{\prime}}\leq
Ce^{-\lambda t},\qquad t>0,$ (4.35)

with

$\tilde{d}^{(2)}(t\,;\mu,z_{1},z_{2})=\bar{d}^{(2)}(t\,;\mu,z_{1},z_{2})-q_{\infty}^{(2)}(t\,;\mu,z_{1},z_{2})p^{\prime}(t\,;\mu).$

This completes the proof of (4.28).

Second Step. We now turn to the proof of (4.27), which is quite similar to the
first step. By Theorem 3.2, we know that, for any $\mu\in{\mathcal{Q}}_{\eta}$
and $\nu\in{\mathcal{P}}(\mathbb{T})$, $(m^{(1)}(t\,;\mu,\nu))_{t\geq 0}$
solves the same equation (4.30) but with $\nu-\mu$ as initial condition.
Therefore, we have the bound for $m^{(1)}(t\,;\mu,\nu)$ in (4.27), using item
(2) in Proposition 4.9, with $k=0$ in (3.17). Also, we have the same for
$\tilde{m}^{(1)}(t\,;\mu,\nu)$, with an obvious definition for the latter.

We now turn to $(m^{(2)}(t\,;\mu,\nu_{1},\nu_{2}))_{t\geq 0}$. For
$t\in[0,1]$, the bound follows from Proposition 3.7, recalling that there is
no need for ergodicity in finite time. In order to address the case $t>1$, we
write the analogue of (4.33) but for
$(\partial_{x}m^{(1)}(t\,;\mu,\nu))_{t\geq 0}$ and the analogue of $r$ in
(4.34). We have $\|r(t)\|_{-1,\infty}\leq C\exp(-\lambda t)$. By (2) in
Proposition 4.9, with $k=1$ in (3.17), we recover (4.35) but for
$\tilde{m}^{(2)}(t\,;\mu,\nu_{1},\nu_{2})$ (with an obvious definition for
it), with $2-\beta$ being replaced by $0$. We get a bound for
$\|m^{(2)}(t\,;\mu,\nu_{1},\nu_{2})\|_{(0,\infty)^{\prime}}$, for $t\geq 1$.∎

In the rest of the subsection, ${\mathcal{U}}$ denotes the same functional as
in (2.6). We start with the following lemma.

######  Lemma 4.11.

Let $\Phi:{\mathcal{P}}(\mathbb{T})\rightarrow{\mathbb{R}}$ be a rotation-
invariant function that satisfies _( Diff-$\Phi$)_ $($see Definition
$\ref{def:rotation:invariant:function})$. Then, for any probability measure
$\mu\in{\mathcal{P}}(\mathbb{T})$ and any distribution
$q\in(W^{1,\infty}(\mathbb{T}))^{\prime}$ with $\langle
q,{\mathbbm{1}}\rangle=0:$

$\begin{split}&\frac{\delta\Phi}{\delta
m}(\mu)\bigl{(}\mu^{\prime}\bigr{)}=0,\qquad\textrm{and}\qquad\frac{\delta^{2}\Phi}{\delta
m^{2}}(\mu)\bigl{(}\mu^{\prime},q\bigr{)}+\frac{\delta\Phi}{\delta
m}(\mu)\bigl{(}q^{\prime}\bigr{)}=0,\end{split}$ (4.36)

where $\mu^{\prime}$ and $q^{\prime}$ are the derivatives, in the sense of
distributions, of $\mu$ and $q$.

###### Proof.

We start from the very definition of rotation-invariant function. It says
that, for any $\psi\in\mathbb{T}$,

$\frac{\mathrm{d}}{\mathrm{d}\psi}_{|\psi=0}\Phi\bigl{(}\mu\circ\tau_{\psi}^{-1}\bigr{)}=0.$

The left-hand side writes in the form $-\langle[\delta\Phi/\delta
m](\mu),\mu^{\prime}\rangle$. We deduce the first identity in the statement.

Assume now that $\mu$ has a (strictly) positive continuous density
$\mathrm{d}\mu/\mathrm{d}x$ and consider an element $q\in
L^{\infty}(\mathbb{T})$ such that $\langle q,{\mathbbm{1}}\rangle=0$. Then,
for $\varepsilon$ small enough, $\mathrm{d}\mu/\mathrm{d}x+\varepsilon q$ may
be regarded as a density on $\mathbb{T}$. With a slight abuse of notation, we
write $\mu+\varepsilon q\in{\mathcal{P}}(\mathbb{T})$. Replacing $\mu$ by
$\mu+\varepsilon q$ in the first identity in the statement and then taking the
derivative with respect to $\varepsilon$ at $\varepsilon=0$, we deduce that
the second identity in the statement holds at any pair $(\mu,q)$ satisfying
the prescribed conditions. Using the density of $L^{\infty}(\mathbb{T})$ in
$(W^{1,\infty})^{\prime}(\mathbb{T})$ together with the fact that $\Phi$
satisfies (Diff-$\Phi$), the second identity also holds at any pair $(\mu,q)$,
with $\mu$ as before and $q\in(W^{1,\infty})^{\prime}(\mathbb{T})$ such that
$\langle q,{\mathbbm{1}}\rangle=0$. Approximating (for the $1$-Wasserstein
topology) any $\mu\in{\mathcal{P}}(\mathbb{T})$ by probability measures with a
positive density and then invoking again the regularity properties of $\Phi$,
we finally obtain that the second identity is satisfied at any
$\mu\in{\mathcal{P}}(\mathbb{T})$ and
$q\in(W^{1,\infty}(\mathbb{T}))^{\prime}$ with $\langle
q,{\mathbbm{1}}\rangle=0$. ∎

We deduce the following important proposition.

######  Proposition 4.12.

Assume that $\Phi$ satisfies _( Reg-$\Phi$-(${4,3}$))_. Then, for any
$\alpha\in(0,1)$ and $\eta\in(0,1)$, there exist two positive constants
$\lambda$ and $C$, with $\lambda$ only depending on $\kappa$, $\alpha$ and
$\eta$, and $C$ only depending on $\kappa$, $\alpha$, $\eta$ and the bounds in
_( Reg-$\Phi$-(${\alpha,2}$))_, such that, for any
$\mu\in{\mathcal{Q}}_{\eta}$, any $t>0$ and any $z_{1},z_{2}\in\mathbb{T}$,

$\displaystyle\Bigl{|}\frac{\delta^{2}{\mathcal{U}}}{\delta
m^{2}}(t,\mu)(z_{1},z_{2})\Big{|}\leq C,\quad t\geq 0,$ (4.37)
$\displaystyle\min(1,t^{1-\alpha/4})\Bigl{|}\partial_{z_{2}}\partial_{z_{1}}\frac{\delta^{2}{\mathcal{U}}}{\delta
m^{2}}(t,\mu)(z_{1},z_{2})\Big{|}\leq Ce^{-\lambda t},\quad t>0.$ (4.38)

###### Proof.

First Step. We start with the proof of (4.37). It is a mere consequence of the
representation formula in Theorem 3.2 and the bound (4.27) in the statement of
Proposition 4.10.

Second Step. We now turn to the proof of (4.38). We recall the following
formula from Theorem 3.5:

$\partial_{z_{2}}\partial_{z_{1}}\frac{\delta^{2}{\mathcal{U}}}{\delta
m^{2}}(t,\mu)(z_{1},z_{2})=\frac{\delta^{2}{\Phi}}{\delta
m^{2}}\bigl{(}m(t\,;\mu)\bigr{)}\Big{(}d^{(1)}(t\,;\mu,z_{1}),d^{(1)}(t\,;\mu,z_{2})\Big{)}+\frac{\delta{\Phi}}{\delta
m}\bigl{(}m(t\,;\mu)\bigr{)}\Big{(}d^{(2)}(t\,;\mu,z_{1},z_{2})\Big{)},$

for $\mu\in{\mathcal{P}}(\mathbb{T})$, $t\geq 0$ and
$z_{1},z_{2}\in\mathbb{T}$. Using the same notation as in the statement of
Proposition 4.10, we rewrite the above identity in the form

$\begin{split}&\partial_{z_{2}}\partial_{z_{1}}\frac{\delta^{2}{\mathcal{U}}}{\delta
m^{2}}(t,\mu)(z_{1},z_{2})=\frac{\delta^{2}{\Phi}}{\delta
m^{2}}\bigl{(}m(t\,;\mu)\bigr{)}\Big{(}\tilde{d}^{(1)}(t\,;\mu,z_{1}),d^{(1)}(t\,;\mu,z_{2})\Big{)}\\\
&\hskip 15.0pt+q_{\infty}^{(1)}(\mu,z_{2})\frac{\delta^{2}{\Phi}}{\delta
m^{2}}\bigl{(}m(t\,;\mu)\bigr{)}\Big{(}p^{\prime}(t\,;\mu),d^{(1)}(t\,;\mu,z_{2})\Big{)}+\frac{\delta{\Phi}}{\delta
m}\bigl{(}m(t\,;\mu)\bigr{)}\Big{(}d^{(2)}(t\,;\mu,z_{1},z_{2})\Big{)}.\end{split}$

With the same notation as in (4.32) and (4.33), we get

$\begin{split}\partial_{z_{2}}\partial_{z_{1}}\frac{\delta^{2}{\mathcal{U}}}{\delta
m^{2}}(t,\mu)(z_{1},z_{2})&=\frac{\delta^{2}{\Phi}}{\delta
m^{2}}\bigl{(}m(t\,;\mu)\bigr{)}\Big{(}\tilde{d}^{(1)}(t\,;\mu,z_{1}),d^{(1)}(t\,;\mu,z_{2})\Big{)}\\\
&\hskip 15.0pt-q_{\infty}^{(1)}(\mu,z_{2})\frac{\delta{\Phi}}{\delta
m}\bigl{(}m(t\,;\mu)\bigr{)}\Big{(}\partial_{x}d^{(1)}(t\,;\mu,z_{2})\Big{)}+\frac{\delta{\Phi}}{\delta
m}\bigl{(}m(t\,;\mu)\bigr{)}\Big{(}d^{(2)}(t\,;\mu,z_{1},z_{2})\Big{)}\\\
&=\frac{\delta^{2}{\Phi}}{\delta
m^{2}}\bigl{(}m(t\,;\mu)\bigr{)}\Big{(}\tilde{d}^{(1)}(t\,;\mu,z_{1}),d^{(1)}(t\,;\mu,z_{2})\Big{)}+\frac{\delta{\Phi}}{\delta
m}\bigl{(}m(t\,;\mu)\bigr{)}\Big{(}\bar{d}^{(2)}(t\,;\mu,z_{1},z_{2})\Bigr{)}.\end{split}$

By the first identity in (4.36), we can remove for free
$q_{\infty}^{(2)}(\mu,z_{1},z_{2})p^{\prime}(t\,;\mu)$ in the second term on
the last line. Using again the notation from the statement of Proposition
4.10, we obtain

$\begin{split}\partial_{z_{2}}\partial_{z_{1}}\frac{\delta^{2}{\mathcal{U}}}{\delta
m^{2}}(t,\mu)(z_{1},z_{2})&=\frac{\delta^{2}{\Phi}}{\delta
m^{2}}\bigl{(}m(t\,;\mu)\bigr{)}\Big{(}\tilde{d}^{(1)}(t\,;\mu,z_{1}),d^{(1)}(t\,;\mu,z_{2})\Big{)}+\frac{\delta{\Phi}}{\delta
m}\bigl{(}m(t\,;\mu)\bigr{)}\Big{(}\tilde{d}^{(2)}(t\,;\mu,z_{1},z_{2})\Bigr{)}.\end{split}$

The end of the proof is similar to the derivation of (3.22), by using (4.28)
and combining with the shape of (Reg-$\Phi$-(${\alpha,2}$)) (see, in
particular, (2.16)). ∎

###  4.4. Semi-group generated by the empirical distribution

We now address the weak error, as in the statement of Theorem 4.3. By the same
regularisation argument as in the proof of Theorem 3.8, we can assume that
$\Phi$ satisfies (Reg-$\Phi$-(${4,3}$)). Following the statement of
Proposition 4.12, we must then prove that the constant $C$ that we obtain in
the end in the main inequality of Theorem 4.3 only depends on $\Phi$ through
the bounds in (Reg-$\Phi$-(${\alpha,2}$)), for a fixed value of $\alpha$ as in
the statement of Theorem 4.3.

Throughout the proof, we make use of the semi-group generated by the empirical
distribution. We hence call ${\mathcal{P}}_{N}(\mathbb{T})$ the collection of
probability measures $\mu$ that are uniformly distributed on some finite state
$\\{x_{1},\cdots,x_{N}\\}$, with $(x_{1},\cdots,x_{N})\in{\mathbb{T}}^{N}$. In
particular, we have that, with probability 1, $\mu^{N}$ is in
${\mathcal{P}}_{N}(\mathbb{T})$.

With this notation, we can define the analogue of the mapping ${\mathcal{U}}$,
but for the particle system (1.1), namely:

$\overline{\mathcal{U}}^{N}(t,\mu)={\mathbb{E}}\bigl{[}\Phi\bigl{(}\mu_{t}^{N}\bigr{)}\,\big{|}\,\mu^{N}_{0}=\mu\bigr{]},\quad\mu\in{\mathcal{P}}_{N}(\mathbb{T}).$

This definition requires a modicum of care, since we must check that
$\overline{\mathcal{U}}^{N}(t,\cdot)$ is indeed a function of
$\mu\in{\mathcal{P}}_{N}(\mathbb{T})$ (in other words, it is a symmetric
function of $(x_{1},\cdots,x_{N})$). This follows from the exchangeability of
the particle system (1.1): for any permutation $\sigma$ on $\\{1,\cdots,N\\}$,
$(Y_{t}^{1,N},\cdots,Y_{t}^{N,N})_{t\geq 0}$ starting from
$(x_{1},\cdots,x_{N})$ has the same law as
$(Y_{t}^{1,N},\cdots,Y_{t}^{N,N})_{t\geq 0}$ starting from
$(x_{\sigma(1)},\cdots,x_{\sigma(N)})$. Accordingly, the function
$\overline{\mathcal{U}}^{N}(t,\cdot)$ is a symmetric function of
$(x_{1},\cdots,x_{N})$ and hence may be regarded as a function of
$\sum_{i=1}^{N}\delta_{x_{i}}/N$. Notice in particular that, by the Markov
property of $(Y_{t}^{1,N},\cdots,Y_{t}^{N,N})_{t\geq 0}$, for any $0\leq s\leq
t$, we have

${\mathbb{E}}\bigl{[}\Phi\bigl{(}{\mu}_{t}^{N}\bigr{)}|{\mathcal{F}}_{s}\bigr{]}=\overline{\mathcal{U}}^{N}\bigl{(}t-s,(Y^{1,N}_{s},\cdots,Y^{N,N}_{s})\bigr{)}=\overline{\mathcal{U}}^{N}\bigl{(}t-s,\mu^{N}_{s}\bigr{)},$
(4.39)

with the notation in the middle term being rather abusive, but making sense
thanks to the symmetry of $\overline{\mathcal{U}}^{N}$.

We then proceed step by step. The first step is dictated by the analysis of
the strong error in [27] and consists of a preliminary form of the result up
until time $\exp(N^{1/2})$, for $\mu$ close enough to
${\mathcal{I}}=\\{p_{\psi},\psi\in{\mathbb{T}}\\}$.

######  Lemma 4.13.

There exist $\delta>0$, only depending on $\kappa$, and a constant $C\geq 0$,
only depending on $\delta$, $\kappa$ and the bounds for $\Phi$ in _(
Reg-$\Phi$-(${\alpha,2}$))_, such that, for any $N\geq 1$ and any
$\mu\in{\mathcal{P}}_{N}(\mathbb{T})$ satisfying $\textrm{\rm
dist}_{\|\cdot\|_{-1,2}}(\mu,{\mathcal{I}})\leq\delta$,

$\begin{split}\forall
t\in\bigl{[}0,\exp(N^{1/2})\bigr{]},\quad\Bigl{|}\overline{\mathcal{U}}^{N}(t,\mu)-{\mathcal{U}}(t,\mu)\Bigr{|}\leq\frac{C}{N},\end{split}$

where $\textrm{\rm
dist}_{\|\cdot\|_{-1,2}}(\mu,{\mathcal{I}})=\inf_{\psi\in\mathbb{T}}\|\mu-
p_{\psi}\|_{-1,2}$. Here, we may choose $\delta$ such that $\\{\mu:\textrm{\rm
dist}_{\|\cdot\|_{-1,2}}(\mu,{\mathcal{I}})\leq\delta\Bigr{\\}}\subset{\mathcal{Q}}_{1/2}$.

###### Proof.

The proof relies on some auxiliary results obtained in [27].

First Step. We apply [27, (98-99)]. It says that there exist a time $T>0$, a
real $\varepsilon>0$ and a constant $C$, all independent of $N$ and the
initial conditions, such that, for any integer $n\geq 1$,

${\mathbb{P}}\biggl{(}\sup_{0\leq t\leq nT}\textrm{\rm
dist}_{\|\cdot\|_{-1,2}}(\mu^{N}_{t},{\mathcal{I}})\leq\varepsilon\,\big{|}\,\bigl{\|}\mu^{N}_{0}-p_{\textrm{proj}(\mu^{N}_{0})}\bigr{\|}_{-1,2}\leq\frac{\varepsilon}{2}\biggr{)}\geq\Bigl{[}1-\exp\bigl{(}-CN)\Bigr{]}^{n},$
(4.40)

where $\textrm{\rm proj}:{\mathcal{P}}(\mathbb{T})\rightarrow\mathbb{T}$ is
the projection mapping defined in [49, Lemma 2.8] and [27, Lemma 3.5] (with
the small difference444There is another technical difference. In [27, 49], the
dual space is not $W^{-1,2}(\mathbb{T})$, but $H^{-1}_{0}=\\{f\in
W^{1,2}(\mathbb{T}):\langle f,{\mathbbm{1}}\rangle=0\\}^{\prime}$. Of course,
any element of $W^{-1,2}(\mathbb{T})$ may be canonically projected onto an
element of $H^{-1}_{0}$ by discarding its constant Fourier mode. that the
function $\mathbb{T}$ is defined here as ${\mathbb{R}}/{\mathbb{Z}}$, whilst
it is defined as ${\mathbb{R}}/(2\pi{\mathbb{Z}})$ in [27, 49]). Notice that
the conditioning in the left-hand side is implicitly required in [27] (see
(84)). More importantly, there is no need to assume that $\mu_{0}^{N}$ is the
$N$-sample of a common distribution and the result holds for an arbitrary
initial condition $\mu_{0}^{N}=\mu\in{\mathcal{P}}_{N}(\mathbb{T})$.

On ${\mathcal{I}}$, the projection proj reduces to the trivial mapping
$\textrm{\rm proj}(p_{\psi})=\psi$, $\psi\in\mathbb{T}$. Therefore, if some
probability measure $\nu\in{\mathcal{P}}(\mathbb{T})$ is close to
${\mathcal{I}}$ (for $\|\cdot\|_{-1,2}$), then it is close to some
$p_{\psi}\in{\mathcal{I}}$ and, by continuity of proj with respect to
$W^{-1,2}(\mathbb{T})$, $\textrm{\rm proj}(\nu)$ is close to $\psi$. In the
end, $\nu$ is close to $p_{\textrm{\rm proj}(\nu)}$. This continuity result,
combined with a standard compactness argument, may be formulated as follows.
For the same $\varepsilon$ as above, we can find some $\delta_{\varepsilon}>0$
such that $\textrm{\rm
dist}_{\|\cdot\|_{-1,2}}(\nu,{\mathcal{I}})\leq\delta_{\varepsilon}$ implies
$\|\nu-p_{\textrm{\rm proj}(\nu)}\|_{-1,2}\leq\varepsilon/2$. Hence, taking
$\mu\in{\mathcal{P}}_{N}(\mathbb{T})$ with $\textrm{\rm
dist}_{\|\cdot\|_{-1,2}}(\mu,{\mathcal{I}})\leq\delta_{\varepsilon}$, we get

${\mathbb{P}}\biggl{(}\sup_{0\leq t\leq nT}\textrm{\rm
dist}_{\|\cdot\|_{-1,2}}(\mu^{N}_{t},{\mathcal{I}})\leq\varepsilon\,\Big{|}\,\mu_{0}^{N}=\mu\biggr{)}\geq\Bigl{[}1-\exp\bigl{(}-CN)\Bigr{]}^{n}.$

Following [27, (100)], we deduce that, for $n=\exp(N^{1/2})$ and for a new
value of $C$,

${\mathbb{P}}\biggl{(}\sup_{0\leq t\leq nT}\textrm{\rm
dist}_{\|\cdot\|_{-1,2}}(\mu^{N}_{t},{\mathcal{I}})\leq\varepsilon\,\Big{|}\,\mu_{0}^{N}=\mu\biggr{)}\geq
1-\frac{C}{N}.$

This may be rewritten in the form

${\mathbb{P}}\Bigl{(}\tau_{N}\leq\exp(N^{1/2})\,\big{|}\,\mu_{0}^{N}=\mu\Bigr{)}\leq\frac{C}{N},$
(4.41)

where $\tau_{N}$ is the stopping time

$\tau_{N}=\inf\Bigl{\\{}t\geq 0:\textrm{\rm
dist}_{\|\cdot\|_{-1,2}}(\mu^{N}_{t},{\mathcal{I}})\geq\varepsilon\Bigr{\\}}.$

Second Step. Following (2.8), we have the following expansion, which holds for
any $t\geq 0$,

$\begin{split}{\mathbb{E}}\bigl{[}\Phi\bigl{(}\mu^{N}_{t}\bigr{)}|\,\mu_{0}^{N}=\mu\bigr{]}-{\mathcal{U}}(t,\mu)&={\mathbb{E}}\bigl{[}\Phi\bigl{(}\mu^{N}_{t}\bigr{)}|\,\mu_{0}^{N}=\mu\bigr{]}-{\mathbb{E}}\bigl{[}{\mathcal{U}}\bigl{(}t-t\wedge\tau_{N},\mu^{N}_{t\wedge\tau_{N}}\bigr{)}|\,\mu_{0}^{N}=\mu\bigr{]}\\\
&\hskip
15.0pt+{\mathbb{E}}\Bigl{[}{\mathcal{U}}\bigl{(}t-t\wedge\tau_{N},\mu^{N}_{t\wedge\tau_{N}}\bigr{)}-{\mathcal{U}}\bigl{(}t,\mu^{N}_{0}\bigr{)}|\,\mu_{0}^{N}=\mu\Bigr{]}.\end{split}$
(4.42)

If we restrict ourselves to $t\in[0,\exp(N^{1/2})]$, then (4.41) says that the
first difference in the right-hand side is bounded by $C/N$, by recalling that
$\Phi$ and thus ${\mathcal{U}}$ are bounded and allowing $C$ to depend on
$\|\Phi\|_{\infty}$.

As for the second term in the right-hand side of (4.42), we may follow (2.9),
by expanding the term $({\mathcal{U}}(t-s,\mu^{N}_{s}))_{s\geq 0}$ up until
time $t\wedge\tau_{N}$. Using the fact that, up until time $\tau_{N}$, the
$\|\cdot\|_{-1,2}$-distance between $\mu^{N}$ and ${\mathcal{I}}$ remains less
than $\varepsilon$ and assuming without any loss of generality that
$\varepsilon$ is small enough so that $\\{\mu:\textrm{\rm
dist}_{\|\cdot\|_{-1,2}}(\mu,{\mathcal{I}})\leq\varepsilon\\}\subset{\mathcal{Q}}_{1/2}$,
we get from Proposition 4.12 that this term is also bounded by $C/N$, with $C$
now depending on $\kappa$ and the bounds for $\Phi$ in
(Reg-$\Phi$-(${\alpha,2}$)). ∎

The result is now extended to initial conditions
$\mu\in{\mathcal{P}}_{N}(\mathbb{T})\cap{\mathcal{Q}}_{\eta}$, for any
$\eta>0$, but on a (slightly) smaller time scale.

######  Proposition 4.14.

For any $\eta\in(0,1)$, there exists a constant $C$, only depending on $\eta$,
$\kappa$ and the bounds for $\Phi$ in _( Reg-$\Phi$-(${\alpha,2}$))_, such
that, for any $N\geq 1$ and
$\mu\in{\mathcal{Q}}_{\eta}\cap{\mathcal{P}}_{N}(\mathbb{T})$,

$\forall
t\in[0,\exp(N^{1/4})],\quad\bigl{|}\overline{\mathcal{U}}^{N}(t,\mu)-{\mathcal{U}}(t,\mu)\bigr{|}\leq\frac{C}{N}.$
(4.43)

###### Proof.

First step. Lemma 4.13 says that, for $\delta$ as therein, the result holds up
until time $\exp(N^{1/2})$ if $\textrm{\rm
dist}_{\|\cdot\|_{-1,2}}(\mu,{\mathcal{I}})\leq\delta$. Then, for any
$\mu\in{\mathcal{Q}}_{\eta}\cap{\mathcal{P}}_{N}(\mathbb{T})$, Proposition 4.1
says that there exists a (fixed hence independent of $N$) time $T$, only
depending on $\delta$, $\eta$ and $\kappa$, such that

$\forall t\geq T,\quad\textrm{\rm
dist}_{\|\cdot\|_{-1,2}}\bigl{(}m(t\,;\mu),{\mathcal{I}}\bigr{)}\leq\frac{\delta}{2}.$
(4.44)

Next, we claim that (the proof is given in the third step below)

${\mathbb{P}}\Bigl{(}\bigl{\|}m(T\,;\mu)-\mu^{N}_{T}\bigr{\|}_{-1,2}>\frac{\delta}{2}\,\big{|}\,\mu_{0}^{N}=\mu\Bigr{)}\leq\frac{C}{N},$
(4.45)

where $C$ only depends on $\eta$, $\kappa$ and the bounds for $\Phi$ in
(Reg-$\Phi$-(${\alpha,2}$)), from which we deduce that

${\mathbb{P}}\Bigl{(}\textrm{\rm
dist}_{\|\cdot\|_{-1,2}}\bigl{(}\mu^{N}_{T},{\mathcal{I}}\bigr{)}>\delta\,\big{|}\,\mu_{0}^{N}=\mu\Bigr{)}\leq\frac{C}{N}.$
(4.46)

By (4.39), we have, for $t\geq T$,

$\overline{\mathcal{U}}^{N}(t,\mu)={\mathbb{E}}\bigl{[}\overline{\mathcal{U}}^{N}\bigl{(}t-T,\mu^{N}_{T}\bigr{)}\big{|}\,\mu_{0}^{N}=\mu\bigr{]},$

when $\mu^{N}_{0}=\mu$. Hence, by (4.43) (for $\textrm{\rm
dist}_{\|\cdot\|_{-1,2}}(\mu,{\mathcal{I}})\leq\delta$) and (4.46), for
$t\in[T,\exp(N^{1/2})/2]$,

$\begin{split}&\Bigl{|}\overline{\mathcal{U}}^{N}(t,\mu)-{\mathbb{E}}\Bigl{[}{\mathcal{U}}\bigl{(}t-T,\mu^{N}_{T}\bigr{)}{\mathbf{1}}_{\\{\textrm{\rm
dist}_{\|\cdot\|_{-1,2}}(\mu^{N}_{T},{\mathcal{I}})\leq\delta\\}}\big{|}\,\mu_{0}^{N}=\mu\Bigr{]}\Bigr{|}\\\
&\leq C{\mathbb{P}}\Bigl{(}\textrm{\rm
dist}_{\|\cdot\|_{-1}}\bigl{(}\mu^{N}_{T},{\mathcal{I}}\bigr{)}>\delta\,\big{|}\,\mu_{0}^{N}=\mu\Bigr{)}+{\mathbb{E}}\Bigl{[}\Bigl{|}{\mathcal{U}}^{N}\bigl{(}t-T,\mu^{N}_{T}\bigr{)}-\overline{\mathcal{U}}\bigl{(}t-T,\mu^{N}_{T}\bigr{)}\Bigr{|}{\mathbf{1}}_{\\{\textrm{\rm
dist}_{\|\cdot\|_{-1,2}}(\mu^{N}_{T},{\mathcal{I}})\leq\delta\\}}\big{|}\,\mu_{0}^{N}=\mu\Bigr{]}\\\
&\leq\frac{C}{N},\end{split}$

where we used the fact that $\overline{\mathcal{U}}^{N}$ is bounded by
$\|\Phi\|_{\infty}$. On the first line, we can replace the indicator function
(inside the expectation) by $\Xi(\mu_{T}^{N})$ for some function
$\Xi:{\mathcal{P}}(\mathbb{T})\rightarrow[0,1]$ that is equal to $1$ on the
ball $\\{\nu\in{\mathcal{P}}(\mathbb{T}):\textrm{\rm
dist}_{\|\cdot\|_{-1,2}}(\nu,{\mathcal{I}})\leq\delta\\}$. By (4.46) again, we
deduce that, for
$\mu\in{\mathcal{Q}}_{\eta}\cap{\mathcal{P}}_{N}(\mathbb{T})$,

$\Bigl{|}\overline{\mathcal{U}}^{N}(t,\mu)-{\mathbb{E}}\bigl{[}{\mathcal{U}}\bigl{(}t-T,\mu^{N}_{T}\bigr{)}\Xi(\mu^{N}_{T})|\,\mu_{0}^{N}=\mu\bigr{]}\Bigr{|}\leq\frac{C}{N}.$
(4.47)

Second Step. We complete the proof of (4.43), recalling that the latter is
already known to hold on $[0,T]$ (since there is no need for any ergodic
estimates in finite time). In order to proceed, assume for a while that we are
given a function $\Xi:{\mathcal{P}}({\mathbb{T}})\rightarrow[0,1]$ such that

  1. (1)

$\Xi$ matches 1 on ${\mathcal{Q}}_{1/2}$ (the latter containing
$\\{\nu\in{\mathcal{P}}(\mathbb{T}):\textrm{\rm
dist}_{\|\cdot\|_{-1,2}}(\nu,{\mathcal{I}})\leq\delta\\}$) and $0$ outside
${\mathcal{Q}}_{3/4}$;

  2. (2)

$\Xi$ is a smooth functional of $\mu\in{\mathcal{P}}(\mathbb{T})$.

Then, since ${\mathcal{Q}}_{1/2}$ contains
$\\{\nu\in{\mathcal{P}}(\mathbb{T}):\textrm{\rm
dist}_{\|\cdot\|_{-1,2}}(\nu,{\mathcal{I}})\leq\delta\\}$, we can apply
(4.47). Since $\Xi$ is zero outside ${\mathcal{Q}}_{3/4}$, we know from
Proposition 4.12 that ${\mathcal{U}}(t-T,\cdot)\Xi(\cdot)$ satisfies
(Reg-$\Phi$-(${\alpha,2}$)) with explicit bounds that are uniform with respect
to $t\geq T$. We then apply the finite-horizon version of (4.43) to
$\Phi(\cdot)={\mathcal{U}}(t-T,\cdot)\Xi(\cdot)$. By (4.47), we get, for any
$t\geq T$ and $\mu\in{\mathcal{Q}}_{\eta}\cap{\mathcal{P}}_{N}(\mathbb{T})$,

$\Bigl{|}\overline{\mathcal{U}}^{N}(t,\mu)-{\mathcal{U}}\bigl{(}t-T,m(T\,;\mu)\bigr{)}\Xi\bigl{(}m(T\,;\mu)\bigr{)}\Bigr{|}\leq\frac{C}{N},$

for $C$ only depending on $\eta$, $\kappa$ and the bounds for $\Phi$ in
(Reg-$\Phi$-(${\alpha,2}$)). Noticing that
${\mathcal{U}}(t-T,m(T\,;\mu))={\mathcal{U}}(t,\mu)$ and that
$\Xi(m(T\,;\mu))=1$ for our choice of $T$ (see (4.44)), we get the announced
result.

The function $\Xi$ is constructed as in the statement of Proposition 4.4. We
consider a smooth non-decreasing cut-off function
$\varphi:[0,1]\rightarrow[0,1]$ that is equal to $1$ on $[0,1/2]$ and to $0$
on $[3/4,1]$. We then let $\Xi(\mu)=\varphi(|\mu^{1}|)$.

Third Step. We now prove (4.45). It is again a consequence of the finite time
horizon of (4.43), but with a difference of choosing (temporarily) the
functional $\Phi$ in the definition of ${\mathcal{U}}$ as in Proposition 2.4
with $(d+\alpha)/2=1$ and $\nu_{0}=\mu_{T}$ therein. The result then follows
from Markov’s inequality. ∎

Our last step is to extend the previous result to times greater than
$\exp(N^{1/4})$. The key idea is that, in long time, the empirical measure
necessarily visits the set ${\mathcal{Q}}_{\eta}$ quite often, for any
$\eta\in(0,1)$.

######  Proposition 4.15.

For any $\eta\in(0,1)$, there exists a constant $C$, only depending on $\eta$,
$\kappa$ and the bounds for $\Phi$ in _( Reg-$\Phi$-(${\alpha,2}$))_, such
that, for any $N\geq 1$ and
$\mu\in{\mathcal{Q}}_{\eta}\cap{\mathcal{P}}_{N}(\mathbb{T})$,

$\forall t\geq
0,\quad\bigl{|}\overline{\mathcal{U}}^{N}(t,\mu)-{\mathcal{U}}(t,\mu)\bigr{|}\leq\frac{C}{N}.$
(4.48)

###### Proof.

By Proposition 4.14, it suffices to prove the result for $t\geq\exp(N^{1/4})$.
For $T_{N}=\exp(N^{1/4})/2$, we call

$\tau_{N}:=\inf\biggl{\\{}s>0:\biggl{|}\int_{\mathbb{T}}e^{-i2\pi\theta}\mu^{N}_{s+t-T_{N}}(\mathrm{d}\theta)\biggr{|}\geq\eta\biggr{\\}}.$

We prove below (see Lemma 4.16) that ${\mathbb{P}}\bigl{(}\tau_{N}\geq
N^{1/4}\bigr{)}\leq C/{N}$, for $C$ as in the statement and for $\eta$ small
enough. By the strong version of the Markov property (4.39) (noticing that
$N^{1/4}\leq\exp(N^{1/4})/2$ for any $N\geq 1$),

$\overline{\mathcal{U}}^{N}(t,\mu)={\mathbb{E}}\Bigl{[}\overline{\mathcal{U}}^{N}\Bigl{(}T_{N}+T_{N}\wedge\tau_{N},\mu^{N}_{t-(T_{N}+T_{N}\wedge\tau_{N})}\Bigr{)}\big{|}\,\mu_{0}^{N}=\mu\Bigr{]},$

and then, assuming without any loss of generality that $\eta$ is small enough,

$\begin{split}&\Bigl{|}\overline{\mathcal{U}}^{N}(t,\mu)-{\mathbb{E}}\Bigl{[}\overline{\mathcal{U}}^{N}\Bigl{(}T_{N}+T_{N}\wedge\tau_{N},\mu^{N}_{t-(T_{N}+T_{N}\wedge\tau_{N})}\Bigr{)}{\mathbbm{1}}_{\\{\tau_{N}<T_{N}\\}}\big{|}\,\mu_{0}^{N}=\mu\Bigr{]}\Bigr{|}\leq\frac{C}{N}.\end{split}$

On the event $\\{\tau_{N}<T_{N}\\}$,
$\overline{\mu}^{N}_{t-(T_{N}+T_{N}\wedge\tau_{N})}\in{\mathcal{Q}}_{\eta}$.
Therefore, by Proposition 4.14, we can replace $\overline{\mathcal{U}}^{N}$ by
${\mathcal{U}}$ inside the expectation and hence get

$\begin{split}&\Bigl{|}\overline{\mathcal{U}}^{N}(t,\mu)-{\mathbb{E}}\Bigl{[}{\mathcal{U}}\Bigl{(}T_{N}+T_{N}\wedge\tau_{N},\mu^{N}_{t-(T_{N}+T_{N}\wedge\tau_{N})}\Bigr{)}{\mathbbm{1}}_{\\{\tau_{N}<T_{N}\\}}\big{|}\,\mu_{0}^{N}=\mu\Bigr{]}\Bigr{|}\leq\frac{C}{N}.\end{split}$
(4.49)

By Proposition 4.1, we know, for $s\geq T_{N}$ and
$\nu\in{\mathcal{Q}}_{\eta}$,

$\begin{split}{\mathcal{U}}(s,\nu)=\Phi\bigl{(}{\mathcal{L}}(X_{s}|X_{0}\sim\nu)\bigr{)}&=\Phi({\mathcal{I}})+O\bigl{(}\frac{1}{N}\bigr{)}={\mathcal{U}}(t,\mu)+O\bigl{(}\frac{1}{N}\bigr{)},\end{split}$
(4.50)

with the Landau symbol $O(\cdot)$ being independent of $\nu$, from which we
deduce that we can replace the expectation in (4.49) by
${\mathcal{U}}(t,\mu)$. This completes the proof. In the above, we used the
slightly abusive notation ${\mathcal{L}}(X_{s}|X_{0}\sim\nu)$ to denote the
law of $X_{s}$ in (1.3) when the law of $X_{0}$ is $\nu$. ∎

###  4.5. End of the proof of Theorem 4.3

We now have all the ingredients to complete the proof of Theorem 4.3. For an
$N$-sample $(Y^{i,N}_{0})_{i=1,\cdots,N}$ with law
${\mu_{\text{init}}}\in{\mathcal{Q}}_{\eta}$, for some $\eta>0$, we let
$\mu_{0}^{N}=(1/N)\sum_{i=1}^{N}\delta_{Y_{0}^{i,N}}$.

###### Proof of Theorem 4.3.

First Step. We start with the following (quite standard) computation:

$\begin{split}{\mathbb{E}}\Bigl{[}\|\mu_{0}-\mu^{N}_{0}\|_{-1,2}^{2}\Bigr{]}&=\sum_{n\geq
0}\frac{1}{(1+n^{2})}{\mathbb{E}}\biggl{[}\biggl{|}\int_{\mathbb{T}}e^{-{\mathrm{i}}2\pi
n\theta}\bigl{(}\mu_{0}-\mu^{N}_{0}\bigr{)}(\mathrm{d}\theta)\biggr{|}^{2}\biggr{]}\leq\frac{1}{N}\sum_{n\geq
0}\frac{1}{(1+n^{2})}\leq\frac{c}{N},\end{split}$ (4.51)

for some universal constant $c\geq 0$, where we used the fact that, for any
bounded measurable function $f:\mathbb{T}\rightarrow\mathbb{R}$,

$\begin{split}&{\mathbb{E}}\biggl{[}\biggl{|}\int_{\mathbb{T}}f(\theta)\bigl{(}\mu_{0}-\mu^{N}_{0}\bigr{)}(\mathrm{d}\theta)\biggr{|}^{2}\biggr{]}=\frac{1}{N}\int_{\mathbb{T}}\biggl{|}f(\theta)-\int_{\mathbb{T}}f(\theta^{\prime})\mathrm{d}\theta^{\prime}\biggr{|}^{2}d\theta\leq\frac{\|f\|_{\infty}^{2}}{N}.\end{split}$

We deduce that, for any $\varrho>0$,

${\mathbb{P}}\Bigl{(}\bigl{\|}\mu_{0}-\mu^{N}_{0}\bigr{\|}_{-1,2}\geq\varrho\Bigr{)}\leq\frac{c}{\varrho^{2}N}.$

Now, we can indeed choose $\varrho$ such that, for any two probability
measures $\nu_{1},\nu_{2}\in{\mathcal{P}}(\mathbb{T})$ with
$\|\nu_{1}-\nu_{2}\|_{-1,2}\leq\varrho$, it holds that
$|\nu_{1}^{1}-\nu_{2}^{1}|\leq\eta/2$ (we recall that $\nu_{1}^{1}$ and
$\nu_{2}^{1}$ are the 1-Fourier modes of $\nu_{1}$ and $\nu_{2}$), from which
we get that, for a constant $C$ depending on $\eta$,

${\mathbb{P}}\Bigl{(}\mu^{N}_{0}\in{\mathcal{Q}}_{\eta/2}\Bigr{)}\leq\frac{C}{N}.$

Therefore, by Proposition 4.15, there exists a constant $C$, only depending on
$\kappa$, $\eta$ and the bounds for $\Phi$ in (Reg-$\Phi$-(${\alpha,2}$)),
such that

$\forall t\geq
0,\quad\Bigl{|}{\mathbb{E}}\bigl{[}\overline{\mathcal{U}}^{N}\bigl{(}t,\mu_{0}^{N}\bigr{)}\bigr{]}-{\mathbb{E}}\bigl{[}{\mathcal{U}}\bigl{(}t,\mu_{0}^{N}\bigr{)}{\mathbf{1}}_{\\{\mu_{0}^{N}\in{\mathcal{Q}}_{\eta/2}\\}}\bigr{]}\Bigr{|}\leq\frac{C}{N}.$

We deduce that it suffices to show that

$\Bigl{|}{\mathbb{E}}\Bigl{[}\Bigl{(}{\mathcal{U}}(t,{\mu_{\text{init}}})-{\mathcal{U}}\bigl{(}t,\mu_{0}^{N}\bigr{)}\Bigr{)}{\mathbf{1}}_{\\{\mu_{0}^{N}\in{\mathcal{Q}}_{\eta/2}\\}}\Bigr{]}\Bigr{|}\leq\frac{C}{N}.$
(4.52)

Second Step. In order to prove (4.52), we may argue as in the second step of
the proof of Proposition 4.14. Indeed, we can consider a smooth function
$\Xi:{\mathcal{P}}({\mathbb{T}})\rightarrow[0,1]$ such that $\Xi$ is 1 on the
set ${\mathcal{Q}}_{\eta/2}$ and 0 outside the set ${\mathcal{Q}}_{\eta/4}$.
Then, instead of proving (4.52), it suffices to show that

$\Bigl{|}{\mathbb{E}}\Bigl{[}{\mathcal{U}}(t,{\mu_{\text{init}}})\Xi({\mu_{\text{init}}})-{\mathcal{U}}\bigl{(}t,\mu_{0}^{N}\bigr{)}\Xi\bigl{(}\mu_{0}^{N}\bigr{)}\Bigr{]}\Bigr{|}\leq\frac{C}{N}.$
(4.53)

Thanks to the cut-off function $\Xi$, the function
$(t,\mu)\mapsto{\mathcal{U}}(t,\mu)\Xi(\mu)$ satisfies the conclusion of
Corollary 4.12, even though $\mu\not\in{\mathcal{Q}}_{\eta}$. This suffices to
apply (2.10) with
${\mathcal{U}}(t,{\mu_{\text{init}}})\Xi({\mu_{\text{init}}})$ instead of
${\mathcal{U}}(t,{\mu_{\text{init}}})$ therein. ∎

The proof of Theorem 4.3 is hence completed provided that we prove the
following lemma, which we invoked in the proof of Lemma 4.15:

######  Lemma 4.16.

There exist a constant $\eta\in(0,1)$ and constant $C$, both independent of
$N$, such that, for any initial distribution $\mu\in{\mathcal{P}}(\mathbb{T})$
and any $t\geq 0$, the distribution of the stopping time
$\tau_{N}:=\inf\\{s>0:|\int_{\mathbb{T}}e^{-{\mathrm{i}}2\pi\theta}\mu^{N}_{s+t}(\mathrm{d}\theta)|\geq\eta\\}$
satisfies ${\mathbb{P}}\bigl{(}\tau_{N}\geq N^{1/4}\bigr{)}\leq C/N$.

###### Proof.

First Step. We go back to the shape of particle system (1.1) with $b$ as in
(4.1):

$\begin{split}dY_{t}^{j,N}&=-\frac{2\pi\kappa}{N}\sum_{k=1}^{N}\sin\Bigl{(}2\pi\bigl{(}Y_{t}^{j,N}-Y_{t}^{k,N}\bigr{)}\Bigr{)}\mathrm{d}t+\mathrm{d}W_{t}^{j},\quad
t\geq 0.\end{split}$

Let $E_{t}^{j,\ell,N}=\exp({\mathrm{i}}2\pi\ell Y_{t}^{j,N})$. Then, recalling
the notation $\overline{z}$ for denoting the complex conjugate of a complex
number $z\in{\mathbb{C}}$, we obtain that

$\begin{split}\mathrm{d}E_{t}^{j,\ell,N}&=-\frac{2\pi^{2}\kappa\ell}{N}E_{t}^{j,\ell,N}\sum_{k=1}^{N}\Bigl{(}E_{t}^{j,1,N}\overline{E}_{t}^{k,1,N}-\overline{E}_{t}^{j,1,N}E_{t}^{k,1,N}\Bigr{)}dt-2\pi^{2}\ell^{2}E_{t}^{j,\ell,N}\mathrm{d}t+{\mathrm{i}}2\pi\ell
E_{t}^{j,\ell,N}\mathrm{d}W_{t}^{j}\\\
&=-2\pi^{2}\kappa\ell\Bigl{(}E_{t}^{j,\ell+1,N}\mu_{t}^{1,N}-\overline{E}_{t}^{j,\ell-1,N}\overline{\mu}_{t}^{1,N}\Bigr{)}\mathrm{d}t-2\pi^{2}\ell^{2}E_{t}^{j,\ell,N}\mathrm{d}t+{\mathrm{i}}2\pi\ell
E_{t}^{j,\ell,N}\mathrm{d}W_{t}^{j},\end{split}$

where $\mu_{t}^{\ell,N}=N^{-1}\sum_{j=1}^{N}\overline{E}_{t}^{j,\ell,N}$ is
the $\ell$-Fourier mode of $\mu_{t}^{N}$. Taking the mean over
$j\in\\{1,\cdots,N\\}$, we get

$\begin{split}\mathrm{d}\overline{\mu}_{t}^{\ell,N}&=-2\pi^{2}\kappa\ell\Bigl{(}\overline{\mu}_{t}^{\ell+1,N}\mu_{t}^{1,N}-\mu_{t}^{\ell-1,N}\overline{\mu}_{t}^{1,N}\Bigr{)}\mathrm{d}t-2\pi^{2}\ell^{2}\overline{\mu}_{t}^{\ell,N}\mathrm{d}t+{\mathrm{i}}2\pi\ell\frac{1}{N}\sum_{j=1}^{N}E_{t}^{j,\ell,N}\mathrm{d}W_{t}^{j}.\end{split}$

Choosing $\ell=1$ and using the fact that $\kappa>1$, we deduce that there
exists a constant $c>1$, only depending on $\kappa$, such that

$\mathrm{d}\bigl{[}|\mu_{t}^{1,N}|^{2}\bigr{]}=\bigl{(}c^{-1}-c|\mu_{t}^{2,N}|\bigr{)}|\mu_{t}^{1,N}|^{2}\mathrm{d}t+\frac{c^{-1}}{{N}}\mathrm{d}t+\mathrm{d}K_{t}^{1}+\mathrm{d}M_{t}^{1},$
(4.54)

where $(K_{t}^{1})_{t\geq 0}$ is a non-decreasing absolutely continuous
process and $(M_{t}^{1})_{t\geq 0}$ is a martingale satisfying
$[\mathrm{d}/\mathrm{d}t]\langle M^{1}\rangle_{t}\leq c/N$. Similarly,
choosing $\ell=2$, we get

$\mathrm{d}\bigl{[}|\mu_{t}^{2,N}|^{2}\bigr{]}=\bigl{(}c|\mu_{t}^{1,N}||\mu_{t}^{2,N}|-c^{-1}|\mu_{t}^{2,N}|^{2}\bigr{)}\mathrm{d}t-\mathrm{d}K_{t}^{2}+\frac{c}{{N}}\mathrm{d}t+\mathrm{d}M_{t}^{2},$
(4.55)

where $(K_{t}^{2})_{t\geq 0}$ is a non-decreasing absolutely continuous
process and $(M_{t}^{2})_{t\geq 0}$ is a martingale satisfying
$[\mathrm{d}/\mathrm{d}t]\langle M^{2}\rangle_{t}\leq c/N$. We now let

$\Lambda_{t}^{N}=|\mu_{t}^{2,N}|^{2}-c^{4}|\mu_{t}^{1,N}|^{2},\quad t\geq 0.$
(4.56)

Using the expansions (4.54) and (4.55), we obtain

$\begin{split}\mathrm{d}\Lambda_{t}^{N}=\Bigl{[}\bigl{(}c|\mu_{t}^{1,N}||\mu_{t}^{2,N}|-c^{-1}|\mu_{t}^{2,N}|^{2}\bigr{)}-c^{4}\bigl{(}c^{-1}-c|\mu_{t}^{2,N}|\bigr{)}|\mu_{t}^{1,N}|^{2}\Bigr{]}\mathrm{d}t-\frac{c^{3}-c}{{N}}\mathrm{d}t-\mathrm{d}K_{t}+\mathrm{d}M_{t},\end{split}$
(4.57)

where $(K_{t})_{t\geq 0}$ is a non-decreasing absolutely continuous process
and $(M_{t})_{t\geq 0}$ is a martingale satisfying
$(\mathrm{d}/\mathrm{d}t)\langle M\rangle_{t}\leq C(c)/N$, in which $C(c)$ is
a constant that only depends on $c$. We then consider the same stopping time
$\tau_{N}$ as in the statement, but with $\eta=c^{-4}/4$. As long as
$t\leq\tau_{N}$, we have (notice that the term below is nothing but the first
term in the expansion of $\mathrm{d}\Lambda_{t}^{N}$)

$\begin{split}&\bigl{(}c|\mu_{t}^{1,N}||\mu_{t}^{2,N}|-c^{-1}|\mu_{t}^{2,N}|^{2}\bigr{)}-c^{4}\bigl{(}c^{-1}-c|\mu_{t}^{2,N}|\bigr{)}|\mu_{t}^{1,N}|^{2}\\\
&=-c^{3}|\mu_{t}^{1,N}|^{2}-c^{-1}|\mu_{t}^{2,N}|^{2}+c|\mu_{t}^{1,N}||\mu_{t}^{2,N}|+c^{5}|\mu_{t}^{2,N}||\mu_{t}^{1,N}|^{2}\\\
&\leq-c^{3}|\mu_{t}^{1,N}|^{2}-c^{-1}|\mu_{t}^{2,N}|^{2}+\frac{5}{4}c|\mu_{t}^{1,N}||\mu_{t}^{2,N}|\\\
&=-\frac{3}{8}c^{-1}\Bigl{(}|\mu_{t}^{2,N}|^{2}+c^{4}|\mu_{t}^{1,N}|^{2}\Bigr{)}-\frac{5}{8}\Bigl{(}c^{3/2}|\mu_{t}^{1,N}|-c^{-1/2}|\mu_{t}^{2,N}|\Bigr{)}^{2}\leq-\frac{3}{8}c^{-1}\Lambda_{t}^{N}.\end{split}$

By modifying the definition of $(K_{t})_{t\geq 0}$ in (4.57) and by assuming
that $c^{3}\geq 2c$, we then get

$\begin{split}\mathrm{d}\Lambda_{t}^{N}=-\frac{3}{8}c^{-1}\Lambda_{t}^{N}\mathrm{d}t-\frac{c}{N}\mathrm{d}t-\mathrm{d}K_{t}+\mathrm{d}M_{t}.\end{split}$

Therefore,

$\begin{split}\Lambda_{t}^{N}&=\exp\Bigl{(}-\frac{3}{8}c^{-1}t\Bigr{)}\biggl{[}\Lambda_{0}^{N}-\frac{c}{N}\int_{0}^{t}\exp\Bigl{(}\frac{3}{8}c^{-1}s\Bigr{)}\mathrm{d}s-\int_{0}^{t}\exp\Bigl{(}\frac{3}{8}c^{-1}s\Bigr{)}\mathrm{d}K_{s}+\int_{0}^{t}\exp\Bigl{(}\frac{3}{8}c^{-1}s\Bigr{)}\mathrm{d}M_{s}\biggr{]}\\\
&\leq\exp\Bigl{(}-\frac{3}{8}c^{-1}t\Bigr{)}\biggl{[}\Lambda_{0}^{N}+\int_{0}^{t}\exp\Bigl{(}\frac{3}{8}c^{-1}s\Bigr{)}\mathrm{d}M_{s}\biggr{]}.\end{split}$
(4.58)

We now observe that, for any integer $n\geq 0$,

$\begin{split}\sup_{n\leq t\leq
n+1}\biggl{[}\exp\Bigl{(}-\frac{3}{8}c^{-1}t\Bigr{)}\biggl{|}\int_{0}^{t}\exp\Bigl{(}\frac{3}{8}c^{-1}s\Bigr{)}\mathrm{d}M_{s}\biggr{|}\biggr{]}&\leq\sup_{n\leq
t\leq
n+1}\biggl{[}\biggl{|}\int_{0}^{t}\exp\Bigl{(}\frac{3}{8}c^{-1}(s-n)\Bigr{)}\mathrm{d}M_{s}\biggr{|}\biggr{]}.\end{split}$

Accordingly, by Burkholder-Davis-Gundy inequalities, we deduce that, for any
integer $p\geq 1$, there exists a constant $C_{p}(c)$, depending on $p$ and
$c$, such that

$\begin{split}&{\mathbb{E}}\biggl{[}\sup_{n\leq t\leq
n+1}\biggl{[}\exp\Bigl{(}-\frac{3}{8}c^{-1}t\Bigr{)}\biggl{|}\int_{0}^{t}\exp\Bigl{(}\frac{3}{8}c^{-1}s\Bigr{)}\mathrm{d}M_{s}\biggr{|}^{p}\biggr{]}\biggr{]}\\\
&\leq\frac{1}{N^{p/2}}C_{p}(c)\biggl{(}\int_{n}^{n+1}\exp\Bigl{(}\frac{3}{4}c^{-1}\bigl{(}s-n)\Bigr{)}\mathrm{d}s\biggr{)}^{p/2}=\frac{1}{N^{p/2}}C_{p}(c)\biggl{(}\int_{0}^{1}\exp\Bigl{(}\frac{3}{4}c^{-1}s\Bigr{)}\mathrm{d}s\biggr{)}^{p/2}.\end{split}$

In turn, by Markov’s inequality, we deduce that, for any $\varepsilon>0$,

${\mathbb{P}}\biggl{(}\biggl{\\{}\sup_{n\leq t\leq
n+1}\biggl{[}\exp\Bigl{(}-\frac{3}{8}c^{-1}t\Bigr{)}\biggl{|}\int_{0}^{t}\exp\Bigl{(}\frac{3}{8}c^{-1}s\Bigr{)}\mathrm{d}M_{s}\biggr{|}\biggr{]}\geq\varepsilon\biggr{\\}}\biggr{)}\leq\frac{C_{p}(c)}{\varepsilon^{p}N^{p/2}},$

for a new value of the constant $C_{p}(c)$, and then

${\mathbb{P}}\biggl{(}\bigcup_{n=0}^{\lfloor
N^{1/4}\rfloor}\biggl{\\{}\sup_{n\leq t\leq
n+1}\biggl{[}\exp\Bigl{(}-\frac{3}{8}c^{-1}t\Bigr{)}\biggl{|}\int_{0}^{t}\exp\Bigl{(}\frac{3}{8}c^{-1}s\Bigr{)}\mathrm{d}M_{s}\biggr{|}\biggr{]}\geq\varepsilon\biggr{\\}}\biggr{)}\leq\frac{C_{p}(c)}{\varepsilon^{p}N^{p/2-1/4}},$

where $\lfloor N^{1/4}\rfloor$ denotes the floor of $N^{1/4}$. Choosing $p$
large enough and using a new value of the constant $C(c)$, we end up with

${\mathbb{P}}\biggl{(}\biggl{\\{}\sup_{0\leq t\leq
N^{1/4}}\biggl{[}\exp\Bigl{(}-\frac{3}{8}c^{-1}t\Bigr{)}\biggl{|}\int_{0}^{t}\exp\Bigl{(}\frac{3}{8}c^{-1}s\Bigr{)}\mathrm{d}M_{s}\biggr{|}\biggr{]}\geq\varepsilon\biggr{\\}}\biggr{)}\leq\frac{C(c)}{\varepsilon^{p}N}.$

The value of $p$ that appears in the right-hand side is hence fixed. This
prompts us to introduce the event

$A_{N}(\varepsilon):=\biggl{\\{}\sup_{0\leq t\leq
N^{1/4}}\biggl{[}\exp\Bigl{(}-\frac{3}{8}c^{-1}t\Bigr{)}\biggl{|}\int_{0}^{t}\exp\Bigl{(}\frac{3}{8}c^{-1}s\Bigr{)}dM_{s}\biggr{|}\biggr{]}\geq\varepsilon\biggr{\\}}.$

On the latter event, we have, by (4.58),

$\begin{split}\forall
t\in\bigl{[}0,N^{1/4}\wedge\tau_{N}\bigr{]},\quad\Lambda_{t}^{N}&\leq\exp\Bigl{(}-\frac{3}{8}c^{-1}t\Bigr{)}\Lambda_{0}^{N}+\varepsilon.\end{split}$

Back to the definition of (4.56), this yields

$\forall
t\in\bigl{[}0,N^{1/4}\wedge\tau_{N}\bigr{]},\quad|\mu_{t}^{2,N}|^{2}-c^{4}|\mu_{t}^{1,N}|^{2}\leq\exp\Bigl{(}-\frac{3}{8}c^{-1}t\Bigr{)}\Lambda_{0}^{N}+\varepsilon,$

that is

$\forall
t\in\bigl{[}0,N^{1/4}\wedge\tau_{N}\bigr{]},\quad|\mu_{t}^{2,N}|^{2}\leq\exp\Bigl{(}-\frac{3}{8}c^{-1}t\Bigr{)}\Lambda_{0}^{N}+\varepsilon+c^{4}|\mu_{t}^{1,N}|^{2}\leq\exp\Bigl{(}-\frac{3}{8}c^{-1}t\Bigr{)}+\varepsilon+c^{4}|\mu_{t}^{1,N}|^{2},$

where we used the obvious inequality $\Lambda_{0}^{N}\leq 1$.

Second Step. We thus introduce the following times. First, we call
$t_{0}(\varepsilon)$ the smallest (deterministic) time such that
$\exp(-(3/8)c^{-1}t_{0}(\varepsilon))\leq\varepsilon$. Secondly, we let

$\begin{split}&\sigma_{N}(\varepsilon):=\inf\bigl{\\{}t\geq
t_{0}:|\mu^{1,N}_{t}|\geq\varepsilon c^{-4}\bigr{\\}}.\end{split}$

Then, assuming that $\varepsilon\in(0,1/4)$ and recalling that $\eta=c^{-4}/4$
in the definition of $\tau_{N}$, we obviously have
$\sigma_{N}(\varepsilon)\leq\tau_{N}\vee t_{0}(\varepsilon)$, which implies

$\forall
t\in\bigl{[}t_{0}(\varepsilon),N^{1/4}\wedge\sigma_{N}(\varepsilon)\wedge\tau_{N}\bigr{]},\quad|\mu_{t}^{2,N}|^{2}\leq
3\varepsilon,$

at least if the above interval is not empty. Subsequently, plugging the latter
into (4.54), we obtain

$\forall
t\in\bigl{[}t_{0}(\varepsilon),N^{1/4}\wedge\sigma_{N}(\varepsilon)\wedge\tau_{N}\bigr{]},\quad\mathrm{d}\bigl{[}|\mu_{t}^{1,N}|^{2}\bigr{]}=\bigl{(}c^{-1}-\sqrt{3}c\varepsilon^{1/2}\bigr{)}|\mu_{t}^{1,N}|^{2}\mathrm{d}t+\frac{c^{-1}}{{N}}\mathrm{d}t+\mathrm{d}\widetilde{K}_{t}^{1}+\mathrm{d}M_{t}^{1},$

for a new non-decreasing absolutely continuous process
$(\widetilde{K}_{t}^{1})_{t\geq 0}$.

So far, $\varepsilon$ has been a free parameter. Now, we can choose it such
that $c^{-1}-\sqrt{3}c\varepsilon^{1/2}=c^{-1}/2$. For this given value of
$\varepsilon$ (which is now frozen in terms of $c$), we get

$\forall
t\in\bigl{[}t_{0}(\varepsilon),N^{1/4}\wedge\sigma_{N}(\varepsilon)\wedge\tau_{N}\bigr{]},\quad\mathrm{d}\bigl{[}|\mu_{t}^{1,N}|^{2}\bigr{]}=\frac{c^{-1}}{2}|\mu_{t}^{1,N}|^{2}\mathrm{d}t+\frac{c^{-1}}{{N}}\mathrm{d}t+\mathrm{d}\widetilde{K}_{t}^{1}+\mathrm{d}M_{t}^{1}.$

Next,

$\forall
t\in\bigl{[}t_{0}(\varepsilon),N^{1/4}\wedge\sigma_{N}(\varepsilon)\wedge\tau_{N}\bigr{]},\quad|\mu_{t}^{1,N}|^{2}\geq\exp\Bigl{(}\frac{c^{-1}}{2}t\Bigr{)}\biggl{[}\frac{c^{-1}}{{N}}\int_{t_{0}(\varepsilon)}^{t}\exp\bigl{(}-\frac{c^{-1}}{2}s\bigr{)}\mathrm{d}s+\int_{t_{0}(\varepsilon)}^{t}\exp\bigl{(}-\frac{c^{-1}}{2}s\bigr{)}\mathrm{d}M_{s}^{1}\biggr{]}.$

In particular, for all $t\geq 0$,

$\begin{split}&{\mathbb{E}}\Bigl{[}{\mathbbm{1}}_{\\{\tau_{N}>t_{0}(\varepsilon)\\}}\bigl{|}\mu_{t_{0}(\varepsilon)\vee(t\wedge
N^{1/4}\wedge\sigma_{N}(\varepsilon))}^{1,N}\bigr{|}^{2}\Bigr{]}\\\
&\geq\frac{c^{-1}}{{N}}{\mathbb{E}}\biggl{[}{\mathbbm{1}}_{\\{\tau_{N}>t_{0}(\varepsilon)\\}}\int_{t_{0}(\varepsilon)}^{t_{0}(\varepsilon)\vee(t\wedge
N^{1/4}\wedge\sigma_{N}(\varepsilon))}\exp\Bigl{(}\frac{c^{-1}}{2}(t-s)\Bigr{)}\mathrm{d}s\biggr{]}\\\
&=\frac{2}{{N}}{\mathbb{E}}\biggl{[}{\mathbbm{1}}_{\\{\tau_{N}>t_{0}(\varepsilon)\\}}\biggl{(}\exp\Bigl{(}\frac{c^{-1}}{2}\bigl{(}t-t_{0}(\varepsilon)\bigr{)}\Bigr{)}-\exp\Bigl{(}\frac{c^{-1}}{2}\bigl{(}t-t_{0}(\varepsilon)\vee\bigl{[}t\wedge
N^{1/4}\wedge\sigma_{N}(\varepsilon)\bigr{]}\bigr{)}\Bigr{)}\biggr{)}\biggr{]}.\end{split}$

Choosing $t=N^{1/4}$, we obtain that

$\begin{split}1&\geq\frac{2}{{N}}\exp\Bigl{(}\frac{c^{-1}}{2}\bigl{(}N^{1/4}-t_{0}(\varepsilon)\bigr{)}\Bigr{)}{\mathbb{P}}\Bigl{(}\bigl{\\{}\sigma_{N}(\varepsilon)\geq
N^{1/4},\tau_{N}>t_{0}(\varepsilon)\bigr{\\}}\Bigr{)},\end{split}$

at least if $N^{1/4}\geq t_{0}(\varepsilon)$, which yields

${\mathbb{P}}\Bigl{(}\bigl{\\{}\sigma_{N}(\varepsilon)\geq
N^{1/4},\tau_{N}>t_{0}(\varepsilon)\bigr{\\}}\Bigr{)}\leq\frac{N}{2}\exp\Bigl{(}-\frac{c^{-1}}{2}\bigl{(}N^{1/4}-t_{0}(\varepsilon)\bigr{)}\Bigr{)}.$

Recalling that $\tau_{N}\geq\sigma_{N}(\varepsilon)$ if
$\tau_{N}>t_{0}(\varepsilon)$, with $\eta=c^{-4}/4$ in the definition of
$\tau_{N}$, this completes the proof. ∎

##  5\. appendix

We introduce here a new type of regularisation for a function $\Phi$ defined
on the space of probability measures on $\mathbb{T}^{d}$. The statement below
is tailored to our needs, but we strongly believe that the method of proof,
which consists in convoluting $\Phi$ when expressed in terms of the Fourier
coefficients of the measure argument, is of a wider scope.

######  Theorem 5.1.

Let $\Phi:{\mathcal{P}}(\mathbb{T}^{d})\rightarrow\mathbb{R}$ satisfy _(
Reg-$\Phi$-(${\alpha,2}$))_ for $\alpha\in[0,1)$. Then there exists a sequence
of smooth functions $(\Phi_{n})_{n\geq 1}$, satisfying _(
Reg-$\Phi$-(${4,3}$))_ for each $n\geq 1$, and converging to $\Phi$, uniformly
on $\mathbb{T}^{d}$, such that the bounds satisfied by $(\Phi_{n})_{n\geq 1}$
in _( Reg-$\Phi$-(${\alpha,2}$))_ are uniform in $n\geq 1$.

The proof relies on the following reformulation of Bochner-Herglotz’ theorem
on the torus (see the proof of [56, Proposition 5]).

######  Lemma 5.2.

Let $\mu$ be a probability measure on ${\mathbb{T}}^{d}$, with Fourier
coefficients
$(\widehat{\mu}^{\bm{n}}=\int_{{\mathbb{T}}^{d}}e^{-{\mathrm{i}}2\pi{\bm{n}}\cdot\theta}\mu(\mathrm{d}\theta))_{{\bm{n}}\in{\mathbb{Z}}^{d}}$,
where the index ${\bm{n}}$ is written below in the form
${\bm{n}}=(n_{1},\cdots,n_{d})$. Then, for any integer $N\geq 1$, the function

$f_{N}(\theta;\mu)=\frac{1}{N^{d}}\sum_{\underset{\large
j\in\\{1,\cdots,d\\}}{1\leq n_{j},n_{j}^{\prime}\leq
N}}\widehat{\mu}^{{\bm{n}}-{\bm{n}}^{\prime}}\exp\Bigl{(}{\mathrm{i}}2\pi({\bm{n}}-{\bm{n}}^{\prime})\cdot\theta\Bigr{)},\quad\theta\in\mathbb{T}^{d},$

is a density on $\mathbb{T}^{d}$. Moreover, the Fourier coefficients of
$f_{N}$ are given by

$\widehat{f}_{N}^{\bm{n}}(\mu)=\left\\{\begin{array}[]{ll}\widehat{\mu}^{\bm{n}}\prod_{j=1}^{d}\bigl{(}1-\frac{|n_{j}|}{N}\bigr{)},&\max_{j=1,\cdots,d}|n_{j}|<N\\\
0&\textrm{\rm otherwise}\end{array}\right.,\quad{\bm{n}}\in{\mathbb{Z}}^{d}.$

Below, we let

$m_{N}(\mu):=f_{N}(\cdot\,;\mu)\cdot\textrm{\rm Leb}_{\mathbb{T}^{d}}.$ (5.1)

We have the following very useful lemma:

######  Lemma 5.3.

With the same notation as in (5.1),
$\displaystyle\lim_{N\rightarrow\infty}\sup_{\mu\in{\mathcal{P}}(\mathbb{T}^{d})}{\mathcal{W}}_{1}\bigl{(}m_{N}(\mu),\mu\bigr{)}=0$.

###### Proof.

(Lemma 5.3.) We notice from the expression of the coefficients
$(\widehat{f}_{N}^{\bm{n}}(\mu))_{{\bm{n}}\in{\mathbb{Z}}^{d}}$ that, for any
$\mu\in\mathcal{P}(\mathbb{T}^{d})$,

$\begin{split}&\bigl{\|}m_{N}(\mu)-\mu\bigr{\|}_{-(d/2+1),2}^{2}\\\
&=\sum_{{\bm{n}}\in{\mathbb{Z}}^{d}}\frac{1}{\bigl{(}1+|{\bm{n}}|^{2}\bigr{)}^{d/2+1}}\bigl{|}\widehat{m_{N}(\mu)}^{\bm{n}}-\widehat{\mu}^{\bm{n}}\bigr{|}^{2}\\\
&\leq\sum_{\max_{i}|n_{i}|\leq
N}\frac{1}{\bigl{(}1+|{\bm{n}}|^{2}\bigr{)}^{d/2+1}}\bigl{|}\widehat{m_{N}(\mu)}^{\bm{n}}-\widehat{\mu}^{\bm{n}}\bigr{|}^{2}+\sum_{\exists
i:|n_{i}|\geq
N}\frac{1}{\bigl{(}1+|{\bm{n}}|^{2}\bigr{)}^{d/2+1}}\bigl{|}\widehat{m_{N}(\mu)}^{\bm{n}}-\widehat{\mu}^{\bm{n}}\bigr{|}^{2}\\\
&\leq\sum_{\max_{i}|n_{i}|\leq
N}\frac{1}{\bigl{(}1+|{\bm{n}}|^{2}\bigr{)}^{d/2+1}}\Bigl{[}\prod_{j=1}^{d}\bigl{(}1-\frac{|n_{j}|}{N}\bigr{)}-1\Bigr{]}^{2}+\sum_{\exists
i:|n_{i}|\geq
N}\frac{1}{\bigl{(}1+|{\bm{n}}|^{2}\bigr{)}^{d/2+1}},\end{split}$

where $|{\bm{n}}|^{2}=n_{1}^{2}+\cdots+n_{d}^{2}$, from which we deduce that

$\lim_{N\rightarrow\infty}\sup_{\mu\in{\mathcal{P}}(\mathbb{T}^{d})}\bigl{\|}m_{N}(\mu)-\mu\bigr{\|}_{-(d/2+1),2}=0.$

By a standard convolution argument, we note that, for any $\varepsilon>0$,
there exists a constant $C_{\varepsilon}>0$ such that, for any function $f\in
W^{1,\infty}(\mathbb{T}^{d})$, with $\|f\|_{1,\infty}\leq 1$, we can find
another function $f_{\varepsilon}\in W^{d/2+1,2}(\mathbb{T}^{d})$ satisfying

$\bigl{\|}f-f_{\varepsilon}\bigr{\|}_{\infty}\leq\varepsilon,\quad\bigl{\|}f_{\varepsilon}\bigr{\|}_{d/2+1,2}\leq
C_{\varepsilon}.$

Therefore,

$\begin{split}\sup_{\|f\|_{1,\infty}\leq
1}\bigl{\langle}f,m^{N}(\mu)-\mu\rangle\leq
2\varepsilon+\sup_{\|f\|_{d/2+1,2}\leq
C_{\varepsilon}}\bigl{\|}m_{N}(\mu)-\mu\bigr{\|}_{-(d/2+1),2}.\end{split}$

For $N$ that is large enough (depending on $\varepsilon$), the right-hand side
is less than $3\varepsilon$. This completes the proof. ∎

We now turn to the proof of Theorem 5.1.

###### Proof of Theorem 5.1.

First step. For a given ${\bm{x}}=(x^{\bm{n}})_{{\bm{n}}:|n_{j}|\leq
N-1}\in{\mathbb{R}}^{(2N-1)^{d}}$ with the constraint that $x^{\bm{0}}=1$
(with ${\bm{0}}=(0,\cdots,0)\in{\mathbb{Z}}^{d}$) and $\varepsilon>0$, we call

$\begin{split}f_{N,\varepsilon}(\theta;{\bm{x}})&:=\varepsilon+(1-\varepsilon){\mathfrak{R}}\biggl{[}\frac{1}{N^{d}}\sum_{\underset{\large
j\in\\{1,\cdots,d\\}}{1\leq n_{j},n_{j}^{\prime}\leq
N}}x^{{\bm{n}}-{\bm{n}}^{\prime}}\exp\Bigl{(}{\mathrm{i}}2\pi({\bm{n}}-{\bm{n}}^{\prime})\cdot\theta\Bigr{)}\biggr{]}\\\
&=1+(1-\varepsilon){\mathfrak{R}}\biggl{[}\frac{1}{N^{d}}\sum_{\underset{\large
j\in\\{1,\cdots,d\\}}{1\leq n_{j},n_{j}^{\prime}\leq
N},{\bm{n}}\not={\bm{n}}^{\prime}}x^{{\bm{n}}-{\bm{n}}^{\prime}}\exp\Bigl{(}{\mathrm{i}}2\pi({\bm{n}}-{\bm{n}}^{\prime})\cdot\theta\Bigr{)}\biggr{]}.\end{split}$

We observe from the statement of Lemma 5.2 that, for any probability measure
$\mu$ on $\mathbb{T}^{d}$,

$f_{N,\varepsilon}\Bigl{(}\theta;\bigl{(}\widehat{\mu}^{\bm{n}}\bigr{)}_{{\bm{n}}:|n_{j}|\leq
N-1}\Bigr{)}=\varepsilon+(1-\varepsilon)f_{N}(\theta;{\bm{\mu}}),$

from which we have the obvious lower bound:

$\forall\theta\in\mathbb{T}^{d},\quad
f_{N,\varepsilon}\Bigl{(}\theta;\bigl{(}\widehat{\mu}^{\bm{n}}\bigr{)}_{{\bm{n}}:|n_{j}|\leq
N-1}\Bigr{)}\geq\varepsilon.$

We then deduce that there exists a real $\eta_{\varepsilon,N}>0$ such that,
for any choice of $\mu$ and for any tuple ${\bm{x}}$ with the property that
$\sup_{{\bm{n}}:|n_{j}|\leq
N-1}|{\bm{x}}^{\bm{n}}-\widehat{\mu}^{\bm{n}}|_{\infty}\leq\eta_{\varepsilon,N}$,
$f_{N,\varepsilon}(\cdot\,;{\bm{x}})$ has positive values and is hence a
density. Following (5.1), we then let

$m_{N,\varepsilon}({\bm{x}})=f_{N,\varepsilon}(\cdot\,;{\bm{x}})\cdot\textrm{\rm
Leb}_{\mathbb{T}^{d}}.$

In order to define our mollified version of $\Phi$, we are also given a smooth
density $\varrho_{\varepsilon,N}$ on ${\mathbb{R}}$ with support included in
$[-(\varepsilon\wedge\eta_{\varepsilon,N})/2,(\varepsilon\wedge\eta_{\varepsilon,N})/2]$.
This allows us to let

$\Phi_{N,\varepsilon}(\mu):=\int_{{\mathbb{R}}^{(2N-1)^{d}}}\Phi\Bigl{(}m_{N,\varepsilon}({\bm{x}})\Bigr{)}\prod_{{\bm{n}}\not=0}\varrho_{N,\varepsilon}\Bigl{(}\widehat{\mu}^{\bm{n}}-x^{\bm{n}}\Bigr{)}\delta_{1}(\mathrm{d}x^{\bm{0}})\prod_{{\bm{n}}\not=0}\mathrm{d}x^{\bm{n}},$
(5.2)

where ${\bm{n}}$ in the above two products is an element of ${\mathbb{Z}}^{d}$
satisfying $|n_{j}|\leq N-1$ for $j\in\\{1,\cdots,d\\}$. Interestingly,

$\begin{split}\Phi_{N,\varepsilon}(\mu)&=\int_{{\mathbb{R}}^{(2N-1)^{d}}}\Phi\biggl{(}m_{N,\varepsilon}\Bigl{(}\bigl{(}\widehat{\mu}^{\bm{n}}-x^{\bm{n}}\bigr{)}_{{\bm{n}}:|n_{j}|\leq
N-1}\Bigr{)}\biggr{)}\prod_{{\bm{n}}\not=0}\varrho_{N,\varepsilon}\bigl{(}x^{\bm{n}}\bigr{)}\,\delta_{1}(\mathrm{d}x^{\bm{0}})\prod_{{\bm{n}}\not=0}\mathrm{d}x^{\bm{n}}\\\
&=\int_{{\mathbb{R}}^{(2N-1)^{d}}}\Phi\biggl{(}\varepsilon\textrm{\rm
Leb}_{\mathbb{T}^{d}}+(1-\varepsilon)\Bigl{(}m_{N}(\mu)-m_{N,0}({\bm{x}})\Bigr{)}\biggr{)}\prod_{{\bm{n}}\not=0}\varrho_{N,\varepsilon}\bigl{(}x^{\bm{n}}\bigr{)}\,\delta_{1}(\mathrm{d}x^{\bm{0}})\prod_{{\bm{n}}\not=0}dx^{\bm{n}},\end{split}$
(5.3)

where we used the fact that

$\begin{split}f_{N,\varepsilon}\Bigl{(}\theta;\bigl{(}\widehat{\mu}^{n}-{\bm{x}}^{n}\bigr{)}_{{\bm{n}}:|n_{j}|\leq
N-1}\Bigr{)}=\varepsilon+(1-\varepsilon)\Bigl{[}f_{N}\bigl{(}\theta;\mu\bigr{)}-f_{N,0}\bigl{(}\theta;{\bm{x}}\bigr{)}\Bigr{]}.\end{split}$

By Lemma 5.3, we clearly have that $\Phi_{N,\varepsilon}$ converges to $\Phi$,
as $(N,\varepsilon)$ converges to $(\infty,0)$.

Second step. The fact that $\Phi_{N,\varepsilon}$ is a smooth functional of
$\mu$ is a direct consequence of (5.2). The Fourier coefficient
$\widehat{\mu}^{\bm{n}}$ is a mere linear functional of $\mu$ and the function
$\varrho_{N,\varepsilon}$ is smooth. Obviously, the function $\Phi$ in (5.2)
plays no main role in the smoothness of $\Phi_{N,\varepsilon}$ except for the
fact that, as it is bounded, it does not yield any integrability problem.

Third step. We now check that the bounds satisfied by $(\Phi_{n})_{n\geq 1}$
in (Reg-$\Phi$-(${\alpha,2}$)) are uniform in $n\geq 1$. Obviously, we have
$\|\Phi_{N,\varepsilon}\|_{\infty}\leq\|\Phi\|_{\infty}$. Next, we compute the
first order derivatives $\delta\Phi_{N,\varepsilon}/\delta m$. In order to do
so, we notice that, for ${\bm{x}}\in{\mathbb{R}}^{(2N-1)^{d}}$ and
$\mu,\nu\in{\mathcal{P}}(\mathbb{T}^{d})$,

$\begin{split}&\Phi\biggl{(}\varepsilon\textrm{\rm
Leb}_{\mathbb{T}^{d}}+(1-\varepsilon)\Bigl{(}m_{N}(\nu)-m_{N,0}({\bm{x}})\Bigr{)}\biggr{)}-\Phi\biggl{(}\varepsilon\textrm{\rm
Leb}_{\mathbb{T}^{d}}+(1-\varepsilon)\Bigl{(}m_{N}(\mu)-m_{N,0}({\bm{x}})\Bigr{)}\biggr{)}\\\
&=(1-\varepsilon)\int_{0}^{1}\int_{\mathbb{T}^{d}}\frac{\delta\Phi}{\delta
m}\biggl{(}\varepsilon\textrm{\rm
Leb}_{\mathbb{T}^{d}}+(1-\varepsilon)\Bigl{(}\lambda
m_{N}(\nu)+(1-\lambda)m_{N}(\mu)-m_{N,0}({\bm{x}})\Bigr{)}\biggr{)}(y)\,m_{N}(\nu-\mu)(\mathrm{d}y),\end{split}$

with the obvious notation (although $\nu-\mu$ is a not a probability measure)
$m_{N}(\nu-\mu)=f_{N}(\cdot\,;\nu-\mu)\cdot\textrm{\rm Leb}_{\mathbb{T}^{d}}$
and $f_{N}(\theta;\nu-\mu)$ as in the statement of Lemma 5.2.

As a result, using the continuity of $\delta\Phi/\delta m$, we deduce that

$\begin{split}&\Phi\biggl{(}\varepsilon\textrm{\rm
Leb}_{\mathbb{T}^{d}}+(1-\varepsilon)\Bigl{(}m_{N}(\nu)-m_{N,0}({\bm{x}})\Bigr{)}\biggr{)}-\Phi\biggl{(}\varepsilon\textrm{\rm
Leb}_{\mathbb{T}^{d}}+(1-\varepsilon)\Bigl{(}m_{N}(\mu)-m_{N,0}({\bm{x}})\Bigr{)}\biggr{)}\\\
&=(1-\varepsilon)\int_{0}^{1}\int_{\mathbb{T}^{d}}\frac{\delta\Phi}{\delta
m}\biggl{(}\varepsilon\textrm{\rm
Leb}_{\mathbb{T}^{d}}+(1-\varepsilon)m_{N}(\mu)-m_{N,0}({\bm{x}})\Bigr{)}\biggr{)}(y)\bigl{(}m_{N}(\nu-\mu)\bigr{)}(\mathrm{d}y)+o\bigl{(}\textrm{\rm
dist}_{\textrm{\rm TV}}(\mu,\nu)\bigr{)},\end{split}$

which suffices to identify $\delta\Phi_{N,\varepsilon}/\delta m$ through the
linear functional that appears in the above right-hand side (see, for
instance, [19, Remark 5.47]). By Lemma 5.2 and Parseval’s identity,

$\begin{split}&\int_{\mathbb{T}^{d}}\frac{\delta\Phi}{\delta
m}\biggl{(}\varepsilon\textrm{\rm
Leb}_{\mathbb{T}^{d}}+(1-\varepsilon)\Bigl{(}m_{N}(\mu)-m_{N,0}({\bm{x}})\Bigr{)}\biggr{)}(y)\bigl{(}m_{N}(\nu-\mu)\bigr{)}(\mathrm{d}y)\\\
&=\sum_{{\bm{n}}\in{\mathbb{Z}}^{d}}\biggl{[}\int_{\mathbb{T}^{d}}\frac{\delta\Phi}{\delta
m}\biggl{(}\varepsilon\textrm{\rm
Leb}_{\mathbb{T}^{d}}+(1-\varepsilon)\Bigl{(}m_{N}(\mu)-m_{N,0}({\bm{x}})\Bigr{)}\biggr{)}(y)e^{{\mathrm{i}}2\pi{\bm{n}}\cdot
y}\mathrm{d}y\biggr{]}\widehat{m_{N}(\nu-\mu)}^{\bm{n}}\\\
&=\sum_{{\bm{n}}:|n_{j}|<N}\biggl{[}\int_{\mathbb{T}^{d}}\frac{\delta\Phi}{\delta
m}\biggl{(}\varepsilon\textrm{\rm
Leb}_{\mathbb{T}^{d}}+(1-\varepsilon)\Bigl{(}m_{N}(\mu)-m_{N,0}({\bm{x}})\Bigr{)}\biggr{)}(y)e^{{\mathrm{i}}2\pi{\bm{n}}\cdot
y}\mathrm{d}y\biggr{]}\bigl{(}\nu^{\bm{n}}-\mu^{\bm{n}}\bigr{)}\prod_{j=1}^{d}\bigl{(}1-\frac{|n_{j}|}{N}\bigr{)}\\\
&=\sum_{{\bm{n}}:|n_{j}|<N}\int_{\mathbb{T}^{d}}\biggl{[}\int_{\mathbb{T}^{d}}\frac{\delta\Phi}{\delta
m}\biggl{(}\varepsilon\textrm{\rm
Leb}_{\mathbb{T}^{d}}+(1-\varepsilon)\Bigl{(}m_{N}(\mu)-m_{N,0}({\bm{x}})\Bigr{)}\biggr{)}(y)e^{-{\mathrm{i}}2\pi{\bm{n}}\cdot(\theta-y)}\mathrm{d}y\biggr{]}\prod_{j=1}^{d}\bigl{(}1-\frac{|n_{j}|}{N}\bigr{)}\mathrm{d}\bigl{(}\nu-\mu\bigr{)}(\theta),\end{split}$

which shows that

$\frac{\delta\Phi_{N,\varepsilon}}{\delta
m}(\mu)(\theta)=\sum_{{\bm{n}}:|n_{j}|<N}\prod_{j=1}^{d}\bigl{(}1-\frac{|n_{j}|}{N}\bigr{)}\int_{\mathbb{T}^{d}}\frac{\delta\Phi}{\delta
m}\biggl{(}\varepsilon\textrm{\rm
Leb}_{\mathbb{T}^{d}}+(1-\varepsilon)\Bigl{(}m_{N}(\mu)-m_{N,0}({\bm{x}})\Bigr{)}\biggr{)}(\theta-y)e^{-{\mathrm{i}}2\pi{\bm{n}}\cdot
y}\mathrm{d}y.$

By reverting the computations, we get

$\begin{split}\frac{\delta\Phi_{N,\varepsilon}}{\delta
m}(\mu)(\theta)&=\int_{\mathbb{T}^{d}}\frac{\delta\Phi}{\delta
m}\biggl{(}\varepsilon\textrm{\rm
Leb}_{\mathbb{T}^{d}}+(1-\varepsilon)\Bigl{(}m_{N}(\mu)-m_{N,0}({\bm{x}})\Bigr{)}\biggr{)}(y)\bigl{\\{}m_{N}(\delta_{\theta})\bigr{\\}}(\mathrm{d}y)\\\
&=\int_{\mathbb{T}^{d}}\frac{\delta\Phi}{\delta
m}\biggl{(}\varepsilon\textrm{\rm
Leb}_{\mathbb{T}^{d}}+(1-\varepsilon)\Bigl{(}m_{N}(\mu)-m_{N,0}({\bm{x}})\Bigr{)}\biggr{)}(y)f_{N}\bigl{(}y-\theta,\delta_{0}\bigr{)}\mathrm{d}y.\end{split}$
(5.4)

The $\alpha$-Hölder norm of the right-hand side with respect to $\theta$ can
be easily estimated in terms of the $\alpha$-Hölder norm of $\delta\Phi/\delta
m$, which fits our objective. Differentiating once again with respect to
$\mu$, we may prove a similar result for the $\alpha$-Hölder regularity of
$[\delta^{2}\Phi_{N,\varepsilon}/\delta m^{2}](\mu)(\theta,\theta^{\prime})$
with respect to $\theta,\theta^{\prime}$. ∎

## References

  * [1] L. Ambrosio, N. Gigli, and G. Savaré.  Gradient flows in metric spaces and in the space of probability measures.  Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2005\.
  * [2] M. Arnaudon and P. Del Moral.  A second order analysis of Mckean–Vlasov semigroups.  Ann. Appl. Probab., 30(6):2613–2664, 12 2020.
  * [3] S. Benachour, B. Roynette, D. Talay, and P. Vallois.  Nonlinear self-stabilizing processes. I. Existence, invariant probability, propagation of chaos.  Stochastic Process. Appl., 75(2):173–201, 1998.
  * [4] S. Benachour, B. Roynette, and P. Vallois.  Nonlinear self-stabilizing processes. II. Convergence to invariant probability.  Stochastic Process. Appl., 75(2):203–224, 1998.
  * [5] O. Bencheikh and B. Jourdain.  Bias behaviour and antithetic sampling in mean-field particle approximations of SDEs nonlinear in the sense of McKean.  ESAIM: Proceedings and Surveys, 65:219–235, 2019.
  * [6] D. Benedetto, E. Caglioti, J. A. Carrillo, and M. Pulvirenti.  A non-Maxwellian steady distribution for one-dimensional granular media.  J. Statist. Phys., 91(5-6):979–990, 1998.
  * [7] L. Bertini, G. Giacomin, and K. Pakdaman.  Dynamical aspects of mean field plane rotators and the Kuramoto model.  J. Stat. Phys., 138(1-3):270–290, 2010.
  * [8] L. Bertini, G. Giacomin, and C. Poquet.  Synchronization and random long time dynamics for mean-field plane rotators.  Probab. Theory Related Fields, 160(3-4):593–653, 2014.
  * [9] V. I. Bogachev, M. Röckner, and S. V. Shaposhnikov.  On convergence to stationary distributions for solutions of nonlinear Fokker-Planck-Kolmogorov equations.  J. Math. Sci. (N.Y.), 242(1, Problems in Mathematical Analysis. No. 98):69–84, 2019.
  * [10] F. Bolley, I. Gentil, and A. Guillin.  Uniform convergence to equilibrium for granular media.  Arch. Ration. Mech. Anal., 208(2):429–445, 2013.
  * [11] H. Brezis and P. Mironescu.  Gagliardo-Nirenberg inequalities and non-inequalities: the full story.  Ann. Inst. H. Poincaré Anal. Non Linéaire, 35(5):1355–1376, 2018.
  * [12] R. Buckdahn, J. Li, S. Peng, and C. Rainer.  Mean-field stochastic differential equations and associated PDEs.  The Annals of Probability, 45(2):824–878, 2017.
  * [13] A. Budhiraja, P. Dupuis, and M. Fischer.  Large deviation properties of weakly interacting processes via weak convergence methods.  The Annals of Probability, pages 74–102, 2012.
  * [14] O.A. Butkovsky.  On ergodic properties of nonlinear Markov chains and stochastic McKean-Vlasov equations.  Theory Probab. Appl., 58(4):661–674, 2014.
  * [15] P. Cardaliaguet.  Notes on mean field games.  Technical report, Technical report, 2010.
  * [16] P. Cardaliaguet, F. Delarue, J.-M. Lasry, and P.-L. Lions.  The Master Equation and the Convergence Problem in Mean Field Games:(AMS-201), volume 201.  Princeton University Press, 2019.
  * [17] P. Cardaliaguet, J.-M. Lasry, P.-L. Lions, and A. Porretta.  Long time average of mean field games with a nonlocal coupling.  SIAM Journal on Control and Optimization, 51(5):3558–3591, 2013\.
  * [18] R. Carmona.  Lectures on BSDEs, Stochastic Control, and Stochastic Differential Games with Financial Applications, volume 1.  SIAM, 2016.
  * [19] R. Carmona and F. Delarue.  Probabilistic Theory of Mean Field Games with Applications. I, volume 83 of Probability Theory and Stochastic Modelling.  Springer, Cham, 2018.  Mean field FBSDEs, control, and games.
  * [20] R. Carmona and F. Delarue.  Probabilistic Theory of Mean Field Games with Applications. II, volume 84 of Probability Theory and Stochastic Modelling.  Springer, Cham, 2018.  Mean field games with common noise and master equations.
  * [21] J. A. Carrillo, R. S. Gvalani, G. A. Pavliotis, and A. Schlichting.  Long-time behaviour and phase transitions for the McKean-Vlasov equation on the torus.  Arch. Ration. Mech. Anal., 235(1):635–690, 2020.
  * [22] J.A. Carrillo, R.J. McCann, and C. Villani.  Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates.  Rev. Mat. Iberoamericana, 19(3):971–1018, 2003.
  * [23] P. Cattiaux, A. Guillin, and F. Malrieu.  Probabilistic approach for granular media equations in the non-uniformly convex case.  Probab. Theory Related Fields, 140(1-2):19–40, 2008.
  * [24] J.-F. Chassagneux, D. Crisan, and F. Delarue.  A probabilistic approach to classical solutions of the master equation for large population equilibria.  arXiv preprint arXiv:1411.3009, 2014.
  * [25] J.-F. Chassagneux, L. Szpruch, and A. Tse.  Weak quantitative propagation of chaos via differential calculus on the space of measures.  arXiv preprint arXiv:1901.02556, 2019.
  * [26] P. Constantin, I. Kevrekidis, and E.S. Titi.  Remarks on a Smoluchowski equation.  Discrete Contin. Dyn. Syst., 11(1):101–112, 2004.
  * [27] F. Coppini.  Long time dynamics for interacting oscillators on graphs.  arXiv preprint arXiv:1908.01520, 2019.
  * [28] D. Dawson and J. Gärtner.  Large deviations from the McKean-Vlasov limit for weakly interacting diffusions.  Stochastics: An International Journal of Probability and Stochastic Processes, 20(4):247–308, 1987.
  * [29] P. Degond, A. Frouvelle, and J.-G. Liu.  Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics.  Arch. Ration. Mech. Anal., 216(1):63–115, 2015.
  * [30] F. Delarue, D. Lacker, and K. Ramanan.  From the master equation to mean field game limit theory: A central limit theorem.  Electron. J. Probab., 24:54 pp., 2019.
  * [31] S. Dereich, M. Scheutzow, and R. Schottstedt.  Constructive quantization: Approximation by empirical measures.  In Annales de l’IHP Probabilités et Statistiques, pages 1183–1203, 2013.
  * [32] E. Di Nezza, G. Palatucci, and E. Valdinoci.  Hitchhiker’s guide to the fractional Sobolev spaces.  Bull. Sci. Math., 136(5):521–573, 2012.
  * [33] A. Durmus, A. Eberle, A. Guillin, and R. Zimmer.  An elementary approach to uniform in time propagation of chaos.  arXiv preprint arXiv:1805.11387, 2018.
  * [34] A. Eberle, A. Guillin, and R. Zimmer.  Quantitative Harris-type theorems for diffusions and McKean-Vlasov processes.  Trans. Amer. Math. Soc., 371(10):7135–7173, 2019.
  * [35] N. Fournier and A. Guillin.  On the rate of convergence in Wasserstein distance of the empirical measure.  Probability Theory and Related Fields, 162(3-4):707–738, 2015.
  * [36] A. Friedman.  Partial differential equations of parabolic type.  Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.
  * [37] N. Frikha and P.-E. Chaudru de Raynal.  Well-posedness for some non-linear diffusion processes and related PDE on the Wasserstein space.  arXiv preprint arXiv:1811.06904, 2018.
  * [38] N. Frikha and P.-E. Chaudru de Raynal.  From the backward Kolmogorov PDE on the Wasserstein space to propagation of chaos for Mckean-Vlasov SDEs.  arXiv preprint arXiv:1907.01410, 2019.
  * [39] M. G. Garroni and J.-L. Menaldi.  Green functions for second order parabolic integro-differential problems, volume 275 of Pitman Research Notes in Mathematics Series.  Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992.
  * [40] J. Gärtner.  On the Mckean-Vlasov limit for interacting diffusions.  Mathematische Nachrichten, 137(1):197–248, 1988.
  * [41] G. Giacomin, K. Pakdaman, and X. Pellegrin.  Global attractor and asymptotic dynamics in the Kuramoto model for coupled noisy phase oscillators.  Nonlinearity, 25(5):1247–1273, 2012.
  * [42] S. Herrmann and J. Tugaut.  Non-uniqueness of stationary measures for self-stabilizing processes.  Stochastic Process. Appl., 120(7):1215–1246, 2010.
  * [43] P.-E. Jabin and Z. Wang.  Quantitative estimates of propagation of chaos for stochastic systems with $W^{-1,\infty}$ kernels.  Invent. Math., 214(1):523–591, 2018.
  * [44] B. Jourdain.  Diffusions with a nonlinear irregular drift coefficient and probabilistic interpretation of generalized Burgers’ equations.  ESAIM Probab. Statist., 1:339–355, 1995/97.
  * [45] V.N. Kolokoltsov.  Nonlinear Markov Processes and Kinetic Equations, volume 182.  Cambridge University Press, 2010.
  * [46] Y. Kuramoto.  Rhythms and turbulence in populations of chemical oscillators.  Phys. A, 106(1-2):128–143, 1981.
  * [47] D. Lacker.  On a strong form of propagation of chaos for McKean-Vlasov equations.  Electron. Commun. Probab., 23:Paper No. 45, 11, 2018.
  * [48] G.M. Lieberman.  Second order parabolic differential equations.  World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
  * [49] E. Luçon and C. Poquet.  Long time dynamics and disorder-induced traveling waves in the stochastic Kuramoto model.  Ann. Inst. Henri Poincaré Probab. Stat., 53(3):1196–1240, 2017\.
  * [50] F. Malrieu.  Convergence to equilibrium for granular media equations and their Euler schemes.  Ann. Appl. Probab., 13(2):540–560, 2003.
  * [51] V. Marx.  A Bismut-Elworthy inequality for a Wasserstein diffusion on the circle.  arXiv preprint arXiv:2005.04972, 2020.
  * [52] S. Méléard.  Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models.  In Probabilistic Models for Nonlinear Partial Differential Equations, pages 42–95. Springer, 1996.
  * [53] S. Mischler and C. Mouhot.  Kac’s program in kinetic theory.  Inventiones Mathematicae, 193(1):1–147, 2013.
  * [54] S. Mischler, C. Mouhot, and B. Wennberg.  A new approach to quantitative propagation of chaos for drift, diffusion and jump processes.  Probability Theory and Related Fields, 161(1-2):1–59, 2015.
  * [55] Y. S. Mishura and A. Y. Veretennikov.  Existence and uniqueness theorems for solutions of McKean-Vlasov stochastic equations.  arXiv preprint arXiv:1603.02212, 2016.
  * [56] M. M. Rao and R. J. Swift.  Probability Theory with Applications, volume 582 of Mathematics and Its Applications (Springer), pages xviii+527.  Springer, New York, second edition, 2006.
  * [57] P. Ren and F.-Y. Wang.  Derivative Formulas in Measure on Riemannian Manifolds.  arXiv preprint arXiv:1908.03711, 2019.
  * [58] D. Ruelle.  Statistical Mechanics.  World Scientific Publishing Co., Inc., River Edge, NJ; Imperial College Press, London, 1999.  Rigorous results, Reprint of the 1989 edition.
  * [59] H. Sakaguchi, S. Shinomoto, and Y. Kuramoto.  Phase transitions and their bifurcation analysis in a large population of active rotators with mean-field coupling.  Progr. Theoret. Phys., 79(3):600–607, 1988.
  * [60] S. Salem.  A gradient flow approach of uniform in time propagation of chaos for particles in a double well confinement.  arXiv preprint arXiv:1810.08946, 2018.
  * [61] A.-S. Sznitman.  A fluctuation result for nonlinear diffusions.  Infinite Dimensional Analysis and Stochastic Processes, pages 145–160, 1985.
  * [62] A.-S. Sznitman.  Topics in propagation of chaos.  In École d’Été de Probabilités de Saint-Flour XIX—1989, volume 1464 of Lecture Notes in Math., pages 165–251. Springer, Berlin, 1991.
  * [63] Ł. Szpruch and A. Tse.  Antithetic multilevel particle system sampling method for Mckean-Vlasov SDEs.  arXiv preprint arXiv:1903.07063, 2019.
  * [64] H. Tanaka and M. Hitsuda.  Central limit theorem for a simple diffusion model of interacting particles.  Hiroshima Mathematical Journal, 11(2):415–423, 1981.
  * [65] F. Trèves.  Topological Vector Spaces, Distributions and Kernels.  Academic Press, New York-London, 1967.
  * [66] A. Tse.  Higher order regularity of nonlinear Fokker-Planck PDEs with respect to the measure component.  Journal de Mathématiques Pures et Appliquées, 2021.
  * [67] J. Tugaut.  Convergence to the equilibria for self-stabilizing processes in double-well landscape.  Ann. Probab., 41(3A):1427–1460, 2013.
  * [68] C. Villani.  Hypocoercive diffusion operators.  In International Congress of Mathematicians, volume 3, pages 473–498, 2006.