&\hspace{15pt} +
\int_{\mathcal P({\mathbb T}^d)}  \biggl\{ \widehat{\mathcal W}^{k_0}( m )
2 W (m  )
\vert k_0 \vert^{2pd}
\widehat{m}^{k_0} \biggr\}
\varphi\Bigl( \bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr)
d {\mathbb P}(m)
\biggr\vert \leq
C \delta,
\end{split}
\end{equation*}

for a constant $C$ independent of $N_0$ and
depending on $\varphi$ only through
$\| \varphi \|_\infty$ and
\|\partial_{\widehat{m}^{k_0}} \varphi \|_\infty$.

Notice that the role of $N_0$ in the statement is to
adjust the size of the support of the test function

the proof, we fix $\delta >0$ and $c>1$ and then choose
$N_0 \geq 1$, $\vartheta : [0,T] \rightarrow {\mathbb R}$
and $\varphi : {\mathbb R}^{2 \vert F_{N_0}^+ \vert} \rightarrow {\mathbb R}$ as in the statement.
Without any loss of generality, we also do as if the subsequences in assumptions $(ii)$ and
$(iii)$ were the full sequence $(N)_{N \geq 1}$ itself.

By item $(iii)$ in the statement, we get

\begin{equation*}
\begin{split}
\lim_{N \rightarrow \infty}
\int_{{\mathcal P}({\mathbb T}^d)} W^{(N)}\bigl(m*f_{N}\bigr)
\varphi\Bigl( \bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr) d {\mathbb P}(m)
& =
\int_{{\mathcal P}({\mathbb T}^d)} W(m)  \varphi\Bigl( \bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr) d {\mathbb P}(m).
\end{split}
\end{equation*}

Using the fact that the functions
$(W^{(N)})_{N \geq 1}$ are uniformly bounded
to get the first equality and
$(iii)$ in
the statement of
<ref>, the above limit can be reformulated as

\begin{equation}
\label{eq:lem:5:7:step:1}
\begin{split}
& \lim_{N \rightarrow \infty}
\int_{\PP} W^{(N)}\bigl(m*f_{N}\bigr)
\varphi\Bigl( \bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr) d {\mathbb P}_{N}(m)
\int_{{\mathcal P}({\mathbb T}^d)} W(m)    \varphi\Bigl( \bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr) d {\mathbb P}(m).
\end{split}
\end{equation}

Here, we can easily replace
$\varphi$ by its partial derivative
$\partial_{\widehat{m}^{k_0}} \varphi$,
for a given $k_0 \in F_{N_0}^+$,

\begin{equation*}
\begin{split}
& \lim_{N \rightarrow \infty}
\int_{\PP}  W^{(N)}\bigl(m*f_{N}\bigr)   \partial _{\widehat{m}^{k_0}} \varphi\Bigl( \bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr)d {\mathbb P}_{N}(m)
\\
&\hspace{15pt} =
\int_{{\mathcal P}({\mathbb T}^d)} W(m)    \partial _{\widehat{m}^{k_0}} \varphi\Bigl( \bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr) d {\mathbb P}(m).
\end{split}
\end{equation*}

We now recall the form of
${\mathbb P}_{N}$ in
(<ref>)–(<ref>) in order to study the left-hand side.
We have

\begin{equation*}
\begin{split}
&  \int_{\PP}  W^{(N)}\bigl(m*f_{N}\bigr)  \partial _{\widehat{m}^{k_0}} \varphi\Bigl( \bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr)d {\mathbb P}_{N}(m)
\\
&= \frac1{Z_{N}}
\int_{{\mathcal O}_{N}} W^{(N)} \bigl( m*f_{N}\bigr)  \partial _{\widehat{m}^{k_0}} \varphi\Bigl( \bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr)
\exp \Bigl( - \sum_{k \in F_{N}^+} \vert k\vert^{2pd} { \vert \widehat{m}^k \vert^2} \Bigr)
\bigotimes_{k \in F_{N}^+}
d \widehat{m}^k,
\end{split}
\end{equation*}

with $p \geq 5$
(see (<ref>) for the original occurrence of $p$)
and with $m$ in the second line being obviously identified with
${\mathscr I}_{N}((\widehat{m}^k)_{k \in F_{N}^+})$.

Fix now $N \geq 1$. By Lemmas
<ref> in the appendix, we can choose $N_0$
large enough such that,
$N \geq N_0$ and the support of $\varphi$ is included in ${\mathscr I}_{N}^{-1}(A_{N_0})$ (with $A_{N_0}$ as in the statement of
Lemma <ref>),

\begin{equation}
\label{eq:pass:from:N0:N}
{\mathbb P}_{N}
\biggl(
\Bigl\{
(\widehat{m}^k)_{k \in F_{N_0}^+} \in \textrm{\rm support}(\varphi)
\Bigr\}
\cap
\biggl\{
\sum_{k \in F_{N}^+ \setminus F_{N_0}^+}
\vert \widehat{m}^k \vert
\geq  \frac{a_0}{\vert k \vert^{5d/2}} \biggr\} \biggr)
\leq \delta,
\end{equation}

for $a_0$ fixed as in the statement of Lemma
<ref> (in particular, $a_0$ only depends on the dimension). By the same statement, the simple fact that

$${\mathscr I}_{N_0}((\widehat{m}^k)_{k \in F_{N_0}^+}) \in A_{N_0}
\quad \text{and}
\quad \sum_{k \in F_{N}^+ \setminus F_{N_0}^+}
\vert \widehat{m}^k \vert
< \frac{a_0}{\vert k \vert^{5d/2}}$$

that $(\widehat{m}^k)_{k \in F_{N}^+} \in {\mathcal O}_{N}$.
For such a choice, we get that

\begin{equation*}
\begin{split}
&\biggl\vert \int_{\PP}  W^{(N)}\bigl(m*f_{N}\bigr)  \partial _{\widehat{m}^{k_0}} \varphi\Bigl( \bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr)d {\mathbb P}_{N}(m)
\\
&\hspace{15pt} - \frac1{Z_{N}}    \int_{{\mathbb R}^{2 \vert F_{N}^+ \vert}} W^{(N)}\bigl( m*f_{N}\bigr) \partial _{\widehat{m}^{k_0}} \varphi\Bigl( \bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr)
\\
&\hspace{50pt} \times
{\mathbf 1}_{\{
\sum_{k \in F_{N}^+ \setminus F_{N_0}^+}
\vert \widehat{m}^k \vert
< a_0/\vert k \vert^{5d/2}\}}
\exp \Bigl( - \sum_{k \in F_{N}^+} \vert k\vert^{2pd} { \vert \widehat{m}^k \vert^2} \Bigr)
\bigotimes_{k \in F_{N}^+}
d \widehat{m}^k \biggr\vert \leq C \delta,
\end{split}
\end{equation*}

with $C$ depending on
$\varphi$ through
$\|\partial_{\widehat{m}^{k_0}} \varphi \|_\infty$, but being independent of $N$.

Then, we can make an integration by parts in the second term in the left-hand side.
This leads to two terms:
one derivative is acting on   $W^{(N)}$ and leads to
$\widehat{\mathcal W}^{k_0}$ (using item $(i)$ in the statement); $(b)$ the other derivative is acting on the density.
Therefore, we obtain

\begin{equation*}
\begin{split}
&\biggl\vert   \int_{\PP}  W^{(N)}\bigl(m*f_{N}\bigr)  \partial _{\widehat{m}^{k_0}} \varphi\Bigl( \bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr)d {\mathbb P}_{N}(m)
\\
&\hspace{15pt} + \frac1{Z_{N}}
\int_{{\mathbb R}^{2 \vert F_{N}^+ \vert}}  \biggl\{ \partial_{\widehat{m}^{k_0}} W^{(N)}\bigl( m*f_{N}\bigr)
\widehat{f}_{N}^{k_0}  -
2 W^{(N)}\bigl( m*f_{N}\bigr)
\vert k_0 \vert^{2pd}
\widehat{m}^{k_0} \biggr\}
\\
&\hspace{30pt} \times
{\mathbf 1}_{\{
\sum_{k \in F_{N}^+ \setminus F_{N_0}^+}
\vert \widehat{m}^k \vert
< a_0/\vert k \vert^{5d/2}\}}
\varphi\Bigl( \bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr)  \exp \Bigl( - \sum_{k \in F_{N}^+} \vert k\vert^{2pd} { \vert \widehat{m}^k \vert^2} \Bigr)
\bigotimes_{k \in F_{N}^+}
d \widehat{m}^k
\biggr\vert
\\
&\leq C \delta,
\end{split}
\end{equation*}

and then,
using once
(<ref>) together with the
$\sup_{N \geq 1}
\textrm{\rm essup}_{m \in {\mathcal P}_N}
\vert
\partial_{\widehat{m}^{k_0}} W^{(N)}(m) \vert
\sup_{N \geq 1}
\textrm{\rm essup}_{m \in {\mathcal P}_N}
\vert
\widehat{\mathcal W}^{k_0}(m)
\vert < \infty$,

\begin{equation*}
\begin{split}
&\biggl\vert    \int_{\PP} W^{(N)}\bigl(m*f_{N}\bigr)  \partial _{\widehat{m}^{k_0}} \varphi\Bigl( \bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr)d {\mathbb P}_{N}(m)
\\
&\hspace{15pt} +
\int_{{\mathcal P}({\mathbb T}^d)}  \biggl\{ \partial_{\widehat{m}^{k_0}} W^{(N)}\Bigl( m*f_{N}\Bigr)
\widehat{f}_{N}^{k_0}  -
2 W^{(N)}\bigl( m*f_{N}\bigr)
\vert k_0 \vert^{2pd}
\widehat{m}^{k_0} \biggr\}
\varphi\Bigl( \bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr)
d {\mathbb P}_{N}(m)
\biggr\vert
\\
&\leq C \delta,
\end{split}
\end{equation*}

with $C$ now depending on $\varphi$ through $\| \varphi \|_\infty$
$\|\partial_{\widehat{m}^{k_0}} \varphi \|_\infty$.
Recalling again that
$  \partial_{\widehat{m}^{k_0}} W^{(N)}(m*f_{N}) =
\widehat{\mathcal W}^{k_0}(m*f_{N})$
almost everywhere, we can use assumption $(ii)$ in the statement in order to pass to the limit in the term
$\partial_{\widehat{m}^{k_0}} W^{(N)}( m*f_{N})$.
and letting $N$ tend to $\infty$, we eventually get

\begin{equation*}
\begin{split}
\int_{{\mathcal P}({\mathbb T}^d)}  W(m)  \partial _{\widehat{m}^{k_0}} \varphi\Bigl( \bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr)d {\mathbb P}(m)
\\
&\hspace{15pt} +
\int_{\mathcal P({\mathbb T}^d)}  \biggl\{ \widehat{\mathcal W}^{k_0}( m )
2 W (m  )
\vert k_0 \vert^{2pd}
\widehat{m}^{k_0} \biggr\}
\varphi\Bigl( \bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr)
d {\mathbb P}(m)
\biggr\vert \leq
C \delta,
\end{split}
\end{equation*}

which completes the proof.

$({\mathcal W}_1,W_1)$ and
$({\mathcal W}_2,W_2)$ be two pairs of functions
satisfying the assumptions of Lemma
If $W_1$ and $W_2$ are equal almost everywhere under ${\mathbb P}$, then
${\mathcal W}_1$ and ${\mathcal W}_2$
are also equal
almost everywhere under ${\mathbb P}$.

We apply
Lemma <ref> to $({\mathcal W}_i,W_i)$, for $i=1,2$, and then use the fact that $W_1=W_2$.
We deduce that,
for $\delta>0$ and $c>1$, there exists
an integer $N_{c,\delta}$ such that, for $N_0 \geq N_{c,\delta}$,
any $k_0 \in F_{N_0}^+$
any smooth function $\varphi : {\mathbb R}^{2 \vert F_{N_0}^+ \vert} \rightarrow {\mathbb R}$
whose support is included in
$\{ (\widehat{m}^j)_{j \in F_{N_0}^+} \in {\mathbb C}^{\vert F_{N_0}^+\vert}
\simeq {\mathbb R}^{2 \vert F_{N_0}^+ \vert} :
1 + 2 \inf_{x \in {\mathbb T}^d}   \sum_{j \in F_{N_0}^+} \Re [\widehat{m}^j e_{-j}(x)]
\geq 1/c\}$,

\begin{equation}
\label{eq:W1-W2}
\biggl\vert
\int_{{\mathcal P}({\mathbb T}^d)} \Bigl(\widehat{\mathcal W}_1^{k_0}(m) - \widehat{\mathcal W}_2^{k_0}(m)\Bigr)   \varphi\Bigl( \bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr) d {\mathbb P}(m)   \biggr\vert \leq  C \delta,
\end{equation}

for a constant $C$ only depending on $\varphi$ through $\| \varphi \|_\infty$
and $\| \partial_{\widehat{m}^{k_0}} \varphi \|_\infty$.
By an obvious regularization procedure, the same holds if $\varphi$ is merely bounded
Lipschitz continuous, in which case the constant $C$ depends on $\varphi$ through
the bound and the Lipschitz constant of $\varphi$.

For $\vartheta : {\mathbb R} \rightarrow [0,1]$,
a $c$-Lipschitz function
that is equal to $0$ on $]-\infty,1/c[$, let

\begin{equation*}
\Theta
\Bigl( \bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr)
\vartheta \Bigl(
1 + 2
\inf_{x \in {\mathbb T}^d}
\sum_{k \in F_{N_0}^+}
\Re \bigl[ \widehat{m}^k e_{-k}(x)\bigr]
\Bigr), \quad
\bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} \in {\mathbb C}^{\vert F_{N_0}^+ \vert}
\simeq
{\mathbb R}^{2 \vert F_{N_0}^+ \vert}.
\end{equation*}

Observe that
the support of $\Theta$ is included in ${\mathcal O}_{N_0}$.
for $( \widehat{m}^k)_{k \in F_{N_0}^+} \in {\mathbb C}^{\vert F_{N_0}^+ \vert}
\simeq {\mathbb R}^{2 \vert F_{N_0}^+\vert}$
$\widehat{r}^{k_0} \in {\mathbb C}$, with $k_0 \in F_{N_0}^+$, we have

\begin{equation*}
\begin{split}
\Theta
\Bigl( \bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} + {\mathbf 1_{\{k=k_0\}}} \widehat{r}^{k_0}  \Bigr)
\Theta
\Bigl( \bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr)
\Bigr\vert
\\
&\leq 2 c \sup_{x \in {\mathbb T}^d}
\biggl\vert
\sum_{k \in F_{N_0}^+}
\widehat{m}^k e_{-k}(x)
\sum_{k \in F_{N_0}^+} \bigl(
\widehat{m}^k + {\mathbf 1}_{\{k=k_0\}} \widehat{r}^{k_0}
\bigr)
\biggr\vert\leq 2c  \bigl\vert \widehat{r}^{k_0} \bigr\vert,
\end{split}
\end{equation*}

hence proving that $\Theta$ is $2 c$-Lipschitz
with respect to $\widehat{m}^{k_0}$.
In particular, for a continuously differentiable test function $\phi$ on ${\mathcal P}({\mathbb T}^d)$,
we deduce from
that, for $N_0$ large enough,

\begin{equation*}
\begin{split}
\int_{{\mathcal P}({\mathbb T}^d)} \Bigl(\widehat{\mathcal W}_1^{k_0}(m) - \widehat{\mathcal W}_2^{k_0}(m)\Bigr)   \phi\bigl( m* f_{N_0} \bigr)
\vartheta\Bigl(
1 +
2 \inf_{x \in {\mathbb T}^d}
\sum_{k \in F_{N_0}^+}
\Re\bigl[\widehat{m}^k e_{-k}(x)\bigr]
\Bigr)
d {\mathbb P}(m)  \biggr\vert
\leq  C(c) \delta,
\end{split}
\end{equation*}

with $C(c)$ depending on $c$, but being independent of $N_0$ and $\delta$.
Above, $\phi(m*f_{N_0})$ is regarded as a smooth function on ${\mathcal O}_{N_0}$.

As before, $m*f_{N_0}$ converges to $m$ for the Wasserstein distance
as $N_0$ tends to $\infty$.
Moreover, Lemma
below says that,
${\mathbb P}$-almost everywhere, the series $\sum_{k \in {\mathbb Z}^d} \vert \widehat{m}^k \vert$
is absolutely convergent hence proving that
$ 1 +
\inf_{x \in {\mathbb T}^d}
2\sum_{k \in F_{N_0}^+}
\Re[\widehat{m}^k e_{-k}(x)]$
converges to $\inf_{x \in {\mathbb T}^d} m(x)$
with $m$ being identified with its density.
Therefore, we can let $N_0$ tend to $\infty$. We thus get

\begin{equation*}
\begin{split}
\int_{{\mathcal P}({\mathbb T}^d)} \Bigl(\widehat{\mathcal W}_1^{k_0}(m) - \widehat{\mathcal W}_2^{k_0}(m)\Bigr)
\phi(m)
\vartheta \Bigl(
\inf_{x \in {\mathbb T}^d}
\Bigr)
d {\mathbb P}(m)
\end{split}
\end{equation*}

There is no difficulty for sending $c$ to $+\infty$ in the definition of $\vartheta$
since ${\mathbb P}$ almost every $m \in \PP$ has a (strictly) positive density.
We obtain

\begin{equation*}
\int_{{\mathcal P}({\mathbb T}^d)} \Bigl(\widehat{\mathcal W}_1^{k_0}(m) - \widehat{\mathcal W}_2^{k_0}(m)\Bigr)  \phi(m)
d {\mathbb P}(m)
\end{equation*}

for any
continuously differentiable test function $\phi$ on ${\mathcal P}({\mathbb T}^d)$.
By the approximation procedure
defined in Proposition
the same holds true if
$\phi$ is merely continuous on ${\mathcal P}({\mathbb T}^d)$.
And, then
the same is true
$\phi$ is bounded and measurable on ${\mathcal P}({\mathbb T}^d)$, see
This suffices to identify
$\widehat{\mathcal W}_1^{k_0}$ and $\widehat{\mathcal W}_2^{k_0}$.

§.§ Classical solutions to the master equation as weak solutions

We now prove the following result, which legitimates Definition

If $U$ is a classical solution to the master equation,
with $\partial_t U$, $\partial_x U$, $\partial^2_{xx} U$, $\partial_\mu U$, $\partial_y \partial_{\mu} U$
being continuous
(with $y$
denoting the last argument in the derivative $\partial_\mu U(t,x,m)(y)$), then the centered version

\begin{equation*}
\tilde U(t,x,m) := U(t,x,m) - \int_{{\mathbb T}^d} U(t,y,m) dy,
\quad t \in [0,T], \ x \in {\mathbb T}^d, \ m \in \PP,
\end{equation*}

satisfies the
conservative version (<ref>) of the master equation.
In particular, $\tilde{U}$ is a weak solution to the master equation, in the sense of Definition <ref>.

For any starting point $(t,m) \in [0,T] \times {\mathcal P}({\mathbb T}^d)$, we can construct
a solution to the system
(<ref>) by solving
the Fokker-Planck equation

\begin{equation*}
\partial_t m_t(x)
- \textrm{\rm div}_x \Bigl( \partial_p H\bigl(x,\nabla_x U(t,x,m_t)\bigr) m_t(x) \Bigr)
%- \textrm{\rm div} \bigl( \nabla u_t  m_t\bigr) (x)
- \frac12 \Delta_x m_t(x) = 0,
\end{equation*}

and then, by letting, $u_t(x)= U(t,x,m_t)$.
In fact, this solution can be proven to be unique, see for instance

By Proposition
<ref>, the forward component of this unique solution, i.e. $(m_t)_{0 \leq t \leq T}$, must coincide with the (hence unique)
optimal path of the MFCP, when initialized from $(t,m)$.
In turn,
implies, for any $N \geq 1$,

\begin{equation*}
\partial_{\widehat{m}^k} V(t,m) =
\int_{{\mathbb T}^d} U(t,x,m) e^{-2 \i \pi k \cdot x} dx, \quad k \in F_N \setminus \{0\},
\end{equation*}

for almost every $(t,m) \in [0,T] \times {\mathcal P}_N$ under the probability measure $\textrm{\rm Leb}_1 \otimes {\mathbb P}_N$.
By continuity of $U$ in the right-hand side,
we deduce that $V$ is differentiable with respect to $\widehat{m}^k$ on the entire $[0,T] \times
{\mathcal P}_N$.

We deduce from Schwarz' theorem that, for
every $(t,m) \in [0,T] \times {\mathcal P}_N$
$k,\ell \in F_N \setminus \{0\}$,

\begin{equation*}
\int_{{\mathbb T}^d} \partial_{\widehat{m}^\ell} U(t,x,m) e^{-2 \i \pi k \cdot x} dx =
\int_{{\mathbb T}^d} \partial_{\widehat{m}^k} U(t,x,m) e^{-2 \i \pi \ell \cdot x} dx,
\quad k, \ \ell \in F_N \setminus \{0\},
\end{equation*}

which rewrites

\begin{equation*}
\int_{{\mathbb T}^d} \int_{{\mathbb T}^d}
\frac{\delta U}{\delta m}(t,x,m)(y) e^{-2 \i \pi k \cdot x}e^{-2 \i \pi \ell \cdot y} dx dy =
\int_{{\mathbb T}^d}\int_{{\mathbb T}^d} \frac{\delta U}{\delta m}(t,x,m)(y)
e^{-2 \i \pi k \cdot y}
e^{-2 \i \pi \ell \cdot x} dx,
\end{equation*}

$k, \ell \in F_N \setminus \{0\}$.
By invoking the continuity of $U$

and by approximating
any measure $m \in \PP$ by the sequence
$(m*f_N)_{N \geq 1}$,
the above holds true for any $(t,m) \in [0,T] \times \PP$ and any
$k,\ell \in {\mathbb Z}^d \setminus \{0\}$.
This shows

\begin{equation*}
\frac{\delta U}{\delta m}(t,x,m)(y)
\frac{\delta U}{\delta m}(t,y,m)(x)
\psi(y)-\phi(x),
\quad x,y \in {\mathbb T}^d,
\end{equation*}

for two real-valued functions $\phi$ and $\psi$
defined on ${\mathbb T}^d$ (both depending on $m$).
Integrating the above identity in $y$ and then in $x$ with respect to the Lebesgue measure on ${\mathbb T}^d$, we get

\begin{equation*}
\begin{split}
&\int_{{\mathbb T}^d} \frac{\delta U}{\delta m}(t,y,m)(x) dy = \phi(x)
\int_{{\mathbb T}^d} \psi(y) dy,
\\
&\int_{{\mathbb T}^d}
\frac{\delta U}{\delta m}(t,x,m)(y)
\psi(y)-
\int_{{\mathbb T}^d} \phi(x) dx,
\end{split}
\end{equation*}

$x,y \in {\mathbb T}^d$.
Integrating again in $x$ the first line, we get

\begin{equation*}
\int_{{\mathbb T}^d} \psi(y) dy
\int_{{\mathbb T}^d} \phi(x) dx,
\end{equation*}

and then

\begin{equation*}
\frac{\delta U}{\delta m}(t,x,m)(y)
\frac{\delta U}{\delta m}(t,y,m)(x)
\int_{{\mathbb T}^d}
\frac{\delta U}{\delta m}(t,x,m)(y)
\int_{{\mathbb T}^d}
\frac{\delta U}{\delta m}(t,y,m)(x)
\end{equation*}

$x,y \in {\mathbb T}^d$.
We then have

\begin{equation*}
\begin{split}
&\frac{\delta}{\delta m} \bigg\{  \int_{{\mathbb T}^d}
H \bigl( y,  \nabla_y  U(t,y,m) \bigr) d m(y) \biggr\}(x)
\\
H \bigl( x,  \nabla_y  U(t,x,m) \bigr)
\int_{{\mathbb T}^d}
H \bigl( y,  \nabla_y  U(t,y,m) \bigr) dy
\\
&\hspace{15pt} +
\int_{{\mathbb T}^d}
\partial_p H \bigl( y,  \nabla_y  U(t,y,m) \bigr)
\cdot
\partial_y \frac{\delta U}{\delta m}(t,y,m)(x)
d m(y)
\\
H \bigl( x,  \nabla_y  U(t,x,m) \bigr)
\int_{{\mathbb T}^d}
H \bigl( y,  \nabla_y   U(t,y,m) \bigr) d y
\\
&\hspace{15pt} +
\int_{{\mathbb T}^d}
\partial_p H \bigl( y,  \nabla_y  U(t,y,m) \bigr)
\cdot
\partial_y \frac{\delta U}{\delta m}(t,x,m)(y)
d m(y)
\\
&\hspace{15pt} -
\int_{{\mathbb T}^d}  \int_{{\mathbb T}^d}
\partial_p H \bigl( y,  \nabla_y  U(t,y,m) \bigr)
\cdot
\partial_y \frac{\delta U}{\delta m}(t,x,m)(y)
d x \, d m(y),
\end{split}
\end{equation*}

and similarly,

\begin{equation*}
\begin{split}
& \frac{\delta}{\delta m} \bigg\{  \int_{{\mathbb T}^d}
\text{Tr} \Bigl[
\partial^2_y   U(t,y,m) \Bigr] d m(y)   \bigg\}(x)
\\
\text{Tr} \Bigl[
\partial^2_x  U(t,x,m) \Bigr]
\int_{{\mathbb T}^d}
\text{Tr} \Bigl[
\partial^2_y  U(t,y,m) \Bigr] d y
\\
&\hspace{15pt} +
\int_{{\mathbb T}^d}
\text{Tr} \Bigl[
\partial^2_y   \frac{\delta U}{\delta m}(t,y,m)(x) \Bigr] d m(y)
\\
&= \text{Tr} \Bigl[
\partial^2_x   U(t,x,m) \Bigr]
\int_{{\mathbb T}^d}
\text{Tr} \Bigl[
\partial^2_y   U(t,y,m) \Bigr] d y
\\
&\hspace{15pt} +
\int_{{\mathbb T}^d}
\text{Tr} \Bigl[
\partial^2_y   \frac{\delta U}{\delta m}(t,x,m)(y) \Bigr] d m(y)
\int_{{\mathbb T}^d}
\int_{{\mathbb T}^d}
\text{Tr} \Bigl[
\partial^2_y   \frac{\delta U}{\delta m}(t,x,m)(y) \Bigr] d x \,  d m(y).
\end{split}
\end{equation*}

The conclusion easily follows: it suffices to
the equation
$x$ with respect to
the Lebesgue measure, to make the difference between
and the hence integrated version of
and then to insert the latter  two identities, noticing that
$\nabla_x U=\nabla_x \tilde U$.

The last claim then follows since $\tilde{U}$ solves (<ref>) in the classical sense. Indeed, we can argue as in Subsection <ref>, the computations being here legitimate because the solution is smooth.

§ APPENDIX

§.§ Construction of a probability measure on ${\mathcal P}({\mathbb T}^d)$ satisfying
Theorem <ref>.

This (long) subsection is dedicated to the proof of
This comes as a by-product of a generic construction, which makes use (in a quite systematic manner)
of the notations introduced in Subsection
<ref> (we invite the reader to have a new look at them before she/he enters the details of the proof below).
In particular, we recall
and the subsequent notation
${\mathbb P}_N= \Gamma_N \circ {\mathscr I}_N^{-1}$.
Quite often in the analysis, we also make use of the normalization constant

\begin{equation}
\label{eq:cN} c_N :=
\int_{{\mathbb R}^{2\vert F_N^+\vert  }}
\exp \biggl( - \sum_{k \in F_{N}^+ }\vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
\bigotimes_{j \in F_{N}^+ }
d \widehat{m}^j.
\end{equation}

(Notice that $c_N$ is different from $Z_N$ in (<ref>).)
Moreover, we also introduce  a sequence
$(\xi_k)_{k \in {\mathbb N}^d \setminus \{0\}}$
of independent two-dimensional Gaussian random variables on an auxiliary probability space with
${\mathbf P}$ as probability measure, such that each
$\xi_k$ has $I_2$
as covariance matrix (with $I_2$ being the identity matrix of dimension 2).

§.§.§ Properties of the sequence
$({\mathbb P}_N)_{N \geq 1}$

In this paragraph, we obtain a series of lemmas on the properties of
the measures
$({\mathbb P}_N)_{N \geq 1}$. We start with the following lemma:

For any two integers $N_0 \leq N$
and for any $[0,1]$-valued
$\varphi$ defined on ${\mathbb R}^{2 \vert F_{N_0}^+\vert}$ that is equal to zero outside ${\mathcal O}_{N_0}$,

\begin{equation*}
\begin{split}
&\int_{{\mathcal P}({\mathbb T}^d)} \varphi\Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr) d {\mathbb P}_N(m)
\\
&\hspace{15pt} \leq \frac{c_N}{c_{N_0}} \frac1{Z_{N}}
\int_{{\mathcal O}_{N_0}} \varphi \Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr)
\exp \biggl( - \sum_{k \in F_{N_0}^+} \vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
\bigotimes_{j \in  F_{N_0}^+}
d \widehat{m}^j.
\end{split}
\end{equation*}

Moreover, for any $\varepsilon \in (0,1)$,
$\varphi$ is null outside the points
$(\widehat{m}^k)_{k \in F_{N_0}^+}$ such that

\begin{equation}
\label{eq:compact:support}
2 \inf_{x \in {\mathbb T}^d}
\sum_{k \in F_{N_0}^+}
2 \Re \bigl[ \widehat{m}^k e_{-k}(x) \bigr] \geq \varepsilon,
%\inf_{(z^k)_{k \in ({\mathbb N} \setminus \{0\})^d} : \sum_k \vert z^k \vert^2 \leq 1}
%\sum_{k,k' \in ({\mathbb N} \setminus \{0\})^d : k \not = k'} \widehat{m}^{k-k'} z^k \overline z^{k'}
%\Bigr] \leq - 1+ \varepsilon,
\end{equation}

\begin{equation*}
\begin{split}
&\int_{{\mathcal P}({\mathbb T}^d)} \varphi\Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr) d {\mathbb P}_N(m)
\\
&\hspace{15pt} \geq \frac{c_N}{c_{N_0}}  \frac{c(\varepsilon,N_0)}{Z_{N}}
\int_{{\mathcal O}_{N_0}} \varphi \Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr)
\exp \biggl( - \sum_{k \in F_{N_0}^+} \vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
\bigotimes_{j \in  F_{N_0}^+}
d \widehat{m}^j,
\end{split}
\end{equation*}

\begin{equation}
\label{eq:compact:support:2}
c(\varepsilon,N_0) := 1 - \frac{C(p)}{\varepsilon N_0^{(p-5/2)d}},
\end{equation}

for $C(p)$ a constant only depending on $p$ and $d$.

First Step.
Using the same notations as in the statement and recalling that $\varphi$ is zero outside
${\mathcal O}_{N_0}$, we have

\begin{align}
&\int_{{\mathcal P}({\mathbb T}^d)} \varphi \Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr)d {\mathbb P}_N(m)
\nonumber
\\
&= \frac1{Z_N}
\int_{{\mathcal O}_N} \varphi \Bigl( (\widehat{m}^k  )_{k \in F_{N_0}^+ } \Bigr)
\exp \biggl( - \sum_{k \in F_N^+} \vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
\bigotimes_{k \in F_N^+}
d \widehat{m}^k
\nonumber
\\
&= \frac1{Z_{N}}
\int_{{\mathcal O}_{N_0}} \varphi \Bigl( (\widehat{m}^k  )_{k \in F_{N_0}^+ } \Bigr)
\exp \biggl( - \sum_{k \in F_{N_0}^+} \vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
\label{eq:c_d:ON:0}
\\
&\hspace{5pt} \times
\int_{{\mathbb R}^{2(\vert F_N^+\vert - \vert F_{N_0}^+\vert )}}
\biggl[
{\mathbf 1}_{{\mathcal O}_N}\Bigl( (\widehat{m}^k)_{k \in F_{N}^+ } \Bigr)
\exp \biggl( - \sum_{k \in F_{N}^+ \setminus F_{N_0}^+}\vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
\bigotimes_{j \in F_{N}^+ \setminus F_{N_0}^+}
d \widehat{m}^j
\biggr]
\bigotimes_{j \in  F_{N_0}^+}
d \widehat{m}^j. \nonumber
\end{align}

Obviously, we can bound the indicator function in the last line by 1 and then get $c_N/c_{N_0}$
(see (<ref>) for the definition) as bound for the whole term on the last line.
This proves the first inequality in the statement.

Second Step.
Take now $\varphi$ as in the statement and $(\widehat m^k)_{k \in F_{N_0}^+}$ satisfying

If we choose another collection $(\widehat{m}^k)_{k \in F_N  \setminus F_{N_0}}$
(with the usual requirement that $\widehat{m}^{-k} = \overline{\widehat{m}^k}$) such that

\begin{equation*}
\sum_{k \in F_N \setminus F_{N_0}} \vert \widehat{m}^k \vert < \frac12 \varepsilon,
\end{equation*}

\begin{equation*}
1 + \inf_{x \in {\mathbb T}^d} \sum_{k \in F_{N} \setminus \{0\}}
\widehat{m}_k e_{-k}(x)=
1 + 2 \inf_{x \in {\mathbb T}^d} \sum_{k \in F_{N} \setminus \{0\}}
\Re \bigl[ \widehat{m}_k e_{-k}(x) \bigr] \geq \frac12 \varepsilon,
\end{equation*}

and accordingly the left hand side can be regarded as a probability measure  in
${\mathcal P}_N$. For instance, so is the case if
$\vert \widehat{m}^k \vert \leq \kappa_{d}^{-1} \varepsilon
\vert k\vert^{-3d/2}$, for $\kappa_d= 2 \sum_{j \in {\mathbb Z}^d \setminus \{0\}} \vert j \vert^{-3d/2}$.
$m \in {\mathcal P}({\mathbb T}^d)$ such that
$(\widehat{m}^k )_{k \in F_{N_0}^+}$
belongs to the support of $\varphi$,

\begin{equation}
\label{eq:c_d:ON}
\begin{split}
&\int_{{\mathbb R}^{2(\vert F_N^+\vert - \vert F_{N_0}^+\vert) }}
\prod_{k \in F_{N}^+ \setminus F_{N_0}^+}
{\mathbf 1}_{\{\vert \widehat{m}^k \vert < \kappa_d^{-1} \varepsilon \vert k\vert^{-3d/2}\}}
\exp \biggl( - \sum_{k \in F_{N}^+ \setminus F_{N_0}^+}\vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
\bigotimes_{j \in F_{N}^+ \setminus F_{N_0}^+}
d \widehat{m}^j
\\
&\leq \int_{{\mathbb R}^{2(\vert F_N^+\vert - \vert F_{N_0}^+\vert )}}
{\mathbf 1}_{{\mathcal O}_N}\Bigl( (\widehat{m}^k)_{k \in F_{N}^+ } \Bigr)
\exp \biggl( - \sum_{k \in F_{N}^+ \setminus F_{N_0}^+}\vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
\bigotimes_{j \in F_{N}^+ \setminus F_{N_0}^+}
d \widehat{m}^j
\\
\int_{{\mathbb R}^{2(\vert F_N^+\vert - \vert F_{N_0}^+\vert )}}
\exp \biggl( - \sum_{k \in F_{N}^+ \setminus F_{N_0}^+}\vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
\bigotimes_{j \in F_{N}^+ \setminus F_{N_0}^+}
d \widehat{m}^j.
\end{split}
\end{equation}

Multiplying both sides by $c_{N_0}/c_{N}$, we get

\begin{equation}
\label{eq:c_d:ON:2}
\begin{split}
& \prod_{k \in F_{N}^+ \setminus F_{N_0}^+} {\mathbf P} \biggl( \Bigl\{ \frac1{\sqrt{2} \vert k\vert^{pd}} \vert \xi_k\vert <
\frac{\varepsilon}{\kappa_d \vert k \vert^{3d/2}} \Bigr\} \biggr)
\\
\frac{c_{N_0}}{c_N}
\int_{{\mathbb R}^{2(\vert F_N^+\vert - \vert F_{N_0}^+\vert )}}
{\mathbf 1}_{{\mathcal O}_N}\Bigl( (\widehat{m}^k)_{k \in F_{N}^+ } \Bigr)
\exp \biggl( - \sum_{k \in F_{N}^+ \setminus F_{N_0}^+}\vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
\bigotimes_{j \in F_{N}^+ \setminus F_{N_0}^+}
d \widehat{m}^j
\\
&\leq 1,
%\int_{{\mathbb R}^{\vert F_N^+\vert - \vert F_{N_0}^+\vert }}
%\prod_{k \in F_{N}^+ \setminus F_{N_0}^+}
%{\mathbf 1}_{\{\vert \widehat{m}^k \vert <  c_d^{-1} \varepsilon \vert k\vert^{-3d/2}\}}
%\exp \biggl( - \sum_{k \in F_{N}^+ \setminus F_{N_0}^+}\vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
%\prod_{j \in F_{N}^+ \setminus F_{N_0}^+}
%d \widehat{m}^j
%&= \prod_{k \in F_{N}^+ \setminus F_{N_0}^+} {\mathbf P} \Bigl\{ \frac1{\sqrt{2} \vert k\vert^{pd}} \vert \xi_k\vert <
%\frac{\varepsilon}{c_d \vert k \vert^{3d/2}} \Bigr\},
%\prod_{k \in F_{N_2}^+ \setminus F_{N_0}^+} {\mathbf P} \Bigl\{  \vert \xi_k\vert <\sqrt{2} \frac{\varepsilon}{c_d \vert k \vert^{d/2}} \Bigr\}
%\times \prod_{k \in F_{N}^+ \setminus F_{N_2}^+} {\mathbf P} \Bigl\{ \vert \xi_k\vert <
%\sqrt{2} \frac{\varepsilon}{c_d \vert k \vert^{d/2}} \Bigr\},
\end{split}
\end{equation}

where $(\xi_k)_{k \in F_{N}^+ \setminus F_{N_0}^+}$ is defined right below (<ref>).

By Markov inequality, for $p > 3/2$,

\begin{equation}
\label{eq:Markov}
\begin{split}
\prod_{k \in F_{N}^+ \setminus F_{N_0}^+} {\mathbf P} \biggl( \Bigl\{
\frac1{\sqrt{2} \vert k\vert^{pd}} \vert \xi_k\vert <
\frac{\varepsilon}{\kappa_d \vert k \vert^{3d/2}}
\Bigr\}
\biggr)
&\geq  1 - \sum_{k \in F_{N}^+ \setminus F_{N_0}^+} {\mathbf P} \biggl( \Bigl\{
\vert \xi_k\vert \geq
\sqrt{2}
\frac{\varepsilon}{\kappa_d} \vert k \vert^{(p-3/2)d}
\Bigr\} \biggr)
\\
&\geq 1 - C(p) \sum_{k \in F_N^+ \setminus F_{N_0}^+} \frac{1}{\varepsilon \vert k\vert^{(p-3/2)d}}
\\
&\geq 1 - \frac{C(p)}{\varepsilon N_0^{(p-5/2)d}},
\end{split}
\end{equation}

where the value of the constant $C(p)$ is allowed to change from line to line (as long as it only depends on
$p$ and $d$).
The difference on the last line coincides with
$c(\varepsilon,N_0)$ in

Back to
(<ref>), we deduce

\begin{equation*}
\begin{split}
\leq \frac{c_{N_0}}{c_{N}} \int_{{\mathbb R}^{2 (\vert F_N^+\vert - \vert F_{N_0}^+\vert) }}
{\mathbf 1}_{{\mathcal O}_N}\Bigl( (\widehat{m}^k)_{k \in F_{N}^+ } \Bigr)
\exp \biggl( - \sum_{k \in F_{N}^+ \setminus F_{N_0}^+}\vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
\bigotimes_{j \in F_{N}^+ \setminus F_{N_0}^+}
d \widehat{m}^j.
%\leq \frac1{c(\varepsilon,N_0)}\int_{{\mathbb R}^{\vert F_N^+\vert - \vert F_{N_0}^+\vert }}
%{\mathbf 1}_{{\mathcal O}_N}\Bigl( (\widehat{m}^k)_{k \in F_{N}^+ } \Bigr)
%\exp \biggl( - \sum_{k \in F_{N}^+ \setminus F_{N_0}^+}\vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
%\prod_{j \in F_{N}^+ \setminus F_{N_0}^+}
%d \widehat{m}^j.
\end{split}
\end{equation*}

Back to
(<ref>), we get
the second inequality in the statement (and this even if
$c(\varepsilon,N_0) $ is negative).

We get the following lemma:

With the same notations as above,

\begin{equation*}
\lim_{N_0 \rightarrow \infty} \sup_{N \geq N_0}
\Bigl\vert \frac{Z_{N_0}}{Z_N} \frac{c_N}{c_{N_0}} - 1 \Bigr\vert =0,
\end{equation*}

\begin{equation*}
\lim_{N_0 \rightarrow \infty}
\inf_{N \geq N_0} {\mathbb P}_N \biggl( \biggl\{
1 +\inf_{x \in {\mathbb T}^d} \sum_{k \in F_{N_0} \setminus \{0\}}
\widehat{m}^k e_{-k}(x) \geq (2N_0)^{-3d/2}
\biggr\} \biggr) = 1,
\end{equation*}

with the usual convention that
$\widehat{m}^{-k} = \overline{\widehat{m}^k}$ for $k \in F_{N_0}^+$.

First Step.
We choose
$\varphi$ in Lemma <ref> as $\varphi_0$, the indicator function of the set
of Fourier coefficients
$(\widehat{m}^k)_{k \in F_{N_0}^+ }$ such that

\begin{equation}
\label{eq:support:varphi}
1 + \inf_{x \in {\mathbb T}^d}
\sum_{k \in F_{N_0} \setminus \{0\}}
\widehat{m}^k e_{-k}(x) > \varepsilon,
\end{equation}

for some $\varepsilon \in (0,1)$ as in the statement of Lemma <ref> and
with the same convention
as in the statement of
$\widehat{m}^{-k} = \overline{\widehat{m}^k}$.
we have

\begin{equation*}
\begin{split}
\frac{c_N}{c_{N_0}}
\int_{{\mathcal O}_{N_0}} \varphi_0 \Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr)
\exp \biggl( - \sum_{k \in F_{N_0}^+} \vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
\bigotimes_{j \in  F_{N_0}^+}
d \widehat{m}^j
\\
\frac1{Z_N}
\frac{c_N}{c_{N_0}}
\int_{{\mathbb R}^{2\vert F^+_{N_0}\vert}} \varphi_0 \Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr)
\exp \biggl( - \sum_{k \in F_{N_0}^+} \vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
\bigotimes_{j \in  F_{N_0}^+}
d \widehat{m}^j.
\end{split}
\end{equation*}

We then make the change of variable

\begin{equation*}
\widehat{y}^k = \frac1{1-\varepsilon} \widehat{m}^k.
\end{equation*}

Then, the condition
merely says that
$(\widehat{y}^k)_{k \in F_{N_0}^+}$
belongs to ${\mathcal O}_{N_0}$ if
$(\widehat{m}^k)_{k \in F_{N_0}^+}$ belongs to the support of $\varphi_0$.

\begin{equation}
\label{eq:lem:25:new:1}
\begin{split}
\frac{c_N}{c_{N_0}}
\int_{{\mathcal O}_{N_0}} \varphi_0 \Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr)
\exp \biggl( - \sum_{k \in F_{N_0}^+} \vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
\bigotimes_{j \in  F_{N_0}^+}
d \widehat{m}^j
\\
&=\frac{(1-\varepsilon)^{2 \vert F_{N_0}^+ \vert}}{Z_N}
\frac{c_N}{c_{N_0}}
\int_{{\mathcal O}_{N_0}}
\exp \biggl( - (1-\varepsilon)^2 \sum_{k \in F_{N_0}^+} \vert k\vert^{2 p d} { \vert \widehat{y}^k \vert^2} \biggr)
\bigotimes_{j \in  F_{N_0}^+}
d \widehat{y}^j
\\
&\geq (1-\varepsilon)^{2 \vert F^+_{N_0}\vert} \frac{Z_{N_0}}{Z_N} \frac{c_N}{c_{N_0}}.
\end{split}
\end{equation}

By the
second inequality in the statement of Lemma <ref>, we deduce that

\begin{equation*}
(1-\varepsilon)^{2 \vert F^+_{N_0}\vert }  \frac{Z_{N_0}}{Z_{N}} \frac{c_N}{c_{N_0}} \leq 1.
\end{equation*}

Moreover, choosing $\varphi=1$ in
the first inequality in the statement of Lemma <ref>, we also have

\begin{equation*}
\frac{Z_{N_0}}{Z_N} \frac{c_N}{c_{N_0}} \geq 1.
\end{equation*}

Second Step.
We notice from Proposition
<ref> that $2 \vert F_{N_0}^+ \vert = D_{N_0} \leq 2 (2 N_0)^d$. So,
if we choose
$\varepsilon= N_0^{-3d/2}$, then
the shape of
$c(\varepsilon,N_0)$ in
Lemma <ref>
together with the fact that $p \geq 5$

\begin{equation*}
(1-\varepsilon)^{2 \vert F^+_{N_0}\vert}
\geq \Bigl( 1 - \frac{ C(p)}{N_0^{d}} \Bigr)
\exp \Bigl(2  (2N_0)^d \ln( 1 - (N_0)^{-3d/2}) \Bigr).
\end{equation*}

Obviously the right-hand side tends to $1$ as $N_0$ tends to $\infty$.
Combining with the conclusion of the first step, we get

\begin{equation*}
\lim_{N_0 \rightarrow \infty} \sup_{N \geq N_0}
\Bigl\vert \frac{Z_{N_0}}{Z_N} \frac{c_N}{c_{N_0}} - 1 \Bigr\vert =0,
\end{equation*}

which is the first claim in the statement.

Third Step.
By the second inequality in the statement of Lemma <ref>,

\begin{equation*}
\begin{split}
&\int_{{\mathcal P}({\mathbb T}^d)} \varphi_0 \Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr) d {\mathbb P}_N(m)
\\
&\hspace{15pt} \geq \frac{c_N}{c_{N_0}}  \frac{c(\varepsilon,N_0)}{Z_{N}}
\int_{{\mathcal O}_{N_0}} \varphi_0 \Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr)
\exp \biggl( - \sum_{k \in F_{N_0}^+} \vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
\bigotimes_{j \in  F_{N_0}^+}
d \widehat{m}^j,
\end{split}
\end{equation*}

(<ref>) (with $\varepsilon = N_0^{-3d/2}$), we get

\begin{equation*}
\begin{split}
{\mathbb P}_N \biggl( \biggl\{
1 +\inf_{x \in {\mathbb T}^d} \sum_{k \in F_{N_0} \setminus \{0\}}
\widehat{m}^k e_{-k}(x) > N_0^{-3d/2}
\biggr\} \biggr)
&= \int_{{\mathcal P}({\mathbb T}^d)} \varphi_0\Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr) d {\mathbb P}_N(m)
\\
& \geq   c(\varepsilon,N_0) (1-\varepsilon)^{2 \vert F^+_{N_0}\vert} \frac{Z_{N_0}}{Z_N} \frac{c_N}{c_{N_0}},
\end{split}
\end{equation*}

By the second step, we get  that the infimum over $N \geq N_0$ of the right-hand side
tends to $1$ as $N_0$ tends to $\infty$.
This completes the proof.

Here is an application, which is very useful throughout the paper.

Let $(\psi_{N_0})_{N_0 \geq 1}$ be a sequence of $[-1,1]$-valued Borel measurable functions defined on
${\mathcal P}({\mathbb T}^d)$ with the following two properties:

* $\psi_{N_0}$ is measurable with respect to the $\sigma$-algebra generated by the mappings
$(m \in {\mathcal P}({\mathbb T}^d) \mapsto \widehat{m}^k)_{k \in F_{N_0}^+}$;

* $\psi_{N_0}$ is null outside

\begin{equation*}
A_{N_0} := \biggl\{ m \in {\mathcal P}({\mathbb T}^d) :
1+ \inf_{x \in {\mathbb T}^d} \sum_{k \in F_{N_0} \setminus \{0\}}
\widehat{m}^k e_{-k}(x) \geq ( 2N_0)^{-3d/2}
\biggr\}.
\end{equation*}

Then, there exists $a_0>0$, only depending on $d$, such that, for any $a \in (0,a_0]$,

\begin{equation*}
\begin{split}
& A_{N_0} \cap
\bigcap_{ k  \in F_N^+ \setminus F_{N_0}^+}
\Bigl\{  m \in {\mathcal P}({\mathbb T}^d) :
\vert \widehat{m}^k \vert
< \frac{a}{\vert k \vert^{5d/2}}
\Bigr\}
\\
&\hspace{15pt} \subset
\biggl\{  m \in {\mathcal P}({\mathbb T}^d) :
1+ \inf_{x \in {\mathbb T}^d} \sum_{k \in F_{N} \setminus \{0\}}
\widehat{m}^k e_{-k}(x) \geq 2^{-(3d/2+1)} N_0^{-3d/2}
\biggr\},
\end{split}
\end{equation*}

\begin{equation*}
\begin{split}
&\lim_{N_0 \rightarrow \infty} \sup_{N \geq N_0} \biggl\vert
\biggl(
\int_{{\mathcal P}({\mathbb T}^d)}
\psi_{N_0}(m)
d {\mathbb P}_{N_0}(m)
\biggr)^{-1}
\int_{{\mathcal P}({\mathbb T}^d)}
\biggl[
\psi_{N_0}(m)
\prod_{ k  \in F_N^+ \setminus F_{N_0}^+}
{\mathbf 1}_{\{
\vert \widehat{m}^k \vert
< a \vert k \vert^{-5d/2}
\Bigr\}}
\biggr]
d {\mathbb P}_N(m)
1 % \prod_{k \in F_N^+ \setminus F_{N_0}^+}
%{\mathbf P}
%\vert \xi_k \vert \leq
%\sqrt{2} a \vert k \vert^{(p-5/2)d}
\biggr\vert
\\
\end{split}
\end{equation*}

with the
numerator in the first line being necessarily equal  $0$ when
the denominator is $0$ and the convention that
the ratio is then understood as $1$.

In fact, the first claim can be reformulated in the following broader sense.
If $(\widehat m^k)_{k \in F_N^+}$ satisfies, for $N \geq N_0$,

1+ \inf_{x \in {\mathbb T}^d} \sum_{k \in F_{N_0} \setminus \{0\}}
\widehat{m}^k e_{-k}(x) \geq ( 2N_0)^{-3d/2},
\quad
\textrm{and}
\quad
\max_{ k  \in F_N^+ \setminus F_{N_0}^+}
\vert \widehat{m}^k \vert
< \frac{a}{\vert k \vert^{5d/2}},

for $a \in (0,a_0]$,

1+ \inf_{x \in {\mathbb T}^d} \sum_{k \in F_{N} \setminus \{0\}}
\widehat{m}^k e_{-k}(x) \geq 2^{-(3d/2+1)} N_0^{-3d/2}.

In words, there is no need to assume a priori that $(\widehat{m}^k)_{k \in F_N^+}$ are the Fourier coefficients of a  probability measure. The conditions on the Fourier coefficients suffice to prove
it a posteriori.

We first observe that, for

\in A_{N_0} \cap
\bigcap_{ k  \in F_N^+ \setminus F_{N_0}^+}
\Bigl\{
\vert \widehat{m}^k \vert
< \frac{a}{\vert k \vert^{5d/2}}
\Bigr\},$$

it holds

\begin{equation*}
\begin{split}
1+ \inf_{x \in {\mathbb T}^d} \sum_{k \in F_{N} \setminus \{0\}}
\widehat{m}^k e_{-k}(x)
&\geq ( 2N_0)^{-3d/2} - \sum_{k \in F_N \setminus F_{N_0}} \frac{a}{\vert k \vert^{5d/2}}
\\
&\geq ( 2N_0)^{-3d/2} - c_d a (N_0)^{-3d/2},
\end{split}
\end{equation*}

for a constant $c_d$ only depending on $d$.
Therefore, for $c_d a \leq 2^{-(3d/2+1)}$,

\begin{equation*}
\begin{split}
1+ \inf_{x \in {\mathbb T}^d} \sum_{k \in F_{N} \setminus \{0\}}
\widehat{m}^k e_{-k}(x)
&\geq  2^{-(3d/2+1)} N_0^{-3d/2}.
\end{split}
\end{equation*}

We deduce that, for such an $a$ and for $\psi_{N_0}$ as in the statement,

\begin{equation*}
\begin{split}
&\int_{{\mathcal P}({\mathbb T}^d)}
\biggl[
\psi_{N_0}(m)
\prod_{ k  \in F_N^+ \setminus F_{N_0}^+}
{\mathbf 1}_{\{
\vert \widehat{m}^k \vert
< a \vert k \vert^{-5d/2}
\}}
\biggr]
d {\mathbb P}_N(m)
\\
&= \frac1{Z_N}
\int_{{\mathbb R}^{2\vert F_N^+\vert}}
\biggl[ \varphi_{N_0}\Bigl( \bigl(\widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr)
\prod_{k  \in F_N^+ \setminus F_{N_0}^+}
{\mathbf 1}_{\{    \vert \widehat{m}^k \vert
< a \vert k \vert^{-5d/2} \}}
\exp \biggl( - \sum_{k \in F_{N}^+} \vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
\biggr]
\bigotimes_{j \in  F_{N}^+}  d \widehat{m}^j ,
\end{split}
\end{equation*}

$\varphi_{N_0} = \psi_{N_0} \circ {\mathscr I}_{N_0}$, see
the notation in

We rewrite the above equality as

\begin{equation}
\label{eq:lem:25:second:1}
\begin{split}
&\int_{{\mathcal P}({\mathbb T}^d)}
\biggl[
\psi_{N_0}(m)
\prod_{ k  \in F_N^+ \setminus F_{N_0}^+}
{\mathbf 1}_{\{
\vert \widehat{m}^k \vert
< a \vert k \vert^{-5d/2}
\}}
\biggr]
d {\mathbb P}_N(m)
\\
&= \frac1{Z_N}
\int_{{\mathcal O}_{N_0}}
\varphi_{{N_0}}\Bigl( \bigl(\widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr)
\exp \biggl( - \sum_{k \in F_{N_0}^+} \vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
\bigotimes_{j \in  F_{N_0}^+}
d \widehat{m}^j
\\
&\hspace{15pt} \times
\int_{{\mathbb R}^{2 (\vert F^+_N \vert -\vert F^+_{N_0}\vert)}}
\prod_{k  \in F_N^+ \setminus F_{N_0}^+}
{\mathbf 1}_{\{    \vert \widehat{m}^k \vert
< a \vert k \vert^{-5d/2} \}}
\exp \biggl( - \sum_{k \in F_{N}^+ \setminus F_{N_0}^+} \vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
\bigotimes_{j \in  F_{N}^+ \setminus F_{N_0}^+}
d \widehat{m}^j
\\
&= \frac{Z_{N_0}}{Z_N}  \frac{c_N}{c_{N_0}}
\biggl( \int_{{\mathcal P}({\mathbb T}^d)}
\psi_{N_0}(m)
d {\mathbb P}_{N_0}(m)
\biggr)
\prod_{k \in F_N^+ \setminus F_{N_0}^+}
{\mathbf P}
\Bigl(
\Bigl\{
\vert \xi_k \vert \leq
\sqrt{2} a \vert k \vert^{(p-5/2)d}
\Bigr\}
\Bigr).
\end{split}
\end{equation}

\begin{equation*}
\begin{split}
\prod_{k \in F_N^+ \setminus F_{N_0}^+}
{\mathbf P}
\Bigl(
\Bigl\{
\vert \xi_k \vert \leq
\sqrt{2} a \vert k \vert^{(p-5/2)d}
\Bigr\}
\Bigr)
\geq 1 -  \frac{C(p)}{a N_0^{(p-7/2)d}},
\end{split}
\end{equation*}

for a constant $C(p)$ only depending on
$p$ and $d$.
Therefore, by
<ref>, we have

\begin{equation*}
\lim_{N_0 \rightarrow \infty}
\sup_{N \geq N_0}
\biggl\vert
\frac{Z_{N_0}}{Z_N}  \frac{c_N}{c_{N_0}}
\prod_{k \in F_N^+ \setminus F_{N_0}^+}
{\mathbf P}
\Bigl(
\Bigl\{
\vert \xi_k \vert \leq
\sqrt{2} a \vert k \vert^{(p-5/2)d}
\Bigr\}
\Bigr)
1 \biggr\vert =0.
\end{equation*}

Inserting the latter into
(<ref>), we get the conclusion.

We deduce the following lemma:

Let $a_0>0$
be as in the statement of Lemma
<ref>. Then,
for any $\delta >0$, there exists
$N_1 \geq 1$ such that,
for any $N \geq N_0 \geq N_1$,

\begin{equation}
\begin{split}
&{\mathbb P}_N \Bigl( \bigl\{
m >
2^{-(3d/2+1)} N_0^{-3d/2}
\bigr\}
\Bigr)
\geq
{\mathbb P}_N
\biggl( A_{N_0} \cap
\bigcap_{ k  \in F_N^+ \setminus F_{N_0}^+}
\Bigl\{  m \in {\mathcal P}({\mathbb T}^d) :
\vert \widehat{m}^k \vert
< \frac{a_0}{\vert k \vert^{5d/2}}
\Bigr\}
\biggr)
\geq 1 - \delta,
\end{split}
\end{equation}

with $A_{N_0}$
as in

For $\delta$ as in the statement, we know
from Lemma
(with $\psi_{N_0} = {\mathbf 1}_{A_{N_0}}$)
that we can find
$a=a_0>0$, only depending on $d$,
such that, for any
$N_0 \geq 1$,

\begin{equation*}
\begin{split}
\sup_{N \geq N_0} \biggl\vert &{\mathbb P}_N \biggl( A_{N_0} \cap
\bigcap_{ k  \in F_N^+ \setminus F_{N_0}^+}
\Bigl\{
\vert \widehat{m}^k \vert
< \frac{a}{\vert k \vert^{5d/2}}
\Bigr\}
\biggr)
{\mathbb P}_{N_0} \bigl( A_{N_0} \bigr)
%\prod_{k \in F_N^+ \setminus F_{N_0}^+}
%{\mathbf P}
%\vert \xi_k \vert \leq
%\sqrt{2} a \vert k \vert^{(p-5/2)d}
\biggr\vert
\leq \frac{\delta}2.
\end{split}
\end{equation*}

By Lemma
for any $N \geq N_0$,

\begin{equation*}
\begin{split}
& A_{N_0} \cap
\bigcap_{ k  \in F_N^+ \setminus F_{N_0}^+}
\Bigl\{  m \in {\mathcal P}({\mathbb T}^d) :
\vert \widehat{m}^k \vert
< \frac{a}{\vert k \vert^{5d/2}}
\Bigr\}
\\
&\hspace{15pt} \subset
\biggl\{  m \in {\mathcal P}({\mathbb T}^d) :
1+ \inf_{x \in {\mathbb T}^d} \sum_{k \in F_{N_0} \setminus \{0\}}
\widehat{m}^k e_{-k}(x) \geq 2^{-(3d/2+1)} N_0^{-3d/2}
\biggr\}.
\end{split}
\end{equation*}

Therefore, for any $N \geq N_0$,

\begin{equation}
\label{eq:lem:27:1}
\begin{split}
&{\mathbb P}_N
\biggl(
\biggl\{
1+ \inf_{x \in {\mathbb T}^d} \sum_{k \in F_{N_0} \setminus \{0\}}
\widehat{m}^k e_{-k}(x) \geq 2^{-(3d/2+1)} N_0^{-3d/2}
\biggr\}
\biggr)
\\
{\mathbb P}_N
\biggl( A_{N_0} \cap
\bigcap_{ k  \in F_N^+ \setminus F_{N_0}^+}
\Bigl\{  m \in {\mathcal P}({\mathbb T}^d) :
\vert \widehat{m}^k \vert
< \frac{a}{\vert k \vert^{5d/2}}
\Bigr\}
\biggr)
\\
{\mathbb P}_{N_0} \bigl( A_{N_0} \bigr)
%\prod_{k \in F_N^+ \setminus F_{N_0}^+}
%{\mathbf P}
%\vert \xi_k \vert \leq
%\sqrt{2} a \vert k \vert^{(p-5/2)d}
- \tfrac12 \delta.
\end{split}
\end{equation}

By Lemma
<ref>, we know that, for $N_0$ large enough,
$  {\mathbb P}_{N_0} \bigl( A_{N_0} \bigr) \geq 1 - \delta /2$ and, then,
the right-hand side is greater than $1-\delta$.

This completes the proof.

With the same notations as before,

\begin{equation*}
\lim_{N_0 \rightarrow \infty}
\sup_{\varphi : {\mathbb R}^{D_{N_0}} \rightarrow [-1,1]}
\sup_{N \geq N_0}
\biggl\vert \int_{{\mathcal P}({\mathbb T}^d)}
\varphi\Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr) d\bigl( {\mathbb P}_N - {\mathbb P}_{N_0} \bigr)(m)
\biggr\vert
\end{equation*}

where, for any $N_0 \geq 1$, $\varphi$ in the argument of the supremum is required to be measurable, and with the already used notation
$D_{N_0}= 2 \vert F_{N_0}^+\vert$.

We fix $\delta>0$.
By Lemma
we can choose $a_0$ as in Lemma <ref>, $N_0$ large enough
and $\varepsilon \in (0,1)$ small enough
such that, for all $N \geq N_0$,

\begin{equation*}
{\mathbb P}_N \Bigl( \bigl\{
m > \varepsilon
\bigr\}
\Bigr)
\geq
{\mathbb P}_N
\biggl( A_{N_0} \cap
\bigcap_{ k  \in F_N^+ \setminus F_{N_0}^+}
\Bigl\{  m \in {\mathcal P}({\mathbb T}^d) :
\vert \widehat{m}^k \vert
< \frac{a_0}{\vert k \vert^{5d/2}}
\Bigr\}
\biggr)
\geq 1 - \delta.
\end{equation*}

For a function $\varphi$ as in the statement, we then have

\begin{equation}
\label{eq:proof:TV:2ndstep}
\begin{split}
&\sup_{N \geq N_0} \biggl\vert \int_{{\mathcal P}({\mathbb T}^d)}
\varphi\Bigl((\widehat{m}^k)_{k \in F_{N_0}^+}\Bigr)
d{\mathbb P}_N(m)
\\
&\hspace{30pt} -
\int_{{\mathcal P}({\mathbb T}^d)}
\varphi\Bigl((\widehat{m}^k)_{k \in F_{N_0}^+}\Bigr)
\biggl(
{\mathbf 1}_{A_{N_0}}(m)
\prod_{k \in F_N^+ \setminus F_{N_0}^+}
{\mathbf 1}_{\{
\vert \widehat{m}^k \vert < a_0 \vert k\vert^{-5d/2}
\}}
\biggr)
d{\mathbb P}_N(m)
\biggr\vert \leq  \delta.
\end{split}
\end{equation}

We now
apply Lemma
to the function
$m \mapsto
\varphi((\widehat{m}^k)_{k \in F_{N_0}^+})
{\mathbf 1}_{A_{N_0}}(m)$.

For $N_0$ large enough, we get

\begin{equation*}
\begin{split}
&\sup_{N \geq N_0} \biggl\vert \int_{{\mathcal P}({\mathbb T}^d)}
\varphi\Bigl((\widehat{m}^k)_{k \in F_{N_0}^+}\Bigr)
d{\mathbb P}_N(m)
\int_{{\mathcal P}({\mathbb T}^d)}
\varphi\Bigl((\widehat{m}^k)_{k \in F_{N_0}^+}\Bigr)
d{\mathbb P}_{N_0}(m)
\biggr\vert \leq 3 \delta.
\end{split}
\end{equation*}

§.§.§ Convergence of the sequence $({\mathbb P}_N)_{N \geq 1}$

Using the properties proven in the previous paragraph,
we now have all the ingredients to
address the limiting points of the sequence
$({\mathbb P}_N)_{N \geq 1}$.

The following lemma proves
the second claim in item $(ii)$ of
Theorem <ref>.

Let ${\mathbb P}$
be a weak limit of $({\mathbb P}_N)_{N \geq 1}$ on ${\mathcal P}({\mathbb T}^d)$.
For an integer $N_0 \geq 1$,
on ${\mathcal P}({\mathbb T}^d)$,
the (sub-probability) measure
${\mathbb Q}$ by

\begin{equation*}
\frac{d{\mathbb Q}}{d {\mathbb P}}(m)
{\mathbf 1}_{{\mathcal P}_{N_0}}(m)
={\mathbf 1}_{{\mathcal O}_{N_0}}\Bigl(
\bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr), \quad m \in {\mathcal P}({\mathbb T}^d).
\end{equation*}

Then, the image of ${\mathbb Q}$ by the mapping

$$\pi^{(2)}_{N_0} : m \in {\mathcal P}({\mathbb T}^d) \mapsto
\bigl( \widehat{m}^k   \bigr)_{k \in F_{N_0}^+} \in {\mathbb R}^{2 \vert F_{N_0}^+\vert} $$

supported by
${\mathcal O}_{N_0}$
${\mathbb Q} \circ ( \pi^{(2)}_{N_0})^{-1}
({\mathcal O}_{N_0}^{\complement})=0$)
and is
absolutely continuous with respect to the Lebesgue measure.
for any Borel subset $B$ of
${\mathbb R}^{2 \vert F^+_{N_0}\vert}$ that is included
in ${\mathcal O}_{N_0}$,

\begin{equation*}
\begin{split}
&{\mathbb Q} \circ \bigl( \pi^{(2)}_{N_0}\bigr)^{-1}(B)
\\
&=\int_{{\mathcal P}({\mathbb T}^d)} {\mathbf 1}_B \Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr) d {\mathbb P}(m)
\\
&\leq \Bigl( \sup_{N \geq N_0} \frac{Z_{N_0} c_N}{Z_N c_{N_0}} \Bigr) \frac1{Z_{N_0}}
\int_{{\mathcal O}_{N_0}} {\mathbf 1}_B \Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr)
\exp \biggl( - \sum_{k \in F_{N_0}^+} \vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
\bigotimes_{j \in  F_{N_0}^+}
d \widehat{m}^j,
\end{split}
\end{equation*}

where we recall from
<ref> that the first factor in the above right-hand side is finite.

Let $E$ be an open subset of ${\mathcal O}_{N_0}$ (equivalently, $E$ is an open subset of
${\mathbb R}^{2\vert F^+_{N_0}\vert }$ included in ${\mathcal O}_{N_0}$).
We observe that $\{ m \in {\mathcal P}({\mathbb T}^d) :
(\widehat{m}^k)_{k \in F_{N_0}^+ } \in E\}$ is an open subset
of ${\mathcal P}({\mathbb T}^d)$.

Up to a subsequence, we can assume that
$({\mathbb P}_N)_{N \geq 1}$
converges to
${\mathbb P}$.
by Portmanteau theorem,

\begin{equation*}
{\mathbb P}\Bigl( \Bigl\{ (\widehat{m}^k)_{k \in F_{N_0}^+ } \in E \Bigr\}
\Bigr)
\leq
\liminf_{N \rightarrow \infty} {\mathbb P}_N\Bigl( \Bigl\{ (\widehat{m}^k)_{k \in F_{N_0}^+ } \in E \Bigr\}
\Bigr).
\end{equation*}

Now, by
Lemma <ref>
(with $\varphi = {\mathbf 1}_E$), letting $N$ tend to $\infty$ therein, we get

\begin{equation}
\label{eq:lem:30:1}
\begin{split}
&\int_{{\mathcal P}({\mathbb T}^d)} {\mathbf 1}_E \Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr) d {\mathbb P}(m)
\\
&\hspace{15pt} \leq
\Bigl( \sup_{N \geq N_0} \frac{Z_{N_0} c_N}{Z_N c_{N_0}} \Bigr)
\frac1{Z_{N_0}}
\int_{{\mathcal O}_{N_0}} {\mathbf 1}_E \Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr)
\exp \biggl( - \sum_{k \in F_{N_0}^+} \vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
\bigotimes_{j \in  F_{N_0}^+}
d \widehat{m}^j,
\end{split}
\end{equation}

where we recall from
<ref> that the first factor in the above right-hand side is finite.

By definition of ${\mathbb Q}$, the left-hand side can be rewritten as
${\mathbb Q} \circ (\pi^{(2)}_{N_0})^{-1}(E)$.

Take now a Borel subset $B$ of
${\mathbb R}^{2 \vert F^+_{N_0}\vert}$ that is included
in ${\mathcal O}_{N_0}$. By outer-regularity of
the Lebesgue measure, we can find, for any $\delta >0$,  an open subset $E$ of ${\mathcal O}_{N_0}$, containing $B$, such that

\begin{equation*}
\begin{split}
\int_{{\mathcal O}_{N_0}} {\mathbf 1}_E \Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr)
\exp \biggl( - \sum_{k \in F_{N_0}^+} \vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
\bigotimes_{j \in  F_{N_0}^+}
d \widehat{m}^j
\\
\frac1{Z_{N_0}}
\int_{{\mathcal O}_{N_0}} {\mathbf 1}_B \Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr)
\exp \biggl( - \sum_{k \in F_{N_0}^+} \vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
\bigotimes_{j \in  F_{N_0}^+}
d \widehat{m}^j + \delta.
\end{split}
\end{equation*}

By (<ref>),

\begin{equation*}
\begin{split}
&\int_{{\mathcal P}({\mathbb T}^d)} {\mathbf 1}_E \Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr) d {\mathbb P}(m)
\\
\Bigl( \sup_{N \geq N_0} \frac{Z_{N_0} c_N}{Z_N c_{N_0}} \Bigr)
\biggl[
\frac1{Z_{N_0}}
\int_{{\mathcal O}_{N_0}} {\mathbf 1}_B \Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr)
\exp \biggl( - \sum_{k \in F_{N_0}^+} \vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
\bigotimes_{j \in  F_{N_0}^+}
d \widehat{m}^j + \delta
\biggr].
\end{split}
\end{equation*}

The left-hand side writes
${\mathbb Q} \circ (\pi^{(2)}_{N_0})^{-1}(E)$.
This yields

\begin{equation*}
\begin{split}
&{\mathbb Q} \circ \bigl(\pi^{(2)}_{N_0} \bigr)^{-1}(B)
\\
{\mathbb Q} \circ \bigl(\pi^{(2)}_{N_0} \bigr)^{-1}(E)
\\
\Bigl( \sup_{N \geq N_0} \frac{Z_{N_0} c_N}{Z_N c_{N_0}} \Bigr)
\biggl[
\frac1{Z_{N_0}}
\int_{{\mathcal O}_{N_0}} {\mathbf 1}_B \Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr)
\exp \biggl( - \sum_{k \in F_{N_0}^+} \vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
\bigotimes_{j \in  F_{N_0}^+}
d \widehat{m}^j + \delta \biggr].
\end{split}
\end{equation*}

Since this is for any $\delta >0$,
we get the result.

Let ${\mathbb P}$
be a weak limit of $({\mathbb P}_N)_{N \geq 1}$ on ${\mathcal P}({\mathbb T}^d)$.

\begin{equation*}
\lim_{N_0 \rightarrow \infty}
{\mathbb P}(A_{N_0}) = 1,
\end{equation*}

with $A_{N_0}$
as in

By the same argument as in
the proof of
Lemma <ref>, $A_{N_0}$ is a closed subset of ${\mathcal P}({\mathbb T}^d)$.
by Portmanteau
theorem (assuming without any loss of generality that
$({\mathbb P}_N)_{N \geq 1}$ converges to ${\mathbb P}$ in the weak sense),

\begin{equation*}
{\mathbb P}(A_{N_0}) \geq \limsup_{N \rightarrow \infty}
{\mathbb P}_N(A_{N_0}).
\end{equation*}

The conclusion follows from Lemma

We now prove item $(iii)$ in the statement of Theorem

Let ${\mathbb P}$
be a weak limit of $({\mathbb P}_N)_{N \geq 1}$ on ${\mathcal P}({\mathbb T}^d)$.

\begin{equation*}
\lim_{N_0 \rightarrow \infty}
\sup_{\varphi : {\mathbb R}^{{D_{N_0}}} \rightarrow [-1,1]}
\biggl\vert \int_{{\mathcal P}({\mathbb T}^d)}
\varphi\Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr) d\bigl( {\mathbb P} - {\mathbb P}_{N_0} \bigr)(m)
\biggr\vert
\end{equation*}

where, for any $N_0 \geq 1$, $\varphi$ in the argument of the supremum is required to be measurable,
and with the already used notation
$D_{N_0}= 2 \vert F_{N_0}^+\vert$.

First Step.
Introduce the sequence

\begin{equation*}
\eta_{N_0}:=
\sup_{\varphi : {\mathbb R}^{D_{N_0}} \rightarrow [-1,1]}
\sup_{N \geq N_0}
\biggl\vert \int_{{\mathcal P}({\mathbb T}^d)}
\varphi\Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr) d\bigl( {\mathbb P}_N - {\mathbb P}_{N_0} \bigr)(m)
\biggr\vert, \quad N_0 \geq 1.
\end{equation*}

Lemma <ref>,
the sequence $(\eta_{N_0})_{N_0 \geq 1}$ converges to $0$.

Take now $\varphi$
a continuous function from ${\mathbb R}^{D_{N_0}}$ into $[-1,1]$.

Letting $N$ tend to $\infty$,
we get

\begin{equation*}
\begin{split}
&\biggl\vert \int_{{\mathcal P}({\mathbb T}^d)}
\varphi\Bigl((\widehat{m}^k)_{k \in F_{N_0}^+}\Bigr)
d\bigl( {\mathbb P}  - {\mathbb P}_{N_0}\bigr) (m)
\biggr\vert \leq \eta_{N_0}.
\end{split}
\end{equation*}

Second Step.
Assume now that $\varphi$ is merely measurable and $[-1,1]$-valued.
Consider also another sequence
$(\delta_{N_0})_{N_0 \geq 1}$ converging to $0$.
by Lusin's theorem,

we can find, for each $N_0 \geq 1$,
continuous function $\widetilde \varphi$ on
${\mathbb R}^{D_{N_0}}$

with values in $[-1,1]$ such that

\begin{align}
\label{eq:lem:30:2}
% \begin{split}
& \Bigl( \sup_{N \geq N_0} \frac{Z_{N_0} c_N}{Z_N c_{N_0}} \Bigr)
{\mathbb P}_{N_0}\Bigl( \Bigl\{ \varphi
\Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr)
\not =
\widetilde \varphi
\Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr) \Bigr\} \Bigr)
\\
&= \Bigl( \sup_{N \geq N_0} \frac{Z_{N_0} c_N}{Z_N c_{N_0}} \Bigr) \frac1{Z_{N_0}}
\int_{{\mathcal O}_{N_0}} {\mathbf 1}_{\{ \varphi \not = \widetilde{\varphi}\} }  \Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr)
\exp \biggl( - \sum_{k \in F_{N_0}^+} \vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
\bigotimes_{j \in  F_{N_0}^+}
d \widehat{m}^j
\leq \delta_{N_0}.
\nonumber
\end{align}

By Lemma

\begin{equation}
\label{eq:lem:30:3}
\begin{split}
&\int_{{\mathcal P}({\mathbb T}^d)} {\mathbf 1}_{\{\varphi \not = \widetilde \varphi\}} \Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr)
{\mathbf 1}_{{\mathcal O}_{N_0}} \Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr)
d {\mathbb P}(m)
\leq \delta_{N_0}.
\end{split}
\end{equation}

\begin{equation*}
\begin{split}
&\biggl\vert \int_{{\mathcal P}({\mathbb T}^d)}
\varphi\Bigl((\widehat{m}^k)_{k \in F_{N_0}^+}\Bigr)
d\bigl( {\mathbb P}  - {\mathbb P}_{N_0}\bigr) (m)
\int_{{\mathcal P}({\mathbb T}^d)}
\widetilde \varphi \Bigl((\widehat{m}^k)_{k \in F_{N_0}^+}\Bigr)
d\bigl( {\mathbb P}  - {\mathbb P}_{N_0}\bigr) (m)
\biggr\vert
%\biggl\vert \int_{{\mathcal P}({\mathbb T}^d)}
%\phi\Bigl((\widehat{m}^k)_{k \in F_{N_0}^+}\Bigr)
%d\bigl( {\mathbb P}  - {\mathbb P}_{N_0}\bigr) (m)
% \int_{{\mathcal P}({\mathbb T}^d)}
%\widetilde \phi \Bigl((\widehat{m}^k)_{k \in F_{N_0}^+}\Bigr)
%d\bigl( {\mathbb P}  - {\mathbb P}_{N_0}\bigr) (m)
\\
\biggl\vert \int_{{\mathcal P}({\mathbb T}^d)}
\Bigl(\varphi
\widetilde \varphi
\Bigr) \Bigl((\widehat{m}^k)_{k \in F_{N_0}^+}\Bigr)
d {\mathbb P}   (m)
\biggr\vert
\biggl\vert \int_{{\mathcal P}({\mathbb T}^d)}
\Bigl(\varphi
\widetilde \varphi
\Bigr) \Bigl((\widehat{m}^k)_{k \in F_{N_0}^+}\Bigr)
d {\mathbb P}_{N_0}   (m)
\biggr\vert
\\
\biggl\vert \int_{{\mathcal P}({\mathbb T}^d)}
\Bigl(\varphi
\widetilde \varphi
\Bigr) \Bigl((\widehat{m}^k)_{k \in F_{N_0}^+}\Bigr)
{\mathbf 1}_{A_{N_0}}(m)
d {\mathbb P}   (m)
\biggr\vert
\biggl\vert \int_{{\mathcal P}({\mathbb T}^d)}
\Bigl(\varphi
\widetilde \varphi
\Bigr) \Bigl((\widehat{m}^k)_{k \in F_{N_0}^+}\Bigr)
{\mathbf 1}_{A_{N_0}^{\complement}}(m)
d {\mathbb P}   (m)
\biggr\vert
\\
&\hspace{15pt} +
\biggl\vert \int_{{\mathcal P}({\mathbb T}^d)}
\Bigl(\varphi
\widetilde \varphi
\Bigr) \Bigl((\widehat{m}^k)_{k \in F_{N_0}^+}\Bigr)
d {\mathbb P}_{N_0}   (m)
\biggr\vert
\\
&\leq 4 \delta_{N_0} + 2{\mathbb P} \Bigl( A_{N_0}^{\complement} \Bigr),
\end{split}
\end{equation*}

where we used
the obvious implication $m \in A_{N_0} \Rightarrow (\widehat{m}^k)_{k \in F_{N_0}^+}
\in {\mathcal O}_{N_0}$ (see Lemma
By Lemma
we can modify our choice of $(\delta_{N_0})_{N_0 \geq 1}$
such that
the right-hand side is less than $5 \delta_{N_0}$.

Third Step.
By the first and second steps, we get

\begin{equation*}
\begin{split}
&\biggl\vert \int_{{\mathcal P}({\mathbb T}^d)}
\varphi\Bigl((\widehat{m}^k)_{k \in F_{N_0}^+}\Bigr)
d\bigl( {\mathbb P}  - {\mathbb P}_{N_0}\bigr) (m)
\biggr\vert \leq \eta_{N_0} + 5 \delta_{N_0}.
\end{split}
\end{equation*}

The right-hand side tends to $0$ as $N_0$ tends to $\infty$, uniformly with respect to $\varphi$.
This completes the proof.

As a corollary, we deduce

The sequence $({\mathbb P}_N)_{N \geq 1}$ is weakly converging.

Take two weak limits ${\mathbb P}$ and
${\mathbb P}'$ of the sequence
$({\mathbb P}_N)_{N \geq 1}$.
By Lemma

\begin{equation}
\label{eq:uniqueness:P:1}
\lim_{N_0 \rightarrow \infty}
\sup_{\varphi : {\mathbb R}^{{D_{N_0}}} \rightarrow [-1,1]}
\biggl\vert \int_{{\mathcal P}({\mathbb T}^d)}
\varphi\Bigl( (\widehat{m}^k)_{k \in F_{N_0}^+ } \Bigr) d\bigl( {\mathbb P} - {\mathbb P}' \bigr)(m)
\biggr\vert
\end{equation}

where, for any $N_0 \geq 1$, $\varphi$ in the argument of the supremum is required to be measurable.

Therefore, ${\mathbb P}$ and ${\mathbb P}'$ coincide on the $\sigma$-algebra generated by
the mappings
$(m \mapsto
\widehat{m}^k)_{k \in F_{N_0}^+ }$, for any $N_0 \geq 1$.
By Lemma
<ref> together with a standard monotone class argument, we deduce that
${\mathbb P}$ and ${\mathbb P}'$ coincide.

§.§.§ Completion of the proof of

We now prove the first claim in item $(ii)$ of Theorem

Let ${\mathbb P}$
be the weak limit of $({\mathbb P}_N)_{N \geq 1}$ on ${\mathcal P}({\mathbb T}^d)$.
for any integer $N_0 \geq 1$,
the image of ${\mathbb P}$ by the projection mapping

\begin{equation*}
\pi_{N_0}^{(1)} : m
\in {\mathcal P}({\mathbb T}^d)
\mapsto
\bigl( \widehat{m}^k \widehat{f}_{N_0}^k \bigr)_{k \in F_{N_0}^+}
\in
{\mathcal O}_{N_0}
\end{equation*}

is absolutely continuous with respect to the Lebesgue measure.

For $N_0 \geq 1$,
we consider a
$[0,1]$-valued measurable function
$\varphi$ defined on
${\mathbb R}^{2 \vert F^+_{N_0}\vert}$ that is equal to $0$ outside
${\mathcal O}_{N_0}$.

(<ref>), we get, for any $N \geq N_0$,

\begin{equation*}
\begin{split}
&\int_{{\mathcal P}({\mathbb T}^d)} \varphi\Bigl( (\widehat{m}^k \widehat{f}^k_{N_0})_{k \in F_{N_0}^+ } \Bigr) d {\mathbb P}_N(m)
\\
&= \frac1{Z_N}
\int_{{\mathcal O}_N}
\varphi\Bigl( (\widehat{m}^k \widehat{f}^k_{N_0})_{k \in F_{N_0}^+ } \Bigr)
\exp \biggl( - \sum_{k \in F_N^+} \vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
\bigotimes_{k \in F_N^+}
d \widehat{m}^k
\\
&= \frac1{Z_{N}}
\int_{{\mathbb R}^{2  \vert F_{N_0}^+\vert } } \varphi \Bigl( (\widehat{m}^k \widehat{f}^k_{N_0})_{k \in F_{N_0}^+ }  \Bigr)
\exp \biggl( - \sum_{k \in F_{N_0}^+} \vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
\\
&\hspace{5pt} \times
\int_{{\mathbb R}^{2(\vert F_N^+\vert - \vert F_{N_0}^+\vert )}}
\biggl[
{\mathbf 1}_{{\mathcal O}_N}\Bigl( (\widehat{m}^k)_{k \in F_{N}^+ } \Bigr)
\exp \biggl( - \sum_{k \in F_{N}^+ \setminus F_{N_0}^+}\vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr)
\bigotimes_{j \in F_{N}^+ \setminus F_{N_0}^+}
d \widehat{m}^j
\biggr]
\bigotimes_{j \in  F_{N_0}^+}
d \widehat{m}^j.
\end{split}
\end{equation*}

And then,

\begin{equation*}
\begin{split}
&\int_{{\mathcal P}({\mathbb T}^d)} \varphi\Bigl( (\widehat{m}^k \widehat{f}^k_{N_0})_{k \in F_{N_0}^+ } \Bigr) d {\mathbb P}_N(m)
\\
\frac1{Z_{N}} \frac{c_N}{c_{N_0}}
\int_{{\mathbb R}^{2  \vert F_{N_0}^+\vert } } \varphi \Bigl( (\widehat{m}^k \widehat{f}^k_{N_0})_{k \in F_{N_0}^+ }  \Bigr)
\exp \biggl( - \sum_{k \in F_{N_0}^+} \vert k\vert^{2 p d} { \vert \widehat{m}^k \vert^2} \biggr) \bigotimes_{j \in  F_{N_0}^+}
d \widehat{m}^j.
\end{split}
\end{equation*}

We make, in the integral on the second line, the following change of variable:

\begin{equation*}
\widehat{y}^k =
\widehat{m}^k \widehat{f}^k_{N_0}, \quad k \in F_{N_0}^+.
\end{equation*}

Then, since $\vert \widehat{f}^k_{N_0}\vert \leq 1$ for each $k \in F_{N_0}^+$, we get

\begin{equation*}
\begin{split}
&\int_{{\mathcal P}({\mathbb T}^d)} \varphi\Bigl( (\widehat{m}^k \widehat{f}^k_{N_0})_{k \in F_{N_0}^+ } \Bigr) d {\mathbb P}_N(m)
\\
\frac1{Z_{N}} \frac{c_N}{c_{N_0}}
\biggl(  \prod_{k \in F_{N_0}^+}
\bigl\vert \widehat{f}^k_{N_0} \vert
\biggr)^{-1}
\int_{{\mathbb R}^{2  \vert F_{N_0}^+\vert } } \varphi \Bigl( (\widehat{y}^k )_{k \in F_{N_0}^+ }  \Bigr)
\exp \biggl( - \sum_{k \in F_{N_0}^+}
\frac{\vert k\vert^{2 p d} { \vert \widehat{y}^k \vert^2} }{
\vert \widehat{f}^k_{N_0} \vert^{2}
\biggr) \bigotimes_{j \in  F_{N_0}^+}
d \widehat{y}^j
\\
\frac1{Z_{N}} \frac{c_N}{c_{N_0}}
\biggl(  \prod_{k \in F_{N_0}^+}
\bigl\vert \widehat{f}^k_{N_0} \vert
\biggr)^{-1}
\int_{{\mathbb R}^{2  \vert F_{N_0}^+\vert } } \varphi \Bigl( (\widehat{y}^k )_{k \in F_{N_0}^+ }  \Bigr)
\exp \biggl( - \sum_{k \in F_{N_0}^+}
\vert k\vert^{2 p d} { \vert \widehat{y}^k \vert^2}
\biggr) \bigotimes_{j \in  F_{N_0}^+}
d \widehat{y}^j.
\end{split}
\end{equation*}

Since $\varphi$ is zero outside ${\mathcal O}_{N_0}$, the above integral reduces to an integral over
${\mathcal O}_{N_0}$.

\begin{equation*}
\begin{split}
&\int_{{\mathcal P}({\mathbb T}^d)} \varphi\Bigl( (\widehat{m}^k \widehat{f}^k_{N_0})_{k \in F_{N_0}^+ } \Bigr) d {\mathbb P}_N(m)
\\
\frac1{Z_{N}} \frac{c_N}{c_{N_0}}
\biggl(  \prod_{k \in F_{N_0}^+}
\bigl\vert \widehat{f}^k_{N_0} \vert
\biggr)^{-1}
\int_{{\mathcal O}_{N_0} } \varphi \Bigl( (\widehat{y}^k )_{k \in F_{N_0}^+ }  \Bigr)
\exp \biggl( - \sum_{k \in F_{N_0}^+}
\vert k\vert^{2 p d}
\biggr) \bigotimes_{j \in  F_{N_0}^+}
d \widehat{y}^j.
\end{split}
\end{equation*}

By Lemma
<ref>, we can let $N$ tend to $\infty$ in the left-hand side.

\begin{equation*}
\begin{split}
&\int_{{\mathcal P}({\mathbb T}^d)} \varphi\Bigl( (\widehat{m}^k \widehat{f}^k_{N_0})_{k \in F_{N_0}^+ } \Bigr) d {\mathbb P}(m)
\\
&\leq  \Bigl( \sup_{N \geq N_0} \frac{Z_{N_0} c_N}{Z_N c_{N_0}} \Bigr)
\frac1{Z_{N_0}}
\biggl(  \prod_{k \in F_{N_0}^+}
\bigl\vert \widehat{f}^k_{N_0} \vert
\biggr)^{-1}
\int_{{\mathcal O}_{N_0} } \varphi \Bigl( (\widehat{y}^k )_{k \in F_{N_0}^+ }  \Bigr)
\exp \biggl( - \sum_{k \in F_{N_0}^+}
\vert k\vert^{2 p d}
\biggr) \bigotimes_{j \in  F_{N_0}^+}
d \widehat{y}^j,
\end{split}
\end{equation*}

where we recall from
<ref> that the first factor in the above right-hand side is finite.
This completes the proof.

We now prove item $(i)$ in the statement of Theorem

Let ${\mathbb P}$
be the weak limit of $({\mathbb P}_N)_{N \geq 1}$ on ${\mathcal P}({\mathbb T}^d)$.
${\mathbb P}$
has a full support on ${\mathcal P}({\mathbb T}^d)$ equipped with the $1$-Wasserstein distance.

First Step.
We consider
a measure $m_0 \in {\mathcal P}({\mathbb T}^d)$ and a real $\varepsilon \in (0,1)$. We have to prove
that ${\mathbb P}(B(m_0,\varepsilon))>0$ with
$B(m_0,\varepsilon)$ the ball of
${\mathcal P}({\mathbb T}^d)$ of
$m_0$ and of radius $\varepsilon$ for the 1-Wasserstein distance.

By Lemma
<ref>, we notice that, for $N_0$ large enough,

\begin{equation*}
\sum_{k \in F_{N_0}} \widehat{m}_0^k \widehat{f}_{N_0}^k e_{-k} \in B\bigl(m_0,\tfrac14 \varepsilon\bigr),
\end{equation*}

And then,
using the same constant $c_0$ as in the notation
introduced in Section <ref>, we get

\begin{equation}
\label{eq:lem:33:1}
\frac{\varepsilon}{4c_0} +
\bigl( 1 - \frac{\varepsilon}{4c_0} \bigr)
m_0 * {f}_{N_0}
\frac{\varepsilon}{4c_0} +
\bigl( 1 - \frac{\varepsilon}{4c_0} \bigr)
\sum_{k \in F_{N_0}} \widehat{m}_0^k \widehat{f}_{N_0}^k e_{-k}
\in B(m_0,\tfrac34 \varepsilon).
\end{equation}

Recall that there exists a constant $c_d$, only depending on $d$, such that, for any $a>0$,

\begin{equation*}
\sum_{k \not \in F_{N_0}}
\frac{a}{\vert k \vert^{5d/2}}
\leq \frac{c_d a}{{N_0}^{3d/2}}.
\end{equation*}

Therefore, for
$a$ fixed, for
$c_d a/ N_0^{3d/2} < \varepsilon/(8c_0)$
for any
collection $(\widehat{m}^k)_{k \in {\mathbb Z}^d}$, with
$\widehat{m}^{-k} = \overline{\widehat{m}^k}$ for $k \not =0$, and

\begin{equation*}
\begin{cases}
&\displaystyle \sum_{k \in F_{N_0} \setminus \{0\}} \Bigl\vert \widehat{m}^k -
\bigl( 1 - \frac{\varepsilon}{4c_0}  \bigr)
\widehat{m}_0^k \widehat{f}_{N_0}^k
\Bigr\vert \leq \frac{\varepsilon}{8c_0},
\\
&\displaystyle \forall k \not \in F_{N_0}, \quad
\vert \widehat{m}^k
\vert \leq
a \vert k \vert^{-5d/2},
\end{cases}
\end{equation*}

it holds that

\begin{equation*}
\begin{split}
&\sup_{x \in {\mathbb T}^d}
\Bigl\vert
\sum_{k \in F_{N_0}}
\widehat{m}^k
\Bigl(
\frac{\varepsilon}{4c_0}  +
\bigl( 1 - \frac{\varepsilon}{4c_0}  \bigr)
m_0 * {f}_{N_0}(x)
\Bigr)
\Bigr\vert \leq \frac{\varepsilon}{8c_0},
\\
&\sup_{x \in {\mathbb T}^d}
\Bigl\vert m(x) -
\Bigl(
\frac{\varepsilon}{4c_0} +
\bigl( 1 - \frac{\varepsilon}{4c_0} \bigr)
m_0 * {f}_{N_0}(x)
\Bigr)
\Bigr\vert < \frac{\varepsilon}{4c_0},
\end{split}
\end{equation*}

\begin{equation*}
m = \sum_{k \in {\mathbb Z}^d} \widehat{m}^k   e_{-k}.
\end{equation*}

$m  \in B(m_0,\varepsilon)$

\begin{equation}
\label{eq:lem:33:4}
\inf_{x \in {\mathbb T}^d} \biggl(
\sum_{k \in F_{N_0}}
\widehat{m}^k
\biggr) \geq \frac{\varepsilon}{4c_0} +
\bigl( 1 - \frac{\varepsilon}{4c_0} \bigr)
\inf_{x \in {\mathbb T}^d}
m_0 * {f}_{N_0}(x) - \frac{\varepsilon}{8c_0}  \geq \frac{\varepsilon}{8c_0}
c_d a  N_0^{-3d/2}.
\end{equation}

From the fact that
$m  \in B(m_0,\varepsilon)$,
we deduce that,
in order to complete the proof, it suffices to choose $a$ such that, for some $N_0$ large enough,

\begin{equation}
\label{eq:lem:33:2}
{\mathbb P}
\biggl(
\biggl\{
\sum_{k \in F_{N_0} \setminus \{0\}} \bigl\vert \widehat{m}^k -
\bigl( 1 - \frac{\varepsilon}{4c_0} \bigr)
\widehat{m}_0^k \widehat{f}_{N_0}^k
\bigr\vert \leq \frac{\varepsilon}{8c_0}
\biggr\}
\cap
\biggl\{ \forall k \not \in F_{N_0}, \
\vert \widehat{m}^k
\vert \leq
a \vert k \vert^{-5d/2}
\biggr\}
\biggr) > 0.
\end{equation}

Second Step.
We claim that in order to prove (<ref>), it suffices to prove that there exists $c>0$ such that, for some $N_0$ large enough and
for $N \geq N_0$,

\begin{equation}
\label{eq:lem:33:3}
{\mathbb P}_N
\biggl(
\biggl\{
\sum_{k \in F_{N_0} \setminus \{0\}} \bigl\vert \widehat{m}^k -
\bigl( 1 - \frac{\varepsilon}{4c_0}  \bigr)
\widehat{m}_0^k \widehat{f}_{N_0}^k
\bigr\vert \leq \frac{\varepsilon}{8c_0}
\biggr\}
\cap
\biggl\{ \forall k  \in F_N^+ \setminus F_{N_0}^+, \
\vert \widehat{m}^k
\vert \leq
a \vert k \vert^{-5d/2}
\biggr\}
\biggr) \geq c.
\end{equation}

Assume indeed that the above holds true. Then, for a given $N_1 \geq N_0$ and for $N \geq N_1$,
we have

\begin{equation*}
{\mathbb P}_N
\biggl(
\biggl\{
\sum_{k \in F_{N_0} \setminus \{0\}} \bigl\vert \widehat{m}^k -
\bigl( 1 - \frac{\varepsilon}{4c_0}  \bigr)
\widehat{m}_0^k \widehat{f}_{N_0}^k
\bigr\vert \leq \frac{\varepsilon}{8c_0}
\biggr\}
\cap
\biggl\{ \forall k  \in F_{N_1}^+ \setminus F_{N_0}^+, \
\vert \widehat{m}^k
\vert \leq
a \vert k \vert^{-5d/2}
\biggr\}
\biggr) \geq c.
\end{equation*}

Then, by Portmanteau theorem, we can easily replace ${\mathbb P}_N$ by
${\mathbb P}$ in the left-hand side.
Letting $N_1$ tend to $\infty$, we then get

We now prove
To do so, we apply Lemma
Assuming that $a$ right above is greater than $a_0$ in the statement of Lemma
<ref>, it suffices to show that

\begin{equation*}
{\mathbb P}_N
\biggl(
\biggl\{
\sum_{k \in F_{N_0} \setminus \{0\}} \bigl\vert \widehat{m}^k -
\bigl( 1 - \frac{\varepsilon}{4c_0} \bigr)
\widehat{m}_0^k \widehat{f}_{N_0}^k
\bigr\vert \leq \frac{\varepsilon}{8c_0}
\biggr\}
\cap
\biggl\{ \forall k  \in F_N^+ \setminus F_{N_0}^+, \
\vert \widehat{m}^k
\vert \leq
\frac12 a_0 \vert k \vert^{-5d/2}
\biggr\}
\biggr) \geq c.
\end{equation*}

(<ref>), we also notice that, for $a$ large enough (the threshold being independent of $N_0$ and only depending on $d$),

\begin{equation*}
\biggl\{
\sum_{k \in F_{N_0} \setminus \{0\}} \bigl\vert \widehat{m}^k -
\bigl( 1 - \frac{\varepsilon}{4c_0} \bigr)
\widehat{m}_0^k \widehat{f}_{N_0}^k
\bigr\vert \leq \frac{\varepsilon}{8c_0}
\biggr\} \subset A_{N_0},
\end{equation*}

with $A_{N_0}$ as in the statement of Lemma <ref>.
Therefore, Lemma
<ref> says that, for some $N_0$ large enough, for any $N \geq N_0$,

\begin{equation*}
\begin{split}
&{\mathbb P}_N
\biggl(
\biggl\{
\sum_{k \in F_{N_0} \setminus \{0\}} \bigl\vert \widehat{m}^k -
\bigl( 1 - \frac{\varepsilon}{4c_0} \bigr)
\widehat{m}_0^k \widehat{f}_{N_0}^k
\bigr\vert \leq \frac{\varepsilon}{8c_0}
\biggr\}
\cap
\biggl\{ \forall k  \in F_N^+ \setminus F_{N_0}^+, \
\vert \widehat{m}^k
\vert \leq
\frac12 a_0 \vert k \vert^{-5d/2}
\biggr\}
\biggr)
\\
&\geq \frac12 {\mathbb P}_{N_0}
\biggl(
\biggl\{
\sum_{k \in F_{N_0} \setminus \{0\}} \bigl\vert \widehat{m}^k -
\bigl( 1 - \frac{\varepsilon}{4c_0} \bigr)
\widehat{m}_0^k \widehat{f}_{N_0}^k
\bigr\vert \leq \frac{\varepsilon}{8c_0}
\biggr\}
\biggr).
\end{split}
\end{equation*}

$(( 1 -   \varepsilon/(4c_0))
\widehat{m}_0^k \widehat{f}_{N_0}^k)_{k \in F_{N_0}^+ \setminus \{0\}}$ is in
${\mathcal O}_{N_0}$ since $1 + \sum_{k \in F_{N_0} \setminus \{0\}}
( 1 - {\varepsilon}/(4c_0))
\widehat{m}_0^k \widehat{f}_{N_0}^k \geq \varepsilon/(4c_0)$.
Therefore, using the fact that the density of ${\Gamma}_{N_0}$
is strictly positive on ${\mathcal O}_{N_0}$,
the last term in the above inequality is strictly positive.

It remains to see that

Let ${\mathbb P}$
be the weak limit of $({\mathbb P}_N)_{N \geq 1}$ on ${\mathcal P}({\mathbb T}^d)$.
Then, for ${\mathbb P}$-almost every $m \in {\mathcal P}({\mathbb T}^d)$,
$m$ has a strictly positive density
and $\sum_{k \in {\mathbb Z}^d} \vert k\vert \vert \widehat{m}^k \vert < \infty$.

From Lemma <ref>,
we recall that, with $a_0$ as therein, for any $\delta \in (0,1)$, for $N_0$ large enough
and for $N \geq N_1 \geq N_0$,

\begin{equation*}
\begin{split}
&{\mathbb P}_N
\biggl( A_{N_0} \cap
\bigcap_{ k  \in F_{N_1}^+ \setminus F_{N_0}^+}
\Bigl\{  m \in {\mathcal P}({\mathbb T}^d) :
\vert \widehat{m}^k \vert
\leq \frac{a_0}{\vert k \vert^{5d/2}}
\Bigr\}
\biggr)
\geq 1 - \delta,
\end{split}
\end{equation*}

where $A_{N_0}$
is as in the statement of
By arguing as in the proof of
<ref>, we can apply Portmanteau theorem and replace ${\mathbb P}_N$ by
${\mathbb P}$
(with $N_1$ being fixed in the intersection symbol).
Then, letting $N_1$ tend to $\infty$, we get

\begin{equation*}
\begin{split}
&{\mathbb P}
\biggl( A_{N_0} \cap
\bigcap_{ k  \in F_{\infty}^+ \setminus F_{N_0}^+}
\Bigl\{  m \in {\mathcal P}({\mathbb T}^d) :
\vert \widehat{m}^k \vert
\leq \frac{a_0}{\vert k \vert^{5d/2}}
\Bigr\}
\biggr)
\geq 1 - \delta,
\end{split}
\end{equation*}

with $F_{\infty}^+ := \bigcup_{N \geq 1} F_N^+$.
In particular, since $\delta$ is arbitrary,

\begin{equation*}
\begin{split}
&{\mathbb P}
\biggl( \bigcup_{N_0 \geq 1} \biggl( A_{N_0} \cap
\bigcap_{ k  \in F_{\infty}^+ \setminus F_{N_0}^+}
\Bigl\{  m \in {\mathcal P}({\mathbb T}^d) :
\vert \widehat{m}^k \vert
\leq \frac{a_0}{\vert k \vert^{5d/2}}
\Bigr\}
\biggr)\biggr)
\end{split}
\end{equation*}

Take now $m$ in the event that appears in the left-hand side. We can find $N_0 \geq 1$
such that $m$ belongs to $A_{N_0}$ and, for any $N \geq N_0$,

\begin{equation*}
\vert \widehat{m}^k \vert
\leq \frac{a_0}{\vert k \vert^{5d/2}}.
\end{equation*}

By Lemma
<ref>, this implies

\begin{equation*}
1+ \inf_{x \in {\mathbb T}^d} \sum_{k \in F_{N} \setminus \{0\}}
\widehat{m}^k e_{-k}(x) \geq 2^{-(3d/2+1)} N_0^{-3d/2}
\end{equation*}

Given the decay of the Fourier modes of $m$, it is easy to deduce that
$m$ has a continuously differentiable density (which we identify with $m$ itself) and that

\begin{equation*}
\inf_{x \in {\mathbb T}^d}  m(x) \geq 2^{-(3d/2+1)} N_0^{-3d/2},
\end{equation*}

which completes the proof.

§.§ Proof of Theorem
<ref> (Rademacher's theorem on
$({\mathcal P}({\mathbb T}^d),{\mathbb P})$)

§.§.§ First part: existence of the derivative

We follow the main lines of the proof given in <cit.> (with the little difference that the latter is
given for functionals defined on a Banach space, which makes it slightly easier).

We start with the following definition. For an integer $N \geq 1$, we consider the mapping

\begin{equation}
\label{eq:PNinfini}
{\mathscr L}_{N,\infty} : m \in {\mathcal P}({\mathbb T}^d) \mapsto
\bigl( \widehat m^k \bigr)_{k \in F_\infty^+ \setminus F_N^+} \in \ell^2_{\mathbb C}( F_\infty^+ \setminus F_N^+ ):=\bigl\{(\widehat r^k)_{k \in F_\infty^+ \setminus F_N^+} :
\widehat r^k \in {\mathbb C} \bigr\},
\end{equation}

with $F_\infty^+ = \{ k \in {\mathbb Z}^d : \sharp(k) >0\} = \bigcup_{n \geq 1} F_n^+$,
and we denote by ${\textsf P}_{N,\infty}$ the image (on $\ell^2_{\mathbb C}(F_\infty^+ \setminus F_N^+)$) of the measure ${\mathbb P}$
${\mathscr L}_{N,\infty}$ and, then

$$\Bigl({\mathbb P}_{N \vert N,\infty}\bigl( \, \cdot \,  \vert (\widehat r^k)_{k \in F_\infty^+ \setminus F_N^+}\bigr)\Bigr)_{(\widehat r^k)_k \in \ell^2_{\mathbb C}(F_\infty^+ \setminus F_N^+)}$$

a regular conditional probability distribution of ${\mathbb P}$ given ${\mathscr L}_{N,\infty}$, namely,
for any Borel subsets $A$ of ${\mathbb C}^{\vert F_N^+\vert} (\simeq {\mathbb R}^{2 \vert F_N^+\vert})$ and $B$ of
$\ell^2_{\mathbb C}(F_{\infty}^+ \setminus F_N^+)$, it holds that

\begin{equation}
\label{eq:rademacher:0}
\begin{split}
&{\mathbb P}
\Bigl( \bigl\{ m \in {\mathcal P}({\mathbb T}^d) :
(\widehat{m}^k)_{k \in F_N^+}
\in A \ \textrm{\rm and}
\
(\widehat{m}^k)_{k \in F_\infty^+ \setminus F_N^+}
\in B
\bigr\}
\Bigr)
\\
&= \int_{\ell^2_{\mathbb C}(F_\infty^+ \setminus F_N^+)}
\biggl[
{\mathbb P}_{N\vert(N,\infty)}
\Bigl(
\bigl\{ m \in {\mathcal P}({\mathbb T}^d) :
(\widehat{m}^k)_{k \in F_N^+}
\in A \bigr\}
\vert
(\widehat{r}^k)_{k \in F_\infty^+ \setminus F_N^+}
\Bigr)
\\
\times
{\mathbf 1}_B
\Bigl(
(\widehat{r}^k)_{k \in F_\infty^+ \setminus F_N^+}
\Bigr)
\biggr]
d {\textsf P}_{N,\infty}
\Bigl(
(\widehat{r}^k)_{k \in F_\infty^+ \setminus F_N^+}
\Bigr).
\end{split}
\end{equation}

In particular, recalling
from Lemma
<ref> that,
${\mathbb P}$
almost every
$ \sum_{j \in {\mathbb Z}^d} \vert \widehat m^j \vert < \infty$,
we deduce that,
$\textsf{\sf P}_{N,\infty}$ almost every
$(\widehat r^k)_{k \in F_\infty^+ \setminus F_N^+} $,
$\sum_{j \in F_\infty^+ \setminus F_N^+} \vert \widehat{r}^j \vert < \infty$.

We now consider the set

\begin{equation*}
C_N := \bigl\{ m \in {\mathcal P}({\mathbb T}^d) :  \phi \ \textrm{\rm is differentiable at \textit{m} along the directions} \ (\fD[e_k])_{k \in F_N^+,\fD=\Re,\Im} \bigr\}.
\end{equation*}

In clear,
$m \in C_N$
for any $k \in F_N^+$,
for $\fD=\Re,\Im$,
the limit

\begin{equation*}
\lim_{\eta \rightarrow 0} \frac{1}{\eta} \Bigl( \phi\bigl( m + \eta  \fD[e_k]\bigr)
- \phi(m) \Bigr)
{\mathbf 1}_{ {\mathcal P}({\mathbb T}^d)}\bigl(m + \eta   \fD[e_k]\bigr)
\end{equation*}

exists, which proves that
$C_N$ is a Borel subset of
${\mathcal P}({\mathbb T}^d)$.
We now address,
for a given
$(\widehat r^k)_{k \in F_\infty^+ \setminus F_N^+} $ in $\ell^2_{\mathbb C}(F_\infty^+ \setminus F_N^+)$,
the probability
${\mathbb P}_{N\vert(N,\infty)}
( C_N \vert
(\widehat r^k)_{k \in F_\infty^+ \setminus F_N^+}
)$. By definition of a regular conditional probability distribution, we have, for
$\textsf{\sf P}_{N,\infty}$ almost every
$(\widehat r^k)_{k \in F_\infty^+ \setminus F_N^+} $,

\begin{equation*}
\begin{split}
&{\mathbb P}_{N\vert(N,\infty)}
\bigl( C_N \vert
(\widehat r^k)_{k \in F_\infty^+ \setminus F_N^+}
\bigr)
\\
&= {\mathbb P}_{N\vert(N,\infty)}
\Bigl(
\bigcap_{k \in F_N^+}
\bigcap_{\fD=\Re,\Im}
\Bigl\{ m :
\lim_{\eta \rightarrow 0} \frac{1}{\eta} \Bigl( \phi( m + \eta \fD[  e_k])
- \phi(m) \Bigr)
{\mathbf 1}_{ {\mathcal P}({\mathbb T}^d)}(m + \eta \fD[ e_k])
\ \textrm{\rm exists}
\Bigr\}
\\
&\hspace{75pt} \cap
\bigcap_{k \in F_\infty^+ \setminus F_N^+}
\{ \widehat{m}^k =  \widehat{r}^k
\}
\,
\vert
\,
(\widehat r^k)_{k \in F_\infty^+ \setminus F_N^+}
\Bigr).
\end{split}
\end{equation*}

Obviously, on the event
$\bigcap_{k \in F_\infty^+ \setminus F_N^+}
\{ \widehat{m}^k =  \widehat{r}^k
\}$,
we have, for any $k \in F_N \setminus \{0\}$
and for $\fD=\Re,\Im$,

\begin{equation}
\label{eq:rademacher:1}
\begin{split}
& \frac1{\eta} \Bigl( \phi(m + \eta \fD[e_k]) - \phi(m) \Bigr)
{\mathbf 1}_{ {\mathcal P}({\mathbb T}^d)}(m + \eta \fD[e_k])
\\
&= \frac{1}{\eta} \biggl\{ \phi\Bigl( 1 + \sum_{j \in F_N^+ }  2 \Re [\widehat m^j e_{-j}] + \eta   \fD[e_k]
+\sum_{j \in F_\infty^+ \setminus F_N^+}  2 \Re [ \widehat r^j e_{-j}]
\Bigr)
\\
&\hspace{30pt} - \phi
\Bigl( 1 +  \sum_{j \in F_N^+ } 2 \Re[ \widehat m^j e_{-j}]
+ \sum_{j  \in F_\infty^+ \setminus F_N^+} 2 \Re[ \widehat r^j e_{-j}]
\Bigr)
\biggr\}
\\
&\hspace{15pt} \times
{\mathbf 1}_{ {\mathcal P}({\mathbb T}^d)}\Bigl(
\sum_{j \in F_N^+ }  2 \Re [\widehat m^j e_{-j}] + \eta   \fD[e_k]
+\sum_{j \in F_\infty^+ \setminus F_N^+}  2 \Re [ \widehat r^j e_{-j}]
\Bigr).
\end{split}
\end{equation}

For $(\widehat r^j)_{j \in F_\infty^+ \setminus F_N^+}$ in $\ell^2_{\mathbb C}(F_\infty^+ \setminus F_N^+)$ such that $\sum_{j \in F_\infty^+ \setminus F_N^+} \vert \widehat{r}^j \vert < \infty$, we let

\begin{equation*}
\begin{split}
&{\mathcal O}_N\bigl(
(\widehat r^j)_{j \in F_\infty^+ \setminus F_N^+} \bigr)
\\
\biggl\{
(\widehat{m}^j)_{j \in F_N^+}
\in {\mathbb C}^{\vert F_N^+\vert} :
%&\widehat{m}^{-j} = \overline{\widehat{m}^j}, \ j \in F_N \setminus \{0\},
\inf_{x \in {\mathbb T}^d}
\Bigl[ 1 +
\sum_{j \in F_N^+ } 2 \Re[ \widehat m^j e_{-j}(x)]
+ \sum_{j  \in F_\infty^+ \setminus F_N^+} 2 \Re[ \widehat r^j e_{-j}(x)]
\Bigr]
\biggr\}.
\end{split}
\end{equation*}

It is easy to see that
${\mathcal O}_N(
(\widehat r^j)_{j \in F_\infty^+ \setminus F_N^+} )$
is an open convex
of ${\mathbb R}^{2 \vert F_N^+ \vert}$ (regarding the complex coordinates as pairs of reals).
Its closure

\begin{equation*}
\begin{split}
&\overline{{\mathcal O}_N}\bigl(
(\widehat r^j)_{j \in F_\infty^+ \setminus F_N^+} \bigr)
\\
\biggl\{
(\widehat{m}^j)_{j \in F_N^+}
\in {\mathbb C}^{\vert F_N^+\vert} :
%&\widehat{m}^{-j} = \overline{\widehat{m}^j}, \ j \in F_N \setminus \{0\},
\inf_{x \in {\mathbb T}^d}
\Bigl[ 1 +
\sum_{j \in F_N^+ } 2 \Re[ \widehat m^j e_{-j}(x)]
+ \sum_{j  \in F_\infty^+ \setminus F_N^+} 2 \Re[ \widehat r^j e_{-j}(x)]
\Bigr]
\geq 0
\biggr\}.
\end{split}
\end{equation*}

We can consider
$p^N$ the orthogonal projection from
${\mathbb R}^{2\vert F_N^+\vert}$ onto
$\overline{{\mathcal O}_N}(
(\widehat r^j)_{j \in F_\infty^+ \setminus F_N^+} )$ (for simplicity, we do not specify the dependence upon
$(\widehat r^j)_{j \in F_\infty^+ \setminus F_N^+}$ in the notation $p^N$).
We then let

\begin{equation*}
\begin{split}
&\phi^N\Bigl( (\widehat{m}^j)_{j \in F_N^+} \vert
(\widehat r^j)_{j \in F_\infty^+ \setminus F_N^+}
\Bigr)
\\
\biggl( 1 + \sum_{j \in F_N^+ } 2 \Re \Bigl\{   \Bigl[ p^N
\Bigl(
(\widehat{m}^k)_{k \in F_N \setminus \{0\}}
\Bigr)
\Bigr]_j
+ \sum_{  j \in F_\infty^+ \setminus F_N^+} 2 \Re [ \widehat r^j e_{-j}]
\Bigr\} \biggr).
\end{split}
\end{equation*}

Obviously, $\phi^N$ is a Lipschitz function on
${\mathbb R}^{2\vert F_N^+\vert}$. As such, it is differentiable almost everywhere:
we call ${\mathcal D}[\phi^N]$
the set of differentiability points.
Back to
(<ref>), we have, on the event
$\bigcap_{k \in F_\infty^+ \setminus F_N^+}
\{ \widehat{m}^k =  \widehat{r}^k
\}$,
the identity

\begin{equation*}
\begin{split}
& \frac1{\eta} \Bigl( \phi(m + \eta \fD[e_k]) - \phi(m) \Bigr)
{\mathbf 1}_{ {\mathcal P}({\mathbb T}^d)}(m + \eta \fD[e_k])
\\
&= \frac{1}{\eta} \Bigl[ \phi^N\Bigl( (\widehat{m}^j + \eta z_{\fD} {\mathbf 1}_{\{k=j\}})_{j \in F_N^+} \vert
(\widehat r^j)_{j \in F_\infty^+ \setminus F_N^+}
\Bigr)
- \phi^N
\Bigl( (\widehat{m}^j)_{j \in F_N^+} \vert
(\widehat r^j)_{j \in F_\infty^+ \setminus F_N^+}
\Bigr)
\Bigr]
\\
&\hspace{15pt} \times
{\mathbf 1}_{ {\mathcal P}({\mathbb T}^d)}\Bigl(
1 +  \sum_{j \in F_N^+} 2 \Re[\widehat m^j e_{-j}] + \eta\fD[   e_k]
+ \sum_{j \in F_\infty^+ \setminus F_N^+} 2 \Re[\widehat r^j e_{-j}]
\Bigr),
\end{split}
\end{equation*}

with $z_{\fD}=1$ if $\fD=\Re$ and $z_{\fD}=\i$ if $\fD=\Im$.
In particular, if
$(\widehat{m}^j + \eta z_{\fD} {\mathbf 1}_{\{k=j\}})_{j \in F_N^+}$
belongs to
${\mathcal O}_N(
(\widehat r^j)_{j \in F_\infty^+ \setminus F_N^+} )$ and to ${\mathcal D}[\phi^N]$, then
the above right-hand side has a limit as $\eta$ tends to $0$.
for $\textsf{P}_{N,\infty}$-a.e. $(\widehat r^k)_{k \in F_\infty^+ \setminus F_N^+} $,

\begin{equation*}
\begin{split}
&{\mathbb P}_{N\vert(N,\infty)}
\bigl( C_N \vert
(\widehat r^k)_{k \in F_\infty^+ \setminus F_N^+}
\bigr)
\\
&\geq {\mathbb P}_{N\vert(N,\infty)}
\Bigl(
{\mathcal D}[\phi^N] \cap
\Bigl\{
(\widehat{m}^j)_{j \in F_N \setminus \{0\}}
\in
{\mathcal O}_N(
(\widehat r^k)_{k \in F_\infty^+ \setminus F_N^+} )
\Bigr\}
\cap
\bigcap_{\vert k \vert \geq N}
\{ \widehat{m}^k = \widehat{r}^k
\}
\,
\vert
\,
(\widehat r^k)_{k \in F_\infty^+ \setminus F_N^+}
\Bigr).
\end{split}
\end{equation*}

We now prove the following two things:
for $\textsf{P}_{N,\infty}$-a.e. $(\widehat r^k)_{k \in F_\infty^+ \setminus F_N^+} $,

\begin{equation}
\label{eq:rademacher:2}
\begin{split}
&{\mathbb P}_{N\vert(N,\infty)}
\Bigl(
{\mathcal D}[\phi^N]
\,
\vert
\,
(\widehat r^k)_{k \in F_\infty^+ \setminus F_N^+}
\Bigr) = 1,
\\
&{\mathbb P}_{N\vert(N,\infty)}
\Bigl(
\Bigl\{
(\widehat{m}^j)_{j \in F_N^+}
\in
{\mathcal O}_N(
(\widehat r^k)_{k \in F_\infty^+ \setminus F_N^+} )
\Bigr\}
\cap
\bigcap_{k \in F_\infty^+ \setminus F_N^+}
\{ \widehat{m}^k = \widehat{r}^k
\}
\,
\vert
\,
(\widehat r^k)_{k \in F_\infty^+ \setminus F_N^+}
\Bigr)=1.
\end{split}
\end{equation}

Thanks to (<ref>), this is sufficient to prove
${\mathbb P}(C_N)=1$

We start with the proof of the first claim
in (<ref>).
We invoke $(ii)$ in Theorem
Moreover, we introduce the function
${\mathscr R}^N : (\widehat \varrho^j)_{j \in F_N^+} \in {\mathbb C}^{\vert F_N^+ \vert} \simeq
{\mathbb R}^{2 \vert F_N^+ \vert}
\mapsto
(\widehat \varrho^j \widehat f_N^j)_{j \in F_N^+}$.
$\widehat f_N^j$ is a (strictly) positive real for each $j \in F_N^+$,
${\mathscr R}^N$ can be identified with a linear mapping
from $ {\mathbb R}^{2 \vert F_N^+ \vert} $
into itself
with a (strictly) positive determinant. In particular,
$(\widehat m^j)_{j \in F_N^+}$ belongs to
${\mathcal D}[\phi^N]$ if and only if
${\mathscr R}^N( (\widehat m^j)_{j \in F_N^+})$ belongs to
${\mathscr R}^N({\mathcal D}[\phi^N])$
and the complementary of the latter has a zero Lebesgue measure. Here now comes
for $m \in {\mathcal P}({\mathbb T}^d)$,
${\mathscr R}^N( (\widehat m^j)_{j \in F_N^+})$
and the image of
the probability measure
${\mathbb P}$ by $\pi_N^{(1)}$ is absolutely continuous with respect to the Lebesgue measure on ${\mathbb R}^{2 \vert F_N^+ \vert}$, from which we deduce that

\begin{equation*}
{\mathbb P} \Bigl(
\Bigl\{
m :
(\widehat m^j)_{j \in F_N^+} \in
{\mathcal D}[\phi^N]
\Bigr\}
\Bigr)
{\mathbb P} \Bigl(
\Bigl\{
m :
\pi_N^{(1)}(m) \in
{\mathscr R}^N( {\mathcal D}[\phi^N])
\Bigr\}
\Bigr) = 1.
\end{equation*}

In turn, we get
that the first line in
holds true for
$\textsf{P}_{N,\infty}$ almost every $(\widehat r^k)_{k \in F_\infty^+ \setminus F_N^+}$.

We proceed similarly for the second claim in
We know that

\begin{equation*}
{\mathbb P}
\Bigl( \Bigl\{
m :
\inf_{x \in {\mathbb T}^d} m(x) >0
\Bigr\}
\Bigr) = 1.
\end{equation*}

for $\textsf{P}_{N,\infty}$ almost every $(\widehat r^k)_{k \in F_\infty^+ \setminus F_N^+} $,

\begin{equation*}
\begin{split}
&{\mathbb P}_{N\vert(N,\infty)}
\Bigl(
\Bigl\{ m :
\inf_{x \in {\mathbb T}^d}
\Bigl[ 1 +
\sum_{j \in F_N^+ } 2 \Re[ \widehat m^j e_{-j}(x)]
+ \sum_{j  \in F_\infty^+ \setminus F_N^+} 2 \Re[ \widehat r^j e_{-j}(x)]
\Bigr]
\Bigr\}
\\
&\hspace{150pt} \cap
\bigcap_{k \in F_\infty^+ \setminus F_N^+}
\{ \widehat{m}^k = \widehat{r}^k
\}
\,
\vert
\,
(\widehat r^k)_{k \in F_\infty^+ \setminus F_N^+}
\Bigr) = 1,
\end{split}
\end{equation*}

from which we easily deduce that the second line in
is indeed satisfied.

Notice from
that, for any $k_0 \in {\mathbb Z}^d$ and any $N_0 \geq \vert k_0 \vert$, the partial derivative $\partial_{\widehat{m}^{k_0}} \phi(m)$, which exists
${\mathbb P}$ almost every
$m \in {\mathcal P}({\mathbb T}^d)$,
coincides with
the finite dimensional derivative
$\partial_{\widehat{m}^{k_0}} \phi(1 +  \sum_{j \in F_{N_0}^+} 2 \Re[\widehat{m}^j e_{-j} ]+
\sum_{j \in F_\infty^+ \setminus F_{N_0}^+} 2 \Re[\widehat{r}^j e_{-j} ])$
on the event $\cap_{j \in F_\infty^+ \setminus F_{N_0}^+}
\{ \widehat{m}^j =  \widehat{r}^j \}$.

§.§.§ Second part: identification of the derivative with a weak limit

We now turn to the proof of
Generally speaking,
it relies on Lemmas
It also requires a preliminary result: in order to proceed,
we need to identify, for a given integer $N_0 \geq 1$, the conditional measure
$\textsf{\sf P}_{N_0,\infty}$
introduced in the first part, see
(<ref>). Here, we do so, but
on the set

\begin{equation*}
B_{N_0,\infty} := \Bigl\{ m \in {\mathcal P}({\mathbb T}^d) :
\forall  k \in F_\infty^+ \setminus F_{N_0}^+,
\
\vert \widehat{m}^k \vert \leq \frac{a_0}{2\vert k \vert^{5d/2}}
\Bigr\},
\end{equation*}

$a_0$ as in the statement of
The complete identification is given (and proven) in
In this second part,
we take it
for granted and explain how
it applies to
the current problem.

the same notations as in
we consider a smooth function $\vartheta : {\mathbb R}^{2 \vert F_{N_0}^+ \vert} \rightarrow {\mathbb R}$
whose support is included in

$$\biggl\{ (\widehat{m}^j)_{j \in F_{N_0}^+} \in {\mathbb C}^{\vert F_{N_0}^+\vert}
\simeq {\mathbb R}^{2 \vert F_{N_0}^+\vert} :
1 + 2 \inf_{x \in {\mathbb T}^d}  \sum_{j \in F_{N_0}^+} \Re[ \widehat{m}^j e_{-j}(x) ]
\geq \frac1{c} \biggr\},$$

for some $c>1$.
In this framework,
says that

\begin{equation}
\label{eq:integration:by:parts:0:1}
\begin{split}
& \int_{{\mathcal P}({\mathbb T}^d)}  \phi(m)  \partial_{\widehat{m}^{k_0}} \vartheta\Bigl( \bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr)
{\mathbf 1}_{A_{N_0} \cap B_{N_0,\infty}}(m)
d {\mathbb P}(m)
= \frac1{{\mathbb P}(A_{N_0})}
\int_{{\mathcal P}({\mathbb T}^d)}  \Psi(m)
{\mathbf 1}_{A_{N_0}}(m)
d {\mathbb P}(m),
\end{split}
\end{equation}

where $A_{N_0}$ is defined
as in Lemma
and has a (strictly) positive probability for $N_0$ large enough, see
Lemma <ref>, and where

\begin{align}
\Psi(m)
{\mathbf 1}_{B_{N_0,\infty}}(m)
\int_{{\mathcal O}_{N_0}}
\biggl[ \phi
\Bigl(
{\mathscr I}_{N_0}
\bigl( (\widehat{r}^k - \widehat{m}^k)_{k \in F_{N_0}^+}
\bigr)
+ m
\Bigr)
{\mathbf 1}_{A_{N_0}}
\bigl( (\widehat{r}^k)_{k \in F_{N_0}^+}
\bigr) \nonumber
\\
&\hspace{150pt} \times
\partial_{\widehat{m}^{k_0}} \vartheta
\Bigl( \bigl( \widehat{r}^k \bigr)_{k \in F_{N_0}^+} \Bigr)
\biggr]
d \Gamma_{N_0}\Bigl(
\bigl( \widehat{r}^k\bigr)_{k \in F_{N_0}^+}
\Bigr) \nonumber
\\
{\mathbf 1}_{B_{N_0,\infty}}(m)
\int_{{\mathcal O}_{N_0}}
\biggl[ \phi
\Bigl(
{\mathscr I}_{N_0}
\bigl( (\widehat{r}^k
- \widehat{m}^k)_{k \in F_{N_0}^+}
\bigr)
+ m
\Bigr)
\label{eq:Psi:t,m}
\\
&\hspace{150pt} \times
\partial_{\widehat{m}^{k_0}} \vartheta
\Bigl( \bigl( \widehat{r}^k \bigr)_{k \in F_{N_0}^+} \Bigr)
\biggr]
d \Gamma_{N_0}\Bigl(
\bigl( \widehat{r}^k\bigr)_{k \in F_{N_0}^+}
\Bigr) + \epsilon_{N_0}, \nonumber
\end{align}

where $\epsilon_{N_0}$ tends to $0$ as $N_0$ tends to $\infty$, uniformly with respect to
$\| \phi \|_\infty$ and $\| \partial_{\widehat{m}^{k_0}} \vartheta  \|_\infty$.
this identity requires some care:

(a) First,
the notation
${\mathscr I}_{N_0}
( (\widehat{r}^k
- \widehat{m}^k)_{k \in F_{N_0}^+}
)$ is rather abusive since
$ (\widehat{r}^k
- \widehat{m}^k)_{k \in F_{N_0}^+}$ may not belong to
${\mathcal O}_{N_0}$. However,
the definition of
${\mathscr I}_{N_0}$ easily extends to the entire
${\mathbb R}^{2 \vert F_{N_0}^+ \vert}$, see (<ref>).

the notation
${\mathbf 1}_{A_{N_0}}
( (\widehat{r}^k)_{k \in F_{N_0}^+}
)$ is also abusive. Instead,
we should write
${\mathbf 1}_{A_{N_0}} \circ {\mathscr I}_{N_0}
( (\widehat{r}^k)_{k \in F_{N_0}^+}
)$, but this would be obviously heavier.

we recall that
the support of $\vartheta$
is included
in $A_{N_0}$ (up
to the embedding
${\mathscr I}_{N_0}$ and for $N_0$ large enough).

As a result of the latter observation,
we notice that,
$( \widehat{r}^k )_{k \in F_{N_0}^+}$ belongs to the support of
$\vartheta$ and
$( \widehat{m}^k )_{k \in F_\infty^+ \setminus F_{N_0}^+}$
$\vert \widehat{m}^k \vert \leq (a_0/2) \vert k \vert^{-5d/2}$
(which is the case if
$( \widehat{m}^k )_{k \in F_\infty^+ \setminus F_{N_0}^+}$'s are the Fourier coefficients of some $m \in B_{N_0,\infty}$), the distribution

\begin{equation*}
+ 2 \sum_{k \in F_{N_0}^+ }
\Re \bigl( \widehat{r}^k e_{-k} \bigr)
+ 2 \sum_{k \in F_\infty^+ \setminus F_{N_0}^+}
\Re \bigl( \widehat{m}^k e_{-k} \bigr)
\end{equation*}

is a probability measure, see Lemma
<ref> again. Obviously, this probability measure identifies  with
${\mathscr I}_{N_0}
( (\widehat{r}^k
- \widehat{m}^k)_{k \in F_{N_0^+}})
+ m $, which is
the argument of $\phi$
when the
$( \widehat{m}^k )_{k \in F_\infty^+ \setminus F_{N_0}^+}$'s are the Fourier coefficients of some $m \in B_{N_0,\infty}$. In particular,
the integrand
in (<ref>)
is always well-defined under the sole assumption that
$m \in B_{N_0,\infty}$.
Equivalently, we can regard $\Psi$ as a function of
$(\widehat{m}^k)_{k \in F_\infty^+ \setminus F_{N_0}^+}$ provided that
$\vert \widehat{m}^k \vert \leq (a_0/2) \vert k \vert^{-5d /2}$ for any
$k \in F_{\infty}^+ \setminus F_{N_0}^+$.
In order to make this clear, we let

\begin{equation}
\begin{split}
\widetilde{\Psi}\Bigl(
(\widehat{m}^k)_{k \in F_\infty^+ \setminus F_{N_0}^+} \Bigr)
\int_{{\mathcal O}_{N_0}}
\biggl[ \phi
\Bigl(
+ 2 \sum_{k \in F_{N_0}^+ }
\Re \bigl( \widehat{r}^k e_{-k} \bigr)
+ 2 \sum_{k \in F_\infty^+ \setminus F_{N_0}^+}
\Re \bigl( \widehat{m}^k e_{-k} \bigr)
\Bigr)
\label{eq:tildePsi:t,m}
\\
&\hspace{125pt} \times
\partial_{\widehat{m}^{k_0}} \vartheta
\Bigl( \bigl( \widehat{r}^k \bigr)_{k \in F_{N_0}^+} \Bigr)
\biggr]
d \Gamma_{N_0}\Bigl(
\bigl( \widehat{r}^k\bigr)_{k \in F_{N_0}^+}
\Bigr),
\end{split}
\end{equation}

$( \widehat m^k)_{k \in F_\infty^+ \setminus F_N^+} \in \ell^2_{\mathbb C}( F_\infty^+ \setminus F_N^+ )$ such that
$\vert \widehat{m}^k \vert \leq (a_0/2) \vert k \vert^{-5d /2}$ for any
$k \in F_{\infty}^+ \setminus F_{N_0}^+$.

And then, by combining
we obtain

\begin{equation}
\label{eq:integration:by:parts:0:2}
\begin{split}
& \int_{{\mathcal P}({\mathbb T}^d)}  \phi(m)  \partial_{\widehat{m}^{k_0}} \vartheta\Bigl( \bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr)
{\mathbf 1}_{A_{N_0}}(m)
{\mathbf 1}_{B_{N_0,\infty}}(m)
d {\mathbb P}(m)
\\
&= \frac1{{\mathbb P}(A_{N_0})}
\int_{{\mathcal P}({\mathbb T}^d)}  \Psi(m)
d {\mathbb P}(m)   + \epsilon_{N_0}
\\
&= \frac1{{\mathbb P}(A_{N_0})}
\int_{\ell^2_{\mathbb C}( F_\infty^+ \setminus F_N^+ )}  \widetilde{\Psi}\Bigl(
\bigl( \widehat{m}^k \bigr)_{k \in F_\infty^+ \setminus F_{N_0}^+}
\Bigr)
d \textsf{\sf P}_{N_0,\infty}\Bigl(
\bigl( \widehat{m}^k \bigr)_{k \in F_\infty^+ \setminus F_{N_0}^+} \Bigr)  + \epsilon_{N_0},
\end{split}
\end{equation}

where, as before, $\epsilon_{N_0}$ tends to $0$ as $N_0$ tends to $\infty$, uniformly in $\| \phi \|_\infty$ and $\| \partial_{\widehat{m}^{k_0}} \vartheta \|_\infty$.

Now, by the first part of the proof, we know that,
for $\textsf{\sf P}_{N_0,\infty}$-almost every
$( \widehat{m}^k )_{k \in F_\infty^+ \setminus F_{N_0}^+}
\in
\ell^2_{\mathbb C}( F_\infty^+ \setminus F_N^+ )$,
the function

$$( \widehat{r}^k )_{k \in F_{N_0}^+}
\in {\mathcal O}_{N_0}
\mapsto
\phi
\biggl(
+ 2 \sum_{k \in F_{N_0}^+ }
\Re ( \widehat{r}^k e_{-k} )
+ 2 \sum_{k \in F_\infty^+ \setminus F_{N_0}^+}
\Re ( \widehat{m}^k e_{-k} )
\biggr)$$

$\textrm{\rm Leb}_{N_0}$
almost everywhere differentiable along any direction
$k \in F_{N_0}^+$. Recalling the form of $\Gamma_{N_0}$ in
(<ref>), we can make an integration by parts in
(<ref>). We get:

\begin{equation*}
\begin{split}
\widetilde{\Psi}\Bigl(
(\widehat{m}^k)_{k \in F_\infty^+ \setminus F_{N_0}^+} \Bigr)
-  \int_{{\mathcal O}_{N_0}}
\biggl[
\partial_{\widehat{m}^{k_0}}
\phi
\Bigl(
+ 2 \sum_{k \in F_{N_0}^+ }
\Re \bigl( \widehat{r}^k e_{-k} \bigr)
+ 2 \sum_{k \in F_\infty^+ \setminus F_{N_0}^+}
\Re \bigl( \widehat{m}^k e_{-k} \bigr)
\Bigr)
\\
-   2
\vert k_0 \vert^{2pd}
\widehat{m}^{k_0}
\phi
\Bigl(
+ 2 \sum_{k \in F_{N_0}^+ }
\Re \bigl( \widehat{r}^k e_{-k} \bigr)
+ 2 \sum_{k \in F_\infty^+ \setminus F_{N_0}^+}
\Re \bigl( \widehat{m}^k e_{-k} \bigr)
\Bigr)
\biggr]
\\
\times
\vartheta
\Bigl( \bigl( \widehat{r}^k \bigr)_{k \in F_{N_0}^+} \Bigr)
d \Gamma_{N_0}\Bigl(
\bigl( \widehat{r}^k\bigr)_{k \in F_{N_0}^+}
\Bigr).
\end{split}
\end{equation*}

we can revert back the computations in
and write

\begin{equation*}
\begin{split}
\Psi(m)
- {\mathbf 1}_{B_{N_0,\infty}}(m)
\int_{{\mathcal O}_{N_0}}
\biggl[ \partial_{\widehat{m}^{k_0}} \phi
\Bigl(
{\mathscr I}_{N_0}
\bigl( (\widehat{r}^k - \widehat{m}^k)_{k \in F_{N_0}^+}
\bigr)
+ m
\Bigr)
\\
\vert k_0 \vert^{2pd}
\widehat{m}^{k_0}
\phi
\Bigl(
{\mathscr I}_{N_0}
\bigl( (\widehat{r}^k
\widehat{m}^k)_{k \in F_{N_0}^+}
\bigr)
+ m
\Bigr)
\biggr]
\\
&\hspace{100pt} \times
{\mathbf 1}_{A_{N_0}}
\bigl( (\widehat{r}^k)_{k \in F_{N_0^+}}
\bigr)
\vartheta
\Bigl( \bigl( \widehat{r}^k \bigr)_{k \in F_{N_0}^+} \Bigr)
d \Gamma_{N_0}\Bigl(
\bigl( \widehat{r}^k\bigr)_{k \in F_{N_0}^+}
\Bigr)
+ \epsilon_{N_0}.
\end{split}
\end{equation*}

Back to
(<ref>), we get

\begin{equation*}
\begin{split}
&  \int_{{\mathcal P}({\mathbb T}^d)}  \phi(m)   \partial_{\widehat{m}^{k_0}} \vartheta\Bigl( \bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr)
{\mathbf 1}_{A_{N_0} \cap B_{N_0,\infty}}(m)
d {\mathbb P}(m)
\\
&= -
\int_{{\mathcal P}({\mathbb T}^d)}
\Bigl[
\partial_{\widehat{m}^{k_0}} \phi(m)
2 \vert k_0\vert^{2pd}
\widehat{m}^{k_0}
\phi(m)
\Bigr]
\vartheta\Bigl( \bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr)
{\mathbf 1}_{A_{N_0} \cap B_{N_0,\infty}}(m)
d {\mathbb P}(m) + \epsilon_{N_0}.
\end{split}
\end{equation*}

By Lemma
<ref>, we have
${\mathbb P}(A_{N_0} \cap B_{N_0,\infty}) \rightarrow 1$
as $N_0 \rightarrow \infty$,
from which we deduce that

\begin{equation}
\label{eq:integration:by:parts:0:3}
\begin{split}
&  \int_{{\mathcal P}({\mathbb T}^d)}  \phi(m)  \partial_{\widehat{m}^{k_0}} \vartheta\Bigl( \bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr)
d {\mathbb P}(m)
\\
&= -
\int_{{\mathcal P}({\mathbb T}^d)}
\Bigl[
\partial_{\widehat{m}^{k_0}} \phi(m)
2 \vert k_0\vert^{2pd}
\widehat{m}^{k_0}
\phi(m)
\Bigr]
\vartheta\Bigl( \bigl( \widehat{m}^k \bigr)_{k \in F_{N_0}^+} \Bigr)
d {\mathbb P}(m)   + \epsilon_{N_0},
\end{split}
\end{equation}

where, as before, $\epsilon_{N_0}$ tends to $0$ as $N_0$ tends to $\infty$, uniformly in $\| \phi \|_\infty$,
$\|\vartheta \|_\infty$, $\| \partial_{\widehat{m}^{k_0}} \phi \|_\infty$ and $\| \partial_{\widehat{m}^{k_0}} \vartheta \|_\infty$.

For each
$k_0 \in {\mathbb Z}^d \setminus \{0\}$, we now call $\widehat{\Phi}^{k_0}(m)$ a weak limit of the sequence
$(\partial_{\widehat{m}^{k_0}} \phi(m*f_N))_{N > \vert k_0\vert}$ with
$\partial_{\widehat{m}^{k_0}} \phi$ being here understood, for each $N > \vert k_0 \vert$,
as the almost everywhere (for the Lebesgue measure on ${\mathcal P}_N$)
derivative of $\phi$ on ${\mathcal P}_N$.
by combining
(<ref>), we reach the same conclusion as
in the proof of
<ref>, but with
$\widehat{\mathcal W}_1^{k_0}(m)$
therein being replaced by
$\widehat{\mathcal W}_2^{k_0}(m)$
being replaced by
$\partial_{\widehat{m}^{k_0}} \phi(m)$ (as given by the conclusion of the first part).
We hence conclude that,
almost everywhere under ${\mathbb P}$,
$\partial_{\widehat{m}^{k_0}} \phi$
are equal, which proves that
$(\partial_{\widehat{m}^{k_0}} \phi(m*f_N))_{N > \vert k_0\vert}$
converges (in the weak sense) to
$\partial_{\widehat{m}^{k_0}} \phi$.

§.§.§ Third part: auxiliary statement

It now
remains to prove the following lemma, which we invoked in the derivation of the above second part.

For any bounded measurable function
$\varphi : {\mathcal P}({\mathbb T}^d) \rightarrow {\mathbb R}$,

\begin{equation*}
\begin{split}
\varphi(m)
{\mathbf 1}_{A_{N_0} \cap B_{N_0,\infty}}(m)
d {\mathbb P}(m)
\frac1{{\mathbb P}_{N_0}(A_{N_0})}
\int_{\PP}
\psi\bigl(m\bigr) {\mathbf 1}_{A_{N_0}}(m)
d {\mathbb P}(m),
\end{split}
\end{equation*}

with $A_{N_0}$ in the first line being as in Lemma
and with

\begin{equation*}
\begin{split}
& \psi(m)
{\mathbf 1}_{B_{N_0,\infty}}(m)
\int_{{\mathcal O}_{N_0}}
\biggl[ \varphi
\Bigl(
{\mathscr I}_{N_0}
\bigl( (\widehat{r}^k)_{k \in F_{N_0}^+}
\bigr)
{\mathscr I}_{N_0}
\bigl( (\widehat{m}^k)_{k \in F_{N_0}^+}
\bigr)
+ m
\Bigr)
{\mathbf 1}_{A_{N_0}}
\circ
{\mathscr I}_{N_0}
\bigl( (\widehat{r}^k)_{k \in F_{N_0}^+}
\bigr)
\biggr]
\\
&\hspace{250pt} \times
d \Gamma_{N_0}\Bigl(
\bigl( \widehat{r}^k\bigr)_{k \in F_{N_0}^+}
\Bigr),
\end{split}
\end{equation*}

and with the same notation
as in (<ref>) for the map ${\mathscr I}_{N_0}$.

Throughout the proof, we use the same abuse of notations as
in the proof of
<ref>: We

\begin{equation*}
\begin{split}
&{\mathscr I}_{N_0}
\bigl( (\widehat{r}^k
\widehat{m}^k)_{k \in F_{N_0}^+}
\bigr)
\quad \textrm{\rm for}
\quad
{\mathscr I}_{N_0}
\bigl( (\widehat{r}^k)_{k \in F_{N_0}^+}
\bigr)
{\mathscr I}_{N_0}
\bigl( (\widehat{m}^k)_{k \in F_{N_0}^+}
\bigr),
\\
\vartheta
\bigl( (\widehat{r}^k)_{k \in F_{N}^+}
\bigr)
\quad \textrm{\rm for}
\quad
\vartheta
\circ
{\mathscr I}_{N}
\bigl(
(\widehat{r}^k)_{k \in F_{N}^+}
\bigr),
\end{split}
\end{equation*}

when $\vartheta$ is defined on
$\PP$ and
$ (\widehat{r}^k)_{k \in F_{N}^+} \in {\mathcal O}_N$.

First Step.
We start with the following observation, very similar to
an argument used in the proof of Theorem
If, for $N \geq N_0$ being fixed,
we set

\begin{equation*}
B_{N_0,N} :=
\Bigl\{ m \in {\mathcal P}({\mathbb T}^d) :
\forall  k \in F_N^+ \setminus F_{N_0}^+, \
\vert \widehat{m}^k \vert \leq \frac{a_0}{2\vert k \vert^{5d/2}}
\Bigr\},
\end{equation*}

then, for any bounded measurable function
$\varphi : \PP \rightarrow {\mathbb R}$, we have

\begin{align}
\varphi\bigl(m\bigr) {\mathbf 1}_{A_{N_0} \cap B_{N_0,N}}(m)
d {\mathbb P}_N(m)
\label{eq:proof:integration:by:parts:dim:infinie}
\\
\int_{{\mathcal O}_N}
\varphi
%\circ {\mathscr I}_{N}
\bigl( (\widehat{m}^k)_{k \in F_{N}^+}
\bigr)
{\mathbf 1}_{A_{N_0}}
%{\mathscr I}_{N_0}
\bigl( (\widehat{m}^k)_{k \in F_{N_0}^+}
\bigr)
{\mathbf 1}_{B_{N_0,N}}
%{\mathscr I}_{N}
\bigl( (\widehat{m}^k)_{k \in F_{N}^+}
\bigr)
d \Gamma_N\Bigl(
\bigl( \widehat{m}^k\bigr)_{k \in F_N^+}
\Bigr)
\nonumber
\\
\frac{Z_{N_0}}{Z_N}
\int_{{\mathbb R}^{2 \vert F_N^+ \setminus F_{N_0}^+ \vert}}
\biggl[
\int_{{\mathcal O}_{N_0}}
\varphi
%{\mathscr I}_{N}
\bigl( (\widehat{m}^k)_{k \in F_{N}^+}
\bigr)
{\mathbf 1}_{A_{N_0}}
%{\mathscr I}_{N_0}
\bigl( (\widehat{m}^k)_{k \in F_{N_0}^+}
\bigr)
d \Gamma_{N_0}\Bigl(
\bigl( \widehat{m}^k\bigr)_{k \in F_{N_0}^+}
\Bigr)
\biggr]
\nonumber
\\
&\hspace{15pt} \times
{\mathbf 1}_{B_{N_0,N}}
%\circ {\mathscr I}_{N}
\biggl( (0)_{k \in F_{N_0}^+}, (\widehat{m}^k)_{k \in F_{N}^+ \setminus F_{N_0}^+}
\biggr)
\exp \biggl( - \sum_{k \in F_N^+ \setminus F_{N_0}^+}
\vert k \vert^{2pd}
\vert \widehat{m}^k \vert^2 \biggr)  \bigotimes_{k \in F_N^+ \setminus F_{N_0}^+}
d \widehat{m}^k,
\nonumber
\end{align}

where we used Lemma
in order to derive the last equality together with the fact that
${\mathbf 1}_{B_{N_0,N}}
%\circ {\mathscr I}_{N}
( (\widehat{m}^k)_{k \in F_{N}^+})=
{\mathbf 1}_{B_{N_0,N}}
%\circ {\mathscr I}_{N}
( (0)_{k \in F_{N_0}^+}, (\widehat{m}^k)_{k \in F_{N}^+ \setminus F_{N_0}^+})$
only depends on
$(\widehat{m}^k)_{k \in F_N^+ \setminus F_{N_0}^+}$.

In particular, if we let

\begin{equation}
\begin{split}
\label{eq:proof:integration:by:parts:dim:infinie:4}
\int_{{\mathcal O}_{N_0}}
\varphi
%{\mathscr I}_{N}
\Bigl(
(\widehat{r}^k)_{k \in F_{N_0}^+},
(\widehat{m}^k)_{k \in F_N^+ \setminus F_{N_0}^+}
\Bigr)
{\mathbf 1}_{A_{N_0}}
%{\mathscr I}_{N_0}
\bigl( (\widehat{r}^k)_{k \in F_{N_0}^+}
\bigr)
d \Gamma_{N_0}\Bigl(
\bigl( \widehat{r}^k\bigr)_{k \in F_{N_0}^+}
\Bigr),
\end{split}
\end{equation}

$m \in B_{N_0,N}$
and $\psi(m):=0$ otherwise,
then, by Fubini's theorem, $\psi$ is
measurable with respect to the sigma-field generated by
$(m \mapsto \widehat{m}^k)_{k \in F_N^+ \setminus F_{N_0}^+}$

\begin{align}
\varphi\bigl(m\bigr) {\mathbf 1}_{A_{N_0} \cap B_{N_0,N}}(m)
d {\mathbb P}_N(m)
\nonumber
\\
\frac{Z_{N_0}}{Z_N}
\int_{{\mathbb R}^{2 \vert F_N^+ \setminus F_{N_0}^+ \vert}}
\biggl[
\label{eq:proof:integration:by:parts:dim:infinie:2}
\Bigl( \psi {\mathbf 1}_{B_{N_0,N}}
\Bigr)
%\circ {\mathscr I}_{N}
\biggl( (0)_{k \in F_{N_0}^+}, (\widehat{m}^k)_{k \in F_{N}^+ \setminus F_{N_0}^+}
\biggr)
\\
&\hspace{150pt} \times
\exp \biggl( - \sum_{k \in F_N^+ \setminus F_{N_0}^+}
\vert k \vert^{2pd}
\vert \widehat{m}^k \vert^2 \biggr) \biggr] \bigotimes_{k \in F_N^+ \setminus F_{N_0}^+}
d \widehat{m}^k. \nonumber
\end{align}

Notice that
the indicator function
${\mathbf 1}_{B_{N_0,N}}$ in the product
$\psi {\mathbf 1}_{B_{N_0,N}}$ is redundant with the definition of
$\psi$, but we keep it for clarity.

Now, the equality (<ref>), with $\varphi$ being replaced by $\psi$, leads to

\begin{equation*}
\begin{split}
\psi\bigl(m\bigr) {\mathbf 1}_{A_{N_0} \cap B_{N_0,N}}(m)
d {\mathbb P}_N(m)
\\
\frac{Z_{N_0}}{Z_N}
\int_{{\mathbb R}^{2 \vert F_N^+ \setminus F_{N_0}^+ \vert}}
\biggl[
\int_{{\mathcal O}_{N_0}}
{\mathbf 1}_{A_{N_0}}
%{\mathscr I}_{N_0}
\bigl( (\widehat{m}^k)_{k \in F_{N_0^+}}
\bigr)
d \Gamma_{N_0}\Bigl(
\bigl( \widehat{m}^k\bigr)_{k \in F_{N_0}^+}
\Bigr)
\biggr]
\\
&\hspace{15pt} \times
\Bigl( \psi
{\mathbf 1}_{B_{N_0,N}}
\Bigr) %\Bigl(
%\circ {\mathscr I}_{N}
\Bigl(
(0)_{k \in F_{N_0}^+},
(\widehat{m}^k)_{k \in F_{N}^+ \setminus F_{N_0}^+}
\Bigr)
\exp \biggl( - \sum_{k \in F_N^+ \setminus F_{N_0}^+}
\vert k \vert^{2pd}
\vert \widehat{m}^k \vert^2 \biggr)  \bigotimes_{k \in F_N^+ \setminus F_{N_0}^+}
d \widehat{m}^k
\\
{\mathbb P}_{N_0}(A_{N_0})
\frac{Z_{N_0}}{Z_N}
\times
\int_{{\mathbb R}^{2 \vert F_N^+ \setminus F_{N_0}^+ \vert}}
\biggl[
\Bigl( \psi
{\mathbf 1}_{B_{N_0,N}}
\Bigr) %\Bigl(
%\circ {\mathscr I}_{N}
\Bigl(
(0)_{k \in F_{N_0}^+},
(\widehat{m}^k)_{k \in F_{N}^+ \setminus F_{N_0}^+}
\Bigr)
\\
&\hspace{150pt} \times \exp \biggl( - \sum_{k \in F_N^+ \setminus F_{N_0}^+}
\vert k \vert^{2pd}
\vert \widehat{m}^k \vert^2 \biggr)
\biggr] \bigotimes_{k \in F_N^+ \setminus F_{N_0}^+}
d \widehat{m}^k,
\end{split}
\end{equation*}

and then,

\begin{equation}
\label{eq:proof:integration:by:parts:dim:infinie:3}
\begin{split}
\varphi\bigl(m\bigr) {\mathbf 1}_{A_{N_0} \cap B_{N_0,N}}(m)
d {\mathbb P}_N(m)
\frac1{{\mathbb P}_{N_0}(A_{N_0})}
\int_{\PP}
\psi\bigl(m\bigr) {\mathbf 1}_{A_{N_0} \cap B_{N_0,N}}(m)
d {\mathbb P}_N(m),
\end{split}
\end{equation}

where we recall from
<ref>, that
for $N_0$ large enough,
${\mathbb P}_{N_0}(A_{N_0})>0$.

Second Step.
We now send $N \rightarrow \infty$ in
In order to do so,

we first assume that
$\varphi$ is

measurable with respect to $(\widehat{m}^k)_{k \in F_{N_1}^+}$
for $N_1 \geq N_0$.

Then, for $N \geq N_1$,
$\psi$ in

only depends, on the set $B_{N_0,N}$, on $m$ through
$(\widehat{m}^k)_{k \in F_{N_1}^+ \setminus
F_{N_0}^+}$ and is independent of $N$.
Next, by using
Lemma <ref>, we get

\begin{equation*}
\begin{split}
\varphi \bigl(m\bigr) {\mathbf 1}_{A_{N_0} \cap B_{N_0,N}}(m)
d {\mathbb P}(m)
\frac1{{\mathbb P}_{N_0}(A_{N_0})}
\int_{\PP}
\psi \bigl(m\bigr) {\mathbf 1}_{A_{N_0} \cap B_{N_0,N}}(m)
d {\mathbb P}(m) + \varepsilon_N,
\end{split}
\end{equation*}

where $\lim_{N \rightarrow \infty} \varepsilon_N=0$.
It then remains to observe that
the subsets $(B_{N_0,N})_{N \geq N_0}$ are non-increasing and their intersection is precisely $B_{N_0,\infty}$. Then,
splitting $\varphi$ into the difference $\varphi_+ - \varphi_-$, we can easily let $N$ tend to $\infty$ by regarding the left-hand side and the first term in the right-hand side as the masses of
$B_{N_0,N}$ under finite positive measures.
We get

\begin{equation*}
\begin{split}
\varphi\bigl(m\bigr) {\mathbf 1}_{A_{N_0}   \cap B_{N_0,\infty}}(m)
d {\mathbb P}(m)
\frac1{{\mathbb P}_{N_0}(A_{N_0})}
\int_{\PP}
\psi\bigl(m\bigr) {\mathbf 1}_{A_{N_0} \cap B_{N_0,\infty}}(m)
d {\mathbb P}(m),
\end{split}
\end{equation*}

where, on the set
$B_{N_0,\infty}$, we have

\begin{equation*}
\begin{split}
& \psi(m)
\int_{{\mathcal O}_{N_0}}
\biggl[ \varphi
\Bigl(
{\mathscr I}_{N_0}
\bigl( (\widehat{r}^k
- \widehat{m}^k)_{k \in F_{N_0^+}}
\bigr)
+ m
\Bigr)
{\mathbf 1}_{A_{N_0}}
\bigl( (\widehat{r}^k)_{k \in F_{N_0^+}}
\bigr)
\biggr]
d \Gamma_{N_0}\Bigl(
\bigl( \widehat{r}^k\bigr)_{k \in F_{N_0}^+}
\Bigr).
\end{split}
\end{equation*}

By monotone class theorem (together with Lemma  <ref>), we easily extend the result to bounded and measurable functions
$\varphi$ on $\PP$.

§.§ Mollification of McKean-Vlasov equations

The following statement plays an important role in the proof of

Let $W_1$ and $W_2$ be two Lipschitz continuous functions on $[0,T] \times {\mathcal P}({\mathbb T}^d)$,
with ${\mathcal P}({\mathbb T}^d)$ being equipped with $d_{-2}$. Then, for a constant
$c>1$, we can find
an integer $N_c$ such that, for $N \geq N_c$, for $\varepsilon \in (0,1)$ and $\rho$ as in
<ref>, the Fokker-Planck equation

\begin{equation}
\label{eq:FPK:mollified:proof}
\partial_t m_t (x)- \textrm{\rm div} \Bigl(
D {\mathcal H}^{N,\varepsilon,\rho}(t,m_t)(x)  \bigl(m_t*f_N\bigr)(x)
\Bigr) - \frac12 \Delta m_t(x) =0,
\end{equation}

for $t \in [0,T]$,

\begin{equation*}
\begin{split}
&D {\mathcal H}_{\lambda}^{N,\varepsilon,\rho}(t,m)(x) :=
\partial_p H
\Bigl( x, \lambda
\partial_\mu W_1^{N,\varepsilon,\rho}(t,m)(x)
+ (1- \lambda)
\partial_\mu W_2^{N,\varepsilon,\rho}(t,m)(x)
\Bigr),
\\
&D {\mathcal H}^{N,\varepsilon,\rho}(t,m)(x) :=
\int_0^1
D {\mathcal H}_{\lambda}^{N,\varepsilon,\rho}(t,m)(x)
d \lambda,
\end{split}
\end{equation*}

and with
$m_0 \in B_N(c)$ as initial condition,
has a (unique) smooth solution with values in ${\mathcal P}({\mathbb T}^d)$.
It satisfies

\begin{equation*}
m_t \geq 1/c', \quad \| \nabla m_t \|_\infty \leq c', \quad t \in [0,T],
\end{equation*}

for $c' >1$ independent of $N$.

We first recall that the function $(t,m,x) \mapsto D {\mathcal H}^{N,\varepsilon,\rho}(t,m)(x)$ is smooth. It depends on
$m$ through $(\widehat{m}^k)_{k \in F_N}$. We can extend the mapping
to inputs $m$ which are general distributions. It suffices to project the Fourier coefficients on the closure of ${\mathcal O}_N$.
In short, we call $\Pi_N$ the mapping that maps $m=(\widehat m^k)_{k \in {\mathbb Z}^d}$ onto
$\Pi_N(m)$ defined by

\begin{equation*}
\widehat{\Pi_N(m)}^k =
\left\{
\begin{array}{ll}
1 &\quad \textrm{\rm if} \ k =0
\vspace{3pt}
\\
{\pi_N\bigl(({\widehat{m}^j})_{j \in F_N^+}\bigr)}^k
&\quad \textrm{\rm if} \ k \in F_N^+
\vspace{3pt}
\\
\overline{\widehat{\Pi_N(m)}^k}
&\quad \textrm{\rm if} \ -k \in F_N^+
\vspace{3pt}
\\
0 &\quad \textrm{\rm if} \ k \not \in F_N
\end{array}
\right.,
\end{equation*}

where $\pi_N$ is the orthogonal projection from ${\mathbb R}^{2 \vert F_N^+\vert}
\simeq {\mathbb C}^{\vert F_N^+ \vert}$ onto
the closure of
${\mathcal O}_N$.
We then consider
$(t,m,x) \mapsto D {\mathcal H}^{N,\varepsilon,\rho}(t,\Pi_N(m))(x)$. It is Lipschitz
with respect to the Fourier coefficients $(\widehat{m}^k)_{k \in F_N \setminus \{0\}}$.
There is no difficulty for proving that the solution is classical.  However,
the solution to the hence extended version of
(<ref>) may not take values in ${\mathcal O}_N$.

We now consider the local Fokker-Planck equation

\begin{equation}
\label{eq:FPK:mollified:proof:2}
\partial_t \widetilde m_t (x)- \textrm{\rm div} \Bigl( D {\mathcal H}^{N,\varepsilon,\rho}(t,m_t)(x)  \widetilde m_t(x)
\Bigr) - \frac12 \Delta \widetilde m_t(x) =0,
\end{equation}

$\widetilde m_0=m_0 \in B_N(c)$ as initial condition. It is important to notice that, in the non-linear dependence, the argument is
$m_t$ and not $\widetilde m_t$.
Since $W_1$ and $W_2$ are $d_{-2}$-Lipschitz continuous in the measure argument, the field $x \mapsto D {\mathcal H}^{N,\varepsilon,\rho}(t,m_t)(x) $
is bounded and Lipschitz continuous, with a bound and a Lipschitz constant that are independent of $N$, $\varepsilon$, $\rho$: it suffices to combine Corollary
the second claim in
In turn, the solution to the Fokker-Planck equation satisfies

\begin{equation*}
\widetilde m_t \geq 1/c', \quad \| \nabla_x \widetilde m_t \|_\infty, \quad \vvvert \nabla_x \widetilde m_t \vvvert_{\gamma} \leq c',
\quad t \in [0,T],
\end{equation*}

for a constant $c'$ that depends on $c$ but that is independent of $N$, $\varepsilon$ and $\rho$,
and for some $\gamma \in (0,1)$ that is also independent of $N$, $\varepsilon$ and $\rho$.

We then compare $m$ and $\widetilde m$.
Denoting by $p(t,x)$ the standard heat kernel on the torus,
we have

\begin{equation*}
\begin{split}
m_t(x) - \widetilde{m}_t(x)
&= - \int_0^t
\int_{{\mathbb T}^d}
\nabla_x p(t-s,x-y)
D {\mathcal H}^{N,\varepsilon,\rho}(s,m_s)(y)
\bigl( f_N*m(s,y) - \widetilde{m}(s,y)
\bigr) dy ds,
\end{split}
\end{equation*}

from which we deduce that

\begin{equation*}
\begin{split}
\| m_t - \widetilde m_t \|_\infty
&\leq C \int_0^t
\int_{{\mathbb T}^d}
\frac{\| m_s - \widetilde m_s \|_\infty}{\sqrt{t-s}}
ds + \eta_N,
\quad t \in [0,T],
\end{split}
\end{equation*}

for a sequence $(\eta_N)_{N \geq 1}$ that tends to $0$ as $N$ tends to $\infty$ and that only depends on $m_0$ through the value of $c$.
It is standard to deduce that

\begin{equation*}
\| m_t - \widetilde m_t \|_\infty
\leq C \eta_N,
\end{equation*}

which proves that $m_t$ is positive for $N$ large enough. Since $\int_{{\mathbb T}^d} m_t(x) dx=1$, it is a probability measure.

Using the regularity in $y$ of
$D {\mathcal H}^{N,\varepsilon,\rho}(s,y)$ together with the Hölder continuity of $\nabla_x \widetilde m_t$, we can proceed in the same way for
$\|\nabla_x m_t - \nabla_x \widetilde{m}_t\|_\infty$.

[1]
S. Ahuja.
Wellposedness of mean field games with common noise under a weak
monotonicity condition.
SIAM J. Control Optim., 54(1):30–48, 2016.

[2]
S. A. Alimov, R. R. Ashurov, and A. K. Pulatov.
Multiple Fourier series and Fourier integrals [ MR1027847
In Commutative harmonic analysis, IV, volume 42 of Encyclopaedia Math. Sci., pages 1–95. Springer, Berlin, 1992.

[3]
D. M. Ambrose and A. R. Mészáros.
Well-posedness of mean field games master equations involving
non-separable local hamiltonians.
arXiv, https://arxiv.org/abs/2105.03926, 2021.

[4]
L. Ambrosio and W. Gangbo.
Hamiltonian ODEs in the Wasserstein space of probability
Comm. Pure Appl. Math., 61(1):18–53, 2008.

[5]
L. Ambrosio, N. Gigli, and G. Savaré.
Gradient flows in metric spaces and in the space of probability
Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel,
second edition, 2008.

[6]
E. Bandini, A. Cosso, M. Fuhrman, and H. Pham.
Randomized filtering and Bellman equation in Wasserstein space
for partial observation control problem.
Stochastic Process. Appl., 129(2):674–711, 2019.

[7]
E. Bayraktar, A. Cosso, and H. Pham.
Randomized dynamic programming principle and feynman-kac
representation for optimal control of mckean-vlasov dynamics.
Transactions of the American Mathematical Society,
370(3):2115–2160, 2018.

[8]
E. Bayraktar and X. Zhang.
On non-uniqueness in mean field games.
Proc. Amer. Math. Soc., 148(9):4091–4106, 2020.

[9]
E. Bayraktar and X. Zhang.
Corrigendum to “On non-uniqueness in mean field games”.
Proc. Amer. Math. Soc., 149(3):1359–1360, 2021.

[10]
A. Bensoussan, J. Frehse, and S. C. P. Yam.
The master equation in mean field theory.
J. Math. Pures Appl., 103(6):1441–1474, 2015.

[11]
C. Bertucci.
Monotone solutions for mean field games master equations : continuous
state space and common noise.
arXiv, https://arxiv.org/abs/2107.09531, 2021.

[12]
C. Bertucci.
Monotone solutions for mean field games master equations: finite
state space and optimal stopping.
J. Éc. polytech. Math., 8:1099–1132, 2021.

[13]
V. I. Bogachev.
Differentiable measures and the Malliavin calculus, volume
164 of Mathematical Surveys and Monographs.
American Mathematical Society, Providence, RI, 2010.

[14]
A. Briani and P. Cardaliaguet.
Stable solutions in potential mean field game systems.
NoDEA Nonlinear Differential Equations Appl., 25(1):Paper No.
1, 26, 2018.

[15]
M. Burzoni, V. Ignazio, A. M. Reppen, and H. M. Soner.
Viscosity solutions for controlled McKean-Vlasov
SIAM J. Control Optim., 58(3):1676–1699, 2020.

[16]
P. Cannarsa and C. Sinestrari.
Semiconcave functions, Hamilton-Jacobi equations, and
optimal control, volume 58 of Progress in Nonlinear Differential
Equations and their Applications.
Birkhäuser Boston, Inc., Boston, MA, 2004.

[17]
P. Cardaliaguet, M. Cirant, and A. Porretta.
Splitting methods and short time existence for the master equations
in mean field games.
arXiv, https://arxiv.org/abs/2001.10406, 2020.

[18]
P. Cardaliaguet, F. Delarue, J.-M. Lasry, and P.-L. Lions.
The master equation and the convergence problem in mean field
games, volume 201 of Annals of Mathematics Studies.
Princeton University Press, Princeton, NJ, 2019.

[19]
P. Cardaliaguet, J. Graber, A. Porretta, and D. Tonon.
Second order mean field games with degenerate diffusion and local
NoDEA, 22:1287–1317, 2015.

[20]
P. Cardaliaguet and A. Porretta.
An introduction to mean field game theory.
In Mean Field Games, chapter 1, Cetraro, Italy 2019,
Cardaliaguet, Pierre, Porretta, Alessio (Eds.), LNM 2281, pages 203–248.
Springer, 2021.

[21]
P. Cardaliaguet and M. Quincampoix.
Deterministic differential games under probability knowledge of
initial condition.
International Game Theory Review, 10(1).

[22]
P. Cardaliaguet and P. Souganidis.
On first order mean field game systems with a common noise.
To appear on Annals of applied probability, 2020.

[23]
P. Cardaliaguet and P. Souganidis.
Weak solutions of the master equation for mean field games with no
idiosyncratic noise.
arXiv, https://arxiv.org/abs/2109.14911, 2021.

[24]
P. Cardaliaguet and P. Souganidis.
Regularity of the value function and quantitative propagation of
chaos for mean field control problems.

[25]
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.

[26]
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.

[27]
A. Cecchin, P. Dai Pra, M. Fischer, and G. Pelino.
On the convergence problem in mean field games: a two state model
without uniqueness.
SIAM J. Control Optim., 57(4):2443–2466, 2019.

[28]
A. Cecchin and F. Delarue.
Selection by vanishing common noise for potential finite state mean
field games.
Comm. Partial Differential Equations, 47(1):89–168, 2022.

[29]
J.-F. Chassagneux, D. Crisan, and F. Delarue.
A probabilistic approach to classical solutions of the master
equation for large population equilibria.
Memoirs of the AMS, To appear.

[30]
G. Conforti, R. Kraaij, and D. Tonon.
Hamilton–jacobi equations for controlled gradient flows: the
comparison principle.
arXiv, https://arxiv.org/abs/2111.13258, 2021.

[31]
A. Cosso, F. Gozzi, I. Kharroubi, H. Pham, and M. Rosestolato.
Master bellman equation in the wasserstein space: Uniqueness of
viscosity solutions.
arXiv, https://arxiv.org/abs/2107.10535, 2021.

[32]
A. Cosso and H. Pham.
Zero-sum stochastic differential games of generalized
McKean-Vlasov type.
J. Math. Pures Appl. (9), 129:180–212, 2019.

[33]
F. Delarue and R. Foguen Tchuendom.
Selection of equilibria in a linear quadratic mean field game.
Stochastic Processes and their Applications, 130(2):1000–1040,

[34]
F. Delarue and A. Tse.
Uniform in time weak propagation of chaos on the torus.
arXiv, https://arxiv.org/abs/2104.14973, 2021.

[35]
L. Dello Schiavo.
A Rademacher-type theorem on $L^2$-Wasserstein spaces over
closed Riemannian manifolds.
J. Funct. Anal., 278(6):108397, 57, 2020.

[36]
M. F. Djete, D. Possamaï, and X. Tan.
Mckean-vlasov optimal control: the dynamic programming principle.
arXiv preprint arXiv:1907.08860, 2019.

[37]
A. Douglis.
The continuous dependence of generalized solutions of non-linear
partial differential equations upon initial data.
Comm. Pure Appl. Math., 14:267–284, 1961.

[38]
W. Gangbo and A. R. Mészáros.
Global well-posedness of master equations for deterministic
displacement convex potential mean field games.
arXiv, https://arxiv.org/abs/2004.01660, 2020.

[39]
W. Gangbo, A. R. Mészáros, C. Mou, and J. Zhang.
Mean field games master equations with non-separable hamiltonians and
displacement monotonicity.
arXiv, https://arxiv.org/abs/2101.12362, 2021.

[40]
W. Gangbo and A. Świech.
Existence of a solution to an equation arising from the theory of
mean field games.
J. Differential Equations, 259(11):6573–6643, 2015.

[41]
W. Gangbo and A. Tudorascu.
On differentiability in the Wasserstein space and well-posedness
for Hamilton-Jacobi equations.
J. Math. Pures Appl. (9), 125:119–174, 2019.

[42]
B. I. Golubov.
Multiple series and Fourier integrals.
In Mathematical analysis, Vol. 19, pages 3–54, 232. Akad.
Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1982.

[43]
D. A. Gomes and J. a. Saúde.
Mean field games models—a brief survey.
Dyn. Games Appl., 4(2):110–154, 2014.

[44]
M. Huang, P. Caines, and R. Malhamé.
Individual and mass behavior in large population stochastic wireless
power control problems: centralized and Nash equilibrium solutions.
pages 98 – 103, 2003.

[45]
M. Huang, R. P. Malhamé, and P. E. Caines.
Large population stochastic dynamic games: Closed-loop
McKean-Vlasov systems and the Nash certainty equivalence principle.
Commun. Inf. Syst., 6(3):221–251, 2006.

[46]
M. Iseri and J. Zhang.
Set values for mean field games.
arXiv, https://arxiv.org/abs/2107.01661, 2021.

[47]
F. Jean, O. Jerhaoui, and H. Zidani.
Deterministic optimal control on Riemannian manifolds under
probability knowledge of the initial condition.
https://hal-ensta-paris.archives-ouvertes.fr//hal-03564787, Feb. 2022.
working paper or preprint.

[48]
A. Q. M. Jimenez, Chloé; Marigonda.
Optimal control of multiagent systems in the Wasserstein space.
Calc. Var. Partial Differential Equations, 59(2).

[49]
R. Jordan, D. Kinderlehrer, and F. Otto.
The variational formulation of the Fokker-Planck equation.
SIAM J. Math. Anal., 29(1):1–17, 1998.

[50]
S. N. Kružkov.
The Cauchy problem in the large for certain non-linear first order
differential equations.
Soviet Math. Dokl., 1:474–477, 1960.

[51]
S. N. Kružkov.
Generalized solutions of nonlinear equations of the first order with
several independent variables. II.
Mat. Sb. (N.S.), 72 (114):108–134, 1967.

[52]
D. Lacker.
Limit theory for controlled mckean-vlasov dynamics.
SIAM J. Control Optim., 55:1641–1672, 2017.

[53]
D. Lacker, M. Shkolnikov, and J. Zhang.
Superposition and mimicking theorems for conditional
McKean-Vlasov equations.
arXiv, https://arxiv.org/abs/2004.00099, 2020.

[54]
J.-M. Lasry and P.-L. Lions.
Jeux à champ moyen. I. Le cas stationnaire.
C. R. Math. Acad. Sci. Paris, 343(9):619–625, 2006.

[55]
J.-M. Lasry and P.-L. Lions.
Jeux à champ moyen. II. Horizon fini et contrôle optimal.
C. R. Math. Acad. Sci. Paris, 343(10):679–684, 2006.

[56]
J.-M. Lasry and P.-L. Lions.
Mean field games.
Jpn. J. Math., 2(1):229–260, 2007.

[57]
M. Laurière and O. Pironneau.
Dynamic programming for mean-field type control.
J. Optim. Theory Appl., 169(3):902–924, 2016.

[58]
P.-L. Lions.
Cours au collège de france, equations aux dérivées partielles
et applications.

[59]
P.-L. Lions.
Estimées nouvelles pour les équations quasilinéaires.
Seminar in Applied Mathematics at the Collège de France.
1h15.htm, 2014.

[60]
P.-L. Lions.
Generalized solutions of Hamilton-Jacobi equations,
volume 69 of Research Notes in Mathematics.
Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982.

[61]
C. Mou and J. Zhang.
Wellposedness of second order master equations for mean field games
with nonsmooth data.
arXiv, https://arxiv.org/abs/1903.09907, 2019.

[62]
C. Mou and J. Zhang.
Mean field game master equations with anti-monotonicity conditions.
arXiv, https://arxiv.org/abs/2201.10762, 2022.

[63]
A. R. Mészáros and C. Mou.
Mean field games systems under displacement monotonicity.
arXiv, https://arxiv.org/abs/2109.06687, 2021.

[64]
H. Pham and X. Wei.
Bellman equation and viscosity solutions for mean-field stochastic
control problem.
ESAIM Control Optim. Calc. Var., 24(1):437–461, 2018.

[65]
A. Wiśniewski.
The structure of measurable mappings on metric spaces.
Proc. Amer. Math. Soc., 122(1):147–150, 1994.

[66]
C. Wu and J. Zhang.
Viscosity solutions to parabolic master equations and
McKean-Vlasov SDEs with closed-loop controls.
Ann. Appl. Probab., 30(2):936–986, 2020.

[67]
L. Zhizhiashvili.
Trigonometric Fourier series and their conjugates, volume 372
of Mathematics and its Applications.
Kluwer Academic Publishers Group, Dordrecht, 1996.
Revised and updated translation of  Some problems of the theory
of trigonometric Fourier series and their conjugate series (Russian)
[Tbilis. Gos. Univ., Tbilisi, 1993], Translated from the Russian by George

[68]
C. Zălinescu.
On uniformly convex functions.
J. Math. Anal. Appl., 95(2):344–374, 1983.