N\\\ j\neq i\end{subarray}}\lVert D^{3}_{hij}u^{j}\rVert_{\infty}^{2},$

where the implied constant depends only on $C_{\sigma}$ and $d$. By assumption
(MF), $\sum_{1\leqslant h,i\leqslant N,\,i\neq h}\lVert
D^{2}_{hi}f^{i}\rVert_{\infty}^{2}\leqslant L_{f}^{2}$; by Propositions 4.7
and 4.9,

$\sup_{1\leqslant i\leqslant
N}\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}\big{|}D^{2}_{ij}u^{j}\big{|}^{2}\bigg{\rVert}_{\infty}\sum_{\begin{subarray}{c}1\leqslant
h,i\leqslant N\\\ h\neq i\end{subarray}}\bigg{\lVert}\sum_{1\leqslant
j\leqslant
N}\big{|}D^{2}_{hj}u^{i}\big{|}^{2}\bigg{\rVert}_{\infty}\leqslant\frac{C_{3}C_{4}}{N};$

by Propositions 4.6 and 4.7,

$\sum_{\begin{subarray}{c}1\leqslant h,i\leqslant N\\\ i\neq
h\end{subarray}}\lVert D_{hi}u^{i}\rVert_{\infty}^{2}\lVert
D_{ii}u^{i}\rVert_{\infty}^{2}\leqslant C_{2}C_{3}.$

Therefore, summing inequality (4.22) over $h$ and $k=i\neq h$ yields

$\sum_{\begin{subarray}{c}1\leqslant h,i\leqslant N\\\ i\neq
h\end{subarray}}\lVert D(D^{2}_{hi}u^{i})\rVert_{\infty}^{2}\lesssim
se^{4Ms}+e^{2Ms}\sum_{\begin{subarray}{c}1\leqslant h,i\leqslant N\\\ i\neq
h\end{subarray}}\lVert
D(D^{2}_{hi}g^{i})\rVert_{\infty}^{2}+\frac{1}{N}\sum_{\begin{subarray}{c}1\leqslant
h,i,j\leqslant N\\\ j\neq i\end{subarray}}\lVert
D^{3}_{hij}u^{j}\rVert_{\infty}^{2}se^{4Ms},$

where the implied constant (here as well as below) depends only on $\sigma$,
$T$, $M$, $L_{f}$ and $L_{g}$. Exploiting now the fact that

$\sum_{\begin{subarray}{c}1\leqslant h,i,j\leqslant N\\\ j\neq
i\end{subarray}}\lVert D^{3}_{hij}u^{j}\rVert_{\infty}^{2}\leqslant
N\sum_{\begin{subarray}{c}1\leqslant i,j\leqslant N\\\ j\neq
i\end{subarray}}\bigg{\lVert}\sum_{1\leqslant h\leqslant
N}\big{|}D^{3}_{hij}u^{j}\big{|}^{2}\bigg{\rVert}_{\infty}=N\sum_{\begin{subarray}{c}1\leqslant
i,j\leqslant N\\\ j\neq i\end{subarray}}\lVert
D(D_{ij}u^{j})\rVert_{\infty}^{2},$

we obtain, for small $s$,

$\sum_{\begin{subarray}{c}1\leqslant h,i\leqslant N\\\ i\neq
h\end{subarray}}\lVert D(D^{2}_{hi}u^{i})\rVert_{\infty}^{2}\lesssim
1+\sum_{\begin{subarray}{c}1\leqslant h,i\leqslant N\\\ i\neq
h\end{subarray}}\lVert
D(D^{2}_{hi}g^{i})\rVert_{\infty}^{2}=1+\sum_{\begin{subarray}{c}1\leqslant
h,i\leqslant N\\\ i\neq h\end{subarray}}\lVert
D(D^{2}_{hi}u^{i})(T,\cdot)\rVert_{\infty}^{2}.$

Using that $\sum_{1\leqslant h,i\leqslant N,\,i\neq h}\lVert
D(D^{2}_{hi}g^{i})\rVert_{\infty}^{2}\leqslant L_{g}^{2}$ due to assumption
(MF), one concludes by arguing as in the proof of Proposition 4.6.

∎

##  5\. The $\mathscr{D}$*semimonotone case

We prove in this section Theorem 3.6 in the $\mathscr{D}$*semimonotone case.
The following result shows the crucial interplay between the previous
estimates and the $\mathscr{D}$*semimonotonicity of the value functions. It
basically shows that if there exists a left temporal open neighbourhood
$(\tau,T]$ of $T$ in which $u$ is $M$-$\mathscr{D}$*semimonotone for some
suitable $M$, then exploiting the estimates on the skew derivatives one can
show that such a semimonotonicity actually holds with constant $\frac{1}{2}M$
on the whole $[\tau,T]\times({\mathbb{R}}^{d})^{N}$, provided that $N$ is
large.

This implies that if the matrix $(D_{ij}u^{j})_{1\leqslant i,j\leqslant
N}+M\boldsymbol{\mathsf{I}}$ is positive definite near $T$, then it cannot
degenerate at any time in $[0,T]$. In this sense, we can consider this fact as
a (backward) _propagation_ of the semimonotonicity. Such an argument will in
turn imply that all estimates on the derivatives themselves propagate, thus
small-time existence of a $\mathscr{D}$*semimonotone solution to the Nash
system is sufficient to prove global existence on $[0,T]$ of a solution which
satisfies all the above estimates (see Theorem 5.2 below).

######  Proposition 5.1 (Improvement of $\mathscr{D}$*semimonotonicity).

Assume (MF). There exist positive constants $M^{*}$, $M_{f}^{*}$ and
$M_{g}^{*}$ (with $M_{g}^{*}<M^{*}$) depending on $T$, and a natural number
$N_{*}$ depending on $\sigma$, $T$, $L_{g}$ and $L_{f}$ such that for any
$\tau\in[0,T)$ one has

$\begin{rcases}\mathcal{D}[u]\geqslant-M^{*}Q&\text{on}\
(\tau,T]\times({\mathbb{R}}^{d})^{2N}\\\ \mathcal{D}[f]\geqslant-
M_{f}^{*}Q&\text{on}\ ({\mathbb{R}}^{d})^{2N}\\\ \mathcal{D}[g]\geqslant-
M_{g}^{*}Q&\text{on}\ ({\mathbb{R}}^{d})^{2N}\\\ N\geqslant
N_{*}\end{rcases}\quad\implies\quad\mathcal{D}[u(\tau,\mspace{1.8mu}\cdot\mspace{1.8mu})]\geqslant-\frac{1}{2}M^{*}Q\
\ \text{on}\ ({\mathbb{R}}^{d})^{2N}.$

Looking at (5.7) below, one can in fact choose $M_{g}^{*}=(12eT)^{-1}$,
$M_{f}^{*}=(12eT^{2})^{-1}$ and $M^{*}=(2T)^{-1}$.

The proof involves again the method of doubling variables. We could proceed
with a PDE approach as before, but we prefer to follow a “dynamic” approach;
that is, we argue along optimal trajectories. This is probably closer in the
spirit to what is usually done in the MFG theory.

More precisely, we use a synchronous coupling between two solutions of (1.2),
for agents playing optimally. For the benefit of the reader we briefly recall
what one means by such a coupling. Let $\overline{t}\in[0,T)$ and consider the
$({\mathbb{R}}^{d})^{m}$-valued SDE

(5.1)
$\mathrm{d}\boldsymbol{Z}_{t}=\boldsymbol{b}(t,\boldsymbol{Z}_{t})\,\mathrm{d}t+\boldsymbol{\Sigma}(t,\boldsymbol{Z}_{t})\,\mathrm{d}\boldsymbol{\tilde{B}}_{t},\quad
t\in[\overline{t},T],$

where $\boldsymbol{\Sigma}$ is an $md\times\ell d$ matrix and
$\boldsymbol{\tilde{B}}_{t}$ is an $\ell d$-dimensional Brownian motion.
Assume that the drift $\boldsymbol{b}$ and the diffusion coefficient
$\boldsymbol{\Sigma}$ are locally Lipschitz continuous and have sublinear
growth with respect to the state variable, uniformly with respect to time. If
$\boldsymbol{Z}$ solves (5.1) with
$\boldsymbol{Z}_{\overline{t}}=\overline{\boldsymbol{z}}\in({\mathbb{R}}^{d})^{m}$,
then the following _Dynkin’s formula_ holds for any $h\in
C^{1,2}([\overline{t},T]\times({\mathbb{R}}^{d})^{m})$ such that
$\partial_{t}^{\ell}D^{\gamma}h(t,\mspace{1.8mu}\cdot\mspace{1.8mu})$ has
polynomial growth for all $t\in[\overline{t},T]$ and $2\ell+|\gamma|\leqslant
2$:

(5.2)
$\mathbb{E}\bigl{[}h(T,\boldsymbol{Z}_{T})\bigr{]}=h(\overline{t},\overline{\boldsymbol{z}})+\int_{\overline{t}}^{T}\mathbb{E}\bigl{[}\partial_{t}h(t,\boldsymbol{Z}_{t})+Lh(t,\boldsymbol{Z}_{t})\bigr{]}\,\mathrm{d}t,$

where
$L=\tfrac{1}{2}\operatorname{tr}\\!\big{(}\boldsymbol{\Sigma}\boldsymbol{\Sigma}^{\mathsf{T}}\\!D^{2}_{\boldsymbol{z}}\big{)}+\boldsymbol{b}\cdot
D_{\boldsymbol{z}}$ is the differential generator of the process
$\boldsymbol{Z}$; see, e.g., [4, Chapter 9]. We are interested in the
particular case where $m=2N$ and $\boldsymbol{Z}$ is the following coupling of
two solution of the $({\mathbb{R}}^{d})^{N}$-valued SDE

(5.3)
$\mathrm{d}\boldsymbol{X}_{t}=\boldsymbol{\alpha}(t,\boldsymbol{X}_{t})\,\mathrm{d}t+\boldsymbol{\sigma}(t,\boldsymbol{X}_{t})\,\mathrm{d}\boldsymbol{\tilde{B}}_{t},\quad
t\in[\overline{t},T],$

where $\boldsymbol{\sigma}$ is an $Nd\times\ell d$ matrix and
$\boldsymbol{\tilde{B}}_{t}$ is a $\ell d$-dimensional Brownian motion. Given
$\boldsymbol{X}$ and $\boldsymbol{Y}$ which both solve (5.3), with
$\boldsymbol{X}_{\overline{t}}=\overline{\boldsymbol{x}}$ and
$\boldsymbol{Y}_{\overline{t}}=\overline{\boldsymbol{y}}$, respectively, we
say that $\boldsymbol{Z}=(\boldsymbol{X},\boldsymbol{Y})^{\mathsf{T}}$ with
$\boldsymbol{Z}_{\overline{t}}\sim\mu$ having marginals
$\delta_{\overline{\boldsymbol{x}}}$ and $\delta_{\overline{\boldsymbol{y}}}$
is a _synchronous coupling_ of $\boldsymbol{X}$ and $\boldsymbol{Y}$. It
solves equation (5.1) with
$\boldsymbol{b}(t,\boldsymbol{x},\boldsymbol{y})=(\boldsymbol{\alpha}(t,\boldsymbol{x}),\boldsymbol{\alpha}(t,\boldsymbol{y}))^{\mathsf{T}}$
and
$\boldsymbol{\Sigma}(t,\boldsymbol{x},\boldsymbol{y})=(\boldsymbol{\sigma}(t,\boldsymbol{x}),\boldsymbol{\sigma}(t,\boldsymbol{y}))^{\mathsf{T}}$,
thus

$\boldsymbol{b}\cdot
D_{\boldsymbol{z}}=\boldsymbol{\alpha}(t,\boldsymbol{x})\cdot
D_{\boldsymbol{x}}+\boldsymbol{\alpha}(t,\boldsymbol{y})\cdot
D_{\boldsymbol{y}},\qquad\boldsymbol{\Sigma}\boldsymbol{\Sigma}^{\mathsf{T}}=\bigg{(}\,\begin{matrix}\boldsymbol{\sigma}(t,\boldsymbol{x})\boldsymbol{\sigma}(t,\boldsymbol{x})^{\mathsf{T}}&\boldsymbol{\sigma}(t,\boldsymbol{x})\boldsymbol{\sigma}(t,\boldsymbol{y})^{\mathsf{T}}\\\
\boldsymbol{\sigma}(t,\boldsymbol{y})\boldsymbol{\sigma}(t,\boldsymbol{x})^{\mathsf{T}}&\boldsymbol{\sigma}(t,\boldsymbol{y})\boldsymbol{\sigma}(t,\boldsymbol{y})^{\mathsf{T}}\end{matrix}\,\bigg{)}.$

In particular, when (5.3) coincides with (1.2) in the equilibrium, which
happens for
$\boldsymbol{\alpha}=\boldsymbol{\alpha}^{*}\vcentcolon=(-D_{i}u^{i})_{1\leqslant
i\leqslant N}$, $\boldsymbol{\tilde{B}}=(\boldsymbol{B},W)^{\mathsf{T}}$ and
$\boldsymbol{\sigma}\equiv\sqrt{2}\,\big{(}\,\sqrt{\sigma}\,\boldsymbol{\mathsf{I}}\mid\sqrt{\beta\,}1_{N}\otimes\mathsf{I}_{d}\,\big{)}$,
one has
$\boldsymbol{\Sigma}\boldsymbol{\Sigma}^{\mathsf{T}}=2(\mathsf{J}_{2}\otimes(\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}}))$
and thus

(5.4)
$L=\operatorname{tr}\\!\big{(}(\mathsf{J}_{2}\otimes(\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}}))D^{2}_{(\boldsymbol{x},\boldsymbol{y})}\big{)}+\boldsymbol{\alpha}^{*}(t,\boldsymbol{x})\cdot
D_{\boldsymbol{x}}+\boldsymbol{\alpha}^{*}(t,\boldsymbol{y})\cdot
D_{\boldsymbol{y}}.$

Clearly, analogous considerations hold if $\boldsymbol{Z}_{\overline{t}}$ is
any random variable with
$\mathbb{E}\bigl{[}|\boldsymbol{Z}_{\overline{t}}|^{k}\bigr{]}<\infty$ for
some $k\in\mathbb{N}$; in this case, (5.2) holds with
$h(\overline{t},\overline{\boldsymbol{z}})$ replaced by
$\mathbb{E}\bigl{[}h(\overline{t},\boldsymbol{Z}_{\overline{t}})\bigr{]}$ and
provided that
$\partial_{t}^{\ell}D^{\gamma}h(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\lesssim
1+|\cdot|^{k}$.

###### Proof of Proposition 5.1.

Let
$w^{i}(t,\boldsymbol{x},\boldsymbol{y})\vcentcolon=D_{i}u^{i}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}_{\boldsymbol{y}}^{\boldsymbol{x}}\cdot(x^{i}-y^{i})$.
We want to use formula (5.2) where $\overline{t}=\tau$, $h=w^{i}$ and
$\boldsymbol{Z}=(\boldsymbol{X},\boldsymbol{Y})$ is a synchronous coupling of
two solutions to (1.2); that is, formula (5.2) with $L$ given by (5.4). We
have

$\begin{multlined}Lw^{i}(t,\boldsymbol{x},\boldsymbol{y})=D_{i}\operatorname{tr}\\!\big{(}(\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})D^{2}u^{i}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}_{\boldsymbol{y}}^{\boldsymbol{x}}\big{)}\cdot(x^{i}-y^{i})\\\
-\sum_{1\leqslant j\leqslant
N}D^{2}_{ij}u^{i}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})D_{j}u^{j}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}_{\boldsymbol{y}}^{\boldsymbol{x}}\cdot(x^{i}-y^{i})-\bigl{|}D_{i}u^{i}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}_{\boldsymbol{y}}^{\boldsymbol{x}}\bigr{|}^{2},\end{multlined}Lw^{i}(t,\boldsymbol{x},\boldsymbol{y})=D_{i}\operatorname{tr}\\!\big{(}(\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})D^{2}u^{i}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}_{\boldsymbol{y}}^{\boldsymbol{x}}\big{)}\cdot(x^{i}-y^{i})\\\
-\sum_{1\leqslant j\leqslant
N}D^{2}_{ij}u^{i}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})D_{j}u^{j}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}_{\boldsymbol{y}}^{\boldsymbol{x}}\cdot(x^{i}-y^{i})-\bigl{|}D_{i}u^{i}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}_{\boldsymbol{y}}^{\boldsymbol{x}}\bigr{|}^{2},$

and, according to (1.1),

$\begin{multlined}\partial_{t}w^{i}(t,\boldsymbol{x},\boldsymbol{y})=-D_{i}\operatorname{tr}\\!\big{(}(\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})D^{2}u^{i}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}_{\boldsymbol{y}}^{\boldsymbol{x}}\big{)}\cdot(x^{i}-y^{i})+\sum_{1\leqslant
j\leqslant
N}D^{2}_{ij}u^{i}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})D_{j}u^{j}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}_{\boldsymbol{y}}^{\boldsymbol{x}}\cdot(x^{i}-y^{i})\\\
+\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}D^{2}_{ij}u^{j}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})D_{j}u^{i}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}_{\boldsymbol{y}}^{\boldsymbol{x}}\cdot(x^{i}-y^{i})-D_{i}f^{i}\big{|}_{\boldsymbol{y}}^{\boldsymbol{x}}\cdot(x^{i}-y^{i}).\end{multlined}\partial_{t}w^{i}(t,\boldsymbol{x},\boldsymbol{y})=-D_{i}\operatorname{tr}\\!\big{(}(\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})D^{2}u^{i}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}_{\boldsymbol{y}}^{\boldsymbol{x}}\big{)}\cdot(x^{i}-y^{i})+\sum_{1\leqslant
j\leqslant
N}D^{2}_{ij}u^{i}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})D_{j}u^{j}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}_{\boldsymbol{y}}^{\boldsymbol{x}}\cdot(x^{i}-y^{i})\\\
+\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}D^{2}_{ij}u^{j}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})D_{j}u^{i}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}_{\boldsymbol{y}}^{\boldsymbol{x}}\cdot(x^{i}-y^{i})-D_{i}f^{i}\big{|}_{\boldsymbol{y}}^{\boldsymbol{x}}\cdot(x^{i}-y^{i}).$

Then equality (5.2) gives

$\mathbb{E}[w^{i}(T,\boldsymbol{X}_{T},\boldsymbol{Y}_{T})]\leqslant
w^{i}(\tau,\overline{\boldsymbol{x}},\overline{\boldsymbol{y}})-\int_{\tau}^{T}\mathbb{E}\Big{[}D_{i}f^{i}\big{|}_{\boldsymbol{Y}_{t}}^{\boldsymbol{X}_{t}}\cdot(X_{t}^{i}-Y_{t}^{i})\Big{]}+\int_{\tau}^{T}\mathbb{E}\,\mathcal{E}^{i}_{t}\,\mathrm{d}t,$

where $\mathcal{E}^{i}_{t}\vcentcolon=\sum_{1\leqslant j\leqslant N,\,j\neq
i}D^{2}_{ij}u^{j}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})D_{j}u^{i}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}_{\boldsymbol{Y}_{t}}^{\boldsymbol{X}_{t}}\cdot(X_{t}^{i}-Y_{t}^{i})$.
Letting $w\vcentcolon=\sum_{1\leqslant i\leqslant N}w^{i}$, it follows that

(5.5)
$w(\tau,\overline{\boldsymbol{x}},\overline{\boldsymbol{y}})\geqslant-M^{*}_{g}\,\mathbb{E}\big{[}|\boldsymbol{X}_{T}-\boldsymbol{Y}_{T}|^{2}\big{]}-M^{*}_{f}\int_{\tau}^{T}\mathbb{E}\big{[}|\boldsymbol{X}_{t}-\boldsymbol{Y}_{t}|^{2}\big{]}\mathrm{d}t-\int_{\tau}^{T}\mathbb{E}\biggl{[}\,\sum_{1\leqslant
i\leqslant N}\mathcal{E}_{t}^{i}\,\biggr{]}\,\mathrm{d}t.$

By the mean value theorem

$D^{2}_{ij}u^{j}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})D_{j}u^{i}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}_{\boldsymbol{Y}_{t}}^{\boldsymbol{X}_{t}}=\bigg{(}\int_{0}^{1}D\big{(}D^{2}_{ij}u^{j}(t,\boldsymbol{Z}_{t}(s))D_{j}u^{i}(t,\boldsymbol{Z}_{t}(s))\big{)}\,\mathrm{d}s\bigg{)}(\boldsymbol{X}_{t}-\boldsymbol{Y}_{t}),$

with $\boldsymbol{Z}_{t}(s)=s\boldsymbol{X}_{t}+(1-s)\boldsymbol{Y}_{t}$; then
note that

$\sum_{1\leqslant i\leqslant
N}\mathcal{E}_{t}^{i}=\int_{0}^{1}\sum_{1\leqslant i,k\leqslant
N}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}\big{(}D^{3}_{ijk}u^{j}(t,\boldsymbol{Z}_{t}(s))D_{j}u^{i}(t,\boldsymbol{Z}_{t}(s))\big{)}(X^{k}_{t}-Y^{k}_{t})\cdot(X_{t}^{i}-Y_{t}^{i})\,\mathrm{d}s\\\
+\int_{0}^{1}\sum_{1\leqslant i,k\leqslant
N}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}\big{(}D^{2}_{ij}u^{j}(t,\boldsymbol{Z}_{t}(s)))D^{2}_{jk}u^{i}(t,\boldsymbol{Z}_{t}(s))\big{)}(X^{k}_{t}-Y^{k}_{t})\cdot(X_{t}^{i}-Y_{t}^{i})\,\mathrm{d}s,$

where

$\biggl{|}\,\int_{0}^{1}\sum_{1\leqslant i,k\leqslant
N}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}\big{(}D^{3}_{ijk}u^{j}(t,\boldsymbol{Z}_{t}(s))D_{j}u^{i}(t,\boldsymbol{Z}_{t}(s))\big{)}(X^{k}_{t}-Y^{k}_{t})\cdot(X_{t}^{i}-Y_{t}^{i})\,\mathrm{d}s\,\biggr{|}\\\\[-7.0pt]
\begin{split}&\leqslant\biggl{\|}\sum_{1\leqslant i,k\leqslant
N}\,\biggl{|}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}D^{3}_{ijk}u^{j}D_{j}u^{i}\biggr{|}^{2}\biggr{\|}_{\infty}^{\frac{1}{2}}|\boldsymbol{X}_{t}-\boldsymbol{Y}_{t}|^{2}\\\
&\leqslant\biggl{(}\sum_{1\leqslant i\leqslant
N}\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}\big{|}D(D^{2}_{ij}u^{j})\big{|}^{2}\bigg{\rVert}_{\infty}\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}\big{|}D_{j}u^{i}\big{|}^{2}\bigg{\rVert}_{\infty}\biggr{)}^{\frac{1}{2}}|\boldsymbol{X}_{t}-\boldsymbol{Y}_{t}|^{2}\end{split}$

and similarly

$\biggl{|}\,\int_{0}^{1}\sum_{1\leqslant i,k\leqslant
N}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}\big{(}D^{2}_{ij}u^{j}(t,\boldsymbol{Z}_{t}(s)))D^{2}_{jk}u^{i}(t,\boldsymbol{Z}_{t}(s))\big{)}(X^{k}_{t}-Y^{k}_{t})\cdot(X_{t}^{i}-Y_{t}^{i})\,\mathrm{d}s\,\biggr{|}\\\\[-7.0pt]
\leqslant\biggl{(}\sum_{1\leqslant i\leqslant
N}\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}\big{|}D^{2}_{ij}u^{j}\big{|}^{2}\bigg{\rVert}_{\infty}\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}\big{|}D(D_{j}u^{i})\big{|}^{2}\bigg{\rVert}_{\infty}\biggr{)}^{\frac{1}{2}}|\boldsymbol{X}_{t}-\boldsymbol{Y}_{t}|^{2}.$

Propositions 4.5 and 4.11 ensure that

$\sum_{1\leqslant i\leqslant
N}\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}\big{|}D(D^{2}_{ij}u^{j})\big{|}^{2}\bigg{\rVert}_{\infty}\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}\big{|}D_{j}u^{i}\big{|}^{2}\bigg{\rVert}_{\infty}\leqslant\frac{c_{1}^{2}C_{5}}{N}$

and Propositions 4.7 and 4.9 give

$\sum_{1\leqslant i\leqslant
N}\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}\big{|}D^{2}_{ij}u^{j}\big{|}^{2}\bigg{\rVert}_{\infty}\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}\big{|}D(D_{j}u^{i})\big{|}^{2}\bigg{\rVert}_{\infty}\leqslant\frac{C_{3}C_{4}}{N}.$

Therefore we have

$\int_{\tau}^{T}\mathbb{E}\biggl{[}\,\sum_{1\leqslant i\leqslant
N}\mathcal{E}_{t}^{i}\,\biggr{]}\leqslant\frac{\overline{C}}{\sqrt{N}}\,\sup_{t\in[\tau,T]}\mathbb{E}\big{[}|\boldsymbol{X}_{t}-\boldsymbol{Y}_{t}|^{2}\big{]}$

for some constant $\overline{C}$ which depends only $\sigma$, $T$, $M^{*}$,
$L_{f}$ and $L_{g}$. On the other hand, by the equation of
$\boldsymbol{X}-\boldsymbol{Y}$,

$\mathrm{d}|\boldsymbol{X}_{t}-\boldsymbol{Y}_{t}|^{2}=-2\sum_{1\leqslant
i\leqslant
N}(D_{i}u^{i}(t,\boldsymbol{X}_{t})-D_{i}u^{i}(t,\boldsymbol{Y}_{t}))\cdot(X^{i}_{t}-Y^{i}_{t})\,\mathrm{d}t\leqslant
2M^{*}|\boldsymbol{X}_{t}-\boldsymbol{Y}_{t}|^{2}\,\mathrm{d}t,$

where the inequality comes from the $\mathscr{D}$*semimonotonicity of $u$, so
that Gronwall’s lemma yields

(5.6)
$\mathbb{E}\big{[}|\boldsymbol{X}_{t}-\boldsymbol{Y}_{t}|^{2}\big{]}\leqslant|\overline{\boldsymbol{x}}-\overline{\boldsymbol{y}}|^{2}e^{2M^{*}(t-\tau)}\quad\forall\,t\in[\tau,T].$

Hence, from (5.5) we have obtained

$w(\tau,\overline{\boldsymbol{x}},\overline{\boldsymbol{y}})\geqslant-e^{2M^{*}T}\Bigl{(}M^{*}_{g}+TM^{*}_{f}+\frac{\overline{C}}{\sqrt{N}\,}\Bigr{)}|\overline{\boldsymbol{x}}-\overline{\boldsymbol{y}}|^{2}.$

Fix now $M^{*}=(2T)^{-1}$; choosing $M^{*}_{g}$ and $M^{*}_{f}$ small enough
and $N_{*}$ large enough so that

(5.7)
$M^{*}_{g}+TM^{*}_{f}+\frac{\overline{C}}{\sqrt{N_{*}}\,}\leqslant\frac{M^{*}}{2e^{2M^{*}T}}=\frac{1}{4eT}$

we see that
$w(\tau,\overline{\boldsymbol{x}},\overline{\boldsymbol{y}})\geqslant-\frac{1}{2}M^{*}|\overline{\boldsymbol{x}}-\overline{\boldsymbol{y}}|^{2}$
holds for any $N\geqslant N_{*}$. The conclusion follows by the arbitrariness
of
$\overline{\boldsymbol{x}},\overline{\boldsymbol{y}}\in({\mathbb{R}}^{d})^{N}$.
∎

As anticipated, such a non-degeneration of semimonotonicity of the solution is
the key ingredient for the propagation of all derivative estimates, and of
semimonotonicity, over the whole time horizon $[0,T]$. This constitutes the
first half of Theorem 3.6.

######  Theorem 5.2 (Estimates on the Nash system with
$\mathscr{D}$*semimonotone data).

Assume that $f$ and $g$ satisfy assumptions (MF) and (DS). Let $T>0$. There
exist positive constants $M_{f}^{*}$ and $M_{g}^{*}$ depending on $T$, and a
natural number $N^{*}$ depending only on $\sigma$, $T$, $L_{f}$ and $L_{g}$,
such that if $M_{g}\leqslant M_{g}^{*}$, $M_{f}\leqslant M_{f}^{*}$ and
$N\geqslant N^{*}$ then any solution $u$ to (1.1) on
$[0,T]\times({\mathbb{R}}^{d})^{N}$ satisfies

$\sup_{1\leqslant i\leqslant N}\bigg{(}\sup_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq i\end{subarray}}\lVert
D_{j}u^{i}\rVert_{\infty}+\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}\big{|}D^{2}_{ij}u^{j}\big{|}^{2}\bigg{\rVert}_{\infty}+\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq i\end{subarray}}\lVert
D(D_{j}u^{i})\rVert_{\infty}^{2}\bigg{)}\lesssim\frac{1}{N}$

and

$\sup_{1\leqslant i\leqslant N}\lVert Du^{i}\rVert_{\infty}+\sup_{1\leqslant
i\leqslant N}\sum_{1\leqslant j\leqslant N}\lVert
D(D_{i}u^{j})\rVert_{\infty}^{2}+\sum_{\begin{subarray}{c}1\leqslant
i,j\leqslant N\\\ j\neq i\end{subarray}}\lVert
D(D_{ij}u^{j})\rVert_{\infty}^{2}\lesssim 1,$

where the implied constants depend only on $\sigma$, $T$, $L_{f}$ and $L_{g}$.
In addition, there exists $M^{*}>0$ (depending only on $T$) such that $u$ is
$M^{*}$-$\mathscr{D}$*semimonotone on $[0,T]$.

###### Proof.

Let $M_{f}^{*}$, $M_{g}^{*}$, $M^{*}$ and $N^{*}$ be given by Proposition 5.1.
By assumption (DS),
$\mathcal{D}[u(T,\mspace{1.8mu}\cdot\mspace{1.8mu})]+M^{*}Q\geqslant\varepsilon
Q$ on $({\mathbb{R}}^{d})^{2N}$, with $\varepsilon=M^{*}-M^{*}_{g}>0$; that
is, by Remark 3.3,
$(D_{ij}^{2}u^{i}(T,\mspace{1.8mu}\cdot\mspace{1.8mu}))_{1\leqslant
i,j\leqslant
N}+M^{*}\boldsymbol{\mathsf{I}}\geqslant\varepsilon\boldsymbol{\mathsf{I}}$ on
$({\mathbb{R}}^{d})^{N}$. Since $D^{2}_{ij}u^{i}$ are assumed to be uniformly
continuous, there exists $\tau\in[0,T)$ (a priori dependent of $N$) such that
$(D_{ij}^{2}u^{i}(t,\mspace{1.8mu}\cdot\mspace{1.8mu}))_{1\leqslant
i,j\leqslant
N}+M^{*}\boldsymbol{\mathsf{I}}\geqslant\frac{\varepsilon}{2}\boldsymbol{\mathsf{I}}$
on $({\mathbb{R}}^{d})^{N}$ for all $t\in(\tau,T]$. Therefore,

$\mathcal{T}\vcentcolon=\big{\\{}s\in[0,T):\ \text{$u$ extends to a solution
on $(s,T]\times({\mathbb{R}}^{d})^{N}$ and $\mathcal{D}[u]>-M^{*}Q$
thereon}\big{\\}}\neq\emptyset$

and $\tau^{*}\vcentcolon=\inf\mathcal{T}\in[0,\tau]$. Seeking for a
contradiction, suppose that $\tau^{*}>0$. Then
$\mathcal{D}[u]\geqslant-M^{*}Q$ on
$[\tau^{*},T]\times({\mathbb{R}}^{d})^{2N}$ and by Proposition 5.1 we have
$\mathcal{D}[u(\tau^{*},\mspace{1.8mu}\cdot\mspace{1.8mu})]\geqslant-\frac{1}{2}M^{*}Q$
on $({\mathbb{R}}^{d})^{2N}$. Repeating the same argument as at the beginning
of the proof, there exists $\tau^{\prime}\in\mathcal{T}$,
$\tau^{\prime}<\tau^{*}$, thus contradicting the definition of $\tau^{*}$.
This proves the $\mathscr{D}$-semimonotonicity of $u$ on $[0,T]$, hence all
estimates on the derivatives are given by Propositions 4.6, 4.7, 4.8, 4.9 and
4.11. ∎

##  6\. The $\mathscr{L}$*semimonotone case

We prove in this section Theorem 3.6 in the $\mathscr{L}$*semimonotone case.
Our observations in Remark 3.3 motivate the following result, which
comprehends all estimates on derivatives we have made in the previous section,
under the assumption of $\mathscr{L}$*semimonotonicity and a lower bound on
the diagonal of the Jacobian matrix of the vector of optimal controls.

######  Proposition 6.1 (Estimates on the derivatives).

Let $\tau\in[0,T)$. Suppose that $\mathcal{L}[u]\geqslant-\kappa Q$ on
$(\tau,T]\times({\mathbb{R}}^{d})^{2N}$ and there exists $\gamma>0$
(independent of $N$), such that

(6.1)
$\boldsymbol{\triangle}_{u(t,\mspace{1.8mu}\cdot\mspace{1.8mu})}\geqslant-\gamma\boldsymbol{\mathsf{I}}\
\ \text{on}\ ({\mathbb{R}}^{d})^{N}\quad\forall\,t\in(\tau,T].$

Then there exists constants $K_{i}$, $i\in\\{1,\dots,5\\}$, depending only on
$\sigma$, $T$, $\kappa$, $\gamma$, $L_{g}$ and $L_{f}$, such that the
following estimates hold for all $t\in[\tau,T]$:

(6.2) $\displaystyle\sup_{\begin{subarray}{c}1\leqslant i,j\leqslant N\\\
j\neq i\end{subarray}}\lVert
D_{j}u^{i}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\rVert_{\infty}\leqslant\frac{K_{1}}{N};$
(6.3) $\displaystyle\sup_{1\leqslant i\leqslant N}\sum_{1\leqslant j\leqslant
N}\lVert
D(D_{i}u^{j})(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\rVert_{\infty}^{2}\leqslant
K_{2};$ (6.4) $\displaystyle\sup_{1\leqslant i\leqslant
N}\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}\big{|}D^{2}_{ij}u^{j}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}^{2}\bigg{\rVert}_{\infty}\leqslant\frac{K_{3}}{N};$
(6.5) $\displaystyle\sup_{1\leqslant i\leqslant
N}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}\lVert
D(D_{j}u^{i})(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\rVert_{\infty}^{2}\leqslant\frac{K_{4}}{N};$
(6.6) $\displaystyle\sum_{\begin{subarray}{c}1\leqslant i,j\leqslant N\\\
j\neq i\end{subarray}}\lVert
D(D_{ij}u^{j})(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\rVert_{\infty}^{2}\leqslant
K_{5}.$

More precisely, each $K_{i}$ is given by the constant $C_{i}$ appearing in
Propositions 4.6, 4.7, 4.8, 4.9 and 4.11 with $M$ replaced by $\kappa+\gamma$.

###### Proof.

As noted in Remark 3.3, we have $\mathcal{D}[u]\geqslant-(\kappa+\gamma)Q$ on
$(\tau,T]\times({\mathbb{R}}^{d})^{2N}$. Then the estimates follows from
Propositions 4.6, 4.7, 4.8, 4.9 and 4.11. ∎

With this piece of information one can show that the pair formed by
$\mathscr{L}$*semimonotonicity and diagonal lower bound (6.1) does not
degenerate, just like displacement semiconvexity did not (cf. Proposition
5.1).

######  Proposition 6.2 (Improvement of $\mathscr{L}$*monotonicity and
diagonal lower bound).

There exist nonnegative constants $\kappa_{f}^{*}$, $\kappa_{g}^{*}$,
$\kappa^{*}$ and $\gamma^{*}$ and a natural number $N_{*}$, all depending only
on $\sigma$, $T$, $L_{g}$ and $L_{f}$, such that $\kappa_{g}^{*}<\kappa^{*}$,
$L_{g}<\gamma^{*}$ and, for any $\tau\in[0,T)$, one has

$\begin{rcases}\mathcal{L}[u]\geqslant-\kappa^{*}Q&\text{on}\
(\tau,T]\times({\mathbb{R}}^{d})^{2N}\\\
\boldsymbol{\triangle}_{u}\geqslant-\gamma^{*}\boldsymbol{\mathsf{I}}&\text{on}\
(\tau,T]\times({\mathbb{R}}^{d})^{N}\\\
\mathcal{L}[f]\geqslant-\kappa_{f}^{*}Q&\text{on}\ ({\mathbb{R}}^{d})^{2N}\\\
\mathcal{L}[g]\geqslant-\kappa_{g}^{*}Q&\text{on}\ ({\mathbb{R}}^{d})^{2N}\\\
N\geqslant
N_{*}\end{rcases}\quad\implies\quad\begin{cases}\mathcal{L}[u(\tau,\mspace{1.8mu}\cdot\mspace{1.8mu})]\geqslant-\dfrac{1}{2}\kappa^{*}Q&\text{on}\
({\mathbb{R}}^{d})^{2N}\vspace{5pt}\\\
\boldsymbol{\triangle}_{u}(\tau,\mspace{1.8mu}\cdot\mspace{1.8mu})\geqslant-\dfrac{1}{2}\gamma^{*}\boldsymbol{\mathsf{I}}&\text{on}\
({\mathbb{R}}^{d})^{N}.\end{cases}$

We are going to split the proof of this result in two propositions. We will
prove that the $\mathscr{L}$*semimonotonicity improves, provided that one has
the diagonal lower bound (see Proposition 6.3). Then, symmetrically, that such
a lower bound improves, provided that one knows that $u$ is
$\mathscr{L}$*semimonotone (see Proposition 6.4).

######  Proposition 6.3 (Improvement of $\mathscr{L}$*semimonotonicity).

Let $\gamma>0$. Then there exist a nonnegative constant $\kappa^{*}$ depending
on $T$ (but not on $\gamma$), and constants $\kappa_{f}^{*}$, $\kappa_{g}^{*}$
(with $\kappa_{g}^{*}<\kappa$) and a natural number $N_{*}^{\prime}$, all
depending only on $\sigma$, $T$, $\gamma$, $L_{g}$ and $L_{f}$, such that for
any $\tau\in[0,T)$ one has

$\begin{rcases}\boldsymbol{\triangle}_{u(t,\cdot)}\geqslant-\gamma\boldsymbol{\mathsf{I}}&\text{on}\
(\tau,T]\times({\mathbb{R}}^{d})^{N}\\\
\mathcal{L}[u]\geqslant-\kappa^{*}Q&\text{on}\
(\tau,T]\times({\mathbb{R}}^{d})^{2N}\\\
\mathcal{L}[f]\geqslant-\kappa_{f}^{*}Q&\text{on}\ ({\mathbb{R}}^{d})^{2N}\\\
\mathcal{L}[g]\geqslant-\kappa_{g}^{*}Q&\text{on}\ ({\mathbb{R}}^{d})^{2N}\\\
N\geqslant
N_{*}^{\prime}\end{rcases}\quad\implies\quad\mathcal{L}[u(\tau,\mspace{1.8mu}\cdot\mspace{1.8mu})]\geqslant-\frac{1}{2}\kappa^{*}Q\
\ \text{on}\ ({\mathbb{R}}^{d})^{2N}.$

Looking at the proof below, one can in fact choose
$\kappa_{g}^{*}=(12Te^{2\gamma T+1})^{-1}$, $\kappa_{f}^{*}=(12T^{2}e^{2\gamma
T+1})^{-1}$ and $\kappa^{*}=(2T)^{-1}$, therefore $\kappa^{*}_{h}$ is actually
independent of $L_{g}$ and $L_{f}$.

###### Proof of Proposition 6.3.

Let
$w(t,\boldsymbol{x},\boldsymbol{y})\vcentcolon=\mathcal{L}[u(t,\mspace{1.8mu}\cdot\mspace{1.8mu})](\boldsymbol{x},\boldsymbol{y})$.
We want to use formula (5.2) where $\overline{t}=\tau$, $h=w$ and
$\boldsymbol{Z}=(\boldsymbol{X},\boldsymbol{Y})$ is a synchronous coupling of
two solutions to (1.2); that is, formula (5.2) with $L$ given by (5.4).
Omitting the dependence on $t$, we have

$Lw(\boldsymbol{x},\boldsymbol{y})=\operatorname{tr}\\!\big{(}(\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})\mathcal{L}[D^{2}u](\boldsymbol{x},\boldsymbol{y})\big{)}-\sum_{1\leqslant
j\leqslant N}\\!\big{(}D_{j}u^{j}(\boldsymbol{x})\cdot
D_{x^{j}}+D_{j}u^{j}(\boldsymbol{y})\cdot
D_{y^{j}}\big{)}\mathcal{L}[u](\boldsymbol{x},\boldsymbol{y}),$

and, according to (1.1),

$\begin{multlined}\partial_{t}w(\boldsymbol{x},\boldsymbol{y})=-\operatorname{tr}\\!\big{(}(\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})\mathcal{L}[D^{2}u](\boldsymbol{x},\boldsymbol{y})\big{)}+\frac{1}{2}\mathcal{L}[|D_{i}u^{i}|^{2}](\boldsymbol{x},\boldsymbol{y})\\\
+\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}\mathcal{L}[D_{j}u^{j}\cdot
D_{j}u](\boldsymbol{x},\boldsymbol{y})-\mathcal{L}[f](\boldsymbol{x},\boldsymbol{y}).\end{multlined}\partial_{t}w(\boldsymbol{x},\boldsymbol{y})=-\operatorname{tr}\\!\big{(}(\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})\mathcal{L}[D^{2}u](\boldsymbol{x},\boldsymbol{y})\big{)}+\frac{1}{2}\mathcal{L}[|D_{i}u^{i}|^{2}](\boldsymbol{x},\boldsymbol{y})\\\
+\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}\mathcal{L}[D_{j}u^{j}\cdot
D_{j}u](\boldsymbol{x},\boldsymbol{y})-\mathcal{L}[f](\boldsymbol{x},\boldsymbol{y}).$

Note that

$\sum_{1\leqslant j\leqslant N}\big{(}D_{j}u^{j}(\boldsymbol{x})\cdot
D_{x^{j}}+D_{j}u^{j}(\boldsymbol{y})\cdot
D_{y^{j}}\big{)}\mathcal{L}[u](\boldsymbol{x},\boldsymbol{y})\\\
=\sum_{1\leqslant i,j\leqslant
N}D_{j}u^{j}(\boldsymbol{x})D_{j}u^{i}(\boldsymbol{x})-\sum_{\begin{subarray}{c}1\leqslant
i,j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}u^{j}(\boldsymbol{x})D_{j}u^{i}(\boldsymbol{x}^{-i},y^{i})-\sum_{1\leqslant
i\leqslant
N}D_{i}u^{i}(\boldsymbol{x})D_{i}u^{i}(\boldsymbol{y}^{-i},x^{i})\\\
-\sum_{1\leqslant i\leqslant
N}D_{i}u^{i}(\boldsymbol{y})D_{i}u^{i}(\boldsymbol{x}^{-i},y^{i})-\sum_{\begin{subarray}{c}1\leqslant
i,j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}u^{j}(\boldsymbol{y})D_{j}u^{i}(\boldsymbol{y}^{-i},x^{i})+\sum_{1\leqslant
i,j\leqslant N}D_{j}u^{j}(\boldsymbol{y})D_{j}u^{i}(\boldsymbol{y})$

and

$\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}\mathcal{L}[D_{j}u^{j}\cdot
D_{j}u](\boldsymbol{x},\boldsymbol{y})\\\\[-6.0pt]
=\sum_{\begin{subarray}{c}1\leqslant i,j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}u^{j}(\boldsymbol{x})D_{j}u^{i}(\boldsymbol{x})-\sum_{\begin{subarray}{c}1\leqslant
i,j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}u^{j}(\boldsymbol{x}^{-i},y^{i})D_{j}u^{i}(\boldsymbol{x}^{-i},y^{i})\\\
\quad\ -\sum_{\begin{subarray}{c}1\leqslant i,j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}u^{j}(\boldsymbol{y}^{-i},x^{i})D_{j}u^{i}(\boldsymbol{y}^{-i},x^{i})+\sum_{\begin{subarray}{c}1\leqslant
i,j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}u^{j}(\boldsymbol{y})D_{j}u^{i}(\boldsymbol{y}),$

where all products between vectors are understood to be Euclidean inner
products. Then easy computations show that

$(\partial_{t}+L)w(\boldsymbol{x},\boldsymbol{y})\vspace{3pt}=\sum_{\begin{subarray}{c}1\leqslant
i,j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}u^{j}\big{|}_{(\boldsymbol{x}^{-i},y^{i})}^{\boldsymbol{x}}\cdot
D_{j}u^{i}(\boldsymbol{x}^{-i},y^{i})+\sum_{\begin{subarray}{c}1\leqslant
i,j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}u^{j}\big{|}_{(\boldsymbol{y}^{-i},x^{i})}^{\boldsymbol{y}}\cdot
D_{j}u^{i}(\boldsymbol{y}^{-i},x^{i})\\\ -\sum_{1\leqslant i\leqslant
N}\big{|}D_{i}u^{i}(\boldsymbol{x})-D_{i}u^{i}(\boldsymbol{y}^{-i},x^{i})\big{|}^{2}-\sum_{1\leqslant
i\leqslant
N}\big{|}D_{i}u^{i}(\boldsymbol{y})-D_{i}u^{i}(\boldsymbol{x}^{-i},y^{i})\big{|}^{2}-\mathcal{L}[f](\boldsymbol{x},\boldsymbol{y}).$

Exploiting the mean value theorem,

$\begin{multlined}\sum_{\begin{subarray}{c}1\leqslant i,j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}u^{j}\big{|}_{(\boldsymbol{x}^{-i},y^{i})}^{\boldsymbol{x}}\cdot
D_{j}u^{i}(\boldsymbol{x}^{-i},y^{i})+\sum_{\begin{subarray}{c}1\leqslant
i,j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}u^{j}\big{|}_{(\boldsymbol{y}^{-i},x^{i})}^{\boldsymbol{y}}\cdot
D_{j}u^{i}(\boldsymbol{y}^{-i},x^{i})\\\
\begin{split}&=\sum_{\begin{subarray}{c}1\leqslant i,j\leqslant N\\\ j\neq
i\end{subarray}}\big{(}D_{j}u^{j}\big{|}_{(\boldsymbol{x}^{-i},y^{i})}^{\boldsymbol{x}}+D_{j}u^{j}\big{|}_{(\boldsymbol{y}^{-i},x^{i})}^{\boldsymbol{y}}\big{)}\cdot
D_{j}u^{i}(\boldsymbol{x}^{-i},y^{i})+\sum_{\begin{subarray}{c}1\leqslant
i,j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}u^{j}\big{|}_{(\boldsymbol{y}^{-i},x^{i})}^{\boldsymbol{y}}\cdot
D_{j}u^{i}\big{|}_{(\boldsymbol{x}^{-i},y^{i})}^{(\boldsymbol{y}^{-i},x^{i})}\\\
&\begin{multlined}=\sum_{\begin{subarray}{c}1\leqslant i,k\leqslant N\\\ k\neq
i\end{subarray}}\int_{0}^{1}\int_{0}^{1}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}D^{3}_{ijk}u^{j}(\boldsymbol{z}^{-i}_{s^{\prime}},z^{i}_{s})D_{j}u^{i}(\boldsymbol{x}^{-i},y^{i})(x^{i}-y^{i})(x^{k}-y^{k})\,\mathrm{d}s\,\mathrm{d}s^{\prime}\\\\[-3.0pt]
+\sum_{1\leqslant i,k\leqslant
N}\int_{0}^{1}\int_{0}^{1}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\
j\neq
i\end{subarray}}D^{2}_{ij}u^{j}(\boldsymbol{y}^{-i},z^{i}_{s})D^{2}_{jk}u^{i}(\boldsymbol{z}^{-i}_{s},z^{i}_{1-s})(x^{i}-y^{i})(x^{k}-y^{k})\,\mathrm{d}s\,\mathrm{d}s^{\prime},\end{multlined}=\sum_{\begin{subarray}{c}1\leqslant
i,k\leqslant N\\\ k\neq
i\end{subarray}}\int_{0}^{1}\int_{0}^{1}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}D^{3}_{ijk}u^{j}(\boldsymbol{z}^{-i}_{s^{\prime}},z^{i}_{s})D_{j}u^{i}(\boldsymbol{x}^{-i},y^{i})(x^{i}-y^{i})(x^{k}-y^{k})\,\mathrm{d}s\,\mathrm{d}s^{\prime}\\\\[-3.0pt]
+\sum_{1\leqslant i,k\leqslant
N}\int_{0}^{1}\int_{0}^{1}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\
j\neq
i\end{subarray}}D^{2}_{ij}u^{j}(\boldsymbol{y}^{-i},z^{i}_{s})D^{2}_{jk}u^{i}(\boldsymbol{z}^{-i}_{s},z^{i}_{1-s})(x^{i}-y^{i})(x^{k}-y^{k})\,\mathrm{d}s\,\mathrm{d}s^{\prime},\\\\[-12.0pt]
\end{split}\end{multlined}\sum_{\begin{subarray}{c}1\leqslant i,j\leqslant
N\\\ j\neq
i\end{subarray}}D_{j}u^{j}\big{|}_{(\boldsymbol{x}^{-i},y^{i})}^{\boldsymbol{x}}\cdot
D_{j}u^{i}(\boldsymbol{x}^{-i},y^{i})+\sum_{\begin{subarray}{c}1\leqslant
i,j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}u^{j}\big{|}_{(\boldsymbol{y}^{-i},x^{i})}^{\boldsymbol{y}}\cdot
D_{j}u^{i}(\boldsymbol{y}^{-i},x^{i})\\\
\begin{split}&=\sum_{\begin{subarray}{c}1\leqslant i,j\leqslant N\\\ j\neq
i\end{subarray}}\big{(}D_{j}u^{j}\big{|}_{(\boldsymbol{x}^{-i},y^{i})}^{\boldsymbol{x}}+D_{j}u^{j}\big{|}_{(\boldsymbol{y}^{-i},x^{i})}^{\boldsymbol{y}}\big{)}\cdot
D_{j}u^{i}(\boldsymbol{x}^{-i},y^{i})+\sum_{\begin{subarray}{c}1\leqslant
i,j\leqslant N\\\ j\neq
i\end{subarray}}D_{j}u^{j}\big{|}_{(\boldsymbol{y}^{-i},x^{i})}^{\boldsymbol{y}}\cdot
D_{j}u^{i}\big{|}_{(\boldsymbol{x}^{-i},y^{i})}^{(\boldsymbol{y}^{-i},x^{i})}\\\
&\begin{multlined}=\sum_{\begin{subarray}{c}1\leqslant i,k\leqslant N\\\ k\neq
i\end{subarray}}\int_{0}^{1}\int_{0}^{1}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}D^{3}_{ijk}u^{j}(\boldsymbol{z}^{-i}_{s^{\prime}},z^{i}_{s})D_{j}u^{i}(\boldsymbol{x}^{-i},y^{i})(x^{i}-y^{i})(x^{k}-y^{k})\,\mathrm{d}s\,\mathrm{d}s^{\prime}\\\\[-3.0pt]
+\sum_{1\leqslant i,k\leqslant
N}\int_{0}^{1}\int_{0}^{1}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\
j\neq
i\end{subarray}}D^{2}_{ij}u^{j}(\boldsymbol{y}^{-i},z^{i}_{s})D^{2}_{jk}u^{i}(\boldsymbol{z}^{-i}_{s},z^{i}_{1-s})(x^{i}-y^{i})(x^{k}-y^{k})\,\mathrm{d}s\,\mathrm{d}s^{\prime},\end{multlined}=\sum_{\begin{subarray}{c}1\leqslant
i,k\leqslant N\\\ k\neq
i\end{subarray}}\int_{0}^{1}\int_{0}^{1}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}D^{3}_{ijk}u^{j}(\boldsymbol{z}^{-i}_{s^{\prime}},z^{i}_{s})D_{j}u^{i}(\boldsymbol{x}^{-i},y^{i})(x^{i}-y^{i})(x^{k}-y^{k})\,\mathrm{d}s\,\mathrm{d}s^{\prime}\\\\[-3.0pt]
+\sum_{1\leqslant i,k\leqslant
N}\int_{0}^{1}\int_{0}^{1}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\
j\neq
i\end{subarray}}D^{2}_{ij}u^{j}(\boldsymbol{y}^{-i},z^{i}_{s})D^{2}_{jk}u^{i}(\boldsymbol{z}^{-i}_{s},z^{i}_{1-s})(x^{i}-y^{i})(x^{k}-y^{k})\,\mathrm{d}s\,\mathrm{d}s^{\prime},\\\\[-12.0pt]
\end{split}$

where products between tensors are understood in the sense of tensor
contraction as discussed in the proof of Proposition 4.7. By the
Cauchy–Schwarz inequality and assumption (LS),

(6.7)
$(\partial_{t}+L)w(\boldsymbol{x},\boldsymbol{y})\leqslant\Bigg{(}{\kappa_{f}}+\bigg{(}\sum_{1\leqslant
i\leqslant N}\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant j,k\leqslant
N\\\ j,k\neq
i\end{subarray}}\big{|}D^{3}_{ijk}u^{j}\big{|}^{2}\bigg{\rVert}_{\infty}\bigg{\rVert}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}\big{|}D_{j}u^{i}\big{|}^{2}\bigg{\rVert}_{\infty}\bigg{)}^{\frac{1}{2}}\\\
+\bigg{(}\sum_{1\leqslant i\leqslant
N}\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}\big{|}D^{2}_{ij}u^{j}\big{|}^{2}\bigg{\rVert}_{\infty}\bigg{\rVert}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}\big{|}D(D_{j}u^{i})\big{|}^{2}\bigg{\rVert}_{\infty}\bigg{)}^{\frac{1}{2}}\Bigg{)}|\boldsymbol{x}-\boldsymbol{y}|^{2}.$

We can now proceed with the proof as we did for the one of Proposition 5.1. By
formula (5.2), estimate (6.7), Proposition 6.1 and estimate (5.6), we obtain

$w(\tau,\overline{\boldsymbol{x}},\overline{\boldsymbol{y}})\geqslant-e^{2(\kappa^{*}+\gamma)T}\Bigl{(}\kappa^{*}_{g}+T\kappa^{*}_{f}+\frac{K}{\sqrt{N}\,}\Bigr{)}|\overline{\boldsymbol{x}}-\overline{\boldsymbol{y}}|^{2},$

where the constant $K$ depend only on $\sigma$, $T$, $\kappa^{*}+\gamma$,
$L_{g}$ and $L_{f}$. Then we conclude by arguing as in the proof of
Proposition 5.1: pick $\kappa^{*}=(2T)^{-1}$, $\kappa^{*}_{g}$ and
$\kappa^{*}_{f}$ small enough and $N_{*}^{\prime}$ large enough so that

$\kappa^{*}_{g}+T\kappa^{*}_{f}+\frac{K}{\sqrt{N_{*}^{\prime}}\,}\leqslant\frac{\kappa^{*}}{2e^{2(\kappa^{*}+\gamma)T}}=\frac{1}{4Te^{2\gamma
T+1}}.\qed$

######  Proposition 6.4 (Improvement of the diagonal lower bound).

Let $\kappa>0$. Then there exists a constant $\gamma^{*}$ depending only on
$\sigma$, $T$, $L_{g}$ and $L_{f}$ (and not on $\kappa$) and a natural number
$N_{*}^{\prime\prime}$ depending only on $\sigma$, $T$, $\kappa$, $L_{g}$ and
$L_{f}$, such that for any $\tau\in[0,T)$ one has

$\begin{rcases}\mathcal{L}[u]\geqslant-\kappa Q&\text{on}\
(\tau,T]\times({\mathbb{R}}^{d})^{N}\\\
\boldsymbol{\triangle}_{u}\geqslant-\gamma^{*}\boldsymbol{\mathsf{I}}&\text{on}\
(\tau,T]\times({\mathbb{R}}^{d})^{N}\\\ N\geqslant
N_{*}^{\prime\prime}\end{rcases}\quad\implies\quad\boldsymbol{\triangle}_{u(\tau,\mspace{1.8mu}\cdot\mspace{1.8mu})}\geqslant-\frac{1}{2}\gamma^{*}\boldsymbol{\mathsf{I}}\
\ \text{on}\ ({\mathbb{R}}^{d})^{N}.$

The explicit constant $\gamma^{*}$ can be found below, see (6.12).

###### Proof of Proposition 6.4.

We are going to use an argument “along optimal trajectories”, but without
doubling variables/coupling. Consider $v=D_{i}u^{i}$; it satisfies the
${\mathbb{R}}^{d}$-valued equation444We point out that we use the natural
notation
$\operatorname{tr}((\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})D^{2}v)$
to denote the vector with coordinates  $\sum_{1\leqslant j,k\leqslant N,\
1\leqslant a,b\leqslant
d}(\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})_{dj+a-d,dk+b-d}D^{3}_{x^{kb}x^{ja}x^{i\ell}}v^{i},\quad\ell=1,\dots,d.$

(6.8)
$-\partial_{t}v-\operatorname{tr}((\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})D^{2}v)+\sum_{1\leqslant
j\leqslant N}D_{j}u^{j}\cdot
D_{j}v=D_{i}f^{i}-\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}D^{2}_{ij}u^{j}D_{j}u^{i}.$

Fix $\boldsymbol{x}\in({\mathbb{R}}^{d})^{N}$, $s\in[\tau,T)$ and let $\rho$
be a flow of probability measures on $({\mathbb{R}}^{d})^{N}$ solving

(6.9)
$\begin{cases}\partial_{t}\rho-\operatorname{tr}((\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})D^{2}\rho)+\mathrm{div}(B\rho)=0&\text{on}\
(s,T)\times({\mathbb{R}}^{d})^{N}\\\
\rho(s)=\delta_{\boldsymbol{x}},\end{cases}$

where $B=(-D_{j}u^{j})_{1\leqslant j\leqslant N}$. Duality between equations
(6.8) and (6.9) gives

$D_{i}u^{i}(s,\boldsymbol{x})=\int_{({\mathbb{R}}^{d})^{N}}D_{i}g^{i}\,\mathrm{d}\rho(T)+\int_{s}^{T}\int_{({\mathbb{R}}^{d})^{N}}D_{i}f^{i}\,\mathrm{d}\rho-\int_{s}^{T}\int_{({\mathbb{R}}^{d})^{N}}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}D^{2}_{ij}u^{j}D_{j}u^{i}\,\mathrm{d}\rho,$

hence by assumption (MF), the Cauchy–Schwarz inequality, estimates (6.2) and
(6.4) and the arbitrariness of $s\in[\tau,T)$,

(6.10) $\lVert D_{i}u^{i}(t,\cdot)\rVert_{\infty}\leqslant
L_{g}+TL_{f}+KN^{-1}\quad\forall\,t\in[\tau,T],$

where the constant $K$ depends only on $\sigma$, $T$, $\kappa$, $\gamma^{*}$
(that will be chosen below), $L_{g}$ and $L_{f}$.

Letting $w=\frac{1}{2}|D_{i}u^{i}|^{2}$, which satisfies

$\begin{multlined}-\partial_{t}w-\operatorname{tr}((\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})D^{2}w)+\sum_{1\leqslant
j\leqslant N}D_{j}u^{j}\cdot D_{j}w+\sigma\\!\sum_{1\leqslant j\leqslant
N}|D_{ij}^{2}u^{i}|^{2}+\beta\bigg{|}\sum_{1\leqslant j\leqslant
N}D_{ij}^{2}u^{i}\bigg{|}^{2}\\\\[2.0pt] =D_{i}u^{i}\cdot
D_{i}f^{i}-\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}D^{2}_{ij}u^{j}D_{j}u^{i}\cdot
D_{i}u^{i},\end{multlined}-\partial_{t}w-\operatorname{tr}((\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})D^{2}w)+\sum_{1\leqslant
j\leqslant N}D_{j}u^{j}\cdot D_{j}w+\sigma\\!\sum_{1\leqslant j\leqslant
N}|D_{ij}^{2}u^{i}|^{2}+\beta\bigg{|}\sum_{1\leqslant j\leqslant
N}D_{ij}^{2}u^{i}\bigg{|}^{2}\\\\[2.0pt] =D_{i}u^{i}\cdot
D_{i}f^{i}-\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}D^{2}_{ij}u^{j}D_{j}u^{i}\cdot D_{i}u^{i},$

again by duality with (6.9) with $s=\tau$ one obtains

$\begin{multlined}\frac{1}{2}|D_{i}u^{i}(\tau,\boldsymbol{x})|^{2}+\sigma\int_{\tau}^{T}\int_{({\mathbb{R}}^{d})^{N}}\sum_{1\leqslant
j\leqslant
N}|D^{2}_{ij}u^{i}|^{2}\,\mathrm{d}\rho+\beta\int_{\tau}^{T}\int_{({\mathbb{R}}^{d})^{N}}\bigg{|}\sum_{1\leqslant
j\leqslant N}D_{ij}^{2}u^{i}\bigg{|}^{2}\mathrm{d}\rho\\\
=\int_{({\mathbb{R}}^{d})^{N}}\frac{1}{2}|D_{i}g^{i}|^{2}\,\mathrm{d}\rho(T)+\int_{\tau}^{T}\int_{({\mathbb{R}}^{d})^{N}}D_{i}u^{i}\cdot
D_{i}f^{i}\,\mathrm{d}\rho-\int_{\tau}^{T}\int_{({\mathbb{R}}^{d})^{N}}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq i\end{subarray}}D^{2}_{ij}u^{j}D_{j}u^{i}\cdot
D_{i}u^{i}\,\mathrm{d}\rho.\end{multlined}\frac{1}{2}|D_{i}u^{i}(\tau,\boldsymbol{x})|^{2}+\sigma\int_{\tau}^{T}\int_{({\mathbb{R}}^{d})^{N}}\sum_{1\leqslant
j\leqslant
N}|D^{2}_{ij}u^{i}|^{2}\,\mathrm{d}\rho+\beta\int_{\tau}^{T}\int_{({\mathbb{R}}^{d})^{N}}\bigg{|}\sum_{1\leqslant
j\leqslant N}D_{ij}^{2}u^{i}\bigg{|}^{2}\mathrm{d}\rho\\\
=\int_{({\mathbb{R}}^{d})^{N}}\frac{1}{2}|D_{i}g^{i}|^{2}\,\mathrm{d}\rho(T)+\int_{\tau}^{T}\int_{({\mathbb{R}}^{d})^{N}}D_{i}u^{i}\cdot
D_{i}f^{i}\,\mathrm{d}\rho-\int_{\tau}^{T}\int_{({\mathbb{R}}^{d})^{N}}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq i\end{subarray}}D^{2}_{ij}u^{j}D_{j}u^{i}\cdot
D_{i}u^{i}\,\mathrm{d}\rho.$

Therefore, using assumption (MF), the Cauchy–Schwarz inequality and estimates
(6.2), (6.4) and (6.10), one obtains in particular

(6.11)
$\int_{\tau}^{T}\int_{({\mathbb{R}}^{d})^{N}}|D^{2}_{ii}u^{i}|^{2}\,\mathrm{d}\rho\leqslant\frac{L_{g}^{2}}{2\sigma}+\frac{TL_{f}}{\sigma}(L_{g}+TL_{f})+K^{\prime}N^{-1},$

where the constant $K^{\prime}$ depends only on $\sigma$, $T$, $\kappa$,
$\gamma^{*}$, $L_{g}$ and $L_{f}$.

As a last step, $\mathsf{V}=D_{ii}^{2}u^{i}$ solves the following
$\mathscr{S}(d)$-valued equation

$\begin{multlined}-\partial_{t}\mathsf{V}-\operatorname{tr}((\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})D^{2}\mathsf{V})+\sum_{1\leqslant
j\leqslant N}D_{j}u^{j}D_{j}\mathsf{V}+\mathsf{V}^{2}\\\
=D_{ii}^{2}f^{i}-\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}D_{ij}^{2}u^{i}D_{ji}^{2}u^{j}-\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}D_{ij}^{2}u^{j}D_{ji}^{2}u^{i}-\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}D_{iij}^{3}u^{j}D_{j}u^{i};\end{multlined}-\partial_{t}\mathsf{V}-\operatorname{tr}((\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})D^{2}\mathsf{V})+\sum_{1\leqslant
j\leqslant N}D_{j}u^{j}D_{j}\mathsf{V}+\mathsf{V}^{2}\\\
=D_{ii}^{2}f^{i}-\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}D_{ij}^{2}u^{i}D_{ji}^{2}u^{j}-\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}D_{ij}^{2}u^{j}D_{ji}^{2}u^{i}-\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq i\end{subarray}}D_{iij}^{3}u^{j}D_{j}u^{i};$

thus duality with equation (6.9) with $s=\tau$ gives

$\begin{multlined}D_{ii}^{2}u^{i}(\tau,\boldsymbol{x})+\int_{\tau}^{T}\int_{({\mathbb{R}}^{d})^{N}}(D^{2}_{ii}u^{i})^{2}\,\mathrm{d}\rho-\int_{({\mathbb{R}}^{d})^{N}}D_{ii}g^{i}\,\mathrm{d}\rho(T)-\int_{\tau}^{T}\int_{({\mathbb{R}}^{d})^{N}}D_{ii}^{2}f^{i}\,\mathrm{d}\rho\\\
=\int_{\tau}^{T}\int_{({\mathbb{R}}^{d})^{N}}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}\\!\big{(}D_{ij}^{2}u^{i}D_{ji}^{2}u^{j}+D_{ij}^{2}u^{j}D_{ji}^{2}u^{i}+D_{iij}^{3}u^{j}D_{j}u^{i}\big{)}\,\mathrm{d}\rho.\end{multlined}D_{ii}^{2}u^{i}(\tau,\boldsymbol{x})+\int_{\tau}^{T}\int_{({\mathbb{R}}^{d})^{N}}(D^{2}_{ii}u^{i})^{2}\,\mathrm{d}\rho-\int_{({\mathbb{R}}^{d})^{N}}D_{ii}g^{i}\,\mathrm{d}\rho(T)-\int_{\tau}^{T}\int_{({\mathbb{R}}^{d})^{N}}D_{ii}^{2}f^{i}\,\mathrm{d}\rho\\\
=\int_{\tau}^{T}\int_{({\mathbb{R}}^{d})^{N}}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}\\!\big{(}D_{ij}^{2}u^{i}D_{ji}^{2}u^{j}+D_{ij}^{2}u^{j}D_{ji}^{2}u^{i}+D_{iij}^{3}u^{j}D_{j}u^{i}\big{)}\,\mathrm{d}\rho.$

By assumption (MF), the Cauchy–Schwarz inequality, estimates in Proposition
6.1 and estimate (6.11) we get

$D_{ii}^{2}u^{i}(\tau,\boldsymbol{x})\geqslant-\Bigl{(}L_{g}+L_{f}+\frac{L_{g}^{2}}{2\sigma}+\frac{TL_{f}}{\sigma}(L_{g}+TL_{f})+\frac{K^{\prime}}{N}+\frac{K^{\prime\prime}}{\sqrt{N}\,}\Bigr{)}\mathsf{I}_{d},$

where the constant $K^{\prime\prime}$ depends only on $\sigma$, $T$, $\kappa$,
$\gamma^{*}$, $L_{g}$ and $L_{f}$. At this point one sees that if

(6.12)
$\gamma^{*}=4\Bigl{(}L_{g}+L_{f}+\frac{L_{g}^{2}}{2\sigma}+\frac{TL_{f}}{\sigma}(L_{g}+TL_{f})\Bigr{)}$

then for any $N$ large enough so that
$K^{\prime}N^{-1}+{K^{\prime\prime}}N^{-\frac{1}{2}}\leqslant\frac{1}{4}\gamma^{*}$
one has
$D_{ii}u^{i}(\tau,\boldsymbol{x})\geqslant-\frac{1}{2}\gamma^{*}\mathsf{I}_{d}$,
uniformly with respect to $i\in\\{1,\dots,N\\}$. The conclusion follows from
the arbitrariness of $\boldsymbol{x}\in({\mathbb{R}}^{d})^{N}$. ∎

For the benefit of the reader, we make explicit how Proposition 6.2 can be now
obtained.

###### Proof of Proposition 6.2.

Let $\kappa^{*}$ be as in Proposition 6.3 and $\gamma^{*}$ be as in
Proposition 6.4 (note that these two do not depend on each other). Then,
Proposition 6.3 applies with $\gamma=\gamma^{*}$, yielding the improvement of
$\mathscr{L}$*semimonotonicity for $N\geqslant N_{*}^{\prime}$. On the other
hand, Proposition 6.4 applies with $\kappa=\kappa^{*}$, so that the diagonal
lower bound improves for $N\geqslant N_{*}^{\prime\prime}$. Taking
$N_{*}=\max\\{N_{*}^{\prime},N_{*}^{\prime\prime}\\}$ gives the assertion. ∎

######  Theorem 6.5 (Estimates on the Nash system with
$\mathscr{L}$*semimonotone data).

Assume that $f$ and $g$ satisfy assumptions (MF) and (LS). Let $T>0$. There
exist nonnegative constants $\kappa_{f}^{*}$ and $\kappa_{g}^{*}$ and a
natural number $N^{*}$, all depending only on $\sigma$, $T$, $L_{f}$ and
$L_{g}$, such that if $\kappa_{g}\leqslant\kappa_{g}^{*}$,
$\kappa_{f}\leqslant\kappa_{f}^{*}$ and $N\geqslant N^{*}$ then any solution
$u$ to (1.1) on $[0,T]\times({\mathbb{R}}^{d})^{N}$ satisfies

$\sup_{1\leqslant i\leqslant N}\bigg{(}\sup_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq i\end{subarray}}\lVert
D_{j}u^{i}(t,\cdot)\rVert_{\infty}+\bigg{\lVert}\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq
i\end{subarray}}\big{|}D^{2}_{ij}u^{j}\big{|}^{2}\bigg{\rVert}_{\infty}+\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq i\end{subarray}}\lVert
D(D_{j}u^{i})\rVert_{\infty}^{2}\bigg{)}\lesssim\frac{1}{N}$

and

$\sup_{1\leqslant i\leqslant N}\lVert
Du^{i}\rVert_{\infty}^{2}+\sup_{1\leqslant i\leqslant N}\sum_{1\leqslant
j\leqslant N}\lVert
D(D_{i}u^{j})\rVert_{\infty}^{2}+\sum_{\begin{subarray}{c}1\leqslant
i,j\leqslant N\\\ j\neq i\end{subarray}}\lVert
D(D_{ij}u^{j})\rVert_{\infty}^{2}\lesssim 1,$

where the implied constants depend only on $\sigma$, $T$, $L_{f}$ and $L_{g}$.
In addition, there exists $\kappa^{*}>0$ (depending only on $T$) such that $u$
is $\kappa^{*}$-$\mathscr{L}$*semimonotone on $[0,T]$.

###### Proof.

The argument is very much analogous to the one of Theorem 5.2, so we omit it.
∎

For the benefit of the following discussion, we explicitly recall that the
estimates on the Nash system we obtained imply

(6.13a) $\displaystyle\sup_{1\leqslant i\leqslant N}\lVert
D_{i}u^{i}\rVert_{\infty}\lesssim 1,\qquad\sup_{1\leqslant i\leqslant
N}\sup_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}\lVert D_{j}u^{i}\rVert_{\infty}\lesssim\frac{1}{N},$ (6.13b)
$\displaystyle\sup_{1\leqslant i\leqslant N}\lVert
D^{2}_{ii}u^{i}\rVert_{\infty}\lesssim 1,\qquad\sup_{1\leqslant i\leqslant
N}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}\lVert D_{j}D_{i}u^{i}\rVert_{\infty}^{2}\lesssim\frac{1}{N}.$

##  7\. Analysis of the $N\to\infty$ limit

This section is devoted to the analysis of the convergence problem for
$u_{N}^{i}$ as $N\to\infty$, where the pedice $N$ is used to highlights the
dimension of the Nash system solved by $u^{i}$. Our main goal is to prove
Theorem 3.10. Recall that, besides the standing assumptions (MF) of Mean-Field
like interactions, and either $\mathscr{D}$*semimonotonicity (DS) or
$\mathscr{L}$*semimonotonicity (LS) that guarantee the estimates obtained in
the first part of the work, we are requiring now the symmetry assumption (S)
on the data $f^{i}$, $g^{i}$, which are moreover actually of the form
$f_{N}(\lambda_{N}^{i},\cdots)$, $g_{N}(\lambda^{i}_{N},\cdots)$ that are
Lipschitz continuous with respect to $\lambda\in\Lambda\vcentcolon=[0,1]$
(assumption (LP)). In this framework, we may focus our attention on
$\mathfrak{u}_{N}(\boldsymbol{\lambda},t,\boldsymbol{x})=u^{1}_{\boldsymbol{\lambda}}(t,\boldsymbol{x})$,
namely on the first solution of the Nash system with parameter vector
$\boldsymbol{\lambda}=(\lambda^{1},\ldots,\lambda^{N})$.

We begin by showing that assumption (LP) reflects in a Lipschitz dependence of
$\mathfrak{u}^{N}$ and $\partial_{x}\mathfrak{u}^{N}$ (recall that
$\partial_{x}\mathfrak{u}^{N}=D_{x^{1}}\mathfrak{u}^{N}$) with respect to
$\boldsymbol{\lambda}=(\lambda,\hat{\boldsymbol{\lambda}})\in\Lambda\times\Lambda^{N-1}=\Lambda^{N}$.

######  Lemma 7.1.

For any $\boldsymbol{\lambda},\boldsymbol{\lambda}^{\prime}\in\Lambda^{N}$,
one has

$\lVert\mathfrak{u}_{N}({\boldsymbol{\lambda}},\mspace{1.8mu}\cdot\mspace{1.8mu})-\mathfrak{u}_{N}({\boldsymbol{\lambda}^{\prime}},\mspace{1.8mu}\cdot\mspace{1.8mu})\rVert_{\infty}+\lVert\partial_{x}\mathfrak{u}_{N}({\boldsymbol{\lambda}},\mspace{1.8mu}\cdot\mspace{1.8mu})-\partial_{x}\mathfrak{u}_{N}({\boldsymbol{\lambda}^{\prime}},\mspace{1.8mu}\cdot\mspace{1.8mu})\rVert_{\infty}\lesssim|\lambda-\lambda^{\prime}|+\frac{|\hat{\boldsymbol{\lambda}}-\hat{\boldsymbol{\lambda}}{}^{\prime}|}{N},$

where the implied constant depends only on $T$, $L_{g}$, $L_{f}$ and
$L_{\Lambda}$.

###### Proof.

All implied constants appearing in this proof depend only on $T$, $L_{g}$ and
$L_{f}$. Let
$\boldsymbol{\lambda},\boldsymbol{\lambda}^{\prime}\in\Lambda^{N}$. Consider
$w^{i}\vcentcolon=u^{i}_{\boldsymbol{\lambda}}-u^{i}_{\boldsymbol{\lambda}^{\prime}}$
that solves the following equation in $(0,T)\times({\mathbb{R}}^{d})^{N}$:

(7.1) $\begin{split}-\partial_{t}w^{i}-\Delta w^{i}+\sum_{1\leqslant
j\leqslant N}D_{j}u^{j}_{\boldsymbol{\lambda}}\cdot
D_{j}w^{i}&=f_{N}\big{|}_{(\boldsymbol{\lambda}^{\prime},\mspace{1.8mu}\cdot\mspace{1.8mu})}^{(\boldsymbol{\lambda},\mspace{1.8mu}\cdot\mspace{1.8mu})}-\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq i\end{subarray}}D_{j}w^{j}\cdot
D_{j}u^{i}_{\boldsymbol{\lambda}^{\prime}}+\frac{1}{2}|D_{i}w^{i}|^{2}\\\
&\lesssim|\lambda^{i}-\lambda^{\prime
i}|+\frac{1}{N}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}|D_{j}w^{j}|+|D_{i}w^{i}|,\end{split}$

where the inequality is obtained by exploiting assumption (LP) and estimates
(6.13a). By Lemma 4.1 we deduce that, for any $\tau\in[0,T]$,

(7.2) $\sup_{t\in[\tau,T]}\lVert
D_{i}w^{i}(t,\cdot)\rVert_{\infty}\lesssim(T-\tau)^{\frac{1}{2}}\Bigl{(}|\lambda^{i}-\lambda^{\prime
i}|+\frac{1}{N}\sum_{\begin{subarray}{c}1\leqslant j\leqslant N\\\ j\neq
i\end{subarray}}\sup_{t\in[\tau,T]}\lVert
D_{j}w^{j}(t,\cdot)\rVert_{\infty}+\sup_{t\in[\tau,T]}\lVert
D_{i}w^{i}(t,\cdot)\rVert_{\infty}\Bigr{)}.$

Since by summing over $i$ we get

$\sum_{1\leqslant i\leqslant N}\sup_{t\in[\tau,T]}\lVert
D_{i}w^{i}(t,\cdot)\rVert_{\infty}\lesssim(T-\tau)^{\frac{1}{2}}\Bigl{(}|\boldsymbol{\lambda}-\boldsymbol{\lambda}^{\prime}|+\sum_{1\leqslant
i\leqslant N}\sup_{t\in[\tau,T]}\lVert
D_{i}w^{i}(t,\cdot)\rVert_{\infty}\Bigr{)},$

possibly by iterating this estimate a finite number of times we obtain

$\sum_{1\leqslant i\leqslant N}\sup_{t\in[0,T]}\lVert
D_{i}w^{i}(t,\cdot)\rVert_{\infty}\lesssim|\boldsymbol{\lambda}-\boldsymbol{\lambda}^{\prime}|;$

then plugging this back into (7.2) (and again possibly iterating the estimate)
we have

$\sup_{t\in[0,T]}\lVert
D_{i}w^{i}(t,\cdot)\rVert_{\infty}\lesssim|\lambda^{i}-\lambda^{\prime
i}|+\frac{|\boldsymbol{\lambda}-\boldsymbol{\lambda}^{\prime}|}{N},$

which is equivalent to
$\lVert\partial_{x}\mathfrak{u}_{N}({\boldsymbol{\lambda}},\mspace{1.8mu}\cdot\mspace{1.8mu})-\partial_{x}\mathfrak{u}_{N}({\overline{\boldsymbol{\lambda}}},\mspace{1.8mu}\cdot\mspace{1.8mu})\rVert_{\infty}\lesssim|\lambda-\lambda^{\prime}|+N^{-1}|\boldsymbol{\lambda}-\hat{\boldsymbol{\lambda}}{}^{\prime}|$.
Now by the maximum principle one deduces that $\lVert
w^{1}\rVert_{\infty}\lesssim|\lambda-\lambda^{\prime}|+N^{-1}|\boldsymbol{\lambda}-\hat{\boldsymbol{\lambda}}{}^{\prime}|$
as well, thus concluding the proof. ∎

######  Remark 7.2.

As a consequence of Lemma 7.1, by Rademacher’s theorem
$\boldsymbol{\lambda}\mapsto\partial_{x}\mathfrak{u}^{N}(\boldsymbol{\lambda},t,\boldsymbol{x})$
is $\mathscr{L}^{N}$-a.e. differentiable on $\Lambda^{N}$ for each
$(t,\boldsymbol{x})\in[0,T]\times({\mathbb{R}}^{d})^{N}$, and we have
$\sup_{1<j\leqslant
N}\lVert\partial_{\lambda^{j}}\partial_{x}\mathfrak{u}\rVert_{\infty}\lesssim
N^{-1}$, where the implied constant depends only on $T$, $L_{g}$, $L_{f}$ and
$L_{\Lambda}$. The same can be said for $\mathfrak{u}$ (clearly, with
differentiability everywhere).

The following result will be instrumental in our study of the limit of the
(solution to the) Nash system as $N\to\infty$; it adapts [12, Theorem 2.1].

######  Lemma 7.3.

Let $(\Gamma,d)$ be a metric space and let
$(E,\lVert\mspace{1.8mu}\cdot\mspace{1.8mu}\rVert)$ be a finite-dimensional
normed vector space. Let $(h_{N})_{N\in\mathbb{N}}$ be a sequence of functions
such that $h_{N}\colon\Gamma\times E^{N}\to{\mathbb{R}}$ is symmetric on
$E^{N}$ and there exist $C>0$ and concave moduli of continuity
$\omega_{\Gamma}$, $\omega_{p}$, all independent of $N$, such that $\lVert
h_{N}\rVert_{\infty}\leqslant C$ and, for some $p\geqslant 1$,

(7.3)
$|h_{N}(\gamma,\boldsymbol{v})-h_{N}(\gamma^{\prime},\boldsymbol{v}^{\prime})|\leqslant\omega_{\Gamma}(d(\gamma,\gamma^{\prime}))+\omega_{p}(W_{p}(m_{\boldsymbol{v}},m_{\boldsymbol{v}^{\prime}}))\quad\forall\,\gamma,\gamma^{\prime}\in\Gamma,\
\forall\,\boldsymbol{v},\boldsymbol{v}^{\prime}\in E^{N}.$

Then there exists $h\in C^{0}(\Gamma\times\mathcal{P}_{p}(E))$ such that, up
to a subsequence, for any compact sets $H\subset\Gamma$ and
$\mathcal{K}\subset\mathcal{P}_{p}(E)$,

$\lim_{N\to\infty}\sup_{\gamma\in H}\sup_{\boldsymbol{v}\in
E^{N}:\,m_{\boldsymbol{v}}\in\mathcal{K}}|h_{N}(\gamma,\boldsymbol{v})-h(\gamma,m_{\boldsymbol{v}})|=0.$

Furthermore, $h$ has the same moduli of continuity with respect to
$\gamma\in\Gamma$ and $m\in\mathcal{P}_{p}(E)$.

###### Proof.

Define $\mathfrak{h}_{N}\colon\Gamma\times\mathcal{P}_{p}(E)\to{\mathbb{R}}$
by

$\mathfrak{h}_{N}(\gamma,m)\vcentcolon=\inf_{\boldsymbol{v}\in
E^{N}}\big{\\{}h_{N}(\gamma,\boldsymbol{v})+\omega_{p}(W_{p}(m_{\boldsymbol{v}},m))\big{\\}}.$

This is well-defined since $h_{N}$ is bounded. By (7.3),
$\mathfrak{h}_{N}(\gamma,m_{\boldsymbol{v}})=h_{N}(\gamma,\boldsymbol{v})$.
Given $\gamma,\gamma^{\prime}\in\Gamma$, $m,m^{\prime}\in\mathcal{P}_{p}(E)$
and $\boldsymbol{v}\in E^{N}$ which is $\varepsilon$-optimal in the definition
of $\mathfrak{h}_{N}(\gamma^{\prime},m^{\prime})$,

$\begin{split}\mathfrak{h}_{N}(\gamma,m)&\leqslant
h_{N}(\gamma,\boldsymbol{v})-\omega_{p}(W_{p}(m_{\boldsymbol{v}},m))\\\
&\leqslant\mathfrak{h}_{N}(\gamma^{\prime},m^{\prime})+\omega_{\Gamma}(d(\gamma,\gamma^{\prime}))+\omega_{p}(W_{p}(m_{\boldsymbol{v}},m))-\omega_{p}(W_{p}(m_{\boldsymbol{v}},m^{\prime}))+\varepsilon\\\
&\leqslant\mathfrak{h}_{N}(\gamma^{\prime},m^{\prime})+\omega_{\Gamma}(d(\gamma,\gamma^{\prime}))+\omega_{p}(W_{p}(m,m^{\prime}))+\varepsilon.\end{split}$

We have used that $\omega_{p}$ is subadditive because it is concave and
$\omega_{p}(0)=0$. Therefore, $(\mathfrak{h}_{N})_{N\in\mathbb{N}}$ is a
sequence of equicontinuous and equibounded functions, so a generalisation of
the Ascoli–Arzelà theorem guarantees the existence of $h\in
C^{0}(\Gamma\times\mathcal{P}_{p}(E))$ such that, up to a subsequence,
$\mathfrak{h}_{N}\to h$ in the topology of compact convergence, that is our
assertion. ∎

The importance of this lemma lies in the fact mentioned in [12, Remark 2.3]
that if $E={\mathbb{R}}^{n}$ and $\sup_{1\leqslant i\leqslant n}\lVert
D_{i}h\rVert_{\infty}\lesssim N^{-1}$, then $\boldsymbol{x}\mapsto
h(\gamma,\boldsymbol{x})$ is Lipschitz continuous with respect to the
$1$-Wasserstein distance of the empirical measures in the sense of (7.3).
Hence we will be able to apply this lemma to $f_{N}$ and $g_{N}$ thanks to the
symmetry (S) and assumptions (MF)–(LP) as well as to $\mathfrak{u}$ and
$\partial_{x}\mathfrak{u}$ thanks to Remark 7.2 and our estimates on the
derivatives.

We formalise a generalisation of the above-mentioned fact in the following
lemma, where we denote by $|\cdot|_{p}$ the standard $p$-norm on
$({\mathbb{R}}^{d})^{N}$.

######  Lemma 7.4.

Let $h\colon({\mathbb{R}}^{d})^{N}\to{\mathbb{R}}$ be symmetric and such that
$\lVert|Dh|_{q}\rVert_{\infty}\leqslant CN^{-\frac{1}{p}}$ with
$\frac{1}{p}+\frac{1}{q}=1$, for some constants $C>0$ and $q\in(1,\infty]$
independent of $N$. Then $|h(\boldsymbol{x})-h(\boldsymbol{y})|\leqslant
CW_{p}(m_{\boldsymbol{x}},m_{\boldsymbol{y}})$ for all
$\boldsymbol{x},\boldsymbol{y}\in({\mathbb{R}}^{d})^{N}$.

###### Proof.

Recall that
$W_{p}(m_{\boldsymbol{x}},m_{\boldsymbol{y}})=N^{-\frac{1}{p}}|\boldsymbol{x}-\boldsymbol{y}^{\sigma}|_{p}$
for some permutation $\sigma$ on $\\{1,\dots,N\\}$. Then by the symmetry of
$h$ and Hölder’s inequality,
$|h(\boldsymbol{x})-h(\boldsymbol{y})|\leqslant\lVert|Dh|_{q}\rVert_{\infty}|\boldsymbol{x}-\boldsymbol{y}^{\sigma}|_{p}$.
The conclusion follows. ∎

We are now ready to prove the existence of a limit of $\mathfrak{u}_{N}$ as
$N\to\infty$.

######  Theorem 7.5.

There exists a map
$U\colon\Lambda\times[0,T]\times{\mathbb{R}}^{d}\times\mathcal{P}_{2}(\Lambda\times{\mathbb{R}}^{d})\to{\mathbb{R}}$
with bounded derivative $\partial_{x}U$ such that $U,\partial_{x}U$ are
Lipschitz continuous on
$\Lambda\times{\mathbb{R}}^{d}\times\mathcal{P}_{2}(\Lambda\times{\mathbb{R}}^{d})$
and $\frac{1}{3}$-Hölder continuous on $[0,T]$, and, up to a subsequence,

(7.4)
$\sup|\partial_{x}^{k}\mathfrak{u}_{N}({\boldsymbol{\lambda}},t,\boldsymbol{x})-\partial_{x}^{k}U(\lambda,t,x,m_{(\hat{\boldsymbol{\lambda}}:\hat{\boldsymbol{x}})})|\,\xrightarrow[N\to\infty]{}\,0,\quad
k\in\\{0,1\\},$

whenever the supremum is taken over any set of the form

$\bigl{\\{}(\boldsymbol{\lambda},t,\boldsymbol{x})\in\Lambda^{N}\times[0,T]\times({\mathbb{R}}^{d})^{N}:\
|x|\leqslant R,\ m_{\hat{\boldsymbol{x}}}\in\mathcal{K}\bigr{\\}}$

for some $R>0$ and some compact set
$\mathcal{K}\subset\mathcal{P}_{2}({\mathbb{R}}^{d})$.

###### Proof.

For the local uniform convergence of a subsequence $\mathfrak{u}_{N}$, we
apply Lemma 7.3 with $\gamma=(t,\lambda,x)$ and
$\boldsymbol{v}=(\hat{\boldsymbol{\lambda}}:\hat{\boldsymbol{x}})$. Indeed,
first note that the $\mathfrak{u}_{N}$ are globally bounded, uniformly in $N$,
by a standard comparison argument (recall that $f^{i}$ and $g^{i}$ are now
assumed to be bounded uniformly). The second inequality in (6.13a) and the
estimate of Lemma 7.1 yield uniform Lipschitz continuity with respect to
$(\hat{\boldsymbol{\lambda}}:\hat{\boldsymbol{x}})$ in $W_{2}$ distance, in
the sense provided by Lemma 7.4. Lipschitz continuity in $(\lambda,x)$ is also
a consequence of Lemma 7.1, and the first estimate in (6.13a). Finally, since
the right-hand side of the equation for $\mathfrak{u}_{N}$

$-\partial_{t}\mathfrak{u}_{N}-\Delta\mathfrak{u}_{N}=-\dfrac{1}{2}|\partial_{x}\mathfrak{u}_{N}|^{2}-\displaystyle\sum_{2\leqslant
j\leqslant N}D_{j}u^{j}\cdot D_{j}\mathfrak{u}_{N}+f_{N}$

is uniformly bounded by (6.13a), and $\mathfrak{u}_{N}$ is uniformly Lipschitz
in the $\boldsymbol{x}$-variable with respect to the weighted norm
$\|\cdot\|^{1}$, again by (6.13a), the desired uniform Hölder regularity in
$t$ follows by Lemma B.1.

The local uniform convergence of $\partial_{x}\mathfrak{u}_{N}$ is analogous.
We argue as before, employing estimates in (6.13b) instead of (6.13a). Time
regularity is obtained from the equation

$-\partial_{t}(\partial_{x}\mathfrak{u}_{N})-\Delta(\partial_{x}\mathfrak{u}_{N})=-\displaystyle\sum_{1\leqslant
j\leqslant N}D_{j}u^{j}\cdot
D_{j}(\partial_{x}\mathfrak{u}_{N})-\displaystyle\sum_{2\leqslant j\leqslant
N}(D^{2}_{j1}u^{j})\cdot D_{j}\mathfrak{u}_{N}+\partial_{x}f_{N},$

whose right-hand side can be controlled using (6.13a), (6.13b). ∎

######  Remark 7.6.

Regularity of $U$ is actually a bit better. First, (6.13a) yields Lipschitz
continuity in the measure variable with respect to the $W_{1}$ distance.
Moreover, looking at the equation for $\mathfrak{u}_{N}$ one observes that
$\lVert\partial_{t}\mathfrak{u}_{N}\rVert_{\infty}\lesssim 1$, that gives
uniform Lipschitz continuity in the variable $t$.

######  Remark 7.7.

In Example 3.11 we discussed the implications of our assumption when taking
the data $h^{N}$ (that is, $f^{N}$ and $g^{N}$) as projections over empirical
measures of given
$\mathfrak{h}_{N}:\Lambda\times{\mathbb{R}}^{d}\times\mathcal{P}_{p}({\mathbb{R}}^{d})\to{\mathbb{R}}$,
and in Remark 3.8 we argued that this can be considered a natural choice.

Furthermore, our standing assumptions force in fact also the convergence of
$h_{N}$ to functions defined on probability measures, at least up to
subsequences. Indeed, assumption (MF) guarantees that $\lVert
D_{j}h_{N}(\lambda,\mspace{1.8mu}\cdot\mspace{1.8mu})\rVert_{\infty}\leqslant
L_{h}N^{-1}$ for all $j>1$. Then by Lemma 7.4 we can apply Lemma 7.3 and say
that there exists a map
$h\in\operatorname{Lip}(\Lambda\times{\mathbb{R}}^{d}\times\mathcal{P}_{1}({\mathbb{R}}^{d}))$
such that, up to a subsequence,

(7.5) $\sup_{\lambda\in\Lambda,\,|x|\leqslant
R,\,m_{\hat{\boldsymbol{x}}}\in\mathcal{K}}|h_{N}(\lambda,\boldsymbol{x})-h(\lambda,x,m_{\hat{\boldsymbol{x}}})|\,\xrightarrow[N\to\infty]{}\,0,$

for any $R>0$ and $\mathcal{K}\subset\mathcal{P}_{1}({\mathbb{R}}^{d})$
compact.

###  7.1. Characterisation of the limit

Henceforth, we will work with the three limit functions $f$, $g$ and $U$
defined in Remark 7.7 and Theorem 7.5. We will adopt the notation
$f^{\lambda}=f(\lambda,\mspace{1.8mu}\cdot\mspace{1.8mu})$ and analogously for
$g^{\lambda}$ and $U^{\lambda}$.

Our aim is to give a characterisation of $U$. As a first step and a starter,
the most basic result we can prove is an application of propagation of chaos
in the standard symmetric setting of MFGs; that is, when
$\boldsymbol{\lambda}=\lambda 1_{N}$ for some $\lambda\in\Lambda$, so that all
$f^{i}$ (and all $g^{i}$) coincide up to a permutation of the coordinates of
$({\mathbb{R}}^{d})^{N}$.

######  Proposition 7.8.

For all $\tau\in[0,T]$,
$\mathfrak{m},\overline{\mathfrak{m}}\in\mathcal{P}_{2}({\mathbb{R}}^{d})$ and
$\lambda\in\Lambda$,

(7.6)
$\begin{multlined}\int_{{\mathbb{R}}^{d}}U^{\lambda}(\tau,\mspace{1.8mu}\cdot\mspace{1.8mu},\delta_{\lambda}\\!\otimes\mathfrak{m})\,\mathrm{d}\overline{\mathfrak{m}}\vspace{-7pt}\\\
=\int_{\tau}^{T}\int_{{\mathbb{R}}^{d}}\Big{(}\,\frac{1}{2}\big{|}\partial_{x}U^{\lambda}(s,\mspace{1.8mu}\cdot\mspace{1.8mu},\delta_{\lambda}\\!\otimes
m_{s})\big{|}^{2}+f^{\lambda}(\mspace{1.8mu}\cdot\mspace{1.8mu},m_{s})\Big{)}\mathrm{d}\overline{m}_{s}\mathrm{d}s+\int_{{\mathbb{R}}^{d}}g^{\lambda}(\mspace{1.8mu}\cdot\mspace{1.8mu},m_{T})\,\mathrm{d}\overline{m}_{T},\end{multlined}\int_{{\mathbb{R}}^{d}}U^{\lambda}(\tau,\mspace{1.8mu}\cdot\mspace{1.8mu},\delta_{\lambda}\\!\otimes\mathfrak{m})\,\mathrm{d}\overline{\mathfrak{m}}\vspace{-7pt}\\\
=\int_{\tau}^{T}\int_{{\mathbb{R}}^{d}}\Big{(}\,\frac{1}{2}\big{|}\partial_{x}U^{\lambda}(s,\mspace{1.8mu}\cdot\mspace{1.8mu},\delta_{\lambda}\\!\otimes
m_{s})\big{|}^{2}+f^{\lambda}(\mspace{1.8mu}\cdot\mspace{1.8mu},m_{s})\Big{)}\mathrm{d}\overline{m}_{s}\mathrm{d}s+\int_{{\mathbb{R}}^{d}}g^{\lambda}(\mspace{1.8mu}\cdot\mspace{1.8mu},m_{T})\,\mathrm{d}\overline{m}_{T},$

where $m$ and $\overline{m}$ solve, respectively,

$\begin{cases}\partial_{t}m-\Delta
m-\mathrm{div}(\partial_{x}U^{\lambda}(t,x,\delta_{\lambda}\\!\otimes
m)m)=0\\\ m|_{t=\tau}=\mathfrak{m}\end{cases}$

and

$\begin{cases}\partial_{t}\overline{m}-\Delta\overline{m}-\mathrm{div}(\partial_{x}U^{\lambda}(t,x,\delta_{\lambda}\\!\otimes
m)\overline{m})=0\\\
\overline{m}|_{t=\tau}=\overline{\mathfrak{m}}.\end{cases}$

for $t\in[\tau,T]$ and $x\in{\mathbb{R}}^{d}$.

###### Proof.

In order to abridge the notation, given
$\mathfrak{m}\in\mathcal{P}_{2}({\mathbb{R}}^{d})$ we will write
$U^{\lambda}(t,x,\mathfrak{m})$ in lieu of
$U^{\lambda}(t,x,\delta_{\lambda}\\!\otimes\mathfrak{m})$. Also, throughout
the proof we will imply all constants which depend only on $T$, $L_{g}$,
$L_{f}$ and $L_{\Lambda}$.

Fix $\lambda\in\Lambda$,
$\overline{m},m_{0}\in\mathcal{P}_{q}({\mathbb{R}}^{d})$, $q>4$, and let
$\boldsymbol{Z}\sim\overline{m}\otimes m_{0}^{\otimes(N-1)}$; then consider
$\boldsymbol{X}^{N}$ and $\boldsymbol{Y}^{N}$ solving, respectively,

$\begin{cases}\mathrm{d}X^{N,i}=-\partial_{x}U^{\lambda}(t,X^{N,i}_{t},m_{\boldsymbol{X}^{N}_{t}})\,\mathrm{d}t+\sqrt{2}\,\mathrm{d}B^{i}_{t},&i\in\\{1,\dots,N\\}\\\
\boldsymbol{X}_{0}^{N}=\boldsymbol{Z}\end{cases}$

and

$\begin{cases}\mathrm{d}Y^{N,i}=-\partial_{x}U^{\lambda}(t,Y^{N,i}_{t},\mathcal{L}_{Y^{N,i}_{t}})\,\mathrm{d}t+\sqrt{2}\,\mathrm{d}B^{i}_{t},&i\in\\{2,\dots,N\\}\\\
\mathrm{d}Y^{N,1}=-\partial_{x}U^{\lambda}(t,Y^{N,1}_{t},\mathcal{L}_{Y^{N,2}_{t}})\,\mathrm{d}t+\sqrt{2}\,\mathrm{d}B^{1}_{t}\\\
\boldsymbol{Y}^{N}_{0}=\boldsymbol{Z}.\end{cases}$

Note that $\mathcal{L}_{Y^{N,i}_{t}}$ is independent of $i\geqslant 2$, so let
$m_{t}\vcentcolon=\mathcal{L}_{Y^{N,i}_{t}}$ if $i\geqslant 2$ and
$\overline{m}_{t}\vcentcolon=\mathcal{L}_{Y^{N,1}_{t}}$. We now divide the
proof into three steps.

Step 1. We prove that

(7.7)
$\begin{multlined}\mathbb{E}\bigg{[}\int_{0}^{T}\Big{(}\,\frac{1}{2}\big{|}\partial_{x}U^{\lambda}(t,X^{N,1}_{t},m_{\boldsymbol{X}^{N}_{t}})\big{|}^{2}+f^{\lambda}(X^{N,1}_{t},m_{\boldsymbol{X}^{N}_{t}})\Big{)}\mathrm{d}t+g^{\lambda}(X^{N,1}_{T},m_{\boldsymbol{X}^{N}_{T}})\bigg{]}\vspace{-7pt}\\\
\xrightarrow[N\to\infty]{}\,\int_{0}^{T}\int_{{\mathbb{R}}^{d}}\Big{(}\,\frac{1}{2}\big{|}\partial_{x}U^{\lambda}(t,\cdot,m_{t})\big{|}^{2}+f^{\lambda}(\cdot,m_{t})\Big{)}\mathrm{d}\overline{m}_{t}\mathrm{d}t+\int_{{\mathbb{R}}^{d}}g^{\lambda}(\cdot,m_{T})\,\mathrm{d}\overline{m}_{T}.\end{multlined}\mathbb{E}\bigg{[}\int_{0}^{T}\Big{(}\,\frac{1}{2}\big{|}\partial_{x}U^{\lambda}(t,X^{N,1}_{t},m_{\boldsymbol{X}^{N}_{t}})\big{|}^{2}+f^{\lambda}(X^{N,1}_{t},m_{\boldsymbol{X}^{N}_{t}})\Big{)}\mathrm{d}t+g^{\lambda}(X^{N,1}_{T},m_{\boldsymbol{X}^{N}_{T}})\bigg{]}\vspace{-7pt}\\\
\xrightarrow[N\to\infty]{}\,\int_{0}^{T}\int_{{\mathbb{R}}^{d}}\Big{(}\,\frac{1}{2}\big{|}\partial_{x}U^{\lambda}(t,\cdot,m_{t})\big{|}^{2}+f^{\lambda}(\cdot,m_{t})\Big{)}\mathrm{d}\overline{m}_{t}\mathrm{d}t+\int_{{\mathbb{R}}^{d}}g^{\lambda}(\cdot,m_{T})\,\mathrm{d}\overline{m}_{T}.$

Subtracting the equations of $\boldsymbol{X}^{N}$ and $\boldsymbol{Y}^{N}$ and
using the Lipschitz continuity of $\partial_{x}U$, we have

$|X^{N,i}_{t}-Y^{N,i}_{t}|\lesssim\int_{0}^{t}\bigl{(}|X^{N,i}-Y^{N,i}|+W_{2}(m_{\boldsymbol{X}^{N}},m_{\boldsymbol{Y}^{N}})+W_{2}(m_{\boldsymbol{Y}^{N}},m)\bigr{)};$

therefore, since due to the boundedness of $\partial_{x}U$ we have555We are
using the notation $m_{\boldsymbol{Y}^{N,-1}}=\frac{1}{N-1}\sum_{1\leqslant
j\leqslant N,\,j\neq 1}\delta_{Y^{N,j}}$, which is coherent to that given in
Section 2; note that in the literature the same empirical measure is often
denoted by $m^{N,1}_{\boldsymbol{Y}}$.
$\mathbb{E}\bigl{[}\sup_{t\in[0,T]}W_{2}(m_{\boldsymbol{Y}^{N,-1}_{t}},m_{\boldsymbol{Y}^{N}_{t}})^{2}\bigr{]}\lesssim
N^{-1}$,

(7.8)
$|X^{N,i}_{t}-Y^{N,i}_{t}|^{2}\lesssim\int_{0}^{t}\Bigl{(}|X^{N,i}-Y^{N,i}|^{2}+\frac{1}{N}\sum_{1\leqslant
j\leqslant
N}|X^{N,j}-Y^{N,j}|^{2}+W_{2}(m_{\boldsymbol{Y}^{N,-1}},m)\Bigr{)}+\frac{1}{N}.$

Averaging over $i$,

$\frac{1}{N}\sum_{1\leqslant i\leqslant
N}|X^{N,i}_{t}-Y^{N,i}_{t}|^{2}\,\lesssim\,\frac{1}{N}+\int_{0}^{t}\Bigl{(}\frac{1}{N}\sum_{1\leqslant
i\leqslant
N}|X^{N,i}-Y^{N,i}|^{2}+W_{2}(m_{\boldsymbol{Y}^{N,-1}},m)\Bigr{)},$

thus by Gronwall’s lemma

$\frac{1}{N}\sum_{1\leqslant i\leqslant
N}|X^{N,i}_{t}-Y^{N,i}_{t}|^{2}\lesssim\frac{1}{N}+\int_{0}^{t}W_{2}(m_{\boldsymbol{Y}^{N,-1}},m)$

and plugging this back into (7.8) one concludes that

(7.9) $\sup_{1\leqslant i\leqslant
N}\mathbb{E}\Big{[}\sup_{t\in[0,T]}\big{|}X^{N,i}_{t}-Y^{N,i}_{t}\big{|}^{2}\Big{]}\lesssim\frac{1}{N}+\int_{0}^{T}\mathbb{E}\big{[}W_{2}(m_{\boldsymbol{Y}^{N,-1}},m)^{2}\big{]},$

where the right-hand side goes to $0$ as $N\to\infty$ by [16, Theorem 5.8]. We
also have

$\begin{multlined}\bigg{|}\big{|}\partial_{x}U^{\lambda}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}^{2}\Big{|}_{(X^{N,i}_{t},m_{\boldsymbol{X}^{N}_{t}})}^{(Y^{N,i}_{t},m_{t})}\bigg{|}+\bigg{|}f^{\lambda}\Big{|}_{(X^{N,i}_{t},m_{\boldsymbol{X}^{N}_{t}})}^{(Y^{N,i}_{t},m_{t})}\bigg{|}+\bigg{|}g^{\lambda}\Big{|}_{(X^{N,i}_{t},m_{\boldsymbol{X}^{N}_{t}})}^{(Y^{N,i}_{t},m_{t})}\bigg{|}\vspace{3pt}\\\
\lesssim\big{|}X^{N,i}_{t}-Y^{N,i}_{t}\big{|}+W_{2}(m_{\boldsymbol{X}^{N}_{t}},m_{t}),\end{multlined}\bigg{|}\big{|}\partial_{x}U^{\lambda}(t,\mspace{1.8mu}\cdot\mspace{1.8mu})\big{|}^{2}\Big{|}_{(X^{N,i}_{t},m_{\boldsymbol{X}^{N}_{t}})}^{(Y^{N,i}_{t},m_{t})}\bigg{|}+\bigg{|}f^{\lambda}\Big{|}_{(X^{N,i}_{t},m_{\boldsymbol{X}^{N}_{t}})}^{(Y^{N,i}_{t},m_{t})}\bigg{|}+\bigg{|}g^{\lambda}\Big{|}_{(X^{N,i}_{t},m_{\boldsymbol{X}^{N}_{t}})}^{(Y^{N,i}_{t},m_{t})}\bigg{|}\vspace{3pt}\\\
\lesssim\big{|}X^{N,i}_{t}-Y^{N,i}_{t}\big{|}+W_{2}(m_{\boldsymbol{X}^{N}_{t}},m_{t}),$

the expectation of the right-hand side going to $0$ as $N\to\infty$ (uniformly
in $t\in[0,T]$) by the triangle inequality (as in the beginning of this step)
and (7.9). Then we deduce (7.7). Note also that we have

(7.10)
$\mathbb{E}\Bigl{[}\,\sup_{t\in[0,T]}W_{2}(m_{\boldsymbol{X}^{N}_{t}},m_{t})\Bigr{]}\to
0\quad\text{as}\ N\to\infty.$

Step 2. We prove that

(7.11) $\mathbb{E}\,\mathfrak{u}_{N}({\lambda
1_{N}},0,\boldsymbol{Z})\,\xrightarrow[N\to\infty]{}\,\int_{0}^{T}\int_{{\mathbb{R}}^{d}}\Big{(}\,\frac{1}{2}\big{|}\partial_{x}U^{\lambda}(t,\mspace{1.8mu}\cdot\mspace{1.8mu},m_{t})\big{|}^{2}+f^{\lambda}(\mspace{1.8mu}\cdot\mspace{1.8mu},m_{t})\Big{)}\mathrm{d}\overline{m}_{t}\mathrm{d}t+\int_{{\mathbb{R}}^{d}}g^{\lambda}(\mspace{1.8mu}\cdot\mspace{1.8mu},m_{T})\,\mathrm{d}\overline{m}_{T}.$

By the representation formula for the value functions solving of the Nash
system (see, e.g., [16, Section 2.1.4]),

(7.12) $\mathbb{E}\,\mathfrak{u}_{N}(\lambda
1_{N},0,\boldsymbol{Z})=\mathbb{E}\biggl{[}\,\int_{0}^{T}\Big{(}\,\frac{1}{2}\big{|}\partial_{x}\mathfrak{u}_{N}(\lambda
1_{N},t,\boldsymbol{\tilde{X}}^{N}_{t})\big{|}^{2}+f_{N}(\lambda,\boldsymbol{\tilde{X}}^{N}_{t})\Big{)}\mathrm{d}t+g_{N}(\lambda,\boldsymbol{\tilde{X}}^{N}_{T})\biggr{]},$

where $\boldsymbol{\tilde{X}}^{N}$ is the vector of optimal trajectories for
the $N$-player game; that is

$\begin{cases}\mathrm{d}\tilde{X}^{N,i}_{t}=-\partial_{x}\mathfrak{u}_{N}({\lambda
1_{N}},t,\tilde{X}^{N,i}_{t},\boldsymbol{\tilde{X}}^{N,-i}_{t})\,\mathrm{d}t+\sqrt{2}\,\mathrm{d}B^{i}_{t},&i\in\\{1,\dots,N\\}\\\
\boldsymbol{X}_{0}^{N}=\boldsymbol{Z}.\end{cases}$

Note that, since
$\mathbb{E}\bigl{[}\sup_{t\in[0,T]}W_{2}(m_{\boldsymbol{\tilde{X}}^{N,-i}_{t}},m_{\boldsymbol{\tilde{X}}^{N}_{t}})^{2}\bigr{]}\lesssim
N^{-1}$,

(7.13)
$\begin{split}|\tilde{X}^{N,i}_{s}-X^{N,i}_{s}|^{2}&\lesssim\begin{multlined}\int_{0}^{s}\Big{(}\big{|}\partial_{x}\mathfrak{u}_{N}({\lambda
1_{N}},t,\tilde{X}^{N,i}_{t},\boldsymbol{\tilde{X}}^{N,-i}_{t})-\partial_{x}\mathfrak{u}_{N}({\lambda
1_{N}},t,X^{N,i}_{t},\boldsymbol{\tilde{X}}^{N,-i}_{t})\big{|}^{2}\vspace{-7pt}\\\
+\big{|}\partial_{x}\mathfrak{u}_{N}(\lambda
1_{N},t,X^{N,i}_{t},\boldsymbol{\tilde{X}}^{N,-i}_{t})-\partial_{x}U^{\lambda}(t,X^{N,i}_{t},m_{\boldsymbol{\tilde{X}}^{N}_{t}})\big{|}^{2}\\\
+\big{|}\partial_{x}U^{\lambda}(t,X^{N,i}_{t},m_{\boldsymbol{\tilde{X}}^{N}_{t}})-\partial_{x}U^{\lambda}(t,X^{N,i}_{t},m_{\boldsymbol{X}^{N}_{t}})\big{|}^{2}\Big{)}\,\mathrm{d}t\vspace{-5pt}\end{multlined}\int_{0}^{s}\Big{(}\big{|}\partial_{x}\mathfrak{u}_{N}({\lambda
1_{N}},t,\tilde{X}^{N,i}_{t},\boldsymbol{\tilde{X}}^{N,-i}_{t})-\partial_{x}\mathfrak{u}_{N}({\lambda
1_{N}},t,X^{N,i}_{t},\boldsymbol{\tilde{X}}^{N,-i}_{t})\big{|}^{2}\vspace{-7pt}\\\
+\big{|}\partial_{x}\mathfrak{u}_{N}(\lambda
1_{N},t,X^{N,i}_{t},\boldsymbol{\tilde{X}}^{N,-i}_{t})-\partial_{x}U^{\lambda}(t,X^{N,i}_{t},m_{\boldsymbol{\tilde{X}}^{N}_{t}})\big{|}^{2}\\\
+\big{|}\partial_{x}U^{\lambda}(t,X^{N,i}_{t},m_{\boldsymbol{\tilde{X}}^{N}_{t}})-\partial_{x}U^{\lambda}(t,X^{N,i}_{t},m_{\boldsymbol{X}^{N}_{t}})\big{|}^{2}\Big{)}\,\mathrm{d}t\vspace{-5pt}\\\
&\lesssim\int_{0}^{s}\big{(}|\tilde{X}^{N,i}_{t}-X^{N,i}_{t}|^{2}+\delta^{N,i}_{t}+W_{2}(m_{\boldsymbol{\tilde{X}}^{N}_{t}},m_{\boldsymbol{X}^{N}_{t}})^{2}\big{)}\,\mathrm{d}t,\end{split}$

where we have set

$\delta^{N,i}_{t}\vcentcolon=\big{|}\partial_{x}\mathfrak{u}_{N}({\lambda
1_{N}},t,X^{N,i}_{t},\boldsymbol{\tilde{X}}^{N,-i}_{t})-\partial_{x}U^{\lambda}(t,X^{N,i}_{t},m_{\boldsymbol{\tilde{X}}^{N,-i}_{t}})\big{|}^{2}+\frac{1}{N}.$

Taking the averages,

$\frac{1}{N}\sum_{1\leqslant i\leqslant
N}|\tilde{X}^{N,i}_{s}-X^{N,i}_{s}|^{2}\lesssim\frac{1}{N}\sum_{1\leqslant
i\leqslant
N}\int_{0}^{T}\delta^{N,i}_{t}\,\mathrm{d}t+\int_{0}^{s}\frac{1}{N}\sum_{1\leqslant
i\leqslant N}|\tilde{X}^{N,i}_{t}-X^{N,i}_{t}|^{2}\,\mathrm{d}t,$

so that Gronwall’s lemma yields

$\sup_{t\in[0,T]}\frac{1}{N}\sum_{1\leqslant i\leqslant
N}|\tilde{X}^{N,i}_{t}-X^{N,i}_{t}|^{2}\lesssim\frac{1}{N}\sum_{1\leqslant
i\leqslant N}\int_{0}^{T}\delta^{N,i}_{t}\,\mathrm{d}t$

and, plugging this back into (7.13),

(7.14) $\sup_{1\leqslant i\leqslant
N}\mathbb{E}\Big{[}\sup_{t\in[0,T]}|\tilde{X}^{N,i}_{t}-X^{N,i}_{t}|^{2}\Big{]}\lesssim\frac{1}{N}\sum_{1\leqslant
i\leqslant
N}\int_{0}^{T}\mathbb{E}\,\delta^{N,i}_{t}\,\mathrm{d}t\,\xrightarrow[N\to\infty]{}\,0;$

in order to justify the convergence to $0$ of the right-hand side of (7.14),
we proceed as follows. First note that, since
$\partial_{x}\mathfrak{u}^{N},D_{x}U$ are bounded uniformly and
$\overline{m},m_{0}\in\mathcal{P}_{q}({\mathbb{R}}^{d})$, $q>4$, it is
standard that $\mathbb{E}|\tilde{X}^{N,i}_{t}|^{q}$ and
$\mathbb{E}|X^{N,i}_{t}|^{q}$ are uniformly bounded (and thus, so is
$N^{-1}\mathbb{E}\sum_{1\leqslant j\leqslant N}|\tilde{X}^{N,j}_{t}|^{q}$).
Then, for $\boldsymbol{x}\in({\mathbb{R}}^{d})^{N}$, let
$M_{q}(\boldsymbol{x})\vcentcolon=N^{-1}\sum_{1\leqslant j\leqslant
N}|x^{j}|^{q}$; since

$\int_{0}^{T}\mathbb{E}\,\delta^{N,i}_{t}=\int_{0}^{T}\mathbb{E}\bigl{[}\boldsymbol{1}_{\\{M_{q}(\boldsymbol{\tilde{X}}^{N}_{t})+|X_{t}^{N,i}|\leqslant
R\\}}\delta^{N,i}_{t}\bigr{]}+\int_{0}^{T}\mathbb{E}\bigl{[}\boldsymbol{1}_{\\{M_{q}(\boldsymbol{\tilde{X}}^{N}_{t})+|X_{t}^{N,i}|>R\\}}\delta^{N,i}_{t}\bigr{]},$

the second term can be made arbitrarily small provided that $R$ is large, by
the aforementioned bounds on $q$-th moments, while the first can be made small
by choosing $N$ large enough ($\delta^{N,i}_{t}$ is bounded uniformly).
Indeed, it is sufficient to apply the uniform convergence (7.4) on the set
$\mathcal{K}$ of all probability measures with $q$-th moment bounded by
$R$,666We are meaning that
$\mathcal{K}\vcentcolon=\\{\nu\in\mathcal{P}_{2}({\mathbb{R}}^{d}):\
\int_{{\mathbb{R}}^{d}}|\boldsymbol{x}|^{q}\,\nu(\mathrm{d}\boldsymbol{x})\leqslant
R\\}$. which is compact in $\mathcal{P}_{2}({\mathbb{R}}^{d})$. Note that the
previous limit is uniform in $i$, since $K$ and the bounds on the moments are
independent of $i$.

Combining (7.7), (7.12) and (7.14) one deduces (7.11).

Step 3. Since $\mathbb{E}\,W_{2}(m_{\boldsymbol{Z}^{-1}},m_{0})\to 0$ again by
[16, Theorem 5.8], from (7.4) and (7.11) we obtain

$\int_{{\mathbb{R}}^{d}}U^{\lambda}(0,\mspace{1.8mu}\cdot\mspace{1.8mu},m_{0})\,\mathrm{d}\overline{m}=\int_{0}^{T}\int_{{\mathbb{R}}^{d}}\Big{(}\,\frac{1}{2}\big{|}\partial_{x}U^{\lambda}(t,\mspace{1.8mu}\cdot\mspace{1.8mu},m_{t})\big{|}^{2}+f^{\lambda}(\mspace{1.8mu}\cdot\mspace{1.8mu},m_{t})\Big{)}\mathrm{d}\overline{m}_{t}\mathrm{d}t+\int_{{\mathbb{R}}^{d}}g^{\lambda}(\mspace{1.8mu}\cdot\mspace{1.8mu},m_{T})\,\mathrm{d}\overline{m}_{T}.$

As $\overline{m},m_{0}$ can be chosen in a dense subset of
$\mathcal{P}_{2}({\mathbb{R}}^{d})$ and the choice of $\tau=0$ as initial time
is arbitrary, (7.6) follows by standard stability of the Fokker-Planck
equation. ∎

We finally extend the characterisation (7.6) to any measure
$\boldsymbol{\mu}\in\mathcal{P}_{2}(\Lambda\times{\mathbb{R}}^{d})$, with a
continuous disintegration with respect to the projection
$\pi_{\Lambda}\colon\Lambda\times{\mathbb{R}}^{d}\to\Lambda$. This will be the
main result of this section. The extension is rather technical, but the main
point is performing the following reduction: the continuity in $\lambda$ (of
the disintegration and of the data) allows to approximate the problem with a
simplest one, where the first marginal of $\mu$ is supported on a finite
subset of $\Lambda$. This is a sort of problem with finitely many populations,
where we apply the previous convergence argument; in other words, we are going
to develop a sort of multiple “localised” propagation of chaos.

######  Theorem 7.9.

Let $\boldsymbol{\mu}\in\mathcal{P}_{2}(\Lambda\times{\mathbb{R}}^{d})$ admit
a disintegration $\mathfrak{m}\colon K\to\mathcal{P}_{2}({\mathbb{R}}^{d})$
with respect to $\pi_{\Lambda}$ which is continuous on some compact subset
$K\supseteq\operatorname{spt}(\pi_{\Lambda}{}_{\sharp}\boldsymbol{\mu})$. Let
$\overline{\boldsymbol{\mathfrak{m}}}\in\mathcal{P}_{2}({\mathbb{R}}^{d})^{\Lambda}$.
Then for all $\tau\in[0,T]$ and $\lambda\in K$,

(7.15)
$\begin{multlined}\int_{{\mathbb{R}}^{d}}U^{\lambda}(\tau,\mspace{1.8mu}\cdot\mspace{1.8mu},\boldsymbol{\mu})\,\mathrm{d}\overline{\mathfrak{m}}^{\lambda}\vspace{-7pt}\\\
=\int_{\tau}^{T}\int_{{\mathbb{R}}^{d}}\Big{(}\,\frac{1}{2}\big{|}\partial_{x}U^{\lambda}(s,\mspace{1.8mu}\cdot\mspace{1.8mu},\mu_{s})\big{|}^{2}+f^{\lambda}(\mspace{1.8mu}\cdot\mspace{1.8mu},{\pi_{{\mathbb{R}}^{d}}}{}_{\sharp}\mu_{s})\Big{)}\,\mathrm{d}\overline{m}^{\lambda}_{s}\mathrm{d}s+\int_{{\mathbb{R}}^{d}}g^{\lambda}(\mspace{1.8mu}\cdot\mspace{1.8mu},{\pi_{{\mathbb{R}}^{d}}}{}_{\sharp}\mu_{T})\,\mathrm{d}\overline{m}^{\lambda}_{T},\end{multlined}\int_{{\mathbb{R}}^{d}}U^{\lambda}(\tau,\mspace{1.8mu}\cdot\mspace{1.8mu},\boldsymbol{\mu})\,\mathrm{d}\overline{\mathfrak{m}}^{\lambda}\vspace{-7pt}\\\
=\int_{\tau}^{T}\int_{{\mathbb{R}}^{d}}\Big{(}\,\frac{1}{2}\big{|}\partial_{x}U^{\lambda}(s,\mspace{1.8mu}\cdot\mspace{1.8mu},\mu_{s})\big{|}^{2}+f^{\lambda}(\mspace{1.8mu}\cdot\mspace{1.8mu},{\pi_{{\mathbb{R}}^{d}}}{}_{\sharp}\mu_{s})\Big{)}\,\mathrm{d}\overline{m}^{\lambda}_{s}\mathrm{d}s+\int_{{\mathbb{R}}^{d}}g^{\lambda}(\mspace{1.8mu}\cdot\mspace{1.8mu},{\pi_{{\mathbb{R}}^{d}}}{}_{\sharp}\mu_{T})\,\mathrm{d}\overline{m}^{\lambda}_{T},$

where $\mu\in C^{0}([\tau,T];\mathcal{P}_{2}(\Lambda\times{\mathbb{R}}^{d}))$
and $\overline{m}\in
C^{0}([\tau,T];\mathcal{P}_{2}({\mathbb{R}}^{d}))^{\Lambda}$ solve,
respectively,

(7.16) $\begin{cases}\partial_{t}m^{\lambda}-\Delta
m^{\lambda}-\mathrm{div}(\partial_{x}U^{\lambda}(t,x,\mu_{t})m^{\lambda})=0\\\
\mu_{t}=\int_{\Lambda}m_{t}^{\lambda}\,\pi_{\Lambda}{}_{\sharp}\boldsymbol{\mu}(\mathrm{d}\lambda)\\\
m^{\lambda}_{\tau}=\mathfrak{m}^{\lambda}\end{cases}$

and

(7.17)
$\begin{cases}\partial_{t}\overline{m}^{\lambda}-\Delta\overline{m}^{\lambda}-\mathrm{div}(\partial_{x}U^{\lambda}(t,x,\mu_{t})\overline{m}^{\lambda})=0\\\
\overline{m}^{\lambda}_{\tau}=\overline{\mathfrak{m}}^{\lambda}\end{cases}$

for $t\in[\tau,T]$, $x\in{\mathbb{R}}^{d}$ and $\lambda\in\Lambda$.

######  Remark 7.10.

Since $\partial_{x}U^{\lambda}$ and ${\rm div}(\partial_{x}U^{\lambda})$ are
globally bounded, $\overline{m}^{\lambda}$ is locally continuous as a function
with values in $L^{1}({{\mathbb{R}}^{d}})$, it solves the Fokker-Planck
equation in the strong sense, and it belongs locally to $W^{2,1}_{p}$ for
every $p$, see for example [10]. Regularity extends up to time $\tau$ provided
that $\overline{\mathfrak{m}}^{\lambda}$ is smooth enough.

Problem (7.17) is simply an uncoupled system of (possibly uncountably many)
Fokker–Planck equations. On the other hand, before proceeding with the proof
of this theorem we show that problem (7.16) is well-posed, as a result of the
following lemma. Also note that it can be written as

(7.18)
$\begin{cases}\partial_{t}\mu-\Delta_{x}\mu-\mathrm{div}_{x}(\partial_{x}U^{\lambda}(t,x,\mu_{t})\mu)=0&\text{on}\
[\tau,T]\times\Lambda\times{\mathbb{R}}^{d}\\\
\mu_{\tau}=\boldsymbol{\mu},\end{cases}$

so that $\mu$ solves an equation of Fokker–Planck with no derivatives with
respect to $\lambda$.

######  Lemma 7.11.

Let
$B\colon\Lambda\times[0,T]\times{\mathbb{R}}^{d}\times\mathcal{P}_{2}(\Lambda\times{\mathbb{R}}^{d})\to{\mathbb{R}}^{d}$
be bounded Lipschitz continuous on
$\Lambda\times{\mathbb{R}}^{d}\times\mathcal{P}_{2}(\Lambda\times{\mathbb{R}}^{d})$
uniformly on $[0,T]$. Let
$\boldsymbol{\mu}\in\mathcal{P}_{2}(\Lambda\times{\mathbb{R}}^{d})$ and let
$\rho\vcentcolon={\pi_{\Lambda}}{}_{\sharp}\boldsymbol{\mu}\in\mathcal{P}(\Lambda)$
be its first marginal. Then problem777With an abuse of notation we write
$\int_{\Lambda}m^{\lambda}_{t}\rho(\mathrm{d}\lambda)$ instead of
$\int_{\Lambda}\hat{m}^{\lambda}_{t}\rho(\mathrm{d}\lambda)$ with
$\hat{m}^{\lambda}_{t}(E)\vcentcolon=m^{\lambda}(\pi_{{\mathbb{R}}^{d}}(E\cap\pi_{\Lambda}^{-1}(\\{\lambda\\})))$
for all Borel sets $E\subseteq\Lambda\times{\mathbb{R}}^{d}$. We will continue
to identify $\hat{m}^{\lambda}$ with $m^{\lambda}$.

(7.19) $\begin{cases}\partial_{t}m^{\lambda}-\Delta
m^{\lambda}+\mathrm{div}(B^{\lambda}(t,x,\mu_{t})m^{\lambda})=0\quad\text{on}\
[0,T]\times{\mathbb{R}}^{d},\ \ \forall\,\lambda\in\Lambda\\\
\mu_{t}=\int_{\Lambda}m_{t}^{\lambda}\rho(\mathrm{d}\lambda)\quad\forall\,t\in[0,T]\\\
\mu_{0}=\boldsymbol{\mu}\end{cases}$

has a unique solution $\mu\in
C^{\frac{1}{2}}([0,T];\mathcal{P}_{2}(\Lambda\times{\mathbb{R}}^{d}))$, whose
disintegration $m$ with respect to $\rho$ is unique as well if one fixes the
disintegration of $\boldsymbol{\mu}$.

Furthermore, if $\mu$ and $\tilde{\mu}$ are the solutions relative to data
$\boldsymbol{\mu}$ and $\boldsymbol{\tilde{\mu}}$ with respective
disintegrations given by $\mathfrak{m}$ and $\tilde{\mathfrak{m}}$,
respectively, one has

(7.20)
$\displaystyle\sup_{t\in[0,T]}W_{2}(\mu_{t},\tilde{\mu}_{t})\,\lesssim\,W_{2}(\boldsymbol{\mu},\boldsymbol{\tilde{\mu}}),$
(7.21)
$\displaystyle\sup_{t\in[0,T]}W_{2}(m^{\lambda}_{t},\tilde{m}^{\lambda^{\prime}}_{t})\,\lesssim\,W_{2}(\mathfrak{m}^{\lambda},\tilde{\mathfrak{m}}^{\lambda^{\prime}})+W_{2}(\boldsymbol{\mu},\boldsymbol{\tilde{\mu}})\quad\forall\,\lambda,\lambda^{\prime}\in\Lambda,$

where the implied constants depend only on $T$ and $\lVert
B\rVert_{\operatorname{Lip}}$.

###### Proof.

For some $A>0$ to be determined, let $\mathfrak{C}$ be the set of all $\mu\in
C^{0}([0,T];\mathcal{P}_{2}(\Lambda\times{\mathbb{R}}^{d}))$ such that
$[\mu]_{0,\frac{1}{2}}\vcentcolon=\sup_{s,t\in[0,T],\,s\neq
t}|s-t|^{-\frac{1}{2}}W_{2}(\mu_{s},\mu_{t})\leqslant A$. Note that
$\mathfrak{C}$ a closed convex subset
$C^{0}([0,T];\mathcal{P}_{2}(\Lambda\times{\mathbb{R}}^{d}))$. Given
$\mu\in\mathfrak{C}$ and $\lambda\in\Lambda$, let $m^{\lambda}\in
C^{0}([0,T];\mathcal{P}_{2}({\mathbb{R}}^{d}))$ be the unique solution to
$\partial_{t}m^{\lambda}-\Delta
m^{\lambda}+\mathrm{div}(B^{\lambda}(t,x,\mu_{t})m^{\lambda})=0$ with
$m^{\lambda}_{0}=\mathfrak{m}^{\lambda}$, where
$(\mathfrak{m}^{\lambda})_{\lambda\in\Lambda}$ is a fixed disintegration of
$\boldsymbol{\mu}$ with respect to $\rho$. Let
$\Phi(\mu)\vcentcolon=\int_{\Lambda}m^{\lambda}\rho(\mathrm{d}\lambda)$. It is
standard (cf. [15, Lemma 1.6]) to show that $[m^{\lambda}]_{0,\frac{1}{2}}$ is
bounded by a constant which increases with $\lVert
B^{\lambda}\rVert_{\infty}$. In addition, if $\pi_{s,t}^{\lambda}$ is an
optimal coupling of $(m^{\lambda}_{s},m^{\lambda}_{t})$, then
$\textstyle{\int_{\Lambda\times\Lambda}}\pi^{\lambda}_{s,t}((\mathrm{id}_{\Lambda},\mathrm{id}_{\Lambda}){}_{\sharp}\rho)(\mathrm{d}\lambda,\mathrm{d}\lambda^{\prime})$
is a coupling of $(\tilde{\mu}_{s},\tilde{\mu}_{t})$, showing that
$W_{2}(\Phi(\mu)_{s},\Phi(\mu)_{t})^{2}\leqslant\int_{\Lambda}W_{2}(m^{\lambda}_{s},m^{\lambda}_{t})^{2}\rho(\mathrm{d}\lambda)$.
Therefore, since $\sup_{\lambda\in\Lambda}\lVert B^{\lambda}\rVert_{\infty}$
is finite, if $A$ is sufficiently large then the map
$\Phi\colon\mathfrak{C}\to\mathfrak{C}$ is well-defined.

The following estimates will prove that $\Phi$ is a contraction for $T$ small,
and thus (by iteration) the existence of a unique solution to (7.19) with
arbitrary $T$. Furthermore, a posteriori, they will also yield the desired
estimates (7.20) and (7.21).

Let $\mu,\tilde{\mu}\in\mathfrak{C}$ and let two initial data
$\boldsymbol{\mu},\boldsymbol{\tilde{\mu}}\in\mathcal{P}(\Lambda\times{\mathbb{R}}^{d})$
be given; let $\rho={\pi_{\Lambda}}{}_{\sharp}\boldsymbol{\mu}$,
$\tilde{\rho}={\pi_{\Lambda}}{}_{\sharp}\boldsymbol{\tilde{\mu}}$ and
$(\mathfrak{m}^{\lambda})_{\lambda\in\Lambda}$ (resp.
$(\tilde{\mathfrak{m}}^{\lambda})_{\lambda\in\Lambda})$ be a disintegration of
$\boldsymbol{\mu}$ (resp. $\boldsymbol{\tilde{\mu}}$) with respect to $\rho$
(resp. $\tilde{\rho}$). Let $\nu\in
C^{0}([0,T];\mathcal{P}_{2}((\Lambda\times{\mathbb{R}}^{d})^{2})$ satisfy

$\partial_{t}\nu-\Delta_{x}\nu-\Delta_{y}\nu+2\sum_{1\leqslant j\leqslant
d}\partial^{2}_{x^{j}y^{j}}\nu+\mathrm{div}_{x}(B^{\lambda}(t,x,\mu_{t})\nu)+\mathrm{div}_{y}(B^{\lambda^{\prime}}(t,y,\tilde{\mu}_{t})\nu)=0,$

so that if $\nu_{0}\in\Pi(\boldsymbol{\mu},\boldsymbol{\tilde{\mu}})$, then
$\nu_{t}$ is a coupling of $(\Phi(\mu)_{t},\Phi(\tilde{\mu})_{t})$. Testing by
$|\lambda-\lambda^{\prime}|^{2}+|x-y|^{2}$ and assuming that $\nu_{0}$ is
optimal for the $2$-Wasserstein distance, we obtain

$\begin{multlined}\int_{(\Lambda\times{\mathbb{R}}^{d})^{2}}(|\lambda-\lambda^{\prime}|^{2}+|x-y|^{2})\,\nu_{t}(\mathrm{d}\lambda,\mathrm{d}x,\mathrm{d}\lambda^{\prime},\mathrm{d}y)\vspace{-5pt}\\\
=W_{2}(\boldsymbol{\mu},\boldsymbol{\tilde{\mu}})^{2}+2\int_{0}^{t}\int_{(\Lambda\times{\mathbb{R}}^{d})^{2}}\big{(}B^{\lambda}(t,x,\mu_{t})-B^{\lambda^{\prime}}(t,y,\tilde{\mu}_{t})\big{)}\cdot(x-y)\,\mathrm{d}\nu_{s}(\mathrm{d}\lambda,\mathrm{d}x,\mathrm{d}\lambda^{\prime},\mathrm{d}y)\,\mathrm{d}s.\end{multlined}\int_{(\Lambda\times{\mathbb{R}}^{d})^{2}}(|\lambda-\lambda^{\prime}|^{2}+|x-y|^{2})\,\nu_{t}(\mathrm{d}\lambda,\mathrm{d}x,\mathrm{d}\lambda^{\prime},\mathrm{d}y)\vspace{-5pt}\\\
=W_{2}(\boldsymbol{\mu},\boldsymbol{\tilde{\mu}})^{2}+2\int_{0}^{t}\int_{(\Lambda\times{\mathbb{R}}^{d})^{2}}\big{(}B^{\lambda}(t,x,\mu_{t})-B^{\lambda^{\prime}}(t,y,\tilde{\mu}_{t})\big{)}\cdot(x-y)\,\mathrm{d}\nu_{s}(\mathrm{d}\lambda,\mathrm{d}x,\mathrm{d}\lambda^{\prime},\mathrm{d}y)\,\mathrm{d}s.$

By the Cauchy–Schwarz and Young’s inequalities,

$\begin{multlined}\int_{(\Lambda\times{\mathbb{R}}^{d})^{2}}(|\lambda-\lambda^{\prime}|^{2}+|x-y|^{2})\,\nu_{t}(\mathrm{d}\lambda,\mathrm{d}x,\mathrm{d}\lambda^{\prime},\mathrm{d}y)\vspace{-10pt}\\\
\leqslant
W_{2}(\boldsymbol{\mu},\boldsymbol{\tilde{\mu}})^{2}+C\int_{0}^{t}W_{2}(\mu_{s},\tilde{\mu}_{s})^{2}\,\mathrm{d}s+C\int_{0}^{t}\int_{(\Lambda\times{\mathbb{R}}^{d})^{2}}(|\lambda-\lambda^{\prime}|^{2}+|x-y|^{2})\,\nu_{s}(\mathrm{d}\lambda,\mathrm{d}x,\mathrm{d}\lambda^{\prime},\mathrm{d}y)\,\mathrm{d}s,\end{multlined}\int_{(\Lambda\times{\mathbb{R}}^{d})^{2}}(|\lambda-\lambda^{\prime}|^{2}+|x-y|^{2})\,\nu_{t}(\mathrm{d}\lambda,\mathrm{d}x,\mathrm{d}\lambda^{\prime},\mathrm{d}y)\vspace{-10pt}\\\
\leqslant
W_{2}(\boldsymbol{\mu},\boldsymbol{\tilde{\mu}})^{2}+C\int_{0}^{t}W_{2}(\mu_{s},\tilde{\mu}_{s})^{2}\,\mathrm{d}s+C\int_{0}^{t}\int_{(\Lambda\times{\mathbb{R}}^{d})^{2}}(|\lambda-\lambda^{\prime}|^{2}+|x-y|^{2})\,\nu_{s}(\mathrm{d}\lambda,\mathrm{d}x,\mathrm{d}\lambda^{\prime},\mathrm{d}y)\,\mathrm{d}s,$

where $C$ depends only on $\lVert B\rVert_{\operatorname{Lip}}$. Apply
Gronwall’s lemma to deduce

(7.22)
$W_{2}(\Phi(\mu)_{t},\Phi(\tilde{\mu})_{t})^{2}\leqslant\biggl{(}W_{2}(\boldsymbol{\mu},\boldsymbol{\tilde{\mu}})^{2}+C\int_{0}^{t}W_{2}(\mu_{s},\tilde{\mu}_{s})^{2}\,\mathrm{d}s\biggr{)}e^{Ct},$

which implies that $\Phi$ is a contraction on $\mathfrak{C}\subset
C^{0}([0,T];\mathcal{P}_{2}(\Lambda\times{\mathbb{R}}^{d}))$ if $CTe^{CT}<1$.
Since this smallness condition on $T$ depends only on $\lVert
B\rVert_{\operatorname{Lip}}$, we can iterate finitely many times the short-
time uniqueness provided by the contraction theorem in order to obtain
uniqueness on an arbitrary horizon.

Now $\Phi(\mu)=\mu$ and $\Phi(\tilde{\mu})=\tilde{\mu}$ in (7.22) above, then
we can apply Gronwall’s lemma once again to get (7.20). With a similar
argument (which considers a coupling of
$(m^{\lambda}_{t},\tilde{m}^{\lambda^{\prime}}_{t})$, where $m^{\lambda}$ is a
solution to (7.19) given $m^{\lambda}_{0}=\mathfrak{m}^{\lambda}$ and
analogously for $\tilde{m}^{\lambda^{\prime}}$) one can also deduce that

$W_{2}(m^{\lambda}_{t},\tilde{m}^{\lambda^{\prime}}_{t})^{2}\,\lesssim\,W_{2}(\mathfrak{m}^{\lambda},\tilde{\mathfrak{m}}^{\lambda^{\prime}})^{2}+\int_{0}^{t}W_{2}(\mu_{s},\tilde{\mu}_{s})^{2}\,\mathrm{d}s.$

Then, using (7.20), one has (7.21). ∎

###### Proof of Theorem 7.9.

We divide the proof into three steps.

Step 1. We are going to show that the proof for general $\boldsymbol{\mu}$ can
be reduced to the case where $\pi_{\Lambda}{}_{\sharp}\boldsymbol{\mu}$ is
finitely supported. The convergence in the latter case of “finitely many”
populations of players will be discussed in the steps below.

Given $\boldsymbol{\mu}\in\mathcal{P}_{2}(\Lambda\times{\mathbb{R}}^{d})$ as
in the statement of the theorem and
$\rho\vcentcolon={\pi_{\Lambda}}{}_{\sharp}\boldsymbol{\mu}$, let
$(\rho_{k})_{k\in\mathbb{N}}\subset\mathcal{P}(K)$ be a sequence of finitely-
supported probability measures such that $\rho_{k}\to\rho$ weakly. Since
$W_{2}$ metrizes the weak convergence of probability measures on $K$ (see,
e.g., [43, Corollary 6.13]), the existence of such a sequence comes from the
Banach–Alaoglu and the Krein–Milman theorems. Moreover, note that one can in
particular choose $\rho_{k}$ in the form specified at the beginning of Step 2
below.

Consider the continuous disintegration $\mathfrak{m}\colon
K\to\mathcal{P}_{2}({\mathbb{R}}^{d})$ of $\boldsymbol{\mu}$ with respect to
$\pi_{\Lambda}$ and define
$\boldsymbol{\mu}_{k}\vcentcolon=\int_{\Lambda}\mathfrak{m}^{\lambda}\rho_{k}(\mathrm{d}\lambda)$;
then, let $m_{k}$ be the corresponding solution to (7.16) (note that
$\pi_{\Lambda}{}_{\sharp}(\mu_{k})_{t}=\rho_{k}$ for all $t$). Thanks to the
continuity of the disintegration $\mathfrak{m}$, the map
$\lambda\mapsto\int_{{\mathbb{R}}^{d}}F(\lambda,x)\mathfrak{m}^{\lambda}(\mathrm{d}x)$
is continuous, so, for any $F\in C(\Lambda\times{\mathbb{R}}^{d})$ with sub-
quadratic growth with respect to $x\in{\mathbb{R}}^{d}$, by the weak
convergence $\rho_{k}\to\rho$ we have

$\int_{\Lambda\times{\mathbb{R}}^{d}}F(\lambda,x)\mathfrak{m}^{\lambda}(\mathrm{d}x)\rho_{k}(\mathrm{d}\lambda)\to\int_{\Lambda\times{\mathbb{R}}^{d}}F(\lambda,x)\mathfrak{m}^{\lambda}(\mathrm{d}x)\rho(\mathrm{d}\lambda);$

that is, $\boldsymbol{\mu}_{k}\to\boldsymbol{\mu}$ in
$\mathcal{P}_{2}(\Lambda\times{\mathbb{R}}^{d})$. Therefore, by (7.20),
$\mu_{k}\to\mu$ in
$C^{0}([\tau,T];\mathcal{P}_{2}(\Lambda\times{\mathbb{R}}^{d}))$, which
implies that
$\pi_{{\mathbb{R}}^{d}}{}_{\sharp}\mu_{k}\to\pi_{{\mathbb{R}}^{d}}{}_{\sharp}\mu$
in $C^{0}([\tau,T];\mathcal{P}_{2}({\mathbb{R}}^{d}))$ as well.

Let now $\lambda\in K$ and assume that
$\lambda\in\bigcap_{k\in\mathbb{N}}\operatorname{spt}(\rho_{k})$. Standard
stability of the Fokker–Planck equation gives the convergence
$\overline{m}^{\lambda}_{k}\to\overline{m}^{\lambda}$ in
$C^{0}([\tau,T];\mathcal{P}({\mathbb{R}}^{d}))$, so we have all the
ingredients to pass the identity (7.15) (that we are assuming to hold when
$\pi_{\Lambda}{}_{\sharp}\boldsymbol{\mu}$ is finitely supported) to the limit
$k\to\infty$.

On the other hand, if
$\lambda\notin\bigcap_{k\in\mathbb{N}}\operatorname{spt}(\rho_{k})$ one can
consider, in place of $\rho_{k}$, measures
$\tilde{\rho}_{k}\vcentcolon=\frac{N-1}{N}\rho_{k}+\frac{1}{N}\delta_{\lambda}$,
so that still $\tilde{\rho}_{k}\to\rho$ weakly and the above argument can be
still performed.

Step 2. We prove now the theorem by assuming that the first marginal of
$\boldsymbol{\mu}$ is finitely supported, that is $\rho=\sum_{1\leqslant
j\leqslant\ell}\theta_{j}\delta_{\lambda^{j}}\in\mathcal{P}(\Lambda)$ for some
(fixed) $\ell\in\mathbb{N}$, $\theta_{j}\in\mathbb{Q}_{+}$ and
$\lambda^{j}\in\Lambda$.

We want to mimic Step 1 of the proof of Proposition 7.8, where the identity is
obtained as the limit of the corresponding identity for value functions of
$N$-players games. We first introduce some notation. Given a vector
$\boldsymbol{\mathfrak{m}}\in\mathcal{P}({\mathbb{R}}^{d})^{\ell}$ define

(7.23)
$U^{\lambda}(t,x,\boldsymbol{\mathfrak{m}})\vcentcolon=U^{\lambda}\bigl{(}t,x,\sum_{1\leqslant
j\leqslant\ell}\theta_{j}\delta_{\lambda^{j}}\\!\otimes\mathfrak{m}^{j}\bigr{)}$

as well as
$h^{\lambda}(x,\boldsymbol{\mathfrak{m}})\vcentcolon=h^{\lambda}\bigl{(}x,\sum_{1\leqslant
k\leqslant\ell}\theta_{k}\mathfrak{m}^{k}\bigr{)}$ for $h\in\\{f,g\\}$. Notice
that in this discrete case (7.16) reduces to the system

(7.24) $\begin{cases}\partial_{t}m^{\lambda^{j}}-\Delta
m^{\lambda^{j}}-\mathrm{div}(\partial_{x}U^{\lambda^{j}}(t,x,\boldsymbol{m}^{\boldsymbol{\lambda}}_{t})m^{\lambda^{j}})=0,&j\in\\{1,\dots,\ell\\}\\\
\boldsymbol{m}^{\boldsymbol{\lambda}}|_{t=\tau}=\boldsymbol{\mathfrak{m}},\end{cases}$

where
$\boldsymbol{m}^{\boldsymbol{\lambda}}\vcentcolon=(m^{\lambda^{j}})_{1\leqslant
j\leqslant\ell}$, and thus, according to notation (7.23),

$\partial_{x}U^{\lambda^{j}}(t,x,\boldsymbol{m}^{\boldsymbol{\lambda}}_{t})=\partial_{x}U^{\lambda^{j}}\Bigl{(}t,x,\sum_{1\leqslant
k\leqslant\ell}\theta_{k}\delta_{\lambda^{k}}\\!\otimes
m^{\lambda^{k}}_{t}\Bigr{)};$

also, for
$\overline{\boldsymbol{\mathfrak{m}}}\in\mathcal{P}({\mathbb{R}}^{d})^{\ell}$,
(7.17) appears as

(7.25)
$\begin{cases}\partial_{t}\overline{m}^{\lambda^{j}}-\Delta\overline{m}^{\lambda^{j}}-\mathrm{div}(\partial_{x}U^{\lambda^{j}}(t,x,\boldsymbol{m}^{\boldsymbol{\lambda}}_{t})\overline{m}^{\lambda^{j}})=0,&j\in\\{1,\dots,\ell\\}\\\
\overline{\boldsymbol{m}}^{\boldsymbol{\lambda}}|_{t=\tau}=\overline{\boldsymbol{\mathfrak{m}}}.\end{cases}$

We now want to construct $N$-player games, where each player is suitably
labelled with $\lambda^{j}$, $j\in\\{1,\dots,\ell\\}$. The distribution of
labels in the limit $N\to\infty$ must be $\rho$. It is elementary to construct
a partition $(I_{j})_{1\leqslant j\leqslant\ell}$ of $\mathbb{N}$ such that
$\mathfrak{d}(I_{j})=\theta_{j}$, where $\mathfrak{d}$ denotes the natural
density;888Recall that the natural density of $S\subset\mathbb{N}$ is defined
as
$\mathfrak{d}(S)\vcentcolon=\lim_{N\to\infty}N^{-1}\\#(S\cap\\{1,\dots,N\\})$,
if the limit exists. indeed, it suffices to consider the minimum natural
number $D$ such that $D\theta_{j}\in\mathbb{N}$ for all
$j\in\\{1,\dots,\ell\\}$, take a partition $\\{\Theta_{j}\\}_{1\leqslant
j\leqslant\ell}$ of $\\{1,\dots,D\\}$ such that $\\#\Theta_{j}=D\theta_{j}$
and set $I_{j}\vcentcolon=\Theta_{j}+D\mathbb{N}$. So we can define
$\overline{\boldsymbol{\lambda}}\in\ell^{\infty}(\mathbb{N};\Lambda)$ by
letting $\overline{\lambda}^{i}\vcentcolon=\lambda^{j}$ if $i\in I_{j}$ and
then $\boldsymbol{\lambda}_{N}\in\Lambda^{N}$ for any $N\in\mathbb{N}$ by
taking the first $N$ coordinates of $\overline{\boldsymbol{\lambda}}$.

Fix now $\boldsymbol{m}_{0}\in\mathcal{P}_{q}({\mathbb{R}}^{d})^{\ell}$,
$q>4$, and consider $\boldsymbol{m}^{\boldsymbol{\lambda}}$ solving (7.24)
with $\boldsymbol{\mathfrak{m}}=\boldsymbol{m}_{0}$. Fix
$\overline{\boldsymbol{\mathfrak{m}}}\in\mathcal{P}_{2}({\mathbb{R}}^{d})^{\ell}$.
Let $\boldsymbol{Z}$ be an $({\mathbb{R}}^{d})^{N}$-valued random vector such
that, with the notation $I^{N}_{j}\vcentcolon=I_{j}\cap\\{1,\dots,N\\}$,

$Z^{i}\sim\begin{cases}\overline{\mathfrak{m}}^{j}&\text{if}\ i=\min
I_{j}^{N}\ \text{(for some $j\in\\{1,\dots,\ell\\}$)}\\\ m_{0}^{j}&\text{if}\
i\in I_{j}^{N}\setminus\bigcup_{1\leqslant k\leqslant\ell}\\{\min
I_{k}^{N}\\}.\end{cases}$

Then we argue as in the proof of Proposition 7.8, with the replacements
$\\{1,\dots,N\\}\rightsquigarrow I^{N}_{j}$ and
$\partial_{x}U^{\lambda}(t,x,m)\rightsquigarrow\partial_{x}U^{\lambda^{j}}(t,x,(\boldsymbol{m}_{t}^{\boldsymbol{\lambda}^{-j}},m))$,
where
$\boldsymbol{m}^{\boldsymbol{\lambda}^{-j}}\vcentcolon=(m^{\lambda^{i}})_{1\leqslant
i\leqslant\ell,\,i\neq j}$; according to the notation introduced in Section 2,
the vector $(\boldsymbol{m}^{\boldsymbol{\lambda}^{-j}},m)$ is that with all
coordinates given by $m^{\lambda^{i}}$ but the $j$-th one, which is $m$, hence
for instance (recall also notation (7.23))

$\partial_{x}U^{\lambda^{j}}(t,x,(\boldsymbol{m}_{t}^{\boldsymbol{\lambda}^{-j}},m))=\partial_{x}U^{\lambda^{j}}\Bigl{(}t,x,\theta_{j}\delta_{\lambda^{j}}\\!\otimes
m+\sum_{\begin{subarray}{c}1\leqslant k\leqslant\ell\\\ k\neq
j\end{subarray}}\theta_{k}\delta_{\lambda^{k}}\\!\otimes
m^{\lambda^{k}}_{t}\Bigr{)}.$

The crucial intermediate step is via “intra-population” propagation of chaos.
Let $\boldsymbol{X}^{I_{j}^{N}}$ be the solution of

$\begin{cases}\mathrm{d}X^{I_{j}^{N},i}=-\partial_{x}U^{\lambda^{j}}\bigl{(}t,X^{I_{j}^{N},i},(\boldsymbol{m}_{t}^{\boldsymbol{\lambda}^{-j}},m_{\boldsymbol{X}^{I^{N}_{j}}_{t}})\bigr{)}\,\mathrm{d}t+\sqrt{2}\,\mathrm{d}B_{t}^{i}\\\
X^{I^{N}_{j},i}_{0}=Z^{i}\end{cases}\quad i\in I^{N}_{j};$

since $\boldsymbol{m}_{t}^{\boldsymbol{\lambda}}$ is a given flow of measures,
each agent $i\in I^{N}_{j}$ is driven by the same drift

$V^{j}\bigl{(}t,X^{I_{j}^{N},i},m_{\boldsymbol{X}^{I^{N}_{j}}_{t}})\vcentcolon=-\partial_{x}U^{\lambda^{j}}\bigl{(}t,X^{I_{j}^{N},i},(\boldsymbol{m}_{t}^{\boldsymbol{\lambda}^{-j}},m_{\boldsymbol{X}^{I^{N}_{j}}_{t}})\bigr{)},$

hence, arguing exactly as in Step 1 of the proof of Proposition 7.8 (note that
$\\#I_{j}^{N}\to\infty$ as $N\to\infty$) we obtain, for $i=\min I^{N}_{j}$,

(7.26)
$\begin{multlined}\begin{multlined}\mathbb{E}\bigg{[}\int_{0}^{T}\Big{(}\,\frac{1}{2}\big{|}\partial_{x}U^{\lambda^{j}}\big{(}t,X^{I_{j}^{N},i}_{t},(\boldsymbol{m}_{t}^{\boldsymbol{\lambda}^{-j}},m_{\boldsymbol{X}^{I_{j}^{N}}_{t}})\big{)}\big{|}^{2}+f^{\lambda^{j}}\big{(}X^{I_{j}^{N},i}_{t},(\boldsymbol{m}_{t}^{\boldsymbol{\lambda}^{-j}},m_{\boldsymbol{X}^{I_{j}^{N}}_{t}})\big{)}\Big{)}\mathrm{d}t\vspace{-7pt}\\\
+g^{\lambda^{j}}\big{(}X^{I_{j}^{N},i}_{T},(\boldsymbol{m}_{T}^{\boldsymbol{\lambda}^{-j}},m_{\boldsymbol{X}^{I_{j}^{N}}_{T}})\big{)}\bigg{]}\end{multlined}\mathbb{E}\bigg{[}\int_{0}^{T}\Big{(}\,\frac{1}{2}\big{|}\partial_{x}U^{\lambda^{j}}\big{(}t,X^{I_{j}^{N},i}_{t},(\boldsymbol{m}_{t}^{\boldsymbol{\lambda}^{-j}},m_{\boldsymbol{X}^{I_{j}^{N}}_{t}})\big{)}\big{|}^{2}+f^{\lambda^{j}}\big{(}X^{I_{j}^{N},i}_{t},(\boldsymbol{m}_{t}^{\boldsymbol{\lambda}^{-j}},m_{\boldsymbol{X}^{I_{j}^{N}}_{t}})\big{)}\Big{)}\mathrm{d}t\vspace{-7pt}\\\
+g^{\lambda^{j}}\big{(}X^{I_{j}^{N},i}_{T},(\boldsymbol{m}_{T}^{\boldsymbol{\lambda}^{-j}},m_{\boldsymbol{X}^{I_{j}^{N}}_{T}})\big{)}\bigg{]}\\\
\xrightarrow[N\to\infty]{}\,\int_{0}^{T}\int_{{\mathbb{R}}^{d}}\Big{(}\,\frac{1}{2}\big{|}\partial_{x}U^{\lambda^{j}}(t,\cdot,\boldsymbol{m}_{t}^{\boldsymbol{\lambda}})\big{|}^{2}+f^{\lambda^{j}}(\cdot,\boldsymbol{m}_{t}^{\boldsymbol{\lambda}})\Big{)}\mathrm{d}\overline{m}_{t}^{\lambda^{j}}\mathrm{d}t+\int_{{\mathbb{R}}^{d}}g^{\lambda^{j}}(\cdot,\boldsymbol{m}_{T}^{\boldsymbol{\lambda}})\,\mathrm{d}\overline{m}^{\lambda^{j}}_{T},\end{multlined}\begin{multlined}\mathbb{E}\bigg{[}\int_{0}^{T}\Big{(}\,\frac{1}{2}\big{|}\partial_{x}U^{\lambda^{j}}\big{(}t,X^{I_{j}^{N},i}_{t},(\boldsymbol{m}_{t}^{\boldsymbol{\lambda}^{-j}},m_{\boldsymbol{X}^{I_{j}^{N}}_{t}})\big{)}\big{|}^{2}+f^{\lambda^{j}}\big{(}X^{I_{j}^{N},i}_{t},(\boldsymbol{m}_{t}^{\boldsymbol{\lambda}^{-j}},m_{\boldsymbol{X}^{I_{j}^{N}}_{t}})\big{)}\Big{)}\mathrm{d}t\vspace{-7pt}\\\
+g^{\lambda^{j}}\big{(}X^{I_{j}^{N},i}_{T},(\boldsymbol{m}_{T}^{\boldsymbol{\lambda}^{-j}},m_{\boldsymbol{X}^{I_{j}^{N}}_{T}})\big{)}\bigg{]}\end{multlined}\mathbb{E}\bigg{[}\int_{0}^{T}\Big{(}\,\frac{1}{2}\big{|}\partial_{x}U^{\lambda^{j}}\big{(}t,X^{I_{j}^{N},i}_{t},(\boldsymbol{m}_{t}^{\boldsymbol{\lambda}^{-j}},m_{\boldsymbol{X}^{I_{j}^{N}}_{t}})\big{)}\big{|}^{2}+f^{\lambda^{j}}\big{(}X^{I_{j}^{N},i}_{t},(\boldsymbol{m}_{t}^{\boldsymbol{\lambda}^{-j}},m_{\boldsymbol{X}^{I_{j}^{N}}_{t}})\big{)}\Big{)}\mathrm{d}t\vspace{-7pt}\\\
+g^{\lambda^{j}}\big{(}X^{I_{j}^{N},i}_{T},(\boldsymbol{m}_{T}^{\boldsymbol{\lambda}^{-j}},m_{\boldsymbol{X}^{I_{j}^{N}}_{T}})\big{)}\bigg{]}\\\
\xrightarrow[N\to\infty]{}\,\int_{0}^{T}\int_{{\mathbb{R}}^{d}}\Big{(}\,\frac{1}{2}\big{|}\partial_{x}U^{\lambda^{j}}(t,\cdot,\boldsymbol{m}_{t}^{\boldsymbol{\lambda}})\big{|}^{2}+f^{\lambda^{j}}(\cdot,\boldsymbol{m}_{t}^{\boldsymbol{\lambda}})\Big{)}\mathrm{d}\overline{m}_{t}^{\lambda^{j}}\mathrm{d}t+\int_{{\mathbb{R}}^{d}}g^{\lambda^{j}}(\cdot,\boldsymbol{m}_{T}^{\boldsymbol{\lambda}})\,\mathrm{d}\overline{m}^{\lambda^{j}}_{T},$

where $\boldsymbol{m}^{\boldsymbol{\lambda}}$ and
$\overline{\boldsymbol{m}}^{\boldsymbol{\lambda}}$ solve, respectively, (7.24)
with $\boldsymbol{\mathfrak{m}}=\boldsymbol{m}_{0}$ and (7.25), both with
initial time $\tau=0$. As it will be needed afterwards, we point out that we
also have (as in (7.10))

(7.27)
$\mathbb{E}\Bigl{[}\,\sup_{t\in[0,T]}W_{2}(m_{\boldsymbol{X}^{I_{j}^{N}}_{t}},m^{\lambda^{j}}_{t})\Bigr{]}\to
0\quad\text{as}\ N\to\infty.$

Step 3. We finally prove that (7.15) holds. (Throughout the proof we will
imply constants if they depend only on $T$, $L_{g}$, $L_{f}$ and
$L_{\Lambda}$.)

By the representation formula for the value functions solving the Nash system
constructed above,

(7.28)
$\mathbb{E}\,\mathfrak{u}_{N}({\boldsymbol{\lambda}_{N}},0,\boldsymbol{Z})=\mathbb{E}\biggl{[}\,\int_{0}^{T}\Big{(}\,\frac{1}{2}\big{|}\partial_{x}\mathfrak{u}_{N}({\boldsymbol{\lambda}_{N}},t,\boldsymbol{\tilde{X}}^{N}_{t})\big{|}^{2}+f_{N}(\overline{\lambda}^{1},\boldsymbol{\tilde{X}}^{N}_{t})\Big{)}\mathrm{d}t+g_{N}(\overline{\lambda}^{1},\boldsymbol{\tilde{X}}^{N}_{T})\biggr{]},$

where $\boldsymbol{\tilde{X}}^{N}$ solves
$\mathrm{d}\tilde{X}^{N,i}=-\partial_{x}\mathfrak{u}_{N}(\lambda_{N}^{i},\boldsymbol{\lambda}_{N}^{-i},t,\tilde{X}^{N,i}_{t},\boldsymbol{\tilde{X}}^{N,-i}_{t})\,\mathrm{d}t+\sqrt{2}\,\mathrm{d}B_{t}^{i}$,
$i\in\\{1,\dots,N\\}$.

In order to pass to the limit the previous identity, we need to compare
$\boldsymbol{\tilde{X}}^{N}$ and $\boldsymbol{X}^{N}$ constructed above. If
$i\in I^{N}_{j}$ we have

(7.29)
$\begin{multlined}\big{|}\tilde{X}^{N,i}_{s}-X^{I^{N}_{j},i}_{s}\big{|}^{2}\\\
\leqslant
2\int_{0}^{s}\Big{(}\big{|}\partial_{x}\mathfrak{u}^{N}(\lambda^{j},\boldsymbol{\lambda}_{N}^{-i},t,\tilde{X}^{N,i}_{t},\boldsymbol{\tilde{X}}^{N,-i}_{t})-\partial_{x}\mathfrak{u}^{N}(\lambda^{j},\boldsymbol{\lambda}_{N}^{-i},t,X^{I^{N}_{j},i}_{t},\boldsymbol{\tilde{X}}^{N,-i}_{t})\big{|}^{2}\vspace{-7pt}\\\
\hskip
45.5pt+\big{|}\partial_{x}\mathfrak{u}^{N}(\lambda^{j},\boldsymbol{\lambda}_{N}^{-i},t,X^{I^{N}_{j},i}_{t},\boldsymbol{\tilde{X}}^{N,-i}_{t})-\partial_{x}U(\lambda^{j},t,X^{I^{N}_{j},i}_{t},m_{(\boldsymbol{\lambda}_{N}:\boldsymbol{\tilde{X}}^{N}_{t})})\big{|}^{2}\\\
+\big{|}\partial_{x}U(\lambda^{j},t,X^{I^{N}_{j},i}_{t},m_{(\boldsymbol{\lambda}_{N}:\boldsymbol{\tilde{X}}^{N}_{t})})-\partial_{x}U^{\lambda^{j}}\bigl{(}t,X^{I^{N}_{j},i}_{t},(\boldsymbol{m}_{t}^{\boldsymbol{\lambda}^{-j}},m_{\boldsymbol{X}^{I^{N}_{j}}_{t}})\bigr{)}\big{|}^{2}\Big{)}\,\mathrm{d}t;\end{multlined}\big{|}\tilde{X}^{N,i}_{s}-X^{I^{N}_{j},i}_{s}\big{|}^{2}\\\
\leqslant
2\int_{0}^{s}\Big{(}\big{|}\partial_{x}\mathfrak{u}^{N}(\lambda^{j},\boldsymbol{\lambda}_{N}^{-i},t,\tilde{X}^{N,i}_{t},\boldsymbol{\tilde{X}}^{N,-i}_{t})-\partial_{x}\mathfrak{u}^{N}(\lambda^{j},\boldsymbol{\lambda}_{N}^{-i},t,X^{I^{N}_{j},i}_{t},\boldsymbol{\tilde{X}}^{N,-i}_{t})\big{|}^{2}\vspace{-7pt}\\\
\hskip
45.5pt+\big{|}\partial_{x}\mathfrak{u}^{N}(\lambda^{j},\boldsymbol{\lambda}_{N}^{-i},t,X^{I^{N}_{j},i}_{t},\boldsymbol{\tilde{X}}^{N,-i}_{t})-\partial_{x}U(\lambda^{j},t,X^{I^{N}_{j},i}_{t},m_{(\boldsymbol{\lambda}_{N}:\boldsymbol{\tilde{X}}^{N}_{t})})\big{|}^{2}\\\
+\big{|}\partial_{x}U(\lambda^{j},t,X^{I^{N}_{j},i}_{t},m_{(\boldsymbol{\lambda}_{N}:\boldsymbol{\tilde{X}}^{N}_{t})})-\partial_{x}U^{\lambda^{j}}\bigl{(}t,X^{I^{N}_{j},i}_{t},(\boldsymbol{m}_{t}^{\boldsymbol{\lambda}^{-j}},m_{\boldsymbol{X}^{I^{N}_{j}}_{t}})\bigr{)}\big{|}^{2}\Big{)}\,\mathrm{d}t;$

note that, according to the notation introduced in Section 2,

$m_{(\boldsymbol{\lambda}_{N}:\boldsymbol{\tilde{X}}^{N}_{t})}=\frac{1}{N}\sum_{1\leqslant
k\leqslant N}\delta_{\lambda^{k}_{N}}\\!\otimes\delta_{\tilde{X}^{N,k}_{t}}.$

By the Lipschitz continuity of $\partial_{x}U$,

(7.30)
$\begin{multlined}\big{|}\partial_{x}U(\lambda^{j},t,X^{I^{N}_{j},i}_{t},m_{(\boldsymbol{\lambda}_{N}:\boldsymbol{\tilde{X}}^{N}_{t})})-\partial_{x}U^{\lambda^{j}}\bigl{(}t,X^{I^{N}_{j},i}_{t},(\boldsymbol{m}_{t}^{\boldsymbol{\lambda}^{-j}},m_{\boldsymbol{X}^{I^{N}_{j}}_{t}})\bigr{)}\big{|}^{2}\\\
\lesssim
W_{2}\Big{(}m_{(\boldsymbol{\lambda}_{N}:\boldsymbol{\tilde{X}}^{N}_{t})},\
\theta_{j}\delta_{\lambda^{j}}\\!\otimes
m_{\boldsymbol{X}^{I^{N}_{j}}_{t}}+\sum_{\begin{subarray}{c}1\leqslant
k\leqslant\ell\\\ k\neq
j\end{subarray}}\theta_{k}\delta_{\lambda^{k}}\\!\otimes
m_{t}^{\lambda^{k}}\Big{)}^{2};\end{multlined}\big{|}\partial_{x}U(\lambda^{j},t,X^{I^{N}_{j},i}_{t},m_{(\boldsymbol{\lambda}_{N}:\boldsymbol{\tilde{X}}^{N}_{t})})-\partial_{x}U^{\lambda^{j}}\bigl{(}t,X^{I^{N}_{j},i}_{t},(\boldsymbol{m}_{t}^{\boldsymbol{\lambda}^{-j}},m_{\boldsymbol{X}^{I^{N}_{j}}_{t}})\bigr{)}\big{|}^{2}\\\
\lesssim
W_{2}\Big{(}m_{(\boldsymbol{\lambda}_{N}:\boldsymbol{\tilde{X}}^{N}_{t})},\
\theta_{j}\delta_{\lambda^{j}}\\!\otimes
m_{\boldsymbol{X}^{I^{N}_{j}}_{t}}+\sum_{\begin{subarray}{c}1\leqslant
k\leqslant\ell\\\ k\neq
j\end{subarray}}\theta_{k}\delta_{\lambda^{k}}\\!\otimes
m_{t}^{\lambda^{k}}\Big{)}^{2};$

on the other hand, recalling the definition of $\boldsymbol{\lambda}_{N}$ and
then controlling the Wasserstein distance with the total variation (cf., e.g.,
[43, Theorem 6.15]), one easily gets

$W_{2}\Big{(}m_{(\boldsymbol{\lambda}_{N}:\boldsymbol{\tilde{X}}^{N}_{t})},\,\sum_{1\leqslant
k\leqslant\ell}\theta_{k}\delta_{\lambda^{k}}\\!\otimes\frac{1}{\\#I^{N}_{k}}\sum_{i\in
I_{k}^{N}}\delta_{\tilde{X}^{N,i}_{t}}\Big{)}^{2}\leqslant 2\sup_{1\leqslant
i\leqslant N}|\tilde{X}^{N,i}_{t}|^{2}\sum_{1\leqslant
k\leqslant\ell}\Big{\lvert}\frac{\\#I^{N}_{k}}{N}-\theta_{k}\Big{\rvert}=\vcentcolon\gamma^{N}_{t},$

where $\mathbb{E}\bigl{[}\sup_{[0,T]}\gamma^{N}\bigr{]}\to 0$ as $N\to\infty$
thanks to the uniform boundedness of $\partial_{x}\mathfrak{u}_{N}$. Then
applying the triangle inequality to the right-hand side of (7.30) we see that
it is controlled by

(7.31) $W_{2}\Big{(}\sum_{1\leqslant
k\leqslant\ell}\theta_{k}\delta_{\lambda^{k}}\\!\otimes\frac{1}{\\#I^{N}_{k}}\sum_{i\in
I_{k}^{N}}\delta_{\tilde{X}^{N,i}_{t}},\
\theta_{j}\delta_{\lambda^{j}}\\!\otimes
m_{\boldsymbol{X}^{I^{N}_{j}}_{t}}+\sum_{\begin{subarray}{c}1\leqslant
k\leqslant\ell\\\ k\neq
j\end{subarray}}\theta_{k}\delta_{\lambda^{k}}\\!\otimes
m_{t}^{\lambda^{k}}\Big{)}^{2}+\gamma^{N}_{t}.$

Now, by the convexity of $W_{2}^{2}$ (cf., e.g., [43, Theorem 4.8]), the first
term in (7.31) is in turn controlled by

$\begin{multlined}\theta_{j}W_{2}\Big{(}\frac{1}{\\#I^{N}_{j}}\sum_{i\in
I_{j}^{N}}\delta_{\tilde{X}^{N,i}_{t}},\
m_{\boldsymbol{X}^{I^{N}_{j}}_{t}}\Big{)}^{2}+\sum_{\begin{subarray}{c}1\leqslant
k\leqslant\ell\\\ k\neq
j\end{subarray}}\theta_{k}W_{2}\Big{(}\,\frac{1}{\\#I^{N}_{k}}\sum_{i\in
I_{k}^{N}}\delta_{\tilde{X}^{N,i}_{t}},\,m^{\lambda^{k}}_{t}\Big{)}^{2}\\\
\leqslant 2\sum_{1\leqslant k\leqslant\ell}\frac{1}{\\#I_{k}^{N}}\sum_{i\in
I_{k}^{N}}\big{|}\tilde{X}^{N,i}_{t}-X^{I^{N}_{k},i}_{t}\big{|}^{2}+2\sum_{1\leqslant
k\leqslant\ell}W_{2}(m_{\boldsymbol{X}^{I_{k}^{N}}_{t}},m^{\lambda^{k}}_{t})^{2}\end{multlined}\theta_{j}W_{2}\Big{(}\frac{1}{\\#I^{N}_{j}}\sum_{i\in
I_{j}^{N}}\delta_{\tilde{X}^{N,i}_{t}},\
m_{\boldsymbol{X}^{I^{N}_{j}}_{t}}\Big{)}^{2}+\sum_{\begin{subarray}{c}1\leqslant
k\leqslant\ell\\\ k\neq
j\end{subarray}}\theta_{k}W_{2}\Big{(}\,\frac{1}{\\#I^{N}_{k}}\sum_{i\in
I_{k}^{N}}\delta_{\tilde{X}^{N,i}_{t}},\,m^{\lambda^{k}}_{t}\Big{)}^{2}\\\
\leqslant 2\sum_{1\leqslant k\leqslant\ell}\frac{1}{\\#I_{k}^{N}}\sum_{i\in
I_{k}^{N}}\big{|}\tilde{X}^{N,i}_{t}-X^{I^{N}_{k},i}_{t}\big{|}^{2}+2\sum_{1\leqslant
k\leqslant\ell}W_{2}(m_{\boldsymbol{X}^{I_{k}^{N}}_{t}},m^{\lambda^{k}}_{t})^{2}$

Therefore, from (7.29) we have

(7.32)
$\big{|}\tilde{X}^{N,i}_{s}-X^{I^{N}_{j},i}_{s}\big{|}^{2}\lesssim\int_{0}^{s}\bigg{(}\sum_{1\leqslant
k\leqslant\ell}\frac{1}{\\#I_{k}^{N}}\sum_{i\in
I_{k}^{N}}\big{|}\tilde{X}^{N,i}_{t}-X^{I^{N}_{k},i}_{t}\big{|}^{2}+\delta^{N,j}_{t}\bigg{)}\,\mathrm{d}t,$

where

$\begin{multlined}\delta^{N,j}_{t}\vcentcolon=\big{|}\partial_{x}\mathfrak{u}_{N}(\lambda^{j},{\boldsymbol{\lambda}_{N}^{-i}},t,X^{I^{N}_{j},i}_{t},\boldsymbol{\tilde{X}}^{N,-i}_{t})-\partial_{x}U(\lambda^{j},t,X^{I^{N}_{j},i}_{t},m_{(\boldsymbol{\lambda}^{N},\boldsymbol{\tilde{X}}^{N}_{t})})\big{|}^{2}+\gamma^{N}_{t}\\\
+\sum_{1\leqslant
k\leqslant\ell}W_{2}(m_{\boldsymbol{X}^{I_{k}^{N}}_{t}},m^{\lambda^{k}}_{t})^{2}\end{multlined}\delta^{N,j}_{t}\vcentcolon=\big{|}\partial_{x}\mathfrak{u}_{N}(\lambda^{j},{\boldsymbol{\lambda}_{N}^{-i}},t,X^{I^{N}_{j},i}_{t},\boldsymbol{\tilde{X}}^{N,-i}_{t})-\partial_{x}U(\lambda^{j},t,X^{I^{N}_{j},i}_{t},m_{(\boldsymbol{\lambda}^{N},\boldsymbol{\tilde{X}}^{N}_{t})})\big{|}^{2}+\gamma^{N}_{t}\\\
+\sum_{1\leqslant
k\leqslant\ell}W_{2}(m_{\boldsymbol{X}^{I_{k}^{N}}_{t}},m^{\lambda^{k}}_{t})^{2}$

is such that $\mathbb{E}\big{[}\sup_{[0,T]}\delta^{N,j}_{t}\big{]}\to 0$ as
$N\to\infty$ (cf. Theorem 7.5, property (7.27); see also the discussion after
equation (7.14)). Averaging (7.32) over $i\in I_{j}$ and summing over
$j\in\\{1,\dots,\ell\\}$ we have

$\sum_{1\leqslant j\leqslant\ell}\frac{1}{\\#I_{j}^{N}}\sum_{i\in
I_{j}^{N}}\big{|}\tilde{X}^{N,i}_{s}-X^{I^{N}_{j},i}_{s}\big{|}^{2}\lesssim\int_{0}^{s}\bigg{(}\ell\sum_{1\leqslant
j\leqslant\ell}\frac{1}{\\#I_{j}^{N}}\sum_{i\in
I_{k}^{N}}\big{|}\tilde{X}^{N,i}_{t}-X^{I^{N}_{j},i}_{t}\big{|}^{2}+\sum_{1\leqslant
j\leqslant\ell}\delta^{N,j}_{t}\bigg{)}\,\mathrm{d}t,$

whence

$\sum_{1\leqslant j\leqslant\ell}\frac{1}{\\#I_{j}^{N}}\sum_{i\in
I_{j}^{N}}\mathbb{E}\Bigl{[}\,\sup_{t\in[0,T]}\big{|}\tilde{X}^{N,i}_{t}-X^{I^{N}_{j},i}_{t}\big{|}^{2}\Bigr{]}\to
0\quad\text{as}\ N\to\infty.$

by Gronwall’s lemma; then, plugging this back into (7.32),

(7.33) $\sup_{1\leqslant j\leqslant\ell}\sup_{i\in
I_{j}^{N}}\mathbb{E}\Bigl{[}\,\sup_{t\in[0,T]}\big{|}\tilde{X}^{N,i}_{t}-X^{I^{N}_{j},i}_{t}\big{|}^{2}\Bigr{]}\to
0\quad\text{as}\ N\to\infty.$

Finally, having fixed $j\in\\{1,\dots,\ell\\}$, note that we can assume
without loss of generality that $1\in I_{j}$ (otherwise, just take a different
partition $\\{\Theta_{j}\\}_{1\leqslant j\leqslant\ell}$ at the beginning of
Step 2); then $\overline{\lambda}^{1}=\lambda^{j}$, and one deduces (7.15)
from (7.26) and (7.28). To motivate this last claim, for the sake of
completeness, we prove that, for $h\in\\{f,g\\}$, one has

$\mathbb{E}\Big{[}\sup_{t\in[0,T]}\big{|}h^{\lambda^{j}}\big{(}X^{I_{j}^{N},i}_{t},(\boldsymbol{m}_{t}^{\boldsymbol{\lambda}^{-j}},m_{\boldsymbol{X}^{I_{j}^{N}}_{t}})\big{)}-h_{N}(\lambda^{j},\boldsymbol{\tilde{X}}^{N}_{t})\big{|}\Big{]}\to
0\quad\text{as}\ N\to\infty$

(the other analogous convergence that we will need, involving
$\partial_{x}\mathfrak{u}_{N}$ and $U$, can be proved in a similar manner).
Under our assumptions, $X^{I^{N}_{j},i}=X^{I^{N}_{j},1}$, so, by the Lipschitz
continuity of $h$ and the triangle inequality,

$\begin{multlined}\big{|}h^{\lambda^{j}}\big{(}X^{I_{j}^{N},i}_{t},(\boldsymbol{m}_{t}^{\boldsymbol{\lambda}^{-j}},m_{\boldsymbol{X}^{I_{j}^{N}}_{t}})\big{)}-h_{N}(\lambda^{j},\boldsymbol{\tilde{X}}^{N}_{t})\big{|}^{2}\\\
\begin{multlined}\lesssim\bigl{|}X^{I^{N}_{j},1}_{t}-\tilde{X}^{N,1}_{t}\bigr{|}^{2}+W_{2}\Bigl{(}\theta_{j}m_{\boldsymbol{X}^{I_{j}^{N}}_{t}}+\sum_{\begin{subarray}{c}1\leqslant
k\leqslant\ell\\\ k\neq
j\end{subarray}}\theta_{k}m^{\lambda^{k}}_{t},\,m_{\boldsymbol{\tilde{X}}^{N}_{t}}\Bigr{)}^{2}\\\
+W_{2}(m_{\boldsymbol{\tilde{X}}^{N}_{t}},m_{\boldsymbol{\tilde{X}}^{N,-1}_{t}})^{2}+\bigl{|}h^{\lambda^{j}}(\tilde{X}^{N,1}_{t},m_{\boldsymbol{\tilde{X}}^{N,-1}_{t}})-h_{N}(\lambda^{j},\boldsymbol{\tilde{X}}^{N}_{t})\bigr{|}^{2};\end{multlined}\lesssim\bigl{|}X^{I^{N}_{j},1}_{t}-\tilde{X}^{N,1}_{t}\bigr{|}^{2}+W_{2}\Bigl{(}\theta_{j}m_{\boldsymbol{X}^{I_{j}^{N}}_{t}}+\sum_{\begin{subarray}{c}1\leqslant
k\leqslant\ell\\\ k\neq
j\end{subarray}}\theta_{k}m^{\lambda^{k}}_{t},\,m_{\boldsymbol{\tilde{X}}^{N}_{t}}\Bigr{)}^{2}\\\
+W_{2}(m_{\boldsymbol{\tilde{X}}^{N}_{t}},m_{\boldsymbol{\tilde{X}}^{N,-1}_{t}})^{2}+\bigl{|}h^{\lambda^{j}}(\tilde{X}^{N,1}_{t},m_{\boldsymbol{\tilde{X}}^{N,-1}_{t}})-h_{N}(\lambda^{j},\boldsymbol{\tilde{X}}^{N}_{t})\bigr{|}^{2};\end{multlined}\big{|}h^{\lambda^{j}}\big{(}X^{I_{j}^{N},i}_{t},(\boldsymbol{m}_{t}^{\boldsymbol{\lambda}^{-j}},m_{\boldsymbol{X}^{I_{j}^{N}}_{t}})\big{)}-h_{N}(\lambda^{j},\boldsymbol{\tilde{X}}^{N}_{t})\big{|}^{2}\\\
\begin{multlined}\lesssim\bigl{|}X^{I^{N}_{j},1}_{t}-\tilde{X}^{N,1}_{t}\bigr{|}^{2}+W_{2}\Bigl{(}\theta_{j}m_{\boldsymbol{X}^{I_{j}^{N}}_{t}}+\sum_{\begin{subarray}{c}1\leqslant
k\leqslant\ell\\\ k\neq
j\end{subarray}}\theta_{k}m^{\lambda^{k}}_{t},\,m_{\boldsymbol{\tilde{X}}^{N}_{t}}\Bigr{)}^{2}\\\
+W_{2}(m_{\boldsymbol{\tilde{X}}^{N}_{t}},m_{\boldsymbol{\tilde{X}}^{N,-1}_{t}})^{2}+\bigl{|}h^{\lambda^{j}}(\tilde{X}^{N,1}_{t},m_{\boldsymbol{\tilde{X}}^{N,-1}_{t}})-h_{N}(\lambda^{j},\boldsymbol{\tilde{X}}^{N}_{t})\bigr{|}^{2};\end{multlined}\lesssim\bigl{|}X^{I^{N}_{j},1}_{t}-\tilde{X}^{N,1}_{t}\bigr{|}^{2}+W_{2}\Bigl{(}\theta_{j}m_{\boldsymbol{X}^{I_{j}^{N}}_{t}}+\sum_{\begin{subarray}{c}1\leqslant
k\leqslant\ell\\\ k\neq
j\end{subarray}}\theta_{k}m^{\lambda^{k}}_{t},\,m_{\boldsymbol{\tilde{X}}^{N}_{t}}\Bigr{)}^{2}\\\
+W_{2}(m_{\boldsymbol{\tilde{X}}^{N}_{t}},m_{\boldsymbol{\tilde{X}}^{N,-1}_{t}})^{2}+\bigl{|}h^{\lambda^{j}}(\tilde{X}^{N,1}_{t},m_{\boldsymbol{\tilde{X}}^{N,-1}_{t}})-h_{N}(\lambda^{j},\boldsymbol{\tilde{X}}^{N}_{t})\bigr{|}^{2};$

it is straightforward to show that, after taking the supremum over $t\in[0,T]$
and then the expectation, the third term in the left-hand side vanished as
$N\to\infty$, while the first one and the last one go to $0$ as well thanks to
(7.33) and Remark 7.7, respectively. As for the second term, by the triangle
inequality and convexity of $W_{2}^{2}$,

$\begin{multlined}W_{1}\Bigl{(}\theta_{j}m_{\boldsymbol{X}^{I_{j}^{N}}_{t}}+\sum_{\begin{subarray}{c}1\leqslant
k\leqslant\ell\\\ k\neq
j\end{subarray}}\theta_{k}m^{\lambda^{k}}_{t},\,m_{\boldsymbol{\tilde{X}}^{N}_{t}}\Bigr{)}\\\
\begin{multlined}\leqslant\theta_{j}W_{2}\Bigl{(}m_{\boldsymbol{X}^{I_{j}^{N}}_{t}},\frac{1}{\\#I^{N}_{j}}\sum_{i\in
I^{N}_{j}}\delta_{\tilde{X}^{N,i}_{t}}\Bigr{)}^{2}+\sum_{\begin{subarray}{c}1\leqslant
k\leqslant\ell\\\ k\neq
j\end{subarray}}\theta_{k}W_{2}\Bigl{(}m^{\lambda^{k}}_{t},\,\frac{1}{\\#I^{N}_{k}}\sum_{i\in
I^{N}_{k}}\delta_{\tilde{X}^{N,i}_{t}}\Bigr{)}^{2}\\\\[-3.0pt]
+W_{2}\Bigl{(}\sum_{1\leqslant
k\leqslant\ell}\frac{\theta_{k}}{\\#I^{N}_{k}}\sum_{i\in
I^{N}_{k}}\delta_{\tilde{X}^{N,i}_{t}},\,m_{\boldsymbol{\tilde{X}}^{N}_{t}}\Bigr{)}^{2};\end{multlined}\leqslant\theta_{j}W_{2}\Bigl{(}m_{\boldsymbol{X}^{I_{j}^{N}}_{t}},\frac{1}{\\#I^{N}_{j}}\sum_{i\in
I^{N}_{j}}\delta_{\tilde{X}^{N,i}_{t}}\Bigr{)}^{2}+\sum_{\begin{subarray}{c}1\leqslant
k\leqslant\ell\\\ k\neq
j\end{subarray}}\theta_{k}W_{2}\Bigl{(}m^{\lambda^{k}}_{t},\,\frac{1}{\\#I^{N}_{k}}\sum_{i\in
I^{N}_{k}}\delta_{\tilde{X}^{N,i}_{t}}\Bigr{)}^{2}\\\\[-3.0pt]
+W_{2}\Bigl{(}\sum_{1\leqslant
k\leqslant\ell}\frac{\theta_{k}}{\\#I^{N}_{k}}\sum_{i\in
I^{N}_{k}}\delta_{\tilde{X}^{N,i}_{t}},\,m_{\boldsymbol{\tilde{X}}^{N}_{t}}\Bigr{)}^{2};\end{multlined}W_{1}\Bigl{(}\theta_{j}m_{\boldsymbol{X}^{I_{j}^{N}}_{t}}+\sum_{\begin{subarray}{c}1\leqslant
k\leqslant\ell\\\ k\neq
j\end{subarray}}\theta_{k}m^{\lambda^{k}}_{t},\,m_{\boldsymbol{\tilde{X}}^{N}_{t}}\Bigr{)}\\\
\begin{multlined}\leqslant\theta_{j}W_{2}\Bigl{(}m_{\boldsymbol{X}^{I_{j}^{N}}_{t}},\frac{1}{\\#I^{N}_{j}}\sum_{i\in
I^{N}_{j}}\delta_{\tilde{X}^{N,i}_{t}}\Bigr{)}^{2}+\sum_{\begin{subarray}{c}1\leqslant
k\leqslant\ell\\\ k\neq
j\end{subarray}}\theta_{k}W_{2}\Bigl{(}m^{\lambda^{k}}_{t},\,\frac{1}{\\#I^{N}_{k}}\sum_{i\in
I^{N}_{k}}\delta_{\tilde{X}^{N,i}_{t}}\Bigr{)}^{2}\\\\[-3.0pt]
+W_{2}\Bigl{(}\sum_{1\leqslant
k\leqslant\ell}\frac{\theta_{k}}{\\#I^{N}_{k}}\sum_{i\in
I^{N}_{k}}\delta_{\tilde{X}^{N,i}_{t}},\,m_{\boldsymbol{\tilde{X}}^{N}_{t}}\Bigr{)}^{2};\end{multlined}\leqslant\theta_{j}W_{2}\Bigl{(}m_{\boldsymbol{X}^{I_{j}^{N}}_{t}},\frac{1}{\\#I^{N}_{j}}\sum_{i\in
I^{N}_{j}}\delta_{\tilde{X}^{N,i}_{t}}\Bigr{)}^{2}+\sum_{\begin{subarray}{c}1\leqslant
k\leqslant\ell\\\ k\neq
j\end{subarray}}\theta_{k}W_{2}\Bigl{(}m^{\lambda^{k}}_{t},\,\frac{1}{\\#I^{N}_{k}}\sum_{i\in
I^{N}_{k}}\delta_{\tilde{X}^{N,i}_{t}}\Bigr{)}^{2}\\\\[-3.0pt]
+W_{2}\Bigl{(}\sum_{1\leqslant
k\leqslant\ell}\frac{\theta_{k}}{\\#I^{N}_{k}}\sum_{i\in
I^{N}_{k}}\delta_{\tilde{X}^{N,i}_{t}},\,m_{\boldsymbol{\tilde{X}}^{N}_{t}}\Bigr{)}^{2};$

by using the estimates previously obtained, the expectation of the supremum
over $t\in[0,T]$ of the right-hand side can be shown to vanish as
$N\to\infty$. ∎

###  7.2. A continuum of MFG systems

Some considerations about the characterisation provided by
(7.15)–(7.16)–(7.17) are now in order.

######  Remark 7.12.

If $U^{\lambda}$ is sufficiently regular on
$[0,T]\times{\mathbb{R}}^{d}\times\mathcal{P}_{2}(\Lambda\times{\mathbb{R}}^{d})$
(e.g., according to Definition 2.4.1 in [14] of a classical solution to the
first-order Master Equation, so, in particular it is differentiable with
respect to $\mu$), one can differentiate
$t\mapsto\int_{{\mathbb{R}}^{d}}U^{\lambda}(t,\mspace{1.8mu}\cdot\mspace{1.8mu},\mu_{t})\,\mathrm{d}m^{\lambda}_{t}$
with $m_{\tau}^{\lambda}=\delta_{x}$ and use (7.15), (7.16), (7.17) to get,
after standard computations,

$\begin{multlined}-\partial_{t}U^{\lambda}(t,x,\boldsymbol{\mu})-\Delta_{x}U^{\lambda}(t,x,\boldsymbol{\mu})-\int_{\Lambda\times{\mathbb{R}}^{d}}\Delta_{y}\frac{\delta}{\delta\boldsymbol{\mu}}U^{\lambda}(t,x,\boldsymbol{\mu},\lambda^{\prime},y)\,\boldsymbol{\mu}(\mathrm{d}\lambda^{\prime},\mathrm{d}y)\\\
+\frac{1}{2}\bigl{|}\partial_{x}U^{\lambda}(t,x,\boldsymbol{\mu})\bigr{|}^{2}+\int_{\Lambda\times{\mathbb{R}}^{d}}D_{y}\frac{\delta}{\delta\boldsymbol{\mu}}U^{\lambda}(t,x,\boldsymbol{\mu},\lambda^{\prime},y)\cdot\partial_{y}U^{\lambda^{\prime}}(t,y,\boldsymbol{\mu})\,\boldsymbol{\mu}(\mathrm{d}\lambda^{\prime},\mathrm{d}y)=f^{\lambda}(x,\pi_{{\mathbb{R}}^{d}}{}_{\sharp}\boldsymbol{\mu}).\end{multlined}-\partial_{t}U^{\lambda}(t,x,\boldsymbol{\mu})-\Delta_{x}U^{\lambda}(t,x,\boldsymbol{\mu})-\int_{\Lambda\times{\mathbb{R}}^{d}}\Delta_{y}\frac{\delta}{\delta\boldsymbol{\mu}}U^{\lambda}(t,x,\boldsymbol{\mu},\lambda^{\prime},y)\,\boldsymbol{\mu}(\mathrm{d}\lambda^{\prime},\mathrm{d}y)\\\
+\frac{1}{2}\bigl{|}\partial_{x}U^{\lambda}(t,x,\boldsymbol{\mu})\bigr{|}^{2}+\int_{\Lambda\times{\mathbb{R}}^{d}}D_{y}\frac{\delta}{\delta\boldsymbol{\mu}}U^{\lambda}(t,x,\boldsymbol{\mu},\lambda^{\prime},y)\cdot\partial_{y}U^{\lambda^{\prime}}(t,y,\boldsymbol{\mu})\,\boldsymbol{\mu}(\mathrm{d}\lambda^{\prime},\mathrm{d}y)=f^{\lambda}(x,\pi_{{\mathbb{R}}^{d}}{}_{\sharp}\boldsymbol{\mu}).$

This is exactly the form one would expect a Master Equation associated to our
non-symmetric game to exhibit. In this sense, the limit game
(7.15)–(7.16)–(7.17) can be considered a _weak formulation of the Master
Equation for a nonsymmetric game_ parametrised by $\lambda\in\Lambda$. Since
$U^{\lambda}$ is Lipschitz continuous with respect to the measure variable,
this formulation is very close to the one proposed in [9]. Moreover, it brings
naturally a continuum of standard MFG systems, as described below.

######  Theorem 7.13.

With the notation of Theorem 7.9, define
$u^{\lambda}(t,x)\vcentcolon=U^{\lambda}(t,x,\mu_{t})$. Then $u$ solves the
following (generalised) Mean-Field system on
$(0,T)\times\Lambda\times{\mathbb{R}}^{d}$:

(7.34)
$\begin{dcases}-\partial_{t}u^{\lambda}-\Delta_{x}u^{\lambda}+\frac{1}{2}|D_{x}u^{\lambda}|^{2}=f^{\lambda}(x,\pi_{{\mathbb{R}}^{d}}{}_{\sharp}\mu_{t})\\\
\partial_{t}\mu-\Delta_{x}\mu-\mathrm{div}_{x}(Du^{\lambda}\mu)=0\\\
u^{\lambda}(T,\cdot)=g^{\lambda}(\cdot,\pi_{{\mathbb{R}}^{d}}{}_{\sharp}\mu_{T}),\quad\mu_{0}=\boldsymbol{\mu},\end{dcases}$

where both the Hamilton–Jacobi and the Fokker–Planck equations are satisfied
in the classical sense.

###### Proof.

Fix $\lambda\in\Lambda$. Note that, by (7.15), for every $\tau$ and smooth
$\overline{\mathfrak{m}}^{\lambda}$,

(7.35)
$\int_{{\mathbb{R}}^{d}}u^{\lambda}(\tau,\cdot)\,\mathrm{d}\overline{\mathfrak{m}}^{\lambda}=\int_{\tau}^{T}\int_{{\mathbb{R}}^{d}}\Big{(}\,\frac{1}{2}\big{|}D_{x}u^{\lambda}(s,\cdot)\big{|}^{2}+f^{\lambda}(\mspace{1.8mu}\cdot\mspace{1.8mu},{\pi_{{\mathbb{R}}^{d}}}{}_{\sharp}\mu_{s})\Big{)}\,\mathrm{d}\overline{m}^{\lambda}_{s}\mathrm{d}s+\int_{{\mathbb{R}}^{d}}g^{\lambda}(\mspace{1.8mu}\cdot\mspace{1.8mu},{\pi_{{\mathbb{R}}^{d}}}{}_{\sharp}\mu_{T})\,\mathrm{d}\overline{m}^{\lambda}_{T}.$

Let $v$ be solution of the linear backward Cauchy problem

$\begin{dcases}-\partial_{t}v-\Delta_{x}v+D_{x}u^{\lambda}\cdot
D_{x}v=f^{\lambda}(x,\pi_{{\mathbb{R}}^{d}}{}_{\sharp}\mu_{t})+\frac{1}{2}\big{|}D_{x}u^{\lambda}\big{|}^{2}\\\
v(T,\cdot)=g^{\lambda}(\cdot,\pi_{{\mathbb{R}}^{d}}{}_{\sharp}\mu_{T}).\end{dcases}$

By standard linear theory [34], $v$ solves the equation a.e., and since
coefficients are in $L^{\infty}$, it is globally bounded and locally in
$W^{2,1}_{p}$ for every $p\in(1,\infty)$. By duality between the equation for
$v$ and the equation for $\overline{m}^{\lambda}$, for every $\tau\in[0,T]$ we
have

$\int_{{\mathbb{R}}^{d}}v(\tau,\cdot)\,\mathrm{d}\overline{\mathfrak{m}}^{\lambda}=\int_{\tau}^{T}\int_{{\mathbb{R}}^{d}}\Big{(}\,\frac{1}{2}\big{|}D_{x}u^{\lambda}(s,\cdot)\big{|}^{2}+f^{\lambda}(\mspace{1.8mu}\cdot\mspace{1.8mu},{\pi_{{\mathbb{R}}^{d}}}{}_{\sharp}\mu_{s})\Big{)}\,\mathrm{d}\overline{m}^{\lambda}_{s}\mathrm{d}s+\int_{{\mathbb{R}}^{d}}g^{\lambda}(\mspace{1.8mu}\cdot\mspace{1.8mu},{\pi_{{\mathbb{R}}^{d}}}{}_{\sharp}\mu_{T})\,\mathrm{d}\overline{m}^{\lambda}_{T}.$

Since the previous equation has the same right-hand side of (7.35), by the
arbitrariness of $\overline{\mathfrak{m}}^{\lambda}$ and $\tau$, we obtain
$v\equiv u^{\lambda}$. Plugging back $D_{x}v\equiv D_{x}u^{\lambda}$ into the
equation for $v$ we obtain the assertion. That $u^{\lambda}$ is a classical
solution follows by standard parabolic regularity. ∎

######  Remark 7.14.

If $f$ and $g$ are constant in $\lambda$, then (7.34) is the classic Mean-
Field system (this case corresponds to the situation in Proposition 7.8). We
also mention that, as a byproduct, under assumptions of monotonicity, one can
then deduce uniqueness of the limit function $U$ by classic uniqueness results
for the Mean-Field system.

##  Appendix A An existence and uniqueness theorem for the Nash system

For the sake of self-containedness, we present an existence theorem for (1.1).

######  Theorem A.1.

Let $f^{i}\in C^{\gamma+\alpha}(({\mathbb{R}}^{d})^{N})$ and $g^{i}\in
C^{2+\gamma+\alpha}(({\mathbb{R}}^{d})^{N})$, for some integer
$\gamma\geqslant 1$ and $\alpha\in(0,1)$. Then for any $T>0$ there exists a
unique bounded solution to the Nash system (1.1) which is of class
$C^{1+\gamma/2+\alpha/2,2+\gamma+\alpha}$.

###### Proof.

Step 1. For $\tau\in[0,T)$, let
$\mathcal{X}\vcentcolon=\mathcal{X}_{\tau}=C([\tau,T];C_{b}^{1}(({\mathbb{R}}^{d})^{N})\,)^{N}$,
with norm $\lVert v\rVert_{\mathcal{X}}\vcentcolon=\sup_{1\leqslant i\leqslant
N}(\lVert v^{i}\rVert_{\infty}+\lVert Dv^{i}\rVert_{\infty})$. For
$v\in\mathcal{X}$, let $u^{i}=\mathcal{S}(v)^{i}$ be the solution of the
parabolic equation

$\begin{cases}-\partial_{t}u^{i}-\operatorname{tr}((\sigma\boldsymbol{\mathsf{I}}+\beta\boldsymbol{\mathsf{J}})D^{2}u^{i})=f^{i}-\dfrac{1}{2}|D_{i}v^{i}|^{2}-\displaystyle\sum_{\begin{subarray}{c}1\leqslant
j\leqslant N\\\ j\neq i\end{subarray}}D_{j}v^{j}\cdot
D_{j}v^{i}\vspace{-0.3cm}\\\ u^{i}|_{t=T}=g^{i}.\end{cases}$