$\displaystyle\int_{\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{2}\times[0,t]}\sigma(x,y,\tilde{\mu}^{\epsilon,N,k}_{s})z_{1}\phi^{\prime}(x)Q^{N,k}(dx,dy,dz,ds)$
$\displaystyle+\int_{\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{2}\times[0,t]}[\tau_{1}(x,y,\tilde{\mu}^{\epsilon,N,k}_{s})z_{1}+\tau_{2}(x,y,\tilde{\mu}^{\epsilon,N,k}_{s})z_{2}]\Phi_{y}(x,y,\tilde{\mu}^{\epsilon,N,k}_{s})\phi^{\prime}(x)Q^{N,k}(dx,dy,dz,ds)$
$\displaystyle=\int_{0}^{t}\frac{1}{N}\sum_{i=1}^{N}\sigma(\tilde{X}^{i,\epsilon,N,k}_{s},\tilde{Y}^{i,\epsilon,N,k}_{s},\tilde{\mu}^{\epsilon,N,k}_{s})\psi^{k}_{1}(s,\tilde{X}^{i,\epsilon,N,k}_{s},\tilde{Y}^{i,\epsilon,N,k}_{s})\phi^{\prime}(\tilde{X}^{i,\epsilon,N,k}_{s})ds$
$\displaystyle+\int_{0}^{t}\frac{1}{N}\sum_{i=1}^{N}\left[\tau_{1}(\tilde{X}^{i,\epsilon,N,k}_{s},\tilde{Y}^{i,\epsilon,N,k}_{s},\tilde{\mu}^{\epsilon,N,k}_{s})\psi^{k}_{1}(s,\tilde{X}^{i,\epsilon,N,k}_{s},\tilde{Y}^{i,\epsilon,N,k}_{s})+\right.$
$\displaystyle\qquad\left.+\tau_{2}(\tilde{X}^{i,\epsilon,N,k}_{s},\tilde{Y}^{i,\epsilon,N,k}_{s},\tilde{\mu}^{\epsilon,N,k}_{s})\psi^{k}_{2}(s,\tilde{X}^{i,\epsilon,N,k}_{s},\tilde{Y}^{i,\epsilon,N,k}_{s})\right]\Phi_{y}(\tilde{X}^{i,\epsilon,N,k}_{s},\tilde{Y}^{i,\epsilon,N,k}_{s},\tilde{\mu}^{\epsilon,N,k}_{s}\phi^{\prime}(\tilde{X}^{i,\epsilon,N,k}_{s})ds$

Then using Proposition 6.4 with

$\displaystyle
F(s,x,y,\mu)=\sigma(x,y,\mu)\psi^{k}_{1}(s,x,y)+[\tau_{1}(x,y,\mu)\psi^{k}_{1}(s,x,y)+\tau_{2}(x,y,\mu)\psi^{k}_{2}(s,x,y)]\Phi_{y}(x,y,\mu)$

using that $s$ only appears as a parameter, in the same way as $x$, so that
the same proof holds (using also the assumed bound on the time derivative of
$\Xi$ in A8)), we get that

$\displaystyle\mathbb{E}\biggl{[}\biggl{|}\int_{0}^{t}\frac{1}{N}\sum_{i=1}^{N}\sigma(\tilde{X}^{i,\epsilon,N,k}_{s},\tilde{Y}^{i,\epsilon,N,k}_{s},\tilde{\mu}^{\epsilon,N,k}_{s})\psi^{k}_{1}(s,\tilde{X}^{i,\epsilon,N,k}_{s},\tilde{Y}^{i,\epsilon,N,k}_{s})\phi^{\prime}(\tilde{X}^{i,\epsilon,N,k}_{s})ds$
$\displaystyle+\int_{0}^{t}\frac{1}{N}\sum_{i=1}^{N}\left[\tau_{1}(\tilde{X}^{i,\epsilon,N,k}_{s},\tilde{Y}^{i,\epsilon,N,k}_{s},\tilde{\mu}^{\epsilon,N,k}_{s})\psi^{k}_{1}(s,\tilde{X}^{i,\epsilon,N,k}_{s},\tilde{Y}^{i,\epsilon,N,k}_{s})+\right.$
$\displaystyle\qquad\left.+\tau_{2}(\tilde{X}^{i,\epsilon,N,k}_{s},\tilde{Y}^{i,\epsilon,N,k}_{s},\tilde{\mu}^{\epsilon,N,k}_{s})\psi^{k}_{2}(s,\tilde{X}^{i,\epsilon,N,k}_{s},\tilde{Y}^{i,\epsilon,N,k}_{s})\right]\Phi_{y}(\tilde{X}^{i,\epsilon,N,k}_{s},\tilde{Y}^{i,\epsilon,N,k}_{s},\tilde{\mu}^{\epsilon,N,k}_{s})\phi^{\prime}(\tilde{X}^{i,\epsilon,N,k}_{s})ds$
$\displaystyle-\int_{0}^{t}\frac{1}{N}\sum_{i=1}^{N}\int_{\mathbb{R}}\sigma(\tilde{X}^{i,\epsilon,N,k}_{s},y,\tilde{\mu}^{\epsilon,N,k}_{s})\psi^{k}_{1}(s,\tilde{X}^{i,\epsilon,N,k}_{s},y)\phi^{\prime}(\tilde{X}^{i,\epsilon,N,k}_{s})$
$\displaystyle\quad+[\tau_{1}(\tilde{X}^{i,\epsilon,N,k}_{s},y,\tilde{\mu}^{\epsilon,N,k}_{s})\psi^{k}_{1}(s,\tilde{X}^{i,\epsilon,N,k}_{s},y)+\tau_{2}(\tilde{X}^{i,\epsilon,N,k}_{s},y,\tilde{\mu}^{\epsilon,N,k}_{s})\psi^{k}_{2}(s,\tilde{X}^{i,\epsilon,N,k}_{s},y)]\times$
$\displaystyle\hskip
142.26378pt\times\Phi_{y}(\tilde{X}^{i,\epsilon,N,k}_{s},y,\tilde{\mu}^{\epsilon,N,k}_{s})\psi(s,\tilde{X}^{i,\epsilon,N,k}_{s})\pi(dy;\tilde{X}^{i,\epsilon,N}_{s},\tilde{\mu}^{\epsilon,N}_{s})ds\biggr{|}\biggr{]}$
$\displaystyle\leq C(T)\epsilon$

Then noting that

$\displaystyle\int_{0}^{t}\frac{1}{N}\sum_{i=1}^{N}\int_{\mathbb{R}}\sigma(\tilde{X}^{i,\epsilon,N,k}_{s},y,\tilde{\mu}^{\epsilon,N,k}_{s})\psi^{k}_{1}(s,\tilde{X}^{i,\epsilon,N,k}_{s},y)\phi^{\prime}(\tilde{X}^{i,\epsilon,N,k}_{s})$
$\displaystyle+[\tau_{1}(\tilde{X}^{i,\epsilon,N,k}_{s},y,\tilde{\mu}^{\epsilon,N,k}_{s})\psi^{k}_{1}(s,\tilde{X}^{i,\epsilon,N,k}_{s},y)+\tau_{2}(\tilde{X}^{i,\epsilon,N,k}_{s},y,\tilde{\mu}^{\epsilon,N,k}_{s})\psi^{k}_{2}(s,\tilde{X}^{i,\epsilon,N,k}_{s},y)]\times$
$\displaystyle\hskip
142.26378pt\times\Phi_{y}(\tilde{X}^{i,\epsilon,N,k}_{s},y,\tilde{\mu}^{\epsilon,N,k}_{s})\phi^{\prime}(\tilde{X}^{i,\epsilon,N,k}_{s})\pi(dy;\tilde{X}^{i,\epsilon,N}_{s},\tilde{\mu}^{\epsilon,N}_{s})ds$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}}\int_{\mathbb{R}}\biggl{(}\sigma(x,y,\tilde{\mu}^{\epsilon,N,k}_{s})\psi^{k}_{1}(s,x,y)\phi^{\prime}(x)+[\tau_{1}(x,y,\tilde{\mu}^{\epsilon,N,k}_{s})\psi^{k}_{1}(s,x,y)+\tau_{2}(x,y,\tilde{\mu}^{\epsilon,N,k}_{s})\psi^{k}_{2}(s,x,y)]\times$
$\displaystyle\hskip
142.26378pt\times\Phi_{y}(x,y,\tilde{\mu}^{\epsilon,N,k}_{s})\phi^{\prime}(x)\biggr{)}\pi(dy;x,\tilde{\mu}^{\epsilon,N}_{s})\tilde{\mu}^{\epsilon,N,k}_{s}(dx)ds$

and using that the integrand of the first two integrals above is bounded by
Assumptions A1),A5),and A6) and continuous in $\mathbb{W}_{2}$ by Assumption
A12), along with the convergence of $\tilde{\mu}^{\epsilon,N}_{s}$ to
$\mathcal{L}(X_{s})$ from Lemma 8.2, we have by dominated convergence theorem
(invoking here Skorokhod’s representation theorem to assume
$\tilde{\mu}^{\epsilon,N}_{s}$ to $\mathcal{L}(X_{s})$ almost surely as in
Proposition 8.3) and Theorem A.3.18 in [DE]:

$\displaystyle\lim_{N\rightarrow\infty}\mathbb{E}\biggl{[}\biggl{|}\int_{0}^{T}\int_{\mathbb{R}}\int_{\mathbb{R}}\biggl{(}\sigma(x,y,\tilde{\mu}^{\epsilon,N,k}_{s})\psi^{k}_{1}(s,x,y)\phi^{\prime}(x)+[\tau_{1}(x,y,\tilde{\mu}^{\epsilon,N,k}_{s})\psi^{k}_{1}(s,x,y)+\tau_{2}(x,y,\tilde{\mu}^{\epsilon,N,k}_{s})\psi^{k}_{2}(s,x,y)]$
$\displaystyle\hskip
142.26378pt\times\Phi_{y}(x,y,\tilde{\mu}^{\epsilon,N,k}_{s})\phi^{\prime}(x)\biggr{)}\pi(dy;x,\tilde{\mu}^{\epsilon,N}_{s})\tilde{\mu}^{\epsilon,N,k}_{s}(dx)ds$
$\displaystyle-\int_{0}^{T}\int_{\mathbb{R}}\int_{\mathbb{R}}\biggl{(}\sigma(x,y,\mathcal{L}(X_{s}))\psi^{k}_{1}(s,x,y)\phi^{\prime}(x)+[\tau_{1}(x,y,\mathcal{L}(X_{s}))\psi^{k}_{1}(s,x,y)+\tau_{2}(x,y,\mathcal{L}(X_{s}))\psi^{k}_{2}(s,x,y)]$
$\displaystyle\hskip
142.26378pt\times\Phi_{y}(x,y,\mathcal{L}(X_{s}))\phi^{\prime}(x)\biggr{)}\pi(dy;x,\mathcal{L}(X_{s}))\mathcal{L}(X_{s})(dx)ds\biggr{|}\biggr{]}=0$

so by triangle inequality, the desired convergence is shown.

Now we seek to establish that $(\tilde{Z}^{k},Q^{k})$ converges to
$(\tilde{Z},Q)$ in $C([0,T];\mathcal{S}_{-w})\times M_{T}(\mathbb{R}^{4})$
where $Q\in P^{*}(\tilde{Z})$ and $Q$ satisfies (71).

We first prove precompactness. We have since $\psi^{k}\rightarrow h$ in
$L^{2}(U\times\mathbb{R}\times\mathbb{R},\nu)$,

$\displaystyle\sup_{k\in\mathbb{N}}\int_{\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{2}\times[0,T]}\left(z_{1}^{2}+z_{2}^{2}\right)Q^{k}(dx,dy,dz,ds)=$
$\displaystyle\quad=\sup_{k\in\mathbb{N}}\int_{0}^{T}\int_{\mathbb{R}}\int_{\mathbb{R}}\left(|\psi^{k}_{1}(s,x,y)|^{2}+|\psi^{k}_{2}(s,x,y)|^{2}\right)\pi(dy;x,\mathcal{L}(X_{s}))\mathcal{L}(X_{s})(dx)ds<\infty.$

Moreover, by ($P^{enumi}$3) and ($P^{enumi}$4)

$\displaystyle\int_{\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{2}\times[0,T]}\left(y^{2}+|x|^{2}\right)Q^{k}(dx,dy,dz,ds)=\int_{0}^{T}\mathbb{E}\biggl{[}\int_{\mathbb{R}}y^{2}\pi(dy;X_{s},\mathcal{L}(X_{s}))+|X_{s}|^{2}\biggr{]}ds<\infty,\forall
k\in\mathbb{N},$

where here we have used that $\pi(\cdot;x,\mu)$ from Equation (20) has bounded
moments of all orders uniformly in
$x\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})$ and that for $X_{s}$ from
Equation (25), $\sup_{s\in[0,T]}\mathbb{E}[|X_{s}|^{2}]<\infty,$ which follows
easily from the fact that $\bar{\gamma}$ and $\bar{D}$ are bounded as per
Assumption A13). Thus, via the same tightness function used for Proposition
7.10, $\\{Q^{k}\\}_{k\in\mathbb{N}}$ is tight in $M_{T}(\mathbb{R}^{4})$.

To see that $\\{\tilde{Z}^{k}\\}_{k\in\mathbb{N}}$ is precompact, we use that
for each $k$ $(\tilde{Z}^{k},Q^{k})$ must satisfy Equation (32). That is, for
$\phi\in\mathcal{S}$ and $t\in[0,T]$:

$\displaystyle\langle\tilde{Z}^{k}_{t},\phi\rangle$
$\displaystyle=\int_{0}^{t}\langle\tilde{Z}^{k}_{s},\bar{L}_{\mathcal{L}(X_{s})}\phi\rangle
ds+\int_{0}^{t}\langle B^{k}_{s},\phi\rangle ds$ $\displaystyle\langle
B^{k}_{t},\phi\rangle$
$\displaystyle\coloneqq\int_{\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{2}}\biggl{(}\sigma(x,y,\mathcal{L}(X_{s}))z_{1}\phi^{\prime}(x)$
$\displaystyle\hskip
56.9055pt+[\tau_{1}(x,y,\mathcal{L}(X_{s}))z_{1}+\tau_{2}(x,y,\mathcal{L}(X_{s}))z_{2}]\Phi_{y}(x,y,\mathcal{L}(X_{s}))\phi^{\prime}(x)\biggr{)}Q_{t}^{k}(dx,dy,dz).$

Here $Q^{k}_{t}\in\mathcal{P}(\mathbb{R}^{4})$ is such that
$Q^{k}(dx,dy,dz,dt)=Q^{k}_{t}(dx,dy,dz)dt$. We can see that
$B^{k}_{t}\in\mathcal{S}_{-(m+2)}$ for almost every $t\in[0,T]$, and in fact

$\displaystyle\sup_{k\in\mathbb{N}}\int_{0}^{T}\mathinner{\\!\left\lVert
B^{k}_{s}\right\rVert}^{2}_{-(m+2)}ds$
$\displaystyle=\sup_{k\in\mathbb{N}}\int_{0}^{T}\sup_{\mathinner{\\!\left\lVert\phi\right\rVert}_{m+2}=1}|\langle
B^{k}_{s},\phi\rangle|^{2}ds$
$\displaystyle\leq\sup_{k\in\mathbb{N}}\int_{0}^{T}\sup_{\mathinner{\\!\left\lVert\phi\right\rVert}_{m+2}=1}\biggl{\\{}\int_{\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{2}}\left(|z_{1}|^{2}+|z_{2}|^{2}\right)Q^{k}_{s}(dx,dy,dz)|\phi|^{2}_{1}\biggr{\\}}ds$
$\displaystyle\leq\sup_{k\in\mathbb{N}}\int_{0}^{T}\sup_{\mathinner{\\!\left\lVert\phi\right\rVert}_{m+2}=1}\biggl{\\{}\int_{\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{2}}\left(|z_{1}|^{2}+|z_{2}|^{2}\right)Q^{k}_{s}(dx,dy,dz)\mathinner{\\!\left\lVert\phi\right\rVert}^{2}_{m+2}\biggr{\\}}ds$
$\displaystyle=\sup_{k\in\mathbb{N}}\int_{0}^{T}\int_{\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{2}}\left(|z_{1}|^{2}+|z_{2}|^{2}\right)Q^{k}_{s}(dx,dy,dz)ds<\infty$

Thus, by the proof of Theorem 2.5.2 in [KalX], it suffices to show that for
fixed $\phi\in\mathcal{S}$, $\langle\tilde{Z}^{k}_{t},\phi\rangle$ is
relatively compact in $C([0,T];\mathbb{R})$, and $\tilde{Z}^{k}$ is uniformly
$(m+2)$-continuous to get precompactness of $\tilde{Z}^{k}$ in
$C([0,T];\mathcal{S}_{-w})$ for $w>m+2$ sufficiently large that the canonical
embedding $\mathcal{S}_{-m-2}\rightarrow\mathcal{S}_{-w}$ is Hilbert-Schmidt
(see Equation (6)).

We have that, in the same way as the proof of Proposition 8.8 (using here that
$\tilde{Z}^{k}\in C([0,T];\mathcal{S}_{-m})$),

$\displaystyle\mathinner{\\!\left\lVert\tilde{Z}^{k}_{t}\right\rVert}_{-(m+2)}^{2}$
$\displaystyle=2\int_{0}^{t}\langle\tilde{Z}^{k}_{s},L^{*}_{\mathcal{L}(X_{s})}\tilde{Z}^{k}_{s}\rangle_{-(m+2)}ds+2\int_{0}^{t}\langle\tilde{Z}^{k}_{s},B^{k}_{s}\rangle_{-(m+2)}ds$
$\displaystyle\leq
C\int_{0}^{t}\mathinner{\\!\left\lVert\tilde{Z}^{k}_{s}\right\rVert}^{2}_{-(m+2)}ds+2\int_{0}^{t}\mathinner{\\!\left\lVert\tilde{Z}^{k}_{s}\right\rVert}_{-(m+2)}\mathinner{\\!\left\lVert
B^{k}_{s}\right\rVert}_{-(m+2)}ds\text{ by Cauchy Schwarz and Lemma
\ref{lemma:4.32BW}}$ $\displaystyle\leq
C\biggl{\\{}\int_{0}^{t}\mathinner{\\!\left\lVert\tilde{Z}^{k}_{s}\right\rVert}^{2}_{-(m+2)}ds+\int_{0}^{t}\mathinner{\\!\left\lVert
B^{k}_{s}\right\rVert}^{2}_{-(m+2)}ds\biggr{\\}}$

so by Gronwall’s inequality,

$\displaystyle\sup_{k\in\mathbb{N}}\sup_{t\in[0,T]}\mathinner{\\!\left\lVert\tilde{Z}^{k}_{t}\right\rVert}^{2}_{-(m+2)}\leq
C(T).$

This gives then that for $t_{1},t_{2}\in[0,T]$ and $\phi\in\mathcal{S}$:

$\displaystyle|\langle\tilde{Z}^{k}_{t_{2}},\phi\rangle-\langle\tilde{Z}^{k}_{t_{1}},\phi\rangle|$
$\displaystyle\leq
2|t_{2}-t_{1}|\biggl{\\{}\int_{0}^{T}|\langle\tilde{Z}^{k}_{s},\bar{L}_{\mathcal{L}(X_{s})}\phi\rangle|^{2}ds+\int_{0}^{T}|\langle
B^{k}_{s},\phi\rangle|^{2}ds\biggr{\\}}$ $\displaystyle\leq
2|t_{2}-t_{1}|\biggl{\\{}\int_{0}^{T}\mathinner{\\!\left\lVert\tilde{Z}^{k}_{s}\right\rVert}_{-(m+2)}^{2}\mathinner{\\!\left\lVert\bar{L}_{\mathcal{L}(X_{s})}\phi\right\rVert}^{2}_{m+2}ds+\int_{0}^{T}\mathinner{\\!\left\lVert
B^{N}_{s}\right\rVert}^{2}_{-(m+2)}\mathinner{\\!\left\lVert\phi\right\rVert}^{2}_{m+2}ds\biggr{\\}}$
$\displaystyle\leq
2|t_{2}-t_{1}|C(T)\mathinner{\\!\left\lVert\phi\right\rVert}^{2}_{m+4}\text{
by Lemma \ref{lem:barLbounded},}$

and precompactness of $\\{\tilde{Z}^{k}\\}_{k\in\mathbb{N}}$ is established.

Taking a convergent subsequence, which we do not relabel in the notation, we
call its limit $(Z,Q)$. The fact that ($P^{enumi}$2)-($P^{enumi}$4) in the
definition of $P^{*}(Z)$ are satisfied follows in the exact same way as
Proposition 10.2. It thus remains to show that $(Z,Q)$ satisfies Equation (32)
with $Q$ given in Equation (71). At this point, by Proposition 8.8, we will
have the limit is uniquely identified for every subsequence, and hence the
lemma is proved. By a density argument, it suffices to show that for each
$\phi\in C^{\infty}_{c}(\mathbb{R})$ and $t\in[0,T]$,

$\displaystyle\lim_{k\rightarrow\infty}\langle\tilde{Z}^{k}_{t},\phi\rangle=\int_{0}^{t}\langle
Z_{s},\bar{L}_{\mathcal{L}(X_{s})}\phi\rangle
ds+\int_{0}^{t}\mathbb{E}\biggl{[}\int_{\mathbb{R}}\biggl{(}\sigma(x,y,\mathcal{L}(X_{s}))h_{1}(s,X_{s},y)\phi^{\prime}(X_{s})$
$\displaystyle+[\tau_{1}(X_{s},y,\mathcal{L}(X_{s}))h_{1}(s,X_{s},y)+\tau_{2}(X_{s},y,\mathcal{L}(X_{s}))h_{2}(s,X_{s},y)]\Phi_{y}(X_{s},y,\mathcal{L}(X_{s}))\phi^{\prime}(X_{s})\biggr{)}\pi(dy;X_{s},\mathcal{L}(X_{s}))\biggr{]}ds.$

We have by dominated convergence theorem, $L^{2}$ convergence of $\psi^{k}$ to
$h$, and that under Assumption A13)
$\bar{L}_{\mathcal{L}(X_{s})}\phi\in\mathcal{S}_{w},\forall s\in[0,T]$:

$\displaystyle\lim_{k\rightarrow\infty}\langle\tilde{Z}^{k}_{t},\phi\rangle=\lim_{k\rightarrow\infty}\biggl{\\{}\int_{0}^{t}\langle\tilde{Z}^{k}_{s},\bar{L}_{\mathcal{L}(X_{s})}\phi\rangle
ds+\int_{0}^{t}\langle B^{k}_{s},\phi\rangle ds\biggr{\\}}$
$\displaystyle=\int_{0}^{t}\lim_{k\rightarrow\infty}\langle\tilde{Z}^{k}_{s},\bar{L}_{\mathcal{L}(X_{s})}\phi\rangle
ds+\lim_{k\rightarrow\infty}\int_{0}^{t}\mathbb{E}\biggl{[}\int_{\mathbb{R}}\biggl{(}\sigma(x,y,\mathcal{L}(X_{s}))\psi^{k}_{1}(s,X_{s},y)\phi^{\prime}(X_{s})$
$\displaystyle+[\tau_{1}(X_{s},y,\mathcal{L}(X_{s}))\psi^{k}_{1}(s,X_{s},y)+\tau_{2}(X_{s},y,\mathcal{L}(X_{s}))\psi^{k}_{2}(s,X_{s},y)]\Phi_{y}(X_{s},y,\mathcal{L}(X_{s}))\phi^{\prime}(X_{s})\biggr{)}\pi(dy;X_{s},\mathcal{L}(X_{s}))\biggr{]}ds$
$\displaystyle=\int_{0}^{t}\langle
Z_{s},\bar{L}_{\mathcal{L}(X_{s})}\phi\rangle
ds+\int_{0}^{t}\mathbb{E}\biggl{[}\int_{\mathbb{R}}\biggl{(}\sigma(x,y,\mathcal{L}(X_{s}))h_{1}(s,X_{s},y)\phi^{\prime}(X_{s})$
$\displaystyle+[\tau_{1}(X_{s},y,\mathcal{L}(X_{s}))h_{1}(s,X_{s},y)+\tau_{2}(X_{s},y,\mathcal{L}(X_{s}))h_{2}(s,X_{s},y)]\Phi_{y}(X_{s},y,\mathcal{L}(X_{s}))\phi^{\prime}(X_{s})\biggr{)}\pi(dy;X_{s},\mathcal{L}(X_{s}))\biggr{]}ds$

as desired. ∎

######  Proposition 10.2.

Under assumptions A1) \- A12) and A’13), $I$ given in Theorems 3.1/3.2 is a
good rate function on $C([0,T];\mathcal{S}_{-r})$ for $r>w+2$ as in Equation
(7).

###### Proof.

We need to show that for any $L>0$,

$\displaystyle\Theta_{L}\coloneqq\\{Z\in
C([0,T];\mathcal{S}_{-r})\mathrel{\mathop{\mathchar 58\relax}}I(Z)\leq L\\}$

is compact in $C([0,T];\mathcal{S}_{-r})$.

Let $\\{Z^{N}\\}_{N\in\mathbb{N}}\subset\Theta_{L}$. Then by the form of $I$,
for each $N\in\mathbb{N}$, there exists $Q^{N}\in P^{*}(Z^{N})$ such that

$\displaystyle\frac{1}{2}\int_{\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{2}\times[0,T]}\left(z_{1}^{2}+z_{2}^{2}\right)Q^{N}(dx,dy,dz,ds)\leq
L+\frac{1}{N}$

and by ($P^{enumi}$3) and ($P^{enumi}$4), we have as with the $Q^{k}$’s in the
proof of Proposition 9.1

$\displaystyle\sup_{N\in\mathbb{N}}\int_{\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{2}\times[0,T]}\left(y^{2}+|x|^{2}\right)Q^{N}(dx,dy,dz,ds)<\infty.$

Thus by the same tightness function used for Proposition 7.10,
$\\{Q^{N}\\}_{N\in\mathbb{N}}$ is tight in $M_{T}(\mathbb{R}^{4})$.

Taking a subsequence of $\\{Q^{N}\\}$ which converges to some $Q\in
M_{T}(\mathbb{R}^{4})$ (which we do not relabel in the notation), define $Z\in
C([0,T];\mathcal{S}_{-w})$ to be the unique solution to Equation (32) with
this choice of $Q$. Here we are using that by the proof of Proposition 10.1
such a solution exists and that by Proposition 8.8 it is unique - see the
discussion before Lemma 4.10 in [BW]. We claim that $(Z^{N},Q^{N})$ converges
to $(Z,Q)$ in $C([0,T];\mathcal{S}_{-r})\times M_{T}(\mathbb{R}^{4})$ and
$Q\in P^{*}(Z)$. At this point we will have that since $Z^{N}$ has a limit,
$\Theta_{L}$ is precompact, and by the version of Fatou’s lemma from Theorem
A.3.12 in [DE]:

$\displaystyle I(Z)$
$\displaystyle\leq\frac{1}{2}\int_{\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{2}\times[0,T]}\left(z_{1}^{2}+z_{2}^{2}\right)Q(dx,dy,dz,ds)\leq\liminf_{N\rightarrow\infty}\frac{1}{2}\int_{\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{2}\times[0,T]}\left(z_{1}^{2}+z_{2}^{2}\right)Q^{N}(dx,dy,dz,ds)\leq
L$

so $\Theta_{L}$ is closed, and hence compact. Note that we have
$I(Z^{N})<\infty$ implies $Z^{N}\in C([0,T];\mathcal{S}_{-w}),\forall
N\in\mathbb{N}$ and by definition $Z\in C([0,T];\mathcal{S}_{-w})$. Thus if we
could show convergence of $Z^{N}\rightarrow Z$ in $C([0,T];\mathcal{S}_{-w})$,
we would have compactness of level sets of $I$ as a rate function on
$C([0,T];\mathcal{S}_{-w})$. However, such convergence is not immediately
obvious, hence the need for the additional assumption A’13).

To see that $\\{Z^{N}\\}_{N\in\mathbb{N}}$ is precompact, we have that since
$Q^{N}\in P^{*}(Z^{N})$, for each $N$ $(Z^{N},Q^{N})$ must satisfy Equation
(32). That is, for $\phi\in\mathcal{S}$ and $t\in[0,T]$:

$\displaystyle\langle Z^{N}_{t},\phi\rangle$
$\displaystyle=\int_{0}^{t}\langle
Z^{N}_{s},\bar{L}_{\mathcal{L}(X_{s})}\phi\rangle ds+\int_{0}^{t}\langle
B^{N}_{s},\phi\rangle ds$ $\displaystyle\langle B^{N}_{t},\phi\rangle$
$\displaystyle\coloneqq\int_{\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{2}}\biggl{(}\sigma(x,y,\mathcal{L}(X_{s}))z_{1}\phi^{\prime}(x)$
$\displaystyle\hskip
56.9055pt+[\tau_{1}(x,y,\mathcal{L}(X_{s}))z_{1}+\tau_{2}(x,y,\mathcal{L}(X_{s}))z_{2}]\Phi_{y}(x,y,\mathcal{L}(X_{s}))\phi^{\prime}(x)\biggr{)}Q_{t}^{N}(dx,dy,dz).$

Thus precompactness of $\\{Z^{N}\\}_{N\in\mathbb{N}}$ in
$C([0,T];\mathcal{S}_{-r})$ follows in the exact same way as precompactness of
$\\{\tilde{Z}^{k}\\}_{k\in\mathbb{N}}$ in $C([0,T];\mathcal{S}_{-w})$ in the
proof of Proposition 10.1, but replacing $m$ by $w$. Note that there we knew
that $\tilde{Z}^{k}$ was in $C([0,T];\mathcal{S}_{-m})$ for each $k$, where
here we only know $Z^{N}\in C([0,T];\mathcal{S}_{-w})$ for each $N$. Along the
way, we get:

$\displaystyle\sup_{N\in\mathbb{N}}\sup_{t\in[0,T]}\mathinner{\\!\left\lVert
Z^{N}_{t}\right\rVert}^{2}_{-(w+2)}\leq C(T).$

To see that $Q\in P^{*}(Z)$, we identify the point-wise limit of $\langle
Z^{N},\phi\rangle$ to satisfy the desired equation, i.e. (32) with our
specific choice of $Q$. This uniquely characterizes the limit along the whole
sequence by Lemma 8.8. This gives ($P^{enumi}$1). ($P^{enumi}$2) follows
immediately from Fatou’s lemma. ($P^{enumi}$3) and ($P^{enumi}$4) follow from
convergence of the measure implying convergence of the marginals and
uniqueness of the decomposition into stochastic kernels (see [DE] Theorems
A.4.2 and A.5.4).

To see (32) with our specific choice of $Q$ holds, we may by a density
argument consider fixed $\phi\in C^{\infty}_{c}(\mathbb{R})$ and $t\in[0,T]$.
Then:

$\displaystyle\langle Z_{t},\phi\rangle$
$\displaystyle=\lim_{N\rightarrow\infty}\langle
Z^{N}_{t},\phi\rangle=\lim_{N\rightarrow\infty}\biggl{\\{}\int_{0}^{t}\langle
Z^{N}_{s},\bar{L}_{\mathcal{L}(X_{s})}\phi\rangle ds+\int_{0}^{t}\langle
B^{N}_{s},\phi\rangle ds\biggr{\\}}$
$\displaystyle=\int_{0}^{t}\lim_{N\rightarrow\infty}\langle
Z^{N}_{s},\bar{L}_{\mathcal{L}(X_{s})}\phi\rangle
ds+\lim_{N\rightarrow\infty}\int_{0}^{t}\langle B^{N}_{s},\phi\rangle ds$
$\displaystyle\text{ (by boundedness of
}\sup_{N\in\mathbb{N}}\sup_{t\in[0,T]}\mathinner{\\!\left\lVert
Z^{N}_{t}\right\rVert}^{2}_{-(w+2)}\text{ and Dominated Convergence Theorem)}$
$\displaystyle=\int_{0}^{t}\langle
Z_{s},\bar{L}_{\mathcal{L}(X_{s})}\phi\rangle ds+\int_{0}^{t}\langle
B_{s},\phi\rangle ds,\text{ since under assumption
\ref{assumption:limitingcoefficientsregularityratefunction}
$\bar{L}_{\mathcal{L}(X_{s})}\phi\in\mathcal{S}_{r},\forall s\in[0,T]$.}$

Here

$\displaystyle\langle B_{t},\phi\rangle$
$\displaystyle\coloneqq\int_{\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{2}}\biggl{(}\sigma(x,y,\mathcal{L}(X_{s}))z_{1}\phi^{\prime}(x)$
$\displaystyle\hskip
113.81102pt+[\tau_{1}(x,y,\mathcal{L}(X_{s}))z_{1}+\tau_{2}(x,y,\mathcal{L}(X_{s}))z_{2}]\Phi_{y}(x,y,\mathcal{L}(X_{s}))\phi^{\prime}(x)\biggr{)}Q_{t}(dx,dy,dz),$

and to pass to the second limit, we use that the integrand appearing in
$\int_{0}^{t}\langle B^{N}_{s},\phi\rangle ds$ is bounded by
$C[|z_{1}|+|z_{2}|]$, and hence is uniformly integrable with respect to
$Q^{N}$. ∎

##  11\. Conclusions and Future Work

In this paper we have derived a moderate deviations principle for the
empirical measure of a fully coupled multiscale system of weakly interacting
particles in the joint limit as number of particles increases and averaging
due to the multiscale structure takes over. Using weak convergence methods we
have derived a variational form of the rate function and have rigorously shown
that the rate function can take equivalent forms analogous to the one derived
in the seminal paper [DG].

In this paper we have assumed that the particles are in dimension one. It is
of great interest to extend this work in the multidimensional case. One source
of difficulty here is that in higher dimensions we would probably have to
consider a different space for the fluctuation process to live in (see, e.g.
[FM] and [LossFromDefault]). This is because in higher dimensions the result
that for each $v$, there is $w\geq v$ such that
$\mathcal{S}_{-v}\rightarrow\mathcal{S}_{-w}$ is Hilbert-Schmidt breaks down,
and the bound (9) no longer holds true. See [DLR] Section 5.1 for a further
discussion of this. The trade-off with using these alternative spaces is that
they often require higher moments of the particles and limiting McKean-Vlasov
Equation in order to establish tightness - see, e.g. Section 4.7 in [FM],
where the proofs depend crucially on Lemma 3.1 (even in one dimension this
would require having bounded $8$’th moments of the controlled particles
$\tilde{X}^{i,N,\epsilon}$, with the required number of moments increasing
with the dimension). This would seem to require strong assumptions on the
coefficients in Equation (1) even in the absence of multiscale structure,
since the controls are a priori only bounded in $L^{2}$.

Another potentially interesting direction is to derive the moderate deviations
principle for the stochastic current. See [Orrieri] for some related results
in the direction of large deviations for an interacting particle system in the
joint mean field and small-noise limit. Also, we are hopeful that the results
of this paper can also be used for the construction of provably-efficient
importance sampling schemes for the computation of rare events for statistics
of weakly interacting diffusions that are relevant to the moderate-deviations
scaling. Lastly, as we also mentioned in the introduction, we believe that the
results of this paper can be used to study dynamical questions related to
phase transitions in the spirit of [Dawson].

##  Appendix A A List of Technical Notation

Here we provide a list of frequently used notation for the various processes,
spaces, operators, ect. used throughout this manuscript for convenient
reference. Other, more standard notation is introduced following Equation (9)
in Section 2.

  * •

$\epsilon$ is the scale separation parameter which decreases to $0$ as
$N\rightarrow\infty$. $N$ is the number of particles. $a(N)$ is moderate
deviations the scaling sequence such that $a(N)\downarrow 0$ and
$a(N)\sqrt{N}\rightarrow\infty$.

  * •

$(X^{i,\epsilon,N},Y^{i,\epsilon,N})$ is the slow-fast system of particles
from Equation (1).

  * •

$\mu^{\epsilon,N}$ from Equation (2) is the empirical measure on the slow
particles $X^{i,\epsilon,N}$ .

  * •

$X_{t}$ is the limiting averaged McKean-Vlasov Equation from Equation (25).
$\mathcal{L}(X_{t})$ denotes its Law.

  * •

$Z^{N}$ is the fluctuations process from Equation (3) for which we derive a
large deviations principle.

  * •

$(\tilde{X}^{i,\epsilon,N},\tilde{Y}^{i,\epsilon,N})$ are the controlled slow-
fast interacting particles from Equation (55).

  * •

$\tilde{\mu}^{\epsilon,N}$ is the empirical measure on the controlled slow
particles $\tilde{X}^{i,\epsilon,N}$ from Equation (54).

  * •

$\tilde{Z}^{N}$ is the controlled fluctuations process from Equation (54).

  * •

$Q^{N}$ are the occupation measures from Equation (56).

  * •

$(\bar{X}^{i,\epsilon},\bar{Y}^{i,\epsilon})$ are the IID slow-fast McKean-
Vlasov Equations from Equation (57). $\bar{X}^{\epsilon}$ is a random process
with law Equal to that of the $\bar{X}^{i,\epsilon}$’s.

  * •

$\bar{\mu}^{\epsilon,N}$ from Equation (58) is the empirical measure on $N$ of
the IID slow particles $\bar{X}^{i,\epsilon}$.

  * •

$\mathcal{P}_{2}(\mathbb{R})$ is the space of square integrable probability
measures with the 2-Wasserstein metric $\mathbb{W}_{2}$ (Definition D.1).

  * •

$M_{T}(\mathbb{R}^{d})$ is the space of measures $Q$ on
$\mathbb{R}^{d}\times[0,T]$ such that $Q(\mathbb{R}^{d}\times[0,t])=t,\forall
t\in[0,T]$ equipped with the topology of weak convergence.

  * •

For $p\in\mathbb{N}$, $\mathcal{S}_{p}$ is the completion of $\mathcal{S}$
with respect to $\mathinner{\\!\left\lVert\cdot\right\rVert}_{p}$ (see
Equation (4)) and $\mathcal{S}_{-p}=\mathcal{S}_{p}^{\prime}$ the dual space
of $\mathcal{S}_{p}$. We prove tightness of
$\\{\tilde{Z}^{N}\\}_{N\in\mathbb{N}}$ in $C([0,T];\mathcal{S}_{-m})$ for the
choice of $m$ found in Equation (5), the Laplace Principle on
$C([0,T];\mathcal{S}_{-w})$ for the choice of $w$ found in Equation (6), and
compactness of level sets of the rate function on $C([0,T];\mathcal{S}_{-r})$
for the choice of $r$ found in Equation (7).

  * •

For $n\in\mathbb{N}$, $|\cdot|_{n}$ is the sup norm defined in Equation (8),
which is related to $\mathinner{\\!\left\lVert\cdot\right\rVert}_{n+1}$ via
Equation (9).

  * •

For $G\mathrel{\mathop{\mathchar
58\relax}}\mathcal{P}_{2}(\mathbb{R})\rightarrow\mathbb{R}$ and
$\nu\in\mathcal{P}_{2}(\mathbb{R})$,
$\partial_{\mu}G(\nu)[\cdot]\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\rightarrow\mathbb{R}$ denotes the Lions derivative of $G$
at the point $\nu$ (Definition D.1) and $\frac{\delta}{\delta
m}G(\nu)[\cdot]\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\rightarrow\mathbb{R}$ denotes the Linear Functional
Derivative of $G$ at the point $\nu$ (Definition D.4).

  * •

For $G\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R})\rightarrow\mathbb{R}$,
we use $D^{(n,l,\beta)}G$ to denote multiple derivatives of $G$ in space and
measure in the multi-index notation of Definition 2.1. Spaces (denoted by
$\mathcal{M}$ with some sub or super-scripts) containing functions with
different regularity of such mixed derivatives are found in Definition 2.4.
When $G\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R})\rightarrow\mathbb{R}$,
polynomial growth of such derivatives in $G$’s second coordinate, denoted by
$q_{G}(n,l,\beta)$ or $\tilde{q}_{G}(n,l,\beta)$, are defined as in Equations
(13) and (15).

  * •

$L_{x,\mu}$ is the frozen generator associated to the fast particles from
Equation (18). $\pi$ denotes its unique associated invariant measure from
Equation (20) and $\Phi$ denotes the solution to the associated Poisson
Equation (22).

  * •

For $\nu\in\mathcal{P}_{2}(\mathbb{R})$ $\bar{L}_{\nu}$ is the linearized
generator of the limiting averaged McKean-Vlasov Equation $X_{t}$ at $\nu$ and
is defined in Equation (32).

  * •

$\bar{\gamma},\bar{D}$ from Equation (24) are the drift and diffusion
coefficients of the limiting averaged McKean-Vlasov Equation $X_{t}$, and are
defined in terms of $\gamma_{1},D_{1},\gamma,D\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R})\rightarrow\mathbb{R}$
from Equation (23).

##  Appendix B A priori Bounds on Moments of the Controlled Process (55)

In this Appendix, we fix any controls satisfying the bound (53) and provide
moment bounds on the fast component of the controlled particles (55). These
are needed, among other places, to handle possible growth lack of boundedness
in $y$ of functions appearing in the remainders in the ergodic-type theorems
of Section 6.

######  Lemma B.1.

Under assumptions A1)\- A2), A4), and A5), we have there is $C\geq 0$ such
that:

$\displaystyle\sup_{N\in\mathbb{N}}\frac{1}{N}\sum_{i=1}^{N}\sup_{t\in[0,T]}\mathbb{E}\biggl{[}|\tilde{Y}^{i,\epsilon,N}_{t}|^{2}\biggr{]}\leq
C+|\eta^{y}|^{2}.$

###### Proof.

By Itô’s formula, we have, letting $C\geq 0$ be any constant independent of
$N$ which may change from line to line and $(i)$ denote the argument
$(\tilde{X}^{i,\epsilon,N}_{s},\tilde{Y}^{i,\epsilon,N}_{s},\tilde{\mu}^{\epsilon,N})$:

$\displaystyle\mathbb{E}\biggl{[}|\tilde{Y}^{i,\epsilon,N}_{t}|^{2}\biggr{]}=|\eta^{y}|^{2}+\int_{0}^{t}\mathbb{E}\biggl{[}\frac{1}{\epsilon^{2}}\biggl{(}2f(i)\tilde{Y}^{i,\epsilon,N}_{s}+\tau_{1}^{2}(i)+\tau_{2}^{2}(i)\biggr{)}\biggr{]}ds+\frac{2}{\epsilon}\int_{0}^{t}\mathbb{E}\biggl{[}g(i)\tilde{Y}^{i,\epsilon,N}_{s}\biggr{]}ds$
$\displaystyle+\frac{2}{\epsilon}\mathbb{E}\biggl{[}\int_{0}^{t}\biggl{(}\tau_{1}(i)\frac{\tilde{u}^{N,1}_{i}(s)}{a(N)\sqrt{N}}+\tau_{2}(i)\frac{\tilde{u}^{N,1}_{i}(s)}{a(N)\sqrt{N}}\biggr{)}\tilde{Y}^{i,\epsilon,N}_{s}ds\biggr{]}+\frac{2}{\epsilon}\mathbb{E}\biggl{[}\int_{0}^{t}\tau_{1}(i)\tilde{Y}^{i,\epsilon,N}_{s}dW^{i}_{s}+\int_{0}^{t}\tau_{2}(i)\tilde{Y}^{i,\epsilon,N}_{s}dB^{i}_{s}\biggr{]}$
$\displaystyle\leq|\eta^{y}|^{2}-\frac{\beta}{\epsilon^{2}}\int_{0}^{t}\mathbb{E}\biggl{[}|\tilde{Y}^{i,\epsilon,N}_{s}|^{2}\biggr{]}ds+\frac{Ct}{\epsilon^{2}}+\frac{2}{\epsilon}\int_{0}^{t}\mathbb{E}\biggl{[}g(i)\tilde{Y}^{i,\epsilon,N}_{s}\biggr{]}ds$
$\displaystyle+\frac{2}{\epsilon}\mathbb{E}\biggl{[}\int_{0}^{t}\biggl{(}\tau_{1}(i)\frac{\tilde{u}^{N,1}_{i}(s)}{a(N)\sqrt{N}}+\tau_{2}(i)\frac{\tilde{u}^{N,1}_{i}(s)}{a(N)\sqrt{N}}\biggr{)}\tilde{Y}^{i,\epsilon,N}_{s}ds\biggr{]}+\frac{2}{\epsilon}\mathbb{E}\biggl{[}\int_{0}^{t}\tau_{1}(i)\tilde{Y}^{i,\epsilon,N}_{s}dW^{i}_{s}+\int_{0}^{t}\tau_{2}(i)\tilde{Y}^{i,\epsilon,N}_{s}dB^{i}_{s}\biggr{]}\text{
by }\eqref{eq:fdecayimplication}$
$\displaystyle\leq|\eta^{y}|^{2}-\frac{\beta}{\epsilon^{2}}\int_{0}^{t}\mathbb{E}\biggl{[}|\tilde{Y}^{i,\epsilon,N}_{s}|^{2}\biggr{]}ds+\frac{Ct}{\epsilon^{2}}+\frac{2}{\epsilon}\int_{0}^{t}\mathbb{E}\biggl{[}|\tilde{Y}^{i,\epsilon,N}_{s}|^{2}\biggr{]}ds+\mathinner{\\!\left\lVert
g\right\rVert}^{2}_{\infty}\frac{t}{\epsilon}$
$\displaystyle+\frac{|\tau_{1}|^{2}_{\infty}\vee|\tau_{2}|^{2}_{\infty}}{\epsilon
a^{2}(N)N}\mathbb{E}\biggl{[}\int_{0}^{T}|\tilde{u}^{N,1}_{i}(s)|^{2}+|\tilde{u}^{N,1}_{i}(s)|^{2}\biggr{]}+\frac{2}{\epsilon}\mathbb{E}\biggl{[}\int_{0}^{t}\tau_{1}(i)\tilde{Y}^{i,\epsilon,N}_{s}dW^{i}_{s}+\int_{0}^{t}\tau_{2}(i)\tilde{Y}^{i,\epsilon,N}_{s}dB^{i}_{s}\biggr{]}$

at which point it becomes clear that applying Burkholder Davis Gundy
inequality, taking $\epsilon$ small enough that the
$\frac{-\beta}{\epsilon^{2}}$ term dominates, and using the bound (53), that
$\mathbb{E}\biggl{[}\int_{0}^{T}|\tilde{Y}^{i,\epsilon,N}_{s}|^{2}\biggr{]}<\infty$
for each $N$, so by boundedness of $\tau_{1},\tau_{2}$ the stochastic
integrals are true martingales, and hence vanish in expectation. Note that in
the above we are using the boundedness of $\tau_{1},\tau_{2}$ from Assumption
A1) and of $g$ from Assumption A5).

So, returning to the initial equality:

$\displaystyle\frac{d}{dt}\mathbb{E}\biggl{[}|\tilde{Y}^{i,\epsilon,N}_{t}|^{2}\biggr{]}=\mathbb{E}\biggl{[}\frac{1}{\epsilon^{2}}\biggl{(}2f(i)\tilde{Y}^{i,\epsilon,N}_{t}+\tau_{1}^{2}(i)+\tau_{2}^{2}(i)\biggr{)}\biggr{]}+\frac{2}{\epsilon}\mathbb{E}\biggl{[}g(i)\tilde{Y}^{i,\epsilon,N}_{t}\biggr{]}$
$\displaystyle+\frac{2}{\epsilon}\mathbb{E}\biggl{[}\biggl{(}\tau_{1}(i)\frac{\tilde{u}^{N,1}_{i}(t)}{a(N)\sqrt{N}}+\tau_{2}(i)\frac{\tilde{u}^{N,1}_{i}(t)}{a(N)\sqrt{N}}\biggr{)}\tilde{Y}^{i,\epsilon,N}_{t}\biggr{]}$
$\displaystyle\leq-\frac{\beta}{\epsilon^{2}}\mathbb{E}\biggl{[}|\tilde{Y}^{i,\epsilon,N}_{t}|^{2}\biggr{]}+\frac{C}{\epsilon^{2}}+\frac{4\mathinner{\\!\left\lVert
g\right\rVert}_{\infty}^{2}}{\beta}+\frac{\beta}{4\epsilon^{2}}\mathbb{E}[|\tilde{Y}^{i,\epsilon,N}_{t}|^{2}]+\frac{4}{\beta
a^{2}(N)N}\mathbb{E}\biggl{[}\biggl{(}\tau_{1}(i)\tilde{u}^{N,1}_{i}(t)+\tau_{2}(i)\tilde{u}^{N,1}_{i}(t)\biggr{)}^{2}\biggr{]}$
$\displaystyle+\frac{\beta}{4\epsilon^{2}}\mathbb{E}[|\tilde{Y}^{i,\epsilon,N}_{t}|^{2}]$
$\displaystyle\leq-\frac{\beta}{2\epsilon^{2}}\mathbb{E}\biggl{[}|\tilde{Y}^{i,\epsilon,N}_{t}|^{2}\biggr{]}+\frac{C}{\epsilon^{2}}+\frac{4\mathinner{\\!\left\lVert
g\right\rVert}_{\infty}^{2}}{\beta}+\frac{8\mathinner{\\!\left\lVert\tau_{1}\right\rVert}_{\infty}^{2}\vee\mathinner{\\!\left\lVert\tau_{2}\right\rVert}^{2}_{\infty}}{\beta
a^{2}(N)N}\mathbb{E}\biggl{[}|\tilde{u}^{N,1}_{i}(t)|^{2}+|\tilde{u}^{N,1}_{i}(t)|^{2}\biggr{]}$

Where in the first inequality we used the consequence (19) of Assumption A2).
Now, recalling that if $g^{\prime}(s)\leq-\gamma g(s)+f(s),\forall s\in[0,t],$
then $g(t)\leq\int_{0}^{t}f(s)e^{-\gamma(t-s)}ds+e^{-\gamma t}g(0)$, we set
$g(t)=\sum_{i=1}^{N}\mathbb{E}\biggl{[}|\tilde{Y}^{i,\epsilon,N}_{t}|^{2}\biggr{]},\gamma=\frac{\beta}{2\epsilon^{2}},f(t)=\frac{CN}{\epsilon^{2}}+\frac{4\mathinner{\\!\left\lVert
g\right\rVert}_{\infty}^{2}N}{\beta}+\frac{8\mathinner{\\!\left\lVert\tau_{1}\right\rVert}_{\infty}^{2}\vee\mathinner{\\!\left\lVert\tau_{2}\right\rVert}^{2}_{\infty}}{\beta
a^{2}(N)}\mathbb{E}\biggl{[}\frac{1}{N}\sum_{i=1}^{N}|\tilde{u}^{N,1}_{i}(t)|^{2}+|\tilde{u}^{N,1}_{i}(t)|^{2}\biggr{]}$
and get

$\displaystyle\sum_{i=1}^{N}\mathbb{E}\biggl{[}|\tilde{Y}^{i,\epsilon,N}_{t}|^{2}\biggr{]}\leq
e^{-\frac{\beta}{2\epsilon^{2}}t}CN(1+\frac{1}{\epsilon^{2}})\int_{0}^{t}e^{\frac{\beta}{2\epsilon^{2}}s}ds$
$\displaystyle\qquad+\frac{C}{a^{2}(N)}e^{-\frac{\beta}{2\epsilon^{2}}t}\int_{0}^{t}\mathbb{E}\biggl{[}\frac{1}{N}\sum_{i=1}^{N}|\tilde{u}^{N,1}_{i}(s)|^{2}+|\tilde{u}^{N,1}_{i}(s)|^{2}\biggr{]}e^{\frac{\beta}{2\epsilon^{2}}s}ds+N|\eta^{y}|^{2}e^{-\frac{\beta}{2\epsilon^{2}}t}$
$\displaystyle\leq
e^{-\frac{\beta}{2\epsilon^{2}}t}CN(1+\frac{1}{\epsilon^{2}})\int_{0}^{t}e^{\frac{\beta}{2\epsilon^{2}}s}ds+\frac{C}{a^{2}(N)}\int_{0}^{t}\mathbb{E}\biggl{[}\frac{1}{N}\sum_{i=1}^{N}|\tilde{u}^{N,1}_{i}(s)|^{2}+|\tilde{u}^{N,1}_{i}(s)|^{2}\biggr{]}ds+N|\eta^{y}|^{2}$
$\displaystyle\leq
e^{-\frac{\beta}{2\epsilon^{2}}t}CN(1+\frac{1}{\epsilon^{2}})\int_{0}^{t}e^{\frac{\beta}{2\epsilon^{2}}s}ds+\frac{C}{a^{2}(N)}+N|\eta^{y}|^{2}\text{
by the bound }\eqref{eq:controlassumptions0}$
$\displaystyle=e^{-\frac{\beta}{2\epsilon^{2}}t}CN(1+\frac{1}{\epsilon^{2}})\frac{2\epsilon^{2}}{\beta}[e^{\frac{\beta}{2\epsilon^{2}}t}-1]+\frac{C}{a^{2}(N)}+N|\eta^{y}|^{2}$
$\displaystyle\leq CN(1+\epsilon^{2})+\frac{C}{a^{2}(N)}+N|\eta^{y}|^{2}$

where $C$ is a constant changing from line to line which is independent of $t$
and $N$. Then

$\displaystyle\frac{1}{N}\sum_{i=1}^{N}\sup_{t\in[0,T]}\mathbb{E}\biggl{[}|\tilde{Y}^{i,\epsilon,N}_{t}|^{2}\biggr{]}$
$\displaystyle\leq C(1+\epsilon^{2}+\frac{1}{a^{2}(N)N})+|\eta^{y}|^{2}$
$\displaystyle\leq C+|\eta^{y}|^{2}$

since $\epsilon\downarrow 0$ and $a(N)\sqrt{N}\rightarrow\infty$. ∎

######  Lemma B.2.

Under assumptions A1)-A2),A4),and A5), we have:

$\displaystyle\frac{1}{N}\sum_{i=1}^{N}\mathbb{E}\biggl{[}\sup_{0\leq t\leq
T}|\tilde{Y}^{i,\epsilon,N}_{t}|^{2}\biggr{]}\leq|\eta^{y}|^{2}+C(\rho)\biggl{[}1+\epsilon^{-\rho}+\frac{1}{a^{2}(N)N}\biggr{]}$

for all $\rho\in(0,2)$.

###### Proof.

The proof follows along the lines of that of Lemma B.4 in [JS] and thus it is
omitted here. Note that this is where the near-Ornstein–Uhlenbeck structure
assumed in (16) plays an important role. ∎

######  Lemma B.3.

Under assumptions A1)-A2),A4), and A5), we have:

$\displaystyle\sup_{N\in\mathbb{N}}\sup_{0\leq t\leq
T}\mathbb{E}\biggl{[}\biggl{(}\frac{1}{N}\sum_{i=1}^{N}|\tilde{Y}^{i,\epsilon,N}_{t}|^{2}\biggr{)}^{2}\biggr{]}<\infty.$

###### Proof.

The proof is very similar to Lemma B.1, but we need in addition to use the
result of Lemma B.2. Because of the similarities, we assume wlog that
$\tau_{2}=g=0$ and label $\tau_{1}$ as $\tau$ and $\tilde{u}^{N,1}_{i}(s)$ and
$\tilde{u}^{N}_{i}(s)$. Then for every $N\in\mathbb{N}$ and $t\in[0,T]$,

$\displaystyle\mathbb{E}\biggl{[}\biggl{(}\frac{1}{N}\sum_{i=1}^{N}|\tilde{Y}^{i,\epsilon,N}_{t}|^{2}\biggr{)}^{2}\biggr{]}=$
$\displaystyle=|\eta^{y}|^{4}+\frac{1}{N}\sum_{i=1}^{N}\biggl{\\{}4\int_{0}^{t}\mathbb{E}\biggl{[}\biggl{(}\frac{1}{N}\sum_{i=1}^{N}|\tilde{Y}^{i,\epsilon,N}_{s}|^{2}\biggr{)}\frac{1}{\epsilon^{2}}\biggl{(}f(i)\tilde{Y}^{i,\epsilon,N}_{s}+\tau^{2}(i)\biggr{)}\biggr{]}ds$
$\displaystyle+4\int_{0}^{t}\mathbb{E}\biggl{[}\biggl{(}\frac{1}{N}\sum_{i=1}^{N}|\tilde{Y}^{i,\epsilon,N}_{s}|^{2}\biggr{)}\frac{1}{\epsilon
a(N)\sqrt{N}}\tau(i)\tilde{u}^{N}_{i}(s)\tilde{Y}^{i,\epsilon,N}_{s}\biggr{]}ds$
$\displaystyle+8\int_{0}^{t}\mathbb{E}\biggl{[}\frac{1}{\epsilon^{2}N}\tau^{2}(i)|\tilde{Y}^{i,\epsilon,N}_{s}|^{2}\biggr{]}ds+\frac{4}{\epsilon}\mathbb{E}\biggl{[}\int_{0}^{t}\biggl{(}\frac{1}{N}\sum_{i=1}^{N}|\tilde{Y}^{i,\epsilon,N}_{s}|^{2}\biggr{)}\tau(i)\tilde{Y}^{i,\epsilon,N}_{s}dW^{i}_{s}\biggr{]}\biggr{\\}},$

where we used the initial conditions are IID. By the same method as in Lemma
B.1, we can see that the martingale term vanishes for each $N$ and $t$. Then
we get

$\displaystyle\frac{d}{dt}\mathbb{E}\biggl{[}\biggl{(}\frac{1}{N}\sum_{i=1}^{N}|\tilde{Y}^{i,\epsilon,N}_{t}|^{2}\biggr{)}^{2}\biggr{]}=\frac{1}{N}\sum_{i=1}^{N}\biggl{\\{}4\mathbb{E}\biggl{[}\biggl{(}\frac{1}{N}\sum_{i=1}^{N}|\tilde{Y}^{i,\epsilon,N}_{t}|^{2}\biggr{)}\frac{1}{\epsilon^{2}}\biggl{(}f(i)\tilde{Y}^{i,\epsilon,N}_{t}+\tau^{2}(i)\biggr{)}\biggr{]}$
$\displaystyle\qquad+4\mathbb{E}\biggl{[}\biggl{(}\frac{1}{N}\sum_{i=1}^{N}|\tilde{Y}^{i,\epsilon,N}_{t}|^{2}\biggr{)}\frac{1}{\epsilon
a(N)\sqrt{N}}\tau(i)\tilde{u}^{N}_{i}(t)\tilde{Y}^{i,\epsilon,N}_{t}\biggr{]}+8\mathbb{E}\biggl{[}\frac{1}{\epsilon^{2}N}\tau^{2}(i)|\tilde{Y}^{i,\epsilon,N}_{t}|^{2}\biggr{]}\biggr{\\}}$
$\displaystyle\leq\frac{1}{N}\sum_{i=1}^{N}\biggl{\\{}\mathbb{E}\biggl{[}\biggl{(}\frac{1}{N}\sum_{i=1}^{N}|\tilde{Y}^{i,\epsilon,N}_{t}|^{2}\biggr{)}\biggl{(}-\frac{\beta}{\epsilon^{2}}|\tilde{Y}^{i,\epsilon,N}_{t}|^{2}+\frac{2}{\epsilon^{2}}\tau^{2}(i)\biggr{)}\biggr{]}$
$\displaystyle\qquad+4\mathbb{E}\biggl{[}\biggl{(}\frac{1}{N}\sum_{i=1}^{N}\tilde{Y}^{i,\epsilon,N}_{t}|^{2}\biggr{)}\frac{1}{\epsilon
a(N)\sqrt{N}}\tau(i)\tilde{u}^{N}_{i}(t)\tilde{Y}^{i,\epsilon,N}_{t}\biggr{]}+8\mathbb{E}\biggl{[}\frac{1}{\epsilon^{2}N}\tau^{2}(i)|\tilde{Y}^{i,\epsilon,N}_{t}|^{2}\biggr{]}\biggr{\\}}$
$\displaystyle\qquad+\frac{C}{\epsilon^{2}}\mathbb{E}\biggl{[}\frac{1}{N}\sum_{i=1}^{N}|\tilde{Y}^{i,\epsilon,N}_{t}|^{2}\biggr{]}$
$\displaystyle\leq-\frac{\beta}{\epsilon^{2}}\mathbb{E}\biggl{[}\biggl{(}\frac{1}{N}\sum_{i=1}^{N}|\tilde{Y}^{i,\epsilon,N}_{t}|^{2}\biggr{)}^{2}\biggr{]}+\frac{4}{\epsilon
a(N)\sqrt{N}}\mathbb{E}\biggl{[}\biggl{(}\frac{1}{N}\sum_{i=1}^{N}|\tilde{Y}^{i,\epsilon,N}_{t}|^{2}\biggr{)}\biggl{(}\frac{1}{N}\sum_{i=1}^{N}\tau(i)\tilde{u}^{N}_{i}(t)\tilde{Y}^{i,\epsilon,N}_{t}\biggr{)}\biggr{]}$
$\displaystyle\qquad+\frac{C}{\epsilon^{2}}\mathbb{E}\biggl{[}\frac{1}{N}\sum_{i=1}^{N}|\tilde{Y}^{i,\epsilon,N}_{t}|^{2}\biggr{]}\biggl{[}1+\frac{1}{N}\biggr{]}$
$\displaystyle\leq-\frac{\beta}{2\epsilon^{2}}\mathbb{E}\biggl{[}\biggl{(}\frac{1}{N}\sum_{i=1}^{N}|\tilde{Y}^{i,\epsilon,N}_{t}|^{2}\biggr{)}^{2}\biggr{]}+\frac{C}{a^{2}(N)N}\mathbb{E}\biggl{[}\biggl{(}\frac{1}{N}\sum_{i=1}^{N}\tau(i)\tilde{u}^{N}_{i}(t)\tilde{Y}^{i,\epsilon,N}_{t}\biggr{)}^{2}\biggr{]}$
$\displaystyle\qquad+\frac{C}{\epsilon^{2}}\mathbb{E}\biggl{[}\frac{1}{N}\sum_{i=1}^{N}|\tilde{Y}^{i,\epsilon,N}_{t}|^{2}\biggr{]}\biggl{[}1+\frac{1}{N}\biggr{]}.$

Here we again used the implication (19) of Assumption A2) and the boundedness
of $\tau$ from A1). Then by the comparison theorem from the proof of Lemma
B.1, we have for all $t\in[0,T],N\in\mathbb{N}$,

$\displaystyle\mathbb{E}\biggl{[}\biggl{(}\frac{1}{N}\sum_{i=1}^{N}|\tilde{Y}^{i,\epsilon,N}_{t}|^{2}\biggr{)}^{2}\biggr{]}\leq$
$\displaystyle\leq|\eta^{y}|^{4}e^{-\frac{\beta}{2\epsilon^{2}}t}+\frac{C}{\epsilon^{2}}e^{-\frac{\beta}{2\epsilon^{2}}t}\biggl{[}1+\frac{1}{N}\biggr{]}\int_{0}^{t}\mathbb{E}\biggl{[}\frac{1}{N}\sum_{i=1}^{N}|\tilde{Y}^{i,\epsilon,N}_{s}|^{2}\biggr{]}e^{\frac{\beta}{2\epsilon^{2}}s}ds$
$\displaystyle\qquad+\frac{C}{a^{2}(N)N}e^{-\frac{\beta}{2\epsilon^{2}}t}\int_{0}^{t}\mathbb{E}\biggl{[}\biggl{(}\frac{1}{N}\sum_{i=1}^{N}\tau(i)\tilde{u}^{N}_{i}(s)\tilde{Y}^{i,\epsilon,N}_{s}\biggr{)}^{2}\biggr{]}e^{\frac{\beta}{2\epsilon^{2}}s}ds$
$\displaystyle\leq|\eta^{y}|^{4}+\frac{C}{\epsilon^{2}}e^{-\frac{\beta}{2\epsilon^{2}}t}\biggl{[}1+\frac{1}{N}\biggr{]}\int_{0}^{t}e^{\frac{\beta}{2\epsilon^{2}}s}ds\sup_{s\in[0,T]}\mathbb{E}\biggl{[}\frac{1}{N}\sum_{i=1}^{N}|\tilde{Y}^{i,\epsilon,N}_{s}|^{2}\biggr{]}$
$\displaystyle\qquad+\frac{C}{a^{2}(N)N}e^{-\frac{\beta}{2\epsilon^{2}}t}\int_{0}^{t}\mathbb{E}\biggl{[}\biggl{(}\frac{1}{N}\sum_{i=1}^{N}\tau(i)\tilde{u}^{N}_{i}(s)\tilde{Y}^{i,\epsilon,N}_{s}\biggr{)}^{2}\biggr{]}e^{\frac{\beta}{2\epsilon^{2}}s}ds$
$\displaystyle\leq|\eta^{y}|^{4}+C\biggl{[}1+\frac{1}{N}\biggr{]}\sup_{s\in[0,T]}\mathbb{E}\biggl{[}\frac{1}{N}\sum_{i=1}^{N}|\tilde{Y}^{i,\epsilon,N}_{s}|^{2}\biggr{]}$
$\displaystyle\qquad+\frac{C}{a^{2}(N)N}e^{-\frac{\beta}{2\epsilon^{2}}t}\int_{0}^{t}\mathbb{E}\biggl{[}\biggl{(}\frac{1}{N}\sum_{i=1}^{N}\tau(i)\tilde{u}^{N}_{i}(s)\tilde{Y}^{i,\epsilon,N}_{s}\biggr{)}^{2}\biggr{]}e^{\frac{\beta}{2\epsilon^{2}}s}ds$
$\displaystyle\leq|\eta^{y}|^{4}+C\biggl{[}1+\frac{1}{N}\biggr{]}\sup_{s\in[0,T]}\mathbb{E}\biggl{[}\frac{1}{N}\sum_{i=1}^{N}|\tilde{Y}^{i,\epsilon,N}_{s}|^{2}\biggr{]}$
$\displaystyle\qquad+\frac{C}{a^{2}(N)N}e^{-\frac{\beta}{2\epsilon^{2}}t}\int_{0}^{t}\mathbb{E}\biggl{[}\biggl{(}\frac{1}{N}\sum_{i=1}^{N}|\tilde{u}^{N}_{i}(s)|^{2}\biggr{)}\biggl{(}\frac{1}{N}\sum_{i=1}^{N}|\tilde{Y}^{i,\epsilon,N}_{s}|^{2}\biggr{)}\biggr{]}e^{\frac{\beta}{2\epsilon^{2}}s}ds$
$\displaystyle\leq|\eta^{y}|^{4}+C\biggl{[}1+\frac{1}{N}\biggr{]}\sup_{s\in[0,T]}\mathbb{E}\biggl{[}\frac{1}{N}\sum_{i=1}^{N}|\tilde{Y}^{i,\epsilon,N}_{s}|^{2}\biggr{]}$
$\displaystyle\qquad+\frac{C}{a^{2}(N)N}\mathbb{E}\biggl{[}\int_{0}^{T}\biggl{(}\frac{1}{N}\sum_{i=1}^{N}|\tilde{u}^{N}_{i}(s)|^{2}\biggr{)}ds\sup_{s\in[0,T]}\biggl{(}\frac{1}{N}\sum_{i=1}^{N}|\tilde{Y}^{i,\epsilon,N}_{s}|^{2}\biggr{)}\biggr{]}$
$\displaystyle\leq|\eta^{y}|^{4}+C\biggl{[}1+\frac{1}{N}\biggr{]}\sup_{s\in[0,T]}\mathbb{E}\biggl{[}\frac{1}{N}\sum_{i=1}^{N}|\tilde{Y}^{i,\epsilon,N}_{s}|^{2}\biggr{]}$
$\displaystyle\qquad+\frac{C}{a^{2}(N)N}\mathbb{E}\biggl{[}\sup_{s\in[0,T]}\biggl{(}\frac{1}{N}\sum_{i=1}^{N}|\tilde{Y}^{i,\epsilon,N}_{s}|^{2}\biggr{)}\biggr{]}\text{
by the bound \eqref{eq:controlassumptions}}$ $\displaystyle\leq
C(\eta^{y},\rho)[1+\frac{1}{N}+\frac{1}{a^{2}(N)N}+\frac{1}{a^{4}(N)N^{2}}+\frac{1}{a^{2}(N)N\epsilon^{\rho}}]$

for any $\rho\in(0,2)$ by Lemmas B.1 and B.2, where the constant $C$ is
independent of $N$. Then taking $\rho\in(0,2)$ such that such that
$\epsilon^{\rho/2}a(N)\sqrt{N}\rightarrow\lambda\in(0,\infty]$ and using
$a(N)\sqrt{N}\rightarrow\infty$, all the terms in the bound which depend on
$N$ are bounded as $N\rightarrow\infty$, so we get a bound independent of $N$
and $t$, and the result is proved. ∎

##  Appendix C Regularity of the Poisson Equations

As discussed in Remark 2.6, there is a current gap in the literature regarding
rates of polynomial growth of derivatives of the Poisson equations used in
Section 6. Nevertheless, it is important to verify that the assumptions
imposed on these solutions in Section 2 are non-empty. For the reasons
outlined in Remark 2.6, we handle the case of the 1D Poisson equations from
Equations (22),(64) and the Multi-Dimensional Poisson Equations (28) and (63),
separately in Subsections C.1 and C.2 below. In Subsection C.3 we provide
specific examples where the Assumptions in Section 2 hold.

###  C.1. Results for the 1-Dimensional Poisson Equation

Throughout this subsection we assume A1) and A2). Recall the frozen generator
$L_{x,\mu}$ from Equation (18), the invariant measure $\pi$ from Equation
(20), the multi-index derivative notation and associated spaces of functions
from Definitions 2.1 and 2.4, and the definition of $a$ from Equation (18).

######  Lemma C.1.

Consider $B\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\times\mathbb{R}\times\mathbb{P}_{2}(\mathbb{R})\rightarrow\mathbb{R}$
continuous such that

$\displaystyle\int_{\mathbb{R}}B(x,y,\mu)\pi(dy;x,\mu)=0,\forall
x\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})$

and $|B(x,y,\mu)|=O(|y|^{q_{B}})$ for $q_{B}\in\mathbb{R}$ uniformly in
$x,\mu$ as $|y|\rightarrow\infty$. Then there exists a unique classical
solution $u\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R})\rightarrow\mathbb{R}$
to

$\displaystyle L_{x,\mu}u(x,y,\mu)=B(x,y,\mu)$

such that $u$ is continuous in $(x,y,\mathbb{W}_{2})$,
$\int_{\mathbb{R}}u(x,y,\mu)\pi(dy;x,\mu)=0$, and $u$ has at most polynomial
growth as $|y|\rightarrow\infty$.

In addition,

$\displaystyle|u(x,y,\mu)|$ $\displaystyle=O(|y|^{q_{B}})\text{ for }q_{B}\neq
0;\text{ if }q_{B}=0,\text{ then }|u(x,y,\mu)|=O(\ln(|y|)$
$\displaystyle|u_{y}(x,y,\mu)|$
$\displaystyle=O(|y|^{q_{B}-1}),\quad|u_{yy}(x,y,\mu)|=O(|y|^{q_{B}})$

as $|y|\rightarrow\infty$ uniformly in $x,y,\mu$.

Furthermore if $B(x,y,\mu)$ is Lipschitz continuous in $y$ uniformly in
$x,\mu$ (so that necessarily $q_{B}\leq 1$), then so are $u,u_{y},u_{yy}$.

###### Proof.

This follows by Proposition A.4 in [GS], since in our setting Condition 2.1
(i) holds with $\alpha=1$ and assumption A.3 holds with $\theta=1$. The
statement about Lipschitz continuity of $u$ and its derivatives follows from
the Lipschitz continuity of $a,f$ under assumptions A1) and A2) and Theorem
9.19 in [GT]. ∎

######  Lemma C.2.

Consider $h\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R})\rightarrow\mathbb{R}$.
Suppose $h$ in jointly continuous in $(x,y,\mathbb{W}_{2})$ and grows at most
polynomially in $y$ uniformly in $x,\mu$. Then

$\displaystyle\int_{\mathbb{R}}h(x,y,\mu_{1})\pi(dy;x,\mu_{1})-\int_{\mathbb{R}}h(x,y,\mu_{2})\pi(dy;x,\mu_{2})=$
(72)
$\displaystyle\qquad=\int_{\mathbb{R}}h(x,y,\mu_{1})-h(x,y,\mu_{2})-[\mathcal{L}_{x,\mu_{1}}-\mathcal{L}_{x,\mu_{2}}]v(x,y,\mu_{2})\pi(dy;x,\mu_{1})$

for all $x\in\mathbb{R},\mu_{1},\mu_{2}\in\mathcal{P}_{2}(\mathbb{R})$, and

$\displaystyle\int_{\mathbb{R}}h(x_{1},y,\mu)\pi(dy;x_{1},\mu)-\int_{\mathbb{R}}h(x_{2},y,\mu)\pi(dy;x_{2},\mu)=$
(73)
$\displaystyle\qquad=\int_{\mathbb{R}}h(x_{1},y,\mu)-h(x_{2},y,\mu)-[\mathcal{L}_{x_{1},\mu}-\mathcal{L}_{x_{2},\mu}]v(x_{2},y,\mu)\pi(dy;x_{1},\mu)$

for all $x_{1},x_{2}\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})$.

Here $v$ solves

(74) $\displaystyle\mathcal{L}_{x,\mu}v(x,y,\mu)$
$\displaystyle=h(x,y,\mu)-\int_{\mathbb{R}}h(x,\bar{y},\mu)\pi(d\bar{y};x,\mu).$

Consider also
$\mathcal{L}^{(k,j,\bm{\alpha}(\bm{p}_{k}))}_{x,\mu}[z_{\bm{p}_{k}}$ the
differential operator acting on $\phi\in C^{2}_{b}(\mathbb{R})$ by

$\displaystyle\mathcal{L}^{(k,j,\bm{\alpha}(\bm{p}_{k}))}_{x,\mu}[z_{\bm{p}_{k}}]\phi(y)=D^{(k,j,\bm{\alpha}(\bm{p}_{k}))}f(x,y,\mu)[z_{\bm{p}_{k}}]\phi^{\prime}(y)+D^{(k,j,\bm{\alpha}(\bm{p}_{k}))}a(x,y,\mu)[z_{\bm{p}_{k}}]\phi^{\prime\prime}(y).$

Assume that for some complete collection of multi-indices $\bm{\zeta}$, that
$h,a,f\in\mathcal{M}_{p}^{\bm{\zeta}}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$,
and that
$v_{y},v_{yy}\in\mathcal{M}_{p}^{\bm{\zeta}^{\prime}}(\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathcal{P}_{2}(\mathbb{R}^{d}))$,
where $\bm{\zeta}^{\prime}$ is obtained from removing any multi-indices which
contain the maximal first and second values from $\bm{\zeta}$. Then for any
multi-index $(n,l,\bm{\beta})\in\bm{\zeta}$:

(75) $\displaystyle
D^{(n,l,\bm{\beta})}\int_{\mathbb{R}}h(x,y,\mu)\pi(dy;x,\mu)[z_{1},...,z_{n}]=\int_{\mathbb{R}}\left(D^{(n,l,\bm{\beta})}h(x,y,\mu)[z_{1},...,z_{n}]-\right.$
$\displaystyle\quad\left.-\sum_{k=0}^{n}\sum_{j=0}^{l}\sum_{\bm{p}_{k}}C_{(\bm{p}_{k},j,n,l)}\mathcal{L}^{(k,j,\bm{\alpha}(\bm{p}_{k}))}_{x,\mu}[z_{\bm{p}_{k}}]D^{(n-k,l-j,\bm{\alpha}(\bm{p}^{\prime}_{n-k}))}v(x,y,\mu)[z_{\bm{p}^{\prime}_{n-k}}]\right)\pi(dy;x,\mu)$

where here $\bm{p}_{k}\in\binom{\\{1,...,n\\}}{k}$ with
$\bm{p}^{\prime}_{n-k}=\\{1,...,n\\}\setminus\bm{p}_{k}$, for
$\bm{p}_{k}=\\{p_{1},...,p_{k}\\}$, the argument $[z_{\bm{p}_{k}}]$ denotes
$[z_{p_{1}},...,z_{p_{k}}]$, and $\bm{\alpha}(\bm{p}_{k})\in\mathbb{N}^{k}$ is
determined by $\bm{\beta}=(\beta_{1},...,\beta_{n})$ by
$\bm{\alpha}(\bm{p}_{k})=(\alpha_{1},...,\alpha_{k})$,
$\alpha_{j}=\beta_{p_{j}},j\in\\{1,...,k\\}$, and similarly for
$\bm{\alpha}(\bm{p}^{\prime}_{n-k})$. Also here $C_{(\bm{p}_{0},0,n,l)}=0$,
and $C_{(\bm{p}_{k},j,n,l)}>0,C_{(\bm{p}_{k},j,n,l)}\in\mathbb{N}$ for
$(k,j)\in\mathbb{N}^{2},(k,j)\neq(0,0)$, (see Remark C.3 for the exact
definition of these constants).

The same result holds replacing $D^{(n,l,\bm{\beta})}$ with
$\bm{\delta}^{(n,l,\bm{\beta})}$ if in addition we assume
$h,a,f\in\mathcal{M}_{\bm{\delta},p}^{\bm{\zeta}}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$,$v_{y},v_{yy}\in\mathcal{M}_{\bm{\delta},p}^{\bm{\zeta}^{\prime}}(\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathcal{P}_{2}(\mathbb{R}^{d}))$.
In this setting we will denote by
$\mathcal{L}^{(k,j,\bm{\alpha}(\bm{p}_{k})),\bm{\delta}}_{x,\mu}[z_{\bm{p}_{k}}]$
is the differential operator acting on $\phi\in C^{2}_{b}(\mathbb{R})$ by

$\displaystyle\mathcal{L}^{(k,j,\bm{\alpha}(\bm{p}_{k})),\bm{\delta}}_{x,\mu}[z_{\bm{p}_{k}}]\phi(y)=\bm{\delta}^{(k,j,\bm{\alpha}(\bm{p}_{k}))}f(x,y,\mu)[z_{\bm{p}_{k}}]\phi^{\prime}(y)+\bm{\delta}^{(k,j,\bm{\alpha}(\bm{p}_{k}))}a(x,y,\mu)[z_{\bm{p}_{k}}]\phi^{\prime\prime}(y).$

###### Proof.

The proofs of (C.2), (C.2) and of (75) for the Lions derivatives is the
content of Lemma A.4 in [BezemekSpiliopoulosAveraging2022].

For the linear functional derivatives, we can use the exact same proof as in
Lemma A.4 of [BezemekSpiliopoulosAveraging2022]. Though it is not immediately
obvious from the definition of the linear functional derivative that standard
properties of derivatives such as chain and product rule apply, we can use
Proposition 5.44/Remark 5.47 along with the representation (5.50) from
Proposition 5.51 in [CD] to see that these properties are inherited from the
Lions derivative (which is defined via lifting and using a Fréchet
derivative). Note that the needed uniform in $\mu$ Lipschitz continuity
assumption for the Lions derivatives of $a,b,h$ needed for Proposition 5.51 is
already implied by the definition of
$\mathcal{M}_{p}^{\bm{\zeta}}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$.

Then we get

$\displaystyle\frac{\delta}{\delta
m}\int_{\mathbb{R}}h(x,y,\mu)\pi(y;x,\mu)dy[z]$
$\displaystyle=\int_{\mathbb{R}}\frac{\delta}{\delta
m}h(x,y,\mu)[z]-\mathcal{L}^{(1,0,0),\bm{\delta}}_{x,\mu}[z]v(x,y,\mu)\pi(dy;x,\mu),$

for all $x,z\in\mathbb{R},\mu\in\mathcal{P}_{2}$, and can induct on $l,n$ in
the same way as is done in Lemma A.4 of [BezemekSpiliopoulosAveraging2022].
The details are omitted for brevity.

∎

######  Remark C.3.

The non-negative integers $C_{(\bm{p}_{k},j,n,l)}$ in the statement of Lemma
C.2 can be iteratively computed according to the following rules:

$\displaystyle C_{(\bm{p}_{0},0,0,0)}$ $\displaystyle=0$

To go up in $l$ (taking an $x$ derivative), we have for any
$l,n\in\mathbb{N}$, $\mathbb{N}\ni j\leq l+1$, $\mathbb{N}\ni k\leq n$, and
$\bm{p}_{k}\in\binom{\\{1,...,n\\}}{k}$:

$\displaystyle C_{(\bm{p}_{k},j,n,l+1)}$
$\displaystyle=\begin{cases}C_{(\bm{p}_{k},l,n,l)},&\text{ if }j=l+1\\\
C_{(\bm{p}_{k},0,n,l)},&\text{ if }j=0\\\ C_{(\bm{p}_{k},1,n,l)}+1,&\text{ if
}j=1\\\ C_{(\bm{p}_{k},j-1,n,l)}+C_{(\bm{p}_{k},j,n,l)},&\text{ otherwise}\\\
\end{cases}$

To go up in $n$ (taking a measure derivative), we have for any
$l,n\in\mathbb{N}$, $\mathbb{N}\ni j\leq l$, $\mathbb{N}\ni k\leq n+1$, and
$\bm{p}_{k}\in\binom{\\{1,...,n+1\\}}{k}$

$\displaystyle C_{(\bm{p}_{k},j,n+1,l)}$
$\displaystyle=\begin{cases}C_{(\\{1,...,n\\},j,n,l)}&\text{ if }k=n+1\\\
C_{(\bm{p}_{0},j,n,l)}&\text{ if }k=0\\\
\mathbbm{1}_{\bm{p}_{1}=\\{n+1\\}}+C_{(\bm{p}_{1},j,n,l)}\mathbbm{1}_{\bm{p}_{1}\neq\\{n+1\\}}&\text{
if }k=1\\\
C_{(\bm{p}_{k}\setminus\\{n+1\\},j,n,l)}\mathbbm{1}_{\\{n+1\\}\in\bm{p}_{k}}+C_{(\bm{p}_{k},j,n,l)}\mathbbm{1}_{\\{n+1\\}\not\in\bm{p}_{k}},&\text{
otherwise}.\\\ \end{cases}$

######  Lemma C.4.

Consider $B\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\times\mathbb{R}\times\mathbb{P}_{2}(\mathbb{R})\rightarrow\mathbb{R}$
continuous such that

$\displaystyle\int_{\mathbb{R}}B(x,y,\mu)\pi(dy;x,\mu)=0,\forall
x\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R}).$

Suppose that for some complete collection of multi-indices $\bm{\zeta}$ that
$B,a,f\in\mathcal{M}_{p}^{\bm{\zeta}}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$.
Then for the unique classical solution $u\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R})\rightarrow\mathbb{R}$
to

$\displaystyle L_{x,\mu}u(x,y,\mu)=B(x,y,\mu)$

such that $u$ is continuous in $(x,y,\mathbb{W}_{2})$,
$\int_{\mathbb{R}}u(x,y,\mu)\pi(dy;x,\mu)=0$, and $u$ has at most polynomial
growth as $|y|\rightarrow\infty$ (which exists by Lemma C.1),

  1. (1)

$u,u_{y},u_{yy}\in\mathcal{M}_{p}^{\bm{\zeta}}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$.

  2. (2)

If
$B,a,f\in\mathcal{M}_{\bm{\delta},p}^{\bm{\zeta}}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))\cap\mathcal{M}_{p}^{\bm{\zeta}}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$,
then
$u,u_{y},u_{yy}\in\mathcal{M}_{\bm{\delta},p}^{\bm{\zeta}}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$.

  3. (3)

If
$B,a,f\in\mathcal{M}_{p,L}^{\bm{\zeta}}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$,
then
$u,u_{y},u_{yy}\in\mathcal{M}_{p,L}^{\bm{\zeta}}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$.

Moreover, if we suppose that for all multi-indices
$(n,l,\bm{\beta})\in\bm{\zeta}$, $q_{f}(n,l,\bm{\beta})\leq 1$ and
$q_{a}(n,l,\bm{\beta})\leq 0$ (using here the notation of (13)), we have
control on the growth rate of the derivatives of $u$ in terms of those of $B$.
In particular, for any $(n,l,\bm{\beta})\in\bm{\zeta}$:

$\displaystyle
q_{u}(n,l,\bm{\beta})\leq\max\\{q_{B}(k,j,\bm{\alpha}(k))\mathrel{\mathop{\mathchar
58\relax}}\alpha(k)\in\binom{\bm{\beta}}{k},k\leq n,j\leq l\\},$

when the right hand side is nonzero, and the corresponding term grows at most
like $\ln(|y|)$ as $|y|\rightarrow\infty$ when the left hand side is zero. In
addition, $q_{u_{y}}(n,l,\bm{\beta})\leq q_{u}(n,l,\bm{\beta})-1$, and
$q_{u_{yy}}(n,l,\bm{\beta})\leq q_{u}(n,l,\bm{\beta})$, for all
$(n,l,\bm{\beta})\in\bm{\zeta}$.

###### Proof.

For 1), the proof essentially uses the same tools and a similar method to
Lemma A.2 in [BezemekSpiliopoulosAveraging2022], so we will only check this in
the case for $(n,l,\bm{\beta})=(0,1,0)$ and then comment on how the rest of
the terms follow. Importantly, Lemma A.2 in [BezemekSpiliopoulosAveraging2022]
only assumes existence and polynomial growth of derivatives of the solution
$u$ up to one order less than the derivative obtained there.

The result for $(n,l,\bm{\beta})=(0,0,0)$ is just another way of writing Lemma
C.1.

The differentiability and continuity of the derivatives is immediate via the
explicit representation for $u$

$\displaystyle v(x,y,\mu)$
$\displaystyle=\int_{-\infty}^{y}\frac{1}{a(x,\bar{y},\mu)\pi(\bar{y};x,\mu)}\biggl{[}\int_{-\infty}^{\bar{y}}B(x,\tilde{y},\mu)\pi(\tilde{y};x,\mu)d\tilde{y}\biggr{]}d\bar{y}$
(76) $\displaystyle\pi(y;x,\mu)$
$\displaystyle=\frac{Z(x,\mu)}{a(x,y,\mu)}\exp\biggl{(}\int_{0}^{y}\frac{f(x,\bar{y},\mu)}{a(x,\bar{y},\mu)}d\bar{y}\biggr{)}$

where
$Z^{-1}(x,\mu)\coloneqq\int_{\mathbb{R}}\frac{1}{a(x,y,\mu)}\exp\biggl{(}\int_{0}^{y}\frac{f(x,\bar{y},\mu)}{a(x,\bar{y},\mu)}d\bar{y}\biggr{)}dy$
is the normalizing constant.

To obtain the rate of polynomial growth of $u_{x}$, we differentiate the
equation that $u$ satisfies to get

$\displaystyle L_{x,\mu}u_{x}(x,y,\mu)$
$\displaystyle=B_{x}(x,y,\mu)-f_{x}(x,y,\mu)u_{y}(x,y,\mu)-a_{x}(x,y,\mu)u_{yy}(x,y,\mu)$
$\displaystyle=B_{x}(x,y,\mu)-L^{(0,1,0)}_{x,\mu}u(x,y,\mu)$

in the notation of Lemma A.2 in [BezemekSpiliopoulosAveraging2022]. But by the
centering condition on $B$, we have that letting $B=h$ in Lemma A.2 in
[BezemekSpiliopoulosAveraging2022], $u=v$ in the statement of that same lemma.
Thus we have

$\displaystyle\int_{\mathbb{R}}\left(B_{x}(x,y,\mu)-L^{(0,1,0)}_{x,\mu}u(x,y,\mu)\right)\pi(dy;x,\mu)$
$\displaystyle=\frac{\partial}{\partial
x}\int_{\mathbb{R}}B(x,y,\mu)\pi(dy;x,\mu)=0,$

and the inhomogeneity of the elliptic PDE that $u_{x}$ solves, in fact obeys
the centering condition, and hence Lemma C.1 applies. From the same lemma we
already know that $q_{u,y}(0,0,0)=q_{B}(0,0,0)-1$ and $q_{yy}=q_{B}(0,0,0)$.
This establishes that $u_{x}$ grows at most polynomially in $y$ uniformly in
$x,\mu$. Under the additional assumptions that $q_{f}(0,1,0)\leq 1$ and
$q_{a}(0,1,0)\leq 0$, we have the inhomogeneity is $O(|y|^{q_{B}(0,0,0)\vee
q_{B}(0,1,0)})$. So by Lemma C.1, $q_{u,x}=q_{B}(0,0,0)\vee
q_{B}(0,1,0),q_{u,x,y}=q_{B}(0,0,0)\vee
q_{B}(0,1,0)-1,q_{u,x,y,y}=q_{B}(0,0,0)\vee q_{B}(0,1,0)$.

All of the bounds work in the same way, with the inhomogeneity of the elliptic
PDE of the desired derivative of $u$ solves being the integrand of the
expression for the corresponding derivative of $\bar{B}(x,y,\mu)$ from Lemma
A.2 in [BezemekSpiliopoulosAveraging2022]. Put explicitly:

$\displaystyle
L_{x,\mu}D^{(n,l,\bm{\beta})}u(x,y,\mu)[z_{1},...,z_{n}]=D^{(n,l,\bm{\beta})}B(x,y,\mu)[z_{1},...,z_{n}]-$
(77) $\displaystyle\hskip
113.81102pt-\sum_{k=0}^{n}\sum_{j=0}^{l}\sum_{\bm{p}_{k}}C_{(\bm{p}_{k},j,n,l)}L^{(k,j,\bm{\alpha}(\bm{p}_{k}))}_{x,\mu}[z_{\bm{p}_{k}}]D^{(n-k,l-j,\bm{\alpha}(\bm{p}^{\prime}_{n-k}))}u(x,y,\mu)[z_{\bm{p}^{\prime}_{n-k}}],$

where the constants $C_{(\bm{p}_{k},j,n,l)}$ are defined inductively in Remark
A.3 in [BezemekSpiliopoulosAveraging2022] and
$L^{(k,j,\bm{\alpha}(\bm{p}_{k}))}_{x,\mu}[z_{\bm{p}_{k}}]$ is the
differential operator acting on $\phi\in C^{2}_{b}(\mathbb{R})$ by

$\displaystyle
L^{(k,j,\bm{\alpha}(\bm{p}_{k}))}_{x,\mu}[z_{\bm{p}_{k}}]\phi(y)=D^{(k,j,\bm{\alpha}(\bm{p}_{k}))}f(x,y,\mu)[z_{\bm{p}_{k}}]\phi^{\prime}(y)+D^{(k,j,\bm{\alpha}(\bm{p}_{k}))}a(x,y,\mu)[z_{\bm{p}_{k}}]\phi^{\prime\prime}(y).$

The first $y$ derivative of a lower order derivative in a parameter of $u$ in
the inhomogeneity is always multiplied by a derivative of $f$, and so if that
derivative of $f$ grows at most linearly in $y$, the growth of that term is at
most that of that lower order derivative of $u$, and same for the second $y$
derivative in a parameter of $u$ in the inhomogeneity, which multiplied by a
bounded lower order derivative of $a$. Thus it is clear the result follows by
proceeding inductively on $n,l$.

The proof for 2) follows in the exact same way. We note here that Lemma A.2 in
[BezemekSpiliopoulosAveraging2022] holds for the linear functional derivatives
$\bm{\delta}^{(n,l,\bm{\beta})}$ in place of the Lions derivatives
$D^{(n,l,\bm{\beta})}$ if in addition we assume
$h,a,f\in\mathcal{M}_{\bm{\delta},p}^{\bm{\zeta}}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$,$v_{y},v_{yy}\in\mathcal{M}_{\bm{\delta},p}^{\bm{\zeta}^{\prime}}(\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathcal{P}_{2}(\mathbb{R}^{d}))$.

The proof for 3) is similar to step 4 in the proof of Theorem 2.1 in
[RocknerFullyCoupled]. For the case $(n,l,\bm{\beta})=(0,0,0)$, we first note
that

$\displaystyle L_{x,\mu_{1}}[u(x,y,\mu_{1})-u(x,y,\mu_{2})]$
$\displaystyle=B(x,y,\mu_{1})-L_{x,\mu_{1}}u(x,y,\mu_{2})$
$\displaystyle=B(x,y,\mu_{1})-B(x,y,\mu_{2})-[L_{x,\mu_{1}}-L_{x,\mu_{2}}]u(x,y,\mu_{2}).$

By the transfer formula in Lemma A.2 of [BezemekSpiliopoulosAveraging2022] we
have

$\displaystyle\int_{\mathbb{R}}\left(B(x,y,\mu_{1})-B(x,y,\mu_{2})-[L_{x,\mu_{1}}-L_{x,\mu_{2}}]u(x,y,\mu_{2})\right)\pi(dy,x,\mu_{1})=$
$\displaystyle\qquad=\int_{\mathbb{R}}B(x,y,\mu_{1})\pi(dy;x,\mu_{1})-\int_{\mathbb{R}}B(x,y,\mu_{2})\pi(dy;x,\mu_{2})=0,$

so in fact the inhomogeneity in the above Poisson equation is centered. Now,
rather than using Lemma C.1, we apply [PV1] Theorem 2 to get there is
$k\in\mathbb{R}$ sufficiently large and $C>0$ such that for all
$x\in\mathbb{R},\mu_{1},\mu_{2}\in\mathcal{P}_{2}(\mathbb{R})$

$\displaystyle\sup_{y\in\mathbb{R}}\frac{|u(x,y,\mu_{1})-u(x,y,\mu_{2})|}{(1+|y|)^{k}}$
$\displaystyle\leq
C\sup_{y\in\mathbb{R}}\frac{\biggl{|}B(x,y,\mu_{1})-B(x,y,\mu_{2})-[L_{x,\mu_{1}}-L_{x,\mu_{2}}]u(x,y,\mu_{2})\biggr{|}}{(1+|y|)^{k}}$
$\displaystyle\leq C\mathbb{W}_{2}(\mu_{1},\mu_{2})$

by the Lipschitz assumptions on $B,f,a$. Thus for all
$x,y\in\mathbb{R},\mu_{1},\mu_{2}\in\mathcal{P}_{2}(\mathbb{R})$,

$\displaystyle|u(x,y,\mu_{1})-u(x,y,\mu_{2})|\leq
C\mathbb{W}_{2}(\mu_{1},\mu_{2})(1+|y|)^{k}.$

To see then that there are
$k^{\prime},k^{\prime\prime}\in\mathbb{R},C^{\prime},C^{\prime\prime}>0$ such
that

$\displaystyle|u_{y}(x,y,\mu_{1})-u_{y}(x,y,\mu_{2})|$ $\displaystyle\leq
C\mathbb{W}_{2}(\mu_{1},\mu_{2})(1+|y|)^{k^{\prime}}$
$\displaystyle|u_{yy}(x,y,\mu_{1})-u_{yy}(x,y,\mu_{2})|$ $\displaystyle\leq
C\mathbb{W}_{2}(\mu_{1},\mu_{2})(1+|y|)^{k^{\prime\prime}},$

we can apply the result of [GS] Lemma B.1 and Remark B.2, and the last line of
Proposition A.4 in the same reference.

The proof with $\mu_{1},\mu_{2}$ replaced by $x_{1},x_{2}$ follows in the same
way.

For the Lipschitz property in $z$, we first recall that for all
$x,y,z\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})$

$\displaystyle
L_{x,\mu}D^{(1,0,0)}u(x,y,\mu)[z]=D^{(1,0,0)}B(x,y,\mu)[z]-L^{(1,0,0)}_{x,\mu}[z]u(x,y,\mu)$

so

$\displaystyle
L_{x,\mu}\biggl{[}D^{(1,0,0)}u(x,y,\mu)[z_{1}]-D^{(1,0,0)}u(x,y,\mu)[z_{2}]\biggr{]}$
$\displaystyle=D^{(1,0,0)}B(x,y,\mu)[z_{1}]-L^{(1,0,0)}_{x,\mu}[z_{1}]u(x,y,\mu)$
$\displaystyle-\biggl{[}D^{(1,0,0)}B(x,y,\mu)[z_{2}]-L^{(1,0,0)}_{x,\mu}[z_{2}]u(x,y,\mu)\biggr{]}.$

By the transfer formula in Lemma A.2 of [BezemekSpiliopoulosAveraging2022] we
have for all $x,z\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})$:

$\displaystyle\int_{\mathbb{R}}\left(D^{(1,0,0)}B(x,y,\mu)[z]-L^{(1,0,0)}_{x,\mu}[z]u(x,y,\mu)\right)\pi(dy;x,\mu)=D^{(0,1,0)}\int_{\mathbb{R}}B(x,y,\mu)\pi(dy;x,\mu)[z]=0,$

so the inhomogeneity in the Poisson equation above in centered. Thus, using
the same argument as for the other Lipschitz continuity as well as the fact
that $D^{(1,0,0)}B,D^{(1,0,0)}f,D^{(1,0,0)}a$ are Lipschitz in $z$ and
$u_{y},u_{yy}$ grow at most polynomially in $y$, we get there is
$K\in\mathbb{R}$ and $C>0$ such that

$\displaystyle\biggl{|}D^{(1,0,0)}u(x,y,\mu)[z_{1}]-D^{(1,0,0)}u(x,y,\mu)[z_{2}]\biggr{|}\leq
C|z_{1}-z_{2}|(1+|y|)^{k},$

and similarly for $D^{(1,0,0)}u_{y}$ and $D^{(1,0,0)}u_{yy}$.

Then using the Poisson equation the derivatives satisfy given in Equation
(C.1), we can iteratively use this same approach, along with the fact that
products and sums of functions in
$\mathcal{M}_{p,L}^{\bm{\zeta}}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$
remain in
$\mathcal{M}_{p,L}^{\bm{\zeta}}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$,
to achieve the full result.

∎

######  Lemma C.5.

Suppose that for some complete collection of multi-indices $\bm{\zeta}$ that
$h,a,f\in\mathcal{M}_{p}^{\bm{\zeta}}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$.
Then
$\int_{\mathbb{R}}h(x,y,\mu)\pi(dy;x,\mu)\in\mathcal{M}_{b}^{\bm{\zeta}}(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$.
If in addition,
$h,a,f\in\mathcal{M}_{\bm{\delta},p}^{\bm{\zeta}}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$,
then
$\int_{\mathbb{R}}h(x,y,\mu)\pi(dy;x,\mu)\in\mathcal{M}_{\bm{\delta},b}^{\bm{\zeta}}(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$.
Further, if we have that
$h,a,f\in\mathcal{M}_{p,L}^{\bm{\zeta}}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$,
then
$\int_{\mathbb{R}}h(x,y,\mu)\pi(dy;x,\mu)\in\mathcal{M}_{b,L}^{\bm{\zeta}}(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$.

###### Proof.

This follows via Lemmas A.2 in [BezemekSpiliopoulosAveraging2022] and C.1 in a
similar way to Lemma C.4. The details are omitted.

∎

###  C.2. Result for the d-Dimensional Poisson Equation

######  Lemma C.6.

Suppose $F\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathcal{P}_{2}(\mathbb{R})\rightarrow\mathbb{R}^{d},G\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathcal{P}_{2}(\mathbb{R})\rightarrow\mathbb{R},\tau\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathcal{P}_{2}(\mathbb{R})\rightarrow\mathbb{R}^{d\times
m}$

$\displaystyle|F(x_{1},y_{1},\mu_{1})-F(x_{2},y_{2},\mu_{2})|+|G(x_{1},y_{1},\mu_{1})-G(x_{2},y_{2},\mu_{2})|+\mathinner{\\!\left\lVert\tau(x_{1},y_{1},\mu_{1})-\tau(x_{2},y_{2},\mu_{2})\right\rVert}$
$\displaystyle\leq
C[|x_{1}-x_{2}|+|y_{1}-y_{2}|+\mathbb{W}_{2}(\mu_{1},\mu_{2})],\forall
x_{1},x_{2},y_{1},y_{2}\in\mathbb{R}^{d},\mu_{1},\mu_{2}\in\mathcal{P}_{2}(\mathbb{R}^{d}),$

and there exists $\beta>0$ such that for all
$x\in\mathbb{R}^{d},\mu\in\mathcal{P}_{2}(\mathbb{R})$:

$\displaystyle 2\langle
F(x,y_{1},\mu)-F(x,y_{2},\mu),(y_{1}-y_{2})\rangle+3\mathinner{\\!\left\lVert\tau(x,y_{1},\mu)-\tau(x,y_{2},\mu)\right\rVert}^{2}$
$\displaystyle\leq-\beta|y_{1}-y_{2}|^{2}.$

Here $\langle\cdot,\cdot\rangle$ is denoting the inner product on
$\mathbb{R}^{d}$ and $\mathinner{\\!\left\lVert\cdot\right\rVert}$ the matrix
norm. Also assume that $\tau$ is bounded, and

$\displaystyle|G(x,y,\mu)|,|F(x,y,\mu)|\leq C(1+|y|),\forall
x\in\mathbb{R}^{d},\mu\in\mathcal{P}_{2}(\mathbb{R}).$

Define the differential operator $\tilde{\mathcal{L}}_{x,\mu}$ which for each
$x\in\mathbb{R}^{d},\mu\in\mathcal{P}_{2}(\mathbb{R})$ acts on $\phi\in
C^{2}_{b}(\mathbb{R})$ by

$\displaystyle\tilde{\mathcal{L}}_{x,\mu}\phi(y)$ $\displaystyle\coloneqq
F(x,y,\mu)\cdot\nabla\phi(y)+\frac{1}{2}\tau\tau^{\top}(x,y,\mu)\mathrel{\mathop{\mathchar
58\relax}}\nabla^{2}\phi(y).$

Then there is a unique invariant measure $\nu(\cdot;x,\mu)$ associated to
$\tilde{\mathcal{L}}_{x,\mu}$ for each $x,\mu$, and we assume the centering
condition on $G$:

$\displaystyle\int_{\mathbb{R}^{d}}G(x,y,\mu)\nu(dy;x,\mu)$
$\displaystyle=0,\forall
x\in\mathbb{R}^{d},\mu\in\mathcal{P}_{2}(\mathbb{R}).$

Finally, we assume the below derivatives all exist, are jointly continuous in
$(x,y,\mathbb{W}_{2})$ and auxiliary variables where applicable, and satisfy:

$\displaystyle\sup_{x\in\mathbb{R}^{d},\mu\in\mathcal{P}_{2}(\mathbb{R})}\max\\{|\partial_{x}G(x,\mu,y_{1})-\partial_{x}G(x,\mu,y_{2})|,|\partial_{y}G(x,\mu,y_{1})-\partial_{y}G(x,\mu,y_{2})|\\}$
$\displaystyle\leq C|y_{1}-y_{2}|$
$\displaystyle\sup_{x\in\mathbb{R}^{d},\mu\in\mathcal{P}_{2}(\mathbb{R})}\max\\{\mathinner{\\!\left\lVert\partial^{2}_{x}G(x,\mu,y_{1})-\partial^{2}_{x}G(x,\mu,y_{2})\right\rVert},\mathinner{\\!\left\lVert\partial^{2}_{y}G(x,\mu,y_{1})-\partial^{2}_{y}G(x,\mu,y_{2})\right\rVert}\\}$
$\displaystyle\leq C|y_{1}-y_{2}|$
$\displaystyle\sup_{x\in\mathbb{R}^{d},\mu\in\mathcal{P}_{2}(\mathbb{R})}\mathinner{\\!\left\lVert\partial_{x}\partial_{y}G(x,\mu,y_{1})-\partial_{x}\partial_{y}G(x,\mu,y_{2})\right\rVert}$
$\displaystyle\leq C|y_{1}-y_{2}|$
$\displaystyle\sup_{x\in\mathbb{R}^{d},\mu\in\mathcal{P}_{2}(\mathbb{R})}\mathinner{\\!\left\lVert\partial_{\mu}\partial_{x}G(x,\mu,y_{1})[\cdot]-\partial_{\mu}\partial_{x}G(x,\mu,y_{2})[\cdot]\right\rVert}_{L^{2}(\mathbb{R},\mu)}$
$\displaystyle\leq C|y_{1}-y_{2}|$
$\displaystyle\sup_{x\in\mathbb{R}^{d},\mu\in\mathcal{P}_{2}(\mathbb{R})}\mathinner{\\!\left\lVert\partial_{z}\partial_{\mu}G(x,\mu,y_{1})[\cdot]-\partial_{z}\partial_{\mu}G(x,\mu,y_{2})[\cdot]\right\rVert}_{L^{2}(\mathbb{R},\mu)}$
$\displaystyle\leq C|y_{1}-y_{2}|$
$\displaystyle\sup_{x\in\mathbb{R}^{d},\mu\in\mathcal{P}_{2}(\mathbb{R})}\mathinner{\\!\left\lVert\partial_{\mu}\partial_{x}G(x,\mu,y_{1})[\cdot]-\partial_{\mu}\partial_{x}G(x,\mu,y_{2})[\cdot]\right\rVert}_{L^{2}(\mathbb{R},\mu)}$
$\displaystyle\leq C|y_{1}-y_{2}|$
$\displaystyle\sup_{x\in\mathbb{R}^{d},\mu\in\mathcal{P}_{2}(\mathbb{R})}\mathinner{\\!\left\lVert\partial_{\mu}\partial_{y}G(x,\mu,y_{1})[\cdot]-\partial_{\mu}\partial_{y}G(x,\mu,y_{2})[\cdot]\right\rVert}_{L^{2}(\mathbb{R},\mu)}$
$\displaystyle\leq C|y_{1}-y_{2}|$
$\displaystyle\sup_{x\in\mathbb{R}^{d},\mu\in\mathcal{P}_{2}(\mathbb{R})}\mathinner{\\!\left\lVert\partial^{2}_{\mu}G(x,\mu,y_{1})[\cdot,\cdot]-\partial^{2}_{\mu}G(x,\mu,y_{2})[\cdot,\cdot]\right\rVert}_{L^{2}(\mathbb{R},\mu)\otimes
L^{2}(\mathbb{R},\mu)}$ $\displaystyle\leq C|y_{1}-y_{2}|$
$\displaystyle\sup_{x,y\in\mathbb{R}^{d},\mu\in\mathcal{P}_{2}(\mathbb{R})}\max\biggl{\\{}\mathinner{\\!\left\lVert\partial_{y}\partial_{x}G(x,\mu,y)\right\rVert},\mathinner{\\!\left\lVert\partial^{2}_{y}G(x,\mu,y)\right\rVert},\mathinner{\\!\left\lVert\partial_{\mu}\partial_{y}G(x,\mu,y)[\cdot]\right\rVert}_{L^{2}(\mathbb{R},\mu)}\biggr{\\}}$
$\displaystyle\leq C$

and same for $G$ replaced by $F$ and $\tau$, and in addition

$\displaystyle\sup_{x,y\in\mathbb{R}^{d},\mu\in\mathcal{P}_{2}(\mathbb{R})}\max\biggl{\\{}\mathinner{\\!\left\lVert\partial^{2}_{x}F(x,\mu,y)\right\rVert},\mathinner{\\!\left\lVert\partial^{2}_{x}\tau(x,\mu,y)\right\rVert},\mathinner{\\!\left\lVert\partial_{z}\partial_{\mu}F(x,\mu,y)[\cdot]\right\rVert}_{L^{2}(\mu,\mathbb{R})},\mathinner{\\!\left\lVert\partial_{z}\partial_{\mu}\tau(x,\mu,y)[\cdot]\right\rVert}_{L^{2}(\mu,\mathbb{R})},$
$\displaystyle\qquad\mathinner{\\!\left\lVert\partial_{\mu}\partial_{x}F(x,\mu,y)[\cdot]\right\rVert}_{L^{2}(\mu,\mathbb{R})},\mathinner{\\!\left\lVert\partial_{\mu}\partial_{x}\tau(x,\mu,y)[\cdot]\right\rVert}_{L^{2}(\mu,\mathbb{R})},\mathinner{\\!\left\lVert\partial^{2}_{\mu}F(x,\mu,y)[\cdot,\cdot]\right\rVert}_{L^{2}(\mu,\mathbb{R})\otimes
L^{2}(\mu,\mathbb{R})},$
$\displaystyle\qquad\mathinner{\\!\left\lVert\partial^{2}_{\mu}\tau(x,\mu,y)[\cdot,\cdot]\right\rVert}_{L^{2}(\mu,\mathbb{R})\otimes
L^{2}(\mu,\mathbb{R})}\biggr{\\}}\leq C.$

Then the partial differential equation

$\displaystyle\tilde{\mathcal{L}}_{x,\mu}\chi(x,y,\mu)$
$\displaystyle=-G(x,y,\mu)$

admits a unique classical solution $\chi\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathcal{P}_{2}(\mathbb{R})\rightarrow\mathbb{R}$
which has all of the above derivatives, and

$\displaystyle\sup_{x\in\mathbb{R}^{d},\mu\in\mathcal{P}_{2}(\mathbb{R})}\max\biggl{\\{}|\chi(x,y,\mu)|,\mathinner{\\!\left\lVert\partial_{x}\chi(x,y,\mu)\right\rVert},\mathinner{\\!\left\lVert\partial_{\mu}\chi(x,y,\mu)[\cdot]\right\rVert}_{L^{2}(\mathbb{R},\mu)},\mathinner{\\!\left\lVert\partial^{2}_{x}\chi(x,y,\mu)\right\rVert},\mathinner{\\!\left\lVert\partial_{z}\partial_{\mu}\chi(x,y,\mu)[\cdot]\right\rVert}_{L^{2}(\mathbb{R},\mu)},$
$\displaystyle\mathinner{\\!\left\lVert\partial_{\mu}\partial_{x}\chi(x,y,\mu)[\cdot]\right\rVert}_{L^{2}(\mathbb{R},\mu)},\mathinner{\\!\left\lVert\partial^{2}_{\mu}\chi(x,y,\mu)[\cdot,\cdot]\right\rVert}_{L^{2}(\mathbb{R},\mu)\otimes
L^{2}(\mathbb{R},\mu)}\biggr{\\}}\leq C(1+|y|),\forall y\in\mathbb{R}^{2},$
$\displaystyle\sup_{x,y\in\mathbb{R}^{d},\mu\in\mathcal{P}_{2}(\mathbb{R})}\max\biggl{\\{}\mathinner{\\!\left\lVert\partial_{y}\chi(x,y,\mu)\right\rVert},\mathinner{\\!\left\lVert\partial^{2}_{y}\chi(x,y,\mu)\right\rVert},\mathinner{\\!\left\lVert\partial_{x}\partial_{y}\chi(x,y,\mu)\right\rVert},\mathinner{\\!\left\lVert\partial_{\mu}\partial_{y}\chi(x,y,\mu)\right\rVert}_{L^{2}(\mathbb{R},\mu)}\leq
C.$

Moreover, if all listed derivatives of $F,G,\tau$ are jointly continuous in
$(x,y,\mathbb{W}_{2})$, then so are listed derivatives of $\chi$.

In the notation of Definition 2.4, this conclusion reads
$\chi\in\tilde{\mathcal{M}}^{\tilde{\bm{\zeta}}}_{p}(\mathbb{R}^{2}\times\mathbb{R}^{2}\times\mathcal{P}_{2}(\mathbb{R}))$,
$\chi_{y}\in\tilde{\mathcal{M}}^{\tilde{\bm{\zeta}}_{1}}_{p}(\mathbb{R}^{2}\times\mathbb{R}^{2}\times\mathcal{P}_{2}(\mathbb{R}))$,
$\chi_{yy}\in\tilde{\mathcal{M}}^{(0,0,0)}_{p}(\mathbb{R}^{2}\times\mathbb{R}^{2}\times\mathcal{P}_{2}(\mathbb{R}))$
with $\tilde{q}_{\chi}(n,l,\bm{\beta})\leq
1,\forall(n,l,\bm{\beta})\in\tilde{\bm{\zeta}}$,
$\tilde{q}_{\chi_{y}}(n,l,\bm{\beta})\leq
0,\forall(n,l,\bm{\beta})\in\tilde{\bm{\zeta}}_{1}$, and
$\tilde{q}_{\chi_{yy}}(0,0,0)\leq 0$ where
$\tilde{\bm{\zeta}},\tilde{\bm{\zeta}}_{1}$ are as in Equation (27).

###### Proof.

The arguments here follow closely those in [RocknerMcKeanVlasov]. Existence
and uniqueness for the invariant measure and strong solution from the Poisson
equation are the subject of the beginning of Section 3.3 and Section 4.1 of
[RocknerMcKeanVlasov]. The bound for $\chi$, $\partial_{y}\chi$,
$\partial_{x}\chi,\partial_{\mu}\chi,\partial^{2}_{x}\chi$, and
$\partial_{z}\partial_{\mu}\chi$ is also the subject of Proposition
4.1/Section 6.3 of [RocknerMcKeanVlasov], where we made the modification that
$\tau,F$ (their $g,f$ respectively) are bounded in $x,\mu$, from which one can
see that the bound on the solution is also uniform in $x,\mu$.

Thus we just need to show the bounds for $\partial^{2}_{y}\chi$,
$\partial_{x}\partial_{y}\chi$, $\partial_{\mu}\partial_{y}\chi$,
$\partial_{\mu}\partial_{x}\chi$, and $\partial^{2}_{\mu}\chi$. The bounds for
$\partial^{2}_{y}\chi$, $\partial_{x}\partial_{y}\chi$ and
$\partial_{\mu}\partial_{y}\chi$ are established in the recent [HLLS]
Proposition 3.1.

For the mixed partial derivative in $x$ and $\mu$ and the second partial
derivative in $\mu$, we can follow the proof of Proposition 4.1 of
[RocknerMcKeanVlasov]. The details are omitted here due to the similarity of
the argument. ∎

###  C.3. Some specific examples for which the assumptions of the paper hold

######  Proposition C.7.

Suppose A1)\- A3) hold. Let $\tilde{\bm{\zeta}},\tilde{\bm{\zeta}}_{1}$ be as
in Equation (27), and consider also:

$\displaystyle\bm{\zeta}$
$\displaystyle\ni\\{(0,j_{1},0),(1,j_{2},j_{3}),(2,j_{4},(j_{5},0)),(3,0,0)\mathrel{\mathop{\mathchar
58\relax}}j_{1}=0,1,2,j_{2}+j_{3}\leq 2,j_{4}+j_{5}\leq 1\\}$
$\displaystyle\bm{\zeta}_{1}$
$\displaystyle\ni\\{(0,2,0),(1,j_{1},j_{2}),(2,0,0)\mathrel{\mathop{\mathchar
58\relax}}j_{1}+j_{2}\leq 1\\}.$

In addition, suppose:

  1. (1)

For $h=\tau_{1},\tau_{2},b$:

$\displaystyle|h(x_{1},y_{1},\mu_{1})-h(x_{2},y_{2},\mu_{2})|\leq
C(|x_{1}-x_{2}|+|y_{1}-y_{2}|+\mathbb{W}_{2}(\mu_{1},\mu_{2})),\forall
x_{1},x_{2},y\in\mathbb{R},\mu_{1},\mu_{2}\in\mathcal{P}_{2}(\mathbb{R}).$

  2. (2)

$a,f,b\in\mathcal{M}^{\bm{\zeta}}_{p}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$,
$a_{y},f_{y},b_{y}\in\mathcal{M}^{\tilde{\bm{\zeta}}}_{p}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$,
$a_{yy},f_{yy},b_{yy}\in\mathcal{M}^{\tilde{\bm{\zeta}}_{1}}_{p}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$,
and $a_{yyy},f_{yyy},b_{yyy}$ are bounded.

  3. (3)

$q_{a}(n,l,\bm{\beta})\leq 0$ for all
$(n,l,\bm{\beta})\in\bm{\zeta},q_{a_{y}}(n,l,\bm{\beta})\leq 0$ for all
$(n,l,\bm{\beta})\in\tilde{\bm{\zeta}},$ and $q_{a_{yy}}(n,l,\bm{\beta})\leq
0$ for all $(n,l,\bm{\beta})\in\tilde{\bm{\zeta}}_{1}$. In addition,
$q_{a}(0,1,0)<0$.

  4. (4)

$q_{f}(n,l,\bm{\beta})\leq 1$ for all
$(n,l,\bm{\beta})\in\bm{\zeta},q_{f}(n,l,\bm{\beta})\leq 0$ for all
$(n,l,\bm{\beta})\in\bm{\zeta}_{1},$ $q_{f_{y}}(n,l,\bm{\beta})\leq 0$ for all
$(n,l,\bm{\beta})\in\tilde{\bm{\zeta}},$ and $q_{f_{yy}}(n,l,\bm{\beta})\leq
0$ for all $(n,l,\bm{\beta})\in\tilde{\bm{\zeta}}_{1}$. In addition,
$q_{f}(0,1,0)<0$.

  5. (5)

$q_{b}(n,l,\bm{\beta})<0$ for all
$(n,l,\bm{\beta})\in\bm{\zeta},q_{b_{y}}(n,l,\bm{\beta})\leq 0$ for all
$(n,l,\bm{\beta})\in\tilde{\bm{\zeta}},$ and $q_{b_{yy}}(n,l,\bm{\beta})\leq
0$ for all $(n,l,\bm{\beta})\in\tilde{\bm{\zeta}}_{1}$.

Then assumptions A7) and A11) hold.

###### Proof.

We first want to show the assumptions of Lemma C.6 with $d=2,m=4$ hold with
$F_{1}(x,y,\mu)=f(x_{1},y_{1},\mu),F_{2}=f(x_{2},y_{2},\mu)$,
$\tau_{11}(x,y,\mu)=\tau_{1}(x_{1},y_{1},\mu)$,$\tau_{12}(x,y,\mu)=\tau_{2}(x_{1},y_{1},\mu),\tau_{23}(x,y,\mu)=\tau_{1}(x_{2},y_{2},\mu)$,
$\tau_{24}(x,y,\mu)=\tau_{2}(x_{2},y_{2},\mu)$, and $\tau_{ij}\equiv 0$
otherwise, and
$G(x,y,\mu)=b(x_{1},y_{1},\mu)\partial_{\mu}\Phi(x_{2},y_{2},\mu)[x_{1}]$ or
$G(x,y,\mu)=b(x_{1},y_{1},\mu)\Phi(x_{2},y_{2},\mu)$.

Under these assumptions we have
$q_{\Phi}(n,l,\bm{\beta}),q_{\Phi_{yy}}(n,l,\bm{\beta})<0$ and
$q_{\Phi_{y}}(n,l,\bm{\beta})<-1$ for all $(n,l,\bf{\beta})\in\bm{\zeta}$ via
Lemma C.4.

The first Lipschitz assumption follows by (1) and A2). The retraction to mean
assumption is immediate from A2). We also have $F$ grows at most linearly in
$|y|$ by A2), and $G$ is in fact bounded by the above assumptions.

Checking the uniform Lipschitz in $y$ assumptions for the derivatives of $G$,
we have, for example, for the $x$ derivative of the first choice, that:

$\displaystyle
b_{x}(x_{1},y_{1},\mu)\partial_{\mu}\Phi(x_{2},y_{2},\mu)[x_{1}]+b(x_{1},y_{1},\mu)\partial_{z}\partial_{\mu}\Phi(x_{2},y_{2},\mu)[x_{1}]$

and

$\displaystyle
b(x_{1},y_{1},\mu)\partial_{\mu}\Phi_{x}(x_{2},y_{2},\mu)[x_{1}]$

need to be Lipschitz in $y$ uniformly in
$x\in\mathbb{R}^{2},\mu\in\mathcal{P}_{2}(\mathbb{R})$. To guarantee that the
product of functions is Lipschitz without any more a priori information on the
structure of each function, we must have that each function is Lipschitz and
bounded. Since we make the assumptions that $q_{b}(0,j,0),q_{b_{y}}(0,j,0)\leq
0,j=0,1$ and assumptions on $b,f,a$ such that
$q_{\Phi}(1,j_{1},j_{2}),q_{\Phi_{y}}(1,j_{1},j_{2})\leq 0,j_{1}+j_{2}\leq 1$,
this assumption holds. This is where the requirement that for many
$(n,l,\bm{\beta})$, $q_{b}(n,l,\bm{\beta})<0$ is coming in to play.

If we differentiate $G$ in the same way to see all the Lipschitz and bounded
assumptions needed on each of $b$, $\Phi$, and $\partial_{\mu}\Phi$’s
derivatives, we see that the only term that requires special care under these
assumptions is $\partial_{\mu}\Phi_{yy}(x_{2},y_{2},\mu)[x_{1}]$. But using
the equation elliptic equation that $\partial_{\mu}\Phi(x,y,\mu)[z]$ satisfies
for each $x,z\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})$ and Lemma C.1,
we see for the second derivative of $\partial_{\mu}\Phi(x,y,\mu)[z]$ to be
uniformly Lipschitz in $y$, it is sufficient for

$\displaystyle\partial_{\mu}b(x,y,\mu)[z]-\partial_{\mu}f(x,y,\mu)[z]\Phi_{y}(x,y,\mu)-\partial_{\mu}a(x,y,\mu)[z]\Phi_{yy}(x,y,\mu)$

to be uniformly Lipschitz in $y$. $\partial_{\mu}b(x,y,\mu)[z]$ is already
assumed to have this property, and $\Phi_{y}(x,y,\mu),\Phi_{yy}(x,y,\mu)$ are
bounded and uniformly Lipschitz in $y$ by assumption. Hence we just need in
addition there that
$q_{f}(1,0,0),q_{f_{y}}(1,0,0),q_{a}(1,0,0),q_{a_{y}}(1,0,0)\leq 0$ as
assumed.

Clearly since we prove the Lipschitz property for each of the derivatives of
$G$ by ensuring each component is Lipschitz and bounded, the needed
boundedness assumption for the mixed derivatives in $y$ of $G$ also holds.

Now to apply Lemma C.6, we just need to make sure that the needed Lipschitz
and bounded assumptions on the derivatives of $F$ and $\tau$ hold. But these
are implied by our assumptions on $f$ and $a$.

Now we just need to improve the result of Lemma C.6 to get
$q_{\chi}(0,j,0)\leq 0,j=0,1$. We turn to [PV1] Theorem 2. We have
$q_{G}(0,0,0)<0$, so $q_{\chi}(0,0,0)\leq 0$ by a direct application of that
Theorem. Then using that, as remarked in the proof of Lemma C.2, the transfer
formula for the $x$ derivatives in the $d$-dimensional case still hold in our
setting and that $q_{\chi_{y}}(0,0,0),q_{\chi_{yy}}(0,0,0)\leq 0$ by Lemma C.6
and $q_{f}(0,1,0),q_{a}(0,1,0)<0$ by assumption, we can get the inhomogeneity
for the Poisson equation which $\partial_{x}\chi$ satisfies also decays
polynomially in $|y|$ as $|y|\rightarrow\infty$ uniformly in
$x\in\mathbb{R}^{2},\mu\in\mathcal{P}_{2}(\mathbb{R})$, so again by [PV1]
Theorem 2, $q_{\chi}(0,1,0)\leq 0$.

Note that, while the sufficient conditions posed here for Assumption A7)
automatically imply those for A11), since Assumption A11) does not require
specific polynomial growth, it can actually be proved under much weaker
sufficient conditions - see Appendix A of [BezemekSpiliopoulosAveraging2022].
∎

######  Proposition C.8.

Suppose the conditions of Proposition C.7 and A5) hold. Let $\bm{\zeta}$ and
$\bm{\zeta}_{1}$ be as in Proposition C.7, consider the collections of multi-
indices from Equation (27), and let, in addition:

$\displaystyle\hat{\bm{\zeta}}_{1}$
$\displaystyle\ni\\{(0,j_{1},0),(1,j_{2},j_{3}),(2,j_{4},(j_{5},j_{6})),(3,j_{7},(j_{8},0,0))$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}j_{1}\in\\{0,1,...5\\},j_{3}\leq 4,j_{2}+j_{3}\leq 5,j_{5}+j_{6}\leq
2,j_{4}+j_{5}+j_{6}\leq 3,j_{7}+j_{8}\leq 1\\}.$
$\displaystyle\mathring{\bm{\zeta}}$
$\displaystyle=\bar{\bm{\zeta}}\cup\bar{\bm{\zeta}}_{w+2}$
$\displaystyle\mathring{\bm{\zeta}}_{1}$
$\displaystyle\ni\\{(j,j_{2},0),(1,j_{2},j_{3})\mathrel{\mathop{\mathchar
58\relax}}j=0,1,2,j_{2}=0,1,j_{3}=1,...,w+2\\}.$

In addition, suppose:

  1. (1)

For $h=\sigma,g,c$:

$\displaystyle|h(x_{1},y_{1},\mu_{1})-h(x_{2},y_{2},\mu_{2})|\leq
C(|x_{1}-x_{2}|+|y_{1}-y_{2}|+\mathbb{W}_{2}(\mu_{1},\mu_{2})),\forall
x_{1},x_{2},y\in\mathbb{R},\mu_{1},\mu_{2}\in\mathcal{P}_{2}(\mathbb{R}).$

  2. (2)

$g,\sigma,\tau_{1},c\in\mathcal{M}_{p,L}^{\hat{\bm{\zeta}}}$ and
$f,a,b\in\mathcal{M}_{p,L}^{\hat{\bm{\zeta}}_{1}}$

  3. (3)

$q_{a}(0,3,0)\leq 0$, $q_{f}(0,3,0)\leq 1$, $q_{b}(0,3,0)\leq 3$

  4. (4)

$q_{\sigma}(n,l,\bm{\beta}),q_{c}(n,l,\bm{\beta})\leq
1,q_{g}(n,l,\bm{\beta}),q_{\tau_{1}}(n,l,\bm{\beta})\leq 2$ for
$(n,l,\bm{\beta})\in\mathring{\bm{\zeta}}_{1}$ and
$q_{\sigma}(n,l,\bm{\beta}),q_{c}(n,l,\bm{\beta})\leq
2,q_{g}(n,l,\bm{\beta}),q_{\tau_{1}}(n,l,\bm{\beta})\leq 3$ for
$(n,l,\bm{\beta})\in\mathring{\bm{\zeta}}$.

  5. (5)

$b,f,\tau_{1},\tau_{2}\in\mathcal{M}_{\bm{\delta},p}^{\mathring{\bm{\zeta}}_{1}}$
and $c,\sigma,g\in\mathcal{M}_{\bm{\delta},p}^{\mathring{\bm{\zeta}}}$.

  6. (6)

$b,f,a\in\mathcal{M}_{p}^{\bm{\zeta}_{x,w+3}},\sigma,\tau_{1},c,g\in\mathcal{M}_{p}^{\bm{\zeta}_{x,w+2}}$,
and

$\displaystyle\mathinner{\\!\left\lVert\frac{\delta}{\delta
m}b(x,y,\mu)[\cdot]\right\rVert}_{w+2},\mathinner{\\!\left\lVert\frac{\delta}{\delta
m}b_{x}(x,y,\mu)[\cdot]\right\rVert}_{w+2},\mathinner{\\!\left\lVert\frac{\delta}{\delta
m}f(x,y,\mu)[\cdot]\right\rVert}_{w+2},\mathinner{\\!\left\lVert\frac{\delta}{\delta
m}f_{x}(x,y,\mu)[\cdot]\right\rVert}_{w+2},\mathinner{\\!\left\lVert\frac{\delta}{\delta
m}a(x,y,\mu)[\cdot]\right\rVert}_{w+2},$
$\displaystyle\mathinner{\\!\left\lVert\frac{\delta}{\delta
m}a_{x}(x,y,\mu)[\cdot]\right\rVert}_{w+2},\mathinner{\\!\left\lVert\frac{\delta}{\delta
m}g(x,y,\mu)[\cdot]\right\rVert}_{w+2},\mathinner{\\!\left\lVert\frac{\delta}{\delta
m}\sigma(x,y,\mu)[\cdot]\right\rVert}_{w+2},\mathinner{\\!\left\lVert\frac{\delta}{\delta
m}\tau_{1}(x,y,\mu)[\cdot]\right\rVert}_{w+2},\mathinner{\\!\left\lVert\frac{\delta}{\delta
m}c(x,y,\mu)[\cdot]\right\rVert}_{w+2}$ $\displaystyle\leq C(1+|y|^{k}),$

uniformly in $x\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})$ for some
$k\in\mathbb{N}.$

  7. (7)

There exists $\bar{\lambda}_{-}>0$ such that
$\bar{D}(x,\mu)\geq\bar{\lambda}_{-}$ for all
$x\in\mathbb{R},\mu\in\mathcal{P}_{2}$.

Then assumptions A4) and A6) \- A13), hold, and A’13) holds if we replace $w$
with $r$ in (6).

###### Proof.

A4) follows from the above Lipschitz properties, writing the system of SDEs
(1) in terms of the empirical projections of the coefficients and using
standard strong existence and uniqueness results (see Proposition A.1 in [BS])
for the weakly interacting system (1) and applying Theorem 2.1 in [Wang] to
the IID McKean-Vlasov system (57).

For the limiting system (25), we have, noting that $\hat{\bm{\zeta}}$ is the
same $\hat{\bm{\zeta}}$ from assumption A10), and $\hat{\bm{\zeta}}_{1}$ is
just $\hat{\bm{\zeta}}$ with one extra $x$ derivative in all spacial
components, that by Lemma C.4,
$\Phi,\Phi_{x},\Phi_{y},\Phi_{xy}\in\mathcal{M}_{p,L}^{\hat{\bm{\zeta}}}$, and
thus under these assumptions
$\gamma,D\in\mathcal{M}_{p,L}^{\hat{\bm{\zeta}}}$, and hence by Lemma C.5
$\bar{\gamma},\bar{D}\in\mathcal{M}_{b,L}^{\hat{\bm{\zeta}}}$. Using that
$x\mapsto\sqrt{x}$ is smooth with bounded derivatives of all orders on bounded
sets in $[\bar{\lambda}_{-},+\infty)$, we get via chain rule that in fact
$\bar{D}^{1/2}\in\mathcal{M}_{b,L}^{\hat{\bm{\zeta}}}$. This immediately
implies A10), and also yields by definition that $\bar{\gamma},\bar{D}$ are
bounded and Lipschitz in $x,\mathbb{W}_{2}$. So again Theorem 2.1 in [Wang]
applies, and we gain strong existence and uniqueness of the averaged McKean-
Vlasov SDE (25). Note that this is the only place where an assumption on the
limiting coefficients, that is (7), is being used, and as per Remark 4.4 this
assumption holds in all but pathological cases.

For assumption A6), we already showed from the assumptions in Proposition C.7,
$q_{\Phi}(n,l,\bm{\beta}),q_{\Phi_{yy}}(n,l,\bm{\beta})<0$ and
$q_{\Phi_{y}}(n,l,\bm{\beta})<-1$ for all $(n,l,\bm{\beta})\in\bm{\zeta}$,
which is much stronger than what we require.

For A7) and A11), we already showed the result in Proposition C.7.

For A8), we can check that for each of the 3 choices of $F$, that
$q_{F},q_{F}(n,l,\bm{\beta})\leq 1$ for
$(n,l,\bm{\beta})\in\mathring{\bm{\zeta}}_{1}$ and $q_{F}(n,l,\bm{\beta})\leq
2$ for $(n,l,\bm{\beta})\in\mathring{\bm{\zeta}}$, so the result follows by
Lemma C.4. In particular, since one of our choices of $F$ is $\gamma$, which
involves $\Phi_{x}$, we use that since $q_{b}(0,3,0)\leq 3,q_{f}(0,3,0)\leq
1,q_{a}(0,3,0)\leq 0$, Lemma C.4 implies $q_{\Phi_{y}}(0,3,0)\leq 2$.

For A9), we use that all the terms in the products involved in each of the
functions are bounded and jointly Lipschitz. The Lipschitz properties in $x,y$
can be extrapolated from boundedness of each of the respective derivatives of
the functions and in $\mathbb{W}_{2}$ follow from the boundedness of the Lions
derivatives by [CD] Remark 5.27.

For A12), we have via Lemma C.4 that
$\Phi,\Phi_{x},\Phi_{y},\Phi_{xy}\in\mathcal{M}_{\bm{\delta},p}^{\mathring{\bm{\zeta}}}$
(by construction of $\mathring{\bm{\zeta}}_{1}$), so that in fact all the
listed functions are in $\mathcal{M}_{\bm{\delta},p}^{\mathring{\bm{\zeta}}}$.
By Lemma C.5, this also implies the continuity for the Linear Functional
Derivatives in A13)/A’13) by definition.

And finally, for A13)/A’13), we have via Lemma C.4 that
$\Phi,\Phi_{x},\Phi_{y},\Phi_{xy}\in\mathcal{M}_{\bm{\delta},p}^{\bm{\zeta}_{x,w+2}}$
(by construction of $\bm{\zeta}_{x,w+3}$). Then in fact
$\gamma,D\in\mathcal{M}_{\bm{\delta},p}^{\bm{\zeta}_{x,w+2}}$, and by Lemma
C.5, we get
$\bar{\gamma},\bar{D}\in\mathcal{M}_{\bm{\delta},b}^{\bm{\zeta}_{x,w+2}}$.

For the regularity of the linear functional derivatives, via the equality (75)
given by Lemma C.2, we see it is sufficient to show

$\displaystyle\mathinner{\\!\left\lVert\frac{\delta}{\delta
m}\Phi(x,y,\mu)[\cdot]\right\rVert}_{w+2},\mathinner{\\!\left\lVert\frac{\delta}{\delta
m}\Phi_{y}(x,y,\mu)[\cdot]\right\rVert}_{w+2},\mathinner{\\!\left\lVert\frac{\delta}{\delta
m}\Phi_{x}(x,y,\mu)[\cdot]\right\rVert}_{w+2},\mathinner{\\!\left\lVert\frac{\delta}{\delta
m}\Phi_{xy}(x,y,\mu)[\cdot]\right\rVert}_{w+2}\leq C(1+|y|^{k}),$

uniformly in $x\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})$ for some
$k\in\mathbb{N}.$ This follows as in the proof of the Lipschitz property in
C.4, iteratively using the equation (C.1) and that the coefficient for the
growth rate in $y$ can be written in terms of the inhomogeneity via [PV1]
Theorem 2 and the assumption (6), then applying [GS] Lemma B1 / Remark B2 to
get the result for the derivatives in $y$ as well. ∎

The examples that follow, examples C.9-C.11, present concrete cases covered by
our assumptions.

######  Example C.9.

(A case with full dependence of the coefficients on $(x,y,\mu)$) Suppose
$\tau_{1},\tau_{2},\sigma>0$ are constant with $\sigma$ large enough that
$\bar{D}(x,\mu)>0,\forall x\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})$,
where $\bar{D}$ is as in Equation (24), and the other coefficients take the
form

$\displaystyle b(x,y,\mu)$
$\displaystyle=q\biggl{(}y-\frac{1}{\kappa}\langle\mu,\phi_{f}\rangle\biggr{)}p_{b}(x),\quad
c(x,y,\mu)=r_{c}(y)+p_{c}(x)+\langle\mu,\phi_{c}\rangle$ $\displaystyle
f(x,y,\mu)$ $\displaystyle=-\kappa y+\langle\mu,\phi_{f}\rangle,\quad
g(x,y,\mu)=r_{g}(y)+p_{g}(x)+\langle\mu,\phi_{g}\rangle,$

where here $\kappa>0$ and
$\langle\phi,\mu\rangle\coloneqq\int_{\mathbb{R}}\phi(z)\mu(dz)$. Suppose also
that $q\in C^{\infty}(\mathbb{R})$ is odd, there is $\beta>0$ such that
$|q(z)|,|q^{\prime}(z)|,|q^{\prime\prime}(z)|,|q^{\prime\prime\prime}(z)|\leq
C(1+|z|)^{-\beta}$, $\mathinner{\\!\left\lVert
r_{c}^{\prime}\right\rVert}_{\infty}\leq C$,
$|r_{g}|_{C_{b}^{1}(\mathbb{R})}\leq C$,
$\phi_{c},\phi_{g},\phi_{f}\in\mathcal{S}_{w+2}$, and $p_{c},p_{g}\in
C_{b}^{w+2}(\mathbb{R}),p_{b}\in C_{b}^{w+3}(\mathbb{R})$. Then assumptions
$\ref{assumption:uniformellipticity}-\ref{assumption:limitingcoefficientsregularity}$
hold, and A’13) holds if $w$ is replaced by $r$.

###### Proof.

A1) and A5) are immediate. A2) follows from noting that
$\eta(x,y,\mu)=\eta(\mu)=\langle\mu,\phi_{f}\rangle$, so
$\partial_{\mu}\eta(\mu)[z]=\phi^{\prime}_{f}(z)$ (see Example 1 in Section
5.2.2 in [CD]) and by Remark 5.27 in [CD],
$|\eta(\mu_{1})-\eta(\mu_{2})|\leq\mathinner{\\!\left\lVert\phi_{f}^{\prime}\right\rVert}_{\infty}\mathbb{W}_{2}(\mu_{1},\mu_{2}).$
In addition,

$\displaystyle
2(f(x,y_{1},\mu)-f(x,y_{2},\mu))(y_{1}-y_{2})+3|\tau_{1}(x,y_{1},\mu)-\tau_{1}(x,y_{2},\mu)|^{2}+3|\tau_{2}(x,y_{1},\mu)-\tau_{2}(x,y_{2},\mu)|^{2}$
$\displaystyle=-2\kappa(y_{1}-y_{2})^{2}.$

For A3), we can find via the explicit form of $\pi$ in Lemma C.2 (or the fact
that the frozen process is given by the Vasicek model and hence the transition
density is an explicitly computable Gaussian) that

$\displaystyle\pi(y;\mu)$ $\displaystyle=\sqrt{\frac{k}{2\pi
a}}\exp\biggl{(}-\frac{\kappa}{2a}[y-\frac{1}{\kappa}\langle\mu,\phi_{f}\rangle]^{2}\biggr{)},$

so

$\displaystyle\int_{\mathbb{R}}b(x,y,\mu)\pi(dy;x,\mu)$
$\displaystyle=p_{b}(x)\sqrt{\frac{k}{2\pi
a}}\int_{\mathbb{R}}q\biggl{(}y-\frac{1}{\kappa}\langle\mu,\phi_{f}\rangle\biggr{)}\exp\biggl{(}-\frac{\kappa}{2a}[y-\frac{1}{\kappa}\langle\mu,\phi_{f}\rangle]^{2}\biggr{)}dy$
$\displaystyle=p_{b}(x)\sqrt{\frac{k}{2\pi
a}}\int_{\mathbb{R}}q(y)\exp\biggl{(}-\frac{\kappa}{2a}y^{2}\biggr{)}dy$
$\displaystyle=0,\forall x\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})$

since the integrand is odd.

For the rest of the assumptions, we use Propositions C.7 and C.8. We have

$\displaystyle\frac{\partial^{j+k}}{\partial y^{j}\partial x^{k}}b(x,y,\mu)$
$\displaystyle=q^{(j)}\biggl{(}y-\frac{1}{\kappa}\langle\mu,\phi_{f}\rangle\biggr{)}p^{(k)}_{b}(x)$
$\displaystyle\partial^{l}_{\mu}\frac{\partial^{j+k}}{\partial y^{j}\partial
x^{k}}b(x,y,\mu)[z_{1},z_{2},...,z_{l}]$
$\displaystyle=q^{(j+l)}\biggl{(}y-\frac{1}{\kappa}\langle\mu,\phi_{f}\rangle\biggr{)}p^{(k)}_{b}(x)(-\kappa^{-1})^{l}\prod_{m=1}^{l}\phi^{\prime}(z_{m})$
$\displaystyle\frac{\delta^{l}}{\delta m^{l}}\frac{\partial^{j+k}}{\partial
y^{j}\partial x^{k}}b(x,y,\mu)[z_{1},z_{2},...,z_{l}]$
$\displaystyle=q^{(j+l)}\biggl{(}y-\frac{1}{\kappa}\langle\mu,\phi_{f}\rangle\biggr{)}p^{(k)}_{b}(x)(-\kappa^{-1})^{l}\prod_{m=1}^{l}\phi(z_{m})$
$\displaystyle\frac{\partial^{l}}{\partial z^{l}}\frac{\delta}{\delta
m}\frac{\partial^{j+k}}{\partial y^{j}\partial x^{k}}b(x,y,\mu)[z]$
$\displaystyle=q^{(j+1)}\biggl{(}y-\frac{1}{\kappa}\langle\mu,\phi_{f}\rangle\biggr{)}p^{(k)}_{b}(x)\phi^{(l)}_{f}(z)$
$\displaystyle\partial_{\mu}f(x,y,\mu)[z]$
$\displaystyle=\phi^{\prime}_{f}(z),\quad\frac{\partial^{l}}{\partial
z^{l}}\frac{\delta}{\delta m}f(x,y,\mu)[z]=\phi^{(l)}_{f}(z)$
$\displaystyle\frac{\partial^{j}}{\partial y^{j}}h(x,y,\mu)$
$\displaystyle=r^{(j)}_{h}(y),\quad\frac{\partial^{k}}{\partial
x^{k}}h(x,y,\mu)=p^{(j)}_{h}(x)$ $\displaystyle\partial_{\mu}h(x,y,\mu)[z]$
$\displaystyle=\phi^{\prime}_{h}(z),\quad\frac{\partial^{l}}{\partial
z^{l}}\frac{\delta}{\delta m}h(x,y,\mu)[z]=\phi^{(l)}_{h}(z)$

for $h=c,g$ and $j,k,l\in\mathbb{N}$ such that the above derivatives are
defined.

From this we can see that $b_{y},b_{x},\partial_{\mu}b(x,y,\mu)[z]$ are all
uniformly bounded, and hence (1) in Proposition C.7 holds.

For (2)-(5), $a$ is constant, and all the considered derivatives of $f$ are
uniformly $0$ except for $f_{y}=-\kappa$,
$\partial^{l}_{z}\partial_{\mu}f(x,y,\mu)[z]=\phi^{(1+l)}_{f}(z),l=0,1,2$, all
of which are uniformly bounded in $y$. All the involved derivatives of $b$ are
seen to be bounded functions of $x,z_{1},z_{2},z_{3}$ multiplied by
$q^{(j)}\biggl{(}y-\frac{1}{\kappa}\langle\mu,\phi_{f}\rangle\biggr{)}$ for
$j\in\\{0,1,2,3\\}$, so since the translation is uniformly bounded in $\mu$,
we see all of the listed
$q_{b}(n,l,\bm{\beta}),q_{b_{y}}(n,l,\bm{\beta}),q_{b_{yy}}(n,l,\bm{\beta})<0$
. So the assumptions of C.7 hold.

Now turning to Proposition C.8, we have
$h_{y},h_{x},\partial_{\mu}h(x,y,\mu)[z]$, $h=g,c$ are all uniformly bounded,
and hence (1) holds.

(2) follows from observing that the desired derivatives in $x$ of $f$ and of
$c,g$ are independent of $\mu$ and bounded Lipschitz in $x$.
$\partial_{z}^{l}\partial_{\mu}p$ for $p=f,c,g$, $l=0,...,4$ only depends on
$z$, and is Lipschitz for all $l$. All the listed derivatives of $b$ can
easily be shown to be Lipschitz in $x,z$ uniformly in $y,\mu$ via the
representations above, and since they take the form of bounded functions in
$x,z_{1},z_{2},z_{3},z_{4}$ multiplied by
$q^{(j)}\biggl{(}y-\frac{1}{\kappa}\langle\mu,\phi_{f}\rangle\biggr{)}$ for
$j\in\\{0,1,2,3,4\\}$, of which the Lions derivative is uniformly bounded, we
have by Remark 5.27 in [CD] that they are Lipschitz in $\mathbb{W}_{2}$
uniformly in $x,y,z$.

$q_{b}(0,3,0)<0$, so (3) holds.

The first and second derivatives of $c,g$ in $x$ are bounded, their first
derivative in $\mu$ is bounded and its derivative in $z$ are bounded, and the
rest of the derivatives of (4) are $0$.

For (5)-(6) $f,c,g$, the listed derivatives do not depend on $y$ or $\mu$, and
are uniformly bounded, with the linear functional derivatives in (6) being in
$\mathcal{S}_{w+2}$ by assumption. For $b$, all the derivatives in
$\tilde{\bm{\zeta}}_{3}$ are bounded by $\mathinner{\\!\left\lVert
q\right\rVert}_{\infty}\mathinner{\\!\left\lVert
p_{b}\right\rVert}_{C_{b}^{w+3}}$, the second linear functional derivatives
are uniformly bounded by their representation above, and
$\mathinner{\\!\left\lVert\frac{\partial^{k}}{\partial x}\frac{\delta}{\delta
m}b(x,y,\mu)[z]\right\rVert}_{w+2}\leq\mathinner{\\!\left\lVert
q^{\prime}\right\rVert}_{\infty}\mathinner{\\!\left\lVert
p_{b}\right\rVert}_{C^{k}_{b}(\mathbb{R})}\mathinner{\\!\left\lVert\phi_{f}\right\rVert}_{w+2}\leq
C,k=0,1$.

Finally, (7) holds by supposition (noting that by the form provided for
$\bar{D}$ in Equation (26) and the fact that $\Phi$ does not depend on
$\sigma$ that such a sufficiently large choice exists).

∎

######  Example C.10.

(A case where $\Phi$ and $\pi$ are independent of $x,\mu$) Consider the case:

$\displaystyle b(x,y,\mu)$ $\displaystyle=b(y),\quad
c(x,y,\mu)=c_{1}(x)+\langle\mu,c_{2}(x-\cdot)\rangle,\quad\sigma(x,y,\mu)\equiv\sigma$
$\displaystyle f(x,y,\mu)$ $\displaystyle=-\kappa y+\eta(y),\quad
g(x,y,\mu)=g_{1}(x)+\langle\mu,g_{2}(x-\cdot)\rangle,\quad\tau_{1}(x,y,\mu)\equiv\tau_{1},\quad\tau_{2}(x,y,\mu)\equiv\tau_{2}.$

Suppose $\eta\in C^{1}_{b}(\mathbb{R})$ with
$\mathinner{\\!\left\lVert\eta^{\prime}\right\rVert}_{\infty}<\kappa$,
$c_{1},g_{1},c_{2},g_{2}\in C_{b}^{w+2}(\mathbb{R})$,
$c_{2},g_{2}\in\mathcal{S}_{w+2}$, $\tau_{1}^{2}+\tau_{2}^{2}>0$, and $b$ is
Lipschitz continuous, $O(|y|^{1/2})$, and satisfies the centering condition
(21). Then Assumptions A1)-A13) hold. Furthermore, if this holds with $w$
replaced by $r$, then Assumption A’13) holds.

###### Proof.

Note that here $\Phi$ does not depend on $x$ or $\mu$, there is no need for
Lemma 6.3, which adds to the simplification of things (we don’t need to check
Assumption A7). In particular, there is no need for the extremely restrictive
assumptions needed to apply Lemma C.6 since, as we will see, an application of
Proposition A.2 from [GS] is sufficient to handle Assumption A11), and the
rest of the Poisson Equations are 1-dimensional.

Assumptions A1),A2), A3), and A5) are immediate.

For A4), we have for $F(x,\mu)=c(x,\mu)$ or $g(x,\mu)$
$F_{x}(x,\mu)=F_{1}^{\prime}(x)+\langle\mu,F^{\prime}(x-\cdot)\rangle$ and
$\partial_{\mu}F(x,\mu)[z]=-F_{2}^{\prime}(x-z)$ are bounded, so for all
coefficients joint Lipschitz continuity in $(x,y,\mathbb{W}_{2})$ holds (using
again Example 1 in Section 5.2.2 and Remark 5.27 in [CD]), and the result
holds in the same way as in Proposition C.8.

For A6), we just need $\Phi(y)$ grows at most linearly in $y$ and
$\Phi^{\prime}(y)$ is bounded. From Lemma C.1, we have in fact $\Phi$ is
$O(|y|^{1/2})$ and $\Phi^{\prime}$ is $O(|y|^{-1/2})$.

For A8), we have
$\gamma(x,y,\mu)=[g_{1}(x)+\langle\mu,g_{2}(x-\cdot)\rangle]\Phi^{\prime}(y)+c_{1}(x)+\langle\mu,c_{2}(x-\cdot)\rangle$
and
$D(x,y,\mu)=D(y)=b(y)\Phi(y)+\sigma\tau_{1}\Phi^{\prime}(y)+\frac{1}{2}\sigma^{2}$.
Then for the case $F=\gamma$,
$\Xi(x,y,\mu)=\tilde{\Xi}(y)[g_{1}(x)+\langle\mu,g_{2}(x-\cdot)\rangle]$ where

$\displaystyle\mathcal{L}\tilde{\Xi}(y)=\Phi^{\prime}(y)-\int_{\mathbb{R}}\Phi^{\prime}(y)\pi(dy).$

$\Phi^{\prime}$ is $O(|y|^{-1/2})$, so by Lemma C.1, $\tilde{\Xi}\in
C^{2}_{b}(\mathbb{R})$. Using $g_{1}$ and $g_{2}$ have two bounded
derivatives, it is plain to see the result holds. A similar proof shows the
result holds with
$F=\sigma\psi_{1}(t,x,y)+[\tau_{1}\psi_{1}(t,x,y)+\tau_{2}\psi_{2}(t,x,y)]\Phi^{\prime}(y),\psi_{1},\psi_{2}\in
C^{\infty}_{c}([0,T]\times\mathbb{R}\times\mathbb{R})$. Since $\Phi$ and $b$
are $O(|y|^{1/2})$, $D$ is $O(|y|)$, so by Lemma C.1 $\Xi(y)$ corresponding to
$F(y)=D(y)$ is $O(|y|)$, with $\Xi^{\prime}$ bounded.

For A9), we use the Lipschitz and boundedness properties of $\Phi^{\prime}(y)$
from Lemma C.1. The result then follows by the previously noted Lipschitz
properties of $c$ and $g$, and hence $\gamma$.

For A10), $\bar{D}$ is constant and $\bar{\gamma}(x,\mu)=\alpha
g(x,\mu)+c(x,\mu)$, for $\alpha=\int_{\mathbb{R}}\Phi^{\prime}(y)\pi(dy)$. The
result thus follows from $g_{1},c_{1}\in C^{5}_{b}(\mathbb{R})$ and
$g_{2},c_{2}\in C^{6}_{b}(\mathbb{R})$.

For A11), we have $\tilde{\chi}(x,y,\mu)=\tilde{\chi}(y)$ grows linearly in
$y$ and $\tilde{\chi}^{\prime}(y)$ is $O(|y|^{3})$ via Proposition A.2 of
[GS].

For A12), none of the listed functions depend on $\mu$ other than $\gamma$,
and $\frac{\delta}{\delta
m}\gamma(x,y,\mu)[z]=g_{2}(x-z)\Phi^{\prime}(y)+c_{2}(x-z)$ is bounded.

For A13), $\bar{D}$ is constant and $\bar{\gamma}(x,\mu)=\alpha
g(x,\mu)+c(x,\mu)$, for $\alpha=\int_{\mathbb{R}}\Phi^{\prime}(y)\pi(dy)$, so
the result follows from $g_{1},g_{2},c_{1},c_{2}\in C^{w+2}_{b}(\mathbb{R})$,
and $g_{2},c_{2}\in\mathcal{S}_{w+2}.$ The proof for extending to A’13)
follows in the same way, replacing $w$ by $r$.

∎

######  Example C.11.

(The case without full-coupling) Consider the case where

$\displaystyle b(x,y,\mu)$ $\displaystyle\equiv
0,\quad\sigma(x,y,\mu)=\sigma(x,\mu).$

In this setting, it is known that when also $g\equiv 0$ and $\tau_{1}\equiv
0$, under sufficient conditions on $c,\sigma,f$ and $\tau_{2}$, we can expect
not only convergence in distribution of
$\bar{X}^{\epsilon}\overset{d}{=}\bar{X}^{i,\epsilon}$ from Equation (57) to
$X$ from Equation (25), but also convergence in $L^{2}$. It is easily seen
that this also holds when $g,\tau_{1}\neq 0$ if they are sufficiently regular.

Note that in the limiting coefficients from Equation (24), we have $\Phi\equiv
0$, so $\bar{\gamma}(x,\mu)=\bar{c}(x,\mu)$ and
$\bar{D}(x,\mu)=\frac{1}{2}\sigma^{2}(x,\mu)$. In this setting, we can see
immediately that there is no need for Assumptions A6), A7), and A11). A8) need
only hold with $F=c$ and $F=\psi\in
C^{\infty}_{c}([0,T]\times\mathbb{R}\times\mathbb{R})$. We will see that,
since we can gain the aforementioned $L^{2}$ averaging, there is no need for
Theorem 7.2, and hence for Assumption A10).

Sufficient conditions for Theorem 3.2 to hold in this case are: A1)\- A3),
A5), A9), A12), and

  1. (1)

$c,a,f\in\mathcal{M}_{p}^{\tilde{\bm{\zeta}}}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$
with $q_{f}(n,l,\bm{\beta})\leq 1$, $q_{a}(n,l,\bm{\beta})\leq 0$,
$q_{c}(n,l,\bm{\beta})\leq 2$, $\forall(n,l,\bm{\beta})\in\tilde{\bm{\zeta}}$
and $q_{c}(n,l,\bm{\beta})\leq 1$,
$\forall(n,l,\bm{\beta})\in\tilde{\bm{\zeta}}_{1}$.

  2. (2)

$\sigma^{2}\in\mathcal{M}_{b}^{\bm{\zeta}_{x,r+2}}(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))\cap\mathcal{M}_{\bm{\delta},b}^{\bar{\bm{\zeta}}_{r+2}}(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$
and
$\sup_{x\in\mathbb{R},\mu\in\mathcal{P}(R)}\mathinner{\\!\left\lVert\frac{\delta}{\delta
m}\sigma^{2}(x,\mu)[\cdot]\right\rVert}_{r+2}<\infty.$

  3. (3)

$f,a,c\in\mathcal{M}_{p}^{\bm{\zeta}_{x,r+2}}$ and

$\displaystyle\mathinner{\\!\left\lVert\frac{\delta}{\delta
m}f(x,y,\mu)[\cdot]\right\rVert}_{r+2},\mathinner{\\!\left\lVert\frac{\delta}{\delta
m}a(x,y,\mu)[\cdot]\right\rVert}_{r+2},\mathinner{\\!\left\lVert\frac{\delta}{\delta
m}c(x,y,\mu)[\cdot]\right\rVert}_{r+2}\leq C(1+|y|^{k}),$

uniformly in $x\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})$ for some
$k\in\mathbb{N}.$

In the above the referenced collections of multi-indices are from Equation
(27).

###### Proof.

A8) holds with $F=c$ and $\psi$ via an application of Lemmas C.4 and C.5,
using (1).

Further, Lemma C.5 gives $\bar{c}(x,\mu)$ is Lipschitz continuous, so A4)
holds in the same way as Examples C.9 and C.10 via the Lipschitz properties
imposed on $c,\sigma,f,\tau_{2}$ from assumptions A2) and A9).

Since A8) holds with $F=c$, one can see that Proposition 4.5 of
[BezemekSpiliopoulosAveraging2022] holds with $F=c$ and $\psi=1$ (noting that
under these assumptions the norm may be moved inside the expectation with
little change to the proof method). Then:

$\displaystyle\mathbb{E}[|\bar{X}^{\epsilon}_{t}-X_{t}|^{2}]$
$\displaystyle\leq
C\biggl{\\{}\mathbb{E}[\biggl{|}\int_{0}^{t}c(\bar{X}^{\epsilon}_{s},Y^{\epsilon}_{s},\mathcal{L}(\bar{X}^{\epsilon}_{s}))-\bar{c}(X_{s},\mathcal{L}(X_{s}))ds\biggr{|}^{2}]+\mathbb{E}[\int_{0}^{t}|\sigma(\bar{X}^{\epsilon}_{s},\mathcal{L}(\bar{X}^{\epsilon}_{s}))-\sigma(X_{s},\mathcal{L}(X_{s}))|^{2}ds]\biggr{\\}}$
$\displaystyle\leq
C\biggl{\\{}\mathbb{E}[\biggl{|}\int_{0}^{t}c(\bar{X}^{\epsilon}_{s},Y^{\epsilon}_{s},\mathcal{L}(\bar{X}^{\epsilon}_{s}))-\bar{c}(\bar{X}^{\epsilon}_{s},\mathcal{L}(\bar{X}^{\epsilon}_{s}))ds\biggr{|}^{2}]+\mathbb{E}[\int_{0}^{t}|\bar{c}(\bar{X}^{\epsilon}_{s},\mathcal{L}(\bar{X}^{\epsilon}_{s}))-\bar{c}(X_{s},\mathcal{L}(X_{s}))|^{2}ds]$
$\displaystyle+\mathbb{E}[\int_{0}^{t}|\sigma(\bar{X}^{\epsilon}_{s},\mathcal{L}(\bar{X}^{\epsilon}_{s}))-\sigma(X_{s},\mathcal{L}(X_{s}))|^{2}ds]\biggr{\\}}$
$\displaystyle\leq
C\biggl{\\{}\mathbb{E}[\biggl{|}\int_{0}^{t}c(\bar{X}^{\epsilon}_{s},Y^{\epsilon}_{s},\mathcal{L}(\bar{X}^{\epsilon}_{s}))-\bar{c}(\bar{X}^{\epsilon}_{s},\mathcal{L}(\bar{X}^{\epsilon}_{s}))ds\biggr{|}^{2}]+\mathbb{E}[\int_{0}^{t}|\bar{c}(\bar{X}^{\epsilon}_{s},\mathcal{L}(\bar{X}^{\epsilon}_{s}))-\bar{c}(X_{s},\mathcal{L}(X_{s}))|^{2}ds]$
$\displaystyle+\mathbb{E}[\int_{0}^{t}|\sigma(\bar{X}^{\epsilon}_{s},\mathcal{L}(\bar{X}^{\epsilon}_{s}))-\sigma(X_{s},\mathcal{L}(X_{s}))|^{2}ds]\biggr{\\}}$
$\displaystyle\leq
C\epsilon^{2}(1+t^{2}+t)+C\int_{0}^{t}\mathbb{E}[|\bar{X}^{\epsilon}_{s}-X_{s}|^{2}]+\mathbb{W}_{2}(\mathcal{L}(\bar{X}^{\epsilon}_{s}),\mathcal{L}(X_{s}))ds$

where in the last step we used Proposition 4.5 of
[BezemekSpiliopoulosAveraging2022] with $F=c$ and $\psi=1$, the assumed
Lipschitz continuity of $\sigma$, and the inherited Lipschitz continuity of
$\bar{c}$ via Lemma C.5. Bounding the 2-Wasserstein distance between the Laws
by the difference in squared expectation of the processes, we get by
Grönwall’s inequality:

$\displaystyle\sup_{t\in[0,T]}\mathbb{E}[|\bar{X}^{\epsilon}_{t}-X_{t}|^{2}]$
$\displaystyle\leq C(T)\epsilon^{2}.$

Thus, in the proof of Lemma 7.6, we can circumnavigate using Theorem 7.2 in
the last line, and instead use

$\displaystyle
a^{2}(N)N\biggl{|}\mathbb{E}[\phi(\bar{X}^{\epsilon}_{t}]-\mathbb{E}[\phi(X_{t})]\biggr{|}^{2}+4a^{2}(N)\mathinner{\\!\left\lVert\phi\right\rVert}^{2}_{\infty}$
$\displaystyle\leq
a^{2}(N)NC(T)\mathbb{E}[|\bar{X}^{\epsilon}_{t}-X_{t}|^{2}]\mathinner{\\!\left\lVert\phi^{\prime}\right\rVert}^{2}_{\infty}+2a(N)\mathinner{\\!\left\lVert\phi\right\rVert}_{\infty}$
$\displaystyle\leq
a^{2}(N)N\epsilon^{2}C(T)\mathinner{\\!\left\lVert\phi^{\prime}\right\rVert}^{2}_{\infty}+2a(N)\mathinner{\\!\left\lVert\phi\right\rVert}_{\infty}.$

This shows we can circumnavigate the need for Assumption A10), and also shows
that the bound from Lemma 7.6 can be improved to $C(T)|\phi|_{1}^{2}$.

Lastly, Assumption A’13) holds immediately for $\bar{D}=\frac{1}{2}\sigma^{2}$
using (2), and can be seen to hold for $\bar{\gamma}=\bar{c}$ using (3) and
the same proof as in Proposition C.8. ∎

######  Remark C.12.

One can in fact see in the setting of Example C.11 that, as noted, the bound
in Lemma 7.6 can be improved to $C(T)|\phi|_{1}^{2}$, and further, that since
$R_{5}^{i}$ in Lemma 7.7 is zero, the bound on $R^{N}_{t}(\phi)$ in the same
Lemma can be improved to $\bar{R}(N,T)|\phi|_{3}$. Moreover, via the bound
above, we can see via triangle inequality and Lemma 7.5 that
$\sup_{s\in[0,T]}\mathbb{E}\biggl{[}\frac{1}{N}\sum_{i=1}^{N}\biggl{|}\tilde{X}^{i,\epsilon,N}_{s}-X^{i}_{s}\biggr{|}^{2}\biggr{]}\leq
C(T)[\epsilon^{2}+\frac{1}{N}+\frac{1}{Na^{2}(N)}],$ where
$\\{X^{i}\\}_{i=1}^{N}$ are IID copies of the limiting McKean-Vlasov Equation
(25) driven by the same Brownian motions as the $\tilde{X}^{i,\epsilon,N}$’s.
This allows for the proof of the Laplace Principle Upper Bound in Proposition
10.1 to go through in the same way as in Subsection 4.4 of [BW], and
eliminates the need for the approximation argument therein. Thus, in fact, the
rate function can be posed on $C([0,T];\mathcal{S}_{-\rho})$ and taken to be
infinite outside of $C([0,T];\mathcal{S}_{-v})$ as in Corollary 3.4. This also
allows us to see that (2) and (3) in Example C.11 can be relaxed by replacing
$r$ with $\rho$.

##  Appendix D On Differentiation of Functions on Spaces of Measures

We will need the following two definitions from [CD]:

######  Definition D.1.

Given a function $u\mathrel{\mathop{\mathchar
58\relax}}\mathcal{P}_{2}(\mathbb{R}^{d})\rightarrow\mathbb{R}$, we may define
a lifting of $u$ to $\tilde{u}\mathrel{\mathop{\mathchar
58\relax}}L^{2}(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}};\mathbb{R}^{d})\rightarrow\mathbb{R}$
via $\tilde{u}(X)=u(\mathcal{L}(X))$ for $X\in
L^{2}(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}};\mathbb{R}^{d})$.
We assume $\tilde{\Omega}$ is a Polish space, $\tilde{\mathcal{F}}$ its Borel
$\sigma$-field, and $\tilde{\mathbb{P}}$ is an atomless probability measure
(since $\tilde{\Omega}$ is Polish, this is equivalent to every singleton
having zero measure).

Here, denoting by
$\mu(|\cdot|^{r})\coloneqq\int_{\mathbb{R}^{d}}|x|^{r}\mu(dx)$ for $r>0$,

$\displaystyle\mathcal{P}_{2}(\mathbb{R}^{d})\coloneqq\\{\mu\in\mathcal{P}(\mathbb{R}^{d})\mathrel{\mathop{\mathchar
58\relax}}\mu(|\cdot|^{2})=\int_{\mathbb{R}^{d}}|x|^{2}\mu(dx)<\infty\\}.$

$\mathcal{P}_{2}(\mathbb{R}^{d})$ is a Polish space under the
$L^{2}$-Wasserstein distance

$\displaystyle\mathbb{W}_{2}(\mu_{1},\mu_{2})\coloneqq\inf_{\pi\in\mathcal{C}_{\mu_{1},\mu_{2}}}\biggl{[}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|x-y|^{2}\pi(dx,dy)\biggr{]}^{1/2},$

where $\mathcal{C}_{\mu_{1},\mu_{2}}$ denotes the set of all couplings of
$\mu_{1},\mu_{2}$.

We say $u$ is L-differentiable or Lions-differentiable at
$\mu_{0}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ if there exists a random variable
$X_{0}$ on some $(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}})$
satisfying the above assumptions, $\mathcal{L}(X_{0})=\mu_{0}$ and $\tilde{u}$
is Fréchet differentiable at $X_{0}$.

The Fréchet derivative of $\tilde{u}$ can be viewed as an element of
$L^{2}(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}};\mathbb{R}^{d})$
by identifying
$L^{2}(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}};\mathbb{R}^{d})$
and its dual. From this, one can find that if $u$ is L-differentiable at
$\mu_{0}\in\mathcal{P}_{2}(\mathbb{R}^{d})$, there is a deterministic
measurable function $\xi\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ such that
$D\tilde{u}(X_{0})=\xi(X_{0})$, and that $\xi$ is uniquely defined
$\mu_{0}$-almost everywhere on $\mathbb{R}^{d}$. We denote this equivalence
class of $\xi\in L^{2}(\mathbb{R}^{d},\mu_{0};\mathbb{R}^{d})$ by
$\partial_{\mu}u(\mu_{0})$ and call
$\partial_{\mu}u(\mu_{0})[\cdot]\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ the Lions derivative of $u$
at $\mu_{0}$. Note that this definition is independent of the choice of
$X_{0}$ and $(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}})$. See
[CD] Section 5.2.

To avoid confusion when $u$ depends on more variables than just $\mu$, if
$\partial_{\mu}u(\mu_{0})$ is differentiable at $z_{0}\in\mathbb{R}^{d}$, we
denote its derivative at $z_{0}$ by
$\partial_{z}\partial_{\mu}u(\mu_{0})[z_{0}]$.

######  Definition D.2.

([CD] Definition 5.83) We say $u\mathrel{\mathop{\mathchar
58\relax}}\mathcal{P}_{2}(\mathbb{R})\rightarrow\mathbb{R}$ is Fully
$\mathbf{C^{2}}$ if the following conditions are satisfied:

  1. (1)

$u$ is $C^{1}$ in the sense of L-differentiation, and its first derivative has
a jointly continuous version
$\mathcal{P}_{2}(\mathbb{R})\times\mathbb{R}\ni(\mu,z)\mapsto\partial_{\mu}u(\mu)[z]\in\mathbb{R}$.

  2. (2)

For each fixed $\mu\in\mathcal{P}_{2}(\mathbb{R})$, the version of
$\mathbb{R}\ni z\mapsto\partial_{\mu}u(\mu)[z]\in\mathbb{R}$ from the first
condition is differentiable on $\mathbb{R}$ in the classical sense and its
derivative is given by a jointly continuous function
$\mathcal{P}_{2}(\mathbb{R})\times\mathbb{R}\ni(\mu,z)\mapsto\partial_{z}\partial_{\mu}u(\mu)[z]\in\mathbb{R}$.

  3. (3)

For each fixed $z\in\mathbb{R}$, the version of
$\mathcal{P}_{2}(\mathbb{R})\ni\mu\mapsto\partial_{\mu}u(\mu)[z]\in\mathbb{R}$
in the first condition is continuously L-differentiable component-by-
component, with a derivative given by a function
$\mathcal{P}_{2}(\mathbb{R})\times\mathbb{R}\times\mathbb{R}\ni(\mu,z,\bar{z})\mapsto\partial^{2}_{\mu}u(\mu)[z][\bar{z}]\in\mathbb{R}$
such that for any $\mu\in\mathcal{P}_{2}(\mathbb{R})$ and $X\in
L^{2}(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}};\mathbb{R})$ with
$\mathcal{L}(X)=\mu$, $\partial^{2}u(\mu)[z][X]$ gives the Fréchet derivative
at $X$ of
$L^{2}(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}};\mathbb{R})\ni
X^{\prime}\mapsto\partial_{\mu}u(\mathcal{L}(X^{\prime}))[z]$ for every
$z\in\mathbb{R}$. Denoting $\partial^{2}_{\mu}u(\mu)[z][\bar{z}]$ by
$\partial^{2}_{\mu}u(\mu)[z,\bar{z}]$, the map
$\mathcal{P}_{2}(\mathbb{R})\times\mathbb{R}\times\mathbb{R}\ni(\mu,z,\bar{z})\mapsto\partial^{2}_{\mu}u(\mu)[z,\bar{z}]$
is also assumed to be continuous in the product topology.

######  Remark D.3.

In this paper we will in fact also look at functions
$u\mathrel{\mathop{\mathchar
58\relax}}\mathcal{P}_{2}(\mathbb{R})\rightarrow\mathbb{R}$ which are required
to have $3$ Lions Derivatives. We will assume such functions are Fully
$\mathbf{C^{2}}$, and satisfy:

  1. (4)

For each each fixed $\mu\in\mathcal{P}_{2}(\mathbb{R})$ the version of
$\mathbb{R}\times\mathbb{R}\ni(z_{1},z_{2})\mapsto\partial^{2}_{\mu}u(\mu)[z_{1},z_{2}]\in\mathbb{R}$
in Definition D.2 (3) is differentiable on $\mathbb{R}^{2}$ in the classical
sense and its derivative is given by a jointly continuous function
$\mathcal{P}_{2}(\mathbb{R})\times\mathbb{R}\times\mathbb{R}\ni(\mu,z_{1},z_{2})\mapsto\nabla_{z}\partial^{2}_{\mu}u(\mu)[z_{1},z_{2}]=(\partial_{z_{1}}\partial^{2}_{\mu}u(\mu)[z_{1},z_{2}],\partial_{z_{2}}\partial^{2}_{\mu}u(\mu)[z_{1},z_{2}])\in\mathbb{R}^{2}$.

  2. (5)

For each fixed $(z_{1},z_{2})\in\mathbb{R}^{2}$, the version of
$\mathcal{P}_{2}(\mathbb{R})\ni\mu\mapsto\partial^{2}_{\mu}u(\mu)[z_{1},z_{2}]\in\mathbb{R}$
in Definition D.2 (3) is continuously L-differentiable component-by-component,
with a derivative given by a function
$\mathcal{P}_{2}(\mathbb{R})\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\ni(\mu,z_{1},z_{2},z_{3})\mapsto\partial^{3}_{\mu}u(\mu)[z_{1},z_{2}][z_{3}]\in\mathbb{R}$
such that for any $\mu\in\mathcal{P}_{2}(\mathbb{R})$ and $X\in
L^{2}(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}};\mathbb{R})$ with
$\mathcal{L}(X)=\mu$, $\partial^{3}u(\mu)[z_{1},z_{2}][X]$ gives the Fréchet
derivative at $X$ of
$L^{2}(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}};\mathbb{R})\ni
X^{\prime}\mapsto\partial^{2}_{\mu}u(\mathcal{L}(X^{\prime}))[z_{1},z_{2}]$
for every $(z_{1},z_{2})\in\mathbb{R}^{2}$. Denoting
$\partial^{3}_{\mu}u(\mu)[z_{1},z_{2}][z_{3}]$ by
$\partial^{2}_{\mu}u(\mu)[z_{1},z_{2},z_{3}]$, the map
$\mathcal{P}_{2}(\mathbb{R})\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\ni(\mu,z_{1},z_{2},z_{3})\mapsto\partial^{3}_{\mu}u(\mu)[z_{1},z_{2},z_{3}]$
is also assumed to be continuous in the product topology.

Though we don’t require higher than 3 Lions derivatives in this paper, when we
state general results for higher Lions derivatives in terms of the spaces from
Definition 2.4, we assume the analogous higher continuity.

We will also make use of another notion of differentiation of functions of
probability measures: the linear functional derivative.

######  Definition D.4.

([CD] Definition 5.43) Let $p\mathrel{\mathop{\mathchar
58\relax}}\mathcal{P}_{2}(\mathbb{R})\rightarrow\mathbb{R}$. We say $p$ has
Linear Functional Derivative $\frac{\delta}{\delta m}p$ if there exists a
function
$(z,\mu)\ni\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R})\mapsto\frac{\delta}{\delta
m}p(\mu)[z]\in\mathbb{R}$ continuous in the product topology on
$\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R})$ such that for any bounded subset
$\mathcal{K}\subset\mathcal{P}_{2}(\mathbb{R})$, the function $\mathbb{R}\ni
z\mapsto\frac{\delta}{\delta m}p(\mu)[z]$ is of at most quadratic growth
uniformly in $\mu$ for $\mu\in\mathcal{K}$, and for all
$\nu_{1},\nu_{2}\in\mathcal{P}_{2}(\mathbb{R}^{d})\mathrel{\mathop{\mathchar
58\relax}}$

$\displaystyle
p(\nu_{2})-p(\nu_{1})=\int_{0}^{1}\int_{\mathbb{R}}\frac{\delta}{\delta
m}p((1-r)\nu_{1}+r\nu_{2})[z](\nu_{2}(dz)-\nu_{1}(dz))dr.$

Note in particular that this implies that $p$ is continuous on
$\mathcal{P}_{2}(\mathbb{R})$.

The second linear functional derivative is said to exist if the linear
functional derivative of $\frac{\delta}{\delta m}p(\mu)[z_{1}]$ as defined
above exists for each $z_{1}\in\mathbb{R}$. For any bounded subset
$\mathcal{K}\subset\mathcal{P}_{2}(\mathbb{R})$, the function
$(z_{1},z_{2})\ni\mathbb{R}\times\mathbb{R}\mapsto\frac{\delta}{\delta
m}\biggl{(}\frac{\delta}{\delta
m}p(\mu)[z_{1}]\biggr{)}[z_{2}]\coloneqq\frac{\delta^{2}}{\delta
m^{2}}p(\mu)[z_{1},z_{2}]\in\mathbb{R}$, is of at most quadratic growth
uniformly in $\mu$ for $\mu\in\mathcal{K}$,
$(z_{1},z_{2},\mu)\ni\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R})\mapsto\frac{\delta^{2}}{\delta
m^{2}}p(\mu)[z_{1},z_{2}]\in\mathbb{R}$ and is assumed to be continuous in the
product topology on
$\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R})$.

######  Remark D.5.

See Section 5.4.1 of [CD] for well-posedness of the above notion of
differentiability and relation to Lions derivative. In particular, under
sufficient regularity on $u\mathrel{\mathop{\mathchar
58\relax}}\mathcal{P}_{2}(\mathbb{R})\rightarrow\mathbb{R}$,
$\partial_{\mu}u(\mu)[z]=\partial_{z}\frac{\delta}{\delta m}u(\mu)[z]$. For a
formal understanding of the linear functional derivative as a Fréchet
Derivative, see p.21 of [CDLL]. Lastly, it is important to note that the
linear functional derivative is only defined up to a constant by definition.
This is usually not of importance, at it normally arises when studying
fluctuations of measures. In particular, applying $\tilde{Z}^{N}_{t}$ as
defined in (3) to a constant function, we of course get $0$ for any
$N\in\mathbb{N}$ and $t\in[0,T]$, so shifting the linear functional derivative
by a constant in Equation (32) does not change the representation of the
limiting process. A common means of fixing this constant for concreteness is
to require that $\langle\mu,\frac{\delta}{\delta
m}u(\mu)[\cdot]\rangle=0,\forall\mu\in\mathcal{P}_{2}(\mathbb{R})$ (see p.31
of [CDLL] or Section 2.2 of [DLR]. However, due to our choice of topology for
the fluctuations process, correcting the constant for the linear functional
derivative may break assumptions A13) and A’13). We thus interpret these
assumptions to mean that there is a choice of constant when defining each of
the linear functional derivatives of the functions in question which makes
them satisfy the desired properties.

We recall a useful connection between the Lions derivative as defined in D.1
and the empirical measure.

######  Proposition D.6.

For $g\mathrel{\mathop{\mathchar
58\relax}}\mathcal{P}_{2}(\mathbb{R}^{d})\rightarrow\mathbb{R}^{d}$ which is
Fully $C^{2}$ in the sense of definition D.2, we can define the empirical
projection of $g$, as $g^{N}\mathrel{\mathop{\mathchar
58\relax}}(\mathbb{R}^{d})^{N}\rightarrow\mathbb{R}^{d}$ given by

$\displaystyle g^{N}(\beta_{1},...,\beta_{N})\coloneqq
g(\frac{1}{N}\sum_{i=1}^{N}\delta_{\beta_{i}}).$

Then $g^{N}$ is twice differentiable on $(\mathbb{R}^{d})^{N}$, and for each
$\beta_{1},..,\beta_{N}\in\mathbb{R}^{d}$, $(i,j)\in\\{1,...,N\\}^{2}$,
$l\in\\{1,...,d\\}$

(78)
$\displaystyle\nabla_{\beta_{i}}g^{N}_{l}(\beta_{1},...,\beta_{N})=\frac{1}{N}\partial_{\mu}g_{l}(\frac{1}{N}\sum_{i=1}^{N}\delta_{\beta_{i}})[\beta_{i}]$

and

(79)
$\displaystyle\nabla_{\beta_{i}}\nabla_{\beta_{j}}g^{N}_{l}(\beta_{1},...,\beta_{N})=\frac{1}{N}\partial_{z}\partial_{\mu}g_{l}(\frac{1}{N}\sum_{i=1}^{N}\delta_{\beta_{i}})[\beta_{i}]\mathbbm{1}_{i=j}+\frac{1}{N^{2}}\partial^{2}_{\mu}g_{l}(\frac{1}{N}\sum_{i=1}^{N}\delta_{\beta_{i}})[\beta_{i},\beta_{j}].$

###### Proof.

This follows from Propositions 5.35 and 5.91 of [CD].

∎

Finally, we provide a Lemma which allows us to couple the interacting
particles (55) to the IID McKean-Vlasov Equations (57) knowing only
information about the growth of the linear functional derivatives of the
coefficients.

######  Lemma D.7.

Suppose $p\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R})\rightarrow\mathbb{R}$
satisfies

$\displaystyle\sup_{x,z\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}|\frac{\delta}{\delta
m}p(x,y,\mu)[z]|+\sup_{x,z,\bar{z}\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}|\frac{\delta^{2}}{\delta
m^{2}}p(x,y,\mu)[z,\bar{z}]|\leq C(1+|y|^{k})$

for some $C>0,k\in\mathbb{N}$ independent of $y\in\mathbb{R}$ and that
$p(x,y,\cdot)\mathrel{\mathop{\mathchar
58\relax}}\mathcal{P}_{2}(\mathbb{R})\rightarrow\mathbb{R}$ is Lipschitz
continuous in $\mathbb{W}_{2}$ for all $x,y\in\mathbb{R}$. Assume A1)-A7) and
A9). Then for $(\bar{X}^{i,\epsilon},\bar{Y}^{i,\epsilon})$ as in Equation
(57) and $\bar{\mu}^{\epsilon,N}_{t}$ as in Equation (58), we have there
exists $C>0$ independent of $N$ such that for all $t\in[0,T]$:

$\displaystyle\mathbb{E}\biggl{[}\biggl{|}p(\bar{X}^{i,\epsilon}_{t},\bar{Y}^{i,\epsilon}_{t},\bar{\mu}^{\epsilon,N}_{t})-p(\bar{X}^{i,\epsilon}_{t},\bar{Y}^{i,\epsilon}_{t},\mathcal{L}(\bar{X}^{\epsilon}_{t}))\biggr{|}^{2}\biggr{]}\leq\frac{C}{N-1}.$

Here $\bar{X}^{\epsilon}\overset{d}{=}\bar{X}^{i,\epsilon},\forall
i,N\in\mathbb{N}$.

###### Proof.

This follows using the same conditional expectation argument as on p.26 in
[DLR] and then following the proof of Lemma 5.10 in the same paper, but where
we only require second order expansions rather than 4th. Since the argument
and assumptions are slightly different, we present the proof here for
completeness.

We first write

$\displaystyle\mathbb{E}\biggl{[}\biggl{|}p(\bar{X}^{i,\epsilon}_{t},\bar{Y}^{i,\epsilon}_{t},\bar{\mu}^{\epsilon,N}_{t})-p(\bar{X}^{i,\epsilon}_{t},\bar{Y}^{i,\epsilon}_{t},\mathcal{L}(\bar{X}^{\epsilon}_{t}))\biggr{|}^{2}\biggr{]}\leq
2\mathbb{E}\biggl{[}\biggl{|}p(\bar{X}^{i,\epsilon}_{t},\bar{Y}^{i,\epsilon}_{t},\bar{\mu}^{\epsilon,N}_{t})-p(\bar{X}^{i,\epsilon}_{t},\bar{Y}^{i,\epsilon}_{t},\bar{\mu}^{\epsilon,N,-i}_{t})\biggr{|}^{2}\biggr{]}$
$\displaystyle\qquad+2\mathbb{E}\biggl{[}\biggl{|}p(\bar{X}^{i,\epsilon}_{t},\bar{Y}^{i,\epsilon}_{t},\bar{\mu}^{\epsilon,N,-i}_{t})-p(\bar{X}^{i,\epsilon}_{t},\bar{Y}^{i,\epsilon}_{t},\mathcal{L}(\bar{X}^{\epsilon}_{t}))\biggr{|}^{2}\biggr{]}$
$\displaystyle\quad\leq
C\mathbb{E}\biggl{[}|\mathbb{W}_{2}(\bar{\mu}^{\epsilon,N}_{t},\bar{\mu}^{\epsilon,N,-i}_{t})|^{2}\biggr{]}+2\mathbb{E}\biggl{[}\biggl{|}p(\bar{X}^{i,\epsilon}_{t},\bar{Y}^{i,\epsilon}_{t},\bar{\mu}^{\epsilon,N,-i}_{t})-p(\bar{X}^{i,\epsilon}_{t},\bar{Y}^{i,\epsilon}_{t},\mathcal{L}(\bar{X}^{\epsilon}_{t}))\biggr{|}^{2}\biggr{]}$

where here $\bar{\mu}^{\epsilon,N,-i}_{t}$ denotes
$\bar{\mu}^{\epsilon,N}_{t}$ with the $i$’th particle removed, i.e.

$\displaystyle\bar{\mu}^{\epsilon,N,-i}_{t}$
$\displaystyle\coloneqq\frac{1}{N-1}\sum_{j=1,j\neq
i}^{N}\delta_{\bar{X}^{j,\epsilon}_{t}}.$

Recall the formula

(80) $\displaystyle\mathbb{W}^{p}_{p}(\mu^{N}_{x},\mu^{N}_{y})$
$\displaystyle=\min_{\sigma}\frac{1}{N}\sum_{i=1}^{N}|x_{i}-\sigma(y)_{i}|^{p}$

for $x,y\in\mathbb{R}^{N}$,
$\mu^{N}_{x}=\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}},\mu^{N}_{x}=\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}$,
and where $\sigma\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}^{N}\rightarrow\mathbb{R}^{N}$ denotes a permutation of
the coordinates of a vector in $\mathbb{R}^{N}$ (see e.g. Equation 2.8 in
[Orrieri]). This suggests that the first term should be bounded due to the
bound $C\mathbb{E}[|\bar{X}^{i,\epsilon}_{t}|^{2}]/N\leq C/N$ from Lemma 8.1.

To see this is indeed true, we take $\mu^{N}_{x},\mu^{N,-i}_{x}$ for any
$x\in\mathbb{R}^{N}$, where here $\mu^{N,-i}_{x}$ is defined in the same way
as $\bar{\mu}^{\epsilon,N,-i}_{t}$, and see

$\displaystyle\mathbb{W}_{2}^{2}(\mu^{N}_{x},\mu^{N,-i}_{x})$
$\displaystyle\leq\int_{\mathbb{R}^{2}}|x-y|^{2}\gamma^{N}(dx,dy)$
$\displaystyle\gamma^{N}(dx,dy)$
$\displaystyle\coloneqq\frac{1}{N}\sum_{j=1,j\neq
i}^{N}\biggl{[}\delta_{x_{j}}(dx)+\frac{1}{N-1}\delta_{x_{i}}(dx)\biggr{]}\delta_{x_{j}}(dy)$

We see that indeed $\gamma^{N}$ is a coupling between
$\mu^{N}_{x},\mu^{N,-i}_{x}$ since it is clearly non-negative,

$\displaystyle\int_{\mathbb{R}^{2}}\gamma^{N}(dx,dy)$
$\displaystyle=\frac{1}{N}[N-1][1+\frac{1}{N-1}]=1,$

and for $f\in C_{b}(\mathbb{R})$,

$\displaystyle\int_{\mathbb{R}^{2}}f(x)\gamma^{N}(dx,dy)$
$\displaystyle=\frac{1}{N}\sum_{j=1,j\neq
i}^{N}\biggl{[}f(x_{j})+\frac{1}{N-1}f(x_{i})\biggr{]}=\biggl{\\{}\frac{1}{N}\sum_{j=1,j\neq
i}^{N}f(x_{j})\biggr{\\}}+\frac{1}{N}f(x_{i})=\int_{\mathbb{R}}f(y)\mu^{N}_{x}(dy)$
$\displaystyle\int_{\mathbb{R}^{2}}f(y)\gamma^{N}(dx,dy)$
$\displaystyle=\frac{1}{N}\biggl{[}1+\frac{1}{N-1}\biggr{]}\sum_{j=1,j\neq
i}^{N}f(x_{j})=\frac{1}{N-1}\sum_{j=1,j\neq
i}^{N}f(x_{j})=\int_{\mathbb{R}}f(y)\mu^{N,-i}_{x}(dy).$

So indeed

$\displaystyle\mathbb{W}_{2}^{2}(\mu^{N}_{x},\mu^{N,-i}_{x})$
$\displaystyle\leq\int_{\mathbb{R}^{2}}|x-y|^{2}\gamma^{N}(dx,dy)=\frac{1}{N}\sum_{j=1,j\neq
i}^{N}\left\\{|x_{j}-x_{j}|^{2}+\frac{1}{N-1}|x_{j}-x_{i}|^{2}\right\\}$
$\displaystyle=\frac{1}{N(N-1)}\sum_{j=1,j\neq i}^{N}|x_{j}-x_{i}|^{2}.$

Now, applying this to the first term we wish to bound,

$\displaystyle
C\mathbb{E}\biggl{[}|\mathbb{W}_{2}(\bar{\mu}^{\epsilon,N}_{t},\bar{\mu}^{\epsilon,N,-i}_{t})|^{2}\biggr{]}$
$\displaystyle\leq\frac{C}{N(N-1)}\sum_{j=1,j\neq
i}^{N}\mathbb{E}\biggl{[}\biggl{|}\bar{X}^{j,\epsilon}_{t}-\bar{X}^{i,\epsilon}_{t}\biggr{|}^{2}\biggr{]}=\frac{C}{N}\biggl{[}\mathbb{E}[|\bar{X}^{\epsilon}_{t}|^{2}]-\mathbb{E}[\bar{X}^{\epsilon}_{t}]^{2}\biggr{]}$
$\displaystyle\leq\frac{C}{N}\mathbb{E}[|\bar{X}^{\epsilon}_{t}|^{2}]\leq\frac{C}{N}$

where in the equality we use that the $\bar{X}^{i,\epsilon}_{t}$’s are IID,
and in the last bound we used Lemma 8.1.

Now we turn to the second term. We have by independence,

$\displaystyle\mathbb{E}\biggl{[}\biggl{|}p(\bar{X}^{i,\epsilon}_{t},\bar{Y}^{i,\epsilon}_{t},\bar{\mu}^{\epsilon,N,-i}_{t})-p(\bar{X}^{i,\epsilon}_{t},\bar{Y}^{i,\epsilon}_{t},\mathcal{L}(\bar{X}^{\epsilon}_{t}))\biggr{|}^{2}\biggr{]}=$
$\displaystyle\qquad=\mathbb{E}\biggl{[}\mathbb{E}\biggl{[}\biggl{|}p(x,y,\bar{\mu}^{\epsilon,N,-i}_{t})-p(x,y,\mathcal{L}(\bar{X}^{\epsilon}_{t}))\biggr{|}^{2}\biggr{]}\Bigg{|}_{(x,y)=(\bar{X}^{i,\epsilon}_{t},\bar{Y}^{i,\epsilon}_{t})}\biggr{]}.$

We will show that for $q\mathrel{\mathop{\mathchar
58\relax}}\mathcal{P}_{2}(\mathbb{R})\rightarrow\mathbb{R}$ with two bounded
Linear Functional Derivatives, that for $\\{\xi_{i}\\}_{i\in\mathbb{N}}$ IID
with $\xi_{i}\sim\mu\in\mathcal{P}_{2}(\mathbb{R})$, that letting
$\xi=(\xi_{1},...,\xi_{N})$ and $\mu^{N}_{\xi}$ be as above with $\xi$ in the
place of $x$:

(81)
$\displaystyle\mathbb{E}\biggl{[}|q(\mu^{N}_{\xi})-q(\mu)|^{2}\biggr{]}\leq\frac{C}{N}\biggl{[}\sup_{z\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}|\frac{\delta}{\delta
m}q(\mu)[z]|^{2}+\sup_{z,\bar{z}\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}|\frac{\delta^{2}}{\delta
m^{2}}q(\mu)[z,\bar{z}]|^{2}\biggr{]}.$

Applying this to the above equality, we have there is
$k\in\mathbb{\mathbb{N}}$ such that

$\displaystyle\mathbb{E}\biggl{[}\biggl{|}p(\bar{X}^{i,\epsilon}_{t},\bar{Y}^{i,\epsilon}_{t},\bar{\mu}^{\epsilon,N,-i}_{t})-p(\bar{X}^{i,\epsilon}_{t},\bar{Y}^{i,\epsilon}_{t},\mathcal{L}(\bar{X}^{\epsilon}_{t}))\biggr{|}^{2}\biggr{]}$
$\displaystyle\leq\frac{C}{N-1}\mathbb{E}\biggl{[}1+|\bar{Y}^{\epsilon}_{t}|^{2k}\biggr{]}\leq\frac{C}{N-1}$

by Lemma 7.1, and the result will have been proved.

We now prove the bound (81). By definition of the linear functional
derivative, we have:

$\displaystyle q(\mu^{N}_{\xi})-q(\mu)$
$\displaystyle=\int_{0}^{1}\int_{\mathbb{R}}\frac{\delta}{\delta
m}q(r\mu^{N}_{\xi}+(1-r)\mu)[z](\mu^{N}_{\xi}(dz)-\mu(dz))dr=S_{1}+S_{2}$

where

$\displaystyle S^{N}_{1}$ $\displaystyle=\int_{\mathbb{R}}\frac{\delta}{\delta
m}q(\mu)[z](\mu^{N}_{\xi}(dz)-\mu(dz)),$ $\displaystyle S^{N}_{2}$
$\displaystyle=\int_{0}^{1}\int_{\mathbb{R}}\biggl{[}\frac{\delta}{\delta
m}q(r\mu^{N}_{\xi}+(1-r)\mu)[z]-\frac{\delta}{\delta
m}q(\mu)[z]\biggr{]}(\mu^{N}_{\xi}(dz)-\mu(dz))dr.$

For $S_{1}$, we have by independence:

$\displaystyle\mathbb{E}\biggl{[}|S_{1}|^{2}\biggr{]}$