$\displaystyle=\mathbb{E}\biggl{[}\biggl{|}\frac{1}{N}\sum_{i=1}^{N}\frac{\delta}{\delta
m}q(\mu)[\xi_{i}]-\mathbb{E}\left[\frac{\delta}{\delta
m}q(\mu)[\xi_{1}]\right]\biggr{|}^{2}\biggr{]}$
$\displaystyle=\frac{1}{N}\biggl{(}\mathbb{E}\biggl{[}\biggl{|}\frac{\delta}{\delta
m}q(\mu)[\xi_{1}]\biggr{|}^{2}\biggr{]}-\mathbb{E}\left[\frac{\delta}{\delta
m}q(\mu)[\xi_{1}]\right]^{2}\biggr{)}$
$\displaystyle\leq\frac{1}{N}\sup_{z\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}|\frac{\delta}{\delta
m}q(\mu)[z]|^{2}.$

Now we set

$\displaystyle\phi^{i}_{r}\coloneqq\frac{\delta}{\delta
m}q(r\mu^{N}_{\xi}+(1-r)\mu)[\xi_{i}]-\frac{\delta}{\delta
m}q(\mu)[\xi_{i}]-\tilde{\mathbb{E}}\biggl{[}\frac{\delta}{\delta
m}q(r\mu^{N}_{\xi}+(1-r)\mu)[\tilde{\xi}]-\frac{\delta}{\delta
m}q(\mu)[\tilde{\xi}]\biggr{]},$

where $\tilde{\xi}$ is an independent copy of the $\xi_{i}$’s, $r\in[0,1]$,
and the expectation $\tilde{\mathbb{E}}$ is taken over the law of
$\tilde{\xi}$.

Then we have $S^{N}_{2}=\frac{1}{N}\sum_{i=1}^{N}\int_{0}^{1}\phi^{i}_{r}dr$,
and

$\displaystyle\mathbb{E}\biggl{[}\biggl{|}S^{N}_{2}\biggr{|}^{2}\biggr{]}$
$\displaystyle\leq\frac{1}{N^{2}}\int_{0}^{1}\mathbb{E}\biggl{[}\biggl{|}\sum_{i=1}^{N}\phi^{i}_{r}\biggr{|}^{2}\biggr{]}dr=\frac{1}{N^{2}}\int_{0}^{1}\sum_{i=1}^{N}\sum_{j=1}^{N}\mathbb{E}\biggl{[}\phi^{i}_{r}\phi^{j}_{r}\biggr{]}dr=S^{N}_{2,1}+S^{N}_{2,2}$

where

$\displaystyle S^{N}_{2,1}$
$\displaystyle=\frac{1}{N^{2}}\int_{0}^{1}\sum_{i=1}^{N}\mathbb{E}\biggl{[}|\phi^{i}_{r}|^{2}\biggr{]}dr,\quad\text{and}\quad
S^{N}_{2,2}=\frac{1}{N^{2}}\int_{0}^{1}\sum_{i=1}^{N}\sum_{j=1,j\neq
i}^{N}\mathbb{E}\biggl{[}\phi^{i}_{r}\phi^{j}_{r}\biggr{]}dr.$

Observing that for all $i\in\mathbb{N},r\in[0,1]$ and $\omega\in\Omega$,
$|\phi^{i}_{r}(\omega)|^{2}\leq
C\sup_{z\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}|\frac{\delta}{\delta
m}q(\mu)[z]|^{2}$, so we have

$\displaystyle
S^{N}_{2,1}\leq\frac{C}{N}\sup_{z\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}|\frac{\delta}{\delta
m}q(\mu)[z]|^{2}.$

For $S^{N}_{2,2}$, we introduce the measures
$\mu^{N,-(i_{1},i_{2})}_{\xi}\coloneqq\frac{1}{N-2}\sum_{j=1,j\neq
i_{1},i_{2}}^{N}\delta_{\xi_{j}}$ for $i_{1},i_{2}\in\\{1,...,N\\}$, and let

$\displaystyle\phi^{i,-(i_{1},i_{2})}_{r}$
$\displaystyle\coloneqq\frac{\delta}{\delta
m}q(r\mu^{N,-(i_{1},i_{2})}_{\xi}+(1-r)\mu)[\xi_{i}]-\frac{\delta}{\delta
m}q(\mu)[\xi_{i}]-\tilde{\mathbb{E}}\biggl{[}\frac{\delta}{\delta
m}q(r\mu^{N,-(i_{1},i_{2})}_{\xi}+(1-r)\mu)[\tilde{\xi}]-\frac{\delta}{\delta
m}q(\mu)[\tilde{\xi}]\biggr{]}.$

Then

$\displaystyle\phi^{i}_{r}\phi^{j}_{r}$
$\displaystyle=[\phi^{i}_{r}-\phi^{i,-(i,j)}_{r}][\phi^{j}_{r}-\phi^{j,-(i,j)}_{r}]+\phi^{j,-(i,j)}_{r}[\phi^{i}_{r}-\phi^{i,-(i,j)}_{r}]+\phi^{i,-(i,j)}_{r}[\phi^{j}_{r}-\phi^{j,-(i,j)}_{r}]+\phi^{i,-(i,j)}_{r}\phi^{j,-(i,j)}_{r},$

so

$\displaystyle S^{N}_{2,2}$
$\displaystyle=S^{N}_{2,2,1}+S^{N}_{2,2,2}+S^{N}_{2,2,3}+S^{N}_{2,2,4}$
$\displaystyle S^{N}_{2,2,1}$
$\displaystyle=\frac{1}{N^{2}}\int_{0}^{1}\sum_{i=1}^{N}\sum_{j=1,j\neq
i}^{N}\mathbb{E}\biggl{[}\phi^{i,-(i,j)}_{r}\phi^{j,-(i,j)}_{r}\biggr{]}dr$
$\displaystyle S^{N}_{2,2,2}$
$\displaystyle=\frac{1}{N^{2}}\int_{0}^{1}\sum_{i=1}^{N}\sum_{j=1,j\neq
i}^{N}\mathbb{E}\biggl{[}\phi^{j,-(i,j)}_{r}[\phi^{i}_{r}-\phi^{i,-(i,j)}_{r}]\biggr{]}dr$
$\displaystyle S^{N}_{2,2,3}$
$\displaystyle=\frac{1}{N^{2}}\int_{0}^{1}\sum_{i=1}^{N}\sum_{j=1,j\neq
i}^{N}\mathbb{E}\biggl{[}\phi^{i,-(i,j)}_{r}[\phi^{j}_{r}-\phi^{j,-(i,j)}_{r}]\biggr{]}dr$
$\displaystyle S^{N}_{2,2,4}$
$\displaystyle=\frac{1}{N^{2}}\int_{0}^{1}\sum_{i=1}^{N}\sum_{j=1,j\neq
i}^{N}\mathbb{E}\biggl{[}[\phi^{i}_{r}-\phi^{i,-(i,j)}_{r}][\phi^{j}_{r}-\phi^{j,-(i,j)}_{r}]\biggr{]}dr.$

For $S^{N}_{2,2,1}$, we have

$\displaystyle\mathbb{E}\biggl{[}\phi^{i,-(i,j)}_{r}\phi^{j,-(i,j)}_{r}\biggr{]}$
$\displaystyle=\mathbb{E}\biggl{[}\mathbb{E}\biggl{[}\phi^{i,-(i,j)}_{r}\phi^{j,-(i,j)}_{r}|\xi_{k},k\neq
i,j\biggr{]}\biggr{]}=\mathbb{E}\biggl{[}\mathbb{E}\biggl{[}\phi^{i,-(i,j),x}_{r}\phi^{j,-(i,j),x}_{r}\biggr{]}\Bigg{|}_{x=\xi}\biggr{]}$
$\displaystyle=\mathbb{E}\biggl{[}\mathbb{E}\biggl{[}\phi^{i,-(i,j),x}_{r}\biggr{]}\Bigg{|}_{x=\xi}\mathbb{E}\biggl{[}\phi^{j,-(i,j),x}_{r}\biggr{]}\Bigg{|}_{x=\xi}\biggr{]}$

where

$\displaystyle\phi^{i,-(i_{1},i_{2}),x}_{r}$
$\displaystyle\coloneqq\frac{\delta}{\delta
m}q(r\mu^{N,-(i_{1},i_{2})}_{x}+(1-r)\mu)[\xi_{i}]-\frac{\delta}{\delta
m}q(\mu)[\xi_{i}]-\tilde{\mathbb{E}}\biggl{[}\frac{\delta}{\delta
m}q(r\mu^{N,-(i_{1},i_{2})}_{x}+(1-r)\mu)[\tilde{\xi}]-\frac{\delta}{\delta
m}q(\mu)[\tilde{\xi}]\biggr{]}$

and same for $j$. Then

$\displaystyle\mathbb{E}\biggl{[}\phi^{i,-(i,j),x}_{r}\biggr{]}\Bigg{|}_{x=\xi}$
$\displaystyle=\biggl{\\{}\mathbb{E}\biggl{[}\frac{\delta}{\delta
m}q(r\mu^{N,-(i_{1},i_{2})}_{x}+(1-r)\mu)[\xi_{i}]-\frac{\delta}{\delta
m}q(\mu)[\xi_{i}]\biggr{]}$
$\displaystyle-\tilde{\mathbb{E}}\biggl{[}\frac{\delta}{\delta
m}q(r\mu^{N,-(i_{1},i_{2})}_{x}+(1-r)\mu)[\tilde{\xi}]-\frac{\delta}{\delta
m}q(\mu)[\tilde{\xi}]\biggr{]}\biggr{\\}}\biggl{|}_{(x=\xi)}=0$

since $\xi_{i}\overset{d}{=}\tilde{\xi}$, and same for
$\mathbb{E}\biggl{[}\phi^{j,-(i,j),x}_{r}\biggr{]}\Bigg{|}_{x=\xi}$. Thus in
fact, $S^{N}_{2,2,1}=0$.

To handle $S_{2,2,2}-S_{2,2,4}$, we need to see how to bound
$|\phi^{i}_{r}-\phi^{i,-(i,j)}_{r}|$. We have that

$\displaystyle\phi^{i}_{r}-\phi^{i,-(i,j)}_{r}$
$\displaystyle=\frac{\delta}{\delta
m}q(r\mu^{N}_{\xi}+(1-r)\mu)[\xi_{i}]-\frac{\delta}{\delta
m}q(r\mu^{N,-(i,j)}_{\xi}+(1-r)\mu)[\xi_{i}]$
$\displaystyle+\tilde{\mathbb{E}}\biggl{[}\frac{\delta}{\delta
m}q(r\mu^{N,-(i,j)}_{\xi}+(1-r)\mu)[\tilde{\xi}]-\frac{\delta}{\delta
m}q(r\mu^{N}_{\xi}+(1-r)\mu)[\tilde{\xi}]\biggr{]}$
$\displaystyle=r\int_{0}^{1}\int_{\mathbb{R}}\frac{\delta^{2}}{\delta
m^{2}}q(rs\mu^{N}_{\xi}+r(1-s)\mu^{N,-(i,j)}_{\xi}+(1-r)\mu)[\xi_{i},\bar{z}][\mu^{N}_{\xi}(d\bar{z})-\mu^{N,-(i,j)}_{\xi}(d\bar{z})]ds$
$\displaystyle+r\tilde{\mathbb{E}}\biggl{[}\int_{0}^{1}\int_{\mathbb{R}}\frac{\delta^{2}}{\delta
m^{2}}q(rs\mu^{N}_{\xi}+r(1-s)\mu^{N,-(i,j)}_{\xi}+(1-r)\mu)[\tilde{\xi},\bar{z}][\mu^{N}_{\xi}(d\bar{z})-\mu^{N,-(i,j)}_{\xi}(d\bar{z})]ds\biggr{]}.$

Then using

$\displaystyle\mu^{N}_{x}-\mu^{N,-(i,j)}_{x}$
$\displaystyle=\frac{1}{N}\sum_{k=1}^{N}\delta_{x_{k}}-\frac{1}{N-2}\sum_{k=1,k\neq
i,j}^{N}\delta_{x_{k}}=\frac{1}{N}\delta_{x_{i}}+\frac{1}{N}\delta_{x_{j}}-\frac{2}{N(N-2)}\sum_{k=1,k\neq
i,j}^{N}\delta_{x_{k}}$

and that $r\in[0,1]$, we get

$\displaystyle|\phi^{i}_{r}(\omega)-\phi^{i,-(i,j)}_{r}(\omega)|$
$\displaystyle\leq\frac{4}{N}\sup_{z,\bar{z}\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}|\frac{\delta^{2}}{\delta
m^{2}}q(\mu)[z,\bar{z}]|$

for all $\omega\in\Omega,r\in[0,1],i,j\in\mathbb{N}.$ This combined with the
fact that $|\phi^{k,-(i,j)}_{r}(\omega)|\leq
C\sup_{z\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}|\frac{\delta}{\delta
m}q(\mu)[z]|,k=i,j$ for any $i,j\in\mathbb{N},r\in[0,1],\omega\in\Omega$
allows us to see:

$\displaystyle S^{N}_{2,2,2}$ $\displaystyle\leq
C\frac{N(N-1)}{N^{2}}\sup_{z\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}|\frac{\delta}{\delta
m}q(\mu)[z]|\frac{1}{N}\sup_{z,\bar{z}\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}|\frac{\delta^{2}}{\delta
m^{2}}q(\mu)[z,\bar{z}]|$
$\displaystyle\leq\frac{C}{N}[\sup_{z\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}|\frac{\delta}{\delta
m}q(\mu)[z]|^{2}+\sup_{z,\bar{z}\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}|\frac{\delta^{2}}{\delta
m^{2}}q(\mu)[z,\bar{z}]|^{2}]$ $\displaystyle S^{N}_{2,2,3}$
$\displaystyle\leq
C\frac{N(N-1)}{N^{2}}\sup_{z\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}|\frac{\delta}{\delta
m}q(\mu)[z]|\frac{1}{N}\sup_{z,\bar{z}\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}|\frac{\delta^{2}}{\delta
m^{2}}q(\mu)[z,\bar{z}]|$
$\displaystyle\leq\frac{C}{N}[\sup_{z\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}|\frac{\delta}{\delta
m}q(\mu)[z]|^{2}+\sup_{z,\bar{z}\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}|\frac{\delta^{2}}{\delta
m^{2}}q(\mu)[z,\bar{z}]|^{2}]$ $\displaystyle S^{N}_{2,2,4}$
$\displaystyle\leq
C\frac{N(N-1)}{N^{2}}\frac{1}{N^{2}}\sup_{z,\bar{z}\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}|\frac{\delta^{2}}{\delta
m^{2}}q(\mu)[z,\bar{z}]|^{2}$
$\displaystyle\leq\frac{C}{N^{2}}\sup_{z,\bar{z}\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}|\frac{\delta^{2}}{\delta
m^{2}}q(\mu)[z,\bar{z}]|^{2}.$

So the bound (81) is proved. ∎

######  Remark D.8.

Note we could have polynomial growth in $x$ for the Linear Functional
Derivatives as well and the result above would still hold, so long as we have
sufficient bounded moments for $\bar{X}^{\epsilon}_{t}$. Also, the result is
independent of the fact that the particles depend on $\epsilon$, and of the
fact that the particles are one-dimensional. See Lemma 5.10 in [DLR] and
Theorem 2.11 in [CST] for similar results in the higher-dimensional setting.

## References