$\mathbb{E}(\mathcal{W}_{2}(\hat{\nu}_{T}^{B},\hat{\mu}_{T}^{B}))\leq\left(\dfrac{1}{B}\sum_{i=1}^{B}\mathbb{E}|z_{T}^{i}-y_{T}^{i}|^{2}\right)^{1/2}\leq
CB^{-1/2+\epsilon}.$ (A.10)

The assertion results from the combination of several lemmas. First, we define

$x_{t}^{i}=y_{t}^{i}-z_{t}^{i}\qquad\text{and}\qquad
p_{t}^{i}=x_{t}^{i}-\overline{x}_{t},\qquad
q_{t}^{i}=z_{t}^{i}-\overline{z}_{t}$

for convenience. We need that the higher moments of $x_{t}^{i}$ and
$p_{t}^{i}$ are bounded for all time.

######  Lemma A.1 (Compare Lemma 5.3 in [9]).

Under the same conditions as in Theorem A.3, for all $2\leq p<\infty$ and
$T>0$, there is a constant $C_{p}$ independent of $B,t$ such that

$\mathbb{E}|x_{t}^{i}|^{p}=\mathbb{E}|x^{1}_{t}|^{p}\leq
C_{p},\qquad\mathbb{E}|p_{t}^{i}|^{p}=\mathbb{E}|p^{1}_{t}|^{p}\leq C_{p}$
(A.11)

for all $1\leq i\leq B$ and $0\leq t\leq T$.

######  Lemma A.2 (Compare Lemma 5.4 in [9]).

Under the same conditions as in Theorem 5, for any $0\leq\alpha<1$ and $T>0$,
if there is a constant $\hat{C}$ independent of $B,t$ such that

$\mathbb{E}|x_{t}^{i}|^{2}\leq\hat{C}B^{-\alpha}$ (A.12)

for all $1\leq i\leq B$ and $0\leq t\leq T$, then for any $0<\epsilon<1/2$ and
$1\leq i\leq B$, there exists a constant $C$ independent of $B,t$ such that

$\mathbb{E}|p_{t}^{i}|^{2}=\mathbb{E}\left|x_{t}^{i}-\dfrac{1}{B}\sum_{k=1}^{B}x_{t}^{k}\right|^{2}\leq\tilde{C}B^{-1/2-\alpha/2+\epsilon}$
(A.13)

for all $1\leq i\leq B$ and $0\leq t\leq T$.

######  Lemma A.3 (Compare Lemma 5.5 in [9]).

Under the same conditions as lemma A.2, we have for any $0<\epsilon<1/2$ and
$T>0$ a constant $\tilde{C}$ independent of $B,t$ such that

$\mathbb{E}|x_{t}^{i}|^{2}\leq\tilde{C}B^{-1/2-\alpha/2+\epsilon}$ (A.14)

for all $1\leq i\leq B$ and $0\leq t\leq T$.

###### Proof of Theorem A.3.

By lemma A.1, (A.12) is satisfied for $\alpha_{0}=0$, yielding

$\mathbb{E}|x_{t}^{i}|^{2}\leq\tilde{C}B^{-1/2+\epsilon}$

by lemma A.3. Hence, (A.12) is satisfied for $\alpha_{1}=1/2-\epsilon$ and
recursively for $\alpha_{n+1}=1/2+\alpha_{n}/2-\epsilon$. The limit of this
sequence is $\lim_{n\rightarrow\infty}\alpha_{n}=1-2\epsilon$, yielding

$\mathbb{E}|z_{t}^{i}-y_{t}^{i}|^{2}=\dfrac{1}{B}\sum_{i=1}^{B}\mathbb{E}|z_{t}^{i}-y_{t}^{i}|^{2}\leq\tilde{C}B^{-1+2\epsilon}.$

∎

It remains to show the three lemmas A.1-A.3, for which we need some more
intermediate results.

######  Lemma A.4 (Compare Lemma 4.1 from [9]).

Let $\\{y_{t}^{i}\\}_{i=1}^{B}$ be the solution of (A.1) with initial
conditions $\\{y_{0}^{i}\\}_{i=1}^{B}$ sampled i.i.d from
$\pi_{0}\in\mathcal{C}^{2}$. If the initial $p$-th moment is finite, i.e.

$\left(\mathbb{E}|y_{0}^{i}|^{p}\right)^{1/p}<M,\quad 1\leq i\leq B$

for $p\geq 2$ and some $M>0$ independent of $B$, then the boundedness also
holds true for $\mathbb{E}|e_{t}^{i}|$, where
$e_{t}^{i}=y_{t}^{i}-\overline{y}_{t}$, namely there is a $C>0$ depending only
on $p$, so that

$\left(\mathbb{E}|e_{t}^{i}|^{p}\right)^{1/p}<2(\kappa(P))^{1/2}M\exp(Ct)$

for all $t\geq 0$ and $1\leq i\leq B$, where
$\kappa(P)=\|P\|_{2}\|P^{-1}\|_{2}$ is the condition number of $P$.

###### Proof of Lemma A.4.

Following the proof of lemma 4.1 in [9], we will show the claim for $2p$ with
$p\geq 1$ and define

$\mathbf{e}_{t}^{i}=\sqrt{P}e_{t}^{i},\quad
V_{p}(\mathbf{e})=\dfrac{1}{B}\sum_{i=1}^{B}\left\langle\mathbf{e}_{t}^{i},\mathbf{e}_{t}^{i}\right\rangle^{p}=\dfrac{1}{B}\sum_{i=1}^{B}|\mathbf{e}_{t}^{i}|^{2p},\quad
h_{p}(t)=\mathbb{E}V_{p}=\mathbb{E}|\mathbf{e}^{1}_{t}|^{2p}.$

Since $\lambda_{\textnormal{min}}(P)>0$, boundedness of
$\mathbb{E}|e_{t}^{i}|^{p}$ follows immediatley from boundedness of
$h_{p}(t)$, which we now show. First, we get

$\displaystyle(h_{p}(0))^{\frac{1}{2p}}$
$\displaystyle=\left(\mathbb{E}|\mathbf{e}^{1}_{0}|^{2p}\right)^{\frac{1}{2p}}\leq\|P\|^{1/2}_{2}\left(\mathbb{E}|e^{1}_{0}|^{2p}\right)^{\frac{1}{2p}}$
$\displaystyle\leq\|P\|^{1/2}_{2}\left(\left(\mathbb{E}|y^{1}_{0}|^{2p}\right)^{\frac{1}{2p}}+\left(\mathbb{E}|\overline{y}_{0}|^{2p}\right)^{\frac{1}{2p}}\right)$
$\displaystyle\leq 2\|P\|_{2}^{1/2}M,$

where we have used the boundedness of the initial moments. Our next
intermediate goal is to show

$h_{p}(t)\leq h_{p}(0)+C(p,D)\int_{0}^{t}h_{p}(s)ds$

with a constant $C(p,D)$ depending on $p$ and $D$. To that end, note that by
(A.1) we have

$\displaystyle\mathrm{d}\overline{y}_{t}$
$\displaystyle=-\mathrm{Cov}_{y_{t}}P(\overline{y}_{t}-y^{\ast})\mathrm{d}t+\dfrac{D+1}{B}\sum_{i=1}^{B}(y_{t}^{i}-\overline{y}_{t})+\sqrt{2\mathrm{Cov}_{y_{t}}}\mathrm{d}\overline{W}_{t}$
(A.15)

and with
$\mathrm{Cov}_{y_{t}}=\mathrm{Cov}_{e_{t}}=\sqrt{P}^{-1}\mathrm{Cov}_{\mathbf{e}_{t}}\sqrt{P}^{-1}$,
we find

$\displaystyle\mathrm{d}\mathbf{e}_{t}^{i}=\mathrm{d}\sqrt{P}(y_{t}^{i}-\overline{y}_{t})$
$\displaystyle=-\sqrt{P}\mathrm{Cov}_{e_{t}}Pe_{t}^{i}\mathrm{d}t+\dfrac{D+1}{B}(\mathbf{e}_{t}^{i}-\overline{\mathbf{e}}_{t})\mathrm{d}t+\sqrt{P}\sqrt{2\mathrm{Cov}_{e_{t}}}\mathrm{d}(W_{t}^{i}-\overline{W}_{t})$
(A.16)
$\displaystyle=-\mathrm{Cov}_{\mathbf{e}_{t}}\mathbf{e}_{t}^{i}\mathrm{d}t+\dfrac{D+1}{B}(\mathbf{e}_{t}^{i}-\overline{\mathbf{e}}_{t})\mathrm{d}t+\sqrt{P}\sqrt{2\mathrm{Cov}_{e_{t}}}\mathrm{d}(W_{t}^{i}-\overline{W}_{t}).$

Noting that

$\displaystyle V_{p}(\mathbf{e}_{t})$
$\displaystyle=\sum_{i=1}^{B}f(\mathbf{e}_{t}^{i}),$ (A.17)

with

$f(x)=\dfrac{1}{B}|x|^{2p},\quad\nabla
f(x)=\dfrac{2p}{B}|x|^{2p-2}x,\quad\mathrm{Hess}_{f}(x)=\dfrac{2p}{B}|x|^{2p-2}I_{d}+\dfrac{4p(p-1)}{B}|x|^{2p-4}xx^{\intercal},$

we apply Itô’s lemma (A.2) to $V_{p}(\mathbf{e}_{t})$ to receive

$\displaystyle\mathrm{d}V_{p}(\mathbf{e}_{t})=$
$\displaystyle-\dfrac{2p}{B}\sum_{i=1}^{B}\left\langle\mathbf{e}_{t}^{i},\mathbf{e}_{t}^{i}\right\rangle^{p-1}\left\langle\mathbf{e}_{t}^{i},\text{Cov}_{\mathbf{e}_{t}}\mathbf{e}_{t}^{i}\right\rangle\mathrm{d}t$
(A.18)
$\displaystyle+\dfrac{2p}{B}\sum_{i=1}^{B}\left\langle\mathbf{e}_{t}^{i},\mathbf{e}_{t}^{i}\right\rangle^{p-1}\left\langle\mathbf{e}_{t}^{i},\sqrt{P}\sqrt{2\text{Cov}_{e_{t}}}\mathrm{d}(W_{t}^{i}-\overline{W}_{t})\right\rangle$
$\displaystyle+\dfrac{4(B-1)p(p-1)}{B^{2}}\sum_{i=1}^{B}\left\langle\mathbf{e}_{t}^{i},\mathbf{e}_{t}^{i}\right\rangle^{p-2}\text{Tr}\left\\{(\mathbf{e}_{t}^{i}\otimes\mathbf{e}_{t}^{i})\text{Cov}_{\mathbf{e}_{t}}\right\\}\mathrm{d}t$
$\displaystyle+\dfrac{2(B-1)p}{B^{2}}\sum_{i=1}^{B}\left\langle\mathbf{e}_{t}^{i},\mathbf{e}_{t}^{i}\right\rangle^{p-1}\text{Tr}\left\\{\text{Cov}_{\mathbf{e}_{t}}\right\\}\mathrm{d}t$
$\displaystyle+\dfrac{2p}{B}\sum_{i=1}^{B}\left\langle\mathbf{e}_{t}^{i},\mathbf{e}_{t}^{i}\right\rangle^{p-1}\left\langle\mathbf{e}_{t}^{i},\dfrac{D+1}{B}\left[\mathbf{e}_{t}^{i}-\sum_{k=1}^{B}\mathbf{e}_{t}^{k}\right]\right\rangle\mathrm{d}t,$

for $p\geq 2$, with the third term vanishing in the case $p=1$. In order to
bound these terms in expectation, note that for $l\in\mathbb{N}$ with $l\leq
p$ we have by Hölder’s inequality

$\displaystyle\mathbb{E}\bigg{[}\left\langle\mathbf{e}_{t}^{j},\mathbf{e}_{t}^{j}\right\rangle^{p-l}\left\langle\mathbf{e}_{t}^{k},\mathbf{e}_{t}^{k}\right\rangle^{l}\bigg{]}$
$\displaystyle\leq\mathbb{E}\bigg{[}\left\langle\mathbf{e}_{t}^{j},\mathbf{e}_{t}^{j}\right\rangle^{p}\bigg{]}^{\frac{p-l}{p}}\mathbb{E}\bigg{[}\left\langle\mathbf{e}_{t}^{j},\mathbf{e}_{t}^{j}\right\rangle^{p}\bigg{]}^{\frac{l}{p}}$
(A.19)
$\displaystyle=\mathbb{E}\bigg{[}\left\langle\mathbf{e}_{t}^{j},\mathbf{e}_{t}^{j}\right\rangle^{p}\bigg{]}.$

Furthermore, we will make use of the inequalities

$\langle x,y\rangle\leq\dfrac{\langle x,x\rangle+\langle
y,y\rangle}{2}\quad\text{and}\quad\langle x,y\rangle^{2}\leq\langle
x,x\rangle\langle y,y\rangle\leq\dfrac{\langle x,x\rangle^{2}+\langle
y,y\rangle^{2}}{2},$ (A.20)

holding for arbitrary elements $x,y$ of a linear space with inner product
$\langle.,.\rangle$. Taking the expectation in (A.18) and noting that the
first term is always nonnegative, we find

$\displaystyle h_{p}(t)-h_{p}(0)\leq$
$\displaystyle~{}\dfrac{4(B-1)p(p-1)}{B^{3}}\int_{0}^{t}\sum_{i,k=1}^{B}\mathbb{E}\left[\left\langle\mathbf{e}_{s}^{i},\mathbf{e}_{s}^{i}\right\rangle^{p-2}\left\langle\mathbf{e}_{s}^{i},\mathbf{e}_{s}^{k}\right\rangle^{2}\right]\mathrm{d}s$
$\displaystyle+\dfrac{2(B-1)p}{B^{3}}\int_{0}^{t}\sum_{i,k=1}^{B}\mathbb{E}\left[\left\langle\mathbf{e}_{s}^{i},\mathbf{e}_{s}^{i}\right\rangle^{p-1}\left\langle\mathbf{e}_{s}^{k},\mathbf{e}_{s}^{k}\right\rangle\right]\mathrm{d}s$
$\displaystyle+\dfrac{2p(D+1)}{B^{2}}\left\\{\int_{0}^{t}\sum_{i=1}^{B}\mathbb{E}\left[\left\langle\mathbf{e}_{s}^{i},\mathbf{e}_{s}^{i}\right\rangle^{p}\right]\mathrm{d}s-\int_{0}^{t}\sum_{i,k=1}^{B}\mathbb{E}\left[\left\langle\mathbf{e}_{s}^{i},\mathbf{e}_{s}^{i}\right\rangle^{p-1}\left\langle\mathbf{e}_{s}^{i},\mathbf{e}_{s}^{k}\right\rangle\right]\mathrm{d}s\right\\}$
$\displaystyle\leq$
$\displaystyle~{}\dfrac{4(B-1)p(p-1)}{B^{3}}\int_{0}^{t}\sum_{i,k=1}^{B}\mathbb{E}\left[\left\langle\mathbf{e}_{s}^{i},\mathbf{e}_{s}^{i}\right\rangle^{p-2}\frac{\left\langle\mathbf{e}_{s}^{i},\mathbf{e}_{s}^{i}\right\rangle^{2}+\left\langle\mathbf{e}_{s}^{k},\mathbf{e}_{s}^{k}\right\rangle^{2}}{2}\right]\mathrm{d}s$
$\displaystyle+\dfrac{2(B-1)p}{B^{3}}\int_{0}^{t}\sum_{i,k=1}^{B}\mathbb{E}\left[\left\langle\mathbf{e}_{s}^{i},\mathbf{e}_{s}^{i}\right\rangle^{p-1}\left\langle\mathbf{e}_{s}^{k},\mathbf{e}_{s}^{k}\right\rangle\right]\mathrm{d}s$
$\displaystyle+\dfrac{2p(D+1)}{B^{2}}\left\\{\int_{0}^{t}\sum_{i=1}^{B}\mathbb{E}\left[\left\langle\mathbf{e}_{s}^{i},\mathbf{e}_{s}^{i}\right\rangle^{p}\right]\mathrm{d}s+\int_{0}^{t}\sum_{i,k=1}^{B}\mathbb{E}\left[\left\langle\mathbf{e}_{s}^{i},\mathbf{e}_{s}^{i}\right\rangle^{p-1}\frac{\left\langle\mathbf{e}_{s}^{i},\mathbf{e}_{s}^{i}\right\rangle+\left\langle\mathbf{e}_{s}^{k},\mathbf{e}_{s}^{k}\right\rangle}{2}\right]\mathrm{d}s\right\\}$
$\displaystyle\leq$
$\displaystyle~{}\dfrac{4(B-1)p(p-1)}{B}\int_{0}^{t}h_{p}(s)\mathrm{d}s+\dfrac{2(B-1)p}{B}\int_{0}^{t}h_{p}(s)\mathrm{d}s$
$\displaystyle+\dfrac{2p(D+1)}{B^{2}}\left\\{B\int_{0}^{t}h_{p}(s)\mathrm{d}s+B^{2}\int_{0}^{t}h_{p}(s)\mathrm{d}s\right\\}$
$\displaystyle\leq$
$\displaystyle~{}\dfrac{4(B-1)p(p-1)+2(B-1)p+2p(D+1)+2Bp(D+1)}{B}\int_{0}^{t}h_{p}(s)\mathrm{d}s$
$\displaystyle\leq$
$\displaystyle~{}(4p(p-1)+2p+2p(D+1)+2p(D+1))\int_{0}^{t}h_{p}(s)\mathrm{d}s\coloneqq
C(p,D)\int_{0}^{t}h_{p}(s)\mathrm{d}s.$

In total we get by Gronwall’s inequality

$h_{p}(t)\leq(2\|B\|_{2}^{1/2}M)^{2p}e^{C(p,D)t}$

and hence

$\displaystyle\left(\mathbb{E}|e^{j}_{t}|^{2p}\right)^{1/2p}$
$\displaystyle=\left(\mathbb{E}|e^{1}_{t}|^{2p}\right)^{1/2p}=\left(\mathbb{E}|\sqrt{P}^{-1}\mathbf{e}^{1}_{t}|^{2p}\right)^{1/2p}$
$\displaystyle\leq\|P^{-1}\|^{1/2}_{2}\left(\mathbb{E}|\mathbf{e}^{1}_{t}|^{2p}\right)^{1/2p}$
$\displaystyle\leq\|P^{-1}\|^{1/2}_{2}\left((2\|P\|_{2}^{1/2}M)^{2p}e^{C(p,D)t}\right)^{1/2p}$
$\displaystyle=2M\kappa(P)^{1/2}e^{C(p,D)t}.$

∎

######  Lemma A.5 (Proposition 4.1 from [9]).

Let the conditions of lemma A.4 be satisfied for $2p$ (i.e. the $2p$-th
initial moments are finite). Then we have

$\left(\mathbb{E}|y_{t}^{i}-\overline{y}_{t}|^{p}\right)^{1/p}\leq
Ce^{Ct}\quad\text{and}\quad(\mathbb{E}\|\mathrm{Cov}_{y_{t}}\|_{2}^{p})^{1/p}\leq
Ce^{Ct}$

as well as

$\left(\mathbb{E}|y^{i}_{t}|^{p}\right)^{1/p}\leq Ce^{Ce^{Ct}}$

with $C>0$ being independent of $B$ and $t.$

###### Proof of Lemma A.5.

The first inequality follows directly from lemma A.4 and Hölder’s inequality.
The second inequality follows from

$\displaystyle(\mathbb{E}\|\text{Cov}_{y_{t}}\|_{2}^{p})^{1/p}$
$\displaystyle\leq\dfrac{1}{B}\sum_{i=1}^{B}\mathbb{E}\left(\|(y_{t}^{i}-\overline{y}_{t})(y_{t}^{i}-\overline{y}_{t})^{\intercal}\|_{2}^{p}\right)^{1/p}$
$\displaystyle=\dfrac{1}{B}\sum_{i=1}^{B}\left(\mathbb{E}|y^{i}_{t}-\overline{y}_{t}|^{2p}\right)^{1/p}$
$\displaystyle=\dfrac{1}{B}\sum_{i=1}^{B}\left(\left(\mathbb{E}|y^{i}_{t}-\overline{y}_{t}|^{2p}\right)^{1/2p}\right)^{2}$
$\displaystyle\leq Ce^{Ct},$

where we have again used lemma A.4 in the last inequality. Similar to the
proof of lemma A.4, we now set

$\mathbf{y}_{t}^{i}=\sqrt{P}y_{t}^{i},\quad
K_{p}(\mathbf{y})=\dfrac{1}{B}\sum_{i=1}^{B}\left\langle\mathbf{y}_{t}^{i},\mathbf{y}_{t}^{i}\right\rangle^{p}=\dfrac{1}{B}\sum_{i=1}^{B}|\mathbf{y}_{t}^{i}|^{2p}$

and

$g_{p}(t)=\mathbb{E}K_{p}=\mathbb{E}|\mathbf{y}^{1}_{t}|^{2p}.$

We will show the remaining inequality for $2p$ instead of $p$, by which
Hölder’s inequality yields the assertion. The dynamics of the scaled particles
$\mathbf{y}_{t}^{i}$ is given by

$\mathrm{d}\mathbf{y}_{t}^{i}=-\mathrm{Cov}_{\mathbf{y}_{t}}(\mathbf{y}_{t}^{i}-\mathbf{y}^{*})\mathrm{d}t+\dfrac{D+1}{B}(\mathbf{y}_{t}^{j}-\overline{\mathbf{y}}_{t})\mathrm{d}t+\sqrt{B}\sqrt{2\mathrm{Cov}_{y_{t}}}\mathrm{d}W^{i}_{t}.$

The proof now follows similar steps to that of lemma A.4. By Itô’s lemma we
get the dynamics of $K_{p}(\mathbf{y}_{t})$ and, by taking the expectation,
that of $g_{p}(t)$, which we then bound by Gronwall’s lemma. More precisely,
Itô’s lemma (A.2) yields

$\displaystyle\mathrm{d}K_{p}(\mathbf{y}_{t})=$
$\displaystyle-\dfrac{2p}{B}\sum_{i=1}^{B}\left\langle\mathbf{y}_{t}^{i},\mathbf{y}_{t}^{i}\right\rangle^{p-1}\left\langle\mathbf{y}_{t}^{i},\mathrm{Cov}_{\mathbf{y}_{t}}(\mathbf{y}_{t}^{i}-\mathbf{y}^{*})\right\rangle\mathrm{d}t$
(A.21)
$\displaystyle+\dfrac{2p}{B}\sum_{i=1}^{B}\left\langle\mathbf{y}_{t}^{i},\mathbf{y}_{t}^{i}\right\rangle^{p-1}\left\langle\mathbf{y}_{t}^{i},\sqrt{B}\sqrt{2\text{Cov}_{y_{t}}}dW^{i}_{t}\right\rangle$
$\displaystyle+\dfrac{4p(p-1)}{B}\sum_{i=1}^{B}\left\langle\mathbf{y}_{t}^{i},\mathbf{y}_{t}^{i}\right\rangle^{p-2}\text{Tr}\left\\{(\mathbf{y}_{t}^{i}\otimes\mathbf{y}_{t}^{i})\text{Cov}_{\mathbf{y}_{t}}\right\\}\mathrm{d}t$
$\displaystyle+\dfrac{2p}{B}\sum_{i=1}^{B}\left\langle\mathbf{y}_{t}^{i},\mathbf{y}_{t}^{i}\right\rangle^{p-1}\text{Tr}\\{\mathrm{Cov}_{\mathbf{y}_{t}}\\}\mathrm{d}t$
$\displaystyle+\dfrac{2p(D+1)}{B^{2}}\sum_{i=1}^{B}\left\langle\mathbf{y}_{t}^{i},\mathbf{y}_{t}^{i}\right\rangle^{p-1}\left\langle\mathbf{y}_{t}^{i},\mathbf{y}_{t}^{i}-\overline{\mathbf{y}}_{t}\right\rangle\mathrm{d}t,$

with the last term corresponding to the ALDI correction term. Using (A.19) and
(A.20), the expectation of this term term can be bounded according to

$\displaystyle~{}\dfrac{2p(D+1)}{B^{2}}\sum_{i=1}^{B}\mathbb{E}\left[\left\langle\mathbf{y}_{t}^{i},\mathbf{y}_{t}^{i}\right\rangle^{p-1}\left\langle\mathbf{y}_{t}^{i},\mathbf{y}_{t}^{i}-\overline{\mathbf{y}}_{t}\right\rangle\right]$
$\displaystyle\leq\dfrac{2p(D+1)}{B}g_{p}(t)+\dfrac{2p(D+1)}{B^{3}}\sum_{i,k=1}^{B}\mathbb{E}\left[\dfrac{\langle\mathbf{y}_{t}^{i},\mathbf{y}_{t}^{i}\rangle^{p}+\langle\mathbf{y}_{t}^{i},\mathbf{y}_{t}^{i}\rangle^{p-1}\langle\mathbf{y}_{t}^{k},\mathbf{y}_{t}^{k}\rangle}{2}\right]$
$\displaystyle\leq\dfrac{2p(D+1)}{B}g_{p}(t)+\dfrac{p(D+1)}{B}g_{p}(t)+\dfrac{p(D+1)}{B}g_{p}(t)$
$\displaystyle=C(p,D)g_{p}(t).$

The second term in (A.21) vanishes in expectation. The first, third and forth
terms can be bounded in the same way as in the proof of Proposition 4.1 in
[9]. The computations are technical but not very insightful and since the
interested reader may follow the arguments in that work step by step, we will
not repeat them here in detail, but simply state the result. For the first
term, we get with Hölder’s inequality and lemma A.4

$\displaystyle-\dfrac{2p}{B}\sum_{i=1}^{B}\mathbb{E}\left[\left\langle\mathbf{y}_{t}^{i},\mathbf{y}_{t}^{i}\right\rangle^{p-1}\left\langle\mathbf{y}_{t}^{i},\mathrm{Cov}_{\mathbf{y}_{t}}(\mathbf{y}_{t}^{i}-\mathbf{y}^{*})\right\rangle\right]$
(A.22) $\displaystyle\leq
2p|\mathbf{y}^{\ast}|\left(\dfrac{1}{B}\sum_{i=1}^{B}\mathbb{E}\langle\mathbf{y}_{t}^{i},\mathbf{y}_{t}^{i}\rangle^{p}\right)^{(p-1/2)/p}\left(\dfrac{1}{B}\sum_{i=1}^{B}\mathbb{E}\langle\mathbf{e}_{t}^{k},\mathbf{e}_{t}^{k}\rangle^{2p}\right)^{1/(2p)}$
$\displaystyle\leq 2p|\mathbf{y}^{\ast}|g_{p}^{(p-1/2)/p}(t)Ce^{C(p,D)t}.$

For the third and forth terms we receive

$\displaystyle\dfrac{4p(p-1)}{B}\sum_{i=1}^{B}\mathbb{E}\left[\left\langle\mathbf{y}_{t}^{i},\mathbf{y}_{t}^{i}\right\rangle^{p-2}\text{Tr}\left\\{(\mathbf{y}_{t}^{i}\otimes\mathbf{y}_{t}^{i})\text{Cov}_{\mathbf{y}_{t}}\right\\}\right]$
(A.23) $\displaystyle\leq
4p(p-1)\left(\dfrac{1}{B}\sum_{i=1}^{B}\mathbb{E}\langle\mathbf{y}_{t}^{i},\mathbf{y}_{t}^{i}\rangle^{p}\right)^{(p-1)/p}\left(\dfrac{1}{B}\sum_{k=1}^{B}\mathbb{E}\langle\mathbf{e}_{t}^{k},\mathbf{e}_{t}^{k}\rangle^{p}\right)^{1/p}$
$\displaystyle\leq 4p(p-1)g_{p}^{(p-1)/p}(t)Ce^{C(p,D)t}$

and

$\displaystyle\dfrac{2p}{B}\sum_{i=1}^{B}\mathbb{E}\left[\left\langle\mathbf{y}_{t}^{i},\mathbf{y}_{t}^{i}\right\rangle^{p-1}\text{Tr}\\{\mathrm{Cov}_{\mathbf{y}_{t}}\\}\right]$
(A.24) $\displaystyle\leq
2p\left(\dfrac{1}{B}\sum_{i=1}^{B}\mathbb{E}\langle\mathbf{y}_{t}^{i},\mathbf{y}_{t}^{i}\rangle^{p}\right)^{(p-1)/p}\left(\dfrac{1}{B}\sum_{k=1}^{B}\mathbb{E}\langle\mathbf{e}_{t}^{k},\mathbf{e}_{t}^{k}\rangle^{p}\right)^{1/p}$
$\displaystyle\leq 2pg_{p}^{(p-1)/p}(t)Ce^{C(p,D)t}.$

In total, we arrive at

$g^{\prime}_{p}(t)\leq
Cg_{p}(t)+Ce^{Ct}\left(g^{(p-1)/p}_{p}(t)+g_{p}^{(p-1/2)/p}(t)\right).$

The result now follows from lemma B.3. ∎

###### Proof of Lemma A.1.

The lemma results directly from boundedness of the higher moments of
$y_{t}^{i}$ (Lemma A.5) and $z_{t}^{i}$ (see Proposition 5.3 in [9]). ∎

###### Proof of Lemma A.2.

First, note that

$\mathbb{E}|p_{t}^{i}|^{2}=\mathbb{E}|x_{t}^{i}-\overline{x}_{t}|^{2}=\mathbb{E}(|x_{t}^{i}|^{2}+|\overline{x}_{t}|^{2}-2\langle
x_{t}^{i},\overline{x}_{t}\rangle)=\mathbb{E}(|x_{t}^{i}|^{2}-|\overline{x}_{t}|^{2})=\mathbb{E}\left(\dfrac{1}{B}\sum_{i=1}^{B}|x_{t}^{i}|^{2}-|\overline{x}_{t}|^{2}\right)$
(A.25)

due to the symmetry between the particles. Using (A.1) and (A.5), we have

$\displaystyle\mathrm{d}x_{t}^{i}=$
$\displaystyle\left(-\mathrm{Cov}_{x_{t}+z_{t}}P(x_{t}^{i}+z_{t}^{i})+\mathrm{Cov}_{\pi(t)}Pz_{t}^{i}\right)\mathrm{d}t+(\mathrm{Cov}_{x_{t}+z_{t}}-\mathrm{Cov}_{\pi(t)})Py^{*}\mathrm{d}t$
(A.26)
$\displaystyle+\dfrac{D+1}{B}\left(y_{t}^{i}-\overline{y}_{t}\right)\mathrm{d}t+\left(\sqrt{2\mathrm{Cov}_{x_{t}+z_{t}}}-\sqrt{2\mathrm{Cov}_{\pi(t)}}\right)\mathrm{d}W_{t}^{i}.$

Now, applying Itô’s formula (A.2) and replacing $\mathrm{Cov}_{\pi(t)}$ in the
second and third terms with $\mathrm{Cov}_{z_{t}}$, we find

$\displaystyle\mathrm{d}|x_{t}^{i}|^{2}=$ $\displaystyle-2\left\langle
x_{t}^{i},\mathrm{Cov}_{x_{t}+z_{t}}Px_{t}^{i}\right\rangle\mathrm{d}t-2\left\langle
x_{t}^{i},(\mathrm{Cov}_{x_{t}+z_{t}}-\mathrm{Cov}_{z_{t}})Pz_{t}^{i}\right\rangle\mathrm{d}t$
(A.27) $\displaystyle+2\left\langle
x_{t}^{i},(\mathrm{Cov}_{x_{t}+z_{t}}-\mathrm{Cov}_{z_{t}}Py^{\ast})\right\rangle\mathrm{d}t+\mathrm{Tr}\left[\left(\sqrt{\mathrm{Cov}_{x_{t}+z_{t}}}-\sqrt{\mathrm{Cov}_{\pi(t)}}\right)^{2}\right]\mathrm{d}t$
$\displaystyle+2\left\langle
x_{t}^{i},\dfrac{D+1}{B}(y_{t}^{i}-\overline{y}_{t})\right\rangle\mathrm{d}t+2\left\langle
x_{t}^{i},\left(\sqrt{2\mathrm{Cov}_{\pi(t)}}-\sqrt{2\mathrm{Cov}_{\pi(t)}}\right)\mathrm{d}W_{t}^{i}\right\rangle+R_{t}^{i}\mathrm{d}t$

and

$\displaystyle\mathrm{d}|\overline{x}_{t}|^{2}=$
$\displaystyle-2\left\langle\overline{x}_{t},\mathrm{Cov}_{x_{t}+z_{t}}P\overline{x}_{t}\right\rangle\mathrm{d}t-2\left\langle\overline{x}_{t},(\mathrm{Cov}_{x_{t}+z_{t}}-\mathrm{Cov}_{z_{t}})P\overline{z}_{t}\right\rangle\mathrm{d}t$
(A.28)
$\displaystyle+2\left\langle\overline{x}_{t},(\mathrm{Cov}_{x_{t}+z_{t}}-\mathrm{Cov}_{z_{t}}Py^{\ast})\right\rangle\mathrm{d}t+\mathrm{Tr}\left[\left(\sqrt{\mathrm{Cov}_{x_{t}+z_{t}}}-\sqrt{\mathrm{Cov}_{\pi(t)}}\right)^{2}\right]\mathrm{d}t$
$\displaystyle+2\sum_{i=1}^{B}\left\langle\overline{x}_{t},\dfrac{D+1}{B^{2}}(y_{t}^{i}-\overline{y}_{t})\right\rangle\mathrm{d}t+2\left\langle\overline{x}_{t},\left(\sqrt{2\mathrm{Cov}_{\pi(t)}}-\sqrt{2\mathrm{Cov}_{\pi(t)}}\right)\mathrm{d}\overline{W}_{t}\right\rangle+\overline{R}_{t}\mathrm{d}t,$

with

$\displaystyle R_{t}^{j}$ $\displaystyle=2\left\langle
x_{t}^{i},(\mathrm{Cov}_{\pi(t)}-\mathrm{Cov}_{z_{t}})Pz_{t}^{i}\right\rangle-2\left\langle
x_{t}^{i},(\mathrm{Cov}_{\pi(t)}-\mathrm{Cov}_{z_{t}})Py^{*}\right\rangle,$
(A.29) $\displaystyle\overline{R}_{t}$
$\displaystyle=2\left\langle\overline{x}_{t},(\mathrm{Cov}_{\pi(t)}-\mathrm{Cov}_{z_{t}})P\overline{z}_{t}\right\rangle-2\left\langle\overline{x}_{t},(\mathrm{Cov}_{\pi(t)}-\mathrm{Cov}_{z_{t}})Py^{*}\right\rangle.$

Combining (A.28) and (A.27), we finally arrive at

$\displaystyle\mathrm{d}\left(\dfrac{1}{B}\sum_{i=1}^{B}|x_{t}^{i}|^{2}-|\overline{x}_{t}|^{2}\right)=$
$\displaystyle-\dfrac{2}{B}\sum_{i=1}^{B}\left\langle
p_{t}^{i},\mathrm{Cov}_{p_{t}+q_{t}}Pp_{t}^{i}\right\rangle\mathrm{d}t-\dfrac{2}{B}\sum_{i=1}^{B}\left\langle
p_{t}^{i},(\mathrm{Cov}_{p_{t}+q_{t}}-\mathrm{Cov}_{q_{t}})Pq^{i}_{t}\right\rangle\mathrm{d}t$
(A.30)
$\displaystyle+2\left(1-\dfrac{1}{B}\right)\mathrm{Tr}\left[\left(\sqrt{\mathrm{Cov}_{x_{t}+z_{t}}}-\sqrt{\mathrm{Cov}_{\pi(t)}}\right)^{2}\right]\mathrm{d}t+\left(\dfrac{1}{B}\sum_{i=1}^{B}R_{t}^{i}-\overline{R}_{t}\right)\mathrm{d}t$
$\displaystyle+\frac{D+1}{B^{2}}\sum_{i=1}^{B}\left[2\left\langle
x_{t}^{i},\left(y_{t}^{i}-\overline{y}_{t}\right)\right\rangle-2\left\langle\overline{x}_{t},\frac{1}{B}\sum_{i=1}^{B}\left(y_{t}^{i}-\overline{y}_{t}\right)\right\rangle\right]\mathrm{d}t$
$\displaystyle+\dfrac{2}{B}\sum_{i=1}^{B}\left\langle\left(x_{t}^{i}-\overline{x}_{t}\right),\left(\sqrt{2\mathrm{Cov}_{x_{t}+z_{t}}}-\sqrt{2\mathrm{Cov}_{\pi(t)}}\right)\mathrm{d}\left(W_{t}^{i}-\overline{W}_{t}\right)\right\rangle.$

The expectation of the last term vanishes by the properties of Brownian
motion. The first four terms have the same form as in equation (5.22) in [9]
and can be treated in a similar way. Note that the terms are not identical to
the terms in that work, since the underlying process $y_{t}$ and hence
$x_{t},p_{t},q_{t}$ are different. They can however be treated in a completely
analogous way, tracing the arguments one by one. Hence, we will only state the
resulting bounds here, and refer to equations (5.22)-(5.27) in [9] for
details. The first two terms can be bounded in expectation according to

$\displaystyle-\dfrac{2}{B}\mathbb{E}\left[\sum_{i=1}^{B}\left\langle
p_{t}^{i},\mathrm{Cov}_{p_{t}+q_{t}}Pp_{t}^{i}\right\rangle\mathrm{d}t+\sum_{i=1}^{B}\left\langle
p_{t}^{i},(\mathrm{Cov}_{p_{t}+q_{t}}-\mathrm{Cov}_{q_{t}})Pq^{i}_{t}\right\rangle\mathrm{d}t\right]$
(A.31) $\displaystyle\leq$
$\displaystyle-\lambda_{\mathrm{min}}(P)\mathbb{E}\left(\|\mathrm{Cov}_{x_{t}+z_{t}}-\mathrm{Cov}_{z_{t}}\|^{2}_{F}\right)+3\|P\|_{F}\mathrm{Var}(\pi(t))\mathbb{E}|p_{t}^{1}|^{2}+CB^{-1/2-\alpha(1-\epsilon)}.$

For the third term, we get

$\displaystyle\mathbb{E}\left[\mathrm{Tr}\left[\left(\sqrt{\mathrm{Cov}_{x_{t}+z_{t}}}-\sqrt{\mathrm{Cov}_{\pi(t)}}\right)^{2}\right]\right]\leq\lambda_{0}(t)^{-1}\mathbb{E}\|\mathrm{Cov}_{p_{t}+q_{t}}-\mathrm{Cov}_{q_{t}}\|_{F}^{2}+C_{\epsilon}B^{-1/2-\alpha/2+\epsilon\alpha/4}$
(A.32)

with a constant $C_{\epsilon}$ depending only on $\epsilon$ and
$\lambda_{0}(t)=(\lambda_{\min}(\mathrm{Cov}_{x_{t}+z_{t}})^{1/2}+\lambda_{\min}(\mathrm{Cov}_{\pi(t)})^{1/2})^{2}$
coming from the Ando-Hemmen inequality (see Theorem 6.2 in [22]).

Finally, for the forth term, we get

$\displaystyle\mathbb{E}\left[\dfrac{1}{B}\sum_{i=1}^{B}R_{t}^{i}-\overline{R}_{t}\right]\leq
C_{\varepsilon}B^{-1/2-\alpha/2+\epsilon\alpha/4},$ (A.33)

again with a constant $C_{\epsilon}$ depending on $\epsilon$. The fifth term
comes from the ALDI correction term. To bound it, we note that

$\displaystyle\left\langle
x_{t}^{i},\left(y_{t}^{i}-\overline{y}_{t}\right)\right\rangle$
$\displaystyle\leq\mathbb{E}|x_{t}^{i}|\left(\mathbb{E}|y_{t}^{i}|+\mathbb{E}|\overline{y}_{t}|\right)$
$\displaystyle\leq
2(\mathbb{E}|x_{t}^{i}|^{2})^{1/2}\mathbb{E}(|y_{t}^{i}|^{2})^{1/2}$
$\displaystyle\leq C_{1}$

with a constant $C_{1}$ independent of $B,t$, where we have used Hölder’s
inequality in the second inequality and Lemmas A.5 and A.1 in the last
inequality. Similarly, we get

$\left\langle\overline{x}_{t},\frac{1}{B}\sum_{i=1}^{B}\left(y_{t}^{j}-\overline{y}_{t}\right)\right\rangle\leq\frac{2}{B}\mathbb{E}|x_{t}^{i}|\sum_{i=1}^{B}\mathbb{E}|y_{t}^{i}|=2\mathbb{E}|x_{t}^{i}|\mathbb{E}|y_{t}^{i}|\leq
C_{1},$

to arrive at

$\displaystyle\mathbb{E}\left[\frac{D+1}{B^{2}}\sum_{i=1}^{B}\left[2\left\langle
x_{t}^{i},\left(y_{t}^{i}-\overline{y}_{t}\right)\right\rangle-2\left\langle\overline{x}_{t},\frac{1}{B}\sum_{i=1}^{B}\left(y_{t}^{i}-\overline{y}_{t}\right)\right\rangle\right]\right]\leq\dfrac{2(D+1)}{B^{2}}BC_{1}=2(D+1)C_{1}B^{-1}.$

Hence, taking the expectation in (A.30), we find

$\displaystyle\dfrac{\mathrm{d}\mathbb{E}|p_{t}^{1}|^{2}}{\mathrm{d}t}\leq$
$\displaystyle
3\|P\|_{2}\mathrm{Var}(\pi(t))\mathbb{E}|p_{t}^{1}|^{2}-\left(\lambda_{\min}(B)-2\lambda_{0}^{-1}\right)\mathbb{E}\|\mathrm{Cov}_{p_{t}+q_{t}}-\mathrm{Cov}_{q_{t}}\|_{F}^{2}$
(A.34)
$\displaystyle+CB^{-1/2-\alpha(1-\varepsilon)}+C_{\epsilon}B^{-1/2-\alpha/2+\epsilon\alpha/4}++2(D+1)C_{1}B^{-1}.$

The second term is always negative by assumption (4.7). Furthermore we note
that since $0\leq\alpha<1$ and $0<\epsilon<1/2$ all exponents of $B$ appearing
in (A.34) are smaller than $-1/2-\alpha/2+\epsilon$, leading to

$\displaystyle\dfrac{\mathrm{d}\mathbb{E}|p_{t}^{1}|^{2}}{\mathrm{d}t}\leq$
$\displaystyle
3\|P\|_{2}\mathrm{Var}(\pi(t))\mathbb{E}|p_{t}^{1}|^{2}+CB^{-1/2-\alpha/2+\epsilon}.$
(A.35)

Now, by equation (2.2) in [6] we have

$\displaystyle\mathrm{Cov}_{\pi(t)}=(1-e^{-2t})P^{-1}+e^{-2t}\mathrm{Cov}_{\pi_{0}}^{-1}$
(A.36)

and hence $\mathrm{Var}(\pi(t))\leq M$ for some $M>0$. Integrating (A.35)
leads to

$\displaystyle\mathbb{E}|p_{t}^{1}|^{2}\leq
C\int_{0}^{t}\mathbb{E}|p_{s}^{1}|^{2}ds+CtB^{-1/2-\alpha/2+\epsilon}$ (A.37)

and applying lemma B.1 yields

$\displaystyle\mathbb{E}|p_{t}^{1}|^{2}\leq
CtB^{-1/2-\alpha/2+\epsilon}+\int_{0}^{t}CsB^{-1/2-\alpha/2+\epsilon}e^{C(t-s)}\mathrm{d}s=C(t)B^{-1/2-\alpha/2+\epsilon},$
(A.38)

finishing the proof.

∎

###### Proof of Lemma A.3.

The proof is the same as that of Lemma 5.5 in [9]. ∎

##  Appendix B Gronwall inequalities

######  Lemma B.1 (Gronwall [19]).

Let $u(t)$ be a nonnegative function satisfying

$u(t)\leq\beta(t)+\int_{0}^{t}c(t)u(t)\mathrm{d}t\qquad\mathrm{for~{}all}\quad
t\geq 0,$

where $\beta$ and $c$ are continuous nonnegative functions for $t\geq 0$. Then

$u(t)\leq\beta(t)+\int_{0}^{t}\beta(s)c(s)e^{(\int_{s}^{t}c(r)\mathrm{d}r)}\mathrm{d}s\quad\mathrm{for~{}all}\quad
t\geq 0.$

###### Proof.

See e.g. Theorem 1 in [10]. ∎

######  Lemma B.2 (Perov, [31] or Theorem 21 in [10]).

Let $u(t)$ be a nonnegative function satisfying

$u^{\prime}(t)\leq c(t)u(t)+a(t)u(t)^{\alpha},$

where $0\leq\alpha<1$ and $c$ and $a$ are countinous nonnegative functions for
$t>0$. Then we have

$u(t)\leq\left[u^{1-\alpha}(0)e^{(1-\alpha)\int_{0}^{t}c(s)ds}+e^{(1-\alpha)\int_{0}^{t}c(s)ds}\int_{0}^{t}(1-\alpha)a(s)e^{(\alpha-1)\int_{0}^{s}a(r)dr}ds\right]^{\frac{1}{1-\alpha}}.$

###### Proof.

Let $v(t)$ solve

$v^{\prime}(t)=c(t)v(t)+a(t)v(t)^{\alpha},\quad v(0)=u(0).$

This is a classic Bernoulli-type ODE, which can be solved by noting that the
ODE becomes linear in $v^{(1-\alpha)}$:

$(v^{1-\alpha})^{\prime}(t)=(1-\alpha)c(t)v^{1-\alpha}(t)+(1-\alpha)a(t).$

Applying the variation of parameters to $v^{1-\alpha}$ we get

$v(t)=\left[u^{1-\alpha}(0)e^{(1-\alpha)\int_{0}^{t}c(s)ds}+\int_{0}^{t}(1-\alpha)a(s)e^{(1-\alpha)\int_{s}^{t}c(r)dr}ds\right]^{\frac{1}{1-\alpha}}.$

The assertion now immediately follows, since $u(t)\leq v(t)$ by construction.
∎

######  Lemma B.3.

Let $u(t)$ be a nonnegative function satisfying

$u^{\prime}(t)\leq c(t)u(t)+a(t)u^{\alpha}(t)+b(t)u^{\beta}(t),$

where $0<\alpha<\beta\leq 1$ and $c,a$ and $b$ are continuous and nonnegative
functions for $t\geq 0$. Then $u(t)$ is bounded for all $t\geq 0$ by

$u(t)\leq\left[u^{1-\alpha}(0)e^{(1-\alpha)\int_{0}^{t}c(s)+b(s)ds}+\int_{0}^{t}(1-\alpha)(a(s)+b(s))e^{(1-\alpha)\int_{s}^{t}c(r)+b(r)dr}ds\right]^{\frac{1}{1-\alpha}}.$
(B.1)

###### Proof.

For $u(t)<1$ we have $u^{\beta}(t)\leq u^{\alpha}(t)$ and hence

$\displaystyle u^{\prime}(t)$ $\displaystyle\leq
c(t)u(t)+(a(t)+b(t))u^{\alpha}(t)$
$\displaystyle\leq(c(t)+b(t))u(t)+(a(t)+b(t))u^{\alpha}(t).$

For $u(t)\geq 1$, we have $u^{\beta}(t)\leq u(t)$ and hence

$\displaystyle u^{\prime}(t)$
$\displaystyle\leq(c(t)+b(t))u(t)+a(t)u^{\alpha}(t)$
$\displaystyle\leq(c(t)+b(t))u(t)+(a(t)+b(t))u^{\alpha}(t).$

In total, we arrive at
$u^{\prime}(t)\leq(c(t)+b(t))u(t)+(a(t)+b(t))u^{\alpha}(t)$ for all $t\geq 0$
and can use lemma B.2. ∎

##  Appendix C Existence and Uniqueness of a solution of ALDI

Existence and uniqueness of a strong solution by means of construction of a
Lyapunov function of the system has already been proven in the original work
on ALDI [16]. We provide an alternative proof, using the same Lyapunov
function that is used in [9] for existence and uniqueness of the EKS solution,
merely for the interest of the reader.

######  Theorem C.1 (Compare Thm 3.2 in [9]).

Suppose $G$ is linear and $(y^{i}_{0})_{i=1}^{B}$ are i.i.d. Then for all
$t\geq 0$ there exists a unique strong solution $(y^{i}_{t})_{i=1}^{B}$ (up to
$\mathbb{P}$-indistinguishability) of the set of coupled SDEs defined by
(2.25).

###### Proof.

Proof of Theorem C.1 We consider the stacked SDE

$\mathrm{d}Y_{t}=F(Y_{t})\mathrm{d}t+G(Y_{t})\mathrm{d}W_{t},$

where $Y_{t}=(y_{t}^{i})_{i=1}^{B}\in\mathbb{R}^{DB}$,
$W_{t}=\left(W_{t}^{i}\right)_{i=1}^{B}$ and

$\displaystyle F(Y_{t})$
$\displaystyle=\left(-\text{Cov}_{y_{t}}P(y_{t}^{i}-u^{*})+\dfrac{D+1}{B}(y_{t}^{j}-\overline{y}_{t})\right)_{i=1}^{B}\in\mathbb{R}^{DB},$
$\displaystyle G(Y_{t})$
$\displaystyle=\text{diag}\left(\sqrt{2\text{Cov}_{y_{t}}}\right)_{i=1}^{B}\in\mathbb{R}^{DB\times
DB},$

where $\text{diag}(D_{i})_{i=1}^{B}$ is a block diagonal matrix with entries
$(D_{i})_{i}^{B}$ on the diagonal.

We prove the assertion by showing existence of a Lyapunov function of the
system, i.e. a function $V\in\mathcal{C}^{2}(\mathbb{R}^{DB},\mathbb{R})$ such
that

  * (1)

there exists a $c>0$ so that

$\mathcal{L}V(Y)\coloneqq\nabla V(Y)\cdot
F(Y)+\frac{1}{2}\text{tr}\left[G(Y)^{\intercal}\text{Hess}_{V}(Y)G(Y)\right]\leq
cV(Y)$

for all $Y\in\mathbb{R}^{DB}$.

  * (2)

we have

$\inf_{|Y|>R}V(Y)\longrightarrow\infty$

as $R\longrightarrow\infty$.

The function we choose is the same that is used in [9] for the EKS:

$V(Y)=V_{1}(Y)+V_{2}(Y)=\dfrac{1}{B}\sum_{i=1}^{B}|y^{j}-\overline{y}|^{2}+|\overline{y}-y^{*}|_{P}^{2}.$

Towards (1), note that

$\displaystyle\nabla V_{1}(Y)$
$\displaystyle=\left(\frac{2}{B}(y^{j}-\overline{y})\right)_{i=1}^{B},$
$\displaystyle\nabla V_{2}(Y)$
$\displaystyle=\left(2P(\overline{y}-y^{*})\right)_{i=1}^{B}$

and that the diagonal blocks of $\text{Hess}_{V_{2}}(Y)$ and
$\text{Hess}_{V_{1}}(Y)$ are given by

$\frac{2}{B}\text{diag}(P)_{i=1}^{B}$

and

$\frac{2}{B}\left(1-\frac{1}{B}\right)\text{diag}\left(I_{d}\right)_{i=1}^{B}$

respectively.

This yields

$\displaystyle\nabla V_{1}(Y)\cdot F(Y)$
$\displaystyle=-\frac{2}{B}\sum_{i=1}^{B}\left\langle
y^{i}-\overline{y},\text{Cov}_{y}B(y^{i}-y^{*})\right\rangle+\frac{2(D+1)}{B^{2}}\sum_{i=1}^{B}\left\langle
y^{i}-\overline{y},y^{i}-\overline{y}\right\rangle$
$\displaystyle=-\frac{2}{B}\sum_{i=1}^{B}\left\langle
y^{i}-\overline{y},\text{Cov}_{y}B(y^{i}-\overline{y})\right\rangle+\frac{2(D+1)}{B}V_{1}(Y)\leq\frac{2(D+1)}{B}V_{1}(Y),$

where we have used that both $\text{Cov}_{y}$ and $B$ are positive semi-
definite, as well as

$\displaystyle\nabla V_{2}(Y)\cdot F(Y)$
$\displaystyle=-2\sum_{i=1}^{B}\left\langle
P(\overline{y}-y^{*}),\text{Cov}_{y}P(y^{j}-y^{*})\right\rangle+\frac{2(D+1)}{B}\sum_{i=1}^{B}\left\langle
P(\overline{y}-y^{*}),y^{i}-\overline{y}\right\rangle,$
$\displaystyle=-2B\left\langle
P(\overline{y}-y^{*}),\text{Cov}_{y}P(\overline{y}-y^{*})\right\rangle+2(D+1)\left\langle
P(\overline{y}-y^{*}),\overline{y}-\overline{y}\right\rangle\leq 0.$

Furthermore, we have

$\displaystyle\frac{1}{2}\text{Tr}\left[G(Y)^{\intercal}\text{Hess}_{V_{1}}(Y)G(Y)\right]$
$\displaystyle=\frac{1}{2}\text{Tr}\left[\text{Hess}_{V_{1}}(Y)G(Y)G(Y)^{\intercal}\right]=\text{Tr}\left[\text{Hess}_{V_{1}}(Y)\text{diag}(\text{Cov}_{y})_{i=1}^{B}\right]$
$\displaystyle=\frac{2}{B}\sum_{i=1}^{B}\left(1-\frac{1}{B}\right)\text{Tr}\left[\text{Cov}_{y}\right]=2\left(1-\frac{1}{B}\right)\text{Tr}\left[\text{Cov}_{y}\right]$
$\displaystyle=2\left(1-\frac{1}{B}\right)\dfrac{1}{B}\sum_{i=1}^{B}(y^{i}-\overline{y})^{\intercal}(y^{i}-\overline{y})=2\left(1-\dfrac{1}{B}\right)V_{1}(Y)$

and

$\displaystyle\frac{1}{2}\text{Tr}\left[G(Y)^{\intercal}\text{Hess}_{V_{2}}(Y)G(Y)\right]$
$\displaystyle=\text{Tr}\left[\text{Hess}_{V_{2}}(Y)\text{diag}(\text{Cov}_{y})_{i=1}^{B}\right]=2\text{Tr}[P\text{Cov}_{y}]$
$\displaystyle=\dfrac{2}{B}\sum_{i=1}^{B}(y^{i}-\overline{y})^{\intercal}P(y^{i}-\overline{y})\leq
2\|P\|V_{1}(Y).$

In total, we get

$\mathcal{L}V(Y)\leq\left(\dfrac{2(D+1)}{B}+2\left(1-\dfrac{1}{B}\right)+2\|P\|\right)V(Y)$

for all $Y$, proving (1). Now, towards (2) assume there is a sequence
$(Y_{n})_{n\in\mathbb{N}}$ in $\mathbb{R}^{DB}$ with $V(Y_{n})<M$ for some
$M>0$ and all $n$. By the construction of $V$, we have

$|y_{n}^{i}-\overline{y}_{n}|<\sqrt{MB},\quad|\overline{y}_{n}-y^{*}|<\sqrt{M}$

and hence

$|Y_{n}|^{2}=\sum_{i=1}^{B}|y_{n}^{i}|^{2}=\sum_{i=1}^{B}|y_{n}^{i}-\overline{y}_{n}+\overline{y}_{n}-y^{*}+y^{*}|^{2}<\sum_{i=1}^{B}\left(\sqrt{M}(\sqrt{B}+1)+|y^{*}|\right)^{2}$

for all $n$, meaning the sequence $(|Y_{n}|)_{n\in\mathbb{N}}$ is also
bounded. ∎

##  Appendix D Wasserstein spaces and convergence

For $p\geq 1$ let $\mathcal{D}_{p}(X)$ denote the set of probability measures
over a Banach space $X$ with norm $\|.\|$ that have finite $p$-th moment. On
$\mathcal{D}_{p}(X)$ we define the $p$-Wasserstein metric as

$\mathcal{W}_{p}(\mu,\nu)^{p}=\inf\limits_{\pi\in\Pi_{p}(\mu,\nu)}\int\limits_{X\times
X}\|x-y\|^{p}\mathrm{d}\pi(x,y),$

with $\Pi_{p}(\mu,\nu)=\\{\pi\in\mathcal{D}_{p}(X\times X)\,|\,\pi(\cdot\times
X)=\mu,\pi(X\times\cdot)=\nu\\}$. Furthermore for $X=\mathbb{R}^{D}$, we
denote with $F_{p}^{k}(X)$ with $k\in\mathbb{N}\cup\\{\infty\\}$ the set of
regular measures with $p$-th moment and a Lebesque density in
$\mathcal{C}^{k}(X)$, in particular

$F_{p}^{k}=\\{\mu\in\mathcal{D}_{p}(\mathbb{R}^{D})|\exists
f:=\mathrm{d}\mu/\mathrm{d}x\text{ and
}f\in\mathcal{C}^{k}(\mathbb{R}^{D})\\}.$

######  Lemma D.1.

Let $p\geq 1$. Then, for any $k\in\mathbb{N}\cup\\{\infty\\}$ the subset
$F_{p}^{k}\subset\mathcal{D}_{p}$ is dense in $\mathcal{D}_{p}$ with respect
to the $\mathcal{W}_{p}$ metric.

###### Proof.

Consider a non-negative convolution kernel
$\rho\in\mathcal{C}_{c}^{\infty}(\mathbb{R}^{D})$ with $\|\rho\|_{1}=1$, where
$\|\cdot\|_{1}$ denotes the $L^{1}$-norm over $\mathbb{R}^{D}$. Then define
$\rho_{\epsilon}(x)=\epsilon^{-D}\rho(\epsilon^{-1}x)$ and

$f_{\epsilon}(x)=(\rho_{\epsilon}*\mu)(x)=\int\limits_{\mathbb{R}^{D}}\rho_{\epsilon}(x-y)\mathrm{d}\mu(y).$

Then $f_{\epsilon}\in\mathcal{C}^{\infty}(\mathbb{R}^{d})$ is a Lebesque
density defining a measure $\mu_{\epsilon}$ on $\mathbb{R}^{D}$. Now consider
the transport plan or coupling

$\pi_{\epsilon}(\mathrm{d}x,\mathrm{d}y)=\rho_{\epsilon}(x-y)\mathrm{d}\mu(y)\mathrm{d}x.$

Then $\pi_{\epsilon}\in\mathcal{W}_{2}(\mathbb{R}^{D}\times\mathbb{R}^{D})$
with marginal projections
$\mu_{\epsilon}(\cdot\times\mathbb{R}^{D})=\mu_{\epsilon}$ and
$\pi_{\epsilon}(\mathbb{R}^{D}\times\cdot)=\mu$. Then it holds

$\displaystyle\mathcal{W}_{p}(\mu,\mu_{\epsilon})^{p}$
$\displaystyle\leq\int\limits_{\mathbb{R}^{D}\times\mathbb{R}^{D}}|x-y|^{p}\mathrm{d}\pi_{\epsilon}$
$\displaystyle=\int\limits_{\mathbb{R}^{D}}\int\limits_{\mathbb{R}^{D}}|x-y|^{p}\rho_{\epsilon}(x-y)\mathrm{d}\mu(y)\mathrm{d}x$
$\displaystyle=\int\limits_{\mathbb{R}^{D}}\int\limits_{\mathbb{R}^{D}}\epsilon^{p}|z|^{p}\rho(z)\mathrm{d}\mu(y)\mathrm{d}z$
$\displaystyle=\epsilon^{p}\int\limits_{\mathbb{R}^{D}}|z|^{p}\rho(z)\mathrm{d}z,$

where we used the substitution $z=\epsilon^{-1}(x-y)$. Now, clearly,
$\mathcal{W}_{p}(\mu,\mu_{\epsilon})\to 0$ as $\epsilon\to 0$. ∎

######  Theorem D.1 (Theorem 1 in [14]).

Let $\mu$ be a probability measure on $\mathbb{R}^{D}$ and let
$\mu^{(B)}=\frac{1}{B}\sum_{i=1}^{B}\delta_{y^{i}}$ be an empirical measure of
i.i.d. samples $\\{y^{i}\\}_{i=1}^{B}$ drawn from $\mu$. Let $p>0$ and assume
that

$M_{q}(\mu)\coloneqq\int_{\mathbb{R}^{D}}|y|^{q}\mathrm{d}\mu(y)<\infty\qquad\textrm{for
some}\quad q>p.$ (D.1)

Then, there exists a constant $C$ depending only on $p,d,q$ such that, for all
$N\geq 1$,

$\displaystyle\mathbb{E}[(\mathcal{W}_{p}(\mu^{(B)},\mu))]\leq
CM^{p/q}_{q}(\mu)\cdot\begin{cases}B^{-1/2}+B^{-(q-p)/q}&\quad\textrm{if}\quad
p>d/2\quad\textrm{and}\quad q=2p,\\\
B^{-1/2}\log(1+B)+B^{-(q-p)/q}&\quad\textrm{if}\quad
p=d/2\quad\textrm{and}\quad q=2p,\\\
B^{-p/d}+B^{-(q-p)/q}&\quad\textrm{if}\quad p\in(0,d/2)\quad\textrm{and}\quad
q=d/(d-p).\end{cases}$ (D.2)


$\displaystyle S_{\rm{OS},\rm{bulk}}^{\rm{(vac)}}$
$\displaystyle=-\frac{L^{9}\pi^{2}}{8G_{N}^{(11)}}\operatorname{vol}(\text{AdS}_{5})\left(\frac{1}{3\epsilon_{u}^{6}}+\frac{1}{2\epsilon_{u}^{4}}+\frac{5}{16\epsilon_{u}^{2}}-\frac{11}{48}\right)+\ldots~{}.$
(C.141)

Combining with the GHY term, we find

$\displaystyle
S_{\rm{OS}}^{\rm{(vac)}}=-\frac{\pi^{2}L^{9}}{8G_{N}^{(11)}}\operatorname{vol}(\text{AdS}_{5})\left(\frac{1}{3\epsilon_{u}^{6}}+\frac{5}{3\epsilon_{u}^{5}}+\frac{1}{2\epsilon_{u}^{4}}+\frac{1}{\epsilon_{u}^{3}}+\frac{5}{16\epsilon_{u}^{2}}+\frac{5}{48\epsilon_{u}}-\frac{11}{48}\right)+\ldots~{}.$
(C.142)

Finally, we note that since $d\star F_{4}=0$ we can introduce $C_{6}$ so that
$dC_{6}=\star_{11}F_{4}$. We can then perform the bulk integral of
$F_{4}\wedge C_{6}$ over the radial cutoff slice at $\epsilon_{u}$ with the
pullback of the six-form potential being given by

$\displaystyle
C_{6}=3L^{6}\left(\frac{1}{3\epsilon_{u}^{6}}+\frac{1}{2\epsilon_{u}^{4}}+\frac{5}{16\epsilon_{u}^{2}}-\frac{11}{48}+\frac{5\epsilon_{u}^{2}}{256}+\frac{\epsilon_{u}^{4}}{512}+\frac{\epsilon_{u}^{6}}{12288}\right)dz\wedge\Upsilon_{\rm
AdS_{5}}.$ (C.143)

Crucially, we have used the residual gauge freedom to fix the six-form
potential to be regular at the origin of AdS7, i.e. we pick a gauge such that
$C_{6}\big{|}_{u=2}=0$. The computation using $C_{6}$ then gives the same
result as above.

###  C.2 Renormalized AdS5 volume

Even accounting for the removal of divergences coming from the asymptotically
AdS7 part of the geometry via background subtraction, we are still left to
deal with the volume of the AdS5 factor in the on-shell action. In order to
regularize the remaining polynomial divergences and read off the universal
log-divergent part of the on-shell action, we will simply treat the intrinsic
parts of the AdS5 geometry using standard counterterms in holographic
renormalization and neglecting any divergences associated with the embedding.
This renormalization scheme is admittedly simplistic as it only treats the set
of counterterms associated with the intrinsic geometry of the AdS5
submanifold. However, since the background subtraction scheme leaves behind
only divergences from the volume of the AdS5 and we choose the boundary
geometry to be $\mathds{S}^{4}\hookrightarrow\mathbb{R}^{6}$, only defect Weyl
anomalies constructed purely from the intrinsic geometry should contribute,
which will be accounted for in the scheme we have chosen. The caveat is that
there may be structures for which we have not accounted in the full set of
$11d$ counterterms, which is difficult to construct, whose pullback to the
AdS5 submanifold contains terms that contribute to the log divergence in a
similar way. Absent a full holographic renormalization scheme for solutions to
SUGRA dual to defects, which would replace background subtraction scheme as
well, this scheme choice constructing counterterms only for the intrinsic
geometry of the AdS5 submanifold is the best tool available.

Moving on, the volume of AdS5 has well known divergences. In order to
systematically remove them and reveal any universal log-divergent terms, we
consider AdS5 in global coordinates with an $\mathds{S}^{4}$ boundary:

$\displaystyle ds_{\rm AdS_{5}}^{2}=dx^{2}+\sinh^{2}(x)~{}d\Omega_{4}^{2}.$
(C.144)

For simplicity, we consider the round metric on $\mathds{S}^{4}$. Computing
the AdS5 volume requires regulating the large $x$ behavior, and so we
introduce a radial cutoff $\Lambda_{x}\equiv-\log\frac{\epsilon_{x}}{2}$ for
$\epsilon_{x}\ll 1$. Then, expanding in small $\epsilon_{x}$

$\displaystyle\text{vol}(\text{AdS}_{5})~{}=\frac{8\pi^{2}}{3}\int_{0}^{\Lambda_{x}}dx\sinh^{4}(x)=\frac{2\pi^{2}}{3\epsilon_{x}^{4}}-\frac{4\pi^{2}}{3\epsilon_{x}^{2}}-\pi^{2}\log\frac{\epsilon_{x}}{2}+\ldots~{}.$
(C.145)

We regulate the volume using covariant counterterms161616To be complete, we
should also fix finite counterterms to ensure that we are in a supersymmetry
preserving scheme, but we will forego addressing this here as it is not
germane to the problem at hand. added on the radial cutoff slice that are
standard in AdS5 holographic renormalization Henningson:1998gx ;
deHaro:2000vlm

$\displaystyle S_{\rm CT,1}$ $\displaystyle=-\frac{1}{4}\int
d\Omega_{4}\sqrt{|g_{\epsilon_{x}}|}=-\frac{2\pi^{2}}{3\epsilon_{x}^{4}}+\frac{2\pi^{2}}{3\epsilon_{x}^{2}}-\frac{\pi^{2}}{4}+\ldots,$
(C.146a) $\displaystyle S_{\rm CT,2}$ $\displaystyle=\frac{1}{48}\int
d\Omega_{4}\sqrt{|g_{\epsilon_{x}}|}{{\mathcal{R}}}_{\epsilon_{x}}=\frac{2\pi^{2}}{3\epsilon_{x}^{2}}-\frac{\pi^{2}}{3}+\ldots,$
(C.146b)

where
$\sqrt{|g_{\epsilon_{x}}|}=(1-\epsilon_{x})^{4}\sqrt{|g_{\mathds{S}^{4}}|}/16\epsilon_{x}^{2}$
and ${{\mathcal{R}}}_{\epsilon_{x}}=12\operatorname{csch}^{2}(\epsilon_{x})$
are the volume form and the intrinsic Ricci scalar on the cutoff slice,
respectively, built from the induced AdS5 metric. Adding these counterterms to
the bulk action, we see that the holographically renormalized volume of the
unit AdS5 takes the well-known form

$\displaystyle\text{vol}(\text{AdS}_{5})~{}=-\pi^{2}\log\frac{\epsilon_{x}}{2}+\ldots~{}.$
(C.147)

To complete the regularization of the on-shell actions for the vacuum
AdS${}_{7}\times\mathds{S}^{4}$ and two-charge solutions and extract the
universal contributions to the defect free energy, we replace
$\text{vol}(\text{AdS}_{5})~{}=-\pi^{2}\log(\epsilon_{x}/2)$ wherever it
appears.

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