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# Mean-field limits for Consensus-Based Optimization and Sampling

Nicolai Jurek Gerbera, Hausdorff Center for Mathematics, Rheinische Friedrich-
Wilhelms-Universität, Bonn, Germany  Franca Hoffmannb, Department of Computing
and Mathematical Sciences, Caltech, USA  Urbain Vaesc, MATHERIALS project-
team, Inria Paris  CERMICS, École des Ponts, France

(3rd September 2024)

###### Abstract

For algorithms based on interacting particle systems that admit a mean-field
description, convergence analysis is often more accessible at the mean-field
level. In order to transpose convergence results obtained at the mean-field
level to the finite ensemble size setting, it is desirable to show that the
particle dynamics converge in an appropriate sense to the corresponding mean-
field dynamics. In this paper, we prove quantitative mean-field limit results
for two related interacting particle systems: Consensus-Based Optimization and
Consensus-Based Sampling. Our approach extends Sznitman’s classical argument:
in order to circumvent issues related to the lack of global Lipschitz
continuity of the coefficients, we discard an event of small probability, the
contribution of which is controlled using moment estimates for the particle
systems.

Keywords. Mean-field limits, Interacting particle systems, Consensus-Based
Optimization, Consensus-Based Sampling, Coupling methods, Wasserstein
stability estimates.

AMS subject classification. 35Q93, 65C35, 70F45, 35K55.

##  1 Introduction

This work focuses on the rigorous derivation of quantitative mean-field limits
for recently proposed algorithms known as Consensus-Based Optimization (CBO)
and Consensus-Based Sampling (CBS), as well as their variants. The former,
initially proposed in [46], is an algorithm for global optimization tasks,
while the latter, proposed in [11], can be employed for both sampling and
optimization purposes. Both methods are based on an interacting particle
system with only locally, not globally, Lipschitz coefficients and interaction
kernels. This introductory section is organized as follows: we first introduce
Consensus-Based Optimization (Subsection 1.1) and Consensus-Based Sampling
(Subsection 1.2), giving pointers to the relevant literature. In Subsection
1.3 we present a brief review of the literature on the use of coupling methods
to prove propagation of chaos for models with non-globally Lipschitz
coefficients. Finally, we summarize the main contributions of this work in
Subsection 1.4, and introduce the notation used throughout the paper in
Subsection 1.5.

Since Consensus-Based Sampling can be viewed as a variant on Consensus-Based
Optimization, we focus predominantly on the former and then briefly discuss
the latter.

###  1.1 Consensus-Based Optimization (CBO)

Given an objective function $f\colon\mathbf{R}^{d}\rightarrow\mathbf{R}$, we
consider the following version of Consensus-Based Optimization (CBO):

$\displaystyle\mathrm{d}X^{(j)}_{t}$
$\displaystyle=-\left(X^{(j)}_{t}-\mathcal{M}_{\beta}\left(\mu^{J}_{t}\right)\right)\,\mathrm{d}t+S\left(X^{(j)}_{t}-\mathcal{M}_{\beta}\left(\mu^{J}_{t}\right)\right)\,\mathrm{d}W^{(j)}_{t},\qquad
t\geqslant 0,\qquad j\in,$ (1.1)

where $J>1$ is a natural number, $:=\\{1,\dots,J\\}$ and
$\mu^{J}_{t}:=\frac{1}{J}\sum_{i=1}^{J}\delta_{X^{(i)}_{t}}$ denotes the
empirical measure associated with the particle system. For a probability
measure $\mu\in\mathcal{P}(\mathbf{R}^{d})$, the notation
$\mathcal{M}_{\beta}(\mu)$ denotes the mean of $\mu$ after reweighting by
$\operatorname{e}^{-\beta f}$. Its precise definition, together with its
expression when $\mu=\mu^{J}_{t}$, is given by

$\displaystyle\mathcal{M}_{\beta}(\mu):=\frac{\int_{\mathbf{R}^{d}}x\operatorname{e}^{-\beta
f(x)}\,\mu(\mathrm{d}x)}{\int_{\mathbf{R}^{d}}\operatorname{e}^{-\beta
f(x)}\,\mu(\mathrm{d}x)},\qquad\mathcal{M}_{\beta}(\mu^{J}_{t}):=\frac{\sum_{j=1}^{J}X^{(j)}\operatorname{e}^{-\beta
f\left(X^{(j)}_{t}\right)}}{\sum_{k=1}^{J}\operatorname{e}^{-\beta
f\left(X^{(k)}_{t}\right)}}.$ (1.2)

The weighted mean $\mathcal{M}_{\beta}(\mu^{J}_{t})$ may be viewed as an
approximation of the particle position $X^{(j)}_{t}$ such that
$f(X^{(j)}_{t})$ is minimal. In (1.1), the processes
$\bigl{(}W^{(j)}_{t}\bigr{)}_{j\in}$ are independent $\mathbf{R}^{d}$-valued
Brownian motions and $S\colon\mathbf{R}^{d}\to\mathbf{R}^{d\times d}$ is a
globally Lipschitz continuous function. The two diffusion types commonly
studied in the literature are isotropic diffusion [46, 10, 23], with
$S(u)=\sqrt{2\theta}\lvert u\rvert\operatorname{I}_{d}$, and anisotropic or
component-wise [13, 24] diffusion with
$S(u)=\sqrt{2\theta}\operatorname{diag}(u)$. Here
$\operatorname{I}_{d}\in\mathbf{R}^{d\times d}$ is the identity matrix and
$\operatorname{diag}\colon\mathbf{R}^{d}\to\mathbf{R}^{d\times d}$ the
operator mapping a vector onto a diagonal matrix with the vector as its
diagonal. It was shown in [13] that CBO with anisotropic diffusion is well-
suited for high-dimensional global optimization problems, for example those
arising in machine learning. Commonly a constant coefficient is included in
front of the drift term in (1.1), but since this coefficient can be removed by
rescaling time upon redefining the diffusion parameter $\theta$, we choose to
omit it.

By design, CBO allows passage to the corresponding mean-field description.
Indeed, since the particles in (1.1) evolve towards the weighted mean (1.2)
instead of the current best particle, as is the case for Particle Swarm
Optimization (PSO) [37], it is possible to formally write down the
corresponding mean-field equation:

$\left\\{\begin{aligned}
\mathrm{d}\overline{X}_{t}&=-\left(\overline{X}_{t}-\mathcal{M}_{\beta}(\overline{\rho}_{t})\right)\,\mathrm{d}t+S\left(\overline{X}_{t}-\mathcal{M}_{\beta}(\overline{\rho}_{t})\right)\,\mathrm{d}W_{t},\\\
\overline{\rho}_{t}&=\operatorname{Law}(\overline{X}_{t})\,,\end{aligned}\right.$
(1.3)

where the weighted mean $\mathcal{M}_{\beta}(\mu)$ for general measures
$\mu\in\mathcal{P}(\mathbf{R}^{d})$ can be interpreted as a smooth
approximation of the ’current best’ that appears in PSO [29]. The connection
between the particle system (1.1) and the dynamics (1.3), which is of
McKean–Vlasov type and comparatively simpler to analyze mathematically,
stimulated theoretical research using tools from stochastic analysis and
partial differential equations [34, 10, 23]. In particular, consensus
formation and convergence of the CBO mean-field dynamics (1.3) to a consensus
point as $t\to\infty$ has been proven in [10]. The distance between the
consensus point and the true global minimizer can be made arbitrarily small by
choosing a large exponential parameter $\beta$. The proof in [10] covers also
non-convex cost functions $f$.

The convergence, in an appropriate sense, of the empirical measure associated
with the CBO finite-size particle system (1.1) to the solution of the mean-
field equation (1.3) in the large particle limit $J\to\infty$ was first proved
rigorously in a recent paper [34] using a compactness argument, as we discuss
further in Subsection 1.3. However, the method of proof employed in [34] does
not provide an explicit rate of convergence with respect to the number of
particles. Obtaining such an explicit rate of convergence is highly desirable,
as it would enable to transfer convergence results concerning the mean-field
equation (1.3) to the finite ensemble size setting (1.1) by means of a
triangle inequality argument, modulo appropriate interfacing of the metrics.
This is the approach proposed in [23, Theorem 13], where the authors obtain a
partial mean-field result in a set of high probability, enabling them to prove
convergence with high probability to the global minimizer in the finite
ensemble size setting. Here, we provide a stronger version of the mean-field
limit result given in [23, Proposition 15], extending to more general
diffusion terms and providing scope for global convergence results in a
stronger metric than convergence in probability.

A number of extensions of CBO have been proposed in the literature. As
mentioned previously, using an anisotropic diffusion term results in a scheme
which is suitable for high-dimensional state spaces, and has been applied
successfully in the training of artificial neural networks [13]. Using
componentwise geometric Brownian motion in this way allows to prove consensus
formation directly at the particle level, which was achieved in [30, 31, 41].
(On this subject, we point out that the results in [31] on the proximity of
the consensus point to the global minimizer concern only the best case
scenario; a more general convergence result in probability or a different
metric is still lacking.) We also mention adaptations with mini-batching [35],
memory effects [49], and Lévy noise [36], as well as extensions to multi-
objective optimization [40, 7] and global optimization with constraints [21,
22, 12, 8, 32, 4]. Moreover, a variant tailored for objective functions with
several minima was proposed in [9], and results establishing a link between
CBO and PSO in the so-called small inertia limit were obtained in [29, 17].
Many of the recent advances concerning the mathematical analysis of CBO and
its extensions were recently overviewed in [48].

###  1.2 Consensus-Based Sampling (CBS)

Consensus-Based Sampling is a sampling method inspired by the Ensemble Kalman
Sampler (EKS) [26] and by consensus-based approaches such as CBO. It is based
on an interacting particle system with a noise term preconditioned by the
square root of the empirical covariance. The method in the finite ensemble
size and continuous time setting reads

$\mathrm{d}X^{(j)}_{t}=-\Bigl{(}X^{(j)}_{t}-\mathcal{M}_{\beta}\left(\mu^{J}_{t}\right)\Bigr{)}\,\mathrm{d}t+\sqrt{2\lambda^{-1}\mathcal{C}_{\beta}\left(\mu^{J}_{t}\right)}\,\mathrm{d}W^{(j)}_{t},\qquad
j\in,$ (1.4)

with parameters $\beta,\lambda>0$. Here, for a probability measure
$\mu\in\mathcal{P}(\mathbf{R}^{d})$, the notation $\mathcal{C}_{\beta}(\mu)$
denotes the covariance of $\mu$ after reweighting by $\operatorname{e}^{-\beta
f}$; see (1.9) in Subsection 1.5 for the precise definition. It is possible to
formally write down the mean-field equation corresponding to (1.4):

$\displaystyle\left\\{\begin{aligned}
\mathrm{d}\overline{X}_{t}&=-\bigl{(}\overline{X}_{t}-\mathcal{M}_{\beta}\left(\overline{\rho}_{t}\right)\bigr{)}\,\mathrm{d}t+\sqrt{2\lambda^{-1}\mathcal{C}_{\beta}(\overline{\rho}_{t})}\,\mathrm{d}W_{t},\\\
\overline{\rho}_{t}&=\operatorname{Law}(\overline{X}_{t}).\end{aligned}\right.$
(1.5)

When the parameters satisfy $\lambda=(1+\beta)^{-1}$, the interacting particle
system (1.4) enables to generate approximate samples from the probability
distribution with density proportional to $\operatorname{e}^{-f}$, that is
from

$\rho_{f}:=\frac{\operatorname{e}^{-f}}{\mathcal{Z}},\qquad\mathcal{Z}:=\int_{\mathbf{R}^{d}}\operatorname{e}^{-f(x)}\,\mathrm{d}x.$

More precisely, the authors of [11] show that for uniformly convex $f$ and
Gaussian initial data $\overline{\rho}_{0}$, the mean-field distribution
$\overline{\rho}_{t}$ converges exponentially in Wasserstein distance as
$t\to\infty$ to a Gaussian $\mathcal{N}(m_{\infty},C_{\infty})$ that can be
made arbitrarily close to the Laplace approximation of $\rho_{f}$ by choosing
a sufficiently large parameter $\beta$. In addition, it was noticed in [11]
that, with $\lambda:=1$, the dynamics (1.4) behaves as a global optimization
method for finding the global minimum of
$f\colon\mathbf{R}^{d}\rightarrow\mathbf{R}$, just like the Consensus-Based
Optimization method proposed in [46]. Since the convergence analysis in [11]
is confined to the mean-field level, it is of great interest to prove
rigorously that the particle dynamics (1.4) converges in an appropriate sense
to the corresponding mean-field dynamics (1.5) as $J\to\infty$. This has so
far been left as an open problem. Recently, consensus-based sampling was
extended to enable sampling of multi-modal distributions in [9], and for rare
event estimation in [2].

###  1.3 Propagation of chaos

The CBO (1.1) and CBS (1.4) dynamics can both be recast as interacting
particle systems of the form

$\mathrm{d}X^{(j)}_{t}=b\left(X^{(j)}_{t},\mu^{J}_{t}\right)\,\mathrm{d}t+\sigma\left(X^{(j)}_{t},\mu^{J}_{t}\right)\,\mathrm{d}W^{(j)},\qquad
j\in\llbracket 1,J\rrbracket.$ (1.6)

with drift coefficient $b$ and diffusion coefficient $\sigma$ depending on
$\mu$ through a convolution with appropriate kernels $K_{b}$ and $K_{\sigma}$:

$b(x,\mu)=\widetilde{b}(x,K_{b}\star\mu),\qquad\sigma(x,\mu)=\widetilde{\sigma}(x,K_{\sigma}\star\mu).$
(1.7)

Although the consensus-based dynamics can be written in the form (1.6), it is
not possible to do so in such a way that
$\widetilde{b},\widetilde{\sigma},K_{b}$ and $K_{\sigma}$ are all globally
Lipschitz continuous and at the same time $K_{b}$ and $K_{\sigma}$ are
globally bounded. Therefore, the classical McKean’s theorem [15, Theorem 3.1]
is not directly applicable, and a different approach is needed. We briefly
discuss hereafter the existing mean-field results for CBO, noting that no
results have been obtained so far for CBS.

  * •

In [34], propagation of chaos for CBO is proved through a compactness argument
based on Prokhorov’s theorem. However, this method comes without convergence
rates in terms of the number of particles $J$.

  * •

Later, propagation of chaos by means of synchronous coupling was shown with
optimal rates in [21] for a variant of CBO constrained to hypersurfaces.

  * •

Recent results in [23] also show a probabilistic mean-field approximation of
the form

$\sup_{t\in[0,T]}\mathbf{E}\left[\left\lvert
X^{(j)}_{t}-\overline{X}^{(j)}_{t}\right\rvert^{2}\mathsf{1}_{\Omega\setminus\Omega_{M}}\right]\leqslant\frac{C}{J},\qquad\Omega_{M}:=\left\\{\sup_{t\in[0,T]}\frac{1}{J}\sum_{j=1}^{J}\max\left\\{\left\lvert
X^{(j)}_{t}\right\rvert^{4},\left\lvert\overline{X}^{(j)}_{t}\right\rvert^{4}\right\\}\geqslant
M\right\\}.$

It is additionally proved in [23], by application of Markov’s inequality, that
the set of realizations $\Omega_{M}$ has probability close to 0 for large $M$.
Specifically, it holds that $\mathbf{P}(\Omega_{M})=\mathcal{O}(M^{-1})$.

  * •

Shortly before completing this work, we found out that propagation of chaos
for CBO had been proved using a coupling approach very recently in [36]. In
that work, the lack of global Lipschitz continuity of the weighted mean is
handled by using stopping times, similarly to the approach we present in
Subsection 5.1. There are, however, several differences between our results
and those presented in [36]. Apart from the assumptions, which are slightly
less restrictive in our work, the main technical differences are that (i) we
improve existing Wasserstein stability estimates on the weighted mean
(Corollary 3.3), (ii) we rely on a powerful result from the statistics
literature [1, 18] to bound one of the terms in Sznitman’s argument (Lemma
3.7) and (iii) we use a non-asymptotic central limit result to bound the
probability that empirical measures exit a compact set (Subsection 2.3).
Combined together, these ingredients enable to obtain mean-field result with
the usual algebraic rate of convergence with respect to $J$, whereas the error
scales as $\ln(\ln(J))^{-1}$ in the mean-field result obtained in [36].

###  1.4 Our contributions

The main contributions of this paper are the following.

  * •

We obtain quantitative results on the mean-field limit for CBO and CBS. More
precisely, we prove results with an explicit rate of convergence with respect
to the number $J$ of particles. Investigating the roles of dimension and other
problem parameters on the value of the constant prefactor is left for future
work.

  * •

We present two approaches for dealing with the weighted mean (in the case of
CBO) and the weighted covariance (in the case of CBS). One of these approaches
is based on the careful use of indicator sets, whereas the other is based on
stopping times. The latter approach is inspired by a method initially proposed
in [33] in order to handle local Lipschitz continuity issues in the context of
numerical analysis for the Euler–Maruyama method.

  * •

In order to prove mean-field results, we generalize existing theorems and
prove new results on the well-posedness of the particle systems and their
mean-field limit. We also present novel stability estimates for the weighted
mean and the weighted covariance.

The rest of this paper is organized as follows. In Section 2, we present our
main results for the consensus-based dynamics (1.1) and (1.4). They are based
on a number of auxiliary results, which are stated precisely and proved in
Section 3. These include Wasserstein stability estimates for the weighted mean
and the weighted covariance, as well as moment estimates for the particle
systems. In Section 4, we prove well-posedness results for the particle
systems and their mean-field limits. Section 5 is reserved for extensions and
perspectives for future work. Additional technical results are presented in
Appendix A. The interdependence of the sections is illustrated in Figure 1.

Main theorems on the mean-field limits for CBO and CBS. Section 2 Wasserstein
stability estimates for weighted moments. Subsection 3.1 Moment estimates for
empirical measures. Subsection 3.2 Convergence of the weighted moments for
i.i.d. samples. Subsection 3.3 Well-posedness result for the mean-field
equations. Section 4 Technical bound on weighted moments. Subsection A.2

Figure 1: Interdependence of the results contained in this paper.

###  1.5 Notation

We let $\mathbf{N}:=\\{0,1,2,3,\dots\\}$ and
$\mathbf{N}^{+}:=\\{1,2,3,\dots\\}$. For symmetric matrices $X$ and $Y$, the
notation $X\succcurlyeq Y$ means that $X-Y$ is positive semidefinite, and the
notation $X\succ Y$ means that $X-Y$ is strictly positive definite. The
notation $\operatorname{I}_{d}\in\mathbf{R}^{d\times d}$ refers to the
identity matrix. The Euclidean norm in $\mathbf{R}^{d}$ is denoted by
$\lvert\mathord{\color[rgb]{0.67,0.67,0.67}\definecolor[named]{pgfstrokecolor}{rgb}{0.67,0.67,0.67}\pgfsys@color@gray@stroke{0.67}\pgfsys@color<EMAIL_ADDRESS>For a matrix $X\in\mathbf{R}^{d\times d}$, the notation $\left\lVert
X\right\rVert$ refers to the operator norm induced by the Euclidean vector
norm, and the notation $\left\lVert X\right\rVert_{\rm F}$ refers to the
Frobenius norm. The set of probability measures over a metric space $E$ is
denoted by $\mathcal{P}(E)$, and $\mathcal{P}_{p}(E)\subset\mathcal{P}(E)$
denotes the set of probability measures with finite moments up to order $p$,
and $\mathcal{P}_{p,R}(E)\subset\mathcal{P}_{p}(E)$ denotes the set of
probability measures $\mu$ such that

$\left(\int_{E}\lvert
x\rvert^{p}\,\mu(\mathrm{d}x)\right)^{\frac{1}{p}}\leqslant R.$

Note that $\mathcal{P}_{q,R}(E)\subset\mathcal{P}_{p,R}(E)$ for all
$0\leqslant p<q$. If the metric space $E$ is not specified, then it is
implicitly assumed that $E=\mathbf{R}^{d}$. For example, we sometimes write
$\mathcal{P}_{p,R}$ as a shorthand notation for
$\mathcal{P}_{p,R}(\mathbf{R}^{d})$. For a probability measure
$\mu\in\mathcal{P}(\mathbf{R}^{d})$ we use the following notation for the mean
and covariance:

$\displaystyle\mathcal{M}(\mu)=\int_{\mathbf{R}^{d}}x\,\mu(\mathrm{d}x),\qquad\mathcal{C}(\mu)=\int_{\mathbf{R}^{d}}\bigl{(}x-\mathcal{M}(\mu)\bigr{)}\otimes\bigl{(}x-\mathcal{M}(\mu)\bigr{)}\,\mu(\mathrm{d}x).$
(1.8)

The notations $\mathcal{M}_{\beta}(\mu)$ and $\mathcal{C}_{\beta}(\mu)$ denote
respectively the mean and covariance after reweighting by
$\operatorname{e}^{-\beta f}$. More precisely, we use the notation

$\mathcal{M}_{\beta}(\mu)=\mathcal{M}(L_{\beta}\mu),\qquad\mathcal{C}_{\beta}(\mu)=\mathcal{C}(L_{\beta}\mu),\qquad
L_{\beta}\mu:=\frac{\operatorname{e}^{-\beta
f}\,\mu}{\int\operatorname{e}^{-\beta f}\,\mathrm{d}\mu}.$ (1.9)

For a probability measure $\mu\in\mathcal{P}(E)$ and a function $f\colon
E\to\mathbf{R}$, we use the short-hand notation

$\mu[f]=\int_{E}f(x)\,\mu(\mathrm{d}x).$

By a slight abuse of notation, we sometimes write $\mu[f(x)]$ instead of
$\mu[f]$ for convenience. For example, for a probability measure
$\mu\in\mathcal{P}_{2}(\mathbf{R})$, the notation $\mu\left[x^{2}\right]$
refers to the second raw moment of $\mu$. Furthermore, for
$J\in\mathbf{N}^{+}$ we denote by
$\mu^{J}:=\frac{1}{J}\sum_{j=1}^{J}\delta_{X^{(j)}}$ and
$\overline{\mu}^{J}:=\frac{1}{J}\sum_{j=1}^{J}\delta_{\overline{X}^{(j)}}$ the
empirical measures of the particles $X^{(1)},\dots,X^{(J)}$ and
$\overline{X}^{(1)},\dots,\overline{X}^{(J)}$, respectively. Finally,
throughout this paper, the notation $\Omega$ refers to the sample space, and
the notation $C$ refers to a constant whose exact value is irrelevant in the
context and may change from line to line.

##  2 Main results

This section is organized as follows. In Subsection 2.1, we present the
assumptions used throughout this work. Then, after stating well-posedness
results for CBO and CBS in Subsection 2.2, we prove in Subsection 2.3 a key
preparatory result for the mean-field limits. We then prove quantitative mean-
field results for Consensus-Based Optimization and Sampling in Subsections 2.4
and 2.5, respectively.

###  2.1 Main assumptions

We assume throughout this paper that the diffusion operator
$S\colon\mathbf{R}^{d}\rightarrow\mathbf{R}^{d\times d}$ appearing in the CBO
dynamics (1.1) is globally Lipschitz continuous. For the objective function
$f$, we use the following assumption.

######  Assumption H.

We assume the following properties for the target function $f$.

  1. (H1)

The function $f\colon\mathbf{R}^{d}\rightarrow\mathbf{R}$ is bounded from
below by $f_{\star}=\inf f$ and satisfies

$\displaystyle\forall x,y\in\mathbf{R}^{d},\qquad|f(x)-f(y)|\leqslant
L_{f}\bigl{(}1+|x|+|y|\bigr{)}^{s}|x-y|,$ (2.1)

for some constants $L_{f}>0$ and $s\geqslant-1$.

  2. (H2)

There exist constants $c_{\ell},c_{u}>0$ and $u\geqslant\ell\geqslant 0$ such
that

$\displaystyle\forall x\in\mathbf{R}^{d},\qquad f(x)-f_{\star}\leqslant
c_{u}\bigl{(}|x|^{u}+1\bigr{)},$ (2.2a) $\displaystyle\forall
x\in\mathbf{R}^{d},\qquad f(x)-f_{\star}\geqslant
c_{\ell}\bigl{(}|x|^{\ell}-1\bigr{)}.$ (2.2b)

######  Remark 2.1.

A few remarks are in order.

  * •

When $\ell=u=0$, assumption (H2) implies that $f$ is bounded from above.

  * •

Observe that (H1) implies (2.2a) with $u=s+1$.

  * •

In the particular case where $s=1$ and either $u=\ell=0$ or $u=\ell=2$, the
assumptions in Assumption H coincide with those in [10]. The well-posedness
and mean-field results proved in this paper generalize beyond this setting, as
explained below.

In the rest of this paper, we denote by $\mathcal{A}(s,\ell,u)$ the set of
objective functions that satisfy Assumption H with these parameters. Our main
results hold only for $f\in\mathcal{A}(s,\ell,\ell)$, i.e. under Assumption H
with $\ell=u$. Many of the results we present hold only for values of $p$ in a
range dictated by the interval of validity of the Wasserstein stability
estimates presented in Subsection 3.1. In order to avoid repetitions, we
introduce the associated notation

$p_{\mathcal{M}}(s,\ell):=\begin{cases}s+2,\,&\text{if $\ell=0$},\\\
1,\,&\text{if $\ell>0$},\end{cases}\qquad
p_{\mathcal{C}}(s,\ell):=\begin{cases}s+3,\,&\text{if $\ell=0$},\\\
1,\,&\text{if $\ell>0$}.\end{cases}$ (2.3)

###  2.2 Well-posedness for CBO and CBS

In this section, we state well-posedness results for CBO and CBS, referring
for the proofs to Section 4. While for CBO, the following theorems are only
generalizations of already existing results [10], the well-posedness proofs
for the CBS particle and mean-field systems are new.

######  Theorem 2.2 (Existence and uniqueness for the particle models).

Assume that $f\colon\mathbf{R}^{d}\rightarrow\mathbf{R}$ is locally Lipschitz
continuous and fix $J\in\mathbf{N}^{+}$. Then, the stochastic differential
equations (1.1) and (1.4) have unique strong solutions
$\bigl{\\{}{\mathbf{X}}_{t}^{(J)}\,|\,t\geqslant 0\bigr{\\}}$ for any initial
condition ${\mathbf{X}}_{0}^{(J)}$ which is independent of the Brownian
motions $\mathbf{W}_{t}=\bigl{(}W^{(1)}_{t},\dots,W^{(J)}_{t}\bigr{)}$. The
solutions are almost surely continuous.

In the case of CBO, Theorem 2.2 was already proved in [10, Theorem 2.1]. We
now present well-posedness results for the mean-field SDEs, beginning with
CBO. For $f\in\mathcal{A}(1,0,0)\cup A(1,2,2)$, the following result
generalizes [10, Theorems 3.1 and 3.2].

######  Theorem 2.3 (CBO: Existence and uniqueness for the mean-field SDE).

Suppose that $f\in\mathcal{A}(s,\ell,\ell)$ with parameters $s\geqslant-1$,
$\ell\geqslant 0$. Assume also that
$\overline{\rho}_{0}\in\mathcal{P}_{p}(\mathbf{R}^{d})$ for $p\geqslant 2\vee
p_{\mathcal{M}}(s,\ell)$, Then, for each final time $T>0$, there exists a
unique process $\overline{X}\in C([0,T],\mathbf{R}^{d})$ satisfying (1.3) in
the strong sense with initial condition $\overline{X}_{0}=x_{0}$. Furthermore,
it holds that

$\displaystyle\mathbf{E}\left[\sup_{t\in[0,T]}\left\lvert\overline{X}_{t}\right\rvert^{p}\right]<\infty\qquad\text{and}\qquad\sup_{t\in[0,T]}\Bigl{\lvert}\mathcal{M}_{\beta}(\overline{\rho}_{t})\Bigr{\rvert}<\infty,\qquad\overline{\rho}_{t}:=\operatorname{Law}(\overline{X}_{t}),$
(2.4)

and the function $t\mapsto\mathcal{M}_{\beta}(\overline{\rho}_{t})$ is
continuous over $[0,T]$. Finally, the function $t\mapsto\overline{\rho}_{t}$
belongs to $C\left([0,T],\mathcal{P}_{p}(\mathbf{R}^{d})\right)$ and satisfies
the following non-local Fokker–Planck equation in the weak sense:

$\partial_{t}\overline{\rho}=\nabla\cdot\Bigl{(}\bigl{(}x-\mathcal{M}_{\beta}(\overline{\rho})\bigr{)}\overline{\rho}\Bigr{)}+\frac{1}{2}\nabla\cdot\nabla\cdot\Bigl{(}D(\overline{\rho},x)\overline{\rho}\Bigr{)}.$
(2.5)

where
$D(\overline{\rho},x):=S\bigl{(}x-\mathcal{M}_{\beta}(\overline{\rho})\bigr{)}S\bigl{(}x-\mathcal{M}_{\beta}(\overline{\rho})\bigr{)}^{\mathsf{T}}$.

######  Theorem 2.4 (CBS: Existence and uniqueness for the mean-field SDE).

Suppose that $f\in\mathcal{A}(s,\ell,\ell)$ with parameters $s\geqslant-1$,
$\ell\geqslant 0$. Assume also that
$\overline{\rho}_{0}\in\mathcal{P}_{p}(\mathbf{R}^{d})$ for $p\geqslant 2\vee
p_{\mathcal{C}}(s,\ell)$, that $C_{\beta}(\overline{\rho}_{0})\succ 0$, and
fix $x_{0}\sim\overline{\rho}_{0}$. Then, for each final time $T>0$, there
exists a unique process $\overline{X}\in C([0,T],\mathbf{R}^{d})$ satisfying
(1.5) in the strong sense with initial condition $\overline{X}_{0}=x_{0}$.
Furthermore, it holds that

$\mathbf{E}\left[\sup_{t\in[0,T]}\left\lvert\overline{X}_{t}\right\rvert^{p}\right]<\infty,\qquad\sup_{t\in[0,T]}\left\lvert\mathcal{M}_{\beta}(\overline{\rho}_{t})\right\rvert<\infty,\qquad\sup_{t\in[0,T]}\left\lVert\mathcal{C}_{\beta}(\overline{\rho}_{t})\right\rVert<\infty,\qquad\overline{\rho}_{t}:=\operatorname{Law}(\overline{X}_{t}),$
(2.6)

and the functions $t\mapsto\mathcal{M}_{\beta}(\overline{\rho}_{t})$,
$t\mapsto\mathcal{C}_{\beta}(\overline{\rho}_{t})$ are continuous over
$[0,T]$. Finally, the function $t\mapsto\overline{\rho}_{t}$ belongs to
$C([0,T],\mathcal{P}_{p}(\mathbf{R}^{d}))$ and satisfies the following non-
local Fokker–Planck equation in the weak sense:

$\displaystyle\partial_{t}\overline{\rho}=\nabla\cdot\Bigl{(}\bigl{(}x-\mathcal{M}_{\beta}(\overline{\rho})\bigr{)}\overline{\rho}+\lambda^{-1}\mathcal{C}_{\beta}(\overline{\rho})\nabla\overline{\rho}\Bigr{)}.$
(2.7)

###  2.3 Key preparatory lemma

We first show a key preparatory lemma which is needed for the mean-field
results. This result is a simple corollary of a standard non-asymptotic
version of the central limit theorem known as Bikelis’ theorem, see for
example [28, Theorem 3.7].

######  Lemma 2.5 (Bound on the probability of large excursions).

Let $(Z_{j})_{j\in\mathbf{N}^{+}}$ be a family of i.i.d. $\mathbf{R}$-valued
random variables such that $\mathbf{E}\left[\left\lvert
Z_{1}\right\rvert^{3}\right]<\infty$. Then for all
$R>\mathbf{E}\bigl{[}\left\lvert Z_{1}\right\rvert\bigr{]}$, there exists a
constant $C>0$ such that

$\forall
J\in\mathbf{N}^{+},\qquad\mathbf{P}\left[\frac{1}{J}\sum_{j=1}^{J}Z_{j}\geqslant
R\right]\leqslant\frac{C}{J^{2}}.$

###### Proof.

Denote by $m$ and $\sigma^{2}$ the mean and variance of $Z_{j}$, respectively.
Bikelis’ theorem provides the following non-asymptotic error estimate:

$\forall
z\in\mathbf{R},\qquad\bigl{\lvert}\phi_{J}(z)-\phi(z)\bigr{\rvert}\leqslant\frac{1}{\sqrt{J}}\,\left(\frac{\mathbf{E}\left[\lvert
Z_{j}\rvert^{3}\right]}{\sigma^{3}(1+\lvert
z\rvert)^{3}}\right),\qquad\phi_{J}(z):=\mathbf{P}\left[\frac{1}{\sqrt{J}}\sum_{j=1}^{J}\frac{Z_{j}-m}{\sigma}\leqslant
z\right],$

where $\phi$ is the cumulative distribution function of the $\mathcal{N}(0,1)$
Gaussian law. Thus, if $R>m$, we obtain

$\displaystyle\mathbf{P}\left[\frac{1}{J}\sum_{j=1}^{J}Z_{j}\geqslant
R\right]$
$\displaystyle=\mathbf{P}\left[\frac{1}{\sqrt{J}}\sum_{j=1}^{J}\frac{Z_{j}-m}{\sigma}\geqslant\sqrt{J}\left(\frac{R-m}{\sigma}\right)\right]$
$\displaystyle=1-\phi_{J}\left(\sqrt{J}k\right),\qquad k:=\frac{R-m}{\sigma}$
$\displaystyle\leqslant
1-\phi\left(\sqrt{J}k\right)+\frac{1}{\sqrt{J}}\,\left(\frac{\mathbf{E}\left[\lvert
Z_{j}\rvert^{3}\right]}{\sigma^{3}(1+\sqrt{J}k)^{3}}\right)=\mathcal{O}(J^{-2}),$

which completes the proof. ∎

###  2.4 Quantitative mean-field limit for CBO

We now present and prove the mean-field results, starting with CBO.
Synchronous coupling for McKean-Vlasov diffusions with drift
$b\colon\mathbf{R}^{d}\times\mathcal{P}(\mathbf{R}^{d})\rightarrow\mathbf{R}^{d}$
and diffusion
$\sigma\colon\mathbf{R}^{d}\times\mathcal{P}(\mathbf{R}^{d})\rightarrow\mathbf{R}^{d\times
d}$ assumes the following setting [14, 15]. Given i.i.d. Brownian motions
$\bigl{(}W^{(j)}\bigr{)}_{j\in\mathbf{N}^{+}}$ and initial conditions
$\bigl{(}X^{(j)}_{0}\bigr{)}_{j\in\mathbf{N}^{+}}$ sampled i.i.d. from some
$\overline{\rho}_{0}\in\mathcal{P}(\mathbf{R}^{d})$, we consider for each
$J\in\mathbf{N}^{+}$ the interacting particle system

$\displaystyle\forall j\in,\qquad
X^{(j)}_{t}=X^{(j)}_{0}+\int_{0}^{t}b\left(X^{(j)}_{s},\mu^{J}_{s}\right)\,\mathrm{d}s+\int_{0}^{t}\sigma\left(X^{(j)}_{s},\mu^{J}_{s}\right)\mathrm{d}W^{(j)}_{s},$
(2.8)

to which we couple the system of i.i.d. mean-field McKean-Vlasov diffusions

$\displaystyle\forall N\in\mathbf{N}^{+},\forall
j\in,\qquad\left\\{\begin{aligned}
\overline{X}^{(j)}_{t}&=X^{(j)}_{0}+\int_{0}^{t}b\left(\overline{X}^{(j)}_{s},\overline{\rho}_{s}\right)\,\mathrm{d}s+\int_{0}^{t}\sigma\left(\overline{X}^{(j)}_{s},\overline{\rho}_{s}\right)\,\mathrm{d}W^{(j)}_{s},\\\
\overline{\rho}_{t}&=\operatorname{Law}\left(\overline{X}^{(j)}_{t}\right),\end{aligned}\right.$
(2.9)

driven by the same Brownian motions and initial conditions as (2.8). In the
case of CBO, the drift and diffusion coefficients for (2.8) and (2.9) are
given by

$\displaystyle
b(x,\mu):=-\bigl{(}x-\mathcal{M}_{\beta}\left(\mu\right)\bigr{)}$ and
$\displaystyle\sigma(x,\mu):=S\bigl{(}x-\mathcal{M}_{\beta}\left(\mu\right)\bigr{)},$
(2.10)

where the diffusion operator
$S\colon\mathbf{R}^{d}\rightarrow\mathbf{R}^{d\times d}$ is, as always in this
work, a globally Lipschitz continuous function.

######  Theorem 2.6 (Mean-field limit for CBO).

Suppose that $f\in\mathcal{A}(s,\ell,\ell)$ with $s,\ell\geqslant 0$ and that
$\overline{\rho}_{0}\in\mathcal{P}_{q}(\mathbf{R}^{d})$, and consider the
systems (2.8) and (2.9) with the coefficients given by (2.10). Then for all
$p\geqslant 2$ such that

$\frac{q}{3}\geqslant p\geqslant p_{\mathcal{M}}(s,\ell),$ (2.11)

where $p_{\mathcal{M}}$ is given in (2.3), there is $C>0$ independent of $J$
such that

$\forall J\in\mathbf{N}^{+},\qquad\forall
j\in,\qquad\mathbf{E}\left[\sup_{t\in[0,T]}\Big{\lvert}X^{(j)}_{t}-\overline{X}^{(j)}_{t}\Big{\rvert}^{p}\right]\leqslant
CJ^{-\min\left\\{\frac{p}{2},2-\frac{2p}{q}\right\\}}.$

######  Remark 2.7.

In particular, we obtain a decay bound of order
$\mathcal{O}\left(\frac{1}{J}\right)$ if $p=2$. If $p\geqslant 4$, then for
each $\varepsilon>0$, decay of order
$\mathcal{O}\left(\frac{1}{J^{2-\varepsilon}}\right)$ holds provided that
$\overline{\rho}_{0}\in\mathcal{P}_{q}(^{d})$ with
$q\geqslant\frac{2p}{\varepsilon}\vee 3p$.

######  Remark 2.8.

Theorem 2.6 implies pathwise Wasserstein-$p$ chaos. More precisely, it holds
that

$W_{p}\left(\rho^{J}_{[0,T]},\overline{\rho}^{\otimes
J}_{[0,T]}\right)\leqslant
CJ^{-\min\left\\{\frac{1}{2},\frac{2}{p}-\frac{2}{q}\right\\}}.$

where $\rho^{J}_{[0,T]}$ is the pathwise law of the microscopic particle
system over $[0,T]$ and $\overline{\rho}_{[0,T]}$ is the pathwise mean-field
law over $[0,T]$. That is to say,

$\left\\{\begin{aligned}
\rho^{J}_{[0,T]}&:=\operatorname{Law}\left(X^{(1)},\dots
X^{(J)}\right)\in\mathcal{P}\Bigl{(}C\bigl{(}[0,T],\mathbf{R}^{d}\bigr{)}^{J}\Bigr{)},\\\
\overline{\rho}_{[0,T]}&:=\operatorname{Law}(\overline{X})\in\mathcal{P}\Bigl{(}C\bigl{(}[0,T],\mathbf{R}^{d}\bigr{)}\Bigr{)}.\end{aligned}\right.$

Here, $W_{p}$ denotes the pathwise Wasserstein-$p$ distance, associated with
the following metric on
$\Bigl{(}C\bigl{(}[0,T],\mathbf{R}^{d}\bigr{)}\Bigr{)}^{J}$:

$d(\mathbf{X},\mathbf{Y})=\sup_{t\in[0,T]}\frac{1}{J}\sum_{j=1}^{J}\left\lvert
X^{(j)}_{t}-Y^{(j)}_{t}\right\rvert,\qquad\mathbf{X},\mathbf{Y}\in\Bigl{(}C\bigl{(}[0,T],\mathbf{R}^{d}\bigr{)}\Bigr{)}^{J}.$

See [14, Definitions 3.1 and 3.5] for more details.

Before presenting the proof, we summarize below a number of useful auxiliary
results which are valid for $f\in\mathcal{A}(s,\ell,\ell)$. These results are
stated precisely and proved in Section 3.

  1. (a)

Wasserstein stability estimate for the weighted mean. We prove in Corollary
3.3 that, for all $R>0$ and for all $p\geqslant p_{\mathcal{M}}(s,\ell)$,
where $p_{\mathcal{M}}(s,\ell)\in^{+}$ is given in (2.3), there exists
$\widetilde{L}_{\mathcal{M}}=\widetilde{L}_{\mathcal{M}}(R,p)>0$ such that

$\displaystyle\forall(\mu,\nu)\in\mathcal{P}_{p,R}\bigl{(}^{d}\bigr{)}\times\mathcal{P}_{p}\bigl{(}^{d}\bigr{)},\qquad\left\lvert\mathcal{M}_{\beta}(\mu)-\mathcal{M}_{\beta}(\nu)\right\rvert\leqslant\widetilde{L}_{\mathcal{M}}\,W_{p}(\mu,\nu).$

  2. (b)

Moment estimate for the particle system. We prove in Lemma 3.5 that, if the
initial condition satisfies $\overline{\rho}_{0}\in\mathcal{P}_{q}(^{d})$,
then there exists a constant $\kappa>0$ independent of $J$ such that

$\forall J\in\mathbf{N}^{+},\qquad\mathbf{E}\left[\sup_{t\in[0,T]}\left\lvert
X_{t}^{(j)}\right\rvert^{q}\right]\quad\vee\quad\mathbf{E}\left[\sup_{t\in[0,T]}\left\lvert\mathcal{M}_{\beta}(\mu^{J}_{t})\right\rvert^{q}\right]\leqslant\kappa.$

  3. (c)

Convergence of the weighted mean for i.i.d. samples. We prove in Lemma 3.7
that, for any reals $r>p>0$ and any $\mu\in\mathcal{P}_{r}(\mathbf{R}^{d})$,
there is $C$ such that

$\mathbf{E}\left\lvert\mathcal{M}_{\beta}(\overline{\mu}^{J})-\mathcal{M}_{\beta}(\mu)\right\rvert^{p}\leqslant
CJ^{-\frac{p}{2}},\qquad\overline{\mu}^{J}:=\frac{1}{J}\sum_{j=1}^{J}\delta_{\overline{X}^{(j)}},\qquad\left\\{\overline{X}^{(j)}\right\\}_{j\in\mathbf{N}}\stackrel{{\scriptstyle\rm{i.i.d.}}}{{\sim}}\mu.$

###### Proof of Theorem 2.6.

Fix $j\in$. As in Sznitman’s argument [15, Theorem 3.1], we begin by writing

$\displaystyle\bigl{\lvert}X_{t}^{(j)}-\overline{X}_{t}^{(j)}\bigr{\rvert}^{p}\leqslant$
$\displaystyle\,2^{p-1}\left\lvert\int_{0}^{t}b\left(X_{s}^{(j)},\mu^{J}_{s}\right)-b\left(\overline{X}_{s}^{(j)},\overline{\rho}_{s}\right)\,\mathrm{d}s\right\rvert^{p}$
$\displaystyle+2^{p-1}\left\lvert\int_{0}^{t}\sigma\left(X_{s}^{(j)},\mu^{J}_{s}\right)-\sigma\left(\overline{X}_{s}^{(j)},\overline{\rho}_{s}\right)\,\mathrm{d}W_{s}\right\rvert^{p}.$

By the Burkholder–Davis–Gundy inequality, see [44, Theorem 7.3], it holds for
any $t\in[0,T]$ that

$\displaystyle\mathbf{E}\left[\sup_{s\in[0,t]}\bigl{\lvert}X_{s}^{(j)}-\overline{X}_{s}^{(j)}\bigr{\rvert}^{p}\right]$
$\displaystyle\leqslant 2^{p-1}\mathbf{E}\left(\int_{0}^{t}\left\lvert
b\left(X_{s}^{(j)},\mu^{J}_{s}\right)-b\left(\overline{X}_{s}^{(j)},\overline{\rho}_{s}\right)\right\rvert\,\mathrm{d}s\right)^{p}$
$\displaystyle\qquad+C_{\rm
BDG}2^{p-1}\mathbf{E}\left(\int_{0}^{t}\left\lVert\sigma\left(X_{s}^{(j)},\mu^{J}_{s}\right)-\sigma\left(\overline{X}_{s}^{(j)},\overline{\rho}_{s}\right)\right\rVert_{\rm
F}^{2}\,\mathrm{d}s\right)^{\frac{p}{2}}$
$\displaystyle\leqslant(2T)^{p-1}\mathbf{E}\int_{0}^{t}\left\lvert
b\left(X_{s}^{(j)},\mu^{J}_{s}\right)-b\left(\overline{X}_{s}^{(j)},\overline{\rho}_{s}\right)\right\rvert^{p}\,\mathrm{d}s$
$\displaystyle\qquad+\,C_{\rm
BDG}2^{p-1}T^{\frac{p}{2}-1}\mathbf{E}\int_{0}^{t}\left\lVert\sigma\left(X_{s}^{(j)},\mu^{J}_{s}\right)-\sigma\left(\overline{X}_{s}^{(j)},\overline{\rho}_{s}\right)\right\rVert_{\rm
F}^{p}\,\mathrm{d}s.$ (2.12)

By the triangle inequality, the first expectation on the right-hand side can
be bounded as follows:

$\displaystyle\mathbf{E}\int_{0}^{t}\left\lvert
b\left(X_{s}^{(j)},\mu^{J}_{s}\right)-b\left(\overline{X}_{s}^{(j)},\overline{\rho}_{s}\right)\right\rvert^{p}\,\mathrm{d}s$
$\displaystyle\leqslant 2^{p-1}\mathbf{E}\int_{0}^{t}\left\lvert
b\left(X_{s}^{(j)},\mu^{J}_{s}\right)-b\left(\overline{X}_{s}^{(j)},\overline{\mu}^{J}_{s}\right)\,\right\rvert^{p}\mathrm{d}s$
$\displaystyle\qquad+2^{p-1}\mathbf{E}\int_{0}^{t}\left\lvert
b\left(\overline{X}_{s}^{(j)},\overline{\mu}^{J}_{s}\right)-b\left(\overline{X}_{s}^{(j)},\overline{\rho}_{s}\right)\right\rvert^{p}\,\mathrm{d}s.$
(2.13)

#### Bounding the first term in (2.13)

From (2.10) we have

$\displaystyle\mathbf{E}\left\lvert
b\left(X_{s}^{(j)},\mu^{J}_{s}\right)-b\left(\overline{X}_{s}^{(j)},\overline{\mu}^{J}_{s}\right)\right\rvert^{p}$
$\displaystyle=\mathbf{E}\left\lvert\Bigl{(}\overline{X}^{(j)}_{s}-X^{(j)}_{s}+\mathcal{M}_{\beta}\left(\mu^{J}_{s}\right)-\mathcal{M}_{\beta}\left(\overline{\mu}^{J}_{s}\right)\Bigr{)}\right\rvert^{p}$
$\displaystyle\leqslant 2^{p-1}\mathbf{E}\left\lvert
X^{(j)}_{s}-\overline{X}^{(j)}_{s}\right\rvert^{p}+2^{p-1}\mathbf{E}\left\lvert\mathcal{M}_{\beta}\left(\mu^{J}_{s}\right)-\mathcal{M}_{\beta}\left(\overline{\mu}^{J}_{s}\right)\right\rvert^{p}.$
(2.14)

In order to deal with the fact that the weighted mean does not have global
Lipschitz properties in the sense of McKean’s framework [15, Theorem 3.1], we
introduce for each $t\in[0,T]$ and $J\in\mathbf{N}^{+}$ the set

$\Omega_{J,t}:=\left\\{\omega\in\Omega:\frac{1}{J}\sum_{j=1}^{J}\left\lvert\overline{X}^{(j)}_{t}(\omega)\right\rvert^{p}\geqslant
R\right\\},$ (2.15)

where $R>\mathbf{E}[Z_{j}]$ is fixed. Define

$Z_{j}=\sup_{t\in[0,T]}\left\lvert\overline{X}^{(j)}_{t}\right\rvert^{p}$

Since $q\geqslant 3p$, Theorem 2.3 implies that $\mathbf{E}\left[\lvert
Z_{j}\rvert^{3}\right]<\infty.$ Therefore, by Lemma 2.5 there exists a
constant $C>0$ such that

$\displaystyle\forall t\in[0,T],\qquad\forall
J\in\mathbf{N}^{+},\qquad\mathbf{P}\left[\Omega_{J,t}\right]\leqslant\mathbf{P}\left[\frac{1}{J}\sum_{j=1}^{J}Z_{j}\geqslant
R\right]\leqslant\frac{C}{J^{2}}.$

We can split the second term on the right-hand of (2.14) as follows:

$\displaystyle\mathbf{E}\left\lvert\mathcal{M}_{\beta}\left(\mu^{J}_{s}\right)-\mathcal{M}_{\beta}\left(\overline{\mu}^{J}_{s}\right)\right\rvert^{p}$
$\displaystyle=\mathbf{E}\left[\left\lvert\mathcal{M}_{\beta}\left(\mu^{J}_{s}\right)-\mathcal{M}_{\beta}\left(\overline{\mu}^{J}_{s}\right)\right\rvert^{p}\mathsf{1}_{\Omega\setminus\Omega_{J,s}}\right]$
$\displaystyle\qquad+\mathbf{E}\left[\left\lvert\mathcal{M}_{\beta}\left(\mu^{J}_{s}\right)-\mathcal{M}_{\beta}\left(\overline{\mu}^{J}_{s}\right)\right\rvert^{p}\mathsf{1}_{\Omega_{J,s}}\right].$
(2.16)

Let us now bound each term on the right-hand side of 2.16 separately.

  * •

Since the $p$-th moment of $\overline{\mu}_{s}^{J}$ is bounded from above by
$R$ for all $\omega\in\Omega\setminus\Omega_{J,s}$, we can use the stability
estimate on the weighted mean given (see Item (a) and Corollary 3.3) to obtain

$\mathbf{E}\left[\left\lvert\mathcal{M}_{\beta}\left(\mu^{J}_{s}\right)-\mathcal{M}_{\beta}\left(\overline{\mu}^{J}_{s}\right)\right\rvert^{p}\mathsf{1}_{\Omega\setminus\Omega_{J,s}}\right]\leqslant
C\mathbf{E}\left[W_{p}\left(\mu^{J}_{s},\overline{\mu}^{J}_{s}\right)^{p}\right]$

for the first term in (2.16). Since in addition

$\mathbf{E}\left[W_{p}\left(\mu^{J}_{s},\overline{\mu}^{J}_{s}\right)^{p}\right]\leqslant\mathbf{E}\left[\frac{1}{J}\sum_{j=1}^{J}\left\lvert
X_{s}^{(j)}-\overline{X}^{(j)}_{s}\right\rvert^{p}\right]=\mathbf{E}\left\lvert
X^{(j)}_{s}-\overline{X}^{(j)}_{s}\right\rvert^{p},$

it follows that

$\mathbf{E}\left[\left\lvert\mathcal{M}_{\beta}\left(\mu^{J}_{s}\right)-\mathcal{M}_{\beta}\left(\overline{\mu}^{J}_{s}\right)\right\rvert^{p}\mathsf{1}_{\Omega\setminus\Omega_{J,s}}\right]\leqslant
C\mathbf{E}\left\lvert X^{(j)}_{s}-\overline{X}^{(j)}_{s}\right\rvert^{p}.$

  * •

We now bound the second term in (2.16). From Item (b) (see Lemma 3.5) and
(2.4) we have

$\mathbf{E}\left[\left\lvert\mathcal{M}_{\beta}\left(\mu^{J}_{s}\right)-\mathcal{M}_{\beta}\left(\overline{\mu}^{J}_{s}\right)\right\rvert^{q}\right]\leqslant
2^{q-1}\left(\mathbf{E}\left\lvert\mathcal{M}_{\beta}\left(\mu^{J}_{s}\right)\right\rvert^{q}+\mathbf{E}\left\lvert\mathcal{M}_{\beta}\left(\overline{\mu}^{J}_{s}\right)\right\rvert^{q}\right)\leqslant
2^{q}\kappa.$

Therefore, we obtain by Hölder’s inequality and Lemma 2.5 that

$\displaystyle\mathbf{E}\left[\left\lvert\mathcal{M}_{\beta}\left(\mu^{J}_{s}\right)-\mathcal{M}_{\beta}\left(\overline{\mu}^{J}_{s}\right)\right\rvert^{p}\mathsf{1}_{\Omega_{J,s}}\right]\leqslant\left(\mathbf{E}\left[\left\lvert\mathcal{M}_{\beta}\left(\mu^{J}_{s}\right)-\mathcal{M}_{\beta}\left(\overline{\mu}^{J}_{s}\right)\right\rvert^{q}\right]\right)^{\frac{p}{q}}\left(\mathbf{P}\left[\Omega_{J,s}\right]\right)^{{}^{\frac{q-p}{q}}}\leqslant
2^{p}\kappa^{\frac{p}{q}}J^{-\frac{2(q-p)}{q}}.$

We have thus shown that

$\mathbf{E}\left\lvert\mathcal{M}_{\beta}\left(\mu^{J}_{s}\right)-\mathcal{M}_{\beta}\left(\overline{\mu}^{J}_{s}\right)\right\rvert^{p}\leqslant
C\mathbf{E}\left\lvert
X^{(j)}_{s}-\overline{X}^{(j)}_{s}\right\rvert^{p}+CJ^{-\frac{2(q-p)}{q}},$

which in turn by (2.14) implies that

$\mathbf{E}\left\lvert
b\left(X_{s}^{(j)},\mu^{J}_{s}\right)-b\left(\overline{X}_{s}^{(j)},\overline{\mu}^{J}_{s}\right)\right\rvert^{p}\leqslant
C\mathbf{E}\left\lvert
X^{(j)}_{s}-\overline{X}^{(j)}_{s}\right\rvert^{p}+CJ^{-\frac{2(q-p)}{q}}.$

#### Bounding the second term in (2.13)

For the second term on the right-hand side of (2.13), we have by Item (c) (see
Lemma 3.7) that

$\mathbf{E}\left\lvert
b\left(\overline{X}_{s}^{(j)},\overline{\mu}^{J}_{s}\right)-b\left(\overline{X}_{s}^{(j)},\overline{\rho}_{s}\right)\right\rvert^{p}=\mathbf{E}\left\lvert\mathcal{M}_{\beta}\left(\overline{\mu}^{J}_{s}\right)-\mathcal{M}_{\beta}\left(\overline{\rho}_{s}\right)\right\rvert^{p}\leqslant
CJ^{-\frac{p}{2}}.$

#### Conclusion

Combining the bounds on both terms in (2.13), we deduce

$\mathbf{E}\left[\int_{0}^{t}\left\lvert
b\left(X_{s}^{(j)},\mu^{J}_{s}\right)-b\left(\overline{X}_{s}^{(j)},\overline{\rho}_{s}\right)\right\rvert^{p}\,\mathrm{d}s\right]\leqslant
CJ^{-\min\left\\{\frac{p}{2},\frac{2(q-p)}{p}\right\\}}+C\int_{0}^{t}\mathbf{E}\left[\sup_{u\in[0,s]}\left\lvert
X^{(j)}_{u}-\overline{X}^{(j)}_{u}\right\rvert^{p}\right]\mathrm{d}s.$

Using that $S\colon\mathbf{R}\rightarrow\mathbf{R}^{d\times d}$ is globally
Lipschitz continuous, we obtain analogously

$\mathbf{E}\left[\int_{0}^{t}\left\lVert\sigma\left(X_{s}^{(j)},\mu^{J}_{s}\right)-\sigma\left(\overline{X}_{s}^{(j)},\overline{\rho}_{s}\right)\right\rVert^{p}\,\mathrm{d}s\right]\leqslant
CJ^{-\min\left\\{\frac{p}{2},\frac{2(q-p)}{p}\right\\}}+C\int_{0}^{t}\mathbf{E}\left[\sup_{u\in[0,s]}\left\lvert
X^{(j)}_{u}-\overline{X}^{(j)}_{u}\right\rvert^{p}\right]\mathrm{d}s.$

Substituting in (2.12), we finally obtain

$\displaystyle\mathbf{E}\left[\sup_{s\in[0,t]}\bigl{\lvert}X_{s}^{(j)}-\overline{X}_{s}^{(j)}\bigr{\rvert}^{p}\right]$
$\displaystyle\leqslant
CJ^{-\min\left\\{\frac{p}{2},\frac{2(q-p)}{p}\right\\}}+C\int_{0}^{t}\mathbf{E}\left[\sup_{u\in[0,s]}\left\lvert
X^{(j)}_{u}-\overline{X}^{(j)}_{u}\right\rvert^{p}\right]\mathrm{d}s.$

The conclusion then follows from Grönwall’s lemma. ∎

######  Remark 2.9.

The proof is similar to that of McKean’s theorem [15, Theorem 3.1], with an
important difference: we eliminate a set of small probability in order to
apply the local stability estimate in Item (a) on its complement.

######  Remark 2.10.

A crucial detail in the proof is that Item (a) only assumes that
$(\mu,\nu)\in\mathcal{P}_{p,R}\times\mathcal{P}_{p}$ and not
$(\mu,\nu)\in\mathcal{P}_{p,R}\times\mathcal{P}_{p,R}$; see Corollary 3.3
below. This allows us to define $\Omega_{J,t}$ in (2.15) only in terms of the
i.i.d. mean-field particles $\overline{X}^{(j)}_{t}$ for which Lemma 2.5 can
be applied; the event $\frac{1}{J}\sum_{j=1}^{J}\left\lvert
X^{(j)}_{t}\right\rvert^{p}\geqslant R$ need not be included in the definition
of $\Omega_{J,t}$ .

######  Remark 2.11.

In his famous lecture notes [47], Sznitman shows for a laboratory example that

$\forall
j\in,\qquad\mathbf{E}\left[\sup_{t\in[0,T]}\Big{\lvert}X^{(j)}_{t}-\overline{X}^{(j)}_{t}\Big{\rvert}^{p}\right]\leqslant
CJ^{-\frac{p}{2}}.$

More precisely, this statement is proved for $p=1$, and the author notes that
the result generalizes to higher moment, which can be achieved by using, among
other technical adaptations of his argument, the Marcinkiewicz–Zygmund
inequality [16]. Although CBO does not fit within Sznitman’s simplified
framework, Item (c) enables to show a bound with the same rate for the second
term on the right-hand side of (2.13),

$\mathbf{E}\biggl{[}\left\lvert
b\left(\overline{X}_{s}^{(j)},\overline{\mu}^{J}_{s}\right)-b\left(\overline{X}_{s}^{(j)},\overline{\rho}_{s}\right)\right\rvert^{p}\mathsf{1}_{\Omega\setminus\Omega_{J,t}}\biggr{]}\leqslant\frac{C}{J^{\frac{p}{2}}}.$

The limiting factor, for the overall rate of convergence in Theorem 2.6, is
the bound on the probability $\mathbf{P}[\Omega_{J,t}]$.

###  2.5 Quantitative mean-field limit for CBS

Using the same reasoning as that in the proof of Theorem 2.6, we can prove a
mean-field result for the CBS dynamics (1.4). We state the theorem and, given
the similarity of the proof with that of Theorem 2.6, only list the key
ingredients needed for the proof.

######  Theorem 2.12 (Mean-field limit for CBS).

Suppose that $f\in\mathcal{A}(s,\ell,\ell)$ with $s\geqslant-1$,
$\ell\geqslant 0$ and that
$\overline{\rho}_{0}\in\mathcal{P}_{q}(\mathbf{R}^{d})$ with
$\mathcal{C}(\overline{\rho}_{0})\succ 0$, and consider the systems (2.8) and
(2.9) with the coefficients given by

$\displaystyle
b(x,\mu):=-\bigl{(}x-\mathcal{M}_{\beta}\left(\mu\right)\bigr{)}$ and
$\displaystyle\sigma(x,\mu):=\sqrt{2\lambda^{-1}\mathcal{C}_{\beta}(\mu)}.$
(2.17)

Then for all $p\geqslant 2$ such that

$\frac{q}{3}\geqslant p\geqslant
2p_{\mathcal{M}}(s,\ell)=\begin{cases}2s+4,\,&\text{if $\ell=0$},\\\
2,\,&\text{if $\ell>0$},\end{cases}$

there is $C>0$ independent of $J$ such that

$\forall J\in\mathbf{N}^{+},\qquad\forall
j\in,\qquad\mathbf{E}\left[\sup_{t\in[0,T]}\left\lvert
X^{(j)}_{t}-\overline{X}^{(j)}_{t}\right\rvert^{p}\right]\leqslant
CJ^{-\min\left\\{\frac{p}{2},2-\frac{2p}{q}\right\\}}.$

We summarize below the additional auxiliary results necessary for the proof,
which are valid for an objective function $f\in\mathcal{A}(s,\ell,\ell)$.
These results are stated precisely and proved in Section 3.

  * •

Wasserstein stability estimate for the weighted covariance. We prove in
Corollary 3.3 that, for all $R>0$ and for all $p\geqslant
2p_{\mathcal{M}}(\ell,s)$, there exists
$\widetilde{L}_{\sqrt{\mathcal{C}}}(R,p)>0$ such that

$\displaystyle\forall(\mu,\nu)\in\mathcal{P}_{p,R}\bigl{(}^{d}\bigr{)}\times\mathcal{P}_{p}\bigl{(}^{d}\bigr{)},\qquad\left\lVert\sqrt{\mathcal{C}_{\beta}(\mu)}-\sqrt{\mathcal{C}_{\beta}(\nu)}\right\rVert\leqslant\widetilde{L}_{\sqrt{\mathcal{C}}}\,W_{p}(\mu,\nu).$

  * •

Moment estimate for the particle system. We prove in Lemma 3.5 that, under the
assumptions of Theorem 2.12, there exists a constant $\kappa>0$ independent of
$J$ such that for all $J\in\mathbf{N}^{+},\qquad$

$\mathbf{E}\left[\sup_{t\in[0,T]}\left\lvert
X_{t}^{(j)}\right\rvert^{q}\right]\quad\vee\quad\mathbf{E}\left[\sup_{t\in[0,T]}\left\lvert\mathcal{M}_{\beta}(\mu^{J}_{t})\right\rvert^{q}\right]\quad\vee\quad\mathbf{E}\left[\sup_{t\in[0,T]}\left\lvert\sqrt{\mathcal{C}_{\beta}(\mu^{J}_{t})}\right\rvert^{q}\right]\leqslant\kappa.$

  * •

Convergence of the square root of the weighted covariance for i.i.d. samples.
We prove in Lemma 3.8 that, for any reals $r>2p>0$ and any
$\mu\in\mathcal{P}_{r}(\mathbf{R}^{d})$ such that
$\mathcal{C}_{\beta}(\mu)\succ 0$, then there is $C$ such that

$\mathbf{E}\left\lVert\sqrt{\mathcal{C}_{\beta}(\overline{\mu}^{J})}-\sqrt{\mathcal{C}_{\beta}(\mu)}\right\rVert^{p}\leqslant
CJ^{-\frac{p}{2}},\qquad\overline{\mu}^{J}:=\frac{1}{J}\sum_{j=1}^{J}\delta_{\overline{X}^{(j)}},\qquad\left\\{\overline{X}^{(j)}\right\\}_{j\in\mathbf{N}}\stackrel{{\scriptstyle\rm{i.i.d.}}}{{\sim}}\mu.$

  * •

No collapse in finite time. In order to apply the third item with
$\mu=\overline{\rho}_{t}$, we prove in Proposition 3.9 that the assumption
$\mathcal{C}(\overline{\rho}_{0})\succ 0$ implies that

$\eta:=\inf_{t\in[0,T]}\lambda_{\text{min}}\bigl{(}\mathcal{C}_{\beta}(\overline{\rho}_{t})\bigr{)}>0.$

##  3 Auxiliary results

This section is organized as follows. In Subsection 3.1, we improve on
existing Wasserstein stability estimates for the weighted mean and prove novel
ones for the weighted covariance and the square root of the weighted
covariance. In Subsection 3.2, we present moment estimates for the empirical
measures associated with the CBO (1.1) and CBS (1.4) dynamics. In Subsection
3.3, we prove convergence estimates for the weighted moments when particle
positions are drawn independently from a given probability distribution, using
results from the statistics literature. Finally, in Subsection 3.4, we prove
that if $\mathcal{C}_{\beta}(\overline{\rho}_{0})\succ 0$, then the covariance
matrix $\mathcal{C}_{\beta}(\overline{\rho}_{t})\succ 0$ associated with the
solution to the mean-field CBS dynamics (1.5) remains strictly positive
definite for all times.

###  3.1 Wasserstein stability estimates for the weighted moments

In this subsection, we generalize [10, Lemma 3.2] to the more general
conditions in Assumption H and, more importantly, to the weighted covariance
and its square root.

######  Proposition 3.1 (Basic stability estimates for the weighted mean and
the weighted covariance).

Suppose that $f\in\mathcal{A}(s,\ell,u)$ with $s\geqslant-1$ and
$u\geqslant\ell\geqslant 0$. Then:

  1. (a)

For all $R>0$ and for all $p\geqslant p_{\mathcal{M}}(s,\ell)$, where
$p_{\mathcal{M}}(s,\ell)$ is given in (2.3), there exists
$L_{\mathcal{M}}=L_{\mathcal{M}}(p,R)>0$ such that

$\forall(\mu,\nu)\in\mathcal{P}_{p,R}\bigl{(}^{d}\bigr{)}\times\mathcal{P}_{p,R}\bigl{(}^{d}\bigr{)},\qquad\lvert\mathcal{M}_{\beta}(\mu)-\mathcal{M}_{\beta}(\nu)\rvert\leqslant
L_{\mathcal{M}}\,W_{p}(\mu,\nu).$ (3.1)

  2. (b)

For all $R>0$ and for all $p\geqslant p_{\mathcal{C}}(s,\ell)$, where
$p_{\mathcal{C}}(s,\ell)$ is given in (2.3), there exists
$L_{\mathcal{C}}=L_{\mathcal{C}}(p,R)>0$ such that

$\forall(\mu,\nu)\in\mathcal{P}_{p,R}\bigl{(}^{d}\bigr{)}\times\mathcal{P}_{p,R}\bigl{(}^{d}\bigr{)},\qquad\Bigl{\lVert}\mathcal{C}_{\beta}(\mu)-\mathcal{C}_{\beta}(\nu)\Bigr{\rVert}_{\rm
F}\leqslant L_{\mathcal{C}}\,W_{p}(\mu,\nu).$ (3.2)

###### Proof.

We prove first (a) and then (b). Both proofs rely on the technical result
Lemma A.1 in the appendix.

#### Proof of (a)

In the notation of Lemma A.1 we can write

$\mathcal{M}_{\beta}(\mu)=\frac{\mu[g]}{\mu[h]},\qquad
g(x):=\operatorname{e}^{-\beta f(x)}x,\qquad h(x):=\operatorname{e}^{-\beta
f(x)}.$

Condition A.1 is satisfied for $\varsigma:=s+1$ if $\ell=0$ or $\varsigma:=0$
if $\ell>0$. Therefore, the claim follows from Lemma A.1.

#### Proof of (b)

First observe that

$\displaystyle\mathcal{C}_{\beta}(\mu)=\frac{\mu[g]}{\mu[h]}-\mathcal{M}_{\beta}(\mu)\otimes\mathcal{M}_{\beta}(\mu),\qquad
g(x):=\operatorname{e}^{-\beta f(x)}(x\otimes x),\qquad
h(x):=\operatorname{e}^{-\beta f(x)}.$

By the triangle inequality, we have that

$\bigl{\lVert}\mathcal{C}_{\beta}(\mu)-\mathcal{C}_{\beta}(\nu)\bigr{\rVert}_{\rm
F}\leqslant\Bigl{\lVert}\mathcal{M}_{\beta}(\mu)\otimes\mathcal{M}_{\beta}(\mu)-\mathcal{M}_{\beta}(\nu)\otimes\mathcal{M}_{\beta}(\nu)\Bigr{\rVert}_{\rm
F}+\left\lVert\frac{\mu[g]}{\mu[h]}-\frac{\nu[g]}{\nu[h]}\right\rVert_{\rm
F}\,.$ (3.3)

We shall use that for all $(x,y)\in\mathbf{R}^{d}\times\mathbf{R}^{d}$, it
holds that

$\displaystyle\left\lVert x\otimes x-y\otimes y\right\rVert_{\rm F}$
$\displaystyle=\left\lVert
x(x-y)^{\mathsf{T}}+(x-y)y^{\mathsf{T}}\right\rVert_{\rm F}$
$\displaystyle\leqslant\left\lVert x(x-y)^{\mathsf{T}}\right\rVert_{\rm
F}+\left\lVert(x-y)y^{\mathsf{T}}\right\rVert_{\rm F}=\lvert
x\rvert\cdot\lvert x-y\rvert+\lvert y\rvert\cdot\lvert
x-y\rvert=\bigl{(}\lvert x\rvert+\lvert y\rvert\bigr{)}\lvert x-y\rvert.$
(3.4)

Using this inequality, that $p_{\mathcal{C}}\geqslant p_{\mathcal{M}}$ and
(a), we obtain for the first term in (3.3) that

$\displaystyle\Bigl{\lVert}\mathcal{M}_{\beta}(\mu)\otimes\mathcal{M}_{\beta}(\mu)-\mathcal{M}_{\beta}(\nu)\otimes\mathcal{M}_{\beta}(\nu)\Bigr{\rVert}_{\rm
F}$
$\displaystyle\leqslant\Bigl{(}\left\lvert\mathcal{M}_{\beta}(\mu)\right\rvert+\left\lvert\mathcal{M}_{\beta}(\nu)\right\rvert\Bigr{)}\cdot\bigl{\lvert}\mathcal{M}_{\beta}(\mu)-\mathcal{M}_{\beta}(\nu)\bigr{\rvert}$
$\displaystyle\leqslant
L_{\mathcal{M}}\,\Bigl{(}\left\lvert\mathcal{M}_{\beta}(\mu)\right\rvert+\left\lvert\mathcal{M}_{\beta}(\nu)\right\rvert\Bigr{)}\,W_{p}(\mu,\nu)$
$\displaystyle\leqslant
L_{\mathcal{M}}^{2}\,\Bigl{(}W_{p}(\mu,\delta_{0})+W_{p}(\nu,\delta_{0})\Bigr{)}\,W_{p}(\mu,\nu)\leqslant
2L_{\mathcal{M}}^{2}R\,W_{p}(\mu,\nu).$

In order to bound the second term in (3.3), we again use (3.4) to deduce that
the function $g$ satisfies the local Lipschitz estimate

$\forall
x,y\in\mathbf{R}^{d},\qquad\bigl{\lvert}g(x)-g(y)\bigr{\rvert}\leqslant
L\bigl{(}1+|x|+|y|\bigr{)}^{\varsigma}\lvert
x-y\rvert,\qquad\varsigma:=\begin{cases}s+2\quad&\text{ if\leavevmode\nobreak\
$\ell=0$},\\\ 0\quad&\text{ if\leavevmode\nobreak\ $\ell>0$}.\end{cases}$

We deduce that the functions $g$ and $h$ satisfy the assumption (A.1) of Lemma
A.1 with this $\varsigma$. Therefore, it holds that

$\displaystyle\forall(\mu,\nu)\in\mathcal{P}_{p,R}\bigl{(}^{d}\bigr{)}\times\mathcal{P}_{p,R}\bigl{(}^{d}\bigr{)},\qquad\left\lVert\frac{\mu[\operatorname{e}^{-\beta
f(x)}x\otimes x]}{\mu[\operatorname{e}^{-\beta
f(x)}]}-\frac{\nu[\operatorname{e}^{-\beta f(x)}x\otimes
x]}{\nu[\operatorname{e}^{-\beta f(x)}]}\right\rVert_{\rm F}\leqslant
CW_{p}(\mu,\nu),$

which completes the proof. ∎

######  Proposition 3.2 (Basic stability estimate for the square root
covariance).

Suppose that $f\in\mathcal{A}(s,\ell,u)$ with $s\geqslant 0$ and
$u\geqslant\ell\geqslant 0$. Then for all $R>0$ and all $p\geqslant
2p_{\mathcal{M}}(s,\ell)$, there exists
$L_{\sqrt{\mathcal{C}}}=L_{\sqrt{\mathcal{C}}}(p,R)>0$ such that

$\displaystyle\forall(\mu,\nu)\in\mathcal{P}_{p,R}\bigl{(}^{d}\bigr{)}\times\mathcal{P}_{p,R}\bigl{(}^{d}\bigr{)},\qquad\left\lVert\sqrt{\mathcal{C}_{\beta}(\mu)}-\sqrt{\mathcal{C}_{\beta}(\nu)}\right\rVert_{\rm
F}\leqslant L_{\sqrt{\mathcal{C}}}\,W_{p}(\mu,\nu).$ (3.5)

###### Proof.

It is sufficient to check the claim for measures $\mu$ and $\nu$ of the form

$\mu^{J}=\frac{1}{J}\sum_{j=1}^{J}\delta_{X^{(j)}},\qquad\nu^{J}=\frac{1}{J}\sum_{j=1}^{J}\delta_{Y^{(j)}},\qquad
J\in\mathbf{N}^{+}.$ (3.6)

Indeed, assume the claim holds for all such pairs of probability measures, fix
$R>0$ and take any pair of probability measures
$(\mu,\nu)\in\mathcal{P}_{p,R}\times\mathcal{P}_{p,R}$. By [51, Theorem 6.18],
there exists a sequence
$\bigl{\\{}(\mu^{J},\nu^{J})\bigr{\\}}_{J\in\mathbf{N}^{+}}$ in
$\mathcal{P}_{p}\times\mathcal{P}_{p}$ such that $W_{p}(\mu^{J},\mu)\to 0$ and
$W_{p}(\nu^{J},\nu)\to 0$ in the limit as $J\to\infty$. Since convergence in
$W_{p}$ implies convergence of the moments of order up to $p$ by [51, Theorem
6.9], we can assume, discarding the beginning of the sequence if necessary,
that $(\mu^{J},\nu^{J})\in\mathcal{P}_{p,2R}$ for all $J$. Then

$\displaystyle\left\lVert\sqrt{\mathcal{C}_{\beta}(\mu})-\sqrt{\mathcal{C}_{\beta}(\nu)}\right\rVert_{\rm
F}\leqslant$
$\displaystyle\left\lVert\sqrt{\mathcal{C}_{\beta}(\mu^{J})}-\sqrt{\mathcal{C}_{\beta}(\nu^{J})}\right\rVert_{\rm
F}$
$\displaystyle\qquad+\left\lVert\sqrt{\mathcal{C}_{\beta}(\nu)}-\sqrt{\mathcal{C}_{\beta}(\nu^{J})}\right\rVert_{\rm
F}+\left\lVert\sqrt{\mathcal{C}_{\beta}(\mu)}-\sqrt{\mathcal{C}_{\beta}(\mu^{J})}\right\rVert_{\rm
F}.$

The first term is bounded from above by $CW_{p}(\mu^{J},\nu^{J})$ by the base
case, while the other two terms converge to 0 in the limit as $J\to\infty$ by
Proposition 3.1. Taking the limit $J\to\infty$, we deduce that

$\left\lVert\sqrt{\mathcal{C}_{\beta}(\mu)}-\sqrt{\mathcal{C}_{\beta}(\nu)}\right\rVert_{\rm
F}\leqslant CW_{p}(\mu,\nu).$

#### Proof of the statement for empirical measures

Fix $R>0$. By [52, p.5], the Wasserstein distance between empirical measures
$\mu^{J}$ and $\nu^{J}$ of the form (3.6) is equal to

$W_{p}(\mu^{J},\nu^{J})=\min_{\sigma\in\mathcal{S}_{J}}\left(\frac{1}{J}\sum_{j=1}^{J}\left\lvert
X^{(j)}-Y^{\sigma(j)}\right\rvert^{p}\right)^{\frac{1}{p}},$

where $\mathcal{S}_{J}$ denotes the set of permutations in $\\{1,\dotsc,J\\}$.
Thus, the claim will follow if we can prove that, for any pair
$(\mu^{J},\nu^{J})\in\mathcal{P}_{p,R}$ of the form (3.6), it holds that

$\left\lVert\sqrt{\mathcal{C}_{\beta}(\mu^{J})}-\sqrt{\mathcal{C}_{\beta}(\nu^{J})}\right\rVert_{\rm
F}\leqslant C\,\left(\frac{1}{J}\sum_{j=1}^{J}\left\lvert
X^{(j)}-Y^{(j)}\right\rvert^{p}\right)^{\frac{1}{p}}.$ (3.7)

We henceforth drop the superscript $J$ in $\mu^{J},\nu^{J}$ for simplicity of
notation, and write $\mathbf{X}=\left(X^{(1)},\dotsc,X^{(J)}\right)$ and
$\mathbf{Y}=\left(Y^{(1)},\dotsc,Y^{(J)}\right)$. First note that

$\mathcal{C}_{\beta}(\mu)=\sum_{j=1}^{J}w_{j}(\mathbf{X})\Bigl{(}X^{(j)}-\mathcal{M}_{\beta}(\mu)\Bigr{)}\otimes\Bigl{(}X^{(j)}-\mathcal{M}_{\beta}(\mu)\Bigr{)},\qquad
w_{j}(\mathbf{X}):=\frac{\operatorname{e}^{-\beta
f(X^{(j)})}}{\sum_{k=1}^{J}\operatorname{e}^{-\beta f(X^{(k)})}}.$

It is easily seen that
$\mathcal{C}_{\beta}(\mu)=M_{\mathbf{X}}M_{\mathbf{X}}^{\mathsf{T}}$, where
$M_{\mathbf{X}}$ is the following $\mathbf{R}^{d\times J}$ matrix

$M_{\mathbf{X}}:=\begin{pmatrix}\sqrt{w_{1}(\mathbf{X})}\left(X^{(1)}-\mathcal{M}_{\beta}(\mu)\right)&\ldots&\sqrt{w_{J}(\mathbf{X})}\left(X^{(J)}-\mathcal{M}_{\beta}(\mu)\right)\end{pmatrix}.$

Proceeding in the same manner, we construct a matrix
$M_{\mathbf{Y}}\in\mathbf{R}^{d\times J}$ such that
$\mathcal{M}_{\beta}(\nu)=M_{\mathbf{Y}}M_{\mathbf{Y}}^{\mathsf{T}}$. A result
by Araki and Yamagami [3], later generalized by Kittaneh [39] and Bhatia [6],
states for any two matrices $A$ and $B$ with the same shape, it holds that

$\left\lVert\sqrt{A^{\mathsf{T}}A}-\sqrt{B^{\mathsf{T}}B}\right\rVert_{\rm
F}\leqslant\sqrt{2}\left\lVert A-B\right\rVert_{\rm F},$

see also [5, Theorem VII.5.7] for a textbook presentation. This yields

$\displaystyle\left\lVert\sqrt{\mathcal{C}_{\beta}(\mu)}-\sqrt{\mathcal{C}_{\beta}(\nu)}\right\rVert_{\rm
F}$
$\displaystyle\qquad\leqslant\sqrt{2}\left(\sum_{j=1}^{J}\left\lvert\sqrt{w_{j}(\mathbf{X})}\left(X^{(j)}-\mathcal{M}_{\beta}(\mu)\right)-\sqrt{w_{j}(\mathbf{Y})}\left(Y^{(j)}-\mathcal{M}_{\beta}(\nu)\right)\right\rvert^{2}\right)^{\frac{1}{2}}$
$\displaystyle\qquad\leqslant\sqrt{2}\left(\sum_{j=1}^{J}\left\lvert\sqrt{w_{j}(\mathbf{X})}X^{(j)}-\sqrt{w_{j}(\mathbf{Y})}Y^{(j)}\right\rvert^{2}\right)^{\frac{1}{2}}+\sqrt{2}\left(\sum_{j=1}^{J}\left\lvert\sqrt{w_{j}(\mathbf{X})}\mathcal{M}_{\beta}(\mu)-\sqrt{w_{j}(\mathbf{Y})}\mathcal{M}_{\beta}(\nu)\right\rvert^{2}\right)^{\frac{1}{2}},$

where we applied the triangle inequality for the Frobenius norm. Let
$\pi\in\Pi(\mu,\nu)\subset\mathcal{P}(\mathbf{R}^{d}\times\mathbf{R}^{d})$
denote the probability measure

$\pi=\frac{1}{J}\sum_{j=1}^{J}\delta_{\left(X^{(j)},Y^{(j)}\right)}$

and set $\gamma=\frac{\beta}{2}$. We may rewrite the previous inequality as

$\displaystyle\frac{1}{\sqrt{2}}\left\lVert\sqrt{\mathcal{C}_{\beta}(\mu)}-\sqrt{\mathcal{C}_{\beta}(\nu)}\right\rVert_{\rm
F}$
$\displaystyle\leqslant\left(\iint_{\mathbf{R}^{d}\times\mathbf{R}^{d}}\left\lvert\frac{\operatorname{e}^{-\gamma
f(x)}x}{\sqrt{\mu\left[\operatorname{e}^{-\beta
f}\right]}}-\frac{\operatorname{e}^{-\gamma
f(y)}y}{\sqrt{\nu\left[\operatorname{e}^{-\beta
f}\right]}}\right\rvert^{2}\,\pi(\mathrm{d}x\,\mathrm{d}y)\right)^{\frac{1}{2}}$
$\displaystyle\qquad+\left(\iint_{\mathbf{R}^{d}\times\mathbf{R}^{d}}\left\lvert\frac{\operatorname{e}^{-\gamma
f(x)}\mathcal{M}_{\beta}(\mu)}{\sqrt{\mu\left[\operatorname{e}^{-\beta
f}\right]}}-\frac{\operatorname{e}^{-\gamma
f(y)}\mathcal{M}_{\beta}(\nu)}{\sqrt{\nu\left[\operatorname{e}^{-\beta
f}\right]}}\right\rvert^{2}\,\pi(\mathrm{d}x\,\mathrm{d}y)\right)^{\frac{1}{2}}.$

The conclusion then follows from Lemma A.2 in the appendix:

  * •

For the first term we let $g(x)=\operatorname{e}^{-\gamma f(x)}x$,
$h(x)=\operatorname{e}^{-\beta f(x)}$, and $\mathcal{S}(\mu)=1$. The
conditions of Lemma A.2 are satisfied for $\varsigma:=s+1$ if $\ell=0$ and
$\varsigma:=0$ if $\ell>0$.

  * •

For the second term let $g(x)=\operatorname{e}^{-\gamma f(x)}$,
$h(x)=\operatorname{e}^{-\beta f(x)}$ and
$\mathcal{S}(\mu)=\mathcal{M}_{\beta}(\mu)$. The conditions of Lemma A.2 are
satisfied for $\varsigma:=s$ if $\ell=0$ and $\varsigma:=0$ if $\ell>0$.

This completes the proof. ∎

The key for our mean-field proofs is the following improvement of Propositions
3.1 and 3.2. Recall that

$\displaystyle
p_{\mathcal{M}}:=p_{\mathcal{M}}(s,\ell):=\begin{cases}s+2,\,&\text{if
$\ell=0$},\\\ 1,\,&\text{if $\ell>0$}.\end{cases}$

######  Corollary 3.3 (Improved stability estimates).

Suppose that $f\in\mathcal{A}(s,\ell,\ell)$ with $s,\ell\geqslant 0$. Then

  1. (a)

For all $R>0$ and for all $p\geqslant p_{\mathcal{M}}$, there exists
$\widetilde{L}_{\mathcal{M}}=\widetilde{L}_{\mathcal{M}}(R,p)>0$ such that

$\displaystyle\forall(\mu,\nu)\in\mathcal{P}_{p,R}(^{d})\times\mathcal{P}_{p}(^{d}),\qquad\left\lvert\mathcal{M}_{\beta}(\mu)-\mathcal{M}_{\beta}(\nu)\right\rvert\leqslant\widetilde{L}_{\mathcal{M}}\,W_{p}(\mu,\nu).$

  2. (b)

For all $R>0$ and $p\geqslant 2p_{\mathcal{M}}$, there exists
$\widetilde{L}_{\mathcal{C}}=\widetilde{L}_{\mathcal{C}}(R,p)>0$ such that

$\displaystyle\forall(\mu,\nu)\in\mathcal{P}_{p,R}(^{d})\times\mathcal{P}_{p}(^{d}),\qquad\Bigl{\lVert}\sqrt{\mathcal{C}_{\beta}(\mu)}-\sqrt{\mathcal{C}_{\beta}(\nu)}\Bigr{\rVert}_{\rm
F}\leqslant\widetilde{L}_{\sqrt{\mathcal{C}}}\,W_{p}(\mu,\nu).$

######  Remark 3.4.

Observe that in contrast to Propositions 3.1 and 3.2, no upper bounds on the
$p$-th moments of $\nu$ are needed in Corollary 3.3, it is only required that
they are finite.

###### Proof.

Let $\mathcal{T}:=\mathcal{M}_{\beta}$ or
$\mathcal{T}:=\sqrt{\mathcal{C}_{\beta}}$, and denote
$p_{\mathcal{T}}=p_{\mathcal{M}}$ if $\mathcal{T}=\mathcal{M}_{\beta}$ or
$p_{\mathcal{T}}=2p_{\mathcal{M}}$ if
$\mathcal{T}=\sqrt{\mathcal{C}_{\beta}}$. Note that the following statements
hold in either case.

  * •

For all $R>0$ and $p\geqslant p_{\mathcal{T}}$, there exists
$L_{\mathcal{T}}(R)>0$ such that

$\forall(\mu,\nu)\in\mathcal{P}_{p,R}(^{d})\times\mathcal{P}_{p,R}(^{d}),\qquad\bigl{\lVert}\mathcal{T}(\mu)-\mathcal{T}(\nu)\bigr{\rVert}\leqslant
L_{\mathcal{T}}(R)\,W_{p}(\mu,\nu).$ (3.8)

Here, $\|\cdot\|$ denotes either $|\cdot|$ or $\|\cdot\|_{\rm F}$. Equation
(3.8) follows from Proposition 3.1 if $\mathcal{T}=\mathcal{M}_{\beta}$ and
Proposition 3.2 if $\mathcal{T}=\sqrt{\mathcal{C}_{\beta}}$.

  * •

For all $p\geqslant p_{\mathcal{T}}$, there is $K>0$ such that

$\forall\mu\in\mathcal{P}_{p}(\mathbf{R}^{d}),\qquad\bigl{\lVert}\mathcal{T}(\mu)\bigr{\rVert}\leqslant
KW_{p}(\mu,\delta_{0}).$ (3.9)

When $\mathcal{T}=\mathcal{M}_{\beta}$, this follows immediately from the
inequality $\operatorname{e}^{-\beta f^{*}}\leqslant\operatorname{e}^{-\beta
f(\mathord{\color[rgb]{0.67,0.67,0.67}\definecolor[named]{pgfstrokecolor}{rgb}{0.67,0.67,0.67}\pgfsys@color@gray@stroke{0.67}\pgfsys@color@gray@fill{0.67}\bullet})}\leqslant\operatorname{e}^{-\beta
f_{*}}$ if $\ell=0$ and from Proposition A.3 if $\ell>0$, where $f^{*}$ the
supremum of $f$ over d. When $\mathcal{T}=\sqrt{\mathcal{C}_{\beta}}$, the
inequality (3.9) follows in the same way after noticing that

$\displaystyle\Bigl{\lVert}\sqrt{\mathcal{C}_{\beta}(\mu)}\Bigr{\rVert}_{\rm
F}^{2}=\operatorname*{tr}\Bigl{(}\mathcal{C}_{\beta}\left(\mu\right)\Bigr{)}=\frac{\int_{\mathbf{R}^{d}}\lvert
x\rvert^{2}\operatorname{e}^{-\beta
f(x)}\,\mu(\mathrm{d}x)}{\int_{\mathbf{R}^{d}}\operatorname{e}^{-\beta
f(x)}\,\mu(\mathrm{d}x)}-\Bigl{\lvert}\mathcal{M}_{\beta}(\mu)\Bigr{\rvert}^{2}\leqslant\frac{\int_{\mathbf{R}^{d}}\lvert
x\rvert^{2}\operatorname{e}^{-\beta
f(x)}\,\mu(\mathrm{d}x)}{\int_{\mathbf{R}^{d}}\operatorname{e}^{-\beta
f(x)}\,\mu(\mathrm{d}x)}$ (3.10)

for all $\mu\in\mathcal{P}_{p}(\mathbf{R}^{d})$.

Now, assume for contradiction that there exists $R>0$ and
$\\{(\mu_{i},\nu_{i})\\}_{i\in\mathbf{N}}$ in
$\mathcal{P}_{p,R}(^{d})\times\mathcal{P}_{p}$ such that

$\displaystyle\forall
i\in\mathbf{N},\qquad\frac{\Bigl{\lVert}\mathcal{T}(\mu_{i})-\mathcal{T}(\nu_{i})\Bigr{\rVert}}{W_{p}\left(\mu_{i},\nu_{i}\right)}\geqslant
i.$ (3.11)

In particular $\nu_{i}[\lvert x\rvert^{p}]\to\infty$ in the limit as
$i\to\infty$, otherwise the left-hand side of 3.11 would be bounded from above
uniformly in $i$ by (3.8). Therefore $W_{p}(\nu_{i},\delta_{0})\to\infty$. On
the other hand, by the triangle inequality,

$\displaystyle W_{p}\left(\mu_{i},\nu_{i}\right)\geqslant
W_{p}\left(\nu_{i},\delta_{0}\right)-W_{p}\left(\mu_{i},\delta_{0}\right)\geqslant
W_{p}\left(\nu_{i},\delta_{0}\right)-R.$

Therefore, by (3.9), we deduce that

$\displaystyle\limsup_{i\to\infty}\frac{\Bigl{\lVert}\mathcal{T}(\mu_{i})-\mathcal{T}(\nu_{i})\Bigr{\rVert}}{W_{p}\left(\mu_{i},\nu_{i}\right)}\leqslant
K\limsup_{i\to\infty}\frac{W_{p}(\nu_{i},\delta_{0})+R}{W_{p}\left(\nu_{i},\delta_{0}\right)-R}=K,$

which contradicts (3.11). ∎

###  3.2 Moment estimates for the empirical measures

######  Lemma 3.5.

Let $f\in\mathcal{A}(s,\ell,\ell)$ with $s,\ell\geqslant 0$, and suppose that
$\overline{\rho}_{0}\in\mathcal{P}_{p}(\mathbf{R}^{d})$ for some $p\geqslant
2$.

  1. (a)

Consider the particle system (1.1) with $\overline{\rho}_{0}^{\otimes
J}$-distributed initial data, and let $\mu^{J}_{t}$ denote the corresponding
empirical measures. Then there exists a constant $\kappa>0$ independent of $J$
such that for all $J\in\mathbf{N}^{+}$

$\displaystyle\mathbf{E}\left[\sup_{t\in[0,T]}\left\lvert
X_{t}^{(j)}\right\rvert^{p}\right]\quad\vee\quad\mathbf{E}\left[\sup_{t\in[0,T]}\mu^{J}_{t}\bigl{[}|x|^{p}\bigr{]}\right]\quad\vee\quad\mathbf{E}\left[\sup_{t\in[0,T]}\left\lvert\mathcal{M}_{\beta}(\mu^{J}_{t})\right\rvert^{p}\right]\leqslant\kappa.$

  2. (b)

Consider the particle system (1.4) with $\overline{\rho}_{0}^{\otimes
J}$-distributed initial data, and let $\mu^{J}_{t}$ denote the corresponding
empirical measures. Then there exists a constant $\kappa>0$ independent of $J$
such that for all $J\in\mathbf{N}^{+}$

$\displaystyle\mathbf{E}\left[\sup_{t\in[0,T]}\left\lvert
X_{t}^{(j)}\right\rvert^{p}\right]\vee\mathbf{E}\left[\sup_{t\in[0,T]}\mu^{J}_{t}\bigl{[}|x|^{p}\bigr{]}\right]\vee\mathbf{E}\left[\sup_{t\in[0,T]}\left\lvert\mathcal{M}_{\beta}(\mu^{J}_{t})\right\rvert^{p}\right]\vee\mathbf{E}\left[\sup_{t\in[0,T]}\left\lVert\sqrt{\mathcal{C}_{\beta}(\mu^{J}_{t})}\right\rVert^{p}\right]\leqslant\kappa.$

###### Proof.

The result can be proved using the same reasoning as in [10, Lemma 3.4] or
[23, Lemma 15]. For convenience, we denote by $C$ any constant depending only
on $(f,\beta,\lambda,p,T)$. We first observe by (3.10) and Proposition A.3
that

$\forall\mu\in\mathcal{P}_{p}(\mathbf{R}^{d}),\qquad\Bigl{\lvert}\mathcal{M}_{\beta}(\mu)\Bigr{\rvert}\vee\Bigl{\lVert}\sqrt{\mathcal{C}_{\beta}(\mu)}\Bigr{\rVert}_{\rm
F}\leqslant C\left(\int_{\mathbf{R}^{d}}\lvert
x\rvert^{p}\,\mathrm{d}\mu\right)^{\frac{1}{p}}.$ (3.12)

Writing the drift and diffusion coefficients for (a) and (b) as in (2.10) and
(2.17), respectively, it holds in both cases that

$\displaystyle\forall
x\in\mathbf{R}^{d},\qquad\forall\mu\in\mathcal{P}(\mathbf{R}^{d}),\qquad\bigl{\lvert}b(x,\mu)\bigr{\rvert}\vee\bigl{\lVert}\sigma(x,\mu)\bigr{\rVert}_{\rm
F}\leqslant C\left(\lvert
x\rvert^{p}+\int_{\mathbf{R}^{d}}|y|^{p}\,\mu(\mathrm{d}y)\right)^{\frac{1}{p}}.$

Arguing in parallel for (a) and (b), we fix $j\in$ and apply the
Burkholder–Davis–Gundy inequality to find for all $t\in[0,T]$ that

$\displaystyle\frac{1}{3^{p-1}}\mathbf{E}\left[\sup_{s\in[0,t]}\left\lvert
X_{s}^{(j)}\right\rvert^{p}\right]$
$\displaystyle\leqslant\mathbf{E}\left\lvert
X_{0}^{(j)}\right\rvert^{p}+T^{p-1}\int_{0}^{t}\mathbf{E}\left\lvert
b\left(X_{s}^{(j)},\mu^{J}_{s}\right)\right\rvert^{p}\,\mathrm{d}s$
$\displaystyle\qquad\qquad+C_{\rm
BDG}T^{\frac{p}{2}-1}\int_{0}^{t}\mathbf{E}\left\lVert\sigma\left(X_{s}^{(j)},\mu^{J}_{s}\right)\right\rVert_{\rm
F}^{p}\,\mathrm{d}s$ $\displaystyle\leqslant\mathbf{E}\left\lvert
X_{0}^{(j)}\right\rvert^{p}+C\int_{0}^{t}\mathbf{E}\left[\left\lvert
X_{s}^{(j)}\right\rvert^{p}+\int_{\mathbf{R}^{d}}|y|^{p}\,\mu^{J}_{s}(\mathrm{d}y)\right]\,\mathrm{d}s.$

Since the common law of $\left(X^{(J)},\dots,X^{(J)}\right)$ is invariant
under permutations of $\llbracket 1,J\rrbracket$, its marginals corresponding
to the particles are all the same. Therefore, given that
$\mu^{J}_{s}=\frac{1}{J}\sum_{j=1}^{J}\delta_{X^{(j)}_{s}}$, we find that

$\mathbf{E}\left[\int_{\mathbf{R}^{d}}|y|^{p}\,\mu^{J}_{s}(\mathrm{d}y)\right]=\mathbf{E}\left\lvert
X^{(j)}_{s}\right\rvert^{p}.$

Consequently, it holds that

$\displaystyle\mathbf{E}\left[\sup_{s\in[0,t]}\lvert
X_{s}^{(j)}\rvert^{p}\right]$ $\displaystyle\leqslant C\mathbf{E}\left\lvert
X_{0}^{(j)}\right\rvert^{p}+C\int_{0}^{t}\mathbf{E}\left[\sup_{u\in[0,s]}\left\lvert
X_{u}^{(j)}\right\rvert^{p}\right]\,\mathrm{d}s.$

Grönwall’s inequality then gives that

$\mathbf{E}\left[\sup_{t\in[0,T]}\bigl{\lvert}X_{t}^{(j)}\bigr{\rvert}^{p}\right]\leqslant\kappa.$

The second bound in (a) and in (b) follows from the inequality

$\mathbf{E}\left[\sup_{t\in[0,T]}\mu^{J}_{t}\bigl{[}\left\lvert
x\right\rvert^{p}\bigr{]}\right]=\mathbf{E}\left[\sup_{t\in[0,T]}\frac{1}{J}\sum_{j=1}^{J}\bigl{\lvert}X^{(j)}_{t}\bigr{\rvert}^{p}\right]\leqslant\mathbf{E}\left[\frac{1}{J}\sum_{j=1}^{J}\sup_{t\in[0,T]}\bigl{\lvert}X^{(j)}_{t}\bigr{\rvert}^{p}\right]=\mathbf{E}\left[\sup_{t\in[0,T]}\bigl{\lvert}X^{(j)}_{t}\bigr{\rvert}^{p}\right],$

while the third bound in (a) as well as the third and fourth bound in (b)
follow from (3.12). ∎

###  3.3 Convergence of weighted moments for i.i.d. samples

For the proof of the lemmas in this section, we will need [18, Theorem 1],
which we recall for the reader’s convenience in Theorem 3.6 below, slightly
adapted to the i.i.d. setting. We also refer to [1, Theorem 2.3], where
similar results for $p=1$ and $p=2$ are proved in the context of importance
sampling, with explicit constant prefactors.

######  Theorem 3.6 ($L^{p}$ convergence of weighted sums).

Let $(w_{j},V_{j})_{j\in\mathbf{N}}$ be i.i.d. random variables with values in
$\mathbf{R}\times\mathbf{R}^{d}$ or $\mathbf{R}\times\mathbf{R}^{d\times d}$
with $w_{j}>0$ almost surely, and set

$\widehat{N}_{J}=\frac{1}{J}\sum_{j=1}^{J}w_{j}V_{j},\qquad\widehat{D}_{J}=\frac{1}{J}\sum_{j=1}^{J}w_{j},\qquad\widehat{R}_{J}=\frac{\widehat{N}_{J}}{\widehat{D}_{J}}.$

Let also $N=\mathbf{E}\left[\widehat{N}_{J}\right]$,
$D=\mathbf{E}\left[\widehat{D}_{J}\right]$, and $R=N/D$. Let $0<p<q$ and
assume that for some $c>0$,

$\left\lVert w_{1}\right\rVert_{L^{q}(\Omega)}\leqslant c,\qquad\left\lVert
V_{1}\right\rVert_{L^{r}(\Omega)}\leqslant c,\qquad\left\lVert
w_{1}V_{1}\right\rVert_{L^{s}(\Omega)}\leqslant c,\qquad
r:=\frac{p(q+2)}{q-p},\qquad s:=\frac{pq}{q-p}.$

Then there is $C$ depending only on $(c,p,q,r,s)$ such that

$\bigl{\lVert}\widehat{R}_{J}-R\bigr{\rVert}_{{L^{p}(\Omega)}}\leqslant
CJ^{-\frac{1}{2}}.$

Using Theorem 3.6, we now prove the convergence of the weighted mean and
weighted covariance for empirical measures formed from i.i.d. samples, in the
limit of infinitely many samples.

######  Lemma 3.7 (Convergence of the weighted mean for i.i.d. samples).

Suppose that $f\colon^{d}\to$ is bounded from below and let $0<p<r$. Then for
all $\mu\in\mathcal{P}_{r}(\mathbf{R}^{d})$, there is $C$ depending only on
$(f,\beta,p,r)$ and the $r$-th moment of $\mu$ such that for all
$J\in\mathbf{N}$

$\mathbf{E}\left\lvert\mathcal{M}_{\beta}\left(\overline{\mu}^{J}\right)-\mathcal{M}_{\beta}\bigl{(}\mu\bigr{)}\right\rvert^{p}\leqslant
CJ^{-\frac{p}{2}},\qquad\overline{\mu}^{J}:=\frac{1}{J}\sum_{j=1}^{J}\delta_{\overline{X}^{(j)}},\qquad\left\\{\overline{X}^{(j)}\right\\}_{j\in\mathbf{N}}\stackrel{{\scriptstyle\rm{i.i.d.}}}{{\sim}}\mu.$

###### Proof of Lemma 3.7.

We apply Theorem 3.6 with $q=\frac{p(r+2)}{r-p}$, in which case
$\frac{p(q+2)}{q-p}=r.$ In our setting

$(w_{j},V_{j})=\left(\operatorname{e}^{-\beta
f\left(\overline{X}^{(j)}\right)},\overline{X}^{(j)}\operatorname{e}^{-\beta
f\left(\overline{X}^{(j)}\right)}\right).$

Since $w_{j}$ is bounded from above almost surely, it obviously holds that
$\left\lVert w_{1}\right\rVert_{L^{q}(\Omega)}<\infty$. Since
$\mu\in\mathcal{P}_{r}(^{d})$, it also holds that $\left\lVert
V_{1}\right\rVert_{L^{r}(\Omega)}<\infty$ and $\left\lVert
w_{1}V_{1}\right\rVert_{L^{s}(\Omega)}<\infty$ since $s:=\frac{pq}{q-p}<r$.
Consequently, the assumptions of Theorem 3.6 are satisfied, and we obtain the
statement. ∎

######  Lemma 3.8 (Convergence of the weighted covariance for i.i.d. samples).

Suppose that $f\colon^{d}\to$ is bounded from below and let $0<2p<r$. Then for
all $\mu\in\mathcal{P}_{r}(\mathbf{R}^{d})$ satisfying
$\mathcal{C}_{\beta}(\mu)\succcurlyeq\eta\operatorname{I}_{d}\succ 0$, there
is $C$ depending only on $(f,\beta,p,r,\eta)$ and the $r$-th moment of $\mu$
such that for all $J\in\mathbf{N}^{+}$

$\mathbf{E}\left\lVert\sqrt{\mathcal{C}_{\beta}(\overline{\mu}^{J})}-\sqrt{\mathcal{C}_{\beta}(\mu)}\right\rVert_{\rm
F}^{p}\leqslant
CJ^{-\frac{p}{2}},\qquad\overline{\mu}^{J}:=\frac{1}{J}\sum_{j=1}^{J}\delta_{X^{(j)}},\qquad\left\\{\overline{X}^{(j)}\right\\}_{j\in\mathbf{N}}\stackrel{{\scriptstyle\rm{i.i.d.}}}{{\sim}}\mu.$

###### Proof.

By assumption it holds that
$\sqrt{\mathcal{C}_{\beta}(\overline{\mu}^{J})}\succcurlyeq\sqrt{\eta}\operatorname{I}_{d}$.
By an inequality due to van Hemmen and Ando [50], see also [5, Problem X.5.5],
it holds that

$\left\lVert\sqrt{\mathcal{C}_{\beta}(\overline{\mu}^{J})}-\sqrt{\mathcal{C}_{\beta}(\mu)}\right\rVert_{\rm
F}\leqslant\frac{1}{\eta}\Bigl{\lVert}\mathcal{C}_{\beta}(\overline{\mu}^{J})-\mathcal{C}_{\beta}(\mu)\Bigr{\rVert}_{\rm
F}.$

Using the triangle inequality, we have that

$\displaystyle\Bigl{\lVert}\mathcal{C}_{\beta}(\overline{\mu}^{J})-\mathcal{C}_{\beta}(\mu)\Bigr{\rVert}_{\rm
F}^{p}$ $\displaystyle=\Bigl{\lVert}L_{\beta}\overline{\mu}^{J}\left[x\otimes
x\right]-L_{\beta}\mu\left[x\otimes
x\right]-\mathcal{M}_{\beta}(\overline{\mu}^{J})\otimes\mathcal{M}_{\beta}(\overline{\mu}^{J})+\mathcal{M}_{\beta}(\mu)\otimes\mathcal{M}_{\beta}(\mu)\Bigr{\rVert}_{\rm
F}^{p}$ $\displaystyle\leqslant
2^{p-1}\Bigl{\lVert}L_{\beta}\overline{\mu}^{J}\left[x\otimes
x\right]-L_{\beta}\mu\left[x\otimes x\right]\Bigr{\rVert}_{\rm F}^{p}$
$\displaystyle\qquad\qquad+2^{p-1}\Bigl{\lVert}\mathcal{M}_{\beta}(\overline{\mu}^{J})\otimes\mathcal{M}_{\beta}(\overline{\mu}^{J})-\mathcal{M}_{\beta}(\mu)\otimes\mathcal{M}_{\beta}(\mu)\Bigr{\rVert}_{\rm
F}^{p}.$

Convergence to zero of the expectation of the first term with rate
$J^{-\frac{p}{2}}$ follows from Theorem 3.6, this time with
$q=\frac{p(r+4)}{r-2p}$, in which case $\frac{p(q+2)}{q-p}=\frac{r}{2}.$ For
the second term, we use (3.4) to obtain that

$\displaystyle\mathbf{E}\Bigl{\lVert}\mathcal{M}_{\beta}(\overline{\mu}^{J})\otimes\mathcal{M}_{\beta}(\overline{\mu}^{J})-\mathcal{M}_{\beta}(\mu)\otimes\mathcal{M}_{\beta}(\mu)\Bigr{\rVert}_{\rm
F}^{p}$
$\displaystyle\qquad\leqslant\mathbf{E}\left[\bigl{(}\left\lvert\mathcal{M}_{\beta}(\overline{\mu}^{J})\right\rvert+\lvert\mathcal{M}_{\beta}(\mu)\rvert\bigr{)}^{p}\lvert\mathcal{M}_{\beta}(\overline{\mu}^{J})-\mathcal{M}_{\beta}(\mu)\rvert^{p}\right]$
$\displaystyle\qquad\leqslant\left(\mathbf{E}\left[\bigl{(}\left\lvert\mathcal{M}_{\beta}(\overline{\mu}^{J})\right\rvert+\lvert\mathcal{M}_{\beta}(\mu)\rvert\bigr{)}^{2p}\right]\mathbf{E}\left[\lvert\mathcal{M}_{\beta}(\overline{\mu}^{J})-\mathcal{M}_{\beta}(\mu)\rvert^{2p}\right]\right)^{\frac{1}{2}}$
$\displaystyle\qquad\leqslant
CJ^{-\frac{p}{2}}\left(\mathbf{E}\left[\bigl{(}\left\lvert\mathcal{M}_{\beta}(\overline{\mu}^{J})\right\rvert+\lvert\mathcal{M}_{\beta}(\mu)\rvert\bigr{)}^{2p}\right]\right)^{\frac{1}{2}},$

where we used Lemma 3.7. Since $\mathcal{M}_{\beta}(\overline{\mu}^{J})$
converges to $\mathcal{M}_{\beta}(\mu)$ in $L^{2p}(\Omega)$ again by Lemma
3.7, this inequality enables to conclude the proof. ∎

###  3.4 No collapse in finite time for CBS

In order to deal with the matrix square root in the CBS dynamics, we prove
that the weighted covariance stays positive definite for finite times.

######  Proposition 3.9 (No collapse in finite time).

Let the cost function $f\colon\mathbf{R}^{d}\rightarrow\mathbf{R}$ be such
that the macroscopic McKean SDE for CBS (1.5) has a solution
$(\overline{X}_{t})_{t\in[0,T]}$ over some finite time interval $[0,T]$. Then

$\displaystyle\forall
t\in[0,T],\qquad\mathcal{C}(\overline{\rho}_{t})\succcurlyeq\mathcal{C}(\overline{\rho}_{0})\operatorname{e}^{-2t}.$
(3.13)

In particular, if $\mathcal{C}(\overline{\rho}_{0})\succ 0$ and if
$t\mapsto\mathcal{C}_{\beta}(\rho_{t})$ is continuous, then

$\displaystyle\inf_{t\in[0,T]}\lambda_{\rm
min}\bigl{(}\mathcal{C}(\overline{\rho}_{t})\bigr{)}>0$ and
$\displaystyle\inf_{t\in[0,T]}\lambda_{\rm
min}\bigl{(}\mathcal{C}_{\beta}(\overline{\rho}_{t})\bigr{)}>0.$ (3.14)

###### Proof.

The evolution of the covariance matrix is governed by [11, Equation (2.18)]

$\displaystyle\partial_{t}\left(\mathcal{C}(\overline{\rho}_{t})\right)=-2\mathcal{C}(\overline{\rho}_{t})+2\lambda^{-1}\mathcal{C}_{\beta}(\overline{\rho}_{t}).$

For a given $y\in\mathbf{R}^{d}$ such that $|y|=1$, define
$g(t):=y^{\mathsf{T}}\mathcal{C}(\overline{\rho}_{t})y$ for $t\in[0,T]$. Then

$\displaystyle
g^{\prime}(t)=-2g(t)+2\lambda^{-1}y^{T}\mathcal{C}_{\beta}(\overline{\rho}_{t})y\geqslant-2g(t),$

since $\mathcal{C}_{\beta}(\overline{\rho}_{t})$ is positive semi-definite.
Applying Grönwall’s inequality to $-g$, we deduce that $g(t)\geqslant
g(0)\operatorname{e}^{-2t}.$ This shows (3.13) and the first inequality in
(3.14). In order to prove the second inequality in (3.14), assume that it was
wrong. By continuity of $t\mapsto\mathcal{C}_{\beta}(\rho_{t})$, we have
$\lambda_{\rm min}\bigl{(}\mathcal{C}_{\beta}(\overline{\rho}_{t})\bigr{)}=0$
for some $t\in[0,T]$. Then
$y^{\mathsf{T}}\mathcal{C}_{\beta}(\overline{\rho}_{t})y=0$ for some
$y\in\mathbf{R}^{d}$ such that $|y|=1$. In particular

$\int_{\mathbf{R}^{d}}\left\lvert
y^{\mathsf{T}}\Bigl{(}x-\mathcal{M}_{\beta}(\overline{\rho}_{t})\Bigr{)}\right\rvert^{2}\operatorname{e}^{-\beta
f(x)}\,\overline{\rho}_{t}(\mathrm{d}x)=0.$

Thus we obtain
$y^{\mathsf{T}}\bigl{(}x-\mathcal{M}_{\beta}(\overline{\rho}_{t})\bigr{)}=0$
for $\overline{\rho}_{t}$-almost every $x\in\mathbf{R}^{d}$, and so
$y^{\mathsf{T}}\mathcal{M}_{\beta}(\overline{\rho}_{t})=y^{\mathsf{T}}\mathcal{M}(\overline{\rho}_{t})$
by integrating against $\overline{\rho}_{t}$. But then,

$y^{\mathsf{T}}\mathcal{C}(\overline{\rho}_{t})y=\int_{\mathbf{R}^{d}}\left\lvert
y^{\mathsf{T}}\Bigl{(}x-\mathcal{M}(\overline{\rho}_{t})\Bigr{)}\right\rvert^{2}\,\overline{\rho}_{t}(\mathrm{d}x)=\int_{\mathbf{R}^{d}}\left\lvert
y^{\mathsf{T}}\Bigl{(}x-\mathcal{M}_{\beta}(\overline{\rho}_{t})\Bigr{)}\right\rvert^{2}\,\overline{\rho}_{t}(\mathrm{d}x)=0.$

which is a contradiction. Thus $C_{\beta}(\overline{\rho}_{t})\succ 0$ for all
$t\in[0,T]$, which implies the second inequality in (3.14). ∎

##  4 Proof of the well-posedness results

###  4.1 Well-posedness for the microscopic models

For a collection of particles $\mathbf{X}\in\mathbf{R}^{dJ}$, the notation
$\lvert\mathbf{X}\rvert$ denotes the usual Euclidean norm; that is

$\lvert\mathbf{X}\rvert:=\sqrt{\sum_{j=1}^{J}\left\lvert
X^{(j)}\right\rvert^{2}}.$

For the proof of Theorem 2.2 we need the following stability result, which
only requires local Lipschitz continuity of $f$, in contrast to the results in
Subsection 3.1.

######  Lemma 4.1.

For all $\mathbf{X}\in\mathbf{R}^{dJ}$ it holds that

$\lvert\mathcal{M}_{\beta}(\mathbf{X})\rvert\leqslant\lvert\mathbf{X}\rvert\qquad\text{and}\qquad\left\lVert\mathcal{C}_{\beta}(\mathbf{X})\right\rVert\leqslant\lvert\mathbf{X}\rvert^{2}.$
(4.1)

Moreover, if $f\colon\mathbf{R}^{d}\rightarrow\mathbf{R}$ is locally Lipschitz
continuous, then for each $R>0$ and $J\in\mathbf{N}^{+}$ there exists a
constant $C_{J,R}>0$ such that

$\displaystyle\lvert\mathcal{M}_{\beta}(\mathbf{X})-\mathcal{M}_{\beta}(\mathbf{Y})\rvert\leqslant
C_{J,R}\lvert\mathbf{X}-\mathbf{Y}\rvert$ and
$\displaystyle\left\lVert\sqrt{\mathcal{C}_{\beta}(\mathbf{X})}-\sqrt{\mathcal{C}_{\beta}(\mathbf{Y})}\right\rVert_{F}\leqslant
C_{J,R}\lvert\mathbf{X}-\mathbf{Y}\rvert$ (4.2)

for all $\mathbf{X},\mathbf{Y}\in\mathbf{R}^{dJ}$ satisfying
$\lvert\textbf{X}\rvert,|\mathbf{Y}|\leqslant R$.

###### Proof.

The bound on $\mathcal{M}_{\beta}(\mathbf{X})$ in (4.1) follows immediately
from the definition: We have that

$\Bigl{\lvert}\mathcal{M}_{\beta}(\mathbf{X})\Bigr{\rvert}^{2}=\left\lvert\sum_{j=1}^{J}w_{j}(\mathbf{X})X^{(j)}\right\rvert^{2}\leqslant\sum_{j=1}^{J}w_{j}(\mathbf{X})\left\lvert
X^{(j)}\right\rvert^{2}\leqslant\sum_{j=1}^{J}\left\lvert
X^{(j)}\right\rvert^{2}=\left\lvert\mathbf{X}\right\rvert^{2},$

where we used convexity of the square function and the fact that the weights,
given hereafter, are bounded from above by 1:

$w_{j}(\mathbf{X}):=\frac{\operatorname{e}^{-\beta
f\left(X^{(j)}\right)}}{\sum_{k=1}^{J}\operatorname{e}^{-\beta
f\left(X^{(k)}\right)}}.$

Similarly, for the weighted covariance, it holds by the triangle inequality
that

$\displaystyle\left\lVert\mathcal{C}_{\beta}(\mathbf{X})\right\rVert_{\rm F}$
$\displaystyle=\left\lVert\sum_{j=1}^{J}w_{j}(\mathbf{X})\Bigl{(}X^{(j)}\otimes
X^{(j)}\Bigr{)}-\mathcal{M}_{\beta}(\mathbf{X})\otimes\mathcal{M}_{\beta}(\mathbf{X})\right\rVert_{\rm
F}$
$\displaystyle\leqslant\left\lVert\sum_{j=1}^{J}w_{j}(\mathbf{X})\Bigl{(}X^{(j)}\otimes
X^{(j)}\Bigr{)}\right\rVert_{\rm
F}\leqslant\sum_{j=1}^{J}w_{j}(\mathbf{X})\left\lVert X^{(j)}\otimes
X^{(j)}\right\rVert_{\rm F}=\sum_{j=1}^{J}w_{j}(\mathbf{X})\left\lvert
X^{(j)}\right\rvert^{2}\leqslant\left\lvert\mathbf{X}\right\rvert^{2}.$

The local Lipschitz estimate on $\mathcal{M}_{\beta}(\mathbf{X})$ in (4.2) is
[10, Lemma 2.1]. In order to prove the local Lipschitz continuity of
$\mathcal{C}_{\beta}(\mathbf{X})$, fix $R>0$ and $J>0$. As in the proof of
Proposition 3.2 we obtain

$\displaystyle\left\lVert\sqrt{\mathcal{C}_{\beta}(\mathbf{X})}-\sqrt{\mathcal{C}_{\beta}(\mathbf{Y})}\right\rVert_{\rm
F}$
$\displaystyle\leqslant\sqrt{2}\left(\sum_{j=1}^{J}\left\lvert\sqrt{w_{j}(\mathbf{X})}\left(X^{(j)}-\mathcal{M}_{\beta}(\mathbf{X})\right)-\sqrt{w_{j}(\mathbf{Y})}\left(Y^{(j)}-\mathcal{M}_{\beta}(\mathbf{Y})\right)\right\rvert^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leqslant\sqrt{2}\sum_{j=1}^{J}\left\lvert\sqrt{w_{j}(\mathbf{X})}\left(X^{(j)}-\mathcal{M}_{\beta}(\mathbf{X})\right)-\sqrt{w_{j}(\mathbf{Y})}\left(Y^{(j)}-\mathcal{M}_{\beta}(\mathbf{Y})\right)\right\rvert.$

Using the triangle inequality, we have that

$\displaystyle\frac{1}{\sqrt{2}}\left\lVert\sqrt{\mathcal{C}_{\beta}(\mathbf{X})}-\sqrt{\mathcal{C}_{\beta}(\mathbf{X})}\right\rVert_{\rm
F}$ $\displaystyle\leqslant\sum_{j=1}^{J}\sqrt{w_{j}(\mathbf{X})}\left\lvert
X^{(j)}-\mathcal{M}_{\beta}(\mathbf{X})-Y^{(j)}+\mathcal{M}_{\beta}(\mathbf{Y})\right\rvert$
$\displaystyle\qquad+\sum_{j=1}^{J}\left\lvert\left(\sqrt{w_{j}(\mathbf{X})}-\sqrt{w_{j}(\mathbf{Y})}\right)\left(Y^{(j)}-\mathcal{M}_{\beta}(\mathbf{Y})\right)\right\rvert.$

Since $\mathbf{X}\mapsto\mathcal{M}_{\beta}(\mathbf{X})$ is a locally
Lipschitz continuous function by [10, Lemma 2.1], it suffices to prove that
$\mathbf{X}\mapsto\sqrt{w_{j}(\mathbf{X})}$ is locally Lipschitz as well. To
this end, observe that

$\displaystyle\sqrt{w_{j}(\mathbf{X})}-\sqrt{w_{j}(\mathbf{Y})}=\frac{w_{j}(\mathbf{X})-w_{j}(\mathbf{Y})}{\sqrt{w_{j}(\mathbf{X})}+\sqrt{w_{j}(\mathbf{Y})}}.$

The denominator can be controlled by noting that

$\forall\lvert\mathbf{X}\rvert\leqslant R,\qquad
w_{j}(\mathbf{X})=\frac{\operatorname{e}^{-\beta
f\left(X^{(j)}\right)}}{\sum_{k=1}^{J}\operatorname{e}^{-\beta
f\left(X^{(k)}\right)}}\geqslant\frac{1}{J}\exp\left(-\beta\left(\max_{\lvert
x\rvert\leqslant R}f(x)-\min_{\lvert x\rvert\leqslant R}f(x)\right)\right).$

Therefore, there exists a constant $c_{R,J}>0$ such that
$\sqrt{w_{j}(\mathbf{X})}\geqslant c_{R,J}$ for all
$\lvert\mathbf{X}\rvert\leqslant R$. Since
$\mathbf{X}\mapsto{w_{j}(\mathbf{X})}$ is locally Lipschitz continuous as a
composition of locally Lipschitz continuous functions, the claim follows. ∎

###### Proof of Theorem 2.2.

We prove only the claim for the CBS dynamics (1.4), since the proof for CBO is
similar. We write the system (1.4) in the form

$\mathrm{d}\mathbf{X}_{t}=F(\mathbf{X}_{t})\,\mathrm{d}t+G(\mathbf{X}_{t})\,\mathrm{d}\mathbf{W}_{t}$

where $\mathbf{X}_{t}:=\Bigl{(}X^{(1)}_{t},\dots,X^{(J)}_{t}\Bigr{)}$. The
drift and diffusion coefficients are given by

$\displaystyle
F(\mathbf{X}_{t}):=-\begin{pmatrix}X^{(1)}_{t}-\mathcal{M}_{\beta}\left(\mu^{J}_{t}\right)\\\
\vdots\\\
X^{(J)}_{t}-\mathcal{M}_{\beta}\left(\mu^{J}_{t}\right)\end{pmatrix}\in\mathbf{R}^{dJ},$

and

$\displaystyle G(\mathbf{X}_{t}):=$
$\displaystyle\sqrt{2\lambda^{-1}}\operatorname{diag}\left(\sqrt{\mathcal{C}_{\beta}\left(\mu^{J}_{t}\right)},\dots,\sqrt{\mathcal{C}_{\beta}\left(\mu^{J}_{t}\right)}\right)\in\mathbf{R}^{dJ\times
dJ}.$

By Lemma 4.1, the functions $F$ and $G$ have sublinear growth and are locally
Lipschitz continuous. Thus, by the non-explosion criterion from stochastic
Lyapunov theory [38, Theorem 3.5], it suffices to find a function $\phi\in
C^{2}\left(\mathbf{R}^{dJ},[0,\infty)\right)$ such that

  * •

The function $\phi$ is coercive: $\phi(\mathbf{X})\to\infty$ as
$|\mathbf{X}|\to\infty$ and

  * •

There exists $c>0$ such that

$\forall\mathbf{X}\in\mathbf{R}^{dJ},\qquad\mathcal{L}\phi(\mathbf{X}):=F(\mathbf{X})\cdot\nabla\phi(\mathbf{X})+\frac{1}{2}\operatorname*{tr}\left(G^{\mathsf{T}}(\mathbf{X})\operatorname{D}^{2}\phi(\mathbf{X})G(X)\right)\leqslant
c\phi(\mathbf{X}).$

Take $\phi(\mathbf{X}):=\tfrac{1}{2}|\mathbf{X}|^{2}$. Then

$\displaystyle\mathcal{L}\phi(\mathbf{X})$
$\displaystyle=-\sum_{j=1}^{J}\left(X^{(j)}-\mathcal{M}_{\beta}\left(\mu^{J}_{t}\right)\right)\cdot
X^{(j)}+J\lambda^{-1}\operatorname*{tr}\Bigl{(}\mathcal{C}_{\beta}\left(\mu^{J}_{t}\right)\Bigr{)}$
$\displaystyle=-\sum_{j=1}^{J}\left(X^{(j)}-\mathcal{M}_{\beta}\left(\mu^{J}_{t}\right)\right)\cdot
X^{(j)}+\lambda^{-1}\sum_{j=1}^{J}w_{j}(\mathbf{X})\left\lvert
X^{(j)}-\mathcal{M}_{\beta}\left(\mu^{J}_{t}\right)\right\rvert^{2}.$

From (4.1) it follows that this is a valid choice, concluding the proof. ∎

###  4.2 Well-posedness for the mean-field SDEs

We adapt the argument from [10], using the more general Assumption H and the
improved stability estimates from Subsection 3.1. We first prove Theorem 2.4,
and then comment on the necessary changes for proving Theorem 2.3 in Remark
4.3. In the following, we endow the vector space
$\mathcal{X}:=\mathbf{R}^{d}\times\mathbf{R}^{d\times d}$ with the norm

$\left\lVert(m,\Gamma)\right\rVert:=\left({|m|^{2}+\left\lVert\Gamma\right\rVert^{2}_{\rm
F}}\right)^{\frac{1}{2}}.$

The Hölder space $C^{0,\frac{1}{2}}\bigl{(}[0,T],\mathcal{X}\bigr{)}$ is given
by

$C^{0,\frac{1}{2}}\bigl{(}[0,T],\mathcal{X}\bigr{)}:=\left\\{\varphi\colon[0,T]\rightarrow\mathcal{X}:\left\lVert\varphi\right\rVert_{C^{0,\frac{1}{2}}}<\infty\right\\},$

where

$\displaystyle\left\lVert\varphi\right\rVert_{C^{0,\frac{1}{2}}}:=\sup_{t\in[0,T]}\bigl{\lVert}\varphi(t)\bigr{\rVert}+[\varphi]_{C^{0,\frac{1}{2}}}<\infty\qquad\text{and}\qquad[\varphi]_{C^{0,\frac{1}{2}}}:=\sup_{0\leqslant
s<t\leqslant
T}\frac{\bigl{\lVert}\varphi(t)-\varphi(s)\bigr{\rVert}}{|t-s|^{\frac{1}{2}}}.$

In order to ensure that certain matrices are positive semidefinite, we use the
map $\mathcal{R}\colon\mathbf{R}^{d\times d}\rightarrow\mathbf{R}^{d\times d}$
defined by $\mathcal{R}(\Gamma)=\sqrt{\Gamma\Gamma^{\mathsf{T}}}$. We will
make use of the following auxiliary lemma.

######  Lemma 4.2.

For $f$ and $\overline{\rho}_{0}$ as in Theorem 2.4, fix $(m,\Gamma)\in
C([0,T],\mathcal{X})$ and $y_{0}\sim\overline{\rho}_{0}$. Then the stochastic
differential equation

$\displaystyle\mathrm{d}Y_{t}=-(Y_{t}-m_{t})\,\mathrm{d}t+\sqrt{2\lambda^{-1}\mathcal{R}(\Gamma_{t})}\,\mathrm{d}W_{t},\qquad
Y_{0}=y_{0}\,,$ (4.3)

admits a unique continuous strong solution $Y\in C([0,T],\mathbf{R}^{d})$.
Moreover, denoting by
$\rho_{t}:=\operatorname{Law}(Y_{t})\in\mathcal{P}(\mathbf{R}^{d})$ the
pointwise law of $Y_{t}$, the functions
$t\mapsto\mathcal{M}_{\beta}(\rho_{t})$ and
$t\mapsto\mathcal{C}_{\beta}(\rho_{t})$ belong to $C([0,T],\mathbf{R}^{d})$
and $C([0,T],\mathbf{R}^{d\times d})$ respectively.

###### Proof.

The existence of a unique continuous strong solution $Y\in
C([0,T],\mathbf{R}^{d})$ to (4.3) follows from classical theory of stochastic
differential equations [45, Theorem 5.2.1]. It remains to prove the continuity
statement for the weighted mean and covariance. For any $0\leqslant r\leqslant
t\leqslant T$, it holds by the Burkholder–Davis–Gundy inequality that

$\displaystyle\mathbf{E}\left[\sup_{u\in[r,t]}\left\lvert
Y_{u}-Y_{r}\right\rvert^{p}\right]$
$\displaystyle\leqslant\bigl{(}2(t-r)\bigr{)}^{p-1}\int_{r}^{t}\mathbf{E}\bigl{\lvert}Y_{u}-m_{u}\bigr{\rvert}^{p}\,\mathrm{d}u$
$\displaystyle\qquad+C_{\rm
BDG}\,2^{p-1}\left(2\lambda^{-1}\right)^{\frac{p}{2}}(t-r)^{\frac{p}{2}-1}\int_{r}^{t}\left\lVert\sqrt{\mathcal{R}\bigl{(}\Gamma(u)\bigr{)}}\right\rVert_{\rm
F}^{p}\,\mathrm{d}u.$ (4.4)

Taking $r=0$ in this inequality and rearranging, we obtain that

$\mathbf{E}\left[\sup_{u\in[0,t]}\left\lvert
Y_{u}\right\rvert^{p}\right]\leqslant C\left(\mathbf{E}\left\lvert
Y_{0}\right\rvert^{p}+\lVert
m\rVert_{L^{\infty}([0,T])}^{p}+\lVert\Gamma\rVert_{L^{\infty}([0,T])}^{\frac{p}{2}}+\int_{0}^{t}\mathbf{E}\lvert
Y_{u}\rvert^{p}\,\mathrm{d}u\right),$

for a constant $C$ depending only on $(T,\lambda,p)$. It follows from this
inequality, by Grönwall’s lemma, that

$\mathbf{E}\left[\sup_{t\in[0,T]}|Y_{t}|^{p}\right]\leqslant
C\left(\mathbf{E}\left\lvert Y_{0}\right\rvert^{p}+\lVert
m\rVert_{L^{\infty}([0,T])}^{p}+\lVert\Gamma\rVert_{L^{\infty}([0,T])}^{\frac{p}{2}}\right)\operatorname{e}^{CT}<\infty.$
(4.5)

Therefore, by dominated convergence, we have for any $t\in[0,T]$ that

$\mathcal{M}_{\beta}(\rho_{r})=\frac{\mathbf{E}\left[Y_{r}\operatorname{e}^{-\beta
f(Y_{r})}\right]}{\mathbf{E}\left[\operatorname{e}^{-\beta
f(Y_{r})}\right]}\xrightarrow[r\to
t]{}\frac{\mathbf{E}\left[Y_{t}\operatorname{e}^{-\beta
f(Y_{t})}\right]}{\mathbf{E}\left[\operatorname{e}^{-\beta
f(Y_{t})}\right]}=\mathcal{M}_{\beta}(\rho_{t}),$

implying that the function $t\mapsto\mathcal{M}_{\beta}(\rho_{t})$ belongs to
$C([0,T],\mathbf{R}^{d})$. An analogous reasoning shows that the function
$t\mapsto\mathcal{C}_{\beta}(\rho_{t})$ belongs to
$C([0,T],\mathbf{R}^{d\times d})$. ∎

###### Proof of Theorem 2.4.

Similarly to the proof of [10, Theorem 3.1], we first construct a map

$\mathcal{T}\colon C\left([0,T],\mathcal{X}\right)\rightarrow
C\left([0,T],\mathcal{X}\right)$

whose fixed points correspond to solutions of (1.5).

#### Step 1: Constructing the map $\mathcal{T}$

Consider the map

$\displaystyle\mathcal{T}\colon C([0,T],\mathcal{X})$
$\displaystyle\rightarrow C([0,T],\mathcal{X})$ $\displaystyle(m,\Gamma)$
$\displaystyle\mapsto\bigl{(}\mathcal{M}_{\beta}(\rho_{t}),\mathcal{C}_{\beta}(\rho_{t})\bigr{)},$

where $\rho_{t}:={\rm Law}(Y_{t})$ and $(Y_{t})_{t\in[0,T]}$ is the unique
solution to (4.3). Thanks to Lemma 4.2, the map $\mathcal{T}$ is well-defined.
Fixed points of $\mathcal{T}$ correspond to solutions of the McKean-Vlasov SDE
(1.5). The existence of fixed points follows from applying the Leray-Schauder
Fixed Point Theorem [27, Chapter 11] in the space
$C\left([0,T],\mathcal{X}\right)$, once we have proved that $\mathcal{T}$ is
compact and that the following set is bounded:

$\displaystyle\Bigl{\\{}(m,\Gamma)\in
C\bigl{(}[0,T],\mathcal{X}\bigr{)}:\exists\xi\in[0,1]\text{ such that
}(m,\Gamma)=\xi\mathcal{T}\bigl{(}(m,\Gamma)\bigr{)}\Bigr{\\}}.$ (4.6)

#### Step 2: Showing that $\mathcal{T}$ is compact

In order to prove that $\mathcal{T}$ is a compact operator, fix $R>0$ and
consider the ball

$B_{R}:=\left\\{(m,\Gamma)\in
C([0,T],\mathcal{X}):\left\lVert(m,\Gamma)\right\rVert_{L^{\infty}([0,T])}\leqslant
R\right\\}.$

By the Arzelà–Ascoli theorem, we have a compact embedding

$\displaystyle C^{0,\frac{1}{2}}\left([0,T],\mathcal{X}\right)\hookrightarrow
C\left([0,T],\mathcal{X}\right).$

Thus, it suffices to show that $\mathcal{T}(B_{R})$ is bounded in
$C^{0,\frac{1}{2}}\left([0,T],\mathcal{X}\right)$. To this end, take
$(m,\Gamma)\in B_{R}$ and consider the corresponding solution $Y_{t}$ of (4.3)
with pointwise law $\rho_{t}$, and let $\varphi=\mathcal{T}(m,\Gamma)$. We
first observe that

$\bigl{\lVert}\varphi(0)\bigr{\rVert}^{2}=\bigl{\lvert}\mathcal{M}_{\beta}(\overline{\rho}_{0})\bigr{\rvert}^{2}+\bigl{\lVert}\mathcal{C}_{\beta}(\overline{\rho}_{0})\bigr{\rVert}_{\rm
F}^{2}<\infty.$

The right-hand side is independent of $m$ and $\Gamma$, and it is indeed
finite by Proposition A.3. Given that

$\left\lVert\varphi\right\rVert_{C^{0,\frac{1}{2}}}\leqslant\left\lVert\varphi(0)\right\rVert+[\varphi]_{C^{0,\frac{1}{2}}}\bigl{(}\sqrt{T}+1\bigr{)},$

it only remains to show that $[\varphi]_{C^{0,\frac{1}{2}}}$ is bounded by a
finite constant depending on $(m,\Gamma)$ only through $R$. In view of (4.4)
and (4.5), there is a constant $L=L(R)$ such that

$\mathbf{E}\left\lvert Y_{t}-Y_{r}\right\rvert^{p}\leqslant L\lvert
t-r\rvert^{\frac{p}{2}}.$

It follows that $W_{p}(\rho_{t},\rho_{r})\leqslant L^{\frac{1}{p}}\lvert
t-r\rvert^{\frac{1}{2}}$, which by Proposition 3.1 implies the Hölder
continuity with exponent $\frac{1}{2}$ of the functions
$t\mapsto\mathcal{M}_{\beta}(\rho_{t})$ and
$t\mapsto\mathcal{C}_{\beta}(\rho_{t})$. This completes the proof that
$\mathcal{T}$ is compact.

#### Step 3: Showing that the set (4.6) is bounded

To this end, let $(m,\Gamma)\in C([0,T],\mathcal{X})$ be such that

$(m,\Gamma)=\xi\mathcal{T}(m,\Gamma)$ (4.7)

for some $\xi\in[0,1]$, and let $(Y_{t})$ denote the corresponding solution to
(4.3). By (4.7), the stochastic process $(Y_{t})$ is also a solution to

$\mathrm{d}Y_{t}=-\Bigl{(}Y_{t}-\xi\mathcal{M}_{\beta}(\rho_{t})\Bigr{)}\,\mathrm{d}t+\sqrt{2\lambda^{-1}\xi\mathcal{C}_{\beta}(\rho_{t})}\,\mathrm{d}W_{t},\qquad\rho_{t}:={\rm
Law}(Y_{t}).$

But then, reasoning as in Subsection 3.2 using a Grönwall-type argument and
Proposition A.3 with $u=\ell$, we find that

$\mathbf{E}\left[\sup_{t\in[0,T]}\bigl{\lvert}Y_{t}\bigr{\rvert}^{p}\right]\leqslant
C\mathbf{E}\lvert Y_{0}\rvert^{p}=C\int_{\mathbf{R}^{d}}\lvert
x\rvert^{p}\,\overline{\rho}_{0}(\mathrm{d}x).$

By Proposition A.3, it follows that $\mathcal{M}_{\beta}(\rho_{t})$ and
$\mathcal{C}_{\beta}(\rho_{t})$ can be bounded uniformly in $[0,T]$ in terms
of $\overline{\rho}_{0}$, implying that the set (4.6) is indeed bounded. Thus,
we established the existence of a fixed point of $\mathcal{T}$, and we obtain
as a byproduct the moment bounds in (2.6).

#### Step 4: Showing uniqueness

For uniqueness, let $(m_{t},\Gamma_{t})$ and
$(\widehat{m}_{t},\widehat{\Gamma}_{t})$ be two fixed points of $\mathcal{T}$
with corresponding solutions $Y_{t},\widehat{Y}_{t}\in C([0,T],\mathcal{X})$
of (4.3) and let $z_{t}:=Y_{t}-\widehat{Y}_{t}$. By definition of
$\mathcal{T}$ and since $\mathcal{C}_{\beta}(\mu)$ is symmetric and positive
semidefinite for all $\mu\in\mathcal{P}_{2}(\mathbf{R}^{d})$, we have
$\mathcal{R}\left(\Gamma_{t}\right)=\Gamma_{t}$ and
$\mathcal{R}\bigl{(}\widehat{\Gamma}_{t}\bigr{)}=\widehat{\Gamma}_{t}$ for all
$t\in[0,T]$. Let $\rho_{t},\widehat{\rho}_{t}\in\mathcal{P}(\mathbf{R}^{d})$
denote the marginal laws of $Y_{t}$ and $\widehat{Y}_{t}$, respectively. By
the Burkholder–Davis–Gundy inequality, we have for all $t\in[0,T]$

$\displaystyle\frac{1}{2^{p-1}}\mathbf{E}\left[\sup_{r\in[0,t]}|z_{r}|^{p}\right]$
$\displaystyle\leqslant T^{p-1}\int_{0}^{t}\mathbf{E}\left\lvert
Y_{r}-\widehat{Y}_{r}-\mathcal{M}_{\beta}(\rho_{r})+\mathcal{M}_{\beta}(\widehat{\rho}_{r})\right\rvert^{p}\,\mathrm{d}r$
$\displaystyle\qquad+C_{\rm
BDG}T^{\frac{p}{2}-1}(2\lambda^{-1})^{\frac{p}{2}}\int_{0}^{t}\left\lVert\sqrt{\mathcal{C}_{\beta}(\rho_{r})}-\sqrt{\mathcal{C}_{\beta}(\widehat{\rho}_{r})}\right\rVert_{\rm
F}^{p}\,\mathrm{d}r.$

By Proposition 3.9, we have that

$\eta:=\inf\left\\{\lambda_{\rm
min}\Bigl{(}\mathcal{C}_{\beta}(\rho_{t})\Bigr{)}:t\in[0,T]\right\\}>0.$

By virtue of the van Hemmen–Ando inequality [50, Proposition 3.2], it holds
for any unitarily invariant norm, thus in particular for the Frobenius norm,
that

$\left\lVert\sqrt{\mathcal{C}_{\beta}(\rho_{r})}-\sqrt{\mathcal{C}_{\beta}(\widehat{\rho}_{r})}\right\rVert_{\rm
F}\leqslant\eta^{-\frac{1}{2}}\left\lVert\mathcal{C}_{\beta}(\rho_{r})-\mathcal{C}_{\beta}(\widehat{\rho}_{r})\right\rVert_{\rm
F}\,.$

Using Proposition 3.1 and noting that
$W_{p}(\rho_{r},\widehat{\rho}_{r})^{p}\leqslant\mathbf{E}\lvert
z_{r}\rvert^{p}$ by definition of the Wasserstein distance, we conclude that

$\mathbf{E}\left[\sup_{r\in[0,t]}\bigl{\lvert}z_{r}\bigr{\rvert}^{p}\right]\leqslant
C\int_{0}^{t}\mathbf{E}\left[\sup_{u\in[0,r]}\bigl{\lvert}z_{u}\bigr{\rvert}^{p}\right]\,\mathrm{d}r.$

By Grönwall’s lemma, and given that $z_{0}=0$, it follows that

$\mathbf{E}\left[\sup_{t\in[0,T]}\lvert z_{t}\rvert^{p}\right]=0,$

which concludes the proof of uniqueness.

#### Step 5: Showing that $\overline{\rho}_{t}$ solves the nonlocal
Fokker–Planck equation (2.7)

This part of the proof is standard, and we present only a formal derivation.
Let $(m,\Gamma)\in C([0,T],\mathcal{X})$ denote the fixed point of
$\mathcal{T}$, in which case $\mathcal{R}(\Gamma_{t})=\Gamma_{t}$, and let
$\overline{Y}_{t}$ denote the corresponding solution to (4.3). Denote also by
$\overline{\rho}_{t}$ the law of $\overline{Y}_{t}$. For a smooth function
$\phi\colon\mathbf{R}^{d}\to\mathbf{R}$ with compact support, it holds by
Itô’s lemma that

$\mathbf{E}\phi\left(\overline{Y}_{t}\right)-\mathbf{E}\phi\left(\overline{Y}_{r}\right)=\mathbf{E}\left[\int_{r}^{t}\mathcal{L}_{u}\phi\left(\overline{Y}_{u}\right)\,\mathrm{d}u\right],$
(4.8)

where
$\mathcal{L}_{u}\phi(y):=-\bigl{(}y-\mathcal{M}_{\beta}\left(\overline{\rho}_{u}\right)\bigr{)}\cdot\nabla\phi(y)+\lambda^{-1}\mathcal{C}_{\beta}\left(\overline{\rho}_{u}\right):\operatorname{D}^{2}\phi(y).$
It follows from (4.8) that $\overline{\rho}_{t}$ satisfies, for all smooth
compactly supported $\phi\colon\mathbf{R}^{d}\to\mathbf{R}$,

$\frac{\mathrm{d}}{\mathrm{d}t}\int_{\mathbf{R}^{d}}\phi(x)\,\overline{\rho}_{t}(\mathrm{d}x)=\int_{\mathbf{R}^{d}}\mathcal{L}_{t}\phi\,\overline{\rho}_{t}(\mathrm{d}x),$
(4.9)

which is indeed a weak formulation of the Fokker–Planck equation (2.5). For
rigorous uniqueness and existence results for the Fokker–Planck equation (4.9)
as well as a more precise discussion of its connection with (4.3), see for
example [25]. ∎

######  Remark 4.3 (Proof of Theorem 2.3).

Observe that the drift coefficients of CBO and CBS are identical and that the
diffusion coefficient of CBO is a globally Lipschitz continuous function of
its drift coefficient. By, among other things, replacing
$\sqrt{2\lambda^{-1}\mathcal{R}(\Gamma_{t})}$ in (4.3) by
$S\left(\overline{X}_{t}-\mathcal{M}_{\beta}(\overline{\rho}_{t})\right)$, the
above proof can be easily adapted to provide a proof for Theorem 2.3. See also
[10], where a weaker version of Theorem 2.3 was already proven in the case
where $p=4$ and $f\in\mathcal{A}(1,0,0)\cup\mathcal{A}(1,2,2)$.

######  Remark 4.4.

Note that in case of $p\geqslant 2p_{\mathcal{M}}$, we can drop the
requirement of $\mathcal{C}(\overline{\rho}_{0})$ to be positive definite by
using Proposition 3.2 instead of Proposition 3.1.

##  5 Extensions and perspectives for future work

This section is organized as follows. First, in Subsection 5.1, we present an
alternative approach based on stopping times for proving our main results
Theorems 2.6 and 2.12. Then, in the following subsections, we discuss three
natural extensions of our results. In Subsection 5.2, we show how our mean-
field limit result for CBS (Theorem 2.12) can be improved when the initial
condition $\overline{\rho}_{0}$ is Gaussian. In Subsection 5.3, we show how
the stopping time approach presented in Subsection 5.1 can be employed in
order to obtain a mean-field result in the setting where
$f\in\mathcal{A}(s,\ell,u)$ with $u>\ell>0$, under which the improved
stability estimates in Corollary 3.3 are no longer valid. Finally, in
Subsection 5.4, we show how Theorems 2.6 and 2.12 can be generalized to
smaller values of $p\geqslant 2$.

###  5.1 Proving mean-field results using stopping times

In this section, we present an alternative approach for proving Theorem 2.6
and Theorem 2.12. This approach is based on stopping times, and inspired by a
method employed in [33] to handle local Lipschitz continuity issues in the
context of numerical analysis for the Euler–Maruyama method. A similar
approach was employed in [36] to obtain a non-quantitative mean-field
estimate. Before presenting the proof, we briefly discuss the main differences
between this approach and the indicator set approach presented in Section 2.

  * •

The main advantage of the approach used in Section 2 is that the event
$\Omega_{J,t}$ for which our stability estimates do not apply can be time-
dependent. For the proof of Theorem 2.6, this flexibility is in fact not
necessary, as we could have employed instead of $\Omega_{J,t}$ in (2.15) the
event

$\Omega_{J}:=\left\\{\omega\in\Omega:\frac{1}{J}\sum_{j=1}^{J}\sup_{t\in[0,T]}\left\lvert\overline{X}^{(j)}_{t}(\omega)\right\rvert^{p}\geqslant
R\right\\}.$ (5.1)

In certain contexts, however, such as the one discussed in Subsection 5.2,
being able to exclude a time-dependent event is very useful.

  * •

The stopping time approach presented hereafter does not enjoy this
flexibility, but is convenient in settings where moments of both $\mu^{J}$ and
$\overline{\mu}^{J}$ need to be controlled in order to apply the stability
estimates obtained in Proposition 3.1 for the weighted moments. This is
illustrated in Subsections 5.3 and 5.4.

We illustrate the stopping times approach only for Theorem 2.12, noting that
Theorem 2.6 can be proved in exactly the same manner.

###### Alternative proof of Theorem 2.12.

Let $R>0$ be such that the stopping time

$\theta_{J}(R):=\inf\Bigl{\\{}t\geqslant
0:\frac{1}{J}\sum_{j=1}^{J}\left\lvert X^{(j)}_{t}\right\rvert^{p}\geqslant
R\Bigr{\\}},$ (5.2)

satisfies $\mathbf{P}[\theta_{J}\leqslant T]=\mathcal{O}(J^{-2}).$ Note that
the event $\\{\theta_{J}\leqslant T\\}$ coincides with the event $\Omega_{J}$
given in (5.1). From Hölder’s inequality it follows that

$\displaystyle\frac{1}{2^{p-1}}\mathbf{E}\left[\sup_{t\in[0,T]}\Big{\lvert}X^{(j)}_{t}-\overline{X}^{(j)}_{t}\Big{\rvert}^{p}\right]$
$\displaystyle\qquad\qquad=\mathbf{E}\left[\sup_{t\in[0,T]}\Big{\lvert}X^{(j)}_{t}-\overline{X}^{(j)}_{t}\Big{\rvert}^{p}\mathsf{1}_{\theta_{J}(R)>T}\right]+\mathbf{E}\left[\sup_{t\in[0,T]}\Big{\lvert}X^{(j)}_{t}-\overline{X}^{(j)}_{t}\Big{\rvert}^{p}\mathsf{1}_{\theta_{J}(R)\leqslant
T}\right]$
$\displaystyle\qquad\qquad\leqslant\mathbf{E}\left[\sup_{t\in[0,T]}\Big{\lvert}X_{t\wedge\theta_{J}}^{(j)}-\overline{X}_{t\wedge\theta_{J}}^{(j)}\Big{\rvert}^{p}\right]+\left(\mathbf{E}\left[\sup_{t\in[0,T]}\Big{\lvert}X^{(j)}_{t}-\overline{X}^{(j)}_{t}\Big{\rvert}^{q}\right]\right)^{\frac{p}{q}}\Bigl{(}\mathbf{P}[\theta_{J}(R)\leqslant
T]\Bigr{)}^{\frac{q-p}{q}}.$ (5.3)

#### Bounding the first term in (5.3)

We again employ an adapted version of Sznitman’s argument. We have

$\displaystyle\left\lvert
X_{t\wedge\theta_{J}}^{(j)}-\overline{X}_{t\wedge\theta_{J}}^{(j)}\right\rvert^{p}\leqslant$
$\displaystyle\left\lvert\int_{0}^{t\wedge\theta_{J}}b\left(X_{s}^{(j)},\mu^{J}_{s}\right)-b\left(\overline{X}_{s}^{(j)},\overline{\rho}_{s}\right)\,\mathrm{d}s\right\rvert^{p}$
$\displaystyle+\left\lvert\int_{0}^{t\wedge\theta_{J}}\sigma\left(X_{s}^{(j)},\mu^{J}_{s}\right)-\sigma\left(\overline{X}_{s}^{(j)},\overline{\rho}_{s}\right)\,\mathrm{d}W_{s}\right\rvert^{p}.$

Let us introduce the martingale

$M_{t}=\int_{0}^{t}\sigma\left(X_{s}^{(j)},\mu^{J}_{s}\right)-\sigma\left(\overline{X}_{s}^{(j)},\overline{\rho}_{s}\right)\,\mathrm{d}W_{s}.$

By Doob’s optional stopping theorem [33, Theorem 3.3], see also [20, Equation
2.29, p.285], the stopped process $(M_{t\wedge\theta_{J}})_{t\geqslant 0}$ is
a martingale, with a quadratic variation process given by the stopped
quadratic variation process of $M_{t}$, i.e. by $(\langle
M\rangle_{t\wedge\theta_{J}})_{t\geqslant 0}$. Therefore, by the
Burkholder–Davis–Gundy inequality, we have for all $t\in[0,T]$ that

$\displaystyle\mathbf{E}\left[\sup_{s\in[0,t]}\bigl{\lvert}X_{s\wedge\theta_{J}}^{(j)}-\overline{X}_{s\wedge\theta_{J}}^{(j)}\bigr{\rvert}^{p}\right]$
$\displaystyle\leqslant(2T)^{p-1}\mathbf{E}\int_{0}^{t\wedge\theta_{J}}\left\lvert
b\left(X_{s}^{(j)},\mu^{J}_{s}\right)-b\left(\overline{X}_{s}^{(j)},\overline{\rho}_{s}\right)\right\rvert^{p}\,\mathrm{d}s$
$\displaystyle\qquad+C_{\rm
BDG}2^{p-1}T^{\frac{p}{2}-1}\mathbf{E}\int_{0}^{t\wedge\theta_{J}}\left\lVert\sigma\left(X_{s}^{(j)},\mu^{J}_{s}\right)-\sigma\left(\overline{X}_{s}^{(j)},\overline{\rho}_{s}\right)\right\rVert_{\rm
F}^{p}\,\mathrm{d}s.$ (5.4)

From the triangle inequality we obtain that

$\displaystyle\frac{1}{2^{p-1}}\mathbf{E}\int_{0}^{t\wedge\theta_{J}}\left\lvert
b\left(X_{s}^{(j)},\mu^{J}_{s}\right)-b\left(\overline{X}_{s}^{(j)},\overline{\rho}_{s}\right)\right\rvert^{p}\,\mathrm{d}s$
$\displaystyle\leqslant\int_{0}^{t}\mathbf{E}\left\lvert
b\left(X_{s\wedge\theta_{J}}^{(j)},\mu^{J}_{s\wedge\theta_{J}}\right)-b\left(\overline{X}_{s\wedge\theta_{J}}^{(j)},\overline{\mu}^{J}_{s\wedge\theta_{J}}\right)\right\rvert^{p}\,\mathrm{d}s$
$\displaystyle\qquad+\int_{0}^{t}\mathbf{E}\left\lvert
b\left(\overline{X}_{s}^{(j)},\overline{\mu}^{J}_{s}\right)-b\left(\overline{X}_{s}^{(j)},\overline{\rho}_{s}\right)\right\rvert^{p}\,\mathrm{d}s.$
(5.5)

Similarly, for the diffusion term, we have

$\displaystyle\frac{1}{2^{p-1}}\mathbf{E}\int_{0}^{t\wedge\theta_{J}}\left\lVert\sigma\left(X_{s}^{(j)},\mu^{J}_{s}\right)-\sigma\left(\overline{X}_{s}^{(j)},\overline{\rho}_{s}\right)\right\rVert_{\rm
F}^{p}\,\mathrm{d}s$
$\displaystyle\leqslant\int_{0}^{t}\mathbf{E}\left\lVert\sigma\left(X_{s\wedge\theta_{J}}^{(j)},\mu^{J}_{s\wedge\theta_{J}}\right)-\sigma\left(\overline{X}_{s\wedge\theta_{J}}^{(j)},\overline{\mu}^{J}_{s\wedge\theta_{J}}\right)\right\rVert_{\rm
F}^{p}\,\mathrm{d}s$
$\displaystyle\qquad+\int_{0}^{t}\mathbf{E}\left\lVert\sigma\left(\overline{X}_{s}^{(j)},\overline{\mu}^{J}_{s}\right)-\sigma\left(\overline{X}_{s}^{(j)},\overline{\rho}_{s}\right)\right\rVert_{\rm
F}^{p}\,\mathrm{d}s.$ (5.6)

Next, we need to bound the terms on the right-hand side of (5.5) and (5.6).
Since the drift terms and the diffusion terms are bounded in a similar manner,
we only present the bounds for the diffusion terms.

  * •

For the first diffusion term, we have that

$\displaystyle\left\lVert\sigma\left(X_{s\wedge\theta_{J}}^{(j)},\mu^{J}_{s\wedge\theta_{J}}\right)-\sigma\left(\overline{X}_{s\wedge\theta_{J}}^{(j)},\overline{\mu}^{J}_{s\wedge\theta_{J}}\right)\right\rVert_{\rm
F}^{p}$
$\displaystyle=\left(2\lambda^{-1}\right)^{\frac{p}{2}}\biggl{\lVert}\sqrt{\mathcal{C}_{\beta}\left(\mu^{J}_{s\wedge\theta_{J}}\right)}-\sqrt{\mathcal{C}_{\beta}\left(\overline{\mu}^{J}_{s\wedge\theta_{J}}\right)}\biggr{\rVert}_{\rm
F}^{p}.$

Using Corollary 3.3, together with the fact

$\mathbf{E}\left[W_{p}\left(\mu^{J}_{s\wedge\theta_{J}},\overline{\mu}^{J}_{s\wedge\theta_{J}}\right)^{p}\right]\leqslant\mathbf{E}\left[\frac{1}{J}\sum_{j=1}^{J}\left\lvert
X_{s\wedge\theta_{J}}^{(j)}-\overline{X}_{s\wedge\theta_{J}}^{(j)}\right\rvert^{p}\right]=\mathbf{E}\left\lvert
X_{s\wedge\theta_{J}}^{(j)}-\overline{X}_{s\wedge\theta_{J}}^{(j)}\right\rvert^{p},$

we deduce that

$\mathbf{E}\left\lVert\sigma\left(X_{s\wedge\theta_{J}}^{(j)},\mu^{J}_{s\wedge\theta_{J}}\right)-\sigma\left(\overline{X}_{s\wedge\theta_{J}}^{(j)},\overline{\mu}^{J}_{s\wedge\theta_{J}}\right)\right\rVert_{\rm
F}^{p}\leqslant C\mathbf{E}\left\lvert
X^{(j)}_{s\wedge\theta_{J}}-\overline{X}^{(j)}_{s\wedge\theta_{J}}\right\rvert^{p}.$

  * •

For the other diffusion term, we use Lemma 3.8. The assumptions of this lemma
are satisfied since, on the one hand,
$\overline{\rho}_{s}\in\mathcal{P}_{q}(\mathbf{R}^{d})$ by Theorem 2.4 and
$2p<q$, and on the other hand, it holds that
$\mathcal{C}_{\beta}(\overline{\rho}_{s})\succ 0$ by virtue of Proposition
3.9. We deduce that

$\mathbf{E}\left\lVert\sigma\left(\overline{X}_{s}^{(j)},\overline{\mu}^{J}_{s}\right)-\sigma\left(\overline{X}_{s}^{(j)},\overline{\rho}_{s}\right)\right\rVert_{\rm
F}^{p}=\left(2\lambda^{-1}\right)^{\frac{p}{2}}\mathbf{E}\left\lVert\sqrt{\mathcal{C}_{\beta}\left(\overline{\mu}^{J}_{s}\right)}-\sqrt{\mathcal{C}_{\beta}\left(\overline{\rho}_{s}\right)}\right\rVert_{\rm
F}^{p}\leqslant CJ^{-\frac{p}{2}}.$

Combining the bounds on both terms, we deduce that

$\mathbf{E}\int_{0}^{t\wedge\theta_{J}}\left\lVert\sigma\left(X_{s}^{(j)},\mu^{J}_{s}\right)-\sigma\left(\overline{X}_{s}^{(j)},\overline{\rho}_{s}\right)\right\rVert_{\rm
F}^{p}\,\mathrm{d}s\leqslant
CJ^{-\frac{p}{2}}+C\int_{0}^{t}\mathbf{E}\left[\sup_{u\in[0,s]}\left\lvert
X^{(j)}_{u\wedge\theta_{J}}-\overline{X}^{(j)}_{u\wedge\theta_{J}}\right\rvert^{p}\right]\,\mathrm{d}s.$

Applying a similar reasoning to the drift terms, we finally obtain from (5.4)
that

$\mathbf{E}\left[\sup_{s\in[0,t]}\Big{\lvert}X_{s\wedge\theta_{J}}^{(j)}-\overline{X}_{s\wedge\theta_{J}}^{(j)}\Big{\rvert}^{p}\right]\leqslant
CJ^{-\frac{p}{2}}+C\int_{0}^{t}\mathbf{E}\left[\sup_{u\in[0,s]}\bigl{\lvert}X_{u\wedge\theta_{J}}^{(j)}-\overline{X}_{u\wedge\theta_{J}}^{(j)}\bigr{\rvert}^{p}\right]\,\mathrm{d}s.$

By Grönwall’s inequality, this implies that

$\mathbf{E}\left[\sup_{t\in[0,T]}\Bigl{\lvert}X_{t\wedge\theta_{J}}^{(j)}-\overline{X}_{t\wedge\theta_{J}}^{(j)}\Bigr{\rvert}^{p}\right]\leqslant
CJ^{-\frac{p}{2}}.$

#### Bounding the second term in (5.3)

We have by Lemma 3.5 and (2.6) that

$\mathbf{E}\left[\sup_{t\in[0,T]}\big{\lvert}X^{(j)}_{t}-\overline{X}^{(t)}_{j}\big{\rvert}^{q}\right]\leqslant
2^{q-1}\left(\mathbf{E}\left[\sup_{t\in[0,T]}\big{\lvert}X^{(j)}_{t}\big{\rvert}^{q}\right]+\mathbf{E}\left[\sup_{t\in[0,T]}\big{\lvert}\overline{X}^{(t)}_{j}\big{\rvert}^{q}\right]\right)\leqslant
2^{q}\kappa.$

In addition, by Lemma 2.5 it holds that

$\mathbf{P}[\theta_{J}(R)\leqslant
T]=\mathbf{P}\left[\sup_{t\in[0,T]}\frac{1}{J}\sum_{j=1}^{J}\left\lvert\overline{X}^{(j)}_{t}\right\rvert^{p}\geqslant
R\right]\leqslant CJ^{-2}.$

#### Conclusion

We finally obtain from (5.3) that

$\mathbf{E}\left[\sup_{t\in[0,T]}\Big{\lvert}X^{(j)}_{t}-\overline{X}^{(t)}_{j}\Big{\rvert}^{p}\right]\leqslant
C\left(J^{-\frac{p}{2}}+J^{-\frac{2(q-p)}{q}}\right),$

which concludes the proof. ∎

###  5.2 Improving the rate of convergence for Gaussian initial conditions.

It is proved in [11] that, if the initial condition to the mean-field equation
1.5 associated with CBS is Gaussian, then at each time $t$ the solution
$\overline{\rho}_{t}$ to the equation is also Gaussian. In this setting, the
probability measure $\overline{\rho}_{0}$ has moments of all orders. Using a
concentration inequality for Gaussian measures (see [43] and [42, Section 2]),
it is possible to prove that the probability of the event

$\Omega_{J,t}:=\left\\{\omega\in\Omega:\frac{1}{J}\sum_{j=1}^{J}\lvert\overline{X}^{(j)}_{t}(\omega)\rvert^{p}\geqslant
R\right\\}$

decreases in fact as $\exp\bigl{(}-CJ^{\frac{p}{2}}\bigr{)}$, for a constant
$C$ depending on $R$ and on the parameters of the Gaussian distribution.
Carrying out the proof of Theorem 2.6 with the sharper bound
$\exp\bigl{(}-CJ^{\frac{p}{2}}\bigr{)}$ instead of $CJ^{-2}$ for
$\mathbf{P}(\Omega_{J,t})$ as in Lemma 2.5, we obtain for all sufficiently
large $p$ a mean-field estimate of the form

$\mathbf{E}\left[\sup_{t\in[0,T]}\Big{\lvert}X^{(j)}_{t}-\overline{X}^{(t)}_{j}\Big{\rvert}^{p}\right]\leqslant
CJ^{-\frac{p}{2}}.$

###  5.3 Handling the case where $u>\ell>0$ in Assumption H

Our main results stand only when $f\in\mathcal{A}(s,\ell,u)$ with $u=\ell$.
This is a convenient setting because it then holds by Proposition A.3 that the
$p$-th moment of the reweighted measure $L_{\beta}\mu$ is controlled by the
$p$-th moment of the $\mu$, which is key in order to prove well-posedness of
the mean-field system, to obtain moment estimates for empirical measures, and,
most importantly, to improve the Wasserstein stability estimates for the
weighted mean and weighted covariance, see Corollary 3.3. In the interest of
simplicity, we refrained from stating precise results for the cases $u>\ell=0$
and $u>\ell>0$. The former remains open, but we note that a quantitative mean-
field estimate can still be obtained in the latter setting. The main
ingredients to this end are the following:

  * •

Moment estimates for the empirical measure and well-posedness of the mean-
field equation can be obtained by using Proposition A.3 together with the
Bihari–LaSalle inequality instead of Grönwall’s inequality, using the same
reasoning as in Lemma 3.5 and Section 4, respectively. In contrast with the
setting $\ell=u$, these results hold only up to some finite time $T<T_{+}$,
where $T_{+}\in(0,\infty)$ is the time at which the bounds through the
Bihari–LaSalle inequality become singular.

  * •

In order to obtain a mean-field result up to some time $T<T_{+}$, the approach
presented in Subsection 5.1 can be used, this time with the stopping time
$\theta_{J}:=\tau_{J}\wedge\overline{\tau}_{J}$, where

$\displaystyle\tau_{J}(R)$ $\displaystyle=\inf\Bigl{\\{}t\geqslant
0:\mu_{t}^{J}\bigl{[}\lvert x\rvert^{p}\bigr{]}\geqslant R\Bigr{\\}},$
$\displaystyle\overline{\tau}_{J}(R)$ $\displaystyle=\inf\Bigl{\\{}t\geqslant
0:\overline{\mu}_{t}^{J}\bigl{[}\lvert x\rvert^{p}\bigr{]}\geqslant
R\Bigr{\\}}.$

Employing this stopping time instead of (5.2) allows for using the basic
stability estimates for the weighted moments of Proposition 3.1, which are
valid even when $\ell<u$, instead of the improved stability estimates in
Corollary 3.3, which require that $\ell=u$. In order to complete the proof as
in Subsection 5.1, it remains to bound the probability
$\mathbf{P}\left[\theta_{J}\leqslant T\right]$, which can be achieved by
noticing that

$\mathbf{P}[\theta_{J}(R)\leqslant T]=\mathbf{P}\left[\tau_{J}(R)\leqslant
T<\overline{\tau}_{J}(R)\right]+\mathbf{P}\left[\overline{\tau}_{J}(R)\leqslant
T\right].$

Using the almost sure continuity of the solution to the interacting particle
system by Theorem 2.2, together with the triangle inequality, we bound the
first probability as follows:

$\displaystyle\mathbf{P}[\tau_{J}\leqslant T<\overline{\tau}_{J}]$
$\displaystyle\leqslant\mathbf{P}\left[\sup_{t\in[0,T]}\frac{1}{J}\sum_{j=1}^{J}\Bigl{\lvert}X^{(j)}_{t\wedge\theta_{J}}\Bigr{\rvert}^{p}=R\right]$
$\displaystyle\leqslant\mathbf{P}\left[\frac{2^{p-1}}{J}\sum_{j=1}^{J}\sup_{t\in[0,T]}\Bigl{\lvert}\overline{X}^{(j)}_{t}\Bigr{\rvert}^{p}+\frac{2^{p-1}}{J}\sum_{j=1}^{J}\sup_{t\in[0,T]}\Bigl{\lvert}X^{(j)}_{t\wedge\theta_{J}}-\overline{X}^{(j)}_{t\wedge\theta_{J}}\Bigr{\rvert}^{p}\geqslant
R\right]$
$\displaystyle\leqslant\mathbf{P}\left[\frac{1}{J}\sum_{j=1}^{J}\sup_{t\in[0,T]}\Bigl{\lvert}\overline{X}^{(j)}_{t}\Bigr{\rvert}^{p}\geqslant\frac{R}{2^{p}}\right]+\mathbf{P}\left[\frac{1}{J}\sum_{j=1}^{J}\sup_{t\in[0,T]}\Bigl{\lvert}X^{(j)}_{t\wedge\theta_{J}}-\overline{X}^{(j)}_{t\wedge\theta_{J}}\Bigr{\rvert}^{p}\geqslant\frac{R}{2^{p}}\right],$

where we used that $\mathbf{P}[A+B\geqslant k]\leqslant\mathbf{P}[A\geqslant
k/2]+\mathbf{P}[B\geqslant k/2]$ for any two real-valued random variables $A$
and $B$, because $\\{A+B\geqslant k\\}\subset\\{A\geqslant
k/2\\}\cup\\{B\geqslant k/2\\}$. A similar trick, of bounding the probability
of an event associated with a stochastic system in terms of a similar event
associated with a simpler system and the discrepancy between the two systems,
is employed in [19, Theorem 2.10]. The probability
$\mathbf{P}[\theta_{J}\leqslant T]$ can then be bounded as follows:

$\displaystyle\mathbf{P}[\theta_{J}(R)\leqslant T]$
$\displaystyle=\mathbf{P}\left[\tau_{J}(R)\leqslant
T<\overline{\tau}_{J}\right]+\mathbf{P}\left[\overline{\tau}_{J}(R)\leqslant
T\right]$ $\displaystyle\leqslant
2\mathbf{P}\left[\frac{1}{J}\sum_{j=1}^{J}\sup_{t\in[0,T]}\Bigl{\lvert}\overline{X}^{(j)}_{t}\Bigr{\rvert}^{p}\geqslant\frac{R}{2^{p}}\right]+\mathbf{P}\left[\frac{1}{J}\sum_{j=1}^{J}\sup_{t\in[0,T]}\Bigl{\lvert}X^{(j)}_{t\wedge\theta_{J}}-\overline{X}^{(j)}_{t\wedge\theta_{J}}\Bigr{\rvert}^{p}\geqslant\frac{R}{2^{p}}\right].$

Using Markov’s inequality for the second term, we then obtain

$\mathbf{P}[\theta_{J}(R)\leqslant T]\leqslant
2\mathbf{P}\left[\frac{1}{J}\sum_{j=1}^{J}\sup_{t\in[0,T]}\Bigl{\lvert}\overline{X}^{(j)}_{t}\Bigr{\rvert}^{p}\geqslant\frac{R}{2^{p}}\right]+\frac{2^{p}}{R}\mathbf{E}\left[\frac{1}{J}\sum_{j=1}^{J}\sup_{t\in[0,T]}\Bigl{\lvert}X^{(j)}_{t\wedge\theta_{J}}-\overline{X}^{(j)}_{t\wedge\theta_{J}}\Bigr{\rvert}^{p}\right].$
(5.7)

The first term is bounded from above by $CJ^{-2}$ by Lemma 2.5, whereas the
second term can be bounded by $CJ^{-\frac{p}{2}}$ as in the proof in
Subsection 5.1. Thus, the probability $\mathbf{P}[\theta_{J}(R)\leqslant T]$
scales as $J^{-\min\\{2,\frac{p}{2}\\}}$, instead of $J^{-2}$ in the case
where $u=\ell$. As a consequence, we obtain a mean-field result of the type

$\forall J\in\mathbf{N}^{+},\qquad\mathbf{E}\left[\sup_{t\in[0,T]}\left\lvert
X_{t}^{(j)}-\overline{X}^{(j)}_{t}\right\rvert^{p}\right]\leqslant
CJ^{-\min\left\\{2,\frac{p}{2}\right\\}\frac{q-p}{q}}.$

###  5.4 Removing the lower bound on $p$

In Theorems 2.6 and 2.12, we assume that $p$ is bounded from below by
$p_{\mathcal{M}}$ and $2p_{\mathcal{M}}$, respectively. It is possible to
replace this restriction by the less restrictive inequality $p\geqslant 2$ in
the case of Theorem 2.6, and $p>2$ in the case of Theorem 2.12, provided that
the initial condition $\overline{\rho}_{0}$ has sufficiently many moments. We
motivate this statement hereafter for the specific case of CBO. Suppose for
simplicity that $\overline{\rho}_{0}$ has finite moments of all orders. Then
by Theorem 2.6, for $f\in\mathcal{A}(s,\ell,\ell)$ there exists an exponent
$p_{\star}\geqslant 2$ such that for all $p\geqslant p_{\star}$ and all
$\varepsilon>0$, it holds for some constant $C$ independent of $J$ that

$\mathbf{E}\left[\sup_{t\in[0,T]}\Big{\lvert}X^{(j)}_{t}-\overline{X}^{(j)}_{t}\Big{\rvert}^{p}\right]\leqslant
CJ^{-(2-\varepsilon)}.$ (5.8)

  * •

First, note that since

$\mathbf{E}\left[\sup_{t\in[0,T]}\Big{\lvert}X^{(j)}_{t}-\overline{X}^{(j)}_{t}\Big{\rvert}^{r}\right]\leqslant\left(\mathbf{E}\left[\sup_{t\in[0,T]}\Big{\lvert}X^{(j)}_{t}-\overline{X}^{(j)}_{t}\Big{\rvert}^{p}\right]\right)^{\frac{r}{p}}$

for any $1\leqslant r\leqslant p$, the existence of a mean-field result of the
type (5.8) with $p=p_{\star}$ immediately implies a similar result for all
$r\in[2,p_{\star}]$, but with a rate that is degraded by a factor
$\frac{r}{p_{\star}}$. In the next bullet point, we present a better approach.

  * •

Fix $r\in[2,p_{\star}]$. From the proof of Proposition 3.1, it is easily
observed that there is $\varrho$ sufficiently large to ensure that for all
$R>0$, it holds that

$\forall(\mu,\nu)\in\mathcal{P}_{\varrho,R}\times\mathcal{P}_{\varrho,R},\qquad\lvert\mathcal{M}_{\beta}(\mu)-\mathcal{M}_{\beta}(\nu)\rvert+\Bigl{\lVert}\mathcal{C}_{\beta}(\mu)-\mathcal{C}_{\beta}(\nu)\Bigr{\rVert}_{\rm
F}\leqslant L\,W_{r}(\mu,\nu),$ (5.9)

for some constant $L=L(R,r)>0$, see Lemma A.6. In order to obtain a mean-field
result for the exponent $r$, we can employ the same approach as in the proof
in Subsection 5.1 with $p$ replaced by $r$, but with the stopping time

$\theta_{J}:=\overline{\tau}_{J}\wedge\tau_{J},\qquad\left\\{\begin{aligned}
\tau_{J}(R)&=\inf\Bigl{\\{}t\geqslant 0:\mu_{t}^{J}\bigl{[}\lvert
x\rvert^{p}\bigr{]}\geqslant R\Bigr{\\}},\\\
\overline{\tau}_{J}(R)&=\inf\Bigl{\\{}t\geqslant
0:\overline{\mu}_{t}^{J}\bigl{[}\lvert x\rvert^{p}\bigr{]}\geqslant
R\Bigr{\\}}.\end{aligned}\right.$

for some $p\geqslant p_{\star}\vee\varrho$. The proof then goes through as
before, except that we use (5.9) instead of Corollary 3.3, and that the
probability $\mathbf{P}[\theta_{J}\leqslant T]$ is bounded from above
differently. Reasoning as in (5.7), it is simple to show that

$\mathbf{P}[\theta_{J}\leqslant T]\leqslant
2\mathbf{P}\left[\frac{1}{J}\sum_{j=1}^{J}\sup_{t\in[0,T]}\Bigl{\lvert}\overline{X}^{(j)}_{t}\Bigr{\rvert}^{p}\geqslant\frac{R}{2^{p}}\right]+\frac{2^{p}}{R}\mathbf{E}\left[\frac{1}{J}\sum_{j=1}^{J}\sup_{t\in[0,T]}\Bigl{\lvert}X^{(j)}_{t}-\overline{X}^{(j)}_{t}\Bigr{\rvert}^{p}\right].$

The first term on the right-hand side decreases as $\mathcal{O}(J^{-2})$ by
Lemma 2.5, whereas the second term decreases as
$\mathcal{O}(J^{-(2-\varepsilon)})$ by (5.8). This enables to prove that

$\forall
J\in\mathbf{N}^{+},\qquad\mathbf{E}\left[\sup_{t\in[0,T]}\Big{\lvert}X^{(j)}_{t}-\overline{X}^{(t)}_{j}\Big{\rvert}^{r}\right]\leqslant
CJ^{-\min\left\\{\frac{r}{2},2-\varepsilon\right\\}}.$ (5.10)

##  Appendix A Further technical results

###  A.1 Stability estimates for ratios of integrals

######  Lemma A.1.

For a real finite-dimensional vector space $\mathcal{V}$ with norm
$\left\lVert\mathord{\color[rgb]{0.67,0.67,0.67}\definecolor[named]{pgfstrokecolor}{rgb}{0.67,0.67,0.67}\pgfsys@color@gray@stroke{0.67}\pgfsys@color<EMAIL_ADDRESS>let $g\colon\mathbf{R}^{d}\rightarrow\mathcal{V}$ and
$h\colon\mathbf{R}^{d}\rightarrow(0,\infty)$ be functions such that the
following condition is satisfied for some $\varsigma\geqslant 0$ and $L>0$:

$\forall
x,y\in\mathbf{R}^{d},\qquad\bigl{\lVert}g(x)-g(y)\bigr{\rVert}\vee\bigl{\lvert}h(x)-h(y)\bigr{\rvert}\leqslant
L\Bigl{(}1+|x|+|y|\Bigr{)}^{\varsigma}\lvert x-y\rvert.$ (A.1)

Then for all $p\geqslant\varsigma+1$ and all $R>0$, there exist a constant
$C=C(p,R)$ such that

$\displaystyle\forall(\mu,\nu)\in\mathcal{P}_{p,R}(^{d})\times\mathcal{P}_{p,R}(^{d}),\qquad\left\lVert\frac{\mu[g]}{\mu[h]}-\frac{\nu[g]}{\nu[h]}\right\rVert\leqslant
CW_{p}(\mu,\nu).\qquad$

###### Proof.

Fix $p\geqslant\varsigma+1$ and $R>0$.

#### Step 1

Let us first generalize [10, Lemma 3.1] by showing that there exists a
constant $\eta=\eta(p,R)>0$ such that

$\forall\mu\in\mathcal{P}_{p,R}(\mathbf{R}^{d}),\qquad\int_{\mathbf{R}^{d}}h\,\mathrm{d}\mu\geqslant\eta.$
(A.2)

Define $B_{K}:=\\{x\in\mathbf{R}^{d}:\lvert x\rvert\leqslant K\\}$ where
$K:=2^{1/p}R$. By Markov’s inequality, we have

$\forall\mu\in\mathcal{P}_{p,R}(\mathbf{R}^{d}),\qquad\mu\left(B_{K}\right)\geqslant
1-\mu\left(\mathbf{R}^{d}\setminus B_{K}\right)\geqslant
1-\frac{R^{p}}{K^{p}}=\frac{1}{2}.$

Therefore we obtain

$\forall\mu\in\mathcal{P}_{p,R}(\mathbf{R}^{d}),\qquad\int_{\mathbf{R}^{d}}h\,\mathrm{d}\mu\geqslant\int_{B_{K}}h\,\mathrm{d}\mu\geqslant\frac{1}{2}\min\bigl{\\{}h(x):x\in
B_{K}\bigr{\\}}=:\eta>0,$

since $h$ is continuous by A.1. This proves (A.2).

#### Step 2

Fix probability measures $\mu,\nu\in\mathcal{P}_{p,R}$. We omit the
integration domains for conciseness. For any coupling $\pi\in\Pi(\mu,\nu)$, we
have by (A.1) and (A.2) that

$\displaystyle\left\lVert\frac{\mu[g]}{\mu[h]}-\frac{\nu[g]}{\nu[h]}\right\rVert$
$\displaystyle\leqslant\left\lVert\frac{\iint
g(x)-g(y)\,\pi(\mathrm{d}x\,\mathrm{d}y)}{\mu[h]}\right\rVert+\left\lVert\int
g\,\mathrm{d}\nu\right\rVert\cdot\left\lvert\frac{\iint
h(x)-h(y)\,\pi(\mathrm{d}x\,\mathrm{d}y)}{\mu[h]\nu[h]}\right\rvert$
$\displaystyle\leqslant L\left(\eta^{-1}+\eta^{-2}\int\left\lVert
g\right\rVert\,\mathrm{d}\nu\right)\iint\Bigl{(}1+|x|+|y|\Bigr{)}^{\varsigma}\lvert
x-y\rvert\,\pi(\mathrm{d}x\,\mathrm{d}y).$

By (A.1), there exists $C>0$ such that $\left\lVert g(x)\right\rVert\leqslant
C\bigl{(}1+|x|\bigr{)}^{\varsigma+1}$ for all $x\in\mathbf{R}^{d}$. The claim
then follows by first applying Hölder’s inequality to obtain the bound

$\iint\Bigl{(}1+|x|+|y|\Bigr{)}^{\varsigma}\lvert
x-y\rvert\,\pi(\mathrm{d}x\,\mathrm{d}y)\leqslant\left(\iint\Bigl{(}1+|x|+|y|\Bigr{)}^{\frac{\varsigma
p}{p-1}}\,\pi(\mathrm{d}x\,\mathrm{d}y)\right)^{\frac{p-1}{p}}\left(\iint\lvert
x-y\rvert^{p}\,\pi(\mathrm{d}x\,\mathrm{d}y)\right)^{\frac{1}{p}},$

noting that $\frac{\varsigma p}{p-1}\leqslant p$ under the assumption that
$p\geqslant\varsigma+1$, and then infimizing over $\pi\in\Pi(\mu,\nu)$. ∎

######  Lemma A.2.

Let $g\colon\mathbf{R}^{d}\rightarrow\mathbf{R}^{m\times n}$,
$h\colon\mathbf{R}^{d}\rightarrow(0,\infty)$ and
$\mathcal{S}\colon\mathcal{P}(\mathbf{R}^{d})\rightarrow\mathbf{R}^{n}$ be
functions such that the following conditions hold.

  1. (a)

There is $\varsigma\geqslant 0$ and $L>0$ such that

$\forall
x,y\in\mathbf{R}^{d},\qquad\bigl{\lVert}g(x)-g(y)\bigr{\rVert}\vee\bigl{\lvert}h(x)-h(y)\bigr{\rvert}\leqslant
L\Bigl{(}1+|x|+|y|\Bigr{)}^{\varsigma}\lvert x-y\rvert.$

  2. (b)

The map
$\mathcal{S}\colon\mathcal{P}(\mathbf{R}^{d})\rightarrow\mathbf{R}^{n}$
satisfies

$\forall(\mu,\nu)\in\mathcal{P}_{p,R}(^{d})\times\mathcal{P}_{p,R}(^{d}),\qquad\lvert\mathcal{S}(\mu)-\mathcal{S}(\nu)\rvert\leqslant
L_{\mathcal{S}}W_{p}(\mu,\nu).$

Then, for all $R>0$ and $p\geqslant 2\varsigma+2$, there exist a constant
$C>0$ such that for all $\mu,\nu\in\mathcal{P}_{p,R}$ and all couplings
$\pi\in\Pi(\mu,\nu)$, it holds that

$\displaystyle\left(\iint_{\mathbf{R}^{d}\times\mathbf{R}^{d}}\left\lVert\frac{g(x)\mathcal{S}(\mu)}{\sqrt{\mu[h]}}-\frac{g(y)\mathcal{S}(\nu)}{\sqrt{\nu[h]}}\right\rVert^{2}\,\pi(\mathrm{d}x\,\mathrm{d}y)\right)^{1/2}\leqslant
C\,\left(\int\lvert
x-y\rvert^{p}\,\pi(\mathrm{d}x\,\mathrm{d}y)\right)^{1/p}.$ (A.3)

###### Proof.

Fix $p\geqslant 2\varsigma+2$ and $R>0$. Again, there is a constant
$\eta=\eta(R,p)>0$ such that (A.2) holds. From the triangle inequality we
obtain