# Distribution dependent SDEs driven by Additive Continuous Noise

Lucio Galeati and Fabian A. Harang and Avi Mayorcas  Lucio Galeati: Institute
of Applied Mathematics, University of Bonn, 53115 Endenicher Allee 60, Bonn,
Germany<EMAIL_ADDRESS>Fabian A. Harang: Department of
Mathematics, University of Oslo, P.O. box 1053, Blindern, 0316, OSLO, Norway.
<EMAIL_ADDRESS>Avi J. Mayorcas: Mathematical Institute, University of
Oxford, Oxford, OX2 6GG, UK<EMAIL_ADDRESS>
###### Abstract.

We study distribution dependent stochastic differential equation driven by a
continuous process, without any specification on its law, following the
approach initiated in [16]. We provide several criteria for existence and
uniqueness of solutions which go beyond the classical globally Lipschitz
setting. In particular we show well-posedness of the equation, as well as
almost sure convergence of the associated particle system, for drifts
satisfying either Osgood-continuity, monotonicity, local Lipschitz or Sobolev
differentiability type assumptions.

###### Key words and phrases:

Additive noise, pathwise approach, McKean–Vlasov equation, mean field limit.

MSC2020: Primary: 60H10, 60F15; Secondary: 60K35, 34F05.
Acknowledgments: FH gratefully acknowledges financial support from the STORM
project 274410, funded by the Research Council of Norway. LG is funded by the
DFG under Germany’s Excellence Strategy - GZ 2047/1, project-id 390685813.

###### Contents

  1. 1 Introduction
    1. 1.1 Notations, conventions and well-known facts
  2. 2 Well-Posedness Under Lipschitz Assumptions
  3. 3 Refined criteria for existence and uniqueness
    1. 3.1 Existence
    2. 3.2 Uniqueness
      1. 3.2.1 Osgood-type condition
      2. 3.2.2 Monotone drifts
      3. 3.2.3 Locally Lipschitz drifts with growth conditions
  4. 4 DDSDEs with convolutional structure
  5. 5 Mean Field Convergence
    1. 5.1 Abstract criteria
    2. 5.2 Applications to particular drifts
      1. 5.2.1 Osgood drifts
      2. 5.2.2 Monotone drifts
      3. 5.2.3 Locally Lipschitz drifts
      4. 5.2.4 Convolutional drifts
  6. A Approximation in metric spaces
  7. B Analytic lemmas and inequalities involving maximal functions

##  1\. Introduction

In this work we consider distribution dependent SDEs (henceforth DDSDEs) of
the form

$X_{t}=\int_{0}^{t}B_{s}(X_{s},\mu_{s})\mathop{}\\!\mathrm{d}s+Y_{t},\quad\mu_{t}=\mathcal{L}(X_{t})$
(1.1)

where
$B:\mathbb{R}_{+}\times\mathbb{R}^{d}\times\mathcal{P}(\mathbb{R}^{d})\rightarrow\mathbb{R}^{d}$,
$X$ is the unknown and $Y$ is a continuous stochastic process with prescribed
law; $\mathcal{L}(X_{t})$ stands for law of the random variable $X_{t}$.

In the literature the process $Y$ is typically sampled as
$Y_{t}=\xi+\sqrt{2\varepsilon}\,W_{t}$, where $\varepsilon\geq 0$, $W$ is a
standard Brownian motion and $\xi$ is an $\mathbb{R}^{d}$-valued random
variable independent of $W$, encoding the initial condition. In this case the
DDSDE is also called a McKean–Vlasov SDE, after the pioneering work [38] where
it was first introduced (setting $\varepsilon=0$ corresponds to the Vlasov
equation).

One reason for the importance of McKean–Vlasov equations is their connection
to systems of $N$ particles subject to a mean field interaction of the form

$X^{i,N}_{t}=\int_{0}^{t}B_{s}\big{(}X^{i,N}_{s},L^{N}(X^{(N)}_{s})\big{)}\mathop{}\\!\mathrm{d}s+Y^{i}_{t},\quad
L^{N}(X^{(N)}_{t}):=\frac{1}{N}\sum_{i=1}^{N}\delta_{X^{i,N}_{t}}$ (1.2)

for all $i=1,\ldots,N$; where $L^{N}(X^{(N)}_{t})$ stands for the empirical
measure of the system at time $t$ and $Y^{i}$ are usually sampled as i.i.d.
copies of $Y$. One expects the DDSDE (1.1) to be the mean field limit of (1.2)
in the sense that, as $N$ goes to infinity, $L^{N}(X^{(N)}_{t})$ converges
weakly to $\mathcal{L}(X_{t})$ with probability $1$. Moreover for
$Y=\xi+\sqrt{2\varepsilon}\,W$, by Itô calculus, the evolution of the marginal
$\mathcal{L}(X_{t})$ is given by the nonlinear Fokker–Planck equation (also
called McKean–Vlasov equation)

$\partial_{t}\rho+\nabla\cdot((B_{t}(\,\cdot\,,\rho)\,\rho)=\varepsilon\Delta\rho,\quad\rho_{0}=\mathcal{L}(\xi).$
(1.3)

In particular, both (1.1) and (1.3) provide a macroscopic, compact description
of system (1.2), reducing its complexity. For this reason, DDSDEs have found
applications in numerous fields, including galaxy formation, plasma physics,
$2$D fluid dynamics, bacteria chemotaxis, agent-based modelling, neuroscience,
flocking and swarming dynamics; we refer the interested reader to [16], the
review [34] and the references therein. In recent years they have also
attracted a lot of attention due to their connection to mean-field games [37].

Classical results concerning the mean-field limit property go back to
Dobrushin [22] for $\varepsilon=0$ and to Sznitman [49] for general
$\varepsilon\geq 0$, under Lipschitz continuity of $B$; let us also mention
the work by Gärtner [29] which remains among the most general on the topic.

Recently the field has received a lot of attention both from the analytic and
probabilistic communities. On one hand, new methods based on entropy
inequalities [27, 35, 10] and modulated energy [46, 47] have allowed for the
rigorous derivation of mean field limits for fairly singular $B$; on the
other, DDSDEs with very irregular drifts are related to the flourishing field
of regularization by noise phenomena, see [3, 41, 44, 19, 36].

Most of the works mentioned above rely on exploiting either the connection to
the PDE (1.3) or stochastic analysis tools related to the Brownian motion $W$.
In this work we instead focus on studying the DDSDE (1.1) with as minimal
assumptions on the noise $Y$ as possible, usually merely continuity and
finiteness of moments. We are strongly inspired by the recent work [16], where
the authors show that in this setting, for globally Lipschitz drifts $B$,
(1.1) is still the mean field limit of (1.2), although the connection to the
PDE (1.3) breaks down; they present many additional results including the use
of common noise, large deviations and a central limit theorem. Let us stress
that the additive structure of the noise $Y$ is such that the DDSDE (1.1) in
integral form is pathwise meaningful, without any need of stochastic
integration; in particular the results apply to naturally non-Markovian, non-
martingale choices of $Y$, e.g. fractional Brownian motion.

A fundamental feature of [16] is the revisitation of a technique first
introduced by Tanaka [50]; it allows, by a clever transformation of the
underlying probability space, to transfer any stability estimate available for
(1.1) into a bound for the corresponding empirical measure $L^{N}(X^{(N)})$ of
the system (1.2). The argument is quite robust and allows one to deduce the
mean field convergence property under very mild assumptions, even when the
family $\\{Y^{i}\\}_{i}$ is not i.i.d. but only requiring

$L^{N}(Y^{(N)})=\frac{1}{N}\sum_{i=1}^{N}\delta_{Y^{i}}\rightharpoonup\mathcal{L}(Y)\quad\mathbb{P}\text{-a.s.}$

together with some finiteness of moments. The i.i.d. assumption indeed is
quite strong, as it implies exchangeability of the system (see [49, Section
I.2] for a detailed discussion) and sometimes one would prefer to avoid it,
see [20, Example II]. In this regard, like [16], our work can be seen as
another attempt to advertise the power of Tanaka’s idea, which has found
applications also in the rough path setting in [14, 2].

As the global Lipschitz assumption considered in [16] is quite restrictive,
our main goal is conduct a deeper analysis of the DDSDE (1.1) providing
sufficient conditions for existence, uniqueness and stability, which will also
imply the mean field limit. As our setting includes in particular the typical
case $Y=\xi+\sqrt{2\varepsilon}\,W$, it also serves as a review of many
standard techniques used in McKean–Vlasov SDEs, including Yamada–Watanabe
principles, compactness arguments and monotonicity assumptions; we believe one
of the merits of this work it to highlight their extreme robustness, as they
apply for any given $Y$. On the downside, since our results all apply in the
case $\varepsilon=0$, our noise can be very degenerate and we can never go
beyond the classical Vlasov setting.

In this regard, another motivation for the current work is to provide a family
of “baseline results” to be compared to the regularizing features of suitable
nondegenerate choices of $Y$; in our second article [28] we will study in
detail the well-posedness of (1.1) for highly irregular $B$ when the noise is
sampled as a fractional Brownian motion.

However, we stress that this work is more than just a survey of known facts;
our aim is also to provide state-of-the-art results. To give an example, for
the rest of the introduction we will present a specific result from the paper
concerning mean field convergence for McKean–Vlasov equations with potentially
degenerate noise. We deal with a drift $B$ of convolutional structure
$B_{t}(x,\mu)=(b_{t}\ast\mu)(x)$ and a noise $Y=\xi+W$ given by independent
variables $(\xi,W)$ such that
$\mathcal{L}(\xi)(\mathop{}\\!\mathrm{d}x)=\rho(x)\mathop{}\\!\mathrm{d}x$ for
some density function $\rho$; we make no assumption on $W$ other than
continuity. The corresponding DDSDE becomes

$X_{t}=\xi+\int_{0}^{t}\int_{\mathbb{R}^{d}}b_{s}(X_{s}-y)\,\mathcal{L}(X_{s})(\mathop{}\\!\mathrm{d}y)+W_{t}.$
(1.4)

We refer to Section 1.1 below for details of some notation used in the next
statement.

######  Theorem 1.1.

Assume $b\in L^{1}_{T}W^{1,q}_{x}$, $\operatorname{div}b\in
L^{1}_{T}L^{\infty}_{x}$ and $\rho\in L^{p}_{x}$ for $(p,q)\in(1,\infty)$
satisfying

$q>d,\quad\frac{1}{p}+\frac{1}{q}<1;$

then there exists a unique solution to (1.4), which moreover satisfies

$\sup_{t\in[0,T]}\|\mathcal{L}(X_{t})\|_{L^{p}_{x}}<\infty.$ (1.5)

Suppose additionally that $\mathcal{L}(\xi+W)\in\mathcal{P}_{r}(C_{T})$ for
all $r<\infty$ and that we are given a sequence $\\{Y^{i},i\in\mathbb{N}\\}$
of $C_{T}$-valued random variables such that, for any $r<\infty$,

$\limsup_{N\to\infty}\|L^{N}(Y^{(N)})(\omega)\|_{r}<\infty,\quad
L^{N}(Y^{N})(\omega)\rightharpoonup\mathcal{L}(\xi+W)\quad\text{for
$\mathbb{P}$-a.e. }\omega;$

then any solution $\\{X^{i,N}\\}$ to (1.2) associated to $\\{Y^{i}\\}_{i}$
satisfies

$L^{N}(X^{N})(\omega)\rightharpoonup\mathcal{L}(X)\quad\text{for
$\mathbb{P}$-a.e. }\omega.$

The above statement applies in particular for the choice $W=0$, i.e. when
$Y^{i}$ is taken as i.i.d. copies of $\xi$ and the result becomes a mean field
limit for the corresponding Vlasov equation. Although singular drifts have
already been considered in such settings, see [31, 33], to the best of our
knowledge no general statements for $b\in L^{1}_{T}W^{1,q}_{x}$ are available
in the literature; in any case, our proof technique is novel and quite simple,
as we illustrate below.

As typical of transport PDEs, the assumption $\operatorname{div}b\in
L^{1}_{T}L^{\infty}_{x}$ is used to propagate the regularity
$\mathcal{L}(X_{0})\in L^{p}_{x}$ and construct a solution $X$ to (1.4)
satisfying (1.5) by compactness arguments; the regularity $b\in
L^{1}_{T}W^{1,q}_{x}$ for $q>d$, together with the one-sided Lipschitz
estimates recently obtained in [12], is then used to show that this solution
$X$ is stable around any other solution $\tilde{X}$, not necessarily
satisfying (1.5). Finally, Tanaka’s argument allows to translate this
information into an estimate for the corresponding particle system and obtain
the conclusion.

We conclude this introduction by outlining the structure of the paper. In
Section 1.1 we briefly recall some notation, conventions and well-known facts
used throughout; Section 2 shortly revisits the theory of DDSDEs with
Lipschitz drifts, in order to derive useful estimates for the sequel. Section
3 constitutes the main body of the paper, establishing sufficient conditions
for existence and uniqueness in several situations for general $B$. Section 4
provides some more refined results in the more specific convolutional case
$B=b\ast\mu$. Finally, in Section 5 we present Tanaka’s idea and apply it to
all the cases previously considered, deriving mean field limit results for
each one. The appendices A and B contain analytic tools used throughout the
paper.

###  1.1. Notations, conventions and well-known facts

Throughout the article we will always work on a finite time interval $[0,T]$,
that may be arbitrarily large; in particular we do not investigate the long-
time behaviour of solutions nor convergence to equilibrium. We will use the
convention that whenever there exists a positive constant $C$ such that $a\leq
Cb$ we write $a\lesssim b$; if the constant $C$ depends on a significant
parameter $p$, we write $a\lesssim_{p}b$.

For a Banach space $E$ we write $C_{T}E:=C([0,T];E)$ for the space of
continuous $E$-valued maps $f:[0,T]\rightarrow E$. Similarly, for $p\geq 1$ we
write $L^{p}_{T}E:=L^{p}(0,T;E)$ for Bochner–Lebesgue space of maps of the
same form. When $E=\mathbb{R}^{d}$ we simply write $C_{T},\,L^{p}_{T}$ etc.

Given a map $f:\mathbb{R}^{d}\rightarrow\mathbb{R}^{m}$ we write $Df$ for the
$d\times m$ matrix of first derivatives. For $p\in[1,\infty]$, we denote by
$L^{p}(\mathbb{R}^{d};\mathbb{R}^{m}),\,W^{1,p}(\mathbb{R}^{d};\mathbb{R}^{m})$
(or simply $L^{p}_{x},\,W^{1,p}_{x}$ when it does not cause confusion) the
classical Lebesgue and Sobolev spaces. We treat the space $W^{1,\infty}_{x}$
as synonymous with the Lipschitz continuous functions. Whenever
$p\in[1,\infty]$ and where it will not cause confusion, we write $p^{\prime}$
to denote the dual exponent to $p$, that is $1/p+1/p^{\prime}=1$, with the
interpretation $p=1\iff p^{\prime}=\infty$. Similarly, we write $C^{0}_{b}$
for the space of bounded, continuous functions from $\mathbb{R}^{d}$ to
$\mathbb{R}^{m}$; $f\in C^{1}_{b}$ if $f$ and $Df$ belong to $C^{0}_{b}$.

Given a separable Banach space $E$, we denote by $\mathcal{P}(E)$ the set of
probability measures over $E$; we write $\mu_{n}\rightharpoonup\mu$ for weak
convergence of measures; i.e. convergence against continuous bounded
functions.

Given $\mu,\nu\in\mathcal{P}(E)$, $\Pi(\mu,\nu)$ stands for the set of all
possible couplings of $(\mu,\nu)$, i.e. the subset of $\mathcal{P}(E\times E)$
with first and second marginals given respectively by $\mu$ and $\nu$. For any
$p\in[1,\infty)$, we define

$d_{p}(\mu,\nu):=\inf_{m\in\Pi(\mu,\nu)}\bigg{(}\int_{E\times
E}\|x-y\|^{p}_{E}\,m(\mathop{}\\!\mathrm{d}x,\mathop{}\\!\mathrm{d}y)\bigg{)}^{1/p}$

which is a well defined quantity (possibly taking value $+\infty$). By [51,
Theorem 4.1], an optimal coupling $\bar{m}\in\Pi(\mu,\nu)$ realizing the above
infimum always exists. Noting that we may equally describe the set
$\Pi(\mu,\nu)$ as the collection of all $E$-valued random variables $X,\,Y$
defined on the same probability space, such that $\mathcal{L}(X)=\mu$ and
$\mathcal{L}(Y)=\nu$, we obtain the frequently used inequality

$d_{p}(\mu,\nu)\leq\mathbb{E}[\|X-Y\|_{E}^{p}]^{1/p}\quad\forall\,(X,Y)\in\Pi(\mu,\nu).$

We define $\mathcal{P}_{p}(E)$ to be set of $p$-integrable probability
measures; that is, $\mu\in\mathcal{P}(E)$ s.t.

$\|\mu\|_{p}:=\bigg{(}\int_{E}\|x\|_{E}^{p}\,\mu(\mathop{}\\!\mathrm{d}x)\bigg{)}^{1/p}<\infty.$

It is well known that $d_{p}(\mu,\nu)<\infty$ for
$\mu,\nu\in\mathcal{P}_{p}(E)$ and that $(\mathcal{P}_{p}(E),d_{p})$ is a
complete metric space, usually referred to as the $p$-Wasserstein space on
$E$; let us stress however that our definition of $d_{p}(\mu,\nu)$ holds for
all $\mu,\nu\in\mathcal{P}(E)$. We recall that, given a sequence
$\\{\mu_{n}\\}_{n}\subset\mathcal{P}_{p}(E)$, $d_{p}(\mu_{n},\mu)\to 0$ is
equivalent to $\mu_{n}\rightharpoonup\mu$ weakly and
$\|\mu_{n}\|_{p}\to\|\mu\|_{p}$, see [51, Theorem 6.9].

Given $\mu\in\mathcal{P}(\mathbb{R}^{d})$, with a slight abuse of notation we
will write $\mu\in L^{q}(\mathbb{R}^{d})$ (or simply $L^{q}_{x}$) for
$q\in[1,\infty]$ to indicate that $\mu$ admits a density
$\mu(\mathop{}\\!\mathrm{d}x)=\rho(x)\mathop{}\\!\mathrm{d}x$ with respect to
the $d$-dim. Lebesgue measure, such that $\rho\in L^{q}(\mathbb{R}^{d})$.

Throughout the article, whenever not mentioned explicitly, we will consider an
underlying probability space $(\Omega,\mathcal{F},\mathbb{P})$, where the
$\sigma$-algebra $\mathcal{F}$ is $\mathbb{P}$-complete, and $\mathcal{L}(X)$
is the push-forward measure of a random variable $X$ on this probability
space, i.e. $\mathcal{L}_{\mathbb{P}}(X)=\mathcal{L}(X)=\mathbb{P}\circ
X^{-1}=:X\\#\mathbb{P}$. For $p\geq 1$, we will frequently consider $E$-valued
random variables in the space $L^{p}(\Omega,\mathcal{F},\mathbb{P};E)$ which
for notational simplicity we will denote by $L^{p}_{\Omega}E$. We say that a
sequence of random variables $\\{X^{n}\\}_{n}$ converges to $X$ in law if
$\mathcal{L}(X^{n})\rightharpoonup\mathcal{L}(X)$.

##  2\. Well-Posedness Under Lipschitz Assumptions

We start by treating DDSDEs with Lipschitz continuous drifts. Although this
case was already treated in [16], we revisit it in order to derive useful a
priori estimates for the sequel; to avoid repetitiveness, we will also present
alternative proofs to those of [16].

In this section we consider the DDSDE

$X_{t}=\int_{0}^{t}B_{s}(X_{s},\mathcal{L}(X_{s}))\mathop{}\\!\mathrm{d}s+Y_{t}$
(2.1)

under the assumption that
$B:[0,T]\times\mathbb{R}^{d}\times\mathcal{P}_{p}(\mathbb{R}^{d})\to\mathbb{R}^{d}$
satisfies

$|B_{t}(x,\mu)-B_{t}(x^{\prime},\nu)|\leq
g_{t}\,(|x-x^{\prime}|+d_{p}(\mu,\nu)),\quad|B_{t}(x,\mu)|\leq
h_{t}\,(1+|x|+\|\mu\|_{p})$ (2.2)

for some $p\in[1,\infty)$ and some $g,h\in L^{1}(0,T;\mathbb{R}^{d})$.

Although the functions $g,h$ could be unified, we keep them separate to
highlight the differing roles they play in the following estimates. Let us
also point out that if $B$ is of the form $B_{t}(x,\mu)=\int
b_{t}(x,y)\mu(\mathop{}\\!\mathrm{d}y)$, then sufficient conditions for (2.2)
to hold for $p=1$ (and thus for any $p\in[1,\infty)$) are

$|b_{t}(x,y)-b_{t}(x^{\prime},y^{\prime})|\leq
g_{t}(|x-x^{\prime}|+|y-y^{\prime}|),\quad|b_{t}(x,y)|\leq h_{t}(1+|x|+|y|).$

We refrain for now from providing a rigorous definition of solution to (2.1),
which will be discussed in detail in Section 3; for now we note that thanks to
the assumptions on $B$, the integral appearing in (2.1) is pathwise meaningful
and thus, given $Y\in
L^{p}(\Omega,\mathcal{F},\mathbb{P};C_{T})=:L^{p}_{\Omega}C_{T}$ for some
probability space $(\Omega,\mathcal{F},\mathbb{P})$, we say that $X\in
L^{p}_{\Omega}C_{T}$ solves (2.1) if the identity holds $\mathbb{P}$-a.s.

######  Proposition 2.1.

Assume $B$ satisfies (2.2); then for any $Y\in L^{p}_{\Omega}C_{T}$ there
exists a unique $X\in L^{p}_{\Omega}C_{T}$ solving (2.1). Moreover given
$Y^{1},Y^{2}$ as above, denoting by $X^{1},X^{2}$ the associated solutions, it
holds that

$\mathbb{E}[\|X^{1}-X^{2}\|_{C_{T}}^{p}]^{1/p}\leq
e^{2\|g\|_{L^{1}_{T}}}\,\mathbb{E}[\|Y^{1}-Y^{2}\|_{C_{T}}^{p}]^{1/p}.$ (2.3)

###### Proof.

Given $Y$ as above, we define a map $\mathcal{I}$ on $L^{p}_{\Omega}C_{T}$, by

$\mathcal{I}(X)_{\,\cdot\,}:=\int_{0}^{\cdot}B_{s}(X_{s},\mathcal{L}(X_{s}))\mathop{}\\!\mathrm{d}s+Y_{\,\cdot\,};$

it is immediate to check, using assumption (2.2), that $\mathcal{I}$ maps
$L^{p}_{\Omega}C_{T}$ into itself. Let us endow $C_{T}$ with the equivalent
norm

$\|x\tilde{\|}_{C_{T}}:=\sup_{t\in[0,T]}\Big{\\{}e^{-4\int_{0}^{t}g_{r}\mathop{}\\!\mathrm{d}r}\,|x_{s}|\Big{\\}},$

and by extension $L^{p}_{\Omega}C_{T}$ with the norm
$\|\cdot\tilde{\|}_{L^{p}C_{T}}$. We claim that $\mathcal{I}$ is a contraction
on $(L^{p}_{\Omega}C_{T},\|\cdot\tilde{\|}_{L^{p}C_{T}})$, which implies
existence and uniqueness of a solution $X$ by the Banach–Caccioppoli theorem.

By the definition of $\|\cdot\tilde{\|}_{C_{T}}$, for any $X^{1},X^{2}\in
L^{p}_{\Omega}C_{T}$,

$d_{p}(\mathcal{L}(X^{1}_{t}),\mathcal{L}(X^{2}_{t}))\leq\mathbb{E}[|X^{1}_{t}-X^{2}_{t}|^{p}]^{1/p}\leq
e^{4\int_{0}^{t}g_{r}\,\mathop{}\\!\mathrm{d}r}\|X^{1}-X^{2}\tilde{\|}_{L^{p}C_{T}};$

therefore for any $X^{1},X^{2}$, by assumption (2.2) it holds that

$\displaystyle
e^{-4\int_{0}^{t}g_{r}\mathop{}\\!\mathrm{d}r}|\mathcal{I}(X^{1})_{t}-\mathcal{I}(X^{2})_{t}|$
$\displaystyle\leq
e^{-4\int_{0}^{t}g_{s}\mathop{}\\!\mathrm{d}s}\int_{0}^{t}|B_{s}(X^{1}_{s},\mathcal{L}(X^{1}_{s}))-B_{s}(X^{2}_{s},\mathcal{L}(X^{2}_{s}))|\mathop{}\\!\mathrm{d}s$
$\displaystyle\leq
e^{-4\int_{0}^{t}g_{r}\mathop{}\\!\mathrm{d}r}\int_{0}^{t}g_{s}\,[|X^{1}_{s}-X^{2}_{s}|+d_{p}(\mathcal{L}(X^{1}_{x}),\mathcal{L}(X^{2}_{s}))]\mathop{}\\!\mathrm{d}s$
$\displaystyle\leq
e^{-4\int_{0}^{t}g_{r}\mathop{}\\!\mathrm{d}r}\bigg{(}\int_{0}^{t}g_{s}\,e^{4\int_{0}^{s}g_{r}\mathop{}\\!\mathrm{d}r}\mathop{}\\!\mathrm{d}s\bigg{)}\,[\|X^{1}-X^{2}\tilde{\|}_{C_{T}}+\|X^{1}-X^{2}\tilde{\|}_{L^{p}C_{T}}]$
$\displaystyle\leq\frac{1}{4}\,[\|X^{1}-X^{2}\tilde{\|}_{C_{T}}+\|X^{1}-X^{2}\tilde{\|}_{L^{p}C_{T}}].$

Taking first the supremum over $t$ on the l.h.s., then the
$L^{p}_{\Omega}$-norm on both sides, we obtain

$\|\mathcal{I}(X^{1})-\mathcal{I}(X^{2})\tilde{\|}_{L^{p}C_{T}}\leq\frac{1}{2}\|X^{1}-X^{2}\tilde{\|}_{L^{p}C_{T}}$

showing that $\mathcal{I}$ is a contraction.

Now let $X^{i}$ be solutions associated to $Y^{i}$ for $i=1,2$. Their
difference satisfies

$\displaystyle|X^{1}_{t}-X^{2}_{t}|$
$\displaystyle\leq\int_{0}^{t}|B_{s}(X^{1}_{s},\mathcal{L}(X^{1}_{s}))-B_{s}(X^{2}_{s},\mathcal{L}(X^{2}_{s}))|\mathop{}\\!\mathrm{d}s+|Y^{1}_{t}-Y^{2}_{t}|$
$\displaystyle\leq\int_{0}^{t}g_{s}\bigg{(}\sup_{r\leq
s}|X^{1}_{r}-X^{2}_{r}|+\mathbb{E}\Big{[}\sup_{r\leq
s}|X^{1}_{r}-X^{2}_{r}|^{p}\Big{]}^{1/p}\bigg{)}\mathop{}\\!\mathrm{d}s+\|Y^{1}-Y^{2}\|_{C_{T}};$

taking the $L^{p}_{\Omega}$-norm on both sides, applying Minkowski’s integral
inequality, we arrive at

$\displaystyle\mathbb{E}\Big{[}\sup_{r\leq
t}|X^{1}_{r}-X^{2}_{r}|^{p}\Big{]}^{1/p}\leq\int_{0}^{t}2g_{s}\,\mathbb{E}\Big{[}\sup_{r\leq
s}|X^{1}_{r}-X^{2}_{r}|^{p}\Big{]}^{1/p}\mathop{}\\!\mathrm{d}s+\|Y^{1}-Y^{2}\|_{C_{T}}.$

Inequality (2.3) then follows from an application of Grönwall’s lemma. ∎

So far we have regarded $X,Y$ as $C_{T}$-valued random variables on an
abstract probability space $(\Omega,\mathcal{F},\mathbb{P})$; as the result
holds on any such space, we can choose it to be
$(C_{T},\mathcal{B}(C_{T}),\mu^{Y})$, where $\mu^{Y}=\mathcal{L}(Y)$ and $Y$
is realized as the canonical process.

Let us define the drift $\bar{b}(s,x):=B_{s}(x,\mathcal{L}(X_{s}))$; by
assumption (2.2), $\bar{b}$ also satisfies a standard Lipschitz-type
condition, so by classical ODE theory, for any $\omega\in C_{T}$ there exists
a unique solution $x\in C_{T}$ to

$x_{\cdot}=\int_{0}^{\cdot}\bar{b}(s,x_{s})\mathop{}\\!\mathrm{d}s+\omega_{\cdot}$

which can be expressed as $x_{t}=\Phi(t,\omega)$ for some continuous map
$\Phi:[0,T]\times C_{T}\to C_{T}$.

Since $X$ is a solution to (2.1), it satisfies

$X_{\cdot}(\omega)=\int_{0}^{\cdot}\bar{b}(s,X_{s}(\omega))\mathop{}\\!\mathrm{d}s+Y_{\cdot}(\omega)\quad\text{
for }\mathbb{P}\text{-a.e. }\omega\in\Omega$

implying that it satisfies $X(\omega)=\Phi(\cdot,Y(\omega))$. In particular,
since $X$ can be expressed as a measurable function of $Y$, its law
$\mathcal{L}(X)=\mu^{X}=\Phi\\#\mu^{Y}$ on $C_{T}$ is entirely determined by
$\mu^{Y}$. As a consequence, we can regard the solution map $S^{B}$ coming
from Proposition 2.1 not only as a map $L^{p}_{\Omega}C_{T}$ to itself, but
also as a map from $\mathcal{P}_{p}(C_{T})$ to itself,
$S^{B}(\mu^{Y})=\mu^{X}$, thus independent of the specific probability space
in consideration.

We then obtain the following estimate, recovering the stability results from
[16].

######  Corollary 2.2.

Assume $B$ satisfies (2.2) and let $S$ be defined as above, then

$d_{p}(S(\mu^{1}),S(\mu^{2}))\leq
e^{2\|g\|_{L^{1}_{T}}}\,d_{p}(\mu^{1},\mu^{2})\quad\forall\,\mu^{1},\mu^{2}\in\mathcal{P}_{p}(C_{T}).$

###### Proof.

Given any coupling $m\in\mathcal{P}_{p}(C_{T}\times C_{T})$ of
$\mu^{1},\mu^{2}$, considering $Y^{1},Y^{2}$ as the canonical variables on
$C_{T}\times C_{T}$, by Proposition 2.1 we have

$d_{p}(S(\mu^{1},S(\mu^{2}))\leq\mathbb{E}[\|X^{1}-X^{2}\|_{C_{T}}^{p}]^{1/p}\leq
e^{2\|g\|_{L^{1}_{T}}}\mathbb{E}[\|Y^{1}-Y^{2}\|_{C_{T}}^{p}]^{1/p};$

the conclusion follows by choosing $m$ to be the optimal coupling for
$d_{p}(\mu^{1},\mu^{2})$. ∎

By the above discussion, in what follows we could always deal with measures
$\mu\in\mathcal{P}_{p}(C_{T})$; however for the sake of computations we prefer
to keep working with random variables $X,Y$ defined on an abstract probability
space (which can be taken as the canonical one if needed).

Given $h\in L^{1}(0,T;\mathbb{R}^{d})$, we denote by $f_{h}$ the associated
modulus of continuity, namely for any $\delta>0$,

$f_{h}(\delta):=\sup_{s,t\in[0,T],|t-s|<\delta}\int_{s}^{t}|h_{r}|\mathop{}\\!\mathrm{d}r;$

let us also define, for any $x\in C_{T}$, the associate seminorm

$\llbracket x\rrbracket_{h}:=\sup_{s,t\in[0,T],s\neq
t}\frac{|x_{t}-x_{s}|}{f_{h}(|t-s|)}$

######  Lemma 2.3.

Let $B$ satisfy (2.2), $Y\in L^{p}_{\Omega}C_{T}$; then there exist a constant
$C$, only depending on $\|h\|_{L^{1}_{T}}$, such that the solution $X$ to
(2.1) satisfies

$\mathbb{E}[\|X\|_{C_{T}}^{p}]^{1/p}\leq
C(1+\mathbb{E}[\|Y\|_{C_{T}}^{p}]^{1/p})$ (2.4)

as well as the pathwise bound

$\|X(\omega)\|_{C_{T}}\leq
C\,\big{(}1+\|Y(\omega)\|_{C_{T}}\big{)}\quad\text{for $\mathbb{P}$-a.e.
}\omega\in\Omega.$ (2.5)

Moreover, $X=Z+Y$ for some $Z\in L^{p}_{\Omega}C_{T}$ satisfying $Z_{0}=0$ and

$\mathbb{E}[\,\llbracket Z\rrbracket_{h}^{p}]^{1/p}\leq
C\,\mathbb{E}[\,\|Y\|_{C_{T}}^{p}]^{1/p}.$ (2.6)

###### Proof.

Let $X$ be a solution, then by assumption (2.2)

$\begin{split}\sup_{r\leq
t}|X_{r}|&\leq\int_{0}^{t}h_{s}(1+|X_{s}|+\|\mathcal{L}(X_{s})\|_{p})\mathop{}\\!\mathrm{d}s+|Y_{t}|\\\
&\leq\int_{0}^{t}h_{s}\bigg{(}1+\sup_{r\leq
s}|X_{r}|+\mathbb{E}\Big{[}\sup_{r\leq
s}|X_{r}|^{p}\Big{]}^{1/p}\bigg{)}\mathop{}\\!\mathrm{d}s+\|Y\|_{C_{T}}.\end{split}$
(2.7)

Taking the $L^{p}_{\Omega}$-norm on both sides, applying Minkowski’s integral
inequality, we find

$\mathbb{E}\Big{[}\sup_{r\leq
t}|X_{r}|^{p}\Big{]}^{1/p}\leq\int_{0}^{t}h_{t}\bigg{(}1+2\mathbb{E}\Big{[}\sup_{r\leq
s}|X_{r}|^{p}\Big{]}^{1/p}\bigg{)}\mathop{}\\!\mathrm{d}s+\mathbb{E}[\|Y\|_{C_{T}}^{p}]^{1/p};$

inequality (2.4) then follows by Grönwall’s lemma, with
$C=\exp(2\|h\|_{L^{1}_{T}})(1+\|h\|_{L^{1}_{T}})$.

On the other hand, we can apply Grönwall’s lemma at a pathwise level to (2.7)
to find

$\displaystyle\|X(\omega)\|_{C_{T}}\leq
e^{\|h\|_{L^{1}_{T}}}\bigg{(}\|h\|_{L^{1}_{T}}+\mathbb{E}[\|X\|_{C_{T}}^{p}]^{1/p}+\|Y(\omega)\|_{C_{T}}\bigg{)}$

which together with (2.4) implies (2.5), for a suitable choice of $C$.

Finally, since $Z_{t}=\int_{0}^{t}B_{s}(X_{s},\mathcal{L}(X_{s}))$, applying
again (2.2) we find

$\displaystyle|Z_{t}-Z_{s}|\leq\int_{s}^{t}|h_{r}|\mathop{}\\!\mathrm{d}r(1+\|X\|_{C_{T}}+\mathbb{E}[\|X\|_{C_{T}}^{p}]^{1/p});$

combined with (2.4), (2.5) this implies the pathwise bound

$\displaystyle\llbracket Z(\omega)\rrbracket\leq
1+\|X(\omega)\|_{C_{T}}+\mathbb{E}[\|X\|_{C_{T}}^{p}]^{1/p}\lesssim_{h}1+\|Y(\omega)\|_{C_{T}}+\mathbb{E}[\|Y\|_{C_{T}}^{p}]^{1/p}.$

Inequality (2.6) readily follows taking the $L^{p}_{\Omega}$-norm on both
sides. ∎

##  3\. Refined criteria for existence and uniqueness

In many applications one cannot expect the drift $B$ to be globally Lipschitz,
thus making the results of Section 2 not applicable. Motivated by this fact,
in this section we present several alternative conditions to establish
existence and/or uniqueness of solutions to (2.1).

The first part of this section is focused on the different concepts of
solution and the relations between them; Section 3.1 provides a general result
of existence of weak solutions under continuity and linear growth assumptions;
Section 3.2 deals with several refinements of the globally Lipschitz condition
which are still sufficient to establish uniqueness. While we treat existence
and uniqueness as separate problems, we provide example well-posedness results
that combine the two. We leave open the possibility that ad-hoc existence or
uniqueness results may be found in more specific cases, complementing the
general results contained here.

The equations considered here are of the general form

$X_{t}=\int_{0}^{t}B_{s}(X_{s},\mathcal{L}(X_{s}))\mathop{}\\!\mathrm{d}s+Y_{t}\quad\forall\,t\in[0,T]$
(3.1)

where
$B:\mathbb{R}_{+}\times\mathbb{R}^{d}\times\mathcal{P}_{p}(\mathbb{R}^{d})\rightarrow\mathbb{R}^{d}$
is a measurable map, for $p\in[1,\infty)$, and $Y$ is a continuous process
with finite $p$-moment. We detail a variety of assumptions on $B$ below.

######  Definition 3.1 (Weak Solutions I).

Given a drift $B$ as above and $\mu^{Y}\in\mathcal{P}_{p}(C_{T})$, we say that
a tuple $(\Omega,\mathcal{F},\mathbb{P};X,Y)$, consisting of a probability
space $(\Omega,\mathcal{F},\mathbb{P})$ and a measurable map
$(X,Y):\Omega\rightarrow C_{T}\times C_{T}$, is a weak solution to the DDSDE
(3.1) on $[0,T]$ if:

  1. (1)

$\mathcal{L}(Y)=\mu^{Y}$;

  2. (2)

$\sup_{t\in[0,T]}\|\mathcal{L}(X_{t})\|_{p}<\infty$;

  3. (3)

$\mathbb{P}$-a.s.
$\int_{0}^{T}|B_{t}(X_{t},\mathcal{L}(X_{t}))|\mathop{}\\!\mathrm{d}t<\infty$;

  4. (4)

the relation (3.1) holds $\mathbb{P}$-a.s., the integral being interpreted in
the Lebesgue sense.

We will refer to both $\mu^{Y}$ and $Y$ as the input data of the DDSDE.

We readapt here the classical concepts of existence and uniqueness for SDEs to
our setting.

######  Definition 3.2 (Uniqueness in Law).

Given $\mu^{Y}\in\mathcal{P}(C_{T})$, we say that weak uniqueness holds for
the DDSDE (3.1) if for any pair of weak solutions
$(\Omega^{i},\mathcal{F}^{i},\mathbb{P}^{i};X^{i},Y^{i})$, $i=1,2$, such that
$\mathcal{L}_{\mathbb{P}^{i}}(Y^{i})=\mu^{Y}$, one has
$\mathcal{L}_{\mathbb{P}^{1}}(X^{1})=\mathcal{L}_{\mathbb{P}^{2}}(X^{2})\in\mathcal{P}(C_{T})$.

######  Definition 3.3 (Pathwise Uniqueness).

Given $\mu^{Y}\in\mathcal{P}(C_{T})$, we say that pathwise uniqueness holds
for the DDSDE (3.1) if for any two solutions $X^{1},X^{2}$ defined on the same
probability space $(\Omega,\mathcal{F},\mathbb{P})$ and w.r.t. to the same
input process $Y$, it holds that $X^{1}=X^{2}$ $\mathbb{P}$-a.s.

######  Definition 3.4 (Strong Existence).

Given $\mu^{Y}\in\mathcal{P}(C_{T})$, we say that strong existence holds for
the DDSDE associated to $\mu^{Y}$ if there exists a solution
$(\Omega,\mathcal{F},\mathbb{P},X,Y)$ such that the process $X$ is adapted to
the filtration $\mathcal{F}^{Y}_{t}=\sigma\\{Y_{s}:s\in[0,t]\\}$.

It follows from the definition that if strong existence holds, we can
construct a solution $X$ on any given probability space
$(\Omega,\mathcal{F},\mathbb{P};Y)$ on which a process with
$\mathcal{L}_{\mathbb{P}}(Y)=\mu^{Y}$ is defined.

We now provide several links between the various concepts of existence and
uniqueness for DDSDEs; we start with a result in the style of the
Yamada–Watanabe theorem [53, Prop. 1], showing that pathwise uniqueness
implies uniqueness in law.

######  Proposition 3.5.

Let $p\in[1,\infty)$, $\mu^{Y}\in\mathcal{P}_{p}(C_{T})$,
$B:[0,T]\times\mathbb{R}^{d}\times\mathcal{P}_{p}(\mathbb{R}^{d})\to\mathbb{R}^{d}$
measurable; if pathwise uniqueness holds for the DDSDE, then uniqueness in law
holds as well.

###### Proof.

Let $(\Omega^{i},\mathcal{F}^{i},\mathbb{P}^{i};X^{i},Y^{i})$, $i=1,2$ be two
weak solutions to (3.1) with $\mathcal{L}(Y^{1})=\mathcal{L}(Y^{2})=\mu^{Y}$.
In order to prove uniqueness in law, it suffices to construct a coupling
$(\tilde{X}^{1},\tilde{X}^{2})$ of $(X^{1},X^{2})$ such that
$(\tilde{X}^{1},\tilde{X}^{2})$ solves (3.1) on the same probability space and
with same input $\tilde{Y}$; indeed once this is done, pathwise uniqueness
implies
$\mathcal{L}_{\mathbb{P}^{1}}(X^{1})=\mathcal{L}_{\tilde{\mathbb{P}}}(\tilde{X}^{1})=\mathcal{L}_{\tilde{\mathbb{P}}}(\tilde{X}^{2})=\mathcal{L}_{\mathbb{P}^{2}}(X^{2})$.

To construct the aforementioned coupling, we follow the proof of [53, Prop. 1]
based on disintegration of measures, for which we refer the reader to [1, Thm.
5.3.1].

Set $\mu^{i}:=\mathcal{L}(X^{i},Y^{i})\in\mathcal{P}(C_{T}\times C_{T})$ for
$i=1,2$; since $C_{T}$ is Polish, we can disintegrate $\mu^{i}$ as
$\\{\mu^{i}_{y}\\}_{y\in C_{T}}\subset\mathcal{P}(C_{T})$, $\mu^{i}_{y}$
representing the law of $X^{i}$ given $Y^{i}=y$. Define a measure
$\bar{\mu}\in\mathcal{P}(C_{T}\times C_{T}\times C_{T})$ by setting

$\mathop{}\\!\mathrm{d}\bar{\mu}(x^{1},x^{2},y):=\mathop{}\\!\mathrm{d}\mu^{1}_{y}(x^{1})\mathop{}\\!\mathrm{d}\mu^{2}_{y}(x^{2})\mathop{}\\!\mathrm{d}\mu^{Y}(y)$

and consider the probability space
$(\bar{\Omega},\bar{\mathcal{F}},\bar{\mu})$ given by $\Omega=C_{T}\times
C_{T}\times C_{T}$, $\bar{\mathcal{F}}$ the $\bar{\mu}$-completion of
$\mathcal{B}(C_{T}\times C_{T}\times C_{T})$. We also define the projection
maps $\pi_{x^{1},x^{2}}(x^{1},x^{2},y)=(x^{1},x^{2})$, and similarly
$\pi_{x^{i},y}$ for $i=1,2$.

We claim that $(\bar{\Omega},\bar{\mathcal{F}},\bar{\mu};\pi_{x^{1},x^{2}})$
is the desired coupling. By construction,

$\mathcal{L}_{\bar{\mathbb{P}}}(\pi_{x^{i},y})=\mathop{}\\!\mathrm{d}\mu^{i}_{y}(x^{i})\mathop{}\\!\mathrm{d}\mu^{Y}(y)=\mathcal{L}_{\mathbb{P}^{i}}(X^{i},Y^{i})\quad\text{for
}i=1,2.$

Moreover the DDSDE relation (3.1) can be written as $Y^{i}=F^{i}(X^{i})$ for
some measurable maps $F^{i}:C_{T}\to C_{T}$ given by
$F^{i}(X^{i})_{\cdot}:=X^{i}-\int_{0}^{\cdot}B_{s}(X^{i}_{s},\mathcal{L}_{\mathbb{P}^{i}}(X^{i}_{s}))\mathop{}\\!\mathrm{d}s$;
the same relations must then hold for $\bar{\mu}$-a.e. $(x^{1},x^{2},y)$,
namely

$\displaystyle
x^{i}_{t}=\int_{0}^{t}B_{s}(x^{i}_{s},\mathcal{L}_{\mathbb{P}^{i}}(X^{i}_{t}))\mathop{}\\!\mathrm{d}s+y_{t}=\int_{0}^{t}B_{s}(x^{i}_{s},\mathcal{L}_{\bar{\mu}}(\pi_{x^{i}_{t}}))\mathop{}\\!\mathrm{d}s+y_{t}\quad\text{for
}i=1,2$

where the property
$\mathcal{L}_{\bar{\mu}}(\pi_{x^{i}_{t}})=\mathcal{L}_{\mathbb{P}^{i}}(X^{i}_{t})$
comes from the identity
$\mathcal{L}_{\bar{\mu}}(\pi_{x^{i},y})=\mathcal{L}_{\mathbb{P}^{i}}(X^{i},Y^{i})$.
Overall this shows that $\pi_{x^{1},x^{2}}$ is a coupling of $X^{1}$, $X^{2}$
solving the DDSDE (3.1) with same input $\pi_{y}$,
$\mathcal{L}_{\bar{\mu}}(\pi_{y})=\mu^{Y}$. ∎

######  Remark 3.6.

The proof above also contains the nontrivial information that, given any two
weak solutions $(\Omega^{i},\mathcal{F}^{i},\mathbb{P}^{i};X^{i},Y^{i})$, we
may construct a coupling of them. This is also true in the case
$\mu^{Y^{1}}=\mathcal{L}_{\mathbb{P}^{1}}(Y^{1})\neq\mathcal{L}_{\mathbb{P}^{2}}(Y^{2})=\mu^{Y^{2}}$,
assuming we already have a coupling $m\in\mathcal{P}(C_{T}\times C_{T})$ of
$(\mu^{Y^{1}},\mu^{Y^{2}})$ available. Following the notation of the proof, in
this case one can take

$\mathop{}\\!\mathrm{d}\bar{\mu}(x^{1},x^{2},y^{1},y^{2}):=\mathop{}\\!\mathrm{d}\mu^{1}_{y^{1}}(x^{1})\mathop{}\\!\mathrm{d}\mu^{2}_{y^{2}}(x^{2})m(\mathop{}\\!\mathrm{d}y^{1},\mathop{}\\!\mathrm{d}y^{2})$

which defines $\bar{\mu}\in\mathcal{P}(C_{T}\times C_{T}\times C_{T}\times
C_{T})$; $(\pi_{x^{1},y^{1}},\pi_{x^{2},y^{2}})$ is then a coupling of
$(X^{1},Y^{1}),(X^{2},Y^{2})$.

The next proposition is in the style of [52, Theorem 6.3].

######  Proposition 3.7.

Let $p\in[1,\infty)$,
$B:[0,T]\times\mathbb{R}^{d}\times\mathcal{P}_{p}(\mathbb{R}^{d})\to\mathbb{R}^{d}$
measurable. Suppose that for any $\mu^{Y}\in\mathcal{P}_{p}(C_{T})$ and any
measurable $\mu:[0,T]\to\mathcal{P}_{p}(\mathbb{R}^{d})$, strong existence and
both pathwise uniqueness and uniqueness in law hold for the SDE

$X_{t}=\int_{0}^{t}B_{s}(X_{s},\mu_{s})\mathop{}\\!\mathrm{d}s+Y_{t}.$ (3.2)

Then weak existence (resp. uniqueness in law) holds for the DDSDE (3.1) if and
only if strong existence (resp. pathwise uniqueness) holds for it.

###### Proof.

Strong existence always implies weak existence and by Proposition 3.5,
pathwise uniqueness implies weak uniqueness.

Suppose now that there exists a weak solution
$(\Omega,\mathcal{F},\mathbb{P};X,Y)$ to the DDSDE, then $X$ is also a
solution to the SDE (3.2) for the choice $\mu_{t}=\mathcal{L}(X_{t})$. By
hypothesis, we can construct a strong solution $\bar{X}$ to the SDE on the
canonical space $(C_{T},\mathcal{B}(C_{T}),\mu^{Y})$; since weak uniqueness
holds for the SDE,
$\mathcal{L}_{\mathbb{P}}(X)=\mathcal{L}_{\mu^{Y}}(\bar{X})$, which implies
that $\bar{X}$ is a strong solution to the DDSDE.

Finally, assume that weak uniqueness holds and let $X^{1}$, $X^{2}$ be two
solutions defined on the same probability space and with same input $Y$. Then
$\mathcal{L}(X^{1}_{t})=\mathcal{L}(X^{2}_{t})=:\mu_{t}$ and they solve the
same SDE with drift $(t,x)\mapsto B_{t}(x,\mu_{t})$, so by pathwise uniqueness
for the SDE it holds $X^{1}=X^{2}$ $\mathbb{P}$-a.s. ∎

###  3.1. Existence

Here we study existence of solutions to (3.1) for continuous drifts $B$
satisfying suitable linear growth conditions.

######  Assumption 3.8.

Given $p\in[1,\infty)$,
$B:[0,T]\times\mathbb{R}^{d}\times\mathcal{P}_{p}(\mathbb{R}^{d})\rightarrow\mathbb{R}^{d}$
is measurable and s.t.:

  * i.

For any $t\in[0,T]$,
$B_{t}:\mathbb{R}^{d}\times\mathcal{P}_{p}(\mathbb{R}^{d})\rightarrow\mathbb{R}^{d}$
is uniformly continuous on bounded sets.

  * ii.

There exists $h\in L^{1}_{T}$ such that for all
$(t,x,\mu)\in[0,T]\times\mathbb{R}^{d}\times\mathcal{P}_{p}(\mathbb{R}^{d})$
one has

$|B_{t}(x,\mu)|\leq h_{t}\left(1+|x|+\|\mu\|_{p}\right).$ (3.3)

######  Remark 3.9.

Let $\delta_{0}$ denote the Dirac delta in $0$, which belongs to
$\mathcal{P}_{p}(\mathbb{R}^{d})$ for any $p\in[1,\infty)$; it’s then easy to
check that $\|\mu\|_{p}=d_{p}(\mu,\delta_{0})$ for any
$\mu\in\mathcal{P}_{p}$. Therefore if $B$ satisfies Assumption 3.8, it also
satisfies Assumption A.1 from Appendix A in the metric space
$E=\mathbb{R}^{d}\times\mathcal{P}_{p}(\mathbb{R}^{d})$.

The next result can be seen as an extension of Peano’s theorem to DDSDEs.

######  Proposition 3.10.

Let $B$ satisfy Assumption 3.8 for some $p\in[1,\infty)$ and
$\mu^{Y}\in\mathcal{P}_{p}(C_{T})$; then there exists a weak solution to (3.1)
in the sense of Definition 3.1. Moreover, any weak solution
$(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}};\tilde{X},\tilde{Y})$
satisfies the pathwise bound

$\|\tilde{X}\|_{C_{T}}\leq
C(1+\|\tilde{Y}\|_{C_{T}})\quad\tilde{\mathbb{P}}\text{-a.s.}$ (3.4)

for some constant $C=C(\|h\|_{L^{1}_{T}})$.

###### Proof.

Estimate (3.4) follows in exactly the same manner as (2.5), since it only
relies on the linear growth of $B$ and Grönwall’s lemma; thus we focus on
establishing existence.

Thanks to Assumption 3.8 and Remark 3.9, we can apply Proposition A.3 from
Appendix A to $B$; we then find a sequence of drifts $\\{B^{n}\\}_{n}$ such
that each
$B^{n}:[0,T]\times\mathbb{R}^{d}\times\mathcal{P}_{p}(\mathbb{R}^{d})\rightarrow\mathbb{R}^{d}$
satisfies condition (2.2) with the same growth function $h_{t}$ and $B^{n}$ is
Lipschitz with constant $g^{n}>0$ independent of $t$. Denote by $Y$ the
canonical process associated to $\mu^{Y}$; since
$\mu^{Y}\in\mathcal{P}_{p}(C_{T})$, we may apply Proposition 2.1 to obtain the
existence of a sequence $\\{X^{n}\\}_{n}$, each $X^{n}$ solving (3.1) with $B$
replaced by $B^{n}$.

Lemma 2.3 furnishes us with suitable a priori estimates for the family
$\\{X^{n}\\}_{n}$: there exists $C=C(\|h\|_{L^{1}_{T}})$ such that

$\mathbb{E}\big{[}\|X^{n}\|_{C_{T}}^{p}\big{]}^{1/p}+\mathbb{E}\big{[}\,\llbracket
Z^{n}\rrbracket_{h}^{p}\big{]}^{1/p}\leq
C\Big{(}1+\mathbb{E}[\|Y\|_{C_{T}}^{p}]^{1/p}\Big{)}$ (3.5)

where the estimate is uniform over $n$ and $Z^{n}$ are given by the relation
$X^{n}=Z^{n}+Y$; we recall that $\llbracket\,\cdot\,\rrbracket_{h}$ was the
seminorm associated to the modulus of continuity $f_{h}$ defined in Section 2.

By Arzelà–Ascoli, subsets of $C_{T}$ of the form $\\{\omega\in
C_{T}:\omega_{0}=0,\llbracket\omega\rrbracket_{h}\leq R\\}$ are compact;
therefore by the moment estimate (3.5), the family of measures
$\\{\mathcal{L}(Z^{n})\\}_{n}$ are tight in $C_{T}$. Since $\mathcal{L}(Y)$ is
also tight on $C_{T}$ (being a probability measure on a Polish space), we
deduce that the family $\\{\mathcal{L}(Y,X^{n})\\}_{n}$ of probability
measures on $C_{T}\times C_{T}$ is tight as well.

We can then apply Prokhorov’s theorem to extract a (not relabelled)
subsequence such that $\\{\mathcal{L}(X^{n},Y)\\}_{n}$ converge weakly to a
limit measure $\mu$. By Skorokhod’s theorem we can construct a new probability
space $(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}})$, carrying a
family of random variables $(\tilde{X}^{n},\tilde{Y}^{n})$,
$(\tilde{X},\tilde{Y})$, such that
$(\tilde{X}^{n}(\tilde{\omega}),\tilde{Y}^{n}(\tilde{\omega}))\to(\tilde{X}(\tilde{\omega}),\tilde{Y}(\tilde{\omega}))$
for $\tilde{\mathbb{P}}$-a.e. $\tilde{\omega}$,
$\mathcal{L}_{\mathbb{P}}(X^{n},Y)=\mathcal{L}_{\tilde{\mathbb{P}}}(\tilde{X}^{n},\tilde{Y}^{n})$
and $\mu=\mathcal{L}_{\tilde{\mathbb{P}}}(\tilde{X},\tilde{Y})$. It remains to
show that
$(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}};\tilde{X},\tilde{Y})$
is a weak solution to (3.1).

First of all, observe that
$\mathcal{L}_{\tilde{\mathbb{P}}}(\tilde{Y}^{n})=\mathcal{L}_{\mathbb{P}}(Y)=\mu^{Y}$
and
$\mathcal{L}_{\tilde{\mathbb{P}}}(\tilde{Y}^{n})\rightharpoonup\mathcal{L}_{\tilde{\mathbb{P}}}(\tilde{Y})$
so that $\mathcal{L}_{\tilde{\mathbb{P}}}(\tilde{Y})=\mu^{Y}$, as well as
$\mathbb{E}_{\tilde{\mathbb{P}}}[\|\tilde{Y}^{n}\|_{C_{T}}^{p}]\to\mathbb{E}_{\tilde{\mathbb{P}}}[\|\tilde{Y}\|_{C_{T}}^{p}]$;
together with the convergence $\tilde{Y}^{n}\to\tilde{Y}$
$\tilde{\mathbb{P}}$-a.s., we find that $\tilde{Y}^{n}\to\tilde{Y}$ in
$L^{p}_{\tilde{\Omega}}C_{T}$. Since $(\tilde{Y}^{n},\tilde{X}^{n})$ have the
same law as $(Y,X^{n})$, they are still solutions to the DDSDEs associated to
$B^{n}$; in particular they satisfy the pathwise bound (2.5). This information
together with the convergence $\tilde{Y}^{n}\to\tilde{Y}$ in
$L^{p}_{\tilde{\Omega}}C_{T}$ implies that $\\{\tilde{X}^{n}\\}_{n}$ is
uniformly $p$-integrable. However, since we also have
$\tilde{X}^{n}\to\tilde{X}$ $\tilde{\mathbb{P}}$-a.s., by Vitali’s convergence
theorem $\tilde{X}^{n}\to\tilde{X}$ in $L^{p}_{\tilde{\Omega}}C_{T}$. In
particular
$d_{p}(\mathcal{L}_{\tilde{\mathbb{P}}}(\tilde{X}^{n}_{t}),\mathcal{L}_{\tilde{\mathbb{P}}}(\tilde{X}_{t}))\to
0$ and overall we have the $\tilde{\mathbb{P}}$-a.s. convergence
$(\tilde{X}_{t}^{n},\mathcal{L}(\tilde{X}_{t}^{n}))\rightarrow(\tilde{X}_{t},\mathcal{L}(\tilde{X}_{t}))$
in $\mathbb{R}^{d}\times\mathcal{P}_{p}(\mathbb{R}^{d})$ for any $t\in[0,T]$.

The above convergence also implies that $\tilde{X}$ satisfies the pathwise
bound (3.4); together with assumption (3.3) this shows that points $(2)$ and
$(3)$ of Definition 3.1 are satisfied. To conclude that identity (3.1) holds
$\tilde{\mathbb{P}}$-a.s. it then suffices to show that
$\int_{0}^{\cdot}B^{n}_{s}(\tilde{X}^{n}_{s},\mathcal{L}(\tilde{X}^{n}_{s}))\mathop{}\\!\mathrm{d}s$
converge to
$\int_{0}^{\cdot}B_{s}(\tilde{X}_{s},\mathcal{L}(\tilde{X}_{s}))\mathop{}\\!\mathrm{d}s$.

Since $B^{n}$ converge to $B$ uniformly on compact sets and
$(\tilde{X}_{t}^{n},\mathcal{L}(\tilde{X}_{t}^{n}))\rightarrow(\tilde{X}_{t},\mathcal{L}(\tilde{X}_{t}))$
$\tilde{\mathbb{P}}$-a.s., it holds
$B^{n}(\tilde{X}_{t}^{n},\mathcal{L}(\tilde{X}_{t}^{n}))\to
B(\tilde{X}_{t},\mathcal{L}(\tilde{X}_{t}))$ $\tilde{\mathbb{P}}$-a.s. as
well. Moreover since $B^{n}$ and $B$ both satisfy the growth assumption (3.3),
we have

$\displaystyle\sup_{t\in[0,T]}\bigg{|}\int_{0}^{t}B^{n}_{s}(\tilde{X}^{n}_{s},\mathcal{L}(\tilde{X}^{n}_{s}))\mathop{}\\!\mathrm{d}s-\int_{0}^{t}B_{s}(\tilde{X}_{s},\mathcal{L}(\tilde{X}_{s})\mathop{}\\!\mathrm{d}s\bigg{|}\leq\int_{0}^{T}|B^{n}_{s}(\tilde{X}^{n}_{s},\mathcal{L}(\tilde{X}^{n}_{s}))-B_{s}(\tilde{X}_{s},\mathcal{L}(\tilde{X}_{s}))|\mathop{}\\!\mathrm{d}s$

with the second integrand satisfying the uniform bound

$\displaystyle|B^{n}_{s}(\tilde{X}^{n}_{s},\mathcal{L}(\tilde{X}^{n}_{s}))-B_{s}(\tilde{X}_{s},\mathcal{L}(\tilde{X}_{s})|$
$\displaystyle\leq\|h\|_{L^{1}_{T}}(2+\|\tilde{X}^{n}\|_{C_{T}}+\|\mathcal{L}(\tilde{X}^{n})\|_{p}+\|\tilde{X}\|_{C_{T}}+\|\mathcal{L}(\tilde{X})\|_{p})$
$\displaystyle\lesssim_{h}1+\|\tilde{Y}^{n}\|_{C_{T}}+\|\tilde{Y}\|_{C_{T}}+\|\mathcal{L}(\tilde{Y})\|_{p}$

where we used again the a priori bound (3.4), as well as
$\mathcal{L}(\tilde{Y}^{n})=\mathcal{L}(\tilde{Y})$. But
$\|\tilde{Y}^{n}\|_{C_{T}}\to\|\tilde{Y}\|_{C_{T}}$ $\tilde{\mathbb{P}}$-a.s.,
thus the previous integrands must be uniformly integrable.

Since $[0,T]$ is finite, we may now apply again Vitali’s convergence theorem
to give that

$\displaystyle\int_{0}^{\cdot}B^{n}(s,\tilde{X}^{n}_{s},\mathcal{L}(\tilde{X}^{n}_{s}))\,\mathop{}\\!\mathrm{d}s\to\int_{0}^{\cdot}B(s,\tilde{X}_{s},\mathcal{L}(\tilde{X}_{s}))\,\mathop{}\\!\mathrm{d}s\quad\tilde{\mathbb{P}}\text{-a.s.}$

which implies the conclusion. ∎

######  Remark 3.11.

Condition i. from Assumption 3.8 can be verified in a variety of situations;
it suffices to check that for any $t\in[0,T]$, the map $(x,\mu)\mapsto
B_{t}(x,\mu)$ is continuous under a weak topology of
$\mathbb{R}^{d}\times\mathcal{P}_{p}(\mathbb{R}^{d})$ for which bounded balls
are compact (as we can then apply Heine–Cantor theorem).

For instance, if $B_{t}(\cdot,\cdot)$ is continuous on
$\mathbb{R}^{d}\times\mathcal{P}_{p}(\mathbb{R}^{d})$, then it’s uniformly
continuous on bounded balls on
$\mathbb{R}^{d}\times\mathcal{P}_{q}(\mathbb{R}^{d})$ for any $q>p$, due to
the compact embedding $\mathcal{P}_{q}\hookrightarrow\mathcal{P}_{p}$.
Similarly, in the case $p=1$ it’s enough to require

$|B_{t}(x,\mu)-B_{t}(y,\nu)|\lesssim_{t}F\Big{(}|x-y|+\inf_{m\in\Pi(\mu,\nu)}\int_{\mathbb{R}^{2d}}|x^{\prime}-y^{\prime}|^{\theta}\,m(\mathop{}\\!\mathrm{d}x^{\prime},\mathop{}\\!\mathrm{d}y^{\prime})\Big{)}$

for some $\theta\in(0,1)$ and some continuous $F$ such that $F(0)=0$. On the
use of more abstract weak topologies guaranteeing sequential compactness of
bounded balls of $\mathcal{P}(\mathbb{R}^{d})$ we also refer to [29, Section
B].

######  Remark 3.12.

The reader might wonder if we can replace the growth condition in (3.3) by a
monotonicity assumption like $\langle B_{t}(x,\mu),x\rangle\lesssim
1+|x|^{2}$; there are several difficulties in doing so. On a technical side,
one would need to control $\int_{0}^{t}\langle X_{s},\dot{Y}_{s}\rangle$,
which require either $Y$ to be of bounded variation or some more refined
integration theory (Itô, rough paths) to be available; recall that here we are
not imposing any assumption on $Y$ other than continuity.

More importantly, already in the case of standard ODEs there are examples of
finite time blow-up for suitable choices of $b$ and continuous $Y$, see [17,
Section 3.3]; the monotonicity assumption can be replaced by more refined
criteria, see [43] and the recent work [9], which however don’t seem to
transfer easily to the distribution-dependent setting.

###  3.2. Uniqueness

In this section we provide various conditions for uniqueness of solutions to
the DDSDE (3.1), thus extending already known results for ODEs and SDEs to the
the distribution dependent case. We do not aim at maximal generality, as the
criteria from Sections 3.2.1-3.2.3 could be combined together to create new
ones; rather we aim to illustrate how many classical assumptions on $B$ still
work perfectly without any need for $Y$ to be Brownian or Markovian.

Whenever possible, we try to keep uniqueness statements separate from their
existence counterparts (which in our setting always comes from Proposition
3.10) and only afterwards unite them together in a well-posedness result.
Depending on the DDSDE in consideration, ad-hoc existence results which go
beyond Assumption 3.8 might be available; they can still be combined with the
uniqueness criteria presented here, which are thus of independent interest.

####  3.2.1. Osgood-type condition

It is well known that classical ODEs admit a unique solution under Osgood type
conditions on the drift (originally proposed in [42]); we include here an
analogous statement for DDSDEs.

######  Assumption 3.13.

It holds that

$|B_{t}(x,\mu)-B_{t}(y,\nu)|\leq h_{t}\,f(|x-y|+d_{1}(\mu,\nu)),$ (3.6)

for all $(x,y)\in\mathbb{R}^{2d}$,
$(\mu,\nu)\in\mathcal{P}_{1}(\mathbb{R}^{d})^{2}$ and $t\in[0,T]$, where $h\in
L^{1}_{T}$ and $f:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ is a modulus of
continuity satisfying

$\int_{0}^{\varepsilon}\frac{1}{f(u)}du=\infty\quad{\rm
for\,\,any}\,\,\varepsilon>0.$ (3.7)

######  Remark 3.14.

By a result of Stechkin which can be found as [39, Lem. A], given any modulus
of continuity $f$, there exists another concave modulus $\tilde{f}$ satisfying
the two-sided bound $\tilde{f}\leq f\leq 2\tilde{f}$. Thus we can always
assume the modulus $f$ appearing in (3.7) to be concave, which also implies it
is increasing and subadditive.

Before proving our uniqueness results, we recall Bihari’s inequality, see e.g.
[4].

######  Proposition 3.15.

Let $f:[a,b]\to(0,+\infty)$ be a continuous, monotone, strictly positive
function and let $x$ and $h$ be two functions on an interval $[t_{0},t_{0}+T]$
such that $x([t_{0},t_{0}+T])\subset[a,b]$, $h\geq 0$. Assume that for some
$\alpha\in[a,b]$ one has

$x_{t}\leq\alpha+\int_{t_{0}}^{t}h_{s}\,f(x_{s})\mathop{}\\!\mathrm{d}s,\quad
t\in[t_{0},t_{0}+T];$

then

$x_{t}\leq
G^{-1}\left(G(a)+\int_{t_{0}}^{t}h_{s}\mathop{}\\!\mathrm{d}s\right),\quad
t\in[t_{0},t_{0}+\tilde{T}],$ (3.8)

where

$G(u):=\int_{x_{0}}^{u}\frac{1}{f(r)}\mathop{}\\!\mathrm{d}r$ (3.9)

and $\tilde{T}\leq T$ is chosen such that

$G(a)+\int_{t_{0}}^{t}h(s)\mathop{}\\!\mathrm{d}s\in{\rm
Dom}(G^{-1}),\quad\forall t\in[t_{0},t_{0}+\tilde{T}].$

######  Remark 3.16.

It’s easy to check, using the definition of $G$, that functions of the form
$a\mapsto G^{-1}(G(a)+\kappa)$ as appears on the r.h.s. of (3.8) do not depend
on the choice of $x_{0}$ (as long as $a$ belongs to the corresponding domain).
Since any concave modulus of continuity $f$ can grow at most linearly at
infinity, under the assumption (3.7) we deduce that $G$ satisfies $\lim_{x\to
0}G(x)=-\infty$, $\lim_{x\to+\infty}G(x)=+\infty$ and ${\rm
Dom}(G^{-1})=\mathbb{R}$, so that we can take $\tilde{T}=T$.

We now provide a stability estimate for solutions defined on the same
probability space; given two drifts
$B^{1},B^{2}:[0,T]\times\mathbb{R}^{d}\times\mathcal{P}_{1}(\mathbb{R}^{d})\to\mathbb{R}^{d}$,
we introduce the notation

$\|B^{1}-B^{2}\|_{\infty}:=\sup_{(s,x,\mu)\in[0,T]\times\mathbb{R}^{d}\times\mathcal{P}_{1}(\mathbb{R}^{d})}|B^{1}_{s}(x,\mu)-B^{2}_{s}(x,\mu)|.$

######  Proposition 3.17.

Let $B^{1}$ and $B^{2}$ be two drifts satisfying Assumption 3.13 for the same
$f$, $h$. Assume $X^{1}$ and $X^{2}$ are two solutions to (3.1) defined on the
same probability space with input $(Y^{1},B^{1})$ and $(Y^{2},B^{2})$
respectively. Then

$\mathbb{E}[\|X^{1}-X^{2}\|_{C_{T}}]\leq
G^{-1}\left(G\left(\mathbb{E}[\|Y^{1}-Y^{2}\|_{C_{T}}]+\|B^{1}-B^{2}\|_{\infty}T\right)+2\|h\|_{L^{1}_{T}}\right)$
(3.10)

where $G$ is defined in (3.9). In particular, pathwise uniqueness and
uniqueness in law hold for the DDSDE.

###### Proof.

We begin by observing that

$X^{1}_{t}-X^{2}_{t}=\int_{0}^{t}B^{1}_{s}(X^{1}_{s},\mathcal{L}(X^{1}_{s}))-B^{2}_{s}(X^{2}_{s},\mathcal{L}(X^{2}_{s}))\mathop{}\\!\mathrm{d}s+Y^{1}_{t}-Y^{2}_{t};$

adding and subtracting $B^{1}_{s}(X^{2}_{s},\mathcal{L}(X^{2}_{s}))$, applying
Assumption 3.13 and the monotonicity of $f$, we see that

$\displaystyle\sup_{r\leq t}|X^{1}_{r}-X^{2}_{r}|$
$\displaystyle\leq\|Y^{1}-Y^{2}\|_{C_{T}}+T\|B^{1}-B^{2}\|_{\infty}+\int_{0}^{t}h_{s}\,f(|X^{1}_{s}-X^{2}_{s}|+d_{1}(\mathcal{L}(X^{1}_{s}),\mathcal{L}(X^{2}_{s}))\,\mathop{}\\!\mathrm{d}s$
$\displaystyle\leq\|Y^{1}-Y^{2}\|_{C_{T}}+T\|B^{1}-B^{2}\|_{\infty}+\int_{0}^{t}h_{s}\,f\bigg{(}\sup_{r\leq
s}|X^{1}_{r}-X^{2}_{r}|+\mathbb{E}\Big{[}\,\sup_{r\leq
s}|X^{1}_{r}-X^{2}_{r}|\Big{]}\bigg{)}\,\mathop{}\\!\mathrm{d}s$

Taking expectation on both sides, using subadditivity and concavity of $f$
along with Jensen’s inequality, we get that

$\displaystyle\mathbb{E}\Big{[}\sup_{r\leq t}|X^{1}_{r}-X^{2}_{r}|\Big{]}$
$\displaystyle\leq\mathbb{E}[\,\|Y^{1}-Y^{2}\|_{C_{T}}]+T\|B^{1}-B^{2}\|_{\infty}+\int_{0}^{t}2h_{s}\,f\bigg{(}\mathbb{E}\Big{[}\sup_{r\leq
s}|X^{1}_{r}-X^{2}_{r}|\Big{]}\bigg{)}\,\mathop{}\\!\mathrm{d}s.$

Applying Proposition 3.15 readily gives estimate (3.10).

Regarding the last statement, observe that for $Y^{1}=Y^{2}$, $B^{1}=B^{2}$
estimate (3.10) implies $X^{1}\equiv X^{2}$, proving pathwise uniqueness.
Uniqueness in law then follows from Proposition 3.5. ∎

Combining Assumptions 3.8 and 3.13 we can obtain strong existence and
uniqueness of solutions; moreover we can recast the stability estimate (3.10)
into a corresponding bound on the Wasserstein distance between the laws of
solutions.

######  Theorem 3.18.

Let $B^{1}$ and $B^{2}$ satisfy Assumption 3.8, with $p=1$ and Assumption 3.13
for the same $f$, $h$ and let
$\mu^{Y^{1}},\,\mu^{Y^{2}}\in\mathcal{P}_{1}(C_{T})$ be two input laws. Then
for $i=1,\,2$ there exist unique weak solutions, $\mu^{X^{i}}$ associated to
$(B^{i},\mu^{Y^{i}})$. Furthermore, there exists a continuous, increasing
function $M:\mathbb{R}_{+}\to\mathbb{R}_{+}$, $M(0)=0$, only depending on $f$
and $h$, such that

$d_{1}(\mu^{X^{1}},\mu^{X^{2}})\leq
M\left(d_{1}(\mu^{Y^{1}},\mu^{Y^{2}})+T\|B^{1}-B^{2}\|_{\infty}\right).$
(3.11)

###### Proof.

Since both $B^{1}$ and $B^{2}$ satisfy Assumption 3.8, weak existence holds by
Proposition 3.10; Assumption 3.13 and Proposition 3.7 then imply strong
existence as well. Pathwise uniqueness and uniqueness in law instead are
consequences of Proposition 3.17.

Now let $(Y^{1},Y^{2})$ be an optimal coupling for
$d_{1}(\mu^{Y^{1}},\mu^{Y^{2}})$, which induces a coupling $(X^{1},X^{2})$ for
$\mu^{X^{1}},\mu^{X^{2}}$ by strong existence and uniqueness in law. Define
$M:(0,+\infty)\to(0,+\infty)$ by

$M(r):=G^{-1}\big{(}G(r)+2\|h\|_{L^{1}_{T}}\big{)}$

for $G$ defined as in (3.9); by the properties $\lim_{x\to 0}G(x)=-\infty$,
$\lim_{x\to-\infty}G^{-1}(x)=0$ we deduce that $M$ extends uniquely and
continuously at $r=0$ with $M(0)=0$.

But then inequality (3.10) and the choice of optimal coupling $(Y^{1},Y^{2})$
readily implies

$\displaystyle
d_{1}(\mu^{X^{1}},\mu^{X^{2}})\leq\mathbb{E}[\|X^{1}-X^{2}\|_{C_{T}}]\leq
M\big{(}\mathbb{E}[\|Y^{1}-Y^{2}\|_{C_{T}}]+T\|B^{1}-B^{2}\|_{\infty}\big{)}$

and the conclusion follows. ∎

Let us provide some relevant examples of drifts $B$ satisfying Assumption 3.13
above.

######  Example 3.19.

Consider the case of a true McKean–Vlasov drift, i.e. $B$ of the form

$B_{t}(x,\mu)=\int_{\mathbb{R}^{d}}b_{t}(x,y)\,\mu(\mathop{}\\!\mathrm{d}y)$

for some measurable function
$b:[0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{R}^{d}$. Then
Assumption 3.13 is satisfied if there exist $h\in L^{1}_{T}$, and a (concave)
modulus of continuity $f$ satisfying (3.7) such that

$|b_{t}(x,z)-b_{t}(y,z^{\prime})|\leq h_{t}\,f(|x-y|+|z-z^{\prime}|).$

Indeed, for any $\mu,\nu\in\mathcal{P}_{1}(\mathbb{R}^{d})$ and any coupling
$m\in\Pi(\mu,\nu)$ it holds

$\displaystyle|B_{t}(x,\mu)-B_{t}(y,\nu)|$
$\displaystyle=\Big{|}\int_{\mathbb{R}^{d}}b_{t}(x,z)\mu(\mathop{}\\!\mathrm{d}z)-\int_{\mathbb{R}^{d}}b_{t}(y,z^{\prime})\nu(\mathop{}\\!\mathrm{d}z^{\prime})\Big{|}$
$\displaystyle\leq\int_{\mathbb{R}^{2d}}|b_{t}(x,z)-b_{t}(y,z^{\prime})|\,m(\mathop{}\\!\mathrm{d}z,\mathop{}\\!\mathrm{d}z^{\prime})$
$\displaystyle\leq
h_{t}\int_{\mathbb{R}^{2d}}f(|x-y|+|z-z^{\prime}|)\,m(\mathop{}\\!\mathrm{d}z,\mathop{}\\!\mathrm{d}z^{\prime})$
$\displaystyle\leq
h_{t}\,\Big{[}f(|x-y|)+\int_{\mathbb{R}^{2d}}f(|z-z^{\prime}|)\,m(\mathop{}\\!\mathrm{d}z,\mathop{}\\!\mathrm{d}z^{\prime})\Big{]}$
$\displaystyle\leq
h_{t}\,\Big{[}f(|x-y|)+f\Big{(}\int_{\mathbb{R}^{2d}}|z-z^{\prime}|\,m(\mathop{}\\!\mathrm{d}z,\mathop{}\\!\mathrm{d}z^{\prime})\Big{)}\Big{]}$

where we used monotonicity, subadditivity and concavity of $f$ as well as
Jensen inequality; minimizing over $m\in\Pi(\mu,\nu)$ gives (3.6).

Of particular relevance are convolutional drifts
$B_{t}(x,\mu)=(b_{t}\ast\mu)(x)=\int
b_{t}(x-z)\,\mu(\mathop{}\\!\mathrm{d}z)$, in which case the above condition
reduces to

$\displaystyle|b_{t}(x)-b_{t}(y)|\leq
h_{t}\,f(|x-y|),\quad\forall\,(t,x,y)\in[0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}.$

####  3.2.2. Monotone drifts

Another classical assumption in ODE theory, that goes beyond the Lipschitz
setting, is a monotonicity condition (sometimes also referred to as one-sided
Lipschitz condition). Similar assumptions in the DDSDE setting have been
employed for example in [29, Sec. 5]; see also the more recent works [52]
(Assumption (H2) therein) and [23]. Finally, let us mention that monotonicity
may be interpreted as a Lyapunov condition on the energy $|\cdot|^{2}$ and is
therefore also close in spirit to the assumptions considered in [40], [30].

We consider the following working assumption for the drift $B$.

######  Assumption 3.20.

Given $p\in(1,\infty)$, there exists an $h\in L^{1}_{T}$ such that

$\langle x-y,B_{t}(x,\mu)-B_{t}(y,\nu)\rangle\leq
h_{t}\big{[}|x-y|^{2}+|x-y|\,d_{p}(\mu,\nu)\big{]}$ (3.12)

uniformly over $x,y\in\mathbb{R}^{d}$, and
$\mu,\nu\in\mathcal{P}_{p}(\mathbb{R}^{d})$.

######  Proposition 3.21.

Let
$B:[0,T]\times\mathbb{R}^{d}\times\mathcal{P}_{p}(\mathbb{R}^{d})\rightarrow\mathbb{R}^{d}$
be a measurable map satisfying Assumption 3.20 for some $p\in(1,\infty)$; then
pathwise uniqueness and uniqueness in law hold for (3.1).

###### Proof.

By virtue of Proposition 3.5, we only need to establish pathwise uniqueness.
Let $X^{1},X^{2}$ be two solutions to (3.1) defined on the same probability
space and with same input $Y$; setting $\mu^{i}_{t}=\mathcal{L}(X^{i}_{t})$
for $i=1,2$, it holds that

$X^{i}_{t}=\int_{0}^{t}B_{s}(X^{i}_{s},\mu^{i}_{s})\,\mathop{}\\!\mathrm{d}s+Y_{t}$

and the difference $X^{1}-X^{2}$ is an absolutely continuous path. By
Assumption 3.20 we have that

$\displaystyle\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}|X^{1}_{t}-X^{2}_{t}|^{p}$
$\displaystyle=p\,|X_{t}^{1}-X_{t}^{2}|^{p-2}\,\langle
X^{1}_{t}-X^{2}_{t},B_{t}(X^{1}_{t},\mu^{1}_{t})-B_{t}(X^{2}_{t},\mu^{2}_{t})\rangle$
$\displaystyle\leq
p\,h_{t}\big{[}|X^{1}_{t}-X^{2}_{t}|^{p}+|X^{1}_{t}-X^{2}_{t}|^{p-1}\,d_{p}(\mu^{1}_{t},\mu^{2}_{t})\big{]}$
$\displaystyle\lesssim_{p}h_{t}\left(|X^{1}_{t}-X^{2}_{t}|^{p}+d_{p}(\mu^{1}_{t},\mu^{2}_{t})^{p}\right)$

where in the last passage we used the basic inequality $ab\lesssim
a^{\frac{p}{p-1}}+b^{p}$. Since $X^{1}_{0}=Y_{0}=X^{2}_{0}$, integrating and
taking expectations we obtain

$\displaystyle\mathbb{E}\big{[}|X^{1}_{t}-X^{2}_{t}|^{p}\big{]}\lesssim\int_{0}^{t}h_{s}\Big{(}\mathbb{E}[|X^{1}_{s}-X^{2}_{s}|^{p}]+d_{p}(\mu^{1}_{s},\mu^{2}_{s})^{p}\Big{)}\mathop{}\\!\mathrm{d}s\lesssim\int_{0}^{t}h_{s}\mathbb{E}[|X^{1}_{s}-X^{2}_{s}|^{p}]\mathop{}\\!\mathrm{d}s$

and by Grönwall we deduce that $\mathbb{E}[|X^{1}_{t}-X^{2}_{t}|^{p}]=0$ for
all $t\in[0,T]$; since $X^{i}$ are continuous paths, pathwise uniqueness
follows. ∎

######  Remark 3.22.

If $p\geq 2$, we can further weaken condition (3.12) by requiring instead

$\langle x-y,B_{t}(x,\mu)-B_{t}(y,\nu)\rangle\leq
h_{t}\big{[}|x-y|^{2}+d_{p}(\mu,\nu)^{2}\big{]}$ (3.13)

uniformly over $t,x,y,\mu,\nu$. Indeed, going through the same computations as
in the proof above, gives

$\displaystyle\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}t}|X^{1}_{t}-X^{2}_{t}|^{p}$
$\displaystyle\leq
p\,h_{t}\big{[}|X^{1}_{t}-X^{2}_{t}|^{p}+|X^{1}_{t}-X^{2}_{t}|^{p-2}\,d_{p}(\mu^{1}_{t},\mu^{2}_{t})^{2}\big{]}$
$\displaystyle\lesssim_{p}h_{t}\big{[}|X^{1}_{t}-X^{2}_{t}|^{p}+d_{p}(\mu^{1}_{t},\mu^{2}_{t})^{p}\big{]}$

where we used the basic inequality $ab\lesssim
a^{\frac{p}{p-2}}+b^{\frac{p}{2}}$ which holds since $p/2\geq 1$.

######  Example 3.23.

It is clear that any Lipschitz $B$ satisfies Assumption 3.20; more generally,
one can take $B_{t}(x,\mu)=F_{t}(x)+G(x,\mu)$ for globally Lipschitz $G$ and
$F$ satisfying

$\langle F_{t}(x)-F_{t}(y),x-y\rangle\leq h_{t}|x-y|^{2}.$ (3.14)

A similar computation holds for $B_{t}(x,\mu)=F_{t}(x)G(\mu)$ with $F$ as
above, once we additionally impose $\|F_{t}\|_{L^{\infty}}\leq h_{t}$ and
$G:\mathcal{P}_{p}(\mathbb{R}^{d})\to\mathcal{P}_{p}(\mathbb{R}^{d})$ is
globally Lipschitz and bounded.

In the case $h\equiv C$ for some $C\in\mathbb{R}$, condition (3.14) is
satisfied if $F=-\nabla V$ for some $V:\mathbb{R}^{d}\to\mathbb{R}$ such that
$x\mapsto V(x)+\lambda|x|^{2}/2$ is convex for some $\lambda\geq 0$;
interesting cases include $F(x)=-xe^{|x|^{2}}$ and $F(x)=-|x|^{\gamma-1}x$ for
$\gamma\geq 0$ where for $\gamma\in[0,1)$, $F$ is not even locally Lipschitz.

Although Assumption 3.20 covers a wide range of drifts, sometimes it’s natural
to further weaken it to an integrated monotonicity assumption.

######  Lemma 3.24.

Suppose
$B:[0,T]\times\mathbb{R}^{d}\times\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}^{d}$
satisfies

$\int_{\mathbb{R}^{2d}}\langle
x-y,B_{t}(x,\mu)-B_{t}(y,\nu)\rangle\,m(\mathop{}\\!\mathrm{d}x,\mathop{}\\!\mathrm{d}y)\leq
h_{t}\,\int_{\mathbb{R}^{2d}}|x-y|^{2}\,m(\mathop{}\\!\mathrm{d}x,\mathop{}\\!\mathrm{d}y)$
(3.15)

uniformly over all possible $t\in[0,T]$,
$\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $m\in\Pi(\mu,\nu)$, for some
integrable $h:[0,T]\to\mathbb{R}$; then pathwise uniqueness and uniqueness in
law hold for the DDSDE.

###### Proof.

By virtue of Proposition 3.5, we only need to establish pathwise uniqueness.
Let $X^{1}$,$X^{2}$ be two solutions defined on the same probability space
$(\Omega,\mathcal{F},\mathbb{P})$ and with same input $Y$. It follows that
$X^{1}-X^{2}$ is a process of bounded variation satisfying $\mathbb{P}$-a.s.

$|X^{1}_{t}-X^{2}_{t}|^{2}=2\int_{0}^{t}\langle
X^{1}_{t}-X^{2}_{t},B_{s}(X^{1}_{s},\mathcal{L}(X^{1}_{s}))-B_{s}(X^{2}_{s},\mathcal{L}(X^{2}_{s}))\rangle\,\mathop{}\\!\mathrm{d}s\quad\forall\,t\in[0,T].$

Taking expectation and applying assumption (3.15) for
$m=\mathbb{P}\circ(X^{1}_{s},X^{2}_{s})^{-1}\in\Pi(\mathcal{L}(X^{1}_{s}),\mathcal{L}(X^{2}_{s}))$,
we arrive at

$\displaystyle\mathbb{E}[|X^{1}_{t}-X^{2}_{t}|^{2}]\leq
2\int_{0}^{t}h_{s}\,\mathbb{E}[|X^{1}_{s}-X^{2}_{s}|^{2}]\,\mathop{}\\!\mathrm{d}s$

and the conclusion follows by Grönwall’s lemma. ∎

######  Example 3.25.

Let $b:\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ be a function
satisfying

$b(x,x^{\prime})=-b(x^{\prime},x),\quad\langle
b(x,x^{\prime})-b(y,y^{\prime}),x-x^{\prime}-y+y^{\prime}\rangle\leq
C(|x-y|^{2}+|x^{\prime}-y^{\prime}|^{2})$

for some constant $C\in\mathbb{R}$, uniformly over
$x,x^{\prime},y,y^{\prime}\in\mathbb{R}^{d}$; then the McKean-Vlasov drift
$B_{t}(x,\mu)=\int_{\mathbb{R}^{d}}b(x,x^{\prime})\,\mu(\mathop{}\\!\mathrm{d}x^{\prime})$
satisfies condition (3.15) with $h_{t}=C$.

Indeed, for any $m\in\Pi(\mu,\nu)$ it holds

$\displaystyle\int_{\mathbb{R}^{2d}}\langle x-y,$ $\displaystyle
B_{t}(x,\mu)-B_{t}(y,\nu)\rangle\,m(\mathop{}\\!\mathrm{d}x,\mathop{}\\!\mathrm{d}y)$
$\displaystyle=\int_{\mathbb{R}^{2d}\times\mathbb{R}^{2d}}\langle
x-y,b(x,x^{\prime})-b(y,y^{\prime})\rangle\,m(\mathop{}\\!\mathrm{d}x,\mathop{}\\!\mathrm{d}y)m(\mathop{}\\!\mathrm{d}x^{\prime},\mathop{}\\!\mathrm{d}y^{\prime})$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}^{2d}\times\mathbb{R}^{2d}}\langle
x-y-x^{\prime}+y^{\prime},b(x,x^{\prime})-b(y,y^{\prime})\rangle\,m(\mathop{}\\!\mathrm{d}x,\mathop{}\\!\mathrm{d}y)m(\mathop{}\\!\mathrm{d}x^{\prime},\mathop{}\\!\mathrm{d}y^{\prime})$

where the last passage follows by exchanging $(x,y)$ and
$(x^{\prime},y^{\prime})$ and applying $b(x^{\prime},x)=-b(x,x^{\prime})$; by
the monotonicity condition on $b$ we readily obtain the conclusion.

For convolutional drifts $B=b\ast\mu$ it suffices to require

$b(-x)=-b(x),\quad\langle x-y,b(x)-b(y)\rangle\leq
2C|x-y|^{2}\quad\forall\,x,y\in\mathbb{R}^{d};$

as in Example 3.23, this is true for instance if $b(x)=-\nabla V(x)$ for some
even function $V:\mathbb{R}^{d}\to\mathbb{R}$ such that $x\mapsto
V(x)+\lambda|x|^{2}$ is convex fr some $\lambda\geq 0$. This class includes
drifts not satisfying the linear growth assumption 3.8, but for which
uniqueness still holds.

Let us mention that the special cases $b(x)=-\lambda|x|^{\gamma-1}x$ for
$\lambda,\gamma>0$ are also the drift terms considered in [52, 26, 13]; the
setting of these works however, also includes a non-linear diffusion term,
associated to kernels of the form $a(x)=|x|^{\gamma-1}(|x|^{2}\,I_{d}-x\otimes
x)$.

It follows from the above results that if $B$ satisfies Assumption 3.8 and one
between Assumption 3.20 or conditions (3.13), (3.15), then for any
$\mu^{Y}\in\mathcal{P}_{p}(C_{T})$ there exists a unique solution
$\mu^{X}\in\mathcal{P}_{p}(C_{T})$; this correspondence defines a solution map
$S^{B}:\mathcal{P}_{p}(C_{T})\to\mathcal{P}_{p}(C_{T})$. Unfortunately we
cannot establish Lipschitz continuity for $S^{B}$, the issue being similar to
that of Remark 3.12 and related to controlling $\int_{0}^{t}\langle
X^{1}_{s}-X^{2}_{s},\mathop{}\\!\mathrm{d}(Y^{1}_{s}-Y^{2}_{s})\rangle$; we
can still at least show continuity.

######  Lemma 3.26.

Let the drift $B$ satisfy the above assumptions, then the solution map
$S^{B}:\mathcal{P}_{p}(C_{T})\to\mathcal{P}_{p}(C_{T})$ is continuous.

###### Proof.

Given a sequence $\\{\mu^{Y^{n}}\\}_{n}$ with $\mu^{Y^{n}}\to\mu^{Y}$ in
$\mathcal{P}_{p}(C_{T})$, we need to show that
$\mu^{X^{n}}=S^{B}(\mu^{Y^{n}})\to S^{B}(\mu^{Y})=\mu^{X}$. By the assumption
and estimate (3.4) it follows that $(\mu^{Y^{n}},\mu^{X^{n}})$ is tight in
$\mathcal{P}(C_{T}\times C_{T})$, so by Prokhorov’s theorem we can extract a
(not relabelled) subsequence s.t.
$(\mu^{Y^{n}},\mu^{X^{n}})\rightharpoonup(\mu^{Y},\mu^{Z})$ for a suitable
$\mu^{Z}\in\mathcal{P}(C_{T})$. By Skorokhod’s theorem, we can work on a
common probability space where all processes $(X^{n},Y^{n})$ and $(Z,Y)$ are
well defined; using the fact that $Y^{n}\to Y\in L^{p}(\Omega;C_{T})$, $X^{n}$
satisfy (3.4) and arguing as in the proof of Proposition 3.10 we deduce that
$X^{n}\to Z$ in $L^{p}(\Omega;C_{T})$ and that $Z$ solves the DDSDE associated
to $Y$. But then by uniqueness $Z=X$ and $\mu^{X^{n}}\to\mu^{X}$ in
$\mathcal{P}_{p}(C_{T})$; as the reasoning works for any possible subsequence,
conclusion follows. ∎

####  3.2.3. Locally Lipschitz drifts with growth conditions

In the classical ODE and SDE setting, it is well known that uniqueness holds
as soon as the drift is merely locally Lipschitz; the same is not true for
DDSDEs, with explicit counterexamples given in [45]. Since global Lipschitz
assumptions are often too restrictive to work with, we provide here some
uniqueness results for drifts with local Lipschitz constant not growing too
fast; the price one has to pay is a strong integrability requirement on the
input $\mu^{Y}$. Similar type of conditions have been proposed in the
classical work [7] and more recently in [52], [24].

######  Assumption 3.27.

Given $\alpha>0$ and $p\geq 1$; the drift
$B:[0,T]\times\mathbb{R}^{d}\times\mathcal{P}(\mathbb{R}^{d})\to\mathbb{R}^{d}$
and input data $\mu^{Y}$ satisfy the following:

  1. i.

$|B_{t}(x,\mu)|\leq h_{t}(1+|x|+\|\mu\|_{p})$,

  2. ii.

$|B_{t}(x,\mu)-B_{t}(y,\nu)|\leq
g_{t}(|x-y|+d_{p}(\mu,\nu))(1+|x|^{\alpha}+|y|^{\alpha}+\|\mu\|_{p}^{\alpha}+\|\nu\|_{p}^{\alpha})$,

  3. iii.

$\int_{C_{T}}\exp(\lambda\|\omega\|_{C_{T}}^{\alpha})\mu^{Y}(\mathop{}\\!\mathrm{d}\omega)<\infty$
for all $\lambda\in\mathbb{R}$,

for integrable functions $h,g\in L^{1}_{T}$.

To obtain a stability estimate under Assumption 3.27, we need the following
simple lemma.

######  Lemma 3.28.

For any $\gamma\in(0,1)$ there exists a $C>0$ such that the map
$\mathbb{R}^{d}\ni x\mapsto e^{(C+|x|)^{\gamma}}$ is convex.

###### Proof.

We reduce to the one dimensional case by arguing component-wise. Then note
that it suffices to chose $C=C(\gamma)>0$ such that $\gamma
C^{\gamma}+\gamma-1\geq 0$ and to check that the resulting second derivative
is non-negative. ∎

######  Proposition 3.29.

Under Assumption 3.27, strong existence, pathwise uniqueness and uniqueness in
law hold for the DDSDE (3.1) associated to $\mu^{Y}$. Moreover for any $q>p$
there exists a constant $C=C(\|h\|_{L^{1}_{T}},\|g\|_{L^{1}_{T}},\alpha,p,q)$
such that, for any two solutions $\mu^{X^{1}}$ and $\mu^{X^{2}}$ driven by the
inputs $\mu^{Y^{1}}$ and $\mu^{Y^{2}}$ respectively, we have the stability
estimate:

$d_{p}(\mu^{X^{1}},\mu^{X^{2}})\leq
C\bigg{(}\int_{C_{T}}\exp(C\|\omega\|_{C_{T}}^{\alpha})\,(\mu^{Y^{1}}+\mu^{Y^{2}})(\mathop{}\\!\mathrm{d}\omega)\bigg{)}\,d_{q}(\mu^{Y^{1}},\mu^{Y^{2}}).$
(3.16)

###### Proof.

Assumption 3.27 implies that the hypothesis of Proposition 3.10 are met, thus
weak existence holds. Moreover any solution $X$ satisfies the a priori
estimate (3.4), namely $\|X\|_{C_{T}}\lesssim_{h}\|Y\|_{C_{T}}$.

Strong existence follows from weak existence and Proposition 3.7; by
Proposition 3.5, weak uniqueness and pathwise uniqueness are equivalent and
will follow from the stability estimate.

Now let $(Y^{1},Y^{2})$ be an optimal coupling for
$d_{q}(\mu^{Y^{1}},\mu^{Y^{2}})$; by Remark 3.6, given any pair of weak
solutions $\mu^{X^{1}},\mu^{X^{2}}$, we can construct an associated coupling
$(X^{1},X^{2})$ and thus work with all variables defined on the same
probability space. Let us set $\mu^{i}_{t}:=\mathcal{L}(X^{i}_{t})$.

By point ii. of Assumption 3.27, the difference $X^{1}-X^{2}$ satisfies

$\displaystyle|X^{1}_{t}-X^{2}_{t}|$
$\displaystyle\leq|Y^{1}_{t}-Y^{2}_{t}|+\int_{0}^{t}g_{s}\,(|X^{1}_{s}-X^{2}_{s}|+d_{p}(\mu^{1}_{s},\mu^{2}_{s}))(1+|X^{1}_{s}|^{\alpha}+|X^{2}_{s}|^{\alpha}+\|\mu^{1}_{s}\|_{p}^{\alpha}+\|\mu^{2}_{s}\|_{p}^{\alpha})\,\mathop{}\\!\mathrm{d}s.$

Setting
$Z_{t}:=1+|X^{1}_{t}|^{\alpha}+|X^{2}_{t}|^{\alpha}+\|\mu^{1}_{t}\|_{p}^{\alpha}+\|\mu^{2}_{t}\|_{p}^{\alpha}$,
it follows from the pathwise bound (3.4) that

$\|Z\|_{C_{T}}\lesssim_{h,\alpha}1+\|Y^{1}\|_{C_{T}}^{\alpha}+\|Y^{2}\|_{C_{T}}^{\alpha}+\|\mu^{Y^{1}}\|_{p}^{\alpha}+\|\mu^{Y^{2}}\|_{p}^{\alpha}\quad\mathbb{P}\text{-a.s.};$
(3.17)

moreover by Grönwall, for any $t\in[0,T]$ we have the $\mathbb{P}$-a.s.
pathwise estimate

$\displaystyle\sup_{r\leq t}|X^{1}_{r}-X^{2}_{r}|$
$\displaystyle\leq\exp\bigg{(}\int_{0}^{t}g_{s}Z_{s}\mathop{}\\!\mathrm{d}s\bigg{)}\left(\|Y^{1}-Y^{2}\|_{C_{T}}+\int_{0}^{t}g_{s}\,d_{p}(\mu^{1}_{s},\mu^{2}_{s})\,Z_{s}\mathop{}\\!\mathrm{d}s\right)$
(3.18)
$\displaystyle\lesssim\exp\Big{(}(1+\|g\|_{L^{1}_{T}})\|Z\|_{C_{T}}\Big{)}\left(\|Y^{1}-Y^{2}\|_{C_{T}}+\int_{0}^{t}g_{s}\,d_{p}(\mu^{1}_{s},\mu^{2}_{s})\,\mathop{}\\!\mathrm{d}s\right)$

Now taking the $L^{p}_{\Omega}$-norm on both sides, using an Hölder inequality
on the r.h.s. (using the fact that $q>p$), we can find another constant
$C^{\prime}=C^{\prime}(\|g\|_{L^{1}_{T}},p,q)$ such that

$\displaystyle\mathbb{E}\Big{[}\,\sup_{r\leq
t}|X^{1}_{r}-X^{2}_{r}|^{p}\Big{]}^{1/p}$
$\displaystyle\lesssim\mathbb{E}[\exp(C^{\prime}\|Z\|_{C_{T}})]\,\left(\mathbb{E}[\|Y^{1}-Y^{2}\|_{C_{T}}^{q}]^{1/q}+\int_{0}^{t}g_{s}\,d_{p}(\mu^{1}_{s},\mu^{2}_{s})\,\mathop{}\\!\mathrm{d}s\right)$
$\displaystyle\lesssim\mathbb{E}[\exp(C^{\prime}\|Z\|_{C_{T}})]\,\left(d_{q}(\mu^{Y^{1}},\mu^{Y^{2}})+\int_{0}^{t}g_{s}\,\mathbb{E}\Big{[}\,\sup_{r\leq
s}|X^{1}_{r}-X^{2}_{r}|^{p}\Big{]}^{1/p}\,\mathop{}\\!\mathrm{d}s\right).$

Applying Grönwall’s lemma again, we deduce that

$\displaystyle
d_{p}(\mu^{X^{1}},\mu^{X^{2}})\leq\mathbb{E}[\|X^{1}-X^{2}\|_{C_{T}}^{p}]^{1/p}\lesssim
e^{\|g\|_{L^{1}_{T}}}\,\mathbb{E}[\exp(C^{\prime}\|Z\|_{C_{T}})]\,d_{q}(\mu^{Y^{1}},\mu^{Y^{2}}).$

Now consider the quantities
$\exp(\|\mu^{Y^{i}}\|^{\alpha}_{p})=\exp(\mathbb{E}[\|Y^{i}\|^{p}_{C_{T}}]^{\alpha/p})$
for $i=1,\,2$. In the case $\alpha>p$ one can directly apply Jensen to give
that
$\exp(\mathbb{E}[\|Y^{i}\|^{p}_{C_{T}}]^{\alpha/p})\leq\mathbb{E}[\exp(\|Y^{i}\|^{\alpha}_{C_{T}})]$.
For $\alpha<p$, we first apply Lemma 3.28 with $\gamma=\alpha/p$, followed by
Jensen to give, for some $C_{\gamma}>0$,

$\displaystyle\exp\left(\mathbb{E}[\|Y^{i}\|^{p}_{C_{T}}]^{\alpha/p}\right)\leq\exp\left((C_{\gamma}+\mathbb{E}[\|Y^{i}\|^{p}_{C_{T}}])^{\alpha/p}\right)$
$\displaystyle\leq\mathbb{E}\left[\exp((C_{\gamma}+\|Y^{i}\|^{p}_{C_{T}})^{\alpha/p})\right]$
$\displaystyle\lesssim_{\alpha,p}\mathbb{E}\left[\exp(\|Y^{i}\|^{\alpha}_{C_{T}})\right].$

In either case we have
$\exp(\|\mu^{Y^{i}}\|^{\alpha}_{p})\lesssim_{\alpha,p}\mathbb{E}[\exp(\|Y^{i}\|^{\alpha}_{C_{T}})]$
and so combined with (3.17) we finally obtain the stability estimate (3.16)
for some new constant $C$.

As the estimate holds for any possible pair of solutions $X^{1},X^{2}$, taking
$Y^{1}=Y^{2}$ we deduce pathwise uniqueness and this completes the proof. ∎

######  Remark 3.30.

It’s clear from the proof above that Assumption 3.27 admits other variants,
for instance we could instead require, i., iii. and

  1. ii′.

$|B_{t}(x,\mu)-B_{t}(y,\nu)|\leq
g_{t}(|x-y|+d_{p}(\mu,\nu))(1+|x|^{\alpha}+|y|^{\alpha}+\langle
e^{c\,|\cdot|^{\alpha}},\mu\rangle+\langle
e^{c\,|\cdot|^{\alpha}},\nu\rangle)$

for some $c>0$. The only difference is that in this case existence of
solutions doesn’t follow from a straightforward application of Proposition
3.10. However, it’s easy to check that, due to the available a priori
estimates, the same compactness argument can be readapted in this setting.

The next example provides concrete choices of $B$ satisfying Assumption 3.27.

######  Example 3.31.

If $Y=\xi+W$, where $\xi$ has Gaussian tails and $W$ is a continuous Gaussian
process (e.g. fractional Brownian motion), then condition iii. is satisfied
for any $\alpha\in[0,2)$.

Suppose $b:[0,T]\times\mathbb{R}^{d}\times\mathbb{R}^{d}\to\mathbb{R}^{d}$
satisfies for some $\alpha>0$

$\displaystyle|b_{t}(x,y)|$ $\displaystyle\leq g_{t}(1+|x|+|y|),$
$\displaystyle|b_{t}(x,x^{\prime})-b_{t}(y,y^{\prime})|$ $\displaystyle\leq
g_{t}(|x-y|+|x^{\prime}-y^{\prime}|)(1+|x|^{\alpha}+|y|^{\alpha}+|x^{\prime}|^{\alpha}+|y^{\prime}|^{\alpha}),$

for some integrable function $g$. Then a few elementary computations reveal
that

$B_{t}(x,\mu):=\int_{\mathbb{R}^{d}}b_{t}(x,x^{\prime})\mu(\mathop{}\\!\mathrm{d}x^{\prime})$

satisfies conditions i., ii. for $p=\alpha+1$. Indeed by Hölder’s and
Minkowski’s inequalities

$\displaystyle|B_{t}(x,\mu)-B_{t}(y,\nu)|$
$\displaystyle\leq\int_{\mathbb{R}^{2d}}|b_{t}(x,x^{\prime})-b_{t}(y,y^{\prime})|m(\mathop{}\\!\mathrm{d}x^{\prime},\mathop{}\\!\mathrm{d}y^{\prime})$
$\displaystyle\leq
g_{t}\bigg{(}|x-y|+\Big{(}\int_{\mathbb{R}^{2d}}|x^{\prime}-y^{\prime}|^{p}\,m(\mathop{}\\!\mathrm{d}x^{\prime},\mathop{}\\!\mathrm{d}y^{\prime})\Big{)}^{1/p}\bigg{)}\times$
$\displaystyle\quad\times\bigg{(}1+|x|^{\alpha}+|y|^{\alpha}+\Big{(}\int_{\mathbb{R}^{2d}}(|x^{\prime}|^{\alpha}+|y^{\prime}|^{\alpha})^{p^{\prime}}\,m(\mathop{}\\!\mathrm{d}x^{\prime},\mathop{}\\!\mathrm{d}y^{\prime})\Big{)}^{1/p^{\prime}}\bigg{)}$
$\displaystyle\leq
g_{t}(|x-y|+d_{p}(\mu,\nu))(1+|x|^{\alpha}+|y|^{\alpha}+\|\mu\|_{p^{\prime}\alpha}^{\alpha}+\|\nu\|_{p^{\prime}\alpha}^{\alpha})$

once we take $m$ to be an optimal coupling for $d_{p}(\mu,\nu)$; choosing
$p=\alpha+1$, so that $p^{\prime}\alpha=p$, we get the desired estimate for
ii. and condition i. can be checked similarly.

The above assumption on $b$ is satisfied for instance if $Db$ satisfies a
growth condition of the form $|Db_{t}(x,x^{\prime})|\lesssim
g_{t}(1+|x|^{\alpha}+|x^{\prime}|^{\alpha})$, for all
$(x,x^{\prime})\in\mathbb{R}^{2d}$.

##  4\. DDSDEs with convolutional structure

In this section we will focus on a particular subcase of (3.1), given by
drifts with the convolutional structure $B_{t}(x,\mu)=(b_{t}\ast\mu)(x)$.
Contrary to the previous sections, we now impose the noise term $Y$ to be of
the form $Y_{t}=\xi+W_{t}$ where $\xi$ and $W$ are independent random
variables taking respectively values in $\mathbb{R}^{d}$ and $C_{T}$; the
reason for this, which will become clearer later on, is to exploit the
transport structure to show that some integrability features of
$\mathcal{L}(\xi)$ are propagated at positive times. Still, no assumption on
$W$ are imposed apart from its continuity.

The DDSDE in consideration therefore becomes

$X_{t}=\xi+\int_{0}^{t}(b_{s}\ast\mathcal{L}(X_{s}))(X_{s})\,\mathrm{d}s+W_{t}=\xi+\int_{0}^{t}\int_{\mathbb{R}^{d}}b_{s}(X_{s}-y)\,\mathcal{L}(X_{s})(\mathop{}\\!\mathrm{d}y)+W_{t}$
(4.1)

for some measurable $b:[0,T]\times\mathbb{R}^{d}\to\mathbb{R}^{d}$.

The convolutional structure arises naturally in the study of particle systems
with pairwise interactions; it also allows to discuss kernels $b$ with poor
spatial regularity, possibly unbounded and merely integrable.

Let us start by providing a suitable notion of solution to equation (4.1); as
usual, we work for simplicity on a finite time interval $[0,T]$.

######  Definition 4.1 (Weak Solutions II).

Let $b:\mathbb{R}_{+}\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ be a
measurable map. We say that a tuple $(\Omega,\mathcal{F},\mathbb{P};X,W,\xi)$,
consisting of a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a
measurable map $(X,W,\xi):\Omega\rightarrow C_{T}\times
C_{T}\times\mathbb{R}^{d}$ is a weak solution to (4.1) on $[0,T]$ if:

  1. (1)

For all $t>0$, $\mathcal{L}(X_{t})=\mathbb{P}\circ X^{-1}_{t}=:\mu_{t}$ is
absolutely continuous w.r.t. Lebesgue.

  2. (2)

There exists a measurable representative of
$(t,x)\mapsto|b_{t}|\ast\mathcal{L}(X_{t})(x)$ such that

$\int_{0}^{T}|b_{s}|\ast\mathcal{L}(X_{s})(X_{s})\mathop{}\\!\mathrm{d}s<\infty,\quad\mathbb{P}\text{-a.s..}$

  3. (3)

The relation (4.1) holds $\mathbb{P}$-a.s., the integral being interpreted in
the Lebesgue sense.

######  Remark 4.2.

Definition 4.1 is insensitive to the choice of measurable representative for
$b_{t}\ast\mathcal{L}(X_{s})$ appearing in $(2)$. Consider any two measurable
maps $f,\tilde{f}:[0,T]\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ such
that $f(t,x)=\tilde{f}(t,x)$ for Lebesgue a.e. $(t,x)$, then by condition
$(1)$ and Fubini we have

$\displaystyle\mathbb{E}\Big{[}\int_{0}^{T}|f(s,X_{s})-\tilde{f}(s,X_{s})|\mathrm{d}t\Big{]}=\int_{0}^{T}\int_{\mathbb{R}^{d}}|f(s,x)-\tilde{f}(s,x)|\,\mathcal{L}(X_{s})(\mathop{}\\!\mathrm{d}x)\mathop{}\\!\mathrm{d}s=0.$

As a consequence, if $\mathbb{P}$-a.s.
$\int_{0}^{T}|f(s,X_{s})|\mathop{}\\!\mathrm{d}s<\infty$, then the same holds
for $\tilde{f}$ and one has that
$\int_{0}^{\cdot}f(s,X_{s})\mathop{}\\!\mathrm{d}s=\int_{0}^{\cdot}\tilde{f}(s,X_{s})\mathop{}\\!\mathrm{d}s$
as $C_{T}$-valued random variables. A similar argument shows that the
definition is insensitive to modifications of the kernel $b$ on Lebesgue
negligible sets; for this reason we can consider kernels belonging to
equivalence classes like $L^{1}_{T}L^{p}_{x}$.

In order for the requirement $(2)$ in Definition 4.1 to be met, it is
desirable to have some information on the integrability properties of
$\mathcal{L}(X_{t})$. To derive it, we need to recall known facts from
classical ODE and transport PDE theory.

Given $x_{0}\in\mathbb{R}^{d}$, a drift
$\bar{b}:[0,T]\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ with continuous
and bounded first derivative and a continuous path $\omega\in C_{T}$, there
exists a unique solution $x\in C_{T}$ to the Cauchy problem

$x_{t}=x_{0}+\int_{0}^{t}\bar{b}(s,x_{s})\mathrm{d}s+\omega_{t}\quad\forall\,t\in[0,T];$
(4.2)

similarly to Section 2, we can define the solution map
$x_{t}=\Phi(t,x_{0};\omega)$ associated to the drift $\bar{b}$, which is a
continuous map $\Phi:[0,T]\times\mathbb{R}^{d}\times C_{T}\to\mathbb{R}^{d}$.
For fixed $(t,\omega)$, the map $x_{0}\mapsto\Phi(t,x_{0};\omega)$ is a
diffeomorphism of $\mathbb{R}^{d}$, with Jacobian given by the formula

$\displaystyle\det
D\Phi(t,x_{0};\omega)=\exp\left(\int_{0}^{t}\operatorname{div}\bar{b}(s,\Phi(s,x_{0};\omega))\mathrm{d}s\right).$

In particular we have the following two-sided estimate independent of
$\omega$:

$0<\exp\big{(}-\|\operatorname{div}\bar{b}\|_{L^{1}_{T}L^{\infty}_{x}}\big{)}\leq\det
D\Phi(t,x_{0};\omega)\leq\exp\big{(}\|\operatorname{div}\bar{b}\|_{L^{1}_{T}L^{\infty}_{x}}\big{)}<\infty;$
(4.3)

similarly for $\det D\Psi(t,x_{0};\omega)$, where $\Psi(t,\,\cdot\,;\omega)$
denotes the inverse of $x_{0}\mapsto\Phi(t,x_{0};\omega)$.

On the other hand, we can consider the solution map as a continuous (thus
measurable) map $(x_{0},\omega)\mapsto\Phi(\,\cdot\,,x_{0};\omega)$ from
$\mathbb{R}^{d}\times C_{T}$ into $C_{T}$; in particular if we now consider a
random variable $(\xi,W)$ defined on a probability space
$(\Omega,\mathcal{F},\mathbb{P})$, we consider the associated variable
$\Phi(\,\cdot\,,\xi;W)$, which amounts to the solution of the random ODE
problem (4.2).

With this considerations in mind, we can prove the following preliminary a
priori estimate.

######  Proposition 4.3.

Let $b\in L^{1}_{T}C^{1}_{b}$, $p\in[1,\infty]$ and $(\xi,W)$ be independent
input data s.t.
$\mathcal{L}(\xi)(\mathop{}\\!\mathrm{d}x)=\rho(x)\mathop{}\\!\mathrm{d}x$
with $\rho\in L^{p}_{x}$ for $p\in[1,\infty]$ and let $X$ denote the unique
solution to (4.1). Then there exists a constant
$C=C(p,\|\operatorname{div}b\|_{L^{1}_{T}L^{\infty}_{x}})$ such that

$\sup_{t\in[0,T]}\|\mathcal{L}(X_{t})\|_{L^{p}_{x}}\leq C\|\rho\|_{L^{p}_{x}}$
(4.4)

Furthermore, only using the assumption that $\rho$ is a probability density,
for any $\varepsilon>0$ there exists a $\delta>0$, only depending on $\rho$
and $\|\operatorname{div}b\|_{L^{1}_{T}L^{\infty}_{x}}$, s.t.

$\sup_{t\in[0,T]}\mathcal{L}(X_{t})(A)\leq\varepsilon\quad\forall\,A\subset\mathbb{R}^{d},\,|A|\leq\delta.$
(4.5)

###### Proof.

Let $X$ be the aforementioned solution, which exists by the results of Section
2. Then $X$ is also a solution to the random ODE (4.2) for the choice
$\bar{b}_{t}(x)=b_{t}\ast\mathcal{L}(X_{t})(x)$ and we can represent it as
$X_{t}=\Phi(t,\xi;W)$; without loss of generality we can take $(\xi,W)$ to be
the canonical variables on the probability space $(\mathbb{R}^{d}\times
C_{T},\mathcal{B}(\mathbb{R}^{d}\times
C_{T}),(\rho\mathop{}\\!\mathrm{d}x)\otimes\mu)$ where $\mu=\mathcal{L}(W)$.
Moreover it holds that

$\|\operatorname{div}\bar{b}\|_{L^{1}_{T}L^{\infty}_{x}}=\|(\operatorname{div}b_{\cdot})\ast\mathcal{L}(X_{\cdot})\|_{L^{1}_{T}L^{\infty}_{x}}\leq\|\operatorname{div}b\|_{L^{1}_{T}L^{\infty}_{x}}.$

For any measurable and bounded $f:\mathbb{R}^{d}\to\mathbb{R}$ and any
$t\in[0,T]$ we have

$\displaystyle\langle f,\mathcal{L}(X_{t})\rangle$
$\displaystyle=\mathbb{E}[f(\Phi(t,\xi;W))]=\int_{C_{T}}\int_{\mathbb{R}^{d}}f(\Phi(t,x;\omega))\rho(x)\mathrm{d}x\,\mu(\mathop{}\\!\mathrm{d}\omega)$
$\displaystyle=\int_{C_{T}}\int_{\mathbb{R}^{d}}f(y)\rho(\Psi(t,x;\omega))\det
D\Psi(t,y;\omega)\mathop{}\\!\mathrm{d}y\,\mu(\mathop{}\\!\mathrm{d}\omega)$
$\displaystyle=\int_{\mathbb{R}^{d}}f(y)\bigg{[}\int_{C_{T}}\rho(\Psi(t,y;\omega))\det
D\Psi(t,y;\omega)\,\mu(\mathop{}\\!\mathrm{d}\omega)\bigg{]}\,\mathop{}\\!\mathrm{d}y.$

As a consequence, $\mathcal{L}(X_{t})$ is absolutely continuous w.r.t.
Lebesgue with density given by

$\rho_{t}(x):=\int_{C_{T}}\rho(\Psi(t,y;\omega))\det
D\Psi(t,y;\omega)\,\mu(\mathop{}\\!\mathrm{d}\omega).$

Given the bounds on the Jacobian of $\Phi$ and $\Psi$ as above, for
$p\in[1,\infty)$ we can estimate $\|\rho_{t}\|_{L^{p}}$ by

$\displaystyle\int_{\mathbb{R}^{d}}|\rho_{t}(x)|^{p}\mathop{}\\!\mathrm{d}x$
$\displaystyle\leq\int_{\mathbb{R}^{d}}\int_{C_{T}}|\rho(\Psi(t,x;\omega))|^{p}|\det
D\Psi(t,y;\omega)|^{p}\,\mu(\mathop{}\\!\mathrm{d}\omega)$
$\displaystyle\lesssim_{\operatorname{div}b}\int_{C_{T}}\int_{\mathbb{R}^{d}}|\rho(\Psi(t,x;\omega))|^{p}\mathop{}\\!\mathrm{d}x\,\mu(\mathop{}\\!\mathrm{d}\omega)$
$\displaystyle=\int_{C_{T}}\int_{\mathbb{R}^{d}}|\rho(y)|^{p}\det
D\Phi(t,y;\omega)\mathop{}\\!\mathrm{d}y\,\mu(\mathop{}\\!\mathrm{d}\omega)\lesssim_{\operatorname{div}b}\int_{\mathbb{R}^{d}}|\rho(y)|^{p}\mathop{}\\!\mathrm{d}y;$

the estimate for $p=\infty$ is similar. To prove the last statement, applying
several times the fact that the incompressibility constants of
$\Phi(t,\cdot;\omega)$ are controlled by
$\exp(\|\operatorname{div}b\|_{L^{1}_{T}L^{\infty}_{x}})$ uniformly in
$\omega\in C_{T}$ for any $t\in[0,T]$ and $A\subset\mathbb{R}^{d}$, see see
(4.3), we have that

$\displaystyle\mathcal{L}(X_{t})(A)\lesssim_{h}\int_{A}\int_{C_{T}}\rho(\Psi(t,y;\omega)\,\mu(\mathop{}\\!\mathrm{d}\omega)\mathop{}\\!\mathrm{d}y\lesssim_{h}\int_{C_{T}}\int\rho(x)\mathds{1}_{\psi(t,A;\omega)}(x)\mathop{}\\!\mathrm{d}x\,\mu(\mathop{}\\!\mathrm{d}\omega);$

the conclusion follows by observing that $\rho\in L^{1}_{x}$ is uniformly
integrable and that $|\psi(t,A;\omega)|\lesssim_{h}|A|$ uniformly in
$t,\omega$. ∎

With the a priori bound (4.4) at hand, we are now ready to prove existence of
solutions to (4.1) for integrable drifts $b$, up to the price of requiring
additional assumptions on $\operatorname{div}b$ and $\rho$.

######  Proposition 4.4.

Suppose $b\in L^{1}_{T}L^{q}_{x}$, $\operatorname{div}b\in
L^{1}_{T}L^{\infty}_{x}$ and
$\mathcal{L}(\xi)(\mathop{}\\!\mathrm{d}x):=\rho(x)\mathop{}\\!\mathrm{d}x$
for a probability density $\rho\in L^{p}_{x}$ with $p,q\in[1,\infty]$ such
that

$\frac{1}{2q}+\frac{1}{p}\leq 1.$ (4.6)

Then there exists at least one weak solution $X_{t}$ to (4.1) in the sense of
Definition 4.1, which moreover satisfies
$\sup_{t\in[0,T]}\|\mathcal{L}(X_{t})\|_{L^{p}_{x}}<\infty$.

###### Proof.

The proof is based on classical compactness arguments; since $\rho\in
L^{1}_{x}\cap L^{p}_{x}$, by interpolation w.l.o.g. we can assume equality
holds in (4.6).

Let $\\{b^{n}\\}_{n}$ be a sequence in $L^{1}_{T}C^{1}_{b}$ satisfying
$\|b^{n}\|_{L^{1}_{T}L^{q}_{x}}\leq\|b\|_{L^{1}_{T}L^{q}_{x}}$,
$\|\operatorname{div}b^{n}\|_{L^{1}_{T}L^{\infty}_{x}}\leq\|\operatorname{div}b\|_{L^{1}_{T}L^{\infty}_{x}}$
(take for instance $b^{n}_{s}:=b_{s}\ast\varphi_{1/n}$ with
$\\{\varphi_{\varepsilon}\\}_{\varepsilon>0}$ standard mollifiers); by Section
2, for each $n\geq 1$ there exists a unique strong solution
$(X^{n},\mathcal{L}(X^{n}))\in C_{T}\times\mathcal{P}(C_{T})$ to the DDSDE
associated to $b^{n}$, with input data $(\xi,W)$. By Proposition 4.3, we can
find some $C=C(\|\operatorname{div}b\|_{L^{1}_{T}L^{\infty}_{x}})$ such that

$\sup_{t\in[0,T]}\|\mathcal{L}(X^{n}_{t})\|_{L^{p}}\leq
C\|\rho\|_{L^{p}_{x}}\quad\forall\,n\in\mathbb{N}.$

Set
$Z_{t}^{n}:=\int_{0}^{t}b^{n}_{s}\ast\mathcal{L}(X^{n}_{s})(X^{n}_{s})\,\mathop{}\\!\mathrm{d}s=X^{n}_{t}-\xi-
W_{t}$. Our first goal is to establish uniform bounds on $\\{Z_{n}\\}_{n}$. It
holds that

$\displaystyle\mathbb{E}[\|Z^{n}\|_{W^{1,1}_{T}}]$
$\displaystyle\leq\mathbb{E}\bigg{[}\int_{0}^{T}|b_{s}^{n}|\ast\mathcal{L}(X^{n}_{s})(X^{n}_{s})\,\mathop{}\\!\mathrm{d}s\bigg{]}$
$\displaystyle\leq\int_{0}^{T}\langle|b^{n}_{s}|\ast\mathcal{L}(X^{n}_{s}),\mathcal{L}(X^{n}_{s})\rangle\,\mathop{}\\!\mathrm{d}s$
$\displaystyle\leq\|b^{n}\|_{L^{1}_{T}L^{q}_{x}}\sup_{t\in[0,T]}\|\mathcal{L}(X^{n}_{s})\|_{L^{p}_{x}}^{2}\lesssim\|b\|_{L^{1}_{T}L^{q}_{x}}\,\|\rho\|_{L^{p}_{x}}^{2}$

where in the third passage we applied Hölder’s and Young’s inequalities.

Then Helly’s selection theorem, together with the fact that $W^{1,1}_{T}$
embeds compactly in $L^{r}_{T}$ for any $r<\infty$ as well as in the topology
of pointwise convergence, implies that $\\{Z^{n}\\}_{n}$ is tight in
$L^{r}_{T}$ and $\\{(\xi,W,Z^{n})\\}_{n}$ is tight in $\mathbb{R}^{d}\times
C_{T}\times L^{r}_{T}$.

We can therefore apply Prokhorov’s and Skorohod’s theorems to construct
another probability space
$(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}})$, carrying a family
of random variables $(\tilde{\xi}^{n},\tilde{W}^{n},\tilde{Z}^{n})$ converging
$\tilde{\mathbb{P}}$-a.s. to $(\tilde{\xi},\tilde{W},\tilde{Z})$ in
$\mathbb{R}^{d}\times C_{T}\times L^{r}_{T}$, such that
$\mathcal{L}_{\tilde{\mathbb{P}}}(\tilde{\xi}^{n},\tilde{W}^{n},\tilde{Z}^{n})=\mathcal{L}_{\mathbb{P}}(\xi,W,Z^{n})$.
From this we deduce as before that $\tilde{\mathbb{P}}$-a.s.
$\tilde{Z}^{n}\to\tilde{Z}$ also in the sense of pointwise convergence,
similarly for $\tilde{X}^{n}:=\tilde{\xi}^{n}+\tilde{W}^{n}+\tilde{Z}^{n}$;
moreover $\tilde{X}^{n}$ are still solutions to the DDSDE associated to
$b^{n}$ and
$\tilde{Z}^{n}_{t}=\int_{0}^{t}b^{n}_{s}\ast\mathcal{L}_{\tilde{\mathbb{P}}}(\tilde{X}^{n}_{s})(\tilde{X}^{n}_{s})\mathop{}\\!\mathrm{d}s$.

In order to conclude, it remains to show that
$(\tilde{X},\tilde{W},\tilde{\xi})$ is a weak solution to the DDSDE associated
to $b$, on the probability space
$(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}})$. For notational
simplicity we will drop the tildes from now on.

We first treat the case $p>1$. Since $\mathbb{P}$-a.s. $X^{n}_{t}\to X^{n}$,
$\mathcal{L}(X^{n}_{t})\rightharpoonup\mathcal{L}(X_{t})$ weakly and they are
uniformly bounded in $L^{p}_{x}$ with $p>1$; thus $\mathcal{L}(X_{t})\in
L^{p}_{x}$ and by lower semicontinuity of weak convergence (weak-$\ast$ if
$p=\infty$) it holds that

$\sup_{t\in[0,T]}\|\mathcal{L}(X_{t})\|_{L^{p}_{x}}\leq\sup_{t\in[0,T]}\liminf_{n\to\infty}\|\mathcal{L}(X^{n}_{t})\|_{L^{p}_{x}}\leq
C\|\rho\|_{L^{p}_{x}},$

which checks requirement $(1)$ of Definition 4.1. The verification of $(2)$ is
similar and we are left with showing that $Z^{n}\to Z$ for
$Z_{t}=\int_{0}^{t}b_{s}\ast\mathcal{L}(X_{s})(X_{s})\mathop{}\\!\mathrm{d}s$.

In the case $p>1$, $q<\infty$, we may further assume that $b^{n}\to b$ in
$L^{1}_{T}L^{q}_{x}$. We fix $\varepsilon>0$ and choose $\bar{b}\in
C^{1}_{b}([0,T]\times\mathbb{R}^{d};\mathbb{R}^{d})$ such that
$\|b-\bar{b}\|_{L^{1}_{T}L^{q}_{x}}<\varepsilon$ and split the estimate as

$\displaystyle\|Z^{n}-Z\|_{C_{T}}$
$\displaystyle\leq\int_{0}^{T}|b^{n}_{s}-\bar{b}_{s}|\ast\mathcal{L}(X^{n}_{s})(X^{n}_{s})\mathop{}\\!\mathrm{d}s+\int_{0}^{T}|\bar{b}_{s}\ast\mathcal{L}(X^{n}_{s})(X^{n}_{s})-\bar{b}_{s}\ast\mathcal{L}(X_{s})(X_{s})|\mathop{}\\!\mathrm{d}s$
$\displaystyle\quad+\int_{0}^{T}|\bar{b}_{s}-b_{s}|\ast\mathcal{L}(X_{s})(X_{s})\mathop{}\\!\mathrm{d}s$
$\displaystyle:=I^{n}_{1}+I^{n}_{2}+I_{3}.$

Regarding $I^{n}_{1}$, we can apply the uniform estimates on
$\|\mathcal{L}(X^{n}_{t})\|_{L^{p}_{x}}$ to obtain

$\displaystyle\mathbb{E}[I^{n}_{1}]\lesssim\sup_{s\in[0,T]}\|\mathcal{L}(X^{n}_{s})\|_{L^{p}_{x}}^{2}\|b^{n}-\bar{b}\|_{L^{1}_{T}L^{q}_{x}}\lesssim\|b^{n}-b\|_{L^{1}_{T}L^{q}_{x}}+\|b-\bar{b}\|_{L^{1}_{T}L^{q}_{x}};$

similarly for $I_{3}$. On the other hand, since $\bar{b}\in C^{1}_{b}$ and
$\mathcal{L}(X^{n})\rightharpoonup\mathcal{L}(X)$,
$\bar{b}\ast\mathcal{L}(X^{n})$ converges uniformly to
$\bar{b}\ast\mathcal{L}(X)$; combined with $X^{n}\to X$ $\mathbb{P}$-a.s. and
the uniform bounds, by dominated convergence we have $\mathbb{E}[I^{n}_{2}]\to
0$. Thus overall we obtain

$\displaystyle\limsup_{n\to\infty}\mathbb{E}[\|Z^{n}-Z\|_{C_{T}}]$
$\displaystyle\leq\limsup_{n\to\infty}(\mathbb{E}[I^{n}_{1}]+\mathbb{E}[I^{n}_{2}]+\mathbb{E}[I_{3}])$
$\displaystyle\lesssim\limsup_{n\to\infty}\|b^{n}-b\|_{L^{1}_{T}L^{q}_{x}}+2\|b-\bar{b}\|_{L^{1}_{T}L^{q}_{x}}\lesssim
2\varepsilon;$

by the arbitrariness of $\varepsilon>0$ we deduce that
$\mathbb{E}[\|Z^{n}-Z\|_{C^{T}}]\to 0$ and that $X$ is a weak solution.

In the case $p=1,q=\infty$ we cannot assume
$\|b^{n}-b\|_{L^{1}_{T}L^{\infty}_{x}}\rightarrow 0$, but only that
$b^{n}_{t}(x)\to b_{t}(x)$ for Lebesgue a.e.
$(t,x)\in[0,T]\times\mathbb{R}^{d}$. Since $X^{n}_{t}\rightharpoonup X_{t}$,
the sequence $\\{\mathcal{L}(X^{n}_{t})\\}_{n}$ is tight, which together with
(4.5) implies its equiintegrability. But then by Dunford–Pettis theorem
$\mathcal{L}(X_{t})\in L^{1}_{x}$ and
$\mathcal{L}(X^{n}_{t})\rightharpoonup\mathcal{L}(X_{t})$ weakly in
$L^{1}_{x}$, checking requirement (1) of Definition 4.1; (2) follows trivially
since $b\in L^{1}_{T}L^{\infty}_{x}$. Finally, since
$\mathcal{L}(X^{n}_{t})\rightharpoonup\mathcal{L}(X_{t})$ weakly in
$L^{1}_{x}$, we can apply Lemma B.1 from Appendix B at any fixed $t\in[0,T]$
to conclude that $\mathbb{E}[\|Z^{n}-Z\|_{C_{T}}]\to 0$ still holds. ∎

Having established existence under suitable integrability assumptions on $b$,
it is natural to establish uniqueness under similar requirements on the
derivative $Db$. We start with a conditional result of uniqueness in a class
of sufficiently regular solutions.

######  Proposition 4.5.

Let $q,\,p\in(1,\infty]$ be such that $\frac{1}{p}+\frac{1}{q}<1$ and assume
$b\in L^{1}_{T}W^{1,q}_{x}$, $\rho\in L^{p}_{x}$. Then both uniqueness in law
and pathwise uniqueness holds for (4.1) in the class of solutions satisfying

$\sup_{t\in[0,T]}\|\mathcal{L}(X_{t})\|_{L^{p}_{x}}<\infty$ (4.7)

in the following sense: if $X^{1}$ and $X^{2}$ are both solutions satisfying
(4.7), then their laws as probabilities on the path space $C_{T}$ coincide; if
$X^{i}$ are defined on the same probability space and solve the DDSDE w.r.t.
same input data $(\xi,W)$, then $X^{1}=X^{2}$ $\mathbb{P}$-a.s.

###### Proof.

Given two solutions $X^{i}$ as above, let us set
$\mu^{i}_{t}=\mathcal{L}(X^{i}_{t})$ and
$b^{i}_{t}(x)=(b_{t}\ast\mu^{i}_{t})(x)$; from Proposition 4.4, for all
$t\in[0,T]$, $\mu^{i}_{t}\in L^{1}_{x}\cap L^{p}_{x}$ for $p>q^{\prime}$, by
interpolation $\mu^{i}\in L^{\infty}_{t}L^{q^{\prime}}_{x}$ and so by Young’s
inequality

$\|b^{i}\|_{L^{1}_{T}W^{1,\infty}_{x}}\lesssim\|b\|_{L^{1}_{T}W^{1,q}}\|\mu^{i}\|_{L^{\infty}_{T}L^{q^{\prime}}_{x}}\lesssim
1.$

In particular, both drifts $b^{i}$ are spatially Lipschitz; moreover by
arguing as in Proposition 3.5, we only need to check pathwise uniqueness for
solutions $X^{i}$ to (4.1) satisfying (4.7).

First observe that the difference $Y:=X^{1}-X^{2}$ satisfies

$\displaystyle|Y_{t}|$
$\displaystyle\leq\int_{0}^{t}|b^{1}_{s}(X^{1}_{s})-b^{1}_{s}(X^{2}_{s})|\mathop{}\\!\mathrm{d}s+\int_{0}^{t}|(b^{1}-b^{2})_{s}(X^{2}_{s})|\mathop{}\\!\mathrm{d}s$
$\displaystyle\leq\int_{0}^{t}\|b_{s}\ast\mu^{1}_{s}\|_{W^{1,\infty}_{x}}|Y_{s}|\mathop{}\\!\mathrm{d}s+\int_{0}^{t}\|b_{s}\ast(\mu^{1}_{s}-\mu^{2}_{s})\|_{L^{\infty}_{x}}\mathop{}\\!\mathrm{d}s$
$\displaystyle\lesssim_{\mu^{i}}\int_{0}^{t}\|b_{s}\|_{W^{1,q}_{x}}(1+|Y_{s}|)\mathop{}\\!\mathrm{d}s;$

applying Grönwall, we find a deterministic constant $C=C(b,\mu^{i})>0$ such
that $\|X^{1}-X^{2}\|_{C_{T}}=\|Y\|_{C_{T}}\leq C$ $\mathbb{P}$-a.s.

As a consequence, we also have that
$d_{r^{\prime}}(\mu^{1}_{t},\mu^{2}_{t})\leq C<\infty$ for any
$r^{\prime}\in(1,\infty)$; in particular, the assumptions of Lemma B.2 from
Appendix B are met, as there exists $r>1$ such that $r/q+1/p\leq 1$. It
follows that

$\displaystyle\|b_{s}\ast(\mu^{1}_{s}-\mu^{2}_{s})\|_{L^{\infty}_{x}}\lesssim\|b_{s}\|_{W^{1,q}}\big{(}\|\mu^{1}_{s}\|_{L^{p}_{x}}^{1/r}+\|\mu^{2}_{s}\|_{L^{p}_{x}}^{1/r}\big{)}d_{r^{\prime}}(\mu^{1}_{s},\mu^{2}_{s})\lesssim_{\mu^{i}}\|b_{s}\|_{W^{1,q}}\|Y_{s}\|_{L^{r^{\prime}}_{\Omega}}.$

We use this new bound to improve the estimates on $Y$, giving:

$|Y_{t}|\lesssim_{\mu^{i}}\int_{0}^{t}\|b_{s}\|_{W^{1,q}_{x}}(|Y_{s}|+\|Y_{s}\|_{L^{r^{\prime}}_{\Omega}})\mathop{}\\!\mathrm{d}s$

and now applying Minkowski’s integral inequality (with $r^{\prime}>1$) we
obtain

$\|Y_{t}\|_{L^{r^{\prime}}_{\Omega}}\lesssim_{\mu^{i}}\int_{0}^{t}\|b_{s}\|_{W^{1,q}_{x}}\,\|Y_{s}\|_{L^{r^{\prime}}_{\Omega}}\,\mathop{}\\!\mathrm{d}s.$

However, by the hypothesis $b\in L^{1}_{T}W^{1,q}_{x}$ and applying Grönwall’s
lemma, we deduce $\|Y_{t}\|_{L^{r^{\prime}}_{\Omega}}=0$ for all $t\in[0,T]$
and so $X^{1}=X^{2}$ $\mathbb{P}$-a.s. ∎

Combining Propositions 4.4 and 4.5 we get the following existence and
uniqueness result.

######  Theorem 4.6.

Let $q,\,p\in(1,\infty]$ be such that $\frac{1}{p}+\frac{1}{q}<1$ and $b\in
L^{1}_{T}W^{1,q}_{x}$ with $\operatorname{div}b\in L^{1}_{T}L^{\infty}_{x}$.
Then for any $\rho\in L^{p}_{x}$ there exists a strong solution $X_{t}$
satisfying $\sup_{t\in[0,T]}\|\mathcal{L}(X_{t})\|_{L^{p}_{x}}<\infty$, which
is unique in this class (in the sense of Proposition 4.5).

###### Proof.

The assumption on $(p,q)$ implies that (4.6) holds, thus by Proposition 4.4
weak existence holds; as observed in the proof of Proposition 4.5, the drift
$\bar{b}_{t}=b_{t}\ast\mathcal{L}(X_{t})\in L^{1}_{T}W^{1,\infty}_{x}$ and so
arguing as in Proposition 3.7 we deduce strong existence. Uniqueness follows
again from Proposition 4.5. ∎

Theorem 4.6, together with the a priori estimates from Proposition 4.3,
guarantees that such solution $X$ is the only possible accumulation point of
solutions $X^{n}$ constructed from mollified drifts
$b^{n}=b\ast\varphi_{1/n}$. It doesn’t however exclude the existence of
different solutions $\tilde{X}$, associated to the same input data $(\xi,W)$,
which fail to meet the requirement
$\sup_{t\in[0,T]}\|\mathcal{L}(\tilde{X}_{t})\|_{L^{p}_{x}}<\infty$.

In order to strengthen the result and truly establish uniqueness in law,
without any regularity assumption on $\mathcal{L}(\tilde{X})_{t}$, we need
more restrictive conditions on $b$; the advantage is that we deduce some
strong stability estimates, in the form of equation (4.8) below. This result
relies on the results of [12] and in particular Lemma B.3.

######  Proposition 4.7.

Let $(p,q,r)\in(1,\infty)^{3}$, $b\in L^{1}_{T}W^{1,q}_{x}$ with
$\operatorname{div}b\in L^{1}_{T}L^{\infty}_{x}$ and

$q>d,\quad\frac{r}{p}+\frac{1}{q}\leq
1,\quad\frac{1}{r}+\frac{1}{r^{\prime}}=1.$

Also assume we are given input data
$\mathcal{L}(\xi^{1},W^{1})=\mu^{\xi^{1}}\otimes\mu^{W^{1}}\in\mathcal{P}_{r^{\prime}}(\mathbb{R}^{d}\times
C_{T})$, such that
$\mu^{\xi^{1}}(\mathop{}\\!\mathrm{d}x)=\rho(x)\mathop{}\\!\mathrm{d}x$ for
some $\rho\in L^{q}_{x}$. Denote by $\mu^{X^{1}}$ the unique solution
associated to $\mu^{\xi^{1}+W^{1}}$, in the sense of Theorem 4.6. Then there
exists a constant $C$, depending on $d,T,p,q,r$,
$\|\operatorname{div}b\|_{L^{1}_{T}L^{\infty}_{x}}$,
$\|b\|_{L^{1}_{T}W^{1,p}_{x}}$ and $\|\rho\|_{L^{q}}$, with the following
property: for any other solution $\mu^{X^{2}}$ to (4.1) with input
$\mu^{Y^{2}}$,

$d_{r^{\prime}}(\mu^{X^{1}},\mu^{X^{2}})\leq
Cd_{r^{\prime}}\big{(}\mu^{\xi^{1}+W^{1}},\mu^{Y^{2}}\big{)}.$ (4.8)

###### Proof.

Since $q>d$, by Sobolev embeddings $b\in L^{1}_{T}C^{0}_{b}$, so that the
DDSDE (4.1) is meaningful in the sense of Definition 3.1, without the need of
any regularity assumption on $\mu^{X^{2}}$. By Remark 3.6, given any coupling
of $\mu^{\xi^{1}+W^{1}},\mu^{Y^{2}}$, we can construct a coupling of the
associated solutions $\mu^{X^{1}},\mu^{X^{2}}$ as well, so from now on we will
work with $X^{i},\xi^{1},W^{1},Y^{2}$ all defined on the same probability
space; also for simplicity we will adopt the compact notation
$Y^{1}=\xi^{1}+W^{1}$. Since $X^{i}$ are solutions to the DDSDE and $b\in
L^{1}_{T}C^{0}_{b}$, we have the trivial estimate

$\mathbb{E}\big{[}\|X^{i}\|_{C_{T}}^{r^{\prime}}\big{]}\lesssim\|b\|_{L^{1}_{T}W^{1,q}_{x}}^{r^{\prime}}+\mathbb{E}\big{[}\|Y^{i}\|_{C_{T}}^{r^{\prime}}\big{]}$

which shows that $\mu^{X^{i}}\in\mathcal{P}_{r^{\prime}}(C_{T})$.

Having made these preparations, we can now pass to the proof of estimate
(4.8). Set $\mu^{i}_{t}=\mathcal{L}(X^{i}_{t})$, then the process
$X^{1}-X^{2}$ satisfies

$|X^{1}_{t}-X^{2}_{t}|\leq\int_{0}^{t}|b_{s}\ast\mu^{1}_{s}(X^{1}_{s})-b_{s}\ast\mu^{2}_{s}(X^{2}_{s})|\mathop{}\\!\mathrm{d}s+|Y^{1}_{t}-Y^{2}_{t}|;$

since $\sup_{t\in[0,T]}\|\mu^{1}_{t}\|_{L^{p}_{x}}<\infty$, we can apply Lemma
B.3 from Appendix B to obtain

$\displaystyle|X^{1}_{t}-X^{2}_{t}|$
$\displaystyle\lesssim\int_{0}^{t}\|b_{s}\|_{W^{1,q}_{x}}(1+\|\mu^{1}_{s}\|_{L^{p}_{x}})\,(|X^{1}_{s}-X^{2}_{s}|+d_{r^{\prime}}(\mu^{1}_{s},\mu^{2}_{s}))\,\mathop{}\\!\mathrm{d}s+|Y^{1}_{t}-Y^{2}_{t}|$
$\displaystyle\lesssim_{\operatorname{div}b,\rho}\int_{0}^{t}\|b_{s}\|_{W^{1,q}_{x}}\,(|X^{1}_{s}-X^{2}_{s}|+d_{r^{\prime}}(\mu^{1}_{s},\mu^{2}_{s}))\,\mathop{}\\!\mathrm{d}s+\|Y^{1}-Y^{2}\|_{C_{T}};$

taking the $L^{r^{\prime}}_{\Omega}$-norm on both sides and applying
Minkowski’s inequality we find

$\|X^{1}_{t}-X^{2}_{t}\|_{L^{r^{\prime}}_{\Omega}}\lesssim\int_{0}^{t}\|b_{s}\|_{W^{1,q}_{x}}\|X^{1}_{s}-X^{2}_{s}\|_{L^{r^{\prime}}_{\Omega}}\mathop{}\\!\mathrm{d}s+\|\|Y^{1}-Y^{2}\|_{C_{T}}\|_{L^{r^{\prime}}_{\Omega}}.$

By Grönwall’s lemma we can find some constant $\kappa$ such that

$\sup_{t\in[0,T]}d_{r^{\prime}}(\mu^{1}_{t},\mu^{2}_{t})\lesssim
e^{\kappa\|b\|_{L^{1}_{T}W^{1,q}_{x}}}\|\|Y^{1}-Y^{2}\|_{C_{T}}\|_{L^{r^{\prime}}_{\Omega}}.$

In order to obtain a bound in the path space $C_{T}$ it suffices to observe
that the previous inequalities also imply, again by Grönwall,

$\|X^{1}-X^{2}\|_{C_{T}}\lesssim_{b}\|b\|_{L^{1}_{T}W^{1,q}_{x}}\sup_{t\in[0,T]}d_{r^{\prime}}(\mu^{1}_{t},\mu^{2}_{t})+\|Y^{1}-Y^{2}\|_{C_{T}};$

inserting the estimate for $d_{r^{\prime}}(\mu^{1}_{t},\mu^{2}_{t})$ and
taking expectation, we find a constant $C$ as above s.t.

$d_{r^{\prime}}(\mu^{X^{1}},\mu^{X^{2}})\leq\mathbb{E}[\|X^{1}-X^{2}\|_{C_{T}}^{r^{\prime}}]^{1/r^{\prime}}\leq
C\,\mathbb{E}[\|Y^{1}-Y^{2}\|_{C_{T}}^{r^{\prime}}]^{1/r^{\prime}}.$

Minimizing over all possible couplings $(Y^{1},Y^{2})$ given the conclusion. ∎

##  5\. Mean Field Convergence

In this section we prove mean field limits for the interacting particle
systems associated to the drifts considered in Sections 3-4. We start by
presenting Tanaka’s idea in a rather abstract fashion and then apply it to our
cases of interest.

###  5.1. Abstract criteria

Let us shortly recall the setup. Given a family of $C_{T}$-valued random
variables $\\{Y^{i},i\in\mathbb{N}\\}$ defined on a probability space
$(\Omega,\mathcal{F},\mathbb{P})$ and a measurable drift
$B:[0,T]\times\mathbb{R}^{d}\times\mathcal{P}(\mathbb{R}^{d})\to\mathbb{R}$
(possibly with $\mathcal{P}(\mathbb{R}^{d})$ replaced by
$\mathcal{P}_{p}(\mathbb{R}^{d})$ for some $p\in[1,\infty)$), we say that a
family $X^{(N)}:=\\{X^{i,N}\\}_{i=1}^{N}$, seen as a $(C_{T})^{N}$-valued
random variable, is a solution to the $N$-particle system if for
$\mathbb{P}$-a.e. $\omega\in\Omega$ it holds

$X^{i,N}_{t}(\omega)=\int_{0}^{t}B_{s}(X^{i,N}_{s}(\omega),L^{N}(X^{(N)}_{s}(\omega)))\,\mathop{}\\!\mathrm{d}s+Y^{i}_{t}\quad\forall\,t\in[0,T],\,i=1,\ldots,N$
(5.1)

where the integral is imposed to be meaningful in the Lebesgue sense for any
such $\omega$ and

$L^{N}(X^{(N)}_{t}(\omega)):=\frac{1}{N}\sum_{i=1}^{N}\delta_{X^{i,N}_{t}(\omega)}$

denotes the empirical measure of the system at time $t$. For notational
simplicity we will assume different particle systems to be defined on the same
probability space, although we could allow $(\Omega,\mathcal{F},\mathbb{P})$
to vary as a function of $N$. In this case we would adopt the notation
$Y^{(N)}:=\\{Y^{i,N}\\}^{N}_{i=1}$ for the sequence of input data below.

We will adopt the following notion of mean field convergence.

######  Definition 5.1 (Mean Field Convergence).

Let $\\{Y^{i}\\}_{i\in\mathbb{N}}$ be a sequence of input data such that,
defining $Y^{(N)}=\\{Y^{i}\\}_{i=1}^{N}$, one has $\mathbb{P}$-a.s.

$L^{N}(Y^{(N)}(\omega))=\frac{1}{N}\sum_{i=1}^{N}\delta_{Y^{i}(\omega)}\rightharpoonup\mu^{Y}\text{
in }\mathcal{P}(C_{T})\text{ as }N\to\infty,$

with $Y$ such that there exists a unique solution, $\mu^{X}$ to the DDSDE
(3.1) with input $\mu^{Y}$.

Then we say that mean field convergence of the particle system (1.2) to the
DDSDE (1.1) holds along $\\{Y^{i}\\}_{i\in\mathbb{N}}$ if for any sequence of
solutions $\\{X^{i,N}\\}$ to the associated $N$-particle, $\mathbb{P}$-a.s. we
have

$L^{N}(X^{(N)}(\omega))\rightharpoonup\mu^{X}\text{ in
}\mathcal{P}(C_{T})\text{ as }N\to\infty.$

Let us remark that although our definition is non-standard, it is well suited
to our purposes below: it doesn’t require the solution $\\{X^{i,N}\\}$ to be
unique, only the candidate limit $\mu^{X}$; similarly we don’t impose the
property to hold for all possible families $\\{Y^{i}\\}$, allowing us some
freedom in their choice. As the applications in the next section will show, it
covers the i.i.d. case for a large class of
$\mathcal{L}(Y)\in\mathcal{P}(C_{T})$.

We are now ready to present Tanaka’s argument; as already mentioned in the
introduction, it can be regarded as a “transfer principle” from the DDSDE to
the particle system.

######  Proposition 5.2.

Consider the DDSDE (3.1) for a given drift $B$. Assume there exists
$\mu^{Y^{1}}\in\mathcal{P}(\mathbb{R}^{d})$ with the following properties:

  1. i.

there exists a unique solution $\mu^{X^{1}}$ associated to $\mu^{Y^{1}}$;

  2. ii.

there exist $p\in[1,\infty)$, a measurable function
$F:\mathcal{P}_{p}(C_{T})\times\mathcal{P}(C_{T})\to[0,+\infty]$ and an
increasing continuous function $M:[0,+\infty)\to[0,+\infty)$ with $M(0)=0$
such that, for any other input $\mu^{Y^{2}}$ and any solution $\mu^{X^{2}}$
associated to $\mu^{Y^{2}}$, it holds that

$d_{p}(\mu^{X^{1}},\mu^{X^{2}})\leq
F(\mu^{Y^{1}},\mu^{Y^{2}})\,M(d_{p}(\mu^{Y^{1}},\mu^{Y^{2}})).$ (5.2)

Then for any solution $\\{X^{i,N}\\}_{i}$ to the $N$-particle system
associated to $\\{Y^{i}\\}_{i}$, $\mathbb{P}$-a.s.

$d_{p}(\mu^{X^{1}},L^{N}(X^{(N)}(\omega))\leq
F(\mu^{Y^{1}},L^{N}(Y^{(N)}(\omega)))\,M\left(d_{p}\left(\mu^{Y^{1}},L^{N}(Y^{(N)}(\omega))\right)\right).$
(5.3)

Before giving the proof, let us explain the meaning of condition (5.2). In the
simplest cases like Section 2, it can be shown that the DDSDE (3.1) is well-
posed for all $\mu^{Y}\in\mathcal{P}(C_{T})$ and defines a continuous solution
map $S^{B}:\mathcal{P}(C_{T})\to\mathcal{P}(C_{T})$ by
$\mu^{Y}\mapsto\mu^{X}$. In this case condition (5.2) is satisfied for
$F\equiv 1$ and $M$ the modulus of continuity of $S^{B}$ around $\mu^{Y^{1}}$;
if $S^{B}$ is globally Lipschitz, then $M(r)=Cr$ for some constant $C>0$. The
assumptions of Proposition 5.2 are just a convenient generalization to the
case in which $S^{B}$ is not well-defined anymore, or maybe not on an open
neighbourhood of $\mu^{Y^{1}}$ in $\mathcal{P}_{p}(C_{T})$ but only on a
strict subset of it (encoded by the presence of $F$ which can take value
$+\infty$).

###### Proof.

Let the family $Y^{(N)}=\\{Y^{i}\\}_{i=1}^{N}$ be defined on a probability
space $(\Omega,\mathcal{F},\mathbb{P})$, together with a solution
$X^{(N)}=\\{X^{i,N}\\}_{i=1}^{N}$ to the $N$-particle system; we can regard
$X^{(N)},\,Y^{(N)}$ as $(C_{T})^{N}$-valued random variables. Define a new
probability space $(\Omega_{N},\mathcal{F}_{N},\mathbb{P}_{N})$ by

$\displaystyle\Omega_{N}:=\\{1,\ldots,N\\},\quad\mathcal{F}_{N}:=2^{\Omega_{N}},\quad\mathbb{P}_{N}:=\frac{1}{N}\sum_{i=1}^{N}\delta_{i}.$

We can identify any element $y^{(N)}=\\{y^{i}\\}_{i=1}^{N}\in(C_{T})^{N}$ as a
random variable $\Omega_{N}\ni i\mapsto y^{i}\in C_{T}$. With this
identification in place, for any $\varphi\in C_{b}(C_{T})$ we have

$\mathbb{E}_{\mathbb{P}_{N}}\Big{[}\varphi\big{(}y^{(N)}\big{)}\Big{]}=\frac{1}{N}\sum_{i=1}^{N}\varphi(y^{i})=\Big{\langle}\varphi,\frac{1}{N}\sum_{i=1}^{N}\delta_{y^{i}}\Big{\rangle}=\langle\varphi,L^{N}(y^{(N)})\rangle;$

that is, $L^{N}(y^{(N)})=\mathcal{L}_{\mathbb{P}_{N}}(y^{(N)})$. Applying the
above to $Y^{(N)}(\omega)$, $X^{(N)}(\omega)$ we deduce that for
$\mathbb{P}$-a.e. $\omega$,

$X^{i,N}_{t}(\omega)=\int_{0}^{t}B_{s}(X^{i,N}_{s}(\omega),\mathcal{L}_{\mathbb{P}_{N}}(X^{(N)}_{s}(\omega))\,\mathop{}\\!\mathrm{d}s+Y^{i}_{t}(\omega)\quad\forall\,i\in\Omega_{N}.$

Namely for any such $\omega$ the solution $X^{(N)}(\omega)$ to the particle
system can be regarded as a solution to the DDSDE (3.1) with input
$Y^{(N)}(\omega)$ with respect to the probability space
$(\Omega_{N},\mathcal{F}_{N},\mathbb{P}_{N})$. Applying assumption (5.2) for
the choice
$\mu^{Y^{2}}=\mathcal{L}_{\mathbb{P}_{N}}(Y^{(N)}(\omega))=L^{N}(Y^{N}(\omega))$,
similarly $\mu^{X^{2}}$, readily gives the conclusion. ∎

######  Remark 5.3.

The proof also reveals the following simple fact: if the DDSDE (3.1) is well-
posed with a globally defined solution map
$S^{B}:\mathcal{P}_{p}(C_{T})\to\mathcal{P}_{p}(C_{T})$, then any solution to
the particle system (5.1) satisfies $X^{(N)}(\omega)=S^{B}(Y^{(N)}(\omega))$.
In particular strong existence, pathwise uniqueness and uniqueness in law hold
for the particle system as well.

We immediately deduce the following.

######  Corollary 5.4.

Suppose $\mu^{Y^{1}}$ satisfies the assumptions of Proposition 5.2. Then the
mean field convergence holds along any sequence $\\{Y^{i}\\}_{i}$ such that
$\mathbb{P}$-a.s.,

$\limsup_{N\to\infty}F\left(\mu^{Y^{1}},L^{N}(Y^{(N)}(\omega))\right)<\infty,\quad\lim_{N\to\infty}d_{p}\left(\mu^{Y^{1}},L^{N}(Y^{(N)}(\omega))\right)=0.$

The above sufficient conditions still need to be checked case-by-case; however
if $F\equiv 1$, the next general lemma shows that they hold in the classical
case of i.i.d. data.

######  Lemma 5.5.

Let $E$ be a separable Banach space, $p\in[1,\infty)$ and $\\{Z^{i}\\}_{i\geq
1}$ be a family of i.i.d. random variables with common law
$\mu\in\mathcal{P}_{p}(E)$. Then

$\lim_{N\rightarrow\infty}d_{p}(L^{N}(Z^{(N)}(\omega)),\mu)=0\quad\text{for
$\mathbb{P}$-a.e. }\omega.$ (5.4)

###### Proof.

The statement is very close to [16, Lem. 54]; the fact that
$L^{N}(X^{N}(\omega))\rightharpoonup\mu$ for $\mathbb{P}$-a.e. $\omega$ is a
classical result, sometimes referred to as Glivenko–Cantelli theorem. Moreover
by the law of large numbers we have

$\displaystyle\|L^{N}(X^{(N)}(\omega))\|_{p}^{p}=\frac{1}{N}\sum_{i=1}^{N}\|X^{i}(\omega)\|_{E}^{p}\to\int_{E}\|x\|_{E}^{p}\,\mu(\mathop{}\\!\mathrm{d}x)=\|\mu\|_{p}^{p}\quad\mathbb{P}\text{-a.s.}$

and combining these facts gives the conclusion. ∎

######  Remark 5.6.

If the data $\\{Y^{i}\\}_{i=1}^{N}$ are exchangeable (which is true in the
i.i.d. case), then so are $\\{X^{i,N}\\}$. It then follows from [49,
Proposition 2.2] that the mean field convergence is equivalent to the property
of propagation of chaos; the proof therein is based on purely combinatorial
arguments and does not rely on any assumption on $Y$, thus can be applied
directly to our setting.

######  Remark 5.7.

Often one is not just interested in establishing the mean field limit property
but also to derive rates of convergence for
$d_{p}(L^{N}(X^{(N)}),\mathcal{L}(X))$ as a function of $N$; thanks to
Proposition 5.2, once the functions $F$ and $M$ are explicit, it suffices to
find similar rates for $d_{p}(L^{N}(Y^{(N)}),\mathcal{L}(Y))$. In the case of
i.i.d. data, this problem has been studied extensively, see for instance [8,
21]; the most advanced result in the case of measures supported on
$\mathbb{R}^{d}$ we are aware of are the ones presented in [25]. However rates
of convergence in $\mathcal{P}_{p}(C_{T})$ are much more difficult to obtain
and only partial results (mostly relying on Sanov’s theorem and Talagrand
inequalities) are available, see [6, 5].

###  5.2. Applications to particular drifts

We are now ready to apply the previous results to the drifts considered in
Sections 3-4.

####  5.2.1. Osgood drifts

We assume that $B$ satisfies Assumption 3.8, with $p=1$ and Assumption 3.13;
it then follows from Theorem 3.18 that the DDSDE has an associated solution
map $S^{B}:\mathcal{P}_{1}(C_{T})\to\mathcal{P}_{1}(C_{T})$ with modulus of
continuity $M$ of the form

$M(r)=G^{-1}\big{(}G(r)+\|h\|_{L^{1}_{T}}\big{)}$

for $G$ given by (3.9). By Remark 5.3, the particle system (5.1) is well-posed
and $X^{(N)}(\omega)=S^{B}(Y^{(N)}(\omega))$.

######  Corollary 5.8.

Under the above assumptions on $B$, for any $\mu^{Y}\in\mathcal{P}_{1}(C_{T})$
we have

$d_{1}(L^{N}(X^{(N)}(\omega)),S^{B}(\mu^{Y}))\leq
M(d_{1}(L^{N}(Y^{(N)}(\omega)),\mu^{Y}))\quad\mathbb{P}\text{-a.s.};$

as a consequence, mean field convergence holds along any sequence
$\\{Y^{i}\\}$ such that

$\displaystyle d_{1}(L^{N}(Y^{(N)}(\omega)),\mu^{Y})\to
0\quad\mathbb{P}\text{-a.s.}$

which is true in particular if $\\{Y^{i}\\}$ are taken as i.i.d. variables
distributed as $\mu^{Y}$.

###### Proof.

The first statement is an immediate consequence of Proposition 5.2 for the
choice $F\equiv 1$ on $\mathcal{P}_{1}(C_{T})$and $M$ as above; the second one
comes from Corollary 5.4. The claim for i.i.d. variables follows from Lemma
5.5. ∎

Let us finally mention that the function $M$ can be explicitly computed from
the modulus of continuity $f$, so one can also obtain more precise information
on the rate of convergence.

####  5.2.2. Monotone drifts

We now assume that $B$ satisfies Assumption 3.8 and one between Assumptions
3.20, (3.13) or (3.15) (for the same $p$); thus Lemma 3.26 is applicable and
we have a continuous solution map
$S^{B}:\mathcal{P}_{p}(C_{T})\to\mathcal{P}_{p}(C_{T})$ associated to the
DDSDE. As before, by Remark 5.3 the particle system (5.1) is also well-posed.

######  Corollary 5.9.

Under the above assumptions on $B$, for any $\mu^{Y}\in\mathcal{P}_{p}(C_{T})$
we have

$d_{p}(L^{N}(X^{(N)}(\omega)),S^{B}(\mu^{Y}))\leq
M^{Y}(d_{p}(L^{N}(Y^{(N)}(\omega)),\mu^{Y}))\quad\mathbb{P}\text{-a.s.}$

once we take $M^{Y}$ to be the modulus of continuity of $S^{B}$ around
$\mu^{Y}$, namely

$M(r):=\sup\Big{\\{}d_{p}(S^{B}(\mu^{Y}),S^{B}(\nu))\ \big{|}\
\nu\in\mathcal{P}_{p}(C_{T}),\,d_{p}(\nu,\mu^{Y^{1}})\leq r\Big{\\}}.$

As a consequence, mean field convergence holds along any sequence
$\\{Y^{i}\\}$ such that

$\displaystyle d_{p}(L^{N}(Y^{(N)}(\omega)),\mu^{Y})\to
0\quad\mathbb{P}\text{-a.s.}$

which is true in particular if $\\{Y^{i}\\}$ are taken as i.i.d. variables
distributed as $\mu^{Y}\in\mathcal{P}_{p}(C_{T})$.

The proof is identical to that of Corollary 5.8 and thus omitted; let us point
out however that in this case only qualitative continuity of $S^{B}$ is
available, thus we can’t obtain quantitative rates of convergence.

####  5.2.3. Locally Lipschitz drifts

Here we impose $B$ to satisfy Assumption 3.27; by Proposition 3.29, we can
talk of stability of solutions, but technically we don’t have a well defined
solution map on all of $\mathcal{P}_{p}(C_{T})$ since the result only applies
to $\mu^{Y}$ belonging to the class

$\displaystyle\mathcal{E}^{\alpha}:=\Big{\\{}\nu\in\mathcal{P}(C_{T}):\int_{C_{T}}e^{\lambda\|\omega\|_{C_{T}}^{\alpha}}\,\nu(\mathop{}\\!\mathrm{d}\omega)<\infty\text{
for all }\lambda\in\mathbb{R}\Big{\\}}.$

Nevertheless, this class always includes convex combinations of Diracs and so
by Remark 5.3 the result transfers to well-posedness of the particle system
(5.1).

######  Corollary 5.10.

Let $B$ satisfy the above assumptions and fix $p,q\in[1,\infty)$ with $p<q$.
Then there exists a constant $C$ such that for any
$\mu^{Y}\in\mathcal{E}^{\alpha}$ we have

$d_{p}(L^{N}(X^{(N)}(\omega)),\mu^{X}))\leq
C\,F(L^{N}(Y^{(N)}(\omega)),\mu^{Y})\,d_{q}(L^{N}(Y^{(N)}(\omega)),\mu^{Y})\quad\mathbb{P}\text{-a.s.}$

for $F$ given by

$F(\nu^{1},\nu^{2}):=\int_{C_{T}}e^{C\|\omega\|_{C_{T}}^{\alpha}}\,\nu^{1}(\mathop{}\\!\mathrm{d}\omega)+\int_{C_{T}}e^{C\|\omega\|_{C_{T}}^{\alpha}}\,\nu^{2}(\mathop{}\\!\mathrm{d}\omega).$

As a consequence, mean field convergence holds along any sequence
$\\{Y^{i}\\}$ such that

$\displaystyle\limsup_{N\to\infty}F(L^{N}(Y^{(N)}(\omega)),\mu^{Y})<\infty,\quad
d_{q}(L^{N}(Y^{(N)}(\omega)),\mu^{Y})\to 0\quad\mathbb{P}\text{-a.s.}$

which is true in particular if $\\{Y^{i}\\}$ are taken as i.i.d. variables
distributed as $\mu^{Y}\in\mathcal{E}^{\alpha}$.

###### Proof.

The first statement is a consequence of Propositions 3.29 and 5.2; technically
here contrary to (5.2) we have $p<q$, but the proof can be readapted to cover
this case. The second statement is analogous to Corollary 5.4 and so it only
remains to cover the i.i.d. claim. Convergence of $L^{N}(Y^{(N)})$ to
$\mu^{Y}$ in $\mathcal{P}_{q}$ follows from Lemma 5.5; we have

$\displaystyle
F(L^{N}(Y^{(N)}(\omega)),\mu^{Y^{1}})=\frac{1}{N}\sum_{i=1}^{N}e^{C\|Y^{i}\|_{C_{T}}^{\alpha}}+\int_{C_{T}}e^{C\|\omega\|_{C_{T}}^{\alpha}}\,\mu^{Y^{1}}(\mathop{}\\!\mathrm{d}\omega)$

and by SLLN it holds

$\displaystyle\lim_{N\to\infty}\frac{1}{N}\sum_{i=1}^{N}e^{C\|Y^{i}\|_{C_{T}}^{\alpha}}=\int_{C_{T}}e^{C\|\omega\|_{C_{T}}^{\alpha}}\,\mu^{Y^{1}}(\mathop{}\\!\mathrm{d}\omega)\quad\mathbb{P}\text{-a.s.}$

implying the conclusion. ∎

####  5.2.4. Convolutional drifts

Let us now assume $B=b\ast\mu$ for $b$ satisfying the assumptions of
Proposition 4.7; recall in particular that $q>d$ and
$L^{1}_{T}W^{1,q}_{x}\hookrightarrow L^{1}_{T}C^{\alpha}_{x}$ for some
$\alpha>0$. As a consequence, the equation the particle system is always
meaningful and existence of weak solutions to (5.1) holds by standard
arguments; uniqueness however does not need to hold.

Recall that here the input is of the form $Y=\xi+W$, for independent $\xi,W$
with $\mathcal{L}(\xi)\in L^{p}_{x}$; if we also considered $Y^{(N)}$ to be of
the form $Y^{i}=\xi^{i}+W^{i}$, one would then invoke the more refined results
from [12] to show that there exists a unique solution to the particle system
(5.1) for $\mathbb{P}$-a.e. realisation of $W^{(N)}(\omega)$ and Lebesgue-a.e.
choice of initial data $\xi^{(N)}\in\mathbb{R}^{Nd}$. However we will not need
this fact here and we will therefore not impose such an assumption on
$Y^{(N)}$.

######  Corollary 5.11.

Let $b,\,\rho$ satisfy the assumptions of Proposition 4.7 and denote by
$\mu^{X}$ the unique solution to the DDSDE (4.1) associated to
$\mu^{Y}=\mu^{\xi+W}$. Then for any input $Y^{(N)}$ and any solution $X^{(N)}$
to the associated particle system (5.1) it holds that

$d_{r^{\prime}}(L^{N}(X^{(N)}(\omega)),\mu^{X^{1}})\leq
Cd_{r^{\prime}}(L^{N}(Y^{(N)}(\omega)),\mu^{Y}))\quad\mathbb{P}\text{-a.s.}$

for the parameter $r^{\prime}\in(1,\infty)$ as defined in Proposition 4.7 and
given by the relation

$\displaystyle\frac{1}{r^{\prime}}=1-\frac{q^{\prime}}{p},\quad\frac{1}{q}+\frac{1}{q^{\prime}}=1.$

We deduce that mean field convergence holds along any sequence $\\{Y^{i}\\}$
such that $\mathbb{P}$-a.s.

$\displaystyle d_{r^{\prime}}(L^{N}(Y^{(N)}(\omega)),\mu^{Y^{1}})\to 0$

which includes the case of $\\{Y^{i}\\}$ i.i.d. distributed as
$\mu^{\xi+W}\in\mathcal{P}_{r^{\prime}}(C_{T})$.

###### Proof.

The proof follows the same line of arguments as the previous corollaries, this
time based on Propositions 5.2 and 4.7. ∎

In particular Corollary 5.11 holds in the case
$\mu^{\xi+W}\in\cap_{n\in\mathbb{N}}\mathcal{P}_{n}(C_{T})$, completing the
proof of Theorem 1.1.

##  Appendix A Approximation in metric spaces

Our main result from Section 3.1 relies on approximating suitable drifts
$B_{t}(\mu,x)$ by more regular ones. For this reason, we provide here an
abstract result on approximation by Lipschitz functions in metric spaces
$(E,d)$; although our main focus is given by
$E=\mathbb{R}^{d}\times\mathcal{P}_{p}(\mathbb{R}^{d})$, the result is true in
general.

Standards results of this kind can be found in [32, Theorem 6.8] and [15,
Section 6]; however, as they are not directly suited for our purpose, we
present the necessary modifications below.

######  Assumption A.1.

Given a metric space $(E,d)$, we say that a measurable map
$f:\mathbb{R}_{+}\times E\rightarrow\mathbb{R}^{d}$, has time inhomogeneous,
linear growth if there exists a locally integrable function
$h:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ and some (w.l.o.g. any) $z_{0}\in
E$ such that

$|f_{t}(z)|\leq h_{t}\,(1+d(z,z_{0}))\quad\forall\,t\geq 0,\,z\in E.$ (A.1)

In order to prove Proposition A.3 below, we first need the following lemma.

######  Lemma A.2.

Let $g:E\to\mathbb{R}$ be a bounded map, uniformly continuous on bounded sets.
Define

$g^{n}(z)=\inf_{w\in E}\big{\\{}g(w)+nd_{E}(z,w)\big{\\}};$ (A.2)

the sequence $\\{g^{n}\\}_{n}$ has the following properties:

  1. i.

$g^{n}$ is $n$-Lipschitz;

  2. ii.

$g^{n}(z)\leq g^{n+1}(z)\leq g(z)$ and $|g^{n}(z)|\leq\|g\|_{C^{0}}$ for all
$z\in E$ and $n\in\mathbb{N}$;

  3. iii.

$g^{n}\to g$ uniformly on bounded sets.

###### Proof.

Let us set $K=\|g\|_{C^{0}}$; we can assume $K\neq 0$. Point i. and the first
part of ii. follow from [15, Theorem 6.4.1]; by the fact that
$g(w)+nd_{E}(z,w)\geq g(w)\geq-K$ we deduce $g^{n}(z)\geq-K$ and thus
$|g^{n}(z)|\leq K$ for all $z\in E$ and $n\in\mathbb{N}$. The proof of iii. is
mostly the same as [15, Theorem 6.4.1], which however only deals with globally
uniformly continuous functions; for this reason we give it explicitly.

Let us take $n$ big enough so that $2K/n<1$ and fix $z_{0}\in E$, $R>0$; let
$m_{R}$ denote the modulus of continuity of $g$ on $B_{R+1}(z_{0})$, which
exists by assumption. For any $w\in E$ such that $d(z,w)\geq 2K/n$ it holds
that

$\displaystyle g(z)+nd_{E}(z,z)=g(z)\leq K=2K-K\leq g(w)+nd_{E}(z,w)$

which implies that

$\displaystyle g^{n}(z)=\inf\big{\\{}g(w)+nd_{E}(z,w):w\in E,d_{E}(w,z)\leq
2K/n\big{\\}}.$

In particular if $z\in B_{R}(z_{0})$, since $2K/n<1$, the infimum is taken
over a subset of $B_{R+1}(z_{0})$. Fix $\varepsilon>0$, $z\in B_{R}(z_{0})$;
we can find $w_{n}$ s.t. $d_{E}(w_{n},z)\leq 2K/n$ and
$g(w_{n})+nd_{E}(w_{n},z)\leq g^{n}(z)+\varepsilon$. Thus

$\displaystyle g(w_{n})-g^{n}(z)\leq
g(w_{n})-g^{n}(z)+nd_{E}(w_{n},z)\leq\varepsilon$

and

$\displaystyle 0\leq g(z)-g^{n}(z)=g(z)\mp g(w_{n})-g^{n}(z)\leq
m_{R}\Big{(}\frac{2K}{n}\Big{)}+\varepsilon\to\varepsilon\quad\text{ as
}n\to\infty.$

It follows that for all $n$ large enough,

$\displaystyle\sup_{z\in B_{R}(z_{0})}|g(z)-g^{n}(z)|\leq 2\varepsilon;$

by the arbitrariness of $\varepsilon>0$, uniform convergence on bounded sets
follows. ∎

Applying Lemma A.2 to time inhomogeneous functions gives the following result.

######  Proposition A.3.

Let $f:[0,T]\times E\rightarrow\mathbb{R}^{d}$ be as in Assumption A.1 and
such that, for every $t\in[0,T]$, $f_{t}(\,\cdot\,)$ is uniformly continuous
on bounded sets. Then there exists a sequence of functions
$\\{f^{n}\\}_{n}:[0,T]\times E\rightarrow\mathbb{R}^{d}$ such that:

  1. i.

For every $n\in\mathbb{N}$ and $t\in[0,T]$, $f^{n}_{t}(\,\cdot\,)$ is globally
Lipschitz with constant $C=C(n)>0$ independent of $t$.

  2. ii.

For every $n\in\mathbb{N}$, $f^{n}_{t}(\,\cdot\,)$ has time inhomogeneous
linear growth, in the sense of Assumption A.1, with growth function $h_{t}$.

  3. iii.

For every $t\in[0,T]$, $f^{n}_{t}(\,\cdot\,)\rightarrow f_{t}(\,\cdot\,)$
uniformly on bounded sets.

###### Proof.

Define the map

$g_{t}(z):=\frac{f_{t}(z)}{(1+d_{E}(z,z_{0}))};$

for every $t\in[0,T]$, by assumption $g_{t}(\,\cdot\,)$ is bounded and
uniformly continuous on bounded sets, so we can apply Lemma A.2; this way we
obtain a sequence $g^{n}_{t}(x)$, defined for fixed $t$ by equation (A.2).
Next, consider $\rho\in C^{\infty}_{c}(\mathbb{R};[0,1])$ such that
$\rho(x)\equiv 1$ for $|x|\leq 1$ and $\rho(x)\equiv 0$ for $|x|\geq 2$, and
set

$f^{n}_{t}(z):=g^{n}_{t}(z)\left(1+d_{E}(z,z_{0})\right)\rho\left(\frac{d_{E}(z,z_{0})}{n}\right).$

We claim that $f^{n}$ has the desired properties. Indeed, since
$|g^{n}_{t}(z)|\leq|g_{t}(z)|\leq h_{t}$, we also have
$|f^{n}_{t}(z)|\leq|g^{n}_{t}(z)|(1+d_{E}(z,z_{0}))\leq
h_{t}(1+d_{E}(z,z_{0}))$, so that ii. is satisfied. Item i. is clear from the
construction of $f^{n}$, while the convergence of item iii. follows from iii.
of Lemma A.2. ∎

##  Appendix B Analytic lemmas and inequalities involving maximal functions

We start by presenting a useful weak-strong convergence lemma.

######  Lemma B.1.

Let $X^{n}$ be a sequence of $\mathbb{R}^{d}$-valued r.v. defined on the same
probability space such that $X^{n}\to X$ in probability; assume that
$\mu^{n}:=\mathcal{L}(X^{n}),\mu:=\mathcal{L}(X)$ all belong to $L^{1}_{x}$
and $\mu^{n}\rightharpoonup\mu$ weakly in $L^{1}_{x}$. Finally, consider a
sequence $b^{n}\in L^{\infty}_{x}$ such that $\|b^{n}\|_{L^{\infty}_{x}}\leq
C$ and $b^{n}(x)\to b(x)$ for Lebesgue-a.e. $x$. Then it holds that

$\lim_{n\to\infty}\mathbb{E}[|(b^{n}\ast\mu^{n})(X^{n})-(b\ast\mu)(X)|]=0.$

###### Proof.

We split the estimate as follows:

$\displaystyle\mathbb{E}[|(b^{n}\ast\mu^{n})(X^{n})-(b\ast\mu)(X)|]$
$\displaystyle\leq\mathbb{E}[(|b^{n}-b|\ast\mu^{n})(X^{n})]+\mathbb{E}[|b\ast(\mu^{n}-\mu)|(X^{n})]$
$\displaystyle\quad+\mathbb{E}[|(b\ast\mu)(X^{n})-(b\ast\mu)(X)]$
$\displaystyle=:I_{1}^{n}+I_{2}^{n}+I_{3}^{n}.$

To handle $I^{n}_{1}$, we recall the following classical fact, which can be
checked using Egorov’s theorem: since $|b^{n}-b|$ is a uniformly bounded
sequence converging Lebesgue a.e. to $0$ and $\mu^{n}$ is equiintegrable,
$|b^{n}-b|\ast\mu^{n}$ converges pointwise to $0$; but then iterating the
procedure to $f^{n}:=|b^{n}-b|\ast\mu^{n}$ in place of $|b^{n}-b|$, we can
conclude that $I_{1}^{n}=\int f^{n}\,\mathop{}\\!\mathrm{d}\mu^{n}\to 0$ as
well.

Convergence of $I_{2}^{n}$ follows from similar arguments.

Finally, since $b\in L^{\infty}_{x}$ and $\mu\in L^{1}_{x}$, $b\ast\mu\in
C^{0}_{b}$ and so convergence of $I^{n}_{3}$ follows from the uniform bounds
and the convergence $X^{n}\to X$ in probability. ∎

The remainder of this appendix is devoted to recalling some classical and more
recent properties of maximal functions and their use in estimates involving
Wasserstein distances.

Maximal functions play a fundamental role in ODEs driven by Sobolev drifts,
see for instance [18] for their application; let us recall some fundamental
facts, which can be found in [48].

Given $b\in L^{p}(\mathbb{R}^{d})$, $p\in[1,\infty]$, its maximal function
$Mb$ is defined by

$Mb(x):=\sup_{r>0}\frac{1}{\lambda_{d}\,r^{d}}\int_{B(x,r)}|b(y)|\,\mathop{}\\!\mathrm{d}y$

where $\lambda_{d}$ stands for the Lebesgue measure of $B(0,1)$ in
$\mathbb{R}^{d}$. It is well known that if $p\in(1,\infty]$, then $Mf\in
L^{p}(\mathbb{R}^{d})$ and

$\|Mb\|_{p}\leq c_{d,p}\|b\|_{p}$

for some constant $c_{d,p}>0$; similar definition and properties hold in the
case of vector-valued drifts $b\in L^{p}(\mathbb{R}^{d};\mathbb{R}^{m})$ (in
which case $c=c_{d,p,m}$).

If $b\in W^{1,p}(\mathbb{R}^{d};\mathbb{R}^{d})$, then there exists a
Lebesgue-negligible set $N\subset\mathbb{R}^{d}$ and a constant $c_{d}>0$ such
that the Hajlasz inequality holds:

$|b(x)-b(y)|\leq
c_{d}\,|x-y|\,(MDb(x)+MDb(y))\quad\forall\,x,y\in\mathbb{R}^{d}\setminus N.$
(B.1)

The above and similar inequalities allows one to control the map $\mu\mapsto
b\ast\mu$ in Wasserstein spaces; these results are relevant for applications
in Section 4.

######  Lemma B.2.

Let $(p,q,r)\in(1,\infty)^{3}$ be such that $b\in W^{1,p}_{x}$, $\mu,\nu\in
L^{q}_{x}$ and

$\displaystyle\frac{r}{p}+\frac{1}{q}\leq 1.$

Then there exists a constant $C=C(d,p,q,r)$ such that

$\|b\ast(\mu-\nu)\|_{\infty}\leq
C\|b\|_{W^{1,p}_{x}}\,\big{(}\|\mu\|_{L^{q}_{x}}^{1/r}+\|\nu\|_{L^{q}_{x}}^{1/r}\big{)}\,d_{r^{\prime}}(\mu,\nu)$

###### Proof.

If $d_{r^{\prime}}(\mu,\nu)=+\infty$ the inequality is trivially true (recall
our definition of $d_{r^{\prime}}(\mu,\nu)$, which is valid for any
$\mu,\nu\in\mathcal{P}(\mathbb{R}^{d})$), so we can assume
$d_{r^{\prime}}(\mu,\nu)<\infty$. Moreover by hypothesis $\mu,\nu\in
L^{\tilde{q}}_{x}$ for any $\tilde{q}\in[1,q]$, therefore w.l.o.g. we can
assume $r/p+1/q=1$.

Let $m\in\Pi(\mu,\nu)$ be any coupling of $(\mu,\nu)$ and let
$N\subset\mathbb{R}^{d}$ be as in (B.1); since $\mu,\nu$ are absolutely
continuous w.r.t. Lebesgue, it holds that
$m((N\times\mathbb{R}^{d})\cup(\mathbb{R}^{d}\times N))=0$. Therefore applying
(B.1), for any fixed $x\in\mathbb{R}^{d}$, gives that

$\displaystyle|b\ast(\mu-\nu)(x)|$
$\displaystyle=\bigg{|}\int_{\mathbb{R}^{2d}}[b(x-y)-b(x-z)]\,m(\mathop{}\\!\mathrm{d}y,\mathop{}\\!\mathrm{d}z)]\bigg{|}$
$\displaystyle\lesssim\int_{((\mathbb{R}^{d}\times
N)\cup(N\times\mathbb{R}^{d}))^{c}}|y-z|\,(MDb(x-y)+MDb(x-z))\,m(\mathop{}\\!\mathrm{d}y,\mathop{}\\!\mathrm{d}z)$