#  Lipschitz Continuity of the Schrödinger Map
in Entropic Optimal Transport

Guillaume Carlier Ceremade, Université Paris Dauphine, PSL, 75775, Paris, and
Inria-Paris, Mokaplan Lénaïc Chizat Institute of Mathematics, École
polytechnique fédérale de Lausanne (EPFL), Station Z, CH-1015 Lausanne Maxime
Laborde Université Paris Cité and Sorbonne Université, CNRS, Laboratoire
Jacques-Louis Lions (LJLL), F-75006 Paris, France

###### Abstract

The function that maps a family of probability measures to the solution of the
dual entropic optimal transport problem is known as the Schrödinger map. We
prove that when the cost function is $\mathcal{C}^{k+1}$ with
$k\in\mathbb{N}^{*}$ then this map is Lipschitz continuous from the
$L^{2}$-Wasserstein space to the space of $\mathcal{C}^{k}$ functions. Our
result holds on compact domains and covers the multi-marginal case. As
applications, we prove displacement smoothness of the entropic optimal
transport cost and the well-posedness of certain Wasserstein gradient flows
involving this functional, including the Sinkhorn divergence and a multi-
species system.

Keywords— Entropic Optimal Transport, Schrödinger map, Wasserstein gradient
flows.
MSC Classification— 49Q22, 49K40, 35A15.

##  1 Introduction

The goal of this paper is to show strong stability results for the multi-
marginal entropic optimal transport problem. For clarity of presentation, let
us first present the context and our results in the classical two marginals
case. Let $\mathcal{X}_{1},\mathcal{X}_{2}\subset\mathbb{R}^{d}$ be two
compact convex sets,
$\operatorname{\boldsymbol{\mu}}=(\mu_{1},\mu_{2})\in\mathcal{P}(\mathcal{X}_{1})\times\mathcal{P}(\mathcal{X}_{2})$
two probability measures and
$c\in\mathcal{C}^{k}(\mathcal{X}_{1}\times\mathcal{X}_{2})$ a $k$-times
continuously differentiable _cost_ function. We consider the _entropic optimal
transport_ problem defined as

$\displaystyle
E(\mu_{1},\mu_{2})\coloneqq\min_{\gamma\in\Pi(\mu_{1},\mu_{2})}\int
c(x_{1},x_{2})\mathrm{d}\gamma(x_{1},x_{2})+H(\gamma|\mu_{1}\otimes\mu_{2})$
(1)

where $\Pi(\mu_{1},\mu_{2})$ is the set of _transport plans_ between $\mu_{1}$
and $\mu_{2}$, that is probability measures on
$\mathcal{X}_{1}\times\mathcal{X}_{2}$ with marginals $\mu_{1}$ and $\mu_{2}$,
and $H$ is the _relative entropy_ defined as
$H(\mu|\nu)=\int\log(\mathrm{d}\mu/\mathrm{d}\nu)\mathrm{d}\mu$ if $\mu$ is
absolutely continuous w.r.t. $\nu$ and $+\infty$ otherwise.

This problem can be seen as a regularization of the optimal transport problem
[Vil09, San15] that benefits from improved computational [KY94, Cut13] and
statistical properties [GCBCP19, MN19, BGLN22], at the expense of an
approximation error that can be quantified [Pal19, Wee18, CRLVP20, CT21,
CPT22, EN22]. It is also tightly related to the Schrödinger bridge problem
[Sch32, Léo12], which is a modification of Eq. (1) obtained by replacing
$\mu_{1}\otimes\mu_{2}$ by the product Lebesgue measure in the relative
entropy term.

###  1.1 Schrödinger Map

Eq. (1) defines a convex optimization problem which admits a dual concave
maximization formulation

$\displaystyle
E(\mu_{1},\mu_{2})=\max_{\begin{subarray}{c}\phi_{1}\in\mathcal{C}^{0}(\mathcal{X}_{1})\\\
\phi_{2}\in\mathcal{C}^{0}(\mathcal{X}_{2})\end{subarray}}\int_{\mathcal{X}_{1}}\\!\\!\phi_{1}\mathrm{d}\mu_{1}+\int_{\mathcal{X}_{2}}\\!\\!\phi_{2}\mathrm{d}\mu_{2}+1-\int_{\mathcal{X}_{1}\times\mathcal{X}_{2}}\\!\\!\\!\\!e^{\phi_{1}(x_{1})+\phi_{2}(x_{2})-c(x_{1},x_{2})}\mathrm{d}\mu_{1}(x_{1})\mathrm{d}\mu_{2}(x_{2}).$

This dual problem admits solutions which satisfy, for $\mu_{1}\otimes\mu_{2}$
almost every $(x_{1},x_{2})$, the following first order optimality conditions,
known as the _Schrödinger system_ :

$\displaystyle\left\\{\begin{aligned}
\phi_{1}(x_{1})&=-\log\int_{\mathcal{X}_{2}}e^{\phi_{2}(x_{2})-c(x_{1},x_{2})}\mathrm{d}\mu_{2}(x_{2})\\\
\phi_{2}(x_{2})&=-\log\int_{\mathcal{X}_{1}}e^{\phi_{1}(x_{1})-c(x_{1},x_{2})}\mathrm{d}\mu_{1}(x_{1})\end{aligned}\right..$

In this paper, our main object of interest is the particular solution
$(\phi_{1},\phi_{2})$ which satisfies the Schrödinger system _for every_
$(x_{1},x_{2})\in\mathcal{X}_{1}\times\mathcal{X}_{2}$. It is not difficult to
see that this solution inherits the $\mathcal{C}^{k}$ regularity of $c$ and is
unique in the quotient space
$\tilde{\mathcal{C}}^{k}\coloneqq\mathcal{C}^{k}(\mathcal{X}_{1})\times\mathcal{C}^{k}(\mathcal{X}_{2})/\sim$
where the equivalence relation

$(\phi_{1},\phi_{2})\sim(\psi_{1},\psi_{2})\quad\Leftrightarrow\quad\exists\kappa\in\mathbb{R}\text{
such that }\phi_{1}=\psi_{1}+\kappa\text{ and }\phi_{2}=\psi_{2}-\kappa$

captures the trivial invariance of the dual problem. This particular choice of
solution $(\phi_{1},\phi_{2})$ is arguably the most natural to consider thanks
to its stability. It is also useful in many contexts because it represents the
differential of the functional $E$ [Fey+19]. We refer to this special solution
$(\phi_{1},\phi_{2})$ as the _Schrödinger potentials_ and we define the
_Schödinger map_
$S:\mathcal{P}(\mathcal{X}_{1})\times\mathcal{P}(\mathcal{X}_{2})\to\tilde{\mathcal{C}}^{k}$
as

$\displaystyle
S:\operatorname{\boldsymbol{\mu}}=(\mu_{1},\mu_{2})\mapsto\operatorname{\boldsymbol{\phi}}=(\phi_{1},\phi_{2}).$
(2)

###  1.2 Main result in the two-marginals case

Our main contribution is a proof that the Schrödinger map $S$ is Lipschitz
continuous with respect to the following distances:

  * •

we endow $\mathcal{P}(\mathcal{X}_{i})$ with the Wasserstein metric defined
for two probability measures $\mu,\nu\in\mathcal{P}(\mathcal{X}_{i})$ by

$W_{2}(\mu,\nu)\coloneqq\left(\min_{\gamma\in\Pi(\mu,\nu)}\int_{\mathcal{X}_{i}\times\mathcal{X}_{i}}\|y-x\|^{2}\mathrm{d}\gamma(x,y)\right)^{\frac{1}{2}}.$

and then we endow
$\mathcal{P}(\mathcal{X}_{1})\times\mathcal{P}(\mathcal{X}_{2})$ with the
product Wasserstein $\operatorname{\boldsymbol{W_{2}}}$ metric given for
$\operatorname{\boldsymbol{\mu}}=(\mu_{1},\mu_{2}),\,\boldsymbol{\nu}=(\nu_{1},\nu_{2})\in\mathcal{P}(\mathcal{X}_{1})\times\mathcal{P}(\mathcal{X}_{2})$

$\operatorname{\boldsymbol{W_{2}}}(\operatorname{\boldsymbol{\mu}},\boldsymbol{\nu})\coloneqq\left(W_{2}(\mu_{1},\nu_{1})^{2}+W_{2}(\mu_{2},\nu_{2})^{2}\right)^{\frac{1}{2}}.$

  * •

we endow $\tilde{\mathcal{C}}^{k}$ with the product, quotient supremum
$\mathcal{C}^{k}$ norm defined from the usual $\mathcal{C}^{k}$ norm
$\|\cdot\|_{\mathcal{C}^{k}}$ as

$\|(\phi_{1},\phi_{2})\|_{\tilde{\mathcal{C}}^{k}}\coloneqq\inf_{\kappa\in\mathbb{R}}\|\phi_{1}-\kappa\|_{\mathcal{C}^{k}}+\|\phi_{2}+\kappa\|_{\mathcal{C}^{k}}.$

Our Lipschitz stability result for the Schrödinger map – which is a particular
case of the more general results Thm. 2.3 and Cor. 2.4 that cover the multi-
marginal case and finer regularity results – reads as follows.

######  Theorem 1.1.

If $c\in\mathcal{C}^{k+1}(\mathcal{X}_{1}\times\mathcal{X}_{2})$ for
$k\in\mathbb{N}^{*}$, then there exists $C>0$ that only depends on
$\|c\|_{\mathcal{C}^{k+1}}$ such that for all
$\operatorname{\boldsymbol{\mu}},\operatorname{\boldsymbol{\mu}}^{\prime}\in\mathcal{P}(\mathcal{X}_{1})\times\mathcal{P}(\mathcal{X}_{2})$,

$\|S(\operatorname{\boldsymbol{\mu}})-S(\operatorname{\boldsymbol{\mu}}^{\prime})\|_{\tilde{\mathcal{C}}^{k}}\leq
C\operatorname{\boldsymbol{W_{2}}}(\operatorname{\boldsymbol{\mu}},\operatorname{\boldsymbol{\mu}}^{\prime}).$

This translates into a useful regularity result for the functional $E$ (see
Thm. 4.1).

######  Corollary 1.2.

If $c\in\mathcal{C}^{2}(\mathcal{X}_{1}\times\mathcal{X}_{2})$, then given
$(\mu^{t}_{1})_{t\in[0,1]}$ and $(\mu^{t}_{2})_{t\in[0,1]}$ two Wasserstein
geodesics, the map $h:t\mapsto E(\mu_{1}^{t},\mu_{2}^{t})$ is differentiable
and its derivative satisfies

$|h^{\prime}(t)-h^{\prime}(s)|\leq
C|t-s|\operatorname{\boldsymbol{W_{2}}}(\operatorname{\boldsymbol{\mu}}^{0},\operatorname{\boldsymbol{\mu}}^{1})$

where $\operatorname{\boldsymbol{\mu}}^{t}=(\mu_{1}^{t},\mu_{2}^{t})$ and
$C>0$ only depends on $\|c\|_{\mathcal{C}^{2}}$. In particular, $E$ and $-E$
are displacement semi-convex.

As an application of these results, we will prove the well-posedness of
Wasserstein gradient flows for several energies involving the functional $E$
in Section 4, and also establish exponential convergence to equilibrium in
some cases. Since the Wasserstein gradient of $E$ is $\nabla S_{1},\nabla
S_{2}$ where $S=(S_{1},S_{2})$ is defined in (2), we have for example the
following result (see Prop. 4.5), where $H(\mu)\coloneqq H(\mu|\mathrm{Leb})$
is the (convex) differential entropy.

######  Corollary 1.3.

Let $c\in\mathcal{C}^{2}(\mathcal{X}_{1}\times\mathcal{X}_{2})$ and
$\operatorname{\boldsymbol{\mu}}^{0}=(\mu_{1}^{0},\mu_{2}^{0})\in\mathcal{P}(\mathcal{X}_{1})\times\mathcal{P}(\mathcal{X}_{2})$.
Then the functional $F$ defined by

$F(\operatorname{\boldsymbol{\mu}}):=E(\operatorname{\boldsymbol{\mu}})+H(\mu_{1})+H(\mu_{2}),$

admits a unique Wasserstein gradient flow starting from
$\operatorname{\boldsymbol{\mu}}^{0}$, i.e. there exists a unique absolutely
continuous curve
$\operatorname{\boldsymbol{\mu}}^{t}=(\mu_{1}^{t},\mu_{2}^{t})\in\mathcal{P}(\mathcal{X}_{1})\times\mathcal{P}(\mathcal{X}_{2})$
for $\operatorname{\boldsymbol{W_{2}}}$, satisfying

$\left\\{\begin{aligned} \partial_{t}\mu_{1}&=\nabla\cdot(\mu_{1}\nabla
S_{1}(\operatorname{\boldsymbol{\mu}}))+\Delta\mu_{1}\\\
\partial_{t}\mu_{2}&=\nabla\cdot(\mu_{2}\nabla
S_{2}(\operatorname{\boldsymbol{\mu}}))+\Delta\mu_{2}\\\
\operatorname{\boldsymbol{\mu}}_{|_{t=0}}&=\operatorname{\boldsymbol{\mu}}^{0}\end{aligned}\right.$

with no-flux boundary conditions. If in addition
$H(\mu_{1}^{0}),H(\mu_{2}^{0})<+\infty$, then
$\operatorname{\boldsymbol{\mu}}^{t}$ converges at an exponential rate to the
unique equilibrium $\operatorname{\boldsymbol{\mu}}^{*}$ (see Eq. (25)), in
the sense that there exists $\kappa>0$ independent of
$\operatorname{\boldsymbol{\mu}}^{0}$ such that

$F(\operatorname{\boldsymbol{\mu}}^{t})-F(\operatorname{\boldsymbol{\mu}}^{*})\leq
e^{-\kappa
t}(F(\operatorname{\boldsymbol{\mu}}^{0})-F(\operatorname{\boldsymbol{\mu}}^{*})).$

Let us mention that the system of PDEs in the previous corollary may naturally
appear as an evolution model for cities: $\mu_{1}$ represents the distribution
of agents, $\mu_{2}$ the distribution of firms, $S_{2}$ the wage paid by firms
to agents and the fact that $S_{1}$ and $S_{2}$ are given by (2) captures an
equilibrium condition on the labour market. For more details about such
models, we refer to [Lab20] (for a gradient flow approach without entropic
regularization) and to [BCL21] (in the different context of mean-field games).
For extensions to more than two species and more general functionals
(typically $F+G$ where $G$ is displacement convex), see Section 4. More
applications of Thm. 1.1 are developed in companion papers that study
optimization dynamics for trajectory inference [CZHS22] and regularized
Wasserstein barycenters [Chi], also involving the functional $E$.

###  1.3 Discussion of prior work

Several works have studied the stability of the unregularized optimal
transport problem [DM21, Ber21, Gig11]. In particular, it is known that with
the square-distance cost, the Kantorovich potential from $\mu_{1}$ to
$\mu_{2}$ (i.e. the counterpart of the Schrödinger potential $\phi_{1}$ in
unregularized optimal transport) is a $\frac{1}{2}$-Hölder function of
$\mu_{2}$ from $W_{2}$ to $\dot{H}^{1}(\mu_{1})$ under suitable assumptions on
the fixed reference measure $\mu_{1}$, and that this is the strongest
regularity that one can hope for in general [Gig11].

In [CL20], it is proved by inverse function arguments that $S$ is Lipschitz
continuous and smooth as a map from $L^{\infty}_{++}\to L^{\infty}$, given
some fixed reference measures on the ambient space. Our results will use a
similar strategy but will lead to the stronger conclusion that $S$ is
Lipschitz in Wasserstein distance. As we shall see, this is particularly
useful in the context of gradient flows of energies involving $E$. Let us also
mention that much less is known about stability in the non-compact case, see
e.g. [EN21] that shows Lipschitz continuity of $E$, [NW22] where the
continuity of the Schrödinger map for the topology of convergence in
probability in a certain non-compact setting, see also [GNB21, DDD21] for
further stability results for the primal variable. Our results are finer, but
rely in an essential way on the compact setting.

The rest of the paper is organized as follows. In Section 2, we introduce the
multi-marginal setting and state the full version of our regularity results
for the Schrödinger map. The proofs of those statements can be found in
Section 3. Finally, in Section 4 we study the regularity of the functional $E$
and apply our results to the analysis of certain Wasserstein gradient flows
involving $E$.

##  2 The multi-marginal case

#### Notation and assumptions on the domain

Let $N\geq 2$ be the number of marginals, let
$\mathcal{X}_{i}\subset\mathbb{R}^{d}$ be convex and compact, for
$i\in[N]\coloneqq\\{1,\dots,N\\}$, and let
$\mathcal{X}\coloneqq\prod_{i=1}^{N}\mathcal{X}_{i}$. Given $i\in[N]$, we
denote $\mathcal{X}_{-i}=\prod_{j\neq i}\mathcal{X}_{j}$ and identify
$\mathcal{X}$ to $\mathcal{X}_{i}\times\mathcal{X}_{-i}$, i.e. we denote
$x=(x_{1},\dots,x_{N})$ as $x=(x_{i},x_{-i})$.

For $k\geq 0$, let $\mathcal{C}^{k}(\mathcal{X}_{i})$ be the space of
$k$-times continuously differentiable functions over $\mathcal{X}_{i}$ (that
is, functions defined on $\mathcal{X}_{i}$ that admit a $\mathcal{C}^{k}$
extension on $\mathbb{R}^{d}$) endowed with the usual supremum norm. Using the
multi-index notation, this norm is defined as
$\|f\|_{\mathcal{C}^{k}}\coloneqq\inf_{\tilde{f}}\sup_{|\alpha|\leq
k}\|\tilde{f}^{(\alpha)}\|_{\infty}$ where the infimum is over functions
$\tilde{f}$ that are extensions of $f$ defined on $\mathbb{R}^{d}$. Endowed
with this norm, $\mathcal{C}^{k}(\mathcal{X}_{i})$ is a Banach space, see
[QZ20, Chap. 8, II] for details.

We denote by $\mathcal{P}(\mathcal{X}_{i})$ the space of Borel probability
measures on $\mathcal{X}_{i}$, which we endow with the weak topology,
characterized by its convergent sequences as
$\mu_{n}\rightharpoonup\mu\Leftrightarrow\int\phi\mathrm{d}\mu_{n}\to\int\phi\mathrm{d}\mu$
for all $\phi\in\mathcal{C}^{0}(\mathcal{X}_{i})$. Given
$(\mu_{1},\dots,\mu_{N})\in\prod_{i=1}^{N}\mathcal{P}(\mathcal{X}_{i})$, we
denote by $\operatorname{\boldsymbol{\mu}}$ the $N$-tuple
$(\mu_{1},\dots,\mu_{N})$ and by $\mu$ the product measure
$\otimes_{i=1}^{N}\mu_{i}\in\mathcal{P}(\mathcal{X})$. For
$\mu=\otimes_{i=1}^{N}\mu_{i}\in\mathcal{P}(\mathcal{X})$ we let
$\mu_{-i}\coloneqq\otimes_{j\neq i}\mu_{j}$ and
$\prod_{i=1}^{N}\mathcal{P}(\mathcal{X}_{i})$ is endowed with the product
Wasserstein $\operatorname{\boldsymbol{W_{2}}}$ metric, given, for
$\operatorname{\boldsymbol{\mu}}=(\mu_{1},\dots,\mu_{N}),\,\boldsymbol{\nu}=(\nu_{1},\dots,\nu_{N})\in\prod_{i=1}^{N}\mathcal{P}(\mathcal{X}_{i})$,
by

$\operatorname{\boldsymbol{W_{2}}}(\operatorname{\boldsymbol{\mu}},\boldsymbol{\nu})\coloneqq\left(\sum_{i=1}^{N}W_{2}(\mu_{i},\nu_{i})^{2}\right)^{\frac{1}{2}}.$

In the following, $k\in\mathbb{N}^{*}$ is arbitrary and always denotes the
regularity of the output space $\tilde{\mathcal{C}}^{k}$ of the Schrödinger
map $S$, while the regularity we require for the cost function $c$ varies
across statements.

In the proofs, we use $C,C^{\prime},\dots$ to denote positive constants that
may change from line to line and only depend on general characteristics of the
problem such as $N$ and other quantities that are specified when needed.

###  2.1 Multi-marginal Schrödinger System

The multi-marginal Schrödinger system arises as the optimality conditions for
the multi-marginal _entropic optimal transport_ problem, defined for
$\operatorname{\boldsymbol{\mu}}\in\prod_{i=1}^{N}\mathcal{P}(\mathcal{X}_{i})$
by

$\displaystyle
E(\operatorname{\boldsymbol{\mu}})\coloneqq\min_{\gamma\in\Pi(\operatorname{\boldsymbol{\mu}})}\int_{\mathcal{X}}c(x)\mathrm{d}\gamma(x)+H(\gamma|\mu).$
(3)

where $\Pi(\operatorname{\boldsymbol{\mu}})$ is the set of probability
measures on $\mathcal{X}$ having marginals $(\mu_{1},\dots,\mu_{N})$. This
convex problem admits a concave dual formulation in terms of the Lagrange
multipliers for the marginal constraints

$\displaystyle
E(\operatorname{\boldsymbol{\mu}})=\max_{\operatorname{\boldsymbol{\phi}}\in\prod_{i=1}^{N}\mathcal{C}^{0}(\mathcal{X}^{i})}\sum_{i=1}^{N}\int_{\mathcal{X}_{i}}\phi_{i}(x_{i})\mathrm{d}\mu_{i}(x_{i})+1-\int_{\mathcal{X}}e^{\sum_{i=1}^{N}\phi_{i}(x_{i})-c(x)}\mathrm{d}\mu(x).$
(4)

Notice that these problems are multi-marginals generalizations of those
presented in Section 1. The primal-dual optimality conditions read

$\displaystyle\gamma(\mathrm{d}x)=e^{\sum_{i=1}^{N}\phi_{i}(x_{i})-c(x)}\mu(\mathrm{d}x).$
(5)

We refer to [Léo12, Nut21] for the basic theory of entropic optimal transport
and [CL20, DG20] for the multi-marginal theory. The optimality conditions for
Eq. (4) coincide with the condition that
$\gamma\in\Pi(\operatorname{\boldsymbol{\mu}})$ in Eq. (5), and lead to the
Schrödinger system.

######  Definition 2.1 (Schrödinger system/potentials/map).

Consider the map
$T:\prod_{i=1}^{N}\mathcal{C}(\mathcal{X}_{i})\times\prod_{i=1}^{N}\mathcal{P}(\mathcal{X}_{i})\to\prod_{i=1}^{N}\mathcal{C}(\mathcal{X}_{i})$
defined for $i\in[N]$ and $x_{i}\in\mathcal{X}_{i}$ as $\displaystyle
T_{i}(\operatorname{\boldsymbol{\phi}},\operatorname{\boldsymbol{\mu}})(x_{i})$
$\displaystyle\coloneqq\log\Big{(}\int_{\mathcal{X}_{-i}}e^{\sum_{j=1}^{N}\phi_{j}(x_{j})-c(x_{i},x_{-i})}\mathrm{d}\mu_{-i}(x_{-i})\Big{)}.$
(6) A function $\operatorname{\boldsymbol{\phi}}=(\phi_{1},\dots,\phi_{N})$ is
called a _Schrödinger potential_ associated to
$\operatorname{\boldsymbol{\mu}}$ if it solves the _Schrödinger system_
$\displaystyle
T(\operatorname{\boldsymbol{\phi}},\operatorname{\boldsymbol{\mu}})=0.$ (7)
The _Schrödinger map_ is the function $S$ that maps
$\operatorname{\boldsymbol{\mu}}$ to its Schrödinger potential
$\operatorname{\boldsymbol{\phi}}$, i.e. that satisfies
$T(S(\operatorname{\boldsymbol{\mu}}),\operatorname{\boldsymbol{\mu}})=0$
(Prop. 2.2 states that this map is well-defined in a suitable sense).

Let us stress that we require Eq. (7) to hold in the space of continuous
functions, that is for _every_ $x\in\mathcal{X}$, rather than only $\mu$-a.e.
which is the optimality condition of Eq. (4).

Clearly, if $\operatorname{\boldsymbol{\phi}}=(\phi_{1},\dots,\phi_{N})$
solves (7) for some fixed $\operatorname{\boldsymbol{\mu}}$, then so does
every family of potentials of the form
$(\phi_{1}+\kappa_{1},\dots,\phi_{N}+\kappa_{N})$ where the
$\kappa\in\mathbb{R}^{N}$ satisfies $\sum_{i=1}^{N}\kappa_{i}=0$. This defines
an equivalence relation $\sim$ and we define the quotient space

$\tilde{\mathcal{C}}^{k}\coloneqq\Big{(}\prod_{i=1}^{N}\mathcal{C}^{k}(\mathcal{X}_{i})\Big{)}/\sim.$

Endowed with the quotient norm (the infimum of the norm over all
representatives in the equivalence class), $\tilde{\mathcal{C}}^{k}$ is a
Banach space.

###  2.2 Existence and weak continuity of the Schrödinger map

Let us state some preliminary results about the Schrödinger map.

######  Proposition 2.2.

If $c\in\mathcal{C}^{k}(\mathcal{X})$ for $k\in\mathbb{N}^{*}$ then for any
$\operatorname{\boldsymbol{\mu}}\in\prod_{i=1}^{N}\mathcal{P}(\mathcal{X}_{i})$,
there exists a unique
$\operatorname{\boldsymbol{\phi}}=\operatorname{\boldsymbol{\phi}}^{\operatorname{\boldsymbol{\mu}}}\in\tilde{\mathcal{C}}^{k}$
such that
$T(\operatorname{\boldsymbol{\phi}},\operatorname{\boldsymbol{\mu}})=0$, i.e.
the Schrödinger map
$S:\operatorname{\boldsymbol{\mu}}\mapsto\operatorname{\boldsymbol{\phi}}^{\operatorname{\boldsymbol{\mu}}}$
is well-defined. Moreover,

  * (i)

for $i\in[N]$, $\phi^{\operatorname{\boldsymbol{\mu}}}_{i}$ is
$L_{i}$-Lipschitz continuous, where
$L_{i}=\sup_{x\in\mathcal{X}}\|\nabla_{x_{i}}c(x)\|_{2}$,

  * (ii)

the Schrödinger map
$S:\prod_{i=1}^{N}\mathcal{P}(\mathcal{X}_{i})\to\tilde{\mathcal{C}}^{k}$ is
weakly continuous and is bounded.

The continuity claim $(ii)$ is not needed in the sequel – and is weaker than
Thm. 2.3 – but it is instructive to recall this known result that can be
obtained by elementary means, before delving into more technical proofs.

###### Proof.

The existence of a unique solution to Eq. (7) in
$\prod_{i=1}^{N}L^{\infty}(\mu_{i})/\sim$, i.e. in the $\mu$-almost-everywhere
sense, is proved in [CL20], see also [DG20]. In order to prove the same in
$\tilde{\mathcal{C}}^{0}$, i.e. in the everywhere sense, let us observe that
$T$ can be expressed as $T=\mathrm{Id}+\bar{T}$ with

$\displaystyle\bar{T}_{i}(\operatorname{\boldsymbol{\phi}},\operatorname{\boldsymbol{\mu}})(x_{i})=\log\Big{(}\int_{\mathcal{X}_{-i}}e^{\sum_{j\neq
i}\phi_{j}(x_{j})-c(x_{i},x_{-i})}\mathrm{d}\mu_{-i}(x_{-i})\Big{)}$ (8)

by factorizing $e^{\phi_{i}(x_{i})}$ out in the definition of $T_{i}$. Thus,
given a representer of the $L^{\infty}$ solution
$\operatorname{\boldsymbol{\phi}}^{L^{\infty}}\in\prod_{i=1}^{N}L^{\infty}(\mathcal{X}_{i})$,
one can _define_ a solution
$\operatorname{\boldsymbol{\phi}}^{\mathcal{C}^{k}}\in\prod_{i=1}^{N}\mathcal{C}^{k}(\mathcal{X}_{i})$
in the “everywhere sense” by setting

$\displaystyle\phi_{i}^{\mathcal{C}^{k}}(x_{i})\coloneqq-\log\Big{(}\int_{\mathcal{X}_{-i}}e^{\sum_{j\neq
i}\phi^{L^{\infty}}_{j}(x_{j})-c(x_{i},x_{-i})}\mathrm{d}\mu_{-i}(x_{-i})\Big{)}=-\bar{T}_{i}(\operatorname{\boldsymbol{\phi}}^{L^{\infty}},\operatorname{\boldsymbol{\mu}})(x_{i}).$
(9)

Observe that $\operatorname{\boldsymbol{\phi}}^{\mathcal{C}^{k}}$ inherits the
$\mathcal{C}^{k}$ regularity of $c$. Moreover, by uniqueness in $L^{\infty}$,
any “everywhere” solution must coincide $\mu$-a.e. with an “almost everywhere”
solution and is thus of the form given by Eq. (9). Noticing that
$\bar{T}(\operatorname{\boldsymbol{\phi}}+\kappa,\operatorname{\boldsymbol{\mu}})=\bar{T}(\operatorname{\boldsymbol{\phi}},\operatorname{\boldsymbol{\mu}})-\kappa$
for any family of constants $\kappa\in\mathbb{R}^{N}$ such that
$\sum_{i=1}^{N}\kappa_{i}=0$ and quotienting by $\sim$, it follows that there
exists a unique solution of the Schrödinger system (7) in
$\tilde{\mathcal{C}}^{k}$.

The Lipschitz continuity constant of
$x_{i}\mapsto\phi^{\operatorname{\boldsymbol{\mu}}}_{i}(x_{i})$ can be bounded
by observing that differentiating
$\operatorname{\boldsymbol{\phi}}^{\operatorname{\boldsymbol{\mu}}}=-\bar{T}(\operatorname{\boldsymbol{\phi}}^{\operatorname{\boldsymbol{\mu}}},\operatorname{\boldsymbol{\mu}})$
in $x_{i}$ gives

$\displaystyle\nabla\phi_{i}^{\operatorname{\boldsymbol{\mu}}}(x_{i})=\int_{\mathcal{X}_{-i}}\nabla_{x_{i}}c(x_{i},x_{-i})\mathrm{d}Q_{-i}(x_{-i}|x_{i})$

where $Q_{-i}(\cdot|x_{i})\in\mathcal{P}(\mathcal{X}_{-i})$ is a probability
measure whose expression is given later in Eq. (13).

Finally, to prove weak continuity of $S$, it is enough to prove that if
$\operatorname{\boldsymbol{\mu}}^{n}$ is a sequence weakly converging to
$\operatorname{\boldsymbol{\mu}}$, then
$\operatorname{\boldsymbol{\phi}}^{\operatorname{\boldsymbol{\mu}}^{n}}$
converges to
$\operatorname{\boldsymbol{\phi}}^{\operatorname{\boldsymbol{\mu}}}$. By the
previous point,
$\\{\operatorname{\boldsymbol{\phi}}^{\operatorname{\boldsymbol{\mu}}^{n}}\\}_{n}$
is uniformly Lipschitz continuous and one can choose a uniformly bounded
sequence of representatives so by Ascoli-Arzelà Theorem we can extract a
subsequence $\operatorname{\boldsymbol{\phi}}^{m_{n}}$ which converges in
$\tilde{\mathcal{C}}^{0}$ to some $\operatorname{\boldsymbol{\phi}}^{\infty}$.
Since the map $T$ is jointly continuous, it follows that
$T(\operatorname{\boldsymbol{\phi}}^{\infty},\operatorname{\boldsymbol{\mu}})=\lim_{m}T(\operatorname{\boldsymbol{\phi}}^{\operatorname{\boldsymbol{\mu}}^{m_{n}}},\operatorname{\boldsymbol{\mu}}^{m_{n}})=0$,
hence
$\operatorname{\boldsymbol{\phi}}^{\infty}=\operatorname{\boldsymbol{\phi}}^{\operatorname{\boldsymbol{\mu}}}$.
But this limit is unique, so the full sequence
$(\operatorname{\boldsymbol{\phi}}^{\operatorname{\boldsymbol{\mu}}^{n}})_{n}$
converges to
$\operatorname{\boldsymbol{\phi}}^{\operatorname{\boldsymbol{\mu}}}$ which
proves the weak continuity in $\tilde{\mathcal{C}}^{0}$. Since one also has
$\operatorname{\boldsymbol{\phi}}^{\operatorname{\boldsymbol{\mu}}^{n}}=-\bar{T}(\operatorname{\boldsymbol{\phi}}^{\operatorname{\boldsymbol{\mu}}^{n}},\operatorname{\boldsymbol{\mu}}^{n})$
for all $n$ and $\bar{T}$ is continuous as a function
$\tilde{\mathcal{C}}^{0}\times\prod_{i=1}^{N}\mathcal{P}(\mathcal{X}_{i})\to\tilde{\mathcal{C}}^{k}$,
we have in fact that
$\operatorname{\boldsymbol{\phi}}^{\operatorname{\boldsymbol{\mu}}^{n}}$
converges to
$\operatorname{\boldsymbol{\phi}}^{\operatorname{\boldsymbol{\mu}}}$ in
$\tilde{\mathcal{C}}^{k}$. Boundedness of $S$ finally follows from the fact
that it is weakly continuous on a weakly compact set. ∎

###  2.3 Main result : Lipschitz continuity of the Schrödinger map

In order to study regularity beyond the zero-th order, we bypass the lack of
differentiable structure of $\mathcal{P}(\mathcal{X})$ by considering
parameterized paths generated by transport plans.

Consider $\operatorname{\boldsymbol{\mu}}^{0}=(\mu^{0}_{i})_{i=1}^{N}$ and
$\operatorname{\boldsymbol{\mu}}^{1}=(\mu^{1}_{i})_{i=1}^{N}$ two families of
probability measures in $\prod_{i=1}^{N}\mathcal{P}(\mathcal{X}_{i})$, and a
family of transport plans111To be clear, we _do not_ assume that $\gamma_{i}$
is an _optimal_ transport plan. $\gamma=(\gamma_{i})_{i=1}^{N}$ such that
$\gamma_{i}\in\mathcal{P}(\mathcal{X}_{i}\times\mathcal{X}_{i})$ has marginals
$\mu^{0}_{i}$ and $\mu^{1}_{i}$. These transport plans define interpolations
between $\mu^{0}_{i}$ and $\mu^{1}_{i}$, defined for $t\in[0,1]$ as

$\displaystyle\mu^{t}_{i}=((1-t)\pi^{1}+t\pi^{2})_{\\#}\gamma_{i}$ (10)

where $\pi^{1}$ (resp. $\pi^{2}$) is the projection on the first (resp.
second) factor of $\mathcal{X}_{i}\times\mathcal{X}_{i}$. In other terms,
$\mu^{t}_{i}$ is characterized by

$\int_{\mathcal{X}_{i}}\varphi_{i}(x_{i})\mathrm{d}\mu^{t}_{i}(x_{i})=\int_{\mathcal{X}_{i}\times\mathcal{X}_{i}}\varphi_{i}((1-t)x_{i}+ty_{i})\mathrm{d}\gamma_{i}(x_{i},y_{i}),\qquad\forall\varphi_{i}\in\mathcal{C}(\mathcal{X}_{i}).$

Our main result is as follows.

######  Theorem 2.3.

For $p,k\in\mathbb{N}^{*}$, $p\leq k$, if $c\in\mathcal{C}^{k+p}(\mathcal{X})$
then the parameterized Schrödinger map
$t\mapsto\operatorname{\boldsymbol{\phi}}^{t}\coloneqq
S(\operatorname{\boldsymbol{\mu}}^{t})$ belongs to
$\mathcal{C}^{p}([0,1];\tilde{\mathcal{C}}^{k})$. Moreover, there exists $C>0$
that only depends on $\|c\|_{\mathcal{C}^{k+1}}$ and $N$ such that
$\|\operatorname{\boldsymbol{\phi}}^{t}-\operatorname{\boldsymbol{\phi}}^{s}\|_{\tilde{\mathcal{C}}^{k}}\leq
C|t-s|\sqrt{\operatorname{cost}(\gamma)}$ where
$\operatorname{cost}(\gamma)\coloneqq\sum_{i=1}^{N}\int_{\mathcal{X}_{i}\times\mathcal{X}_{i}}\|y_{i}-x_{i}\|^{2}\mathrm{d}\gamma_{i}(x_{i},y_{i})$
is the $L^{2}$-transport cost associated with $\gamma$.

The detailed proof is postponed to the next section, the basic ingredient
being the application of the Implicit Function Theorem to the map
$G(\operatorname{\boldsymbol{\phi}},t):=T(\operatorname{\boldsymbol{\phi}},\operatorname{\boldsymbol{\mu}}^{t})$.
We can make the following comments:

  * •

tracking the constants in the proof, it can be seen that $C$ depends
exponentially on the oscillation of the cost $\sup_{x}c(x)-\inf_{x}c(x)$ and
polynomially on $\|c\|_{\mathcal{C}^{k+1}}$.

  * •

from the primal-dual relation Eq. (5), one could easily deduce stability
results for the primal variable $\gamma$ from this theorem.

  * •

the fact that the map $t\mapsto S(\operatorname{\boldsymbol{\mu}}^{t})$
belongs to $\mathcal{C}^{p}([0,1];\tilde{\mathcal{C}}^{k})$, also holds if
$(\operatorname{\boldsymbol{\mu}}^{t})_{t\in[0,1]}$, instead of being of the
form (10), is of the form $\mu_{i}^{t}=\xi_{i}(\cdot,t)_{\\#}\mu_{i}^{0}$ for
$i\in[N]$, for some $\mu_{i}^{0}\in\mathcal{P}(\mathcal{X}_{i})$ and a
measurable $\xi_{i}(x_{i},\cdot)\in\mathcal{C}^{p}([0,1];\mathcal{X}_{i})$
with a $\mathcal{C}^{p}$ norm uniformly bounded in $x_{i}$. This can be seen
by suitably adapting the proof of Lem 3.4.

Applying Thm. 2.3 by choosing $\gamma_{i}$ as the optimal transport plan
between $\mu_{i}$ and $\nu_{i}$ immediately leads to the following Lipschitz
continuity result for the Schrödinger map.

######  Corollary 2.4.

For $k\in\mathbb{N}^{*}$, assume that $c\in\mathcal{C}^{k+1}(\mathcal{X})$.
The Schrödinger map
$S:\prod_{i=1}^{N}\mathcal{P}(\mathcal{X}_{i})\to\tilde{\mathcal{C}}^{k}$ is
Lipschitz continuous, i.e. there exists $C>0$ such that, for all
$\operatorname{\boldsymbol{\mu}},\operatorname{\boldsymbol{\nu}}\in\prod_{i=1}^{N}\mathcal{P}(\mathcal{X}_{i})$,
letting
$(\operatorname{\boldsymbol{\phi}}^{\operatorname{\boldsymbol{\mu}}},\operatorname{\boldsymbol{\phi}}^{\operatorname{\boldsymbol{\nu}}})=(S(\operatorname{\boldsymbol{\mu}}),S(\operatorname{\boldsymbol{\nu}}))$,
$\|\operatorname{\boldsymbol{\phi}}^{\operatorname{\boldsymbol{\mu}}}-\operatorname{\boldsymbol{\phi}}^{\operatorname{\boldsymbol{\nu}}}\|_{\tilde{\mathcal{C}}^{k}}\leq
C\operatorname{\boldsymbol{W_{2}}}(\operatorname{\boldsymbol{\mu}},\operatorname{\boldsymbol{\nu}}).$

##  3 Proofs

The main tool to prove Thm. 2.3 is the Implicit Function Theorem. We will
apply it to the function
$G:\tilde{\mathcal{C}}^{k}\times[0,1]\to\mathcal{C}^{k}$ defined as

$\displaystyle G(\operatorname{\boldsymbol{\phi}},t)$ $\displaystyle\coloneqq
T(\operatorname{\boldsymbol{\phi}},\operatorname{\boldsymbol{\mu}}^{t})$ (11)

whose expression222the maps $G$ and $T$ take values in $\mathcal{C}^{k}$ but
it will sometimes be convenient to compose them from the left with the
canonical projection $\mathcal{C}^{k}\to\tilde{\mathcal{C}}^{k}$, slightly
abusing notations, we will still denote by $G$ and $T$ these maps with values
in $\tilde{\mathcal{C}}^{k}$. is, using the convention $y_{i}\coloneqq x_{i}$,

$\displaystyle
G_{i}(\operatorname{\boldsymbol{\phi}},t)(x_{i})=\phi_{i}(x_{i})+\log\Big{(}\int_{\mathcal{X}_{-i}\times\mathcal{X}_{-i}}e^{\sum_{j\neq
i}\phi_{j}((1-t)x_{j}+ty_{j})-c((1-t)x+ty)}\mathrm{d}\gamma_{-i}(x_{-i},y_{-i})\Big{)}.$

For this purpose, in the next sections, we study the applications $T$ and $G$.

###  3.1 Invertibility of the differential of $T$

Let us fix
$\operatorname{\boldsymbol{\mu}}=(\mu_{1},\dots,\mu_{N})\in\prod_{i=1}^{N}\mathcal{P}(\mathcal{X}_{i})$
and $\mu=\otimes_{i=1}^{N}\mu_{i}\in\mathcal{P}(\mathcal{X})$ and study the
map $\operatorname{\boldsymbol{\phi}}\mapsto
T(\operatorname{\boldsymbol{\phi}},\operatorname{\boldsymbol{\mu}})$, which is
a self-map of $\tilde{\mathcal{C}}^{k}$. Note that $T$ is of class
$C^{\infty}$ in the first variable and its differential is given, for
$\operatorname{\boldsymbol{h}}\in\tilde{\mathcal{C}}^{k}$, by

$\displaystyle
D_{\phi}T_{i}(\operatorname{\boldsymbol{\phi}},\operatorname{\boldsymbol{\mu}})(\operatorname{\boldsymbol{h}})(x_{i})$
$\displaystyle=h_{i}(x_{i})+\int_{\mathcal{X}_{-i}}\big{(}\sum_{j\neq
i}h_{j}(x_{j})\big{)}q_{-i}(x_{-i}|x_{i})\mathrm{d}\mu_{-i}(x_{-i}).$

where we have introduced the function $q_{-i}$ defined, with the convention
$x^{\prime}_{i}=x_{i}$, by

$\displaystyle q_{-i}(x_{-i}|x_{i})$
$\displaystyle\coloneqq\frac{e^{\sum_{j\neq
i}\phi_{j}(x_{j})-c(x)}}{\int_{\mathcal{X}_{-i}}e^{\sum_{j\neq
i}\phi_{j}(x^{\prime}_{j})-c(x^{\prime})}\mathrm{d}\mu_{-i}(x^{\prime}_{-i})}.$

Note that $q_{-i}$ depends on $\operatorname{\boldsymbol{\phi}}$ and
$\operatorname{\boldsymbol{\mu}}$ although this is not explicit in the
notation. Similarly, let

$\displaystyle q(x)$
$\displaystyle\coloneqq\frac{e^{\sum_{j}\phi_{j}(x_{j})-c(x)}}{\int_{\mathcal{X}}e^{\sum_{j}\phi_{j}(x_{j}^{\prime})-c(x^{\prime})}\mathrm{d}\mu(x^{\prime})},$
$\displaystyle q_{i}(x_{i})$
$\displaystyle\coloneqq\frac{\int_{\mathcal{X}_{-i}}e^{\sum_{j}\phi_{j}(x_{j})-c(x)}\mathrm{d}\mu_{-i}(x_{-i})}{\int_{\mathcal{X}}e^{\sum_{j}\phi_{j}(x^{\prime}_{j})-c(x^{\prime})}\mathrm{d}\mu(x^{\prime})}.$
(12)

Observe that if $\operatorname{\boldsymbol{\phi}}$ and $c$ are of class
$\mathcal{C}^{k}$ then the functions $q,q_{i},q_{-i}$ are of class
$\mathcal{C}^{k}$ as well. These functions are densities of probability
densities in the sense that it holds

$\displaystyle Q$ $\displaystyle\coloneqq q\mu\in\mathcal{P}(\mathcal{X}),$
$\displaystyle Q_{i}$ $\displaystyle\coloneqq
q_{i}\mu_{i}\in\mathcal{P}(\mathcal{X}_{i}),$ $\displaystyle
Q_{-i}(\cdot|x_{i})$ $\displaystyle\coloneqq
q_{-i}(\cdot|x_{i})\mu_{-i}\in\mathcal{P}(\mathcal{X}_{-i}),\;\forall
x_{i}\in\mathcal{X}_{i}.$ (13)

By construction, for each $i$, $Q_{i}$ is the $i$-th marginal of $Q$ on
$\mathcal{X}_{i}$ and $Q_{-i}$ is the disintegration of $Q$ with respect to
this marginal, i.e.:

$\displaystyle\mathrm{d}Q(x_{i},\;x_{-i})=\mathrm{d}Q_{-i}(x_{-i}|x_{i})\mathrm{d}Q_{i}(x_{i}).$
(14)

In the next lemma, we remark that these densities are uniformly bounded from
above and below by positive quantities, a fact which we will often use in the
following.

######  Lemma 3.1.

Let $q$ be defined by (12). Then, for all $x\in\mathcal{X}$,

$e^{-2(N\|\operatorname{\boldsymbol{\phi}}\|_{\tilde{\mathcal{C}}^{0}}+\|c\|_{\mathcal{C}^{0}})}\leq
q(x)\leq
e^{2(N\|\operatorname{\boldsymbol{\phi}}\|_{\tilde{\mathcal{C}}^{0}}+\|c\|_{\mathcal{C}^{0}})}.$

Moreover $q_{i}$ and $q_{-i}$ satisfy the same bounds.

###### Proof.

From the definition of $q$, for all $x\in\mathcal{X}$,

$\displaystyle q(x)$
$\displaystyle=\frac{e^{\sum_{j}\phi_{j}(x_{j})-c(x)}}{\int_{\mathcal{X}}e^{\sum_{j}\phi_{j}(x_{j}^{\prime})-c(x^{\prime})}\mathrm{d}\mu(x^{\prime})}\leq\frac{e^{\sum_{j}\|\phi_{j}\|_{\infty}+\|c\|_{\mathcal{C}^{0}}}}{e^{-\sum_{j}\|\phi_{j}\|_{\infty}-\|c\|_{\mathcal{C}^{0}}}}\leq
e^{2(N\|\operatorname{\boldsymbol{\phi}}\|_{\mathcal{C}^{0}}+\|c\|_{\mathcal{C}^{0}})},$

since $\mu\in\mathcal{P}(\mathcal{X})$. In addition, from the definition of
$q$, we remark that $q$ does not depend of the representative of
$\operatorname{\boldsymbol{\phi}}$ in $\tilde{\mathcal{C}}^{0}$ which gives
the upper bound on $q$. We obtain the lower bound, as well as the result for
$q_{i}$ and $q_{-i}$ with the same arguments. ∎

Let us also remark that in the previous bounds, the norm
$\|c\|_{\mathcal{C}^{0}}$ can be replaced by
$\inf_{\kappa\in\mathbb{R}}\|c+\kappa\|_{\mathcal{C}^{0}}=(\sup c-\inf c)/2$,
i.e. half the _oscillation of $c$_ (in fact, the Schrödinger map is invariant
if $c$ changes by an additive constant). The following lemma is central in our
development.

######  Lemma 3.2.

Let $k\in\mathbb{N}^{*}$ and assume that
$\operatorname{\boldsymbol{\phi}}\in\tilde{\mathcal{C}}^{0}$ and
$c\in\mathcal{C}^{k}(\mathcal{X})$. Then
$D_{\operatorname{\boldsymbol{\phi}}}T(\operatorname{\boldsymbol{\phi}},\operatorname{\boldsymbol{\mu}})$
is an invertible linear self-map of $\tilde{\mathcal{C}}^{k}$. Moreover, there
exists $C>0$ that only depends on $N$, $\|c\|_{\mathcal{C}^{k}}$ and
$\|\operatorname{\boldsymbol{\phi}}\|_{\tilde{\mathcal{C}}^{0}}$ such that

$\|[D_{\operatorname{\boldsymbol{\phi}}}T(\operatorname{\boldsymbol{\phi}},\operatorname{\boldsymbol{\mu}})]^{-1}\|_{\tilde{\mathcal{C}}^{k}\to\tilde{\mathcal{C}}^{k}}\leq
C.$

###### Proof.

We have
$D_{\operatorname{\boldsymbol{\phi}}}T(\operatorname{\boldsymbol{\phi}},\operatorname{\boldsymbol{\mu}})=\mathrm{Id}+L$
with

$\displaystyle L_{i}(\operatorname{\boldsymbol{h}})(x_{i})$
$\displaystyle=\int_{\mathcal{X}_{-i}}\Big{(}\sum_{j\neq
i}h_{j}(x_{j})\Big{)}q_{-i}(x_{-i}|x_{i})\mathrm{d}\mu_{-i}(x_{-i}).$

Observe that since $c\in\mathcal{C}^{k}(\mathcal{X})$,
$L_{i}(\operatorname{\boldsymbol{h}})\in\mathcal{C}^{k}(\mathcal{X}_{i})$ with
its derivatives up to order $k$ equi-continuous when $h$ runs through a
bounded set of $\tilde{\mathcal{C}}^{0}$. It follows, by Arzelà-Ascoli
Theorem, that $L:\tilde{\mathcal{C}}^{0}\to\tilde{\mathcal{C}}^{k}$ is
compact, and a fortiori $L:\tilde{\mathcal{C}}^{k}\to\tilde{\mathcal{C}}^{k}$
is compact too.

_Step 1._ Let us show that $\mathrm{id}+L$ is invertible. Let
$\operatorname{\boldsymbol{h}}\in\prod\mathcal{C}(\mathcal{X}_{i})$ be such
that $\operatorname{\boldsymbol{h}}+L(\operatorname{\boldsymbol{h}})=0$ in
$\tilde{\mathcal{C}}^{0}$, i.e.

$\displaystyle h_{i}(.)+\int_{\mathcal{X}_{-i}}\Big{(}\sum_{j\neq
i}h_{j}(x_{j})\Big{)}\mathrm{d}Q_{-i}(x_{-i}|.)=\lambda_{i},\;i=1,\dots,N,\;\sum_{j=1}^{N}\lambda_{j}=0.$
(15)

Integrating (15) with respect to $Q_{i}$, we deduce from (14), that

$\sum_{k=1}^{N}\int_{\mathcal{X}_{k}}h_{k}\mathrm{d}Q_{k}=\lambda_{i},\;\;i=1,\dots,N$

so that all the $\lambda_{i}$’s are equal to $0$; hence
$\operatorname{\boldsymbol{h}}+L(\operatorname{\boldsymbol{h}})=0$ in
$\mathcal{C}^{0}$ (and not only in the quotient $\tilde{\mathcal{C}}^{0}$).
Then, taking the dot product of $\operatorname{\boldsymbol{h}}$ with
$\operatorname{\boldsymbol{h}}+L(\operatorname{\boldsymbol{h}})$ in $\prod
L^{2}(Q_{i})$ and using (14), it follows, reasoning as in [CL20],

$\displaystyle 0$
$\displaystyle=\sum_{i=1}^{N}\int_{\mathcal{X}_{i}}h_{i}(x_{i})\Big{(}h_{i}(x_{i})+\int_{\mathcal{X}_{-i}}\Big{(}\sum_{j\neq
i}h_{j}(x_{j})\Big{)}\mathrm{d}Q_{-i}(x_{-i}|x_{i})\Big{)}\mathrm{d}Q_{i}(x_{i})$
$\displaystyle=\sum_{i}\int_{\mathcal{X}_{i}}h_{i}(x_{i})^{2}\mathrm{d}Q_{i}(x_{i})+\sum_{i\neq
j}\int_{\mathcal{X}}h_{i}(x_{i})h_{j}(x_{j})\mathrm{d}Q(x)$
$\displaystyle=\int_{\mathcal{X}}\Big{(}\sum_{i}h_{i}(x_{i})\Big{)}^{2}\mathrm{d}Q(x).$

We deduce that $x\mapsto\sum h_{i}(x_{i})$ is equal to $0$ as a function in
$L^{2}(Q)$ and hence in $L^{2}(\mu)$.

Now consider the space
$\tilde{L}^{2}_{\operatorname{\boldsymbol{\mu}}}\coloneqq\prod_{i=1}^{N}L^{2}(\mu_{i})/\sim$
which, endowed with the quotient space structure, is also a Hilbert space. By
Lem. 3.3 (proved hereafter), it follows that
$\operatorname{\boldsymbol{h}}\sim 0$ in
$\tilde{L}^{2}_{\operatorname{\boldsymbol{\mu}}}$, i.e. there exists
$\kappa\in\mathbb{R}^{n}$ such that $\sum\kappa_{i}=0$ and
$h_{i}(x_{i})=\kappa_{i}$ for $\mu_{i}$-a.e. $x_{i}$. It only remains to show
that this equality holds in fact everywhere. Using
$\operatorname{\boldsymbol{h}}=-L(\operatorname{\boldsymbol{h}})$ and
$q_{-i}\mu_{-i}\in\mathcal{P}(\mathcal{X}_{-i})$, it holds for
$x_{i}\in\mathcal{X}_{i}$

$\displaystyle h_{i}(x_{i})$
$\displaystyle=-\int_{\mathcal{X}_{-i}}\Big{(}\sum_{j\neq
i}h_{j}(x_{j})\Big{)}q_{-i}(x_{-i}|x_{i})\mathrm{d}\mu_{-i}(x_{-i})=-\sum_{j\neq
i}\kappa_{j}=\kappa_{i}$

Thus $\operatorname{\boldsymbol{h}}=0$ in $\tilde{\mathcal{C}}^{k}$.
Conversely, any $\operatorname{\boldsymbol{h}}=0$ in $\tilde{\mathcal{C}}^{k}$
also clearly belongs to
$\ker(D_{\operatorname{\boldsymbol{\phi}}}T(\operatorname{\boldsymbol{\phi}},\operatorname{\boldsymbol{\mu}}))$.
Hence
$\ker(D_{\operatorname{\boldsymbol{\phi}}}T(\operatorname{\boldsymbol{\phi}},\operatorname{\boldsymbol{\mu}}))$
is precisely the equivalence class of $0$ i.e.
$D_{\operatorname{\boldsymbol{\phi}}}T(\operatorname{\boldsymbol{\phi}},\operatorname{\boldsymbol{\mu}})$
is injective on $\tilde{\mathcal{C}}^{k}$. Since $L$ is a compact operator of
$\tilde{\mathcal{C}}^{k}$, it follows from the Fredholm Alternative Theorem
[Bré11, Chap. 6] that the range of $\mathrm{Id}+L$ is
$\tilde{\mathcal{C}}^{k}$. Hence
$D_{\operatorname{\boldsymbol{\phi}}}T(\operatorname{\boldsymbol{\phi}},\operatorname{\boldsymbol{\mu}})$
is onto and therefore an invertible linear self-map of
$\tilde{\mathcal{C}}^{k}$.

_Step 2._ Now let us estimate the operator norm of
$D_{\operatorname{\boldsymbol{\phi}}}T(\operatorname{\boldsymbol{\phi}},\operatorname{\boldsymbol{\mu}})^{-1}$
as a self-map of $\tilde{\mathcal{C}}^{k}$. Let
$\operatorname{\boldsymbol{h}}\in\tilde{\mathcal{C}}^{k},\operatorname{\boldsymbol{g}}\in\tilde{\mathcal{C}}^{k}$
be such that
$\operatorname{\boldsymbol{h}}+L(\operatorname{\boldsymbol{h}})=\operatorname{\boldsymbol{g}}$.
Let us choose the representative of $\operatorname{\boldsymbol{g}}$ that
satisfies

$\displaystyle\int_{\mathcal{X}_{1}}g_{1}(x_{1})\mathrm{d}Q_{1}(x_{1})=\dots=\int_{\mathcal{X}_{N}}g_{N}(x_{N})\mathrm{d}Q_{N}(x_{N}).$
(16)

Reasoning as above, it holds

$\displaystyle\sum_{i=1}^{N}\int_{\mathcal{X}_{i}}g_{i}(x_{i})h_{i}(x_{i})\mathrm{d}Q_{i}(x_{i})$
$\displaystyle=\sum_{i}\int_{\mathcal{X}_{i}}h_{i}(x_{i})^{2}\mathrm{d}Q_{i}(x_{i})+\sum_{i,j,\,i\neq
j}\int_{\mathcal{X}}h_{i}(x_{i})h_{j}(x_{j})\mathrm{d}Q(x)$
$\displaystyle=\int_{\mathcal{X}}\Big{(}\sum_{i}h_{i}(x_{i})\Big{)}^{2}\mathrm{d}Q(x)$

where the first integral is unambiguously defined because, thanks to our
choice of representative for $g$, it does not depend on the representative
chosen for $h$. Let us choose the optimal representative for
$\operatorname{\boldsymbol{h}}$ in
$\tilde{L}^{2}_{\operatorname{\boldsymbol{\mu}}}$ appearing in Lem. 3.3. We
have, for some $C,C^{\prime}>0$ that only depends on $N$,
$\|c\|_{\mathcal{C}_{0}}$ and
$\|\operatorname{\boldsymbol{\phi}}\|_{\tilde{\mathcal{C}}_{0}}$:

$\displaystyle\|\operatorname{\boldsymbol{h}}\|^{2}_{\prod L^{2}(\mu_{i})}$
$\displaystyle\overset{(i)}{\leq}N\left\|\oplus_{i=1}^{N}h_{i}\right\|^{2}_{L^{2}(\mu)}$
$\displaystyle\overset{(ii)}{\leq}C\left\|\oplus_{i=1}^{N}h_{i}\right\|^{2}_{L^{2}(Q)}$
$\displaystyle\overset{(iii)}{\leq}C\|\operatorname{\boldsymbol{g}}\|_{\prod
L^{2}(Q_{i})}\|\operatorname{\boldsymbol{h}}\|_{\prod L^{2}(Q_{i})}$
$\displaystyle\overset{(iv)}{\leq}C^{\prime}\|\operatorname{\boldsymbol{g}}\|_{\prod
L^{2}(Q_{i})}\|\operatorname{\boldsymbol{h}}\|_{\prod L^{2}(\mu_{i})}$

where we have used (i) Lem. 3.3 (where the notation $\oplus$ is defined) ,
(ii)&(iv) Lem. 3.1, and (iii) the previous computation and Cauchy-Schwarz
inequality in $\prod L^{2}(Q_{i})$. It follows, invoking once again Lem. 3.3
and Lem. 3.1, that

$\|\operatorname{\boldsymbol{h}}\|_{\tilde{L}^{2}_{\operatorname{\boldsymbol{\mu}}}}=\|\operatorname{\boldsymbol{h}}\|_{\prod
L^{2}(\mu_{i})}\leq C^{\prime}\|\operatorname{\boldsymbol{g}}\|_{\prod
L^{2}(Q_{i})}=C^{\prime}\|\operatorname{\boldsymbol{g}}\|_{\tilde{L}^{2}_{Q}}.$

For the last equality, we have used the fact that the $\tilde{L}^{2}_{Q}$ norm
is precisely the $\prod L^{2}(Q_{i})$ norm of the representative that
satisfies Eq. (16), by Lem. 3.3 (here $\tilde{L}^{2}_{Q}$ is defined similarly
as $\tilde{L}^{2}_{\operatorname{\boldsymbol{\mu}}}$ from the marginals
$Q_{i}$ of $Q$).

_Step 3._ We now improve the $\tilde{L}^{2}_{\operatorname{\boldsymbol{\mu}}}$
control into a $\tilde{\mathcal{C}}^{k}$ control. Restarting from
$\operatorname{\boldsymbol{h}}+L(\operatorname{\boldsymbol{h}})=\operatorname{\boldsymbol{g}}$,
it holds

$\displaystyle
h_{i}(x_{i})=g_{i}(x_{i})-\int_{\mathcal{X}_{-i}}\Big{(}\sum_{j\neq
i}h_{j}(x_{j})\Big{)}q_{-i}(x_{-i}|x_{i})\mathrm{d}\mu_{-i}(x_{-i}).$

Thanks to our control on $\|\operatorname{\boldsymbol{h}}\|_{\prod
L^{2}(\mu_{i})}$ and the assumption that $c\in\mathcal{C}^{k}$, it follows

$\|h_{i}\|_{\tilde{\mathcal{C}}^{k}}\leq\|g_{i}\|_{\tilde{\mathcal{C}}^{k}}+C^{\prime\prime}\sum_{j\neq
i}\|h_{j}\|_{L^{1}(\mu_{j})}\leq\|g_{i}\|_{\tilde{\mathcal{C}}^{k}}+C^{\prime\prime\prime}\|\operatorname{\boldsymbol{g}}\|_{\tilde{L}^{2}_{Q}}$

which leads to the conclusion since
$\|\operatorname{\boldsymbol{g}}\|_{\tilde{L}^{2}_{Q}}$ is controlled by
$\|\operatorname{\boldsymbol{g}}\|_{\tilde{\mathcal{C}}^{0}}$. Thus
$D_{\operatorname{\boldsymbol{\phi}}}T(\operatorname{\boldsymbol{\phi}},\operatorname{\boldsymbol{\mu}})^{-1}$
is bounded as an operator $\tilde{\mathcal{C}}^{k}\to\tilde{\mathcal{C}}^{k}$,
with an operator norm that only depends on $N$, $\|c\|_{\mathcal{C}^{k}}$ and
$\|\operatorname{\boldsymbol{\phi}}\|_{\tilde{\mathcal{C}}^{0}}$. ∎

To end this section, we prove Lem. 3.3 used in the previous proof.

######  Lemma 3.3.

For $\operatorname{\boldsymbol{h}}\in\prod_{i=1}^{N}L^{2}(\mu_{i})$, denoting
$\oplus_{i=1}^{N}h_{i}:x\mapsto\sum_{i=1}^{N}h_{i}(x_{i})$ it holds

$\|\operatorname{\boldsymbol{h}}\|^{2}_{\tilde{L}^{2}_{\operatorname{\boldsymbol{\mu}}}}\leq\left\|\oplus_{i=1}^{N}h_{i}\right\|^{2}_{L^{2}(\mu)}\leq
N\|\operatorname{\boldsymbol{h}}\|^{2}_{\tilde{L}^{2}_{\operatorname{\boldsymbol{\mu}}}}.$

Moreover, the quotient norm is achieved by the unique representative
$\hat{\operatorname{\boldsymbol{h}}}\sim\operatorname{\boldsymbol{h}}$ that
satisfies
$\int_{\mathcal{X}_{1}}\hat{h}_{1}\mathrm{d}\mu_{1}=\dots=\int_{\mathcal{X}_{N}}\hat{h}_{N}\mathrm{d}\mu_{N}$,
i.e. it holds
$\|h\|^{2}_{\tilde{L}^{2}_{\operatorname{\boldsymbol{\mu}}}}=\sum_{i=1}^{N}\|\hat{h}_{i}\|^{2}_{L^{2}(\mu_{i})}$.

###### Proof.

By definition,

$\displaystyle\|\operatorname{\boldsymbol{h}}\|^{2}_{\tilde{L}^{2}_{\operatorname{\boldsymbol{\mu}}}}$
$\displaystyle=\min_{\begin{subarray}{c}\kappa\in\mathbb{R}^{N}\\\
\sum_{i}\kappa_{i}=0\end{subarray}}\sum_{i=1}^{N}\int_{\mathcal{X}_{i}}(h_{i}(x_{i})-\kappa_{i})^{2}\mathrm{d}\mu_{i}(x_{i}).$
(17)

A vector $\kappa\in\mathbb{R}^{N}$ solves this problem iff
$\sum_{i}\kappa_{i}=0$ and there exists a Lagrange multiplier
$\nu\in\mathbb{R}$ such that for $i\in[N]$,

$0=\int_{\mathcal{X}_{i}}(h_{i}(x_{i})-\kappa_{i})\mathrm{d}\mu_{i}(x_{i})-\nu=\mathbf{E}_{\mu_{i}}[h_{i}]-\kappa_{i}-\nu.$

with the shorthand $\mathbf{E}_{\mu}[h]\coloneqq\int h\mathrm{d}\mu$ and
$\operatorname{Var}_{\mu}(h)\coloneqq\mathbf{E}_{\mu}[(h-\mathbf{E}_{\mu}[h])^{2}]$.
It follows that $\nu=\frac{1}{N}\sum_{i=1}^{N}\mathbf{E}_{\mu_{i}}[h_{i}]$ and
as a consequence

$\displaystyle\|\operatorname{\boldsymbol{h}}\|^{2}_{\tilde{L}^{2}_{\operatorname{\boldsymbol{\mu}}}}$
$\displaystyle=\sum_{i=1}^{N}\int_{\mathcal{X}_{i}}(h_{i}(x_{i})-\mathbf{E}_{\mu_{i}}[h_{i}]+\nu)^{2}\mathrm{d}\mu_{i}(x_{i})$
$\displaystyle=\sum_{i=1}^{N}\Big{[}\int_{\mathcal{X}_{i}}(h_{i}(x_{i})-\mathbf{E}_{\mu_{i}}[h_{i}])^{2}\mathrm{d}\mu_{i}(x_{i})+\nu^{2}\Big{]}$
$\displaystyle=\sum_{i=1}^{N}\operatorname{Var}_{\mu_{i}}(h_{i})+\frac{1}{N}\Big{(}\sum_{i=1}^{N}\int
h_{i}\mathrm{d}\mu_{i}\Big{)}^{2}$

where the second equality follows by expanding the square and observing that
the cross-terms vanish. On the other hand, using the fact that
$\sum_{i}\kappa_{i}=0$, it holds

$\displaystyle\left\|\oplus_{i=1}^{N}h_{i}\right\|^{2}_{L^{2}(\mu)}$
$\displaystyle=\int_{\mathcal{X}}\big{(}\sum_{i=1}^{N}(h_{i}(x_{i})-\kappa_{i})\big{)}^{2}\mathrm{d}\mu(x)$
$\displaystyle=\sum_{i,j,i\neq
j}\int_{\mathcal{X}_{i}\times\mathcal{X}_{j}}(h_{i}(x_{i})-\mathbf{E}_{\mu_{i}}[h_{i}]+\nu)(h_{j}(x_{j})-\mathbf{E}_{\mu_{j}}[h_{j}]+\nu)\mathrm{d}\mu_{i}(x_{i})\mathrm{d}\mu_{j}(x_{j})$
$\displaystyle\quad+\sum_{i}\int_{\mathcal{X}_{i}}(h_{i}(x_{i})-\mathbf{E}_{\mu_{i}}[h_{i}])+\nu)^{2}\mathrm{d}\mu_{i}(x_{i})$
$\displaystyle=N(N-1)\nu^{2}+\|\operatorname{\boldsymbol{h}}\|^{2}_{\tilde{L}^{2}_{\operatorname{\boldsymbol{\mu}}}}$
$\displaystyle=\sum_{i=1}^{N}\operatorname{Var}_{\mu_{i}}(h_{i})+\Big{(}\sum_{i=1}^{N}\int_{\mathcal{X}_{i}}h_{i}\mathrm{d}\mu_{i}\Big{)}^{2}$

The first claim follows. For the second claim, observe that this
representative $\hat{h}$ satisfies the optimality condition of Eq. (17). ∎

###  3.2 Differentiability of $G$

Let us first characterize the regularity of the map $G$.

######  Lemma 3.4.

For $p,k\in\mathbb{N}^{*}$, $p\leq k$, if
$c\in\mathcal{C}^{k+p}(\mathcal{X})$, then the map
$G:\mathcal{C}^{k}\times[0,1]\to\mathcal{C}^{k}$ is of class
$\mathcal{C}^{p}$.

###### Proof.

The $i$-th component of $G$ can be expressed as

$G_{i}(\operatorname{\boldsymbol{\phi}},t)(x_{i})=\phi_{i}(x_{i})+\log\Big{(}\int_{\mathcal{X}_{-i}\times\mathcal{X}_{-i}}e^{\sum_{j\neq
i}\phi_{j}((1-t)x_{j}+ty_{j})-c(x_{i},(1-t)x_{-i}+ty_{-i})}\mathrm{d}\gamma_{-i}(x_{-i},y_{-i})\Big{)}$

Fixing $i$ and $(x_{i},y_{-i})\in\mathcal{X}_{-i}\times\mathcal{X}_{-i}$, let
us observe that when $c\in\mathcal{C}^{k+p}(\mathcal{X})$, the curve
$t\in[0,1]\mapsto c(.,(1-t)x_{-i}+ty_{-i})\in\mathcal{C}^{k}(\mathcal{X}_{i})$
is of class $\mathcal{C}^{p}$ and that its derivatives up to order $p$ can be
bounded independently of $(x_{-i},y_{-i})$. Now, for $j\neq i$ (and fixed
$x_{j}$ and $y_{j}$ in $\mathcal{X}_{j}$), consider the real-valued map
$L_{j}:(\phi_{j},t)\in\mathcal{C}^{k}(\mathcal{X}_{j})\times[0,1]\mapsto\phi_{j}(x_{j}+t(y_{j}-x_{j}))$.
For $k=1$, this map admits partial derivatives with respect to $t$ and
$\phi_{j}$ which are given respectively by
$\nabla\phi_{j}(x_{j}+t(y_{j}-x_{j}))^{\top}(y_{j}-x_{j})$ and $L_{j}(.,t)$,
both being continuous (for the $\mathcal{C}^{1}$ norm for $\phi_{j}$) so that
$L_{j}\in\mathcal{C}^{1}(\mathcal{C}^{1}(\mathcal{X}_{j})\times[0,1],\mathbb{R})$
note also that the first-order partial derivatives of $L_{j}$ can be bounded
by a constant depending on the $\mathcal{C}^{1}$ norm of $\phi_{j}$ but not on
$x_{j},y_{j}$.

For $k\geq 2$, we can argue inductively. Indeed, by the previous argument,
showing $k$ times continuous differentiability of $L_{j}$ amounts to showing
$k-1$ times continuous differentiability of $L_{j}$ applied to $t$ and
$\nabla\phi_{j}(.)^{\top}(y_{j}-x_{j})$. This shows that
$L_{j}\in\mathcal{C}^{k}(\mathcal{C}^{k}(\mathcal{X}_{j})\times[0,1],\mathbb{R})$,
with bounds on derivatives up to order $k$ controlled by the $\mathcal{C}^{k}$
norm of $\phi_{j}$ independently of
$(x_{j},y_{j})\in\mathcal{X}_{j}\times\mathcal{X}_{j}$. By the chain rule and
differentiating under the integral sign by dominated convergence, we can
readily conclude that $G$ is of class
$\mathcal{C}^{\min\\{k,p\\}}=\mathcal{C}^{p}$ from
$\mathcal{C}^{k}(\mathcal{X})\times[0,1]$ to $\mathcal{C}^{k}(\mathcal{X})$. ∎

We now give a quantitative regularity estimate for the partial derivative of
$G$ in its real variable $t$.

######  Lemma 3.5.

Let $k\in\mathbb{N}^{*}$ and assume that $c\in\mathcal{C}^{k+1}(\mathcal{X})$.
Given $\operatorname{\boldsymbol{\phi}}\in\tilde{\mathcal{C}}^{1}$, the
partial differential of $G$ in $t$ satisfies

$\|D_{t}G(\operatorname{\boldsymbol{\phi}},t)\|_{\tilde{\mathcal{C}}^{k}}\leq
C\sqrt{\operatorname{cost}(\gamma)}$

where $C>0$ only depends on
$\|\operatorname{\boldsymbol{\phi}}\|_{\tilde{\mathcal{C}}^{1}}$ and
$\|c\|_{\mathcal{C}^{k+1}}$ and $\operatorname{cost}(\gamma)$ is the transport
cost associated with $\gamma$ as in Thm. 2.3.

###### Proof.

By Lem. 3.4, $G$ is differentiable in $t$. Using the shorthand
$x^{t}\coloneqq(1-t)x+ty$ and again the convention $y_{i}=x_{i}$, it holds

$\displaystyle\frac{d}{dt}G_{i}(\operatorname{\boldsymbol{\phi}},t)(x_{i})=\int_{\mathcal{X}_{-i}\times\mathcal{X}_{-i}}\Big{(}\sum_{j\neq
i}(y_{j}-x_{j})^{\top}(\nabla\phi_{j}(x_{j}^{t})-\nabla_{j}c(x^{t}))\Big{)}\mathrm{d}Q^{t}_{-i}(x_{-i},y_{-i}|x_{i})$
(18)

with $Q^{t}_{-i}\coloneqq q^{t}_{-i}\gamma_{-i}\in\prod_{j\neq
i}\mathcal{P}(\mathcal{X}_{j}\times\mathcal{X}_{j})$ and, posing
$(x^{\prime})^{t}_{i}=x_{i}^{t}$,

$q^{t}_{-i}(x_{-i},y_{-i}|x_{i})\coloneqq\frac{e^{\sum_{j\neq
i}\phi_{j}(x^{t}_{j})-c(x^{t})}}{\int_{\mathcal{X}_{-i}\times\mathcal{X}_{-i}}e^{\sum_{j\neq
i}\phi_{j}((x^{\prime})^{t}_{j})-c((x^{\prime})^{t})}\mathrm{d}\gamma_{-i}(x^{\prime}_{-i},y^{\prime}_{-i})}.$

Reasoning as in Lem. 3.1, this function $q^{t}_{-i}$ admits positive upper and
lower bounds only depending on
$\|\operatorname{\boldsymbol{\phi}}\|_{\tilde{\mathcal{C}}_{0}}$ and
$\|c\|_{\mathcal{C}^{0}}$. Let us now control Eq. (18), starting with a
control in uniform norm. First, by Cauchy-Schwarz in
$L^{2}(Q^{t}_{-i}(\cdot,\cdot|x_{i}))$, for $i\in[N]$ and
$x_{i}\in\mathcal{X}_{i}$,

$\Big{|}\frac{d}{dt}G_{i}(\operatorname{\boldsymbol{\phi}},t)(x_{i})\Big{|}^{2}\\\
\leq\left(\int_{\mathcal{X}_{-i}\times\mathcal{X}_{-i}}\\!\\!\|y-x\|^{2}\mathrm{d}Q^{t}_{-i}(x_{-i},y_{-i}|x_{i})\right)\left(\int_{\mathcal{X}_{-i}\times\mathcal{X}_{-i}}\\!\\!\|(\nabla\operatorname{\boldsymbol{\phi}}-\nabla
c)(x^{t})\|^{2}\mathrm{d}Q^{t}_{-i}(x_{-i},y_{-i}|x_{i})\right)$ (19)

where
$\nabla\operatorname{\boldsymbol{\phi}}\coloneqq(\nabla\phi_{1},\dots,\nabla\phi_{N})$.
Observe that the second factor is uniformly bounded for
$x_{i}\in\mathcal{X}_{i}$ because $Q^{t}_{-i}(\cdot,\cdot|x_{i})$ is a
probability measure and both $\operatorname{\boldsymbol{\phi}}$ and $c$ are
continuously differentiable on a compact set. It follows,

$\displaystyle\Big{\|}\frac{d}{dt}G_{i}(\operatorname{\boldsymbol{\phi}},t)\Big{\|}_{\mathcal{C}^{0}}$
$\displaystyle\leq
C\sup_{x_{i}\in\mathcal{X}_{i}}\left(\int_{\mathcal{X}_{-i}\times\mathcal{X}_{-i}}\|y-x\|^{2}\mathrm{d}Q^{t}_{-i}(x_{-i},y_{-i}|x_{i})\right)^{1/2}$
$\displaystyle\leq
C^{\prime}\left(\int_{\mathcal{X}_{-i}\times\mathcal{X}_{-i}}\|y-x\|^{2}\mathrm{d}\gamma_{-i}(x_{-i},y_{-i})\right)^{1/2}$
$\displaystyle\leq C^{\prime}\sqrt{\operatorname{cost}(\gamma)}$

where $C,C^{\prime}$ depend on $\|c\|_{\mathcal{C}^{1}}$ and
$\|\operatorname{\boldsymbol{\phi}}\|_{\tilde{\mathcal{C}}^{1}}$ only.
Moreover, one can further differentiate Eq. (18) in $x_{i}$ and obtain
analogous bounds because this variable only appears in the term $\nabla_{j}c$
which is of regularity $\mathcal{C}^{k}$ and in the factor $q^{t}_{-i}$ which
is of regularity $\mathcal{C}^{k+1}$. With this reasoning, it follows

$\Big{\|}\frac{d}{dt}G_{i}(\operatorname{\boldsymbol{\phi}},t)\Big{\|}_{\tilde{\mathcal{C}}^{k}}\leq
C_{k}\sqrt{\operatorname{cost}(\gamma)}.$

where $C_{k}$ depends on $\|c\|_{\tilde{\mathcal{C}}^{k+1}}$ and
$\|\operatorname{\boldsymbol{\phi}}\|_{\tilde{\mathcal{C}}^{1}}$ only. ∎

###  3.3 Proof of Thm. 2.3

###### Proof.

Let us first observe that $c$ can be extended in a $\mathcal{C}^{k+p}$ way to
a convex open set containing $\mathcal{X}$. One can therefore extend by
extrapolation the definition of $\operatorname{\boldsymbol{\mu}}^{t}$ to an
open time interval $(-\varepsilon,1+\varepsilon)$, for some $\varepsilon>0$
containing $[0,1]$. We shall then apply the Implicit Function Theorem (IFT) to
$G:\tilde{\mathcal{C}}^{k}\times[0,1]\to\tilde{\mathcal{C}}^{k}$ defined in
(11).

The existence and continuity on $[0,1]$ of the Schrödinger map
$t\mapsto\operatorname{\boldsymbol{\phi}}^{t}$ is guaranteed by Prop. 2.2. In
Lem. 3.4, we have shown that $G$ is of class $\mathcal{C}^{p}$ and in Lem.
3.2, we have shown that
$D_{\operatorname{\boldsymbol{\phi}}}G(\operatorname{\boldsymbol{\phi}}^{t},t)=D_{\operatorname{\boldsymbol{\phi}}}T(\operatorname{\boldsymbol{\phi}}^{t},\operatorname{\boldsymbol{\mu}}^{t})$
is an invertible linear self-map of $\tilde{\mathcal{C}}^{k}$. Thus all the
hypotheses are gathered to apply the Implicit Function Theorem, see e.g.
[Die11, Thm. 10.2.1]: the map $t\mapsto\operatorname{\boldsymbol{\phi}}^{t}$
is of class $\mathcal{C}^{p}$ on $[0,1]$ and its derivative is given by

$D_{t}\operatorname{\boldsymbol{\phi}}^{t}=-[D_{\operatorname{\boldsymbol{\phi}}}T(\operatorname{\boldsymbol{\phi}}^{t},\operatorname{\boldsymbol{\mu}}^{t})]^{-1}\circ
D_{t}G(\operatorname{\boldsymbol{\phi}}^{t},t).$

Moreover, we have by respectively Lem. 3.2 and Lem. 3.5 that there exists
$C>0$ only depending on $N$, $\|c\|_{\mathcal{C}^{k+1}}$ and
$\|\operatorname{\boldsymbol{\phi}}^{t}\|_{\tilde{\mathcal{C}}^{1}}$ such that

$\displaystyle\|[D_{\operatorname{\boldsymbol{\phi}}}T(\operatorname{\boldsymbol{\phi}}^{t},\operatorname{\boldsymbol{\mu}}^{t})]^{-1}\|_{\mathrm{op}}\leq
C$ and
$\displaystyle\|D_{t}G(\operatorname{\boldsymbol{\phi}}^{t},t)\|_{\mathrm{op}}\leq
C\sqrt{\operatorname{cost}(\gamma)}.$

Since we know by Prop. 2.2 that
$\|\operatorname{\boldsymbol{\phi}}^{t}\|_{\tilde{\mathcal{C}}^{1}}$ is a
priori bounded by $\|c\|_{\mathcal{C}^{1}}$, it follows that
$\|D_{t}\operatorname{\boldsymbol{\phi}}^{t}\|\leq
C^{\prime}\sqrt{\operatorname{cost}(\gamma)}$ for some $C^{\prime}_{k}>0$ that
only depends on $\|c\|_{\mathcal{C}^{k+1}}$. The Lipschitz estimate in Thm.
2.3 follows by the mean-value inequality. ∎

##  4 Smoothness of Entropic Optimal Transport and Wasserstein Gradient Flows

In this section, we apply our main stability results to the analysis of
Wasserstein gradient flows of functionals involving the entropic optimal
transport functional $E$.

###  4.1 Displacement Smoothness and Gradient Flows

Let
$\operatorname{\boldsymbol{\mu}}^{0},\operatorname{\boldsymbol{\mu}}^{1},\gamma,\operatorname{\boldsymbol{\mu}}^{t}$
and $\operatorname{\boldsymbol{\phi}}^{t}$ be as in the previous section. A
consequence of Thm. 2.3 is that the functional $E$ is as nice as one could
hope for in the Wasserstein space.

######  Theorem 4.1.

If $c\in\mathcal{C}^{2k-1}(\mathcal{X})$ for some $k\geq 1$, then the function
$t\mapsto E(\operatorname{\boldsymbol{\mu}}^{t})$ is of class
$\mathcal{C}^{k}$. Moreover, if $c\in\mathcal{C}^{2}(\mathcal{X})$ then its
derivative is $C\operatorname{cost}(\gamma)$-Lipschitz, for some $C>0$ that
only depends on $N$ and $\|c\|_{\mathcal{C}^{2}}$. In particular both $E$ and
$-E$ are $(-C)$-displacement convex.

###### Proof.

Using the convexity of the entropic optimal transport objective and the fact
that $\operatorname{\boldsymbol{\phi}}^{t}$ is a subgradient of $E$ at
$\operatorname{\boldsymbol{\mu}}^{t}$ (as a solution of the dual problem), it
holds

$\displaystyle\sum_{i}\int_{\mathcal{X}_{i}}\phi_{i}^{t}\mathrm{d}(\mu_{i}^{t+\varepsilon}-\mu_{i}^{t})\leq
E(\operatorname{\boldsymbol{\mu}}^{t+\varepsilon})-E(\operatorname{\boldsymbol{\mu}}^{t})\leq\sum_{i}\int_{\mathcal{X}_{i}}\phi_{i}^{t+\varepsilon}\mathrm{d}(\mu_{i}^{t+\varepsilon}-\mu_{i}^{t}).$
(20)

With the notation $x_{i}^{t}=(1-t)x_{i}+ty_{i}$ as before, remark that

$\displaystyle\int_{\mathcal{X}_{i}}\phi_{i}^{t}\mathrm{d}(\mu_{i}^{t+\varepsilon}-\mu_{i}^{t})$
$\displaystyle=\int_{\mathcal{X}_{i}^{2}}\big{(}\phi_{i}(x_{i}^{t+\varepsilon})-\phi_{i}(x_{i}^{t})\big{)}\mathrm{d}\gamma_{i}(x_{i},y_{i})$
$\displaystyle=\varepsilon\int_{\mathcal{X}_{i}^{2}}(y_{i}-x_{i})^{\top}\nabla\phi_{i}(x_{i}^{t})\mathrm{d}\gamma_{i}(x_{i},y_{i})+o(\varepsilon).$

Thus dividing Eq. (20) by $\varepsilon$ and using that
$\operatorname{\boldsymbol{\phi}}^{t+\varepsilon}\to\operatorname{\boldsymbol{\phi}}^{t}$
in $\tilde{\mathcal{C}}^{1}$ we obtain as $\varepsilon$ tends to $0$:

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}E(\operatorname{\boldsymbol{\mu}}^{t})=\sum_{i=1}^{N}\int_{\mathcal{X}_{i}\times\mathcal{X}_{i}}(y_{i}-x_{i})^{\top}\nabla\phi^{t}_{i}(x^{t}_{i})\mathrm{d}\gamma_{i}(x_{i},y_{i}).$
(21)

From Thm. 2.3, we know that $t\mapsto\operatorname{\boldsymbol{\phi}}^{t}$ is
in $\mathcal{C}^{k-1}([0,1],\tilde{\mathcal{C}}^{k})$ (note that the case
$k=1$ is instead a consequence of Prop. 2.2) and hence
$t\mapsto((x_{i},y_{i})\mapsto\nabla\operatorname{\boldsymbol{\phi}}^{t}_{i}(x_{i}^{t}))$
is in
$\mathcal{C}^{k-1}([0,1],\mathcal{C}^{0}(\mathcal{X}_{i}\times\mathcal{X}_{i}))$.
It follows that $h:t\mapsto
E(\operatorname{\boldsymbol{\mu}}^{t})\in\mathcal{C}^{k}([0,1])$. Notice how
this argument uses the two notions of regularity of the Schrödinger map
(indexed by $p$ and $k$ in Thm. 2.3).

For the Lipschitz regularity of $h^{\prime}$, fixing $s,t\in[0,1]$, one has

$\displaystyle|h^{\prime}(t)-h^{\prime}(s)|$
$\displaystyle\leq\left|\sum_{i=1}^{N}\int_{\mathcal{X}_{i}\times\mathcal{X}_{i}}(y_{i}-x_{i})^{\top}\big{(}\nabla\phi^{t}_{i}(x^{t}_{i})-\nabla\phi^{s}_{i}(x^{s}_{i})\big{)}\mathrm{d}\gamma_{i}(x_{i},y_{i})\right|$
$\displaystyle\leq\sum_{i=1}^{N}\int_{\mathcal{X}_{i}\times\mathcal{X}_{i}}\|y_{i}-x_{i}\|\|\nabla\phi^{t}_{i}(x^{t}_{i})-\nabla\phi^{t}_{i}(x^{s}_{i})\|\mathrm{d}\gamma_{i}(x_{i},y_{i})$
$\displaystyle\qquad+\sum_{i=1}^{N}\int_{\mathcal{X}_{i}\times\mathcal{X}_{i}}\|y_{i}-x_{i}\|\|\nabla\phi^{t}_{i}(x^{s}_{i})-\nabla\phi^{s}_{i}(x^{s}_{i})\|\mathrm{d}\gamma_{i}(x_{i},y_{i}).$

Now, if $c\in\mathcal{C}^{2}(\mathcal{X})$, using the Lipschitz regularity of
$x_{i}\mapsto\nabla\phi_{i}^{t}(x_{i})$ and of
$t\mapsto\nabla\operatorname{\boldsymbol{\phi}}^{t}$, from Thm. 2.3, it
follows that

$\|\nabla\phi^{t}_{i}(x^{t}_{i})-\nabla\phi^{t}_{i}(x^{s}_{i})\|\leq
C|t-s|\|y_{i}-x_{i}\|\text{ and
}\|\nabla\phi^{t}_{i}(x^{s}_{i})-\nabla\phi^{s}_{i}(x^{s}_{i})\|\leq
C|t-s|\sqrt{\operatorname{cost}(\gamma)},$

for some $C$ that only depends on $N$ and $\|c\|_{\mathcal{C}^{2}}$. Then, we
obtain

$\displaystyle|h^{\prime}(t)-h^{\prime}(s)|$ $\displaystyle\leq
C|t-s|\sum_{i=1}^{N}\int_{\mathcal{X}_{i}\times\mathcal{X}_{i}}\|y_{i}-x_{i}\|^{2}\mathrm{d}\gamma_{i}(x_{i},y_{i})$
$\displaystyle\qquad+C|t-s|\sqrt{\operatorname{cost}(\gamma)}\sum_{i=1}^{N}\int_{\mathcal{X}_{i}\times\mathcal{X}_{i}}\|y_{i}-x_{i}\|\mathrm{d}\gamma_{i}(x_{i},y_{i})$
$\displaystyle\leq C|t-s|\operatorname{cost}(\gamma).$

In particular, this implies that $t\in[0,1]\mapsto h(t)+\frac{C}{2}t^{2}$ is
convex hence

$E(\operatorname{\boldsymbol{\mu}}^{t})\leq(1-t)E(\operatorname{\boldsymbol{\mu}}^{0})+tE(\operatorname{\boldsymbol{\mu}}^{1})+\frac{C\operatorname{cost}(\gamma)t(1-t)}{2}$

and displacement semi-convexity follows by choosing $\gamma_{i}$ to be an
optimal transport plan between $\mu_{i}^{0}$ and $\mu_{i}^{1}$ for each
$i\in[N]$ (see [AGS05] for a definition). Displacement semi-convexity of $-E$
is obtained in the same way, observing that
$t\in[0,1]\mapsto-h(t)+\frac{C}{2}t^{2}$ is convex. ∎

######  Proposition 4.2.

If $c\in\mathcal{C}^{1}(\mathcal{X})$, we have that
$S(\operatorname{\boldsymbol{\mu}})$ is the gradient of
$\operatorname{\boldsymbol{\mu}}\mapsto E(\operatorname{\boldsymbol{\mu}})$
and $x\mapsto\nabla_{x}S(\operatorname{\boldsymbol{\mu}})(x)$ is its
Wasserstein gradient, in the sense of [AGS05] i.e. for
$\operatorname{\boldsymbol{\mu}}^{0}$ and
$\operatorname{\boldsymbol{\mu}}^{1}$ in
$\prod_{i=1}^{N}\mathcal{P}(\mathcal{X}_{i})$ and for any $\gamma_{i}$ optimal
plan between $\mu^{0}_{i}$, and $\mu_{i}^{1}$, one has

$E(\operatorname{\boldsymbol{\mu}}^{1})-E(\operatorname{\boldsymbol{\mu}}^{0})=\sum_{i=1}^{N}\int_{\mathcal{X}_{i}\times\mathcal{X}_{i}}(y_{i}-x_{i})^{\top}\nabla\phi_{i}(x_{i})\mathrm{d}\gamma_{i}(x_{i},y_{i})+o(\operatorname{\boldsymbol{W_{2}}}(\operatorname{\boldsymbol{\mu}}^{0},\operatorname{\boldsymbol{\mu}}^{1}))$

where
$\operatorname{\boldsymbol{\phi}}:=S(\operatorname{\boldsymbol{\mu}}^{0})$. If
$c\in\mathcal{C}^{2}(\mathcal{X})$ then the error
$o(\operatorname{\boldsymbol{W_{2}}}(\operatorname{\boldsymbol{\mu}}^{0},\operatorname{\boldsymbol{\mu}}^{1}))$
is in fact
$O(\operatorname{\boldsymbol{W_{2}}}(\operatorname{\boldsymbol{\mu}}^{0},\operatorname{\boldsymbol{\mu}}^{1})^{2})$.

###### Proof.

For the case $c\in\mathcal{C}^{1}(\mathcal{X})$, this follows by integrating
(21) in time and Proposition 2.2-(ii) which guarantees that $S$ is weakly
continuous as a function in $\tilde{\mathcal{C}}^{1}$. For the case
$c\in\mathcal{C}^{2}(\mathcal{X})$, this follows from (21) and Thm. 2.3. ∎

Thm. 4.1 and the above identification of the Wasserstein gradient of $E$
enable us to deduce from [AGS05, Thm. 11.2.1] that $E$ admits a unique
Wasserstein gradient flow, which shows well-posedness of the Cauchy problem
for the system of PDEs

$\left\\{\begin{aligned} \partial_{t}\mu_{i}&=\nabla\cdot(\mu_{i}\nabla
S_{i}(\operatorname{\boldsymbol{\mu}})),\quad i\in\\{1,\ldots,N\\}\\\
\operatorname{\boldsymbol{\mu}}|_{t=0}&=\operatorname{\boldsymbol{\mu}}^{0}\end{aligned}\right.$

This system, as all the PDEs below, is understood in the sense of
distributions with no-flux boundary conditions, i.e. for
$i\in\\{1,\dots,N\\}$, for every
$\psi\in\mathcal{C}^{\infty}_{c}(\mathbb{R}^{d}\times[0,+\infty))$ it holds

$\int_{0}^{\infty}\int_{\mathcal{X}_{i}}\big{(}\partial_{t}\psi(t,x_{i})+\nabla
S_{i}(\operatorname{\boldsymbol{\mu}})(x_{i})^{\top}\nabla_{x_{i}}\psi(t,x_{i})\big{)}\mathrm{d}\mu^{t}_{i}(x_{i})\mathrm{d}t=-\int_{\mathcal{X}_{i}}\psi(0,x_{i})\mathrm{d}\mu_{i}^{0}(x_{i}).$

We also have that the fact that the gradient flow map
$\operatorname{\boldsymbol{\mu}}^{0}\mapsto\operatorname{\boldsymbol{\mu}}^{t}$
satisfies

$\operatorname{\boldsymbol{W_{2}}}(\operatorname{\boldsymbol{\mu}}^{t},\operatorname{\boldsymbol{\nu}}^{t})^{2}\leq
e^{Ct}\operatorname{\boldsymbol{W_{2}}}(\operatorname{\boldsymbol{\mu}}^{0},\operatorname{\boldsymbol{\nu}}^{0})^{2}.$

Of course, adding to $E$ a separable term of the form
$\sum_{i=1}^{N}E_{i}(\mu_{i})$ where each $E_{i}$ is displacement semi-convex,
we can deduce well-posedness for more general systems like

$\partial_{t}\mu_{i}-\alpha_{i}\Delta\mu_{i}-\nabla\cdot(\mu_{i}\nabla
S_{i}(\operatorname{\boldsymbol{\mu}}))=0,i=1,\ldots,N,\quad\operatorname{\boldsymbol{\mu}}|_{t=0}=\operatorname{\boldsymbol{\mu}}^{0}$

or

$\partial_{t}\mu_{i}-\alpha_{i}\Delta\mu_{i}^{m_{i}}-\nabla\cdot(\mu_{i}\nabla
S_{i}(\operatorname{\boldsymbol{\mu}}))=0,i=1,\ldots,N,\quad\operatorname{\boldsymbol{\mu}}|_{t=0}=\operatorname{\boldsymbol{\mu}}^{0}$

with $m_{i}\geq 1$ and $\alpha_{i}\geq 0$. For the sake of concreteness, we
are going to detail three such examples with interesting additional structure
in the next paragraphs.

###  4.2 Wasserstein Gradient Flow of the Sinkhorn Divergence

We consider the Sinkhorn divergence functional [Fey+19], the gradient flow of
which has been previously considered as a numerical method for density
fitting. As a consequence of our analysis and of [AGS05, Thm. 11.2.1] we have
the following result.

######  Proposition 4.3.

Let $\mathcal{X}\subset\mathbb{R}^{d}$ be a compact convex set,
$c\in\mathcal{C}^{2}(\mathcal{X}\times\mathcal{X})$ and let
$\mu^{0},\nu\in\mathcal{P}(\mathcal{X})$. There exists a unique Wasserstein
gradient flow starting from $\mu^{0}$ of the Sinkhorn divergence (from $\nu$)
functional

$\mu\mapsto E(\mu,\nu)-\frac{1}{2}E(\mu,\mu)-\frac{1}{2}E(\nu,\nu).$

Here, a Wasserstein gradient flow is a curve $(\mu^{t})_{t\geq
0}\in\mathcal{P}(\mathcal{X})$ that is absolutely continuous for the $W_{2}$
metric and that satisfies

$\displaystyle\partial_{t}\mu^{t}=\nabla\cdot(v^{t}\mu^{t}),$ $\displaystyle
v^{t}=\nabla S_{1}(\mu^{t},\nu)-\nabla S_{1}(\mu^{t},\mu^{t}),$
$\displaystyle\mu|_{t=0}=\mu^{0}$ (22)

where we recall that $S$ is the Schrödinger map.

An interesting open question is whether this dynamics can be provably shown to
converge to the unique minimizer $\mu^{*}$, which is $\mu^{*}=\nu$ for
suitable choices of costs, e.g. for $c(x,y)=\|y-x\|^{2}$, as proved in
[Fey+19].

###  4.3 Convergence to equilibrium for the Schrödinger Bridge Energy

Let us continue with another simple example that shows that our theory is also
natural to deal with the Lebesgue measure as a reference in the definition of
$E$ Eq. (1), which is the original definition of the Schrödinger bridge
problem. This alternative definition is equivalent (see e.g. [DG20]) to
considering $E+H$ where $H(\mu)\coloneqq\int\log(\mu)\mathrm{d}\mu$ if $\mu$
is absolutely continuous and $+\infty$ otherwise is minus the differential
entropy.

######  Proposition 4.4.

Let $\mathcal{X}\subset\mathbb{R}^{d}$ be a compact convex set,
$c\in\mathcal{C}^{2}(\mathcal{X}\times\mathcal{X})$ and let
$\mu^{0},\nu\in\mathcal{P}(\mathcal{X})$. There exists a unique Wasserstein
gradient flow of

$\mu\mapsto E(\mu,\nu)+H(\mu)$

starting from $\mu^{0}$. Moreover, if $H(\mu^{0})<\infty$ then this gradient
flow converges at an exponential rate to the unique global minimizer
$\mu^{*}$. Specifically, there exists $\kappa>0$ independent of $\mu^{0},\nu$
such that

$F(\mu^{t})-F(\mu^{*})\leq e^{-\kappa t}(F(\mu^{0})-F(\mu^{*})).$

In addition, there exists a constant $C>0$, independent of $\mu^{0}$, $\nu$
such that

$W_{2}(\mu^{t},\mu^{*})^{2}\leq Ce^{-\kappa t}(F(\mu^{0})-F(\mu^{*})).$

For this functional, the Wasserstein gradient flow $(\mu^{t})_{t\geq
0}\in\mathcal{P}(\mathcal{X})$ solves

$\displaystyle\partial_{t}\mu^{t}=\nabla\cdot(v^{t}\mu^{t})+\Delta\mu^{t},$
$\displaystyle v^{t}=\nabla S_{1}(\mu^{t},\nu).$ (23)

###### Proof.

We have semi-convexity along Wasserstein geodesics, by Thm. 4.1 for the first
component and by a standard result [San15] for the $H$ component. Thus the
general well-posedness results from [AGS05] applies. For the exponential
convergence – in function value and in distance – we apply the result from
[Chi22, Thm. 3.2], see also [NWS22] where the same argument was discovered
independently (although stated on $\mathbb{R}^{d}$, the argument goes through
on a compact domain).

The main assumptions to check are that (i) $\mu\mapsto E(\mu,\nu)$ is convex,
which is clear from the dual formulation Eq. (4) which expresses this
functional as a supremum of affine forms, (ii) that a global minimizer
$\mu^{*}$ exists, which is not difficult here since $\mathcal{P}(\mathcal{X})$
is weakly compact, $H$ is weakly lower-semicontinuous and $E$ is weakly
continuous and finally we need to check that the probability measure
$\hat{\mu}_{t}\propto e^{-S_{1}(\mu^{t},\nu)}\in\mathcal{P}(\mathcal{X})$
satisfies a log-Sobolev inequality, uniformly in $t$ (Assumption 3 in
[Chi22]).

Since $\mathcal{X}$ is bounded, the normalized Lebesgue measure satisfies a
log-Sobolev inequality [Led99, Thm. 7.3]. By the Holley-Stroock perturbation
criterion [HS87] $\hat{\mu}_{t}$ satisfies it as well; this criterion applies
here because $\sup_{x}S_{1}(\mu,\nu)(x)-\inf_{x}S_{1}(\mu,\nu)(x)$ is bounded,
uniformly in $\mu,\nu\in\mathcal{P}(\mathcal{X})$ by Prop. 2.2. The
convergence in Wasserstein distance is stated in [Chi22, Cor. 3.3] and follows
from the fact that log-Sobolev inequalities imply Talagrand inequalities
[OV00]. ∎

###  4.4 Convergence to equilibrium in the multi-species case

We now consider Wasserstein gradient flow of333Note that
$F(\operatorname{\boldsymbol{\mu}})$ also be written as the value of an
entropic optimal transport problem but with the Lebesgue measure as reference
measure i.e.
$F(\operatorname{\boldsymbol{\mu}})=\min_{\gamma\in\Pi(\operatorname{\boldsymbol{\mu}})}\int_{\mathcal{X}}c(x)\mathrm{d}\gamma(x)+H(\gamma)$.

$F(\operatorname{\boldsymbol{\mu}}):=E(\operatorname{\boldsymbol{\mu}})+\sum_{i=1}^{N}H(\mu_{i}),\quad\operatorname{\boldsymbol{\mu}}=(\mu_{1},\ldots,\mu_{N})\in\prod_{i=1}^{N}\mathcal{P}(\mathcal{X}_{i})$

that is

$\partial_{t}\mu_{i}-\Delta\mu_{i}-\nabla\cdot(\mu_{i}\nabla
S_{i}(\operatorname{\boldsymbol{\mu}}))=0,\quad
i=1,\ldots,N,\;\operatorname{\boldsymbol{\mu}}|_{t=0}=\operatorname{\boldsymbol{\mu}}^{0}.$
(24)

Note that $F$ has a unique minimizer $\operatorname{\boldsymbol{\mu}}^{*}$ on
$\prod_{i=1}^{N}\mathcal{P}(\mathcal{X}_{i})$, and this minimizer is given by

$\mu_{i}^{*}(x_{i})=\frac{\int_{\mathcal{X}_{-i}}e^{-c(x_{i},x_{-i})}\mathrm{d}x_{-i}}{\int_{\mathcal{X}}e^{-c}\mathrm{d}x}$
(25)

for every $i$ and $x_{i}\in\mathcal{X}_{i}$ and
$F(\operatorname{\boldsymbol{\mu}}^{*})=0$. In the next proposition, we extend
Prop. 4.4 to the multi-species case. In this case, the functional
$\operatorname{\boldsymbol{\mu}}\mapsto E(\operatorname{\boldsymbol{\mu}})$ is
not convex anymore but we can overcome this difficulty taking advantage of the
form of $F$.

######  Proposition 4.5.

Let
$\operatorname{\boldsymbol{\mu}}^{0}\in\prod_{i=1}^{N}\mathcal{P}(\mathcal{X}_{i})$.
Then there exists a unique Wasserstein gradient flow of $F$ starting from
$\operatorname{\boldsymbol{\mu}}^{0}$, that we call
$\operatorname{\boldsymbol{\mu}}^{t}$. Assume that $H(\mu^{0}_{i})<+\infty$
for every $i$, then $\operatorname{\boldsymbol{\mu}}^{t}$ converges at an
exponential rate to the equilibrium $\operatorname{\boldsymbol{\mu}}^{*}$,
defined in Eq. (25), i.e. there exists $\kappa>0$ independent of
$\operatorname{\boldsymbol{\mu}}^{0}$ such that

$F(\operatorname{\boldsymbol{\mu}}^{t})-F(\operatorname{\boldsymbol{\mu}}^{*})\leq
e^{-\kappa
t}(F(\operatorname{\boldsymbol{\mu}}^{0})-F(\operatorname{\boldsymbol{\mu}}^{*})).$

In addition, there exists a constant $C>0$, independent of
$\operatorname{\boldsymbol{\mu}}^{0}$, such that

$\operatorname{\boldsymbol{W_{2}}}(\operatorname{\boldsymbol{\mu}}^{t},\operatorname{\boldsymbol{\mu}}^{*})^{2}\leq
Ce^{-\kappa
t}(F(\operatorname{\boldsymbol{\mu}}^{0})-F(\operatorname{\boldsymbol{\mu}}^{*})).$

###### Proof.

The well-posedness of the Wasserstein gradient flow is proved as previously
using the geodesic semi-convexity of $F$ that follows from Thm. 4.1. For the
convergence, first note the identities

$E(\operatorname{\boldsymbol{\mu}})=\sum_{i=1}^{N}\int_{\mathcal{X}_{i}}S_{i}(\operatorname{\boldsymbol{\mu}})\mathrm{d}\mu_{i},\quad
F(\operatorname{\boldsymbol{\mu}})=\sum_{i=1}^{N}H(\mu_{i}|e^{-S_{i}(\operatorname{\boldsymbol{\mu}})})$

which hold for any
$\operatorname{\boldsymbol{\mu}}\in\prod_{i=1}^{N}\mathcal{P}(\mathcal{X}_{i})$
and easily follow from (4) and (7). Since $H(\mu^{0}_{i})<+\infty$ for every
$i$, and denoting by $\operatorname{\boldsymbol{\mu}}^{t}$ the Wasserstein
gradient flow of $F$ starting from $\operatorname{\boldsymbol{\mu}}^{0}$, we
have by standard arguments

$\frac{\mathrm{d}}{\mathrm{d}t}F(\operatorname{\boldsymbol{\mu}}^{t})=-\sum_{i=1}^{N}I_{i}(\mu_{i}|e^{-S_{i}(\operatorname{\boldsymbol{\mu}}^{t})})$

where $I_{i}(\rho|e^{-V})$, for $\rho\in\mathcal{P}(\mathcal{X}_{i})$, stands
for the relative Fisher information

$I_{i}(\rho|e^{-V}):=\int_{\mathcal{X}_{i}}\left\|\nabla\log\Big{(}\frac{\rho}{e^{-V}}\Big{)}\right\|^{2}\mathrm{d}\rho.$

But since $\mathcal{X}_{i}$ is convex and the oscillation of
$S_{i}(\operatorname{\boldsymbol{\mu}})$ is controlled (uniformly in
$\operatorname{\boldsymbol{\mu}}$) by the oscillation of $c$, it follows from
[HS87], that the measure with density
$e^{-S_{i}(\operatorname{\boldsymbol{\mu}})}$ with respect to the Lebesgue
measure on $\mathcal{X}_{i}$ satisfies a log-Sobolev inequality

$I_{i}(\mu_{i}|e^{-S_{i}(\operatorname{\boldsymbol{\mu}})})\geq-\kappa
H(\mu_{i}|e^{-S_{i}(\operatorname{\boldsymbol{\mu}})})$

with $\kappa>0$ depending only on $\mathcal{X}$ and $c$. We thus have

$\frac{\mathrm{d}}{\mathrm{d}t}F(\operatorname{\boldsymbol{\mu}}^{t})\leq-\kappa\sum_{i=1}^{N}H(\mu_{i}|e^{-S_{i}(\operatorname{\boldsymbol{\mu}}^{t})})=-\kappa
F(\operatorname{\boldsymbol{\mu}}^{t})$

hence

$F(\operatorname{\boldsymbol{\mu}}^{t})\leq e^{-\kappa
t}F(\operatorname{\boldsymbol{\mu}}^{0}).$

Thanks to Talagrand’s inequality, which follows from the log-Sobolev
inequality [OV00], we get, as in the single species case from the previous
paragraph, an exponential decay of
$\operatorname{\boldsymbol{W_{2}}}(\operatorname{\boldsymbol{\mu}}^{t},\operatorname{\boldsymbol{\mu}}^{*})$
for the multi-species system (24). ∎

range

pages31 rangepages42 rangepages9 rangepages1 rangepages28 rangepages10
rangepages10 rangepages9 rangepages36 rangepages14 rangepages20 rangepages97
rangepages42 rangepages17 rangepages40 rangepages1 rangepages42 rangepages15

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