proposition@alttheorem lemma@alttheorem corollary@alttheorem
counterexample@alttheorem definition@alttheorem question@alttheorem
example@alttheorem remark@alttheorem

# Optimal transport with nonlinear mobilities: a deterministic particle
approximation result

S. Di Marino  Università di Genova, DIMA, MaLGa, Via Dodecaneso 35, 16146
Genova (GE), Italy<EMAIL_ADDRESS>,  L. Portinale  Haussdorff
Center for Mathematics, Institute for Applied Mathematics, Endenicher Allee
62, D-53115 Bonn, Germany<EMAIL_ADDRESS>and  E. Radici
Università degli Studi dell’Aquila, DISIM, Via Vetoio 1, 67100 (Coppito)
L’Aquila (AQ), Italy<EMAIL_ADDRESS>
###### Abstract.

We study the discretisation of generalised Wasserstein distances with
nonlinear mobilities on the real line via suitable discrete metrics on the
cone of N ordered particles, a setting which naturally appears in the
framework of deterministic particle approximation of partial differential
equations. In particular, we provide a $\Gamma$-convergence result for the
associated discrete metrics as $N\to\infty$ to the continuous one and discuss
applications to the approximation of one-dimensional conservation laws (of
gradient flow type) via the so-called generalised minimising movements,
proving a convergence result of the schemes at any given discrete time step
$\tau>0$. This the first work of a series aimed at sheding new lights on the
interplay between generalised gradient-flow structures, conservation laws, and
Wasserstein distances with nonlinear mobilities.

###### Contents

  1. 1 Introduction
  2. 2 The discrete and continuous settings
  3. 3 Statements of the main results
  4. 4 Approximation of finite-energy solutions of $\mathsf{CE}$
  5. 5 Proof of the main result: $\Gamma$-convergence.
  6. 6 Proof of the main result: convergence of the JKO schemes
  7. 7 Appendix A. Properties of the action densities

##  1\. Introduction

The optimal transport theory has been a very active research field in the last
twenty years, revealing suprising connections between the classical
Monge–Kantorovich transportation model and various other fields of
mathematics, including the theory of partial differential equations (PDEs)
[JKO98], functional and geometric inequalities [OV00], and metric measure
spaces [LV07, LV09, Stu06]. These connections are fundamentally based on the
ideas of the seminal works of Benamou–Brenier [BB00] and Otto [Ott01]: one can
associate a (formal) infinite-dimensional Riemannian structure to the space of
probability measures using the quadratic transport distance (i.e. the
$2$-Wasserstein–Kantorovich–Rubenstein distance), whose associated gradient
flow of the relative entropy coincides with the heat equation (in
$\mathbb{R}^{d}$). This new perspective opened the door to wide range
applications, not limited to the continuous setting of Riemannian manifolds,
but applicable to discrete and non-commutative frameworks as well [Maa11,
Mie11, CM14].

The main focus of this work is the study of generalised optimal transport
distances [DNS12], [LM10] on the real line, its spatial discretisation using
deterministic particle methods (in the spirit of [DFR15]), and the
applications of these methods to the study of the associated gradient-flow
evolutions.

The class of evolutions we are interested in might include local and/or non
local effects, possibily including prevention of overcrowding. Mathematically,
this translates into considering evolutions with non-linear mobilities,
possibly with bounded support. To these equations, one can associate a
corresponding Wasserstein-like distance that takes into account the
nonlinearity.

Our goal is to study suitable discretisations of these problems, both the
generalised optimal transport distances and the associated gradient flows,
with particular attention to two types of approximations: in space and in
time.

In particular, we introduce, in the scalar case, a space discretisation in the
framework of non-linear mobilities, adopting a Lagrangian point of view. The
generalised transport distances are approximated using systems of $N$-ordered
particles, and a discrete-to-continuum approximation result is provided
(Theorem 3.1). Subsequently, we take advantage of this discrete-in-space
approximation to show the stability at the level of the corresponding
gradient-flow structures, providing a finite-dimensional approximation of the
associated evolution equation (Theorem 3.2).

Our contribution aim towards the understanding of the discrete-to-continuum
time and space commuting diagram regarding the one dimensional evolutionary
PDEs that can be seen as gradient flows with respect to a non-linear mobility
Wasserstein like distance, as described in Figure 1.

discrete in spacecontinuous in timeODEN $\to$ PDEcontinuous in spacecontinuous
in timeJKO $\to$ PDEdiscrete in spacediscrete in timeJKON $\to$ ODEN$\tau\to
0$$(\mathcal{K}_{N},d_{2,m}^{N})$$N\to\infty$$(\mathcal{P}(\mathbb{R}),\mathbb{W}_{2,m})$JKON
$\to$ JKOcontinuous in spacediscrete in time Figure 1. The discrete-to-
continuum diagram.

This work focus on the purple arrows, concerning the convergence of the
discrete-in-time schemes and the discrete distances $d_{2,m}^{N}$ (Section 2)
on the cones $\mathcal{K}_{N}$ as $N\to\infty$. The top, blue arrow concerns
deterministic many particle approximation of PDEs, and it has been already
investigated in several works, see Section 1.4 for a more detailed discussion.
The natural continuation of this manuscript would consist in the analysis of
the two vertical dotted arrows in Figure 1. This is currently under
investigation, and we postpone it to future works, since it seems to require a
different type of analysis.

###  1.1. Optimal transport with non-linear mobilities

The generalised Wasserstein distances were introduced in [DNS12], [LM10].
Given $\mu_{0},\mu_{1}\in\mathscr{P}(\Omega)$ and a mobility
$m:\mathbb{R}_{+}\to\mathbb{R}_{+}$, the generalized Wasserstein psuedo-
distance is defined by

$\displaystyle\mathbb{W}_{2,m}(\mu_{0},\mu_{1})^{2}:=\inf\left\\{\int_{0}^{1}\int_{\mathbb{R}^{d}}|v_{t}|^{2}m(\rho_{t})\,\mathrm{d}\mathscr{L}^{1}\,\mathrm{d}t\
:\ \partial_{t}\mu_{t}+\nabla\cdot(m(\rho_{t})v_{t})=0,\
\mu_{t=i}=\mu_{i}\right\\},$

for $i=0,1$, where in the formula above $\rho_{t}$ denotes the density of
$\mu_{t}$ with respect to the Lebesgue measure $\mathscr{L}^{1}$ on
$\mathbb{R}$.

One of the motivation for the introduction of $\mathbb{W}_{2,m}$ is precisely
to be able to give a variational structure to PDEs of the type

$\displaystyle\partial_{t}\rho_{t}-\nabla\cdot(m(\rho_{t})\nabla(\text{D}\mathcal{E}(\rho_{t}))=0\,,$
(1.1)

which indeed can be formally seen as gradient flows of $\mathcal{E}$ with
respect to $\mathbb{W}_{2,m}$. The linear case $m(\rho)=\rho$ corresponds to
the classical Wasserstein distance $\mathbb{W}_{2}$.

Following the ideas of Jordan–Kinderlehrer–Otto, one rigorous interpretation
of (1.1) as a gradient flow consists in considering the time-discrete
approximation (also known as JKO-scheme from the names of the authors, or,
more generally, minimising movements scheme in the spirit of De Giorgi, see
[AGS08, San17]) of parameter $\tau>0$ given by

$\displaystyle\mu_{\tau}^{(n)}\in\operatorname*{arg\,min}_{\mu\in\mathscr{P}(\mathbb{R}^{d})}\left\\{\mathcal{E}(\mu)+\frac{1}{2\tau}\mathbb{W}_{2,m}(\mu,\mu_{\tau}^{(n-1)})^{2}\right\\}\,.$
(1.2)

The goal is then to recover a solution (1.1) defining a curve
$\bm{\mu}^{\tau}:[0,1]\to\mathscr{P}(\mathbb{R}^{d})$ obtained, for every
$\tau>0$, interpolating the family $\left(\mu_{\tau}^{(n)}\right)_{n}$ and
taking its limit as $\tau\to 0$. This is the content of [JKO98] in the case of
linear mobility and the heat equation in $\mathbb{R}^{d}$.

###  1.2. Main results

The goal of this work is a particle approximation result for the Wasserstein-
like distances with non-linear mobilities $\mathbb{W}_{2,m}$ and its
applications to the convergence of associated gradient-flow structures, in the
scalar case $d=1$. A natural generalisation of this result to the case of
$p$-homogenous energy functionals for $p\neq 2$ is also considered. For the
sake of simplicity, we here restrict the exposition of the main results to the
quadratic case.

The starting point of discussion is the work [DFR15] (together with several
companion works, see Section 1.4 for a more detailed literature review), where
the authors propose a deterministic particle approximation result to entropic
solutions to scalar conservation laws of the form
$\partial_{t}\mu_{t}+\partial_{x}(f[\mu_{t}])=0$. The procedure is based on a
discretisation of the Lagrangian formulation of the evolution, working in
terms of pseudo-inverses of cumulative distributions and their approximation.
For any $\mu\in\mathscr{P}(\mathbb{R})$, one considers the quantile function

$\displaystyle X_{\mu}(z):=\inf\left\\{a\in\mathbb{R}\ :\
\mu(-\infty,a)>z\right\\}\,.$

At least formally, the evolution of $x_{t}:=X_{\mu_{t}}$ is then given by

$\displaystyle\dot{x}_{t}(z)=v[\mu_{t}(x_{t}(z))]\,,$ (1.3)

where $v[\mu]:=f[\mu]/\mu$ is the Eulerian velocity. The above mentioned works
consider the discrete model given by the space of $N+1$ ordered particles
$\mathcal{K}_{N}:=\\{x=(x_{0},\dots,x_{N})\ :\
x_{0}<\dots<x_{N}\\}\subset\mathbb{R}^{N+1}$ and space-discrete evolutions
obtained via a natural discretisation of (1.3). Folloeing [DFR15], for
compactly supported measures $\mu$, we set $x_{j}^{N}:=X_{\mu}(j/N)$ for
$j=0,\dots,N$; then we notice that formally we also have
$\mu_{t}(x_{t}(z))=\partial_{z}x_{t}(z)^{-1}$, so that we can use the discrete
derivative of $x_{t}$ as a recostruction for the density. In the end we
consider the system of ODEs

$\displaystyle\dot{x}_{j}^{N}(t)=v[R_{j}^{N}(t)]\,,\quad
R_{j}^{N}(t)=\frac{1}{N(x_{j+1}^{N}(t)-x_{j}^{N}(t))}\,,\quad
j=0,\dots,N-1\,,$ (1.4)

There is no canonical definition of the rightmost reconstructed density
$R_{N}^{N}(t)$, since $x_{N}^{N}(t)$ have no particles in front. For example,
in [DFR15] the authors set $R_{N}^{N}(t)=\arg\max v(\cdot)$, the so-called
Follow-The-Leader scheme. The name comes from the fact that the last particle
$x^{N}_{N}$ moves with maximum velocity, whilst the other particles "see" the
one in front of them, and they choose their velocity accordingly. It is then
showed that, both the empirical and piecewise constant measures

$\displaystyle\mathcal{E}^{N}(x^{N}(t)):=\frac{1}{N+1}\sum_{j=0}^{N}\delta_{x_{j}^{N}(t)},\quad\mathcal{PC}^{N}(x^{N}(t)):=\sum_{j=0}^{N-1}R_{j}^{N}(t)\mathbbm{1}_{\left[x_{j}^{N}(t),x_{j+1}^{N}(t)\right)}$
(1.5)

converge as $N\to+\infty$ to the (unique) entropy solution of the associated
conservation law.

The main goal of this work is to introduce a suitable transport distance at
the discrete level, on the open cone $\mathcal{K}_{N}$, which plays the role
of a discrete approximation of the optimal transport distance
$\mathbb{W}_{2,m}$ on $\mathscr{P}(\mathbb{R})$, and prove a convergence
result as $N\to\infty$; then we will extend this approach to the gradient flow
structure introducing the JKO scheme at the discrete level. For every given
concave mobility $m:\mathbb{R}_{+}\to\mathbb{R}_{+}$ and $N\in\mathbb{N}$, we
define

$\displaystyle
d_{2,m}^{N}(z,y)^{2}:=\inf\left\\{\int_{0}^{1}\frac{1}{N+1}\sum_{j=0}^{N}\frac{|\dot{x}_{j}(t)|^{2}}{\theta(R_{j}(t))}\,\mathrm{d}t\
:\ x(0)=z,\,x(1)=y\,,\ x(t)\in\mathcal{K}_{N}\right\\}\,,$ (1.6)

where $\theta(\rho):=m(\rho)/\rho$ and $R_{j}(t)$ as in (1.4). Our first
contribution is to show that the discrete (pseudo)distances $d_{2,m}^{N}$ are
a good approximation of the continuous ones $\mathbb{W}_{2,m}$. We define
$\mathbb{W}_{2,m}^{N}:\mathscr{P}(\mathbb{R})\times\mathscr{P}(\mathbb{R})\to\mathbb{R}_{+}\cup\\{+\infty\\}$
as

$\mathbb{W}_{2,m}^{N}(\mu,\nu)=\begin{cases}d_{2,m}^{N}(x,y)\qquad&\text{ if
}\mu=\mathcal{E}^{N}(x)\text{ and }\nu=\mathcal{E}^{N}(y)\text{ for some
}x,y\in\mathcal{K}_{N}\\\ +\infty&\text{ if
}\mu\notin\mathcal{E}^{N}(\mathcal{K}_{N})\text{ or
}\nu\notin\mathcal{E}^{N}(\mathcal{K}_{N})\end{cases}$

First main result: the (pseudo)distances $\mathbb{W}_{2,m}^{N}$
$\Gamma$-converge to $\mathbb{W}_{2,m}$ as $N\to+\infty$, with respect to the
vague and the classical $\mathbb{W}_{2}$-topology.

Note that in the interesting case when m is assumed to have compact support,
the definition of $\mathbb{W}_{2,m}^{N}$ presents an additional singularity,
which in particular makes this convergence less trivial.

The study of the discrete distances opens the doors to possible applications
to gradient flows approximation. In particular, at fixed $N\in\mathbb{N}$, for
a given functional on $F_{N}:\mathcal{K}_{N}\to\mathbb{R}$ and initial value
$x^{(0),N}\in\mathcal{K}_{N}$, one can consider the associated JKO-scheme

$\displaystyle
x_{\tau}^{(n),N}\in\operatorname*{arg\,min}_{x\in\mathcal{K}_{N}}\left\\{F_{N}(x)+\frac{1}{2\tau}d_{2,m}^{N}\big{(}x,x_{\tau}^{(n-1),N}\big{)}^{2}\right\\}.$
(1.7)

Assume that $F_{N}$ suitably converges to some continuous functional
$F:\mathscr{P}(\mathbb{R})\to\mathbb{R}$ and that the empirical measures
$\mu^{(0),N}$ associated to the initial data $x^{(0),N}$ as in (1.5) converge
to a limit $\mu^{(0)}\in\mathscr{P}(\mathbb{R})$. Similarly, denote by
$\mu_{\tau}^{(n),N}=\mathcal{E}^{N}\big{(}x_{\tau}^{(n),N}\big{)}$ the
empirical measures associated to $x_{\tau}^{(n),N}$.

Second main result: for $\tau>0$, under suitable assumptions on $F_{N}\to F$
(see (A), (B), and (3.1)), the steps of the discrete JKO-scheme (1.7) suitably
converge to the steps of the continuous JKO-scheme (1.2), i.e.
$\mu_{\tau}^{(n),N}\to\mu_{\tau}^{(n)}$ for every $n\in\mathbb{N}$, as
$N\to+\infty$.

In particular, we believe that the variational point of view coming from the
gradient-flow structure represents a powerful tool to handle evolution
equations that might present deep mathematical difficulties, due to the lack
of regularity and uniqueness.

###  1.3. Sketch of the proofs

For the convenience of the reader, we include a short sketch of the proofs of
Theorem 3.1 and Theorem 3.2, avoiding the technical details. In what follows,
we denote by $\mathsf{CE}$ the set of solutions of the continuity equation on
$\mathbb{R}$, i.e. the curves $[0,1]\ni
t\mapsto(\mu_{t},\nu_{t})\in\mathscr{P}(\mathbb{R})\times\mathcal{M}(\mathbb{R})$
which solves in the sense of distribution the equation

$\displaystyle\partial_{t}\mu_{t}+\partial_{x}\nu_{t}=0\,,\quad t\in[0,1]\,.$

#### Sketch of the liminf inequality

Consider a sequence $\big{\\{}\big{(}x^{N}(0),x^{N}(1)\big{)}\big{\\}}_{N}$,
with $x^{N}(0),x^{N}(1)\in\mathcal{K}_{N}$. Assume that the piecewise-constant
measures $\mu_{0}^{N}:=\mathcal{PC}^{N}(x^{N}(0))$,
$\mu_{1}^{N}:=\mathcal{PC}^{N}(x^{N}(1))\in\mathscr{P}(\mathbb{R})$ are
vaguely (in duality with $C_{c}(\mathbb{R})$) converging to
$\mu_{0},\mu_{1}\in\mathscr{P}(\mathbb{R})$ as $N\to\infty$. We want to show
the liminf inequality

$\displaystyle\liminf_{N\to\infty}d_{2,m}^{N}(x^{N}(0),x^{N}(1))\geq\mathbb{W}_{2,m}(\mu_{0},\mu_{1})\,.$

Let $x^{N}(\cdot)$ be the discrete geodesic between $x^{N}(0)$ and $x^{N}(1)$
with respect to $d_{2,m}^{N}$, viz. satisfying

$\displaystyle
d_{2,m}^{N}(x^{N}(0),x^{N}(1))=\int_{0}^{1}\sum_{j=0}^{N}\frac{|\dot{x}_{j}^{N}(t)|^{2}}{\theta(R_{j}^{N}(t))}\,\mathrm{d}t\quad\text{and}\quad\sup_{N\in\mathbb{N}}\sup_{t\in[0,1]}\sum_{j=0}^{N}\frac{|\dot{x}_{j}^{N}(t)|^{2}}{\theta(R_{j}^{N}(t))}<\infty\,.$
(1.8)

We define $\mu_{t}^{N}:=\mathcal{PC}^{N}(x^{N}(t))$ and consider the momentum
fields given by

$\displaystyle
j_{t}^{N}(x):=\sum_{i=0}^{N-1}\dot{x}_{i}^{N}(t)R_{i}^{N}(t)\mathbbm{1}_{[x_{i}^{N}(t),x_{i+1}^{N}(t))}\,,\quad\nu_{t}^{N}:=j_{t}^{N}\mathscr{L}^{1}\,.$
(1.9)

Intuitively, because the particles follows the evolution described by
$\dot{x}_{i}^{N}$, we expect $(\mu_{t}^{N},\nu_{t}^{N})$ to be close to a
solution to $\mathsf{CE}$. We show indeed, in the limit as
$N\rightarrow+\infty$, that $\partial_{t}\mu_{t}^{N}+\partial_{x}j_{t}^{N}\to
0$ in the sense of distributions.

The second step is to provide a compactness result for (almost) solutions to
$\mathsf{CE}$ satisfying energy bounds as (1.8), in the same spirit of [DNS12,
Lemma 4.5]: we prove that, up to a non-relabeled subsequence,
$(\mu_{t}^{N},\nu_{t}^{N})\to(\mu_{t},\nu_{t})\in\mathsf{CE}$ in a weak
topology (Proposition 5.2).

The final step is the comparison of the discrete and continuous energy
functionals: by construction, we have that

$\displaystyle\int_{\mathbb{R}}\frac{|j_{t}^{N}|^{2}}{m(\rho_{t}^{N})}\,\mathrm{d}x\leq\frac{N+1}{N}\sum_{j=0}^{N}\frac{|\dot{x}_{j}^{N}(t)|^{2}}{\theta(R_{j}^{N}(t))}\,,\quad
t\in[0,1]\,.$

The proof of the liminf inequality is then concluded by (1.8) and by means of
lower semicontinuity properties of the continuous action (I).

#### Sketch of the limsup inequality

We fix $\mu_{0},\mu_{1}\in\mathscr{P}_{q}(\mathbb{R})$ two probability
measures with $q$-th moment finite and such that
$\mathbb{W}_{2,m}(\mu_{0},\mu_{1})<\infty$. We pick
$(\mu_{t},\nu_{t})\in\mathsf{CE}$ the unique $\mathbb{W}_{2,m}$-geodesic
between $\mu_{0}$ and $\mu_{1}$. The goal is to find a sequence
$\\{(x^{N}(0),x^{N}(1))\\}_{N}$ with $x^{N}(i)\in\mathcal{K}_{N}$ such that we
have the limsup inequality

$\displaystyle\limsup_{N\to\infty}d_{2,m}^{N}(x^{N}(0),x^{N}(1))\leq\mathbb{W}_{2,m}(\mu_{0},\mu_{1})\,.$

The idea is to first regularise the curve $(\mu_{t},\nu_{t})$ in order to
obtain a solution of $\mathsf{CE}$ with compact support and smooth densities
(w.r.t to the Lebesgue measure on $\mathbb{R}$), comparable energy, and whose
density does not assume value at the boundary of the support of $m$ (which
creates singularities in the energy functional). This is the content of
Proposition 4.2. Therefore, without loss of generality, we assume here that
$(\mu_{t},\nu_{t})\in\mathsf{CE}$ has such properties:

  1. i)

for all $t\in[0,1]$, $\mu_{t}$ and $\nu_{t}$ have compact and connected
support, $\mu_{t},\nu_{t}\ll\mathscr{L}^{1}$ and their respective densities
densities $\rho_{t}$, $j_{t}$ are $C^{\infty}$ in their support

  2. ii)

there exist a compact set $K$ and two constants $0<l\leq\tilde{M}<M$ such that

$\operatorname{supp}(\rho_{t}),\,\operatorname{supp}(j_{t})\subset
K\quad\text{and}\quad l<\rho_{t}(x)<\tilde{M}\quad$

for all $x\in\operatorname{supp}(\rho_{t})$ and $t\in[0,1]$.

Therefore, we can consider the flow map $\Psi_{v}$ associated with
$v_{t}:=j_{t}/\rho_{t}$ and define for $t\in[0,1]$

$\displaystyle X_{t}(z):=\Psi_{v}(t,X_{0}(z)),\quad X_{0}:=X_{\mu_{0}},\quad
X_{0}\in C^{1}([0,1]),$

which is nothing but the quantile function $X_{t}=X_{\mu_{t}}$ of $\mu_{t}$.
The sought recovery sequence can then be obtained as

$\displaystyle x_{i}^{N}(t):=X_{t}\left(\frac{i}{N}\right),\quad
t\in[0,1],\;i\in\\{0,...,N\\}\,.$

It is not hard to see that $\mu_{t}^{N}:=\mathcal{PC}^{N}(x^{N}(t))\to\mu_{t}$
in $\mathbb{W}_{q}$ as $N\to\infty$. Moreover, by means of Jensen’s inequality
and by the regularity assumptions on $(\mu_{t},\nu_{t})$, one can show that
the discrete energy of $x^{N}(t)$ converge to the one of $(\mu_{t},\nu_{t})$
as $N\to\infty$, which concludes the proof.

#### Sketch of the JKO-scheme convergence

The convergence of the discrete JKO-schemes relies on the convergence of the
distances. Indeed, under suitable assumptions on the functionals $F_{N}$, one
can reduce the proof to the simpler case of $F_{N}\equiv 0$. In particular, it
suffices to show that, for some $q>1$, the functionals

$\displaystyle
y^{N}\mapsto\mathcal{J}_{N}(y^{N}):=d_{2,m}^{N}(x^{N},y^{N})^{2}\,,\quad\text{for}\;x^{N}\in\mathcal{K}_{N}\;\text{s.t.}\;\begin{cases}\displaystyle\mathcal{E}^{N}(x^{N})\xrightarrow{\mathbb{W}_{q}}\mu^{(0)}\in\mathscr{P}(\mathbb{R}),\\\
\displaystyle\sup_{N\in\mathbb{N}}\mathbb{E}_{\mathcal{E}^{N}(x^{N})}|x|^{2}<\infty,\end{cases}$

are equicoercive and $\Gamma$-converging to the continuous one
$\mu\mapsto\mathcal{J}(\mu):=\mathbb{W}_{2,m}(\mu^{(0)},\mu)$.

In order to have equicoercivity for $\left(\mathcal{J}_{N}\right)_{N}$, and
because we do not want to enforce any coercivity for the functionals
$\left(F_{N}\right)_{N}$ (due to the applications we have in mind), we work
with $q<2$. Indeed, using that balls in $\mathbb{W}_{2}$ are pre-compact in
$\mathbb{W}_{q}$, for every $q<2$ (but not $q=2$!), it is not hard to prove
the sought equicoercivity. On the other hand, working with $q<2$ makes the
$\Gamma$-convergence less trivial.

Let us say we have $\big{(}x^{N}\big{)}_{N}$, $\mu^{(0)}$, $\mu$ given and we
want to find a recovery sequence $\big{(}y^{N}\big{)}_{N}$ for
$\left(\mathcal{J}_{N}\right)_{N}$. A first näive attempt would be to apply
Theorem 3.1, which ensures the existence of a recovery sequence
$\\{(\tilde{x}^{N},\tilde{y}^{N})\\}_{N}$ such that
$d_{2,m}^{N}(\tilde{x}^{N},\tilde{y}^{N})\to\mathbb{W}_{2,m}(\mu^{(0)},\mu)$,
and set $y^{N}:=\tilde{y}^{N}$. Nonetheless, having by assumption that
$\mathcal{E}^{N}(x^{N})\to\mu^{(0)}$ in $\mathbb{W}_{q}$ (not in
$\mathbb{W}_{2}$), a priori we are not able to show that
$d_{2,m}^{N}(x^{N},\tilde{x}^{N})\to 0$.

However, thanks to the fact that $\big{(}x^{N}\big{)}_{N}$ has uniformly
bounded second moments, we are able to provide the existence of the sought
recovery sequence in $\mathbb{W}_{q}$, for $q<2$. This relies on an
quantitative regularisation result (Proposition 4.2, in particular $(iii)$)
and a careful construction of the recovery sequence, based on Theorem 3.1
together with a delicate localisation argument (see proof of Lemma 6.1).

###  1.4. Related literature

The Wasserstein gradient-flow structure is a common feature of several
evolution equations, including Fokker–Planck equations, porous-medium [Ott01],
fourth-order evolutions [MMS09, LMZ16], and more generally can describe
diffusion, aggregation, and advection effects, see [San17] for a general
discussion about various possible combinations.

The JKO scheme as a numerical tool to approximate PDEs has recently been the
focus of several works, such as porous medium equations [BCL16] and the
Patlak–Keller–Segel model [BCC08]. In the setting of non-linear mobilities,
this variational strcture is exploited in a couple of recent works, in the
case of scalar Burger equations [GO13] and thin-films evolutions [LMS12].

In the last decade, a lot of work have been devoted to the study of discrete
approximation of optimal transport distances [GM13, GT20, GKM20, GKMP20,
GKMP21] and of the associated gradient flows [DL15, FMP22]. Nonetheless, these
works deal with a discretisation of the underlying space (here $\mathbb{R}$),
and do not work with deterministic particle methods.

The notion of generalised Wasserstein distances has been introduced in [DNS12,
LM10] and the study of the corresponding gradient-flow structures found
applications to the study of non-linear cross-diffusion systems [BDFPS10] and
fourth-order equations, such as Cahn-Hillard and thin film equations [LMS12],
[Zin17]. See also [CLSS10] for the study of convexity properties of
functionals on the space of measures with respect to such transport distances.

On the other hand, the validation of the continuum evolutive equations is
classically obtained proving the convergence of systems of interacting
particles. The literature involving probabilistic methods is extremely rich
and quite well-understood, see e.g. [DMP91], [SV79]. A first rigorous fully
determistic approximation result for non-linear conservation laws has been
obtained in [DFR15]. More in general, many-particle limit results for scalar
evolutions with non-local interaction have been obtained for both linear
[DRR22, FR21] and non-linear mobilities [DFS19, DFFR19, FR18, RS21, FT22]. In
particular, these works concern the validity of the top, blue arrow in Figure
1.

In all the above mentioned deterministic methods, the particle system is
obtained as a discretised version of the Lagrangian formulation of the
evolution, using the pseudo-inverse formalism. We emphasise that [FT22] shows
the convergence of the particle systems to the corresponding entropy solution,
working directly at the level of the gradient-flow structures at the
continuous-time scale (but with no analysis of the behavior of the discrete
distances nor the time-discrete scheme).

### Organisation of the paper

In Section 2 and 3 we introduce the setting of the problem and state our main
results. Section 4 includes some preliminary results concerning the
approximation of finite-energy solutions to the continuity equation in
$\mathbb{R}$, whereas in Section 5 and 6 we prove our two main results, i.e.
the convergence of the discrete distances and of the gradient-flow schemes.

### Notation

For $d,m\in\mathbb{N}$, $\Omega\subset\mathbb{R}^{m}$, we denote by
$\mathcal{M}(\Omega)$ (resp. $\mathscr{P}(\Omega)$) the set of all signed,
locally finite Radon measures (resp. probability measures) on $\Omega$. Also,
$C(\Omega)$, $C_{b}(\Omega)$, $C_{c}(\Omega)$ denote respectively the space of
continuous, bounded, and compactly supported functions.

For a given sequence of measures
$\left(\nu_{n},\nu\right)_{n}\subset\mathcal{M}(\Omega)$, we write

  1. (i)

$\nu_{n}\to\nu$ vaguely in $\mathcal{M}(\Omega)$ if
$\int\varphi\,\mathrm{d}\nu_{n}\to\int\varphi\,\mathrm{d}\nu$, for every
$\varphi\in C_{c}(\Omega)$.

  2. (ii)

$\nu_{n}\to\nu$ narrowly in $\mathcal{M}(\Omega)$ if
$\int\varphi\,\mathrm{d}\nu_{n}\to\int\varphi\,\mathrm{d}\nu$, for every
$\varphi\in C_{b}(\Omega)$.

We also adopt the following notation:

  * •

For $\mu\in\mathscr{P}(\Omega)$, $s\in\mathbb{R}_{+}$, set
$\mathbb{E}_{\mu}|x|^{s}:=\int|x|^{s}\,\mathrm{d}\mu(x)$ ($s$-th moment).

  * •

For $N\in\mathbb{N}$, $x\in\mathbb{R}^{N+1}$, $p\geq 1$, we write $\left\lVert
x\right\rVert_{\ell_{p,N}}:=\frac{1}{N+1}\sum_{i=0}^{N}|x_{i}|^{p}$.

  * •

$\mathscr{L}^{d}$ denotes the Lebesgue measure on $\mathbb{R}^{d}$,
$d\in\mathbb{N}$.

  * •

For $a,b\in\mathbb{R}$, $a\vee b=\max\\{a,b\\}$.

##  2\. The discrete and continuous settings

In this section, we introduce the general framework of the problem, both in
the continuous and discrete setting.

###  2.1. Optimal transport in $\mathbb{R}$ with nonlinear mobility

We start recalling the definition of solutions of the continuity equation (as
in [DNS12], [LM10]), namely the equation
$\partial_{t}\mu_{t}+\partial_{x}\nu_{t}=0,$ in $(0,1)\times\mathbb{R}$, where
$\mu_{t},\nu_{t}$ are Borel families of measures in $\mathscr{P}(\mathbb{R})$
and $\mathcal{M}(\mathbb{R})$ respectively, and satisfying the equation in the
sense of distributions, namely, $\forall\xi\in
C_{c}^{1}((0,1)\times\mathbb{R})$,

$\displaystyle\int_{0}^{1}\int_{\mathbb{R}}\partial_{t}\xi(t,x)\,\mathrm{d}\mu_{t}(x)\,\mathrm{d}t+\int_{0}^{1}\int_{\mathbb{R}}\partial_{x}\xi(t,x)\,\mathrm{d}\nu_{t}(x)\,\mathrm{d}t=0\,.$
(2.1)

For a given sequence of measures $\nu_{t}$ we consider the corresponding
measure
${\bm{\nu}}:=\int_{0}^{1}\nu_{t}\,\mathrm{d}t\in\mathcal{M}((0,1)\times\mathbb{R})$
defined via disintegration

$\displaystyle<{\bm{\nu}},\xi>=\int_{0}^{1}\Big{(}\int_{\mathbb{R}}\xi(t,x)\,\mathrm{d}\nu_{t}(x)\Big{)}\,\mathrm{d}t,\quad\forall\xi\in
C_{c}((0,1)\times\mathbb{R}).$ (2.2)

Similarly, we write
${\bm{\mu}}:=\int_{0}^{1}\mu_{t}\,\mathrm{d}t\in\mathscr{P}((0,1)\times\mathbb{R})$.

######  Definition  (Solution of continuity equation, [LM10]).

We denote by $\mathsf{CE}$ the set of curves
$\left(\mu_{t},\nu_{t}\right)_{t\in[0,1]}$ such that

  1. (1)

$t\mapsto\mu_{t}$ is vaguely-continuous in $\mathscr{P}(\mathbb{R})$.

  2. (2)

$\left(\nu_{t}\right)_{t}$ is a Borel family of finite Radon measures on
$\mathbb{R}$.

  3. (3)

$({\bm{\mu}},{\bm{\nu}})$ satisfies (2.1).

In this case we write in short $\partial_{t}\mu_{t}+\partial_{x}\nu_{t}=0$ and
$(\mu_{t},\nu_{t})\in\mathsf{CE}$, or $({\bm{\mu}},{\bm{\nu}})\in\mathsf{CE}$.
Whenever we want to emphasis the dependence on the initial and final datum, we
write $(\mu_{t},\nu_{t})\in\mathsf{CE}(\mu_{0},\mu_{1})$.

We now fix the notation concerning the action functionals. See [LM10, Section
2.3].

######  Definition  (Nonlinear mobility).

Given a maximal density $M>0$, a nonlinear mobility is a continuous concave
function $m:[0,M]\to[0,+\infty)$ such that $m(0)=0$; sometimes, if needed we
extend the domain of $m$ to $[0,+\infty)$ letting $m(\rho)=-\infty$ for
$\rho>M$. We moreover define

$\displaystyle\theta:[0,M]\to[0,+\infty)\,,\quad\theta(\rho):=\begin{cases}\frac{m(\rho)}{\rho}&\text{if}\
\rho>0,\\\ {\displaystyle\lim_{\rho\downarrow
0}}\,\frac{m(\rho)}{\rho}&\text{if}\ \rho=0.\end{cases}$

Notice that $\theta$ is an (extended) continuous function in $[0,M]$; in this
work, we often assume that $\theta$ is also bounded. Note that by concavity of
$m$, this is equivalent to the superdifferential of $m$ being not empty in
$\rho=0$.

######  Remark  (Monotonicity of $\theta$).

Note that by concavity of $m$, it follows that $\theta$ is nonincreasing on
$(0,M]$. Indeed, for $0<\rho_{1}<\rho_{2}\leq M$, we have that

$\displaystyle\theta(\rho_{2})-\theta(\rho_{1})=\frac{1}{\rho_{2}}\Big{[}\big{(}m(\rho_{2})-m(\rho_{1})\big{)}-\frac{m(\rho_{1})}{\rho_{1}}(\rho_{2}-\rho_{1})\Big{]}\leq
0$

by concavity of $m$ on $[0,M]$ and $m(0)=0$. In particular,
$\sup\theta=\theta(0)$.

######  Definition  (Action functionals).

Fix a maximal density $M>0$ and $p>1$. We then consider
$\phi_{p,m}:\mathbb{R}\times\mathbb{R}\rightarrow[0,+\infty]$ defined as

$\displaystyle\phi_{p,m}(\rho,j)=\begin{cases}\displaystyle\frac{|j|^{p}}{m(\rho)^{p-1}},&\mathrm{if}\;\
m(\rho)>0,\\\ 0&\mathrm{if}\;\ m(\rho)=0=j,\\\
+\infty,&\mathrm{if}\;\emph{either}\ m(\rho)=0\,,j\neq 0\emph{ or
}\rho>M\,,\end{cases}$

where $m$ is a nonlinear mobility with maximal density $M$. Note that
$\phi_{p,m}$ is a lower semicontinuous, nonnegative, proper, convex function,
with $\mathrm{int}(\mathrm{dom}(\phi_{p,m}))=(0,M)\times\mathbb{R}$. For every
$\mu\in\mathscr{P}(\mathbb{R})$ and $\nu\in\mathcal{M}(\mathbb{R})$, we define
the action functional

$\displaystyle\Phi_{p,m}(\mu,\nu):=\begin{cases}\displaystyle\int_{\mathbb{R}}\phi_{p,m}(\rho,j)\,\mathrm{d}x&\text{if}\quad\,\mathrm{d}\mu=\rho\,\mathrm{d}x\,,\
\,\mathrm{d}\nu=j\,\mathrm{d}x\,,\\\ +\infty&\text{otherwise}\,.\end{cases}$
(2.3)

######  Remark .

As observed in [LM10, Proposition 2.3], if $\mu$, $\nu$ are such that
$\Phi_{p,m}(\mu,\nu)<\infty$ (in particular they are absolutely continuous
w.r.t. d$x$), then $\,\mathrm{d}\mu=\rho\,\mathrm{d}x$ with
$0\leq\rho(\cdot)\leq M$.

######  Definition  ($m$-Wasserstein distances).

Let $\phi_{p,m}$, $\Phi_{p,m}$ be as in Definition 2.1. For any
$\mu_{0},\mu_{1}\in\mathscr{P}_{p}(\mathbb{R})$, we define

$\displaystyle\mathbb{W}_{p,m}(\mu_{0},\mu_{1}):=\inf\left\\{\int_{0}^{1}\Phi_{p,m}(\mu_{t},\nu_{t})\,\mathrm{d}t\
:\ (\mu_{t},\nu_{t})\in\text{CE}(\mu_{0},\mu_{1})\right\\}^{\frac{1}{p}}.$
(2.4)

Note that $\mathbb{W}_{p,m}$ is only a pseudo-distance, given that it can
assume the value $+\infty$. If $m(\rho)=\rho$, then we recover the usual
$p$-Wasserstein distance $\mathbb{W}_{p}$.

######  Remark  (Time regularity of finite energy curves).

Curves of measures with finite energy enjoy good regularity properties in the
time-variable, if $\theta$ is bounded. Precisely, if
$(\mu_{t},\nu_{t})\in\mathsf{CE}$ and

$\displaystyle
E:=\int_{0}^{1}\Phi_{p,m}(\mu_{t},\nu_{t})\,\mathrm{d}t<\infty\,.$

Then $t\mapsto\mu_{t}$ is Lipschitz-continuous with respect to the
$\mathbb{W}_{p}$ distance on $\mathscr{P}_{p}(\mathbb{R})$. Indeed, by the
homogeneity of $\phi$ in the second variable,

$\displaystyle\mathbb{W}_{p}(\mu_{s},\mu_{t})^{p}\leq\|\theta\|_{\infty}\mathbb{W}_{p,m}(\mu_{s},\mu_{t})^{p}\leq\|\theta\|_{\infty}|t-s|^{p}\int_{0}^{1}\Phi_{p,m}(\mu_{h},\nu_{h})\,\mathrm{d}h\leq
E\left\lVert\theta\right\rVert_{\infty}|t-s|^{p}\,,$

as claimed. In particular, by Prokhorov’s theorem, the family of probability
measures $\left(\mu_{t}\right)_{t\in[0,1]}$ is tight in
$\mathscr{P}(\mathbb{R})$.

####  2.1.1. Auxiliary results

In this section we recall some well-known properties of the actions and the
generalised distances. We refer to [DNS12] and [LM10, Section 3&4] for
details. Let $\phi_{p,m}$ be as in Definition 2.1.

  1. (I)

Lower semicontinuity. For every
$\left(\mu_{n}\right)_{n}\in\mathscr{P}(\mathbb{R})$,
$\left(\nu_{n}\right)_{n}\subset\mathcal{M}(\mathbb{R})$ vaguely converging to
$\mu$, $\nu$ respectively, then
$\displaystyle\liminf_{n\to\infty}\Phi_{p,m}(\mu_{n},\nu_{n})\geq\Phi_{p,m}(\mu,\nu)$.

  2. (II)

Convolution. Let $h\in C_{b}(\mathbb{R}\times\mathbb{R})$ be a positive
function with $\int h(x)\,\mathrm{d}x=1$. For $[0,1]\ni
t\mapsto(\mu_{t},\nu_{t})\in\mathscr{P}(\mathbb{R})\times\mathcal{M}(\mathbb{R})$
a curve of measures, we consider their natural extension on $\mathbb{R}$ by
setting

$\displaystyle\mu_{t}=\mu_{0}\ \ \text{if}\ \ t<0\,,\quad\mu_{t}=\mu_{1}\ \
\text{if}\ \ t>1\,,\quad\nu_{t}=0\ \ \text{if}\ \ t\in(0,1)^{c}\,.$

Let $({\bm{\mu}},{\bm{\nu}})$ as in (2.2); define
$\hat{\bm{\mu}}:=h*{\bm{\mu}}$ and $\hat{\bm{\nu}}:=h*{\bm{\nu}}$, where
$h*{\bm{\nu}}$ denotes the measure which is absolutely continuous with respect
to the Lebesgue measure on $\mathbb{R}\times\mathbb{R}$ with density
$\hat{j}(t,x):=\int h(t-s,x-y)\,\mathrm{d}{\bm{\nu}}(s,y)$. Note that if
$(\mu_{t},\nu_{t})\in\mathsf{CE}$, then
$(\hat{\mu}_{t},\hat{\nu}_{t})\in\mathsf{CE}$ as well. Moreover, by $\phi\geq
0$ and $\phi(\cdot,0)=0$, we have that

$\displaystyle\int_{0}^{1}\Phi_{p,m}(\hat{\mu}_{t},\hat{\nu}_{t})\,\mathrm{d}t\leq\int_{0}^{1}\Phi_{p,m}(\mu_{t},\nu_{t})\,\mathrm{d}t\,.$

  3. (III)

$\mathbb{W}_{p,m}$-geodesics. Fix $\mu_{0}$,
$\mu_{1}\in\mathscr{P}(\mathbb{R})$ such that
$\mathbb{W}_{p,m}(\mu_{0},\mu_{1})<\infty$. Then there exists
$(\mu_{t},\nu_{t})\in\mathsf{CE}(\mu_{0},\mu_{1})$ such that
$\mathbb{W}_{p,m}(\mu_{0},\mu_{1})=\int_{0}^{1}\Phi_{p,m}(\mu_{t},\nu_{t})\,\mathrm{d}t$.
Furthermore, $\mu_{t}$ is a constant speed geodesic for $\mathbb{W}_{p,m}$,
i.e. it satisfies
$\mathbb{W}_{p,m}(\mu_{t},\mu_{s})=|t-s|\mathbb{W}_{p,m}(\mu_{0},\mu_{1})$,
for all $s,t\in[0,1]$.

###  2.2. The discrete transport problem

In this section we introduce the discrete setting. Recall that
$M\in(0,+\infty)$ is a fixed positive number. For any $N\in\mathbb{N}$, we
consider the (closed) cone $\mathcal{K}_{N}$ in $\mathbb{R}^{N+1}$ given by

$\displaystyle\mathcal{K}_{N}:=\left\\{x=(x_{0},...,x_{N})\in\mathbb{R}^{N+1}\
:\ x_{i+1}-x_{i}\geq\frac{1}{NM}\,,\;\forall
i\right\\}\subset\mathbb{R}^{N+1}.$

For simplicity, we omit the dependence on $M$ in $\mathcal{K}_{N}$. We
consistently adopt the following notation: for $x\in\mathcal{K}_{N}$, we
denote by

$\displaystyle\Delta x_{i}:=x_{i+1}-x_{i},\quad R_{i}^{N}(x):=\frac{1}{N\Delta
x_{i}}\leq M\,,$ (2.5)

for $i=0,...,N-1$, and $\rho^{*}\in\mathbb{R}_{+}$ a nonnegative number. The
definition of the density reconstruction for $i=N$ admits no canonical
definition (there is no particle in front of $x_{N}$), and several choice are
possible. We define $R_{N}^{N}(x):=\rho_{N}^{*}(x)$, where
$\rho_{N}^{*}:\mathcal{K}_{N}\to\mathbb{R}_{+}$ is a family of continuous
functions satisfying

$\displaystyle\sup_{N\in\mathbb{N}}\|\rho_{N}^{*}\|_{\infty}<M\,.$ (2.6)

######  Remark  (Examples of $\rho_{N}^{*}$).

A simple choice is to set $\rho_{N}^{*}(x)\equiv\rho^{*}:=\arg\max\theta$,
then the last particle motion is not influenced by the rest of the particles
(follow-the-leader scheme, see e.g. [DFR15]). Another option could be to "look
back" to the previous particle, i.e. $\rho_{N}(x)=R_{N-1}^{N}(x)$.

######  Definition  (Discrete distances).

Fix $N\in\mathbb{N}$, $p>1$, $M\in\mathbb{R}_{+}$. Let $\phi_{p,m}$ be as in
Definition 2.1. We define the corresponding discrete action functional
$\Phi_{p,m}^{N}:\text{T}\mathcal{K}_{N}\to\mathbb{R}^{+}\cup\\{+\infty\\}$,

$\displaystyle\Phi_{p,m}^{N}(x,v):=\frac{1}{N+1}\sum_{i=0}^{N}(R_{i}^{N}(x))^{p-1}\phi_{p,m}(R_{i}^{N}(x),v_{i}^{N})=\frac{1}{N+1}\sum_{i=0}^{N}\frac{|v_{i}|^{p}}{\theta(R_{i}^{N}(x))^{p-1}}\,,$
(2.7)

for any $x\in\mathcal{K}_{N}$ and $v\in\text{T}_{x}\mathcal{K}_{N}$. Here
$\text{T}\mathcal{K}_{N}\approx\mathcal{K}_{N}\times\mathbb{R}^{N+1}$ denotes
the tangent bundle of the cone $\mathcal{K}_{N}$. Furthermore, we define the
discrete $\phi_{p,m}$-distance between $x,y\in\mathcal{K}_{N}$ as

$\displaystyle d_{p,m}^{N}(x,y)^{p}:=\inf_{x\in
W^{1,p}(0,1;\mathcal{K}_{N})}\left\\{\int_{0}^{1}\Phi_{p,m}^{N}(x(t),\dot{x}(t))\,\mathrm{d}t\
:\ x(0)=x,\;x(1)=y\right\\},$ (2.8)

where $W^{1,p}(0,1;\mathcal{K}_{N})$ denotes the space of $p$-Sobolev
functions from $(0,1)$ to $\mathcal{K}_{N}$. Note that $d_{p,m}^{N}$ is only a
pseudo-distance, given that it can assume the value $+\infty$.

######  Remark .

For the sake of simplicity, we omit the dependence of $\Phi_{p,m}^{N}$ on
$\rho^{*}$. As we are going to see, the exact value chosen plays no role in
our main results, due to the fact we consider limits as $N\to\infty$. If
$\rho^{*}(x)\geq M$ (and thus $\theta(R_{N}^{N}(x))=0$), then
$\phi_{p,m}(x,v)<\infty$ forces $v_{N}=0$.

######  Example  (Riemannian case).

In the case $p=2$, $d_{2,m}^{N}$ is a Riemannian distance induced by the
Riemannian metric on the cone $\mathcal{K}_{N}$ given by

$\displaystyle
g_{N}\left(v,w\right)_{x}:=\frac{1}{N+1}\sum_{i=0}^{N}\frac{v_{i}w_{i}}{\theta(R_{i}^{N}(x))},\quad\forall
x\in\mathcal{K}_{N},\;\forall v,w\in\text{T}_{x}\mathcal{K}_{N}\,.$ (2.9)

Now we prove the existence of geodesics in $\mathcal{K}_{N}$, which follows by
standard compactness arguments, together with a careful analysis close to the
singularity of $\Phi_{p,m}^{N}$.

######  Lemma  (Existence of discrete geodesics).

Let $\theta\in\emph{L}^{\infty}(0,M)$ and $p>1$. For every
$z,y\in\mathcal{K}_{N}$, there exists $x\in W^{1,p}(0,1;\mathcal{K}_{N})$ such
that $x(0)=z$, $x(1)=y$ and

$\displaystyle
d_{p,m}^{N}(z,y)^{p}=\int_{0}^{1}\Phi_{p,m}^{N}(x(t),\dot{x}(t))\,\mathrm{d}t=\Phi_{p,m}^{N}(x(s),\dot{x}(s))\,,\quad\forall
s\in[0,1]\,.$

###### Proof.

For $z,y\in\mathcal{K}_{N}$, pick a minimising sequence
$\left(x^{n}\right)_{n}\subset W^{1,p}(0,1;\mathcal{K}_{N})$, i.e.

$\displaystyle\lim_{n\to\infty}\int_{0}^{1}\Phi_{p,m}^{N}(x^{n}(t),\dot{x}^{n}(t))\,\mathrm{d}t=d_{p,m}^{N}(z,y)^{p}\,,\quad
x^{n}(0)=z\,,\;x^{n}(1)=y\,.$

With no loss of generality, we assume
$D:=\sup_{n}\int_{0}^{1}\Phi_{p,m}^{N}(x^{n},\dot{x}^{n})\,\mathrm{d}t<\infty$.
In particular

$\displaystyle\sup_{n\in\mathbb{N}}\int_{0}^{1}\left\lVert\dot{x}^{n}(t)\right\rVert_{\ell_{p,N}}^{p}\,\mathrm{d}t\leq
D\left\lVert\theta\right\rVert_{\infty}^{p-1}<\infty\,.$

In particular, up to (a non-relabeled) subsequence, $x^{n}\to x$ weakly in
$W^{1,p}(0,1;\mathcal{K}_{N})$ and strongly in $C(0,1;\mathcal{K}_{N})$ (by
Sobolev embedding, thanks to $p>1$) as $n\to\infty$. To show that $x$ is
indeed a minimiser, we show the lower semicontinuity of the discrete energies
with respect to such convergence.

First we assume that $\theta(M)>0$ and in particular $\theta$ is continuous
and bounded from below. Now, since $R_{i}^{N}(x^{n})$ is converging uniformly
to $R_{i}^{N}(x)$, we also have that $\theta(R_{i}^{N}(x^{n}))^{1-p}$ will
converge uniformly to $\theta(R_{i}^{N}(x))^{1-p}$. Since
$\dot{x}^{n}\rightharpoonup\dot{x}$ weakly in $L^{p}$, we then can conclude,
by classical semicontinuity arguments, that

$\liminf_{n\to\infty}\int_{0}^{1}\frac{|\dot{x}^{n}_{i}(t)|^{p}}{\theta(R_{i}^{N}(x^{n}(t)))^{p-1}}\,\mathrm{d}t\geq\int_{0}^{1}\frac{|\dot{x}_{i}(t)|^{p}}{\theta(R_{i}^{N}(x(t)))^{p-1}}\,\mathrm{d}t.$

Summing up over $i=0,\ldots,N$ and then dividing by $N+1$ we get the actual
semicontinuity of the energy
$\liminf_{n\to\infty}\int_{0}^{1}\Phi_{p,m}^{N}(x^{n},\dot{x}^{n})\,\mathrm{d}t\geq\int_{0}^{1}\Phi_{p,m}^{N}(x,\dot{x})\,\mathrm{d}t$,
hence $x$ is a minimizer.

In order to deal with the general case we consider for $\lambda>1$ a relaxed
mobility $m^{\lambda}(\rho)=\lambda m(\rho/\lambda)$. Of course we have
$\theta^{\lambda}(\rho)=\theta(\rho/\lambda)$. Using the monotonicity of
$\theta$ we have that
$\Phi_{p,m^{\lambda}}^{N}(x^{n},\dot{x}^{n})\leq\Phi_{p,m}^{N}(x^{n},\dot{x}^{n})$.
Moreover, since $\theta^{\lambda}(M)=\theta(M/\lambda)>0$ we can apply the
previous result and we can get

$\liminf_{n\to\infty}\int_{0}^{1}\Phi_{p,m}^{N}(x^{n},\dot{x}^{n})\,\mathrm{d}t\geq\liminf_{n\to\infty}\int_{0}^{1}\Phi_{p,m^{\lambda}}^{N}(x^{n},\dot{x}^{n})\,\mathrm{d}t\geq\int_{0}^{1}\Phi_{p,m^{\lambda}}^{N}(x,\dot{x})\,\mathrm{d}t.$

By monotone convergence we then conclude since
$\Phi_{p,m^{\lambda}}^{N}(x,\dot{x})\uparrow\Phi_{p,m}^{N}(x,\dot{x})$.

The fact that $s\mapsto\Phi_{p,m}^{N}(x(s),\dot{x}(s))$ is constant along
discrete geodesics follows by classical arguments and is consequence of the
$p$-homogeneity of $\phi$. Let us argue by contradiction that it is not true;
in particular by Jensen we have
$\int_{0}^{1}\Phi_{p,m}^{N}(x(s),\dot{x}(s))\,ds>\left(\int_{0}^{1}\Phi_{p,m}^{N}(x(s),\dot{x}(s))^{1/p}\,ds\right)^{p}$.
Now let us consider any $C^{1}$ increasing reparametrization
$\xi:[0,1]\to[0,1]$, and let $y(t)=x(\xi(t))$; we have of course $y(0)=0$ and
$y(1)=1$. Moreover

$\displaystyle\int_{0}^{1}\Phi_{p,m}^{N}(y(t),\dot{y}(t))\,dt$
$\displaystyle=\int_{0}^{1}\Phi_{p,m}^{N}(x(\xi(t)),\xi^{\prime}(t)\dot{x}(\xi(t)))\,dt$
$\displaystyle\geq\left(\int_{0}^{1}\Phi_{p,m}^{N}(x(\xi(t)),\xi^{\prime}(t)\dot{x}(\xi(t)))^{1/p}\,dt\right)^{p}$
$\displaystyle=\left(\int_{0}^{1}\xi^{\prime}(t)\Phi_{p,m}^{N}(x(\xi(t)),\dot{x}(\xi(t)))^{1/p}\,dt\right)^{p}$
$\displaystyle=\left(\int_{0}^{1}\Phi_{p,m}^{N}(x(s),\dot{x}(s))^{1/p}\,dt\right)^{p}.$

We can have equality as long as
$\xi^{\prime}(\xi^{-1}(s))=(\xi^{-1})^{\prime}(s)$ is proportional to
$\Phi_{p,m}^{N}(x(s),\dot{x}(s))^{1/p}$. While this is not guaranteed to
happen for $\xi\in C^{1}$ we can use a sufficiently good approximation in
order to get
$\int_{0}^{1}\Phi_{p,m}^{N}(y(t),\dot{y}(t))\,dt<\int_{0}^{1}\Phi_{p,m}^{N}(x(s),\dot{x}(s))\,ds$,
which is a contradiction.

∎

In order to compare the discrete and the continuous setting, we introduce
suitable embedding maps from the cones $\mathcal{K}_{N}$ to the space of
continuous measures $\mathscr{P}(\mathbb{R})$.

Embedding via piecewise-constant measures. For any $N\in\mathbb{N}$, we
consider the map

$\displaystyle\mathcal{PC}^{N}:\mathcal{K}_{N}\to\mathscr{P}(\mathbb{R})\,,\quad\mathcal{PC}^{N}(x)=\sum_{i=0}^{N-1}R_{i}^{N}(x)\mathbbm{1}_{\left[x_{i},x_{i+1}\right]}(\cdot)\mathscr{L}^{1}\,,$
(2.10)

where $\mathscr{L}^{1}$ denotes the Lebesgue measure on $\mathbb{R}$. Note
that, for every $x\in\mathcal{K}_{N}$, the measures $\mathcal{PC}^{N}(x)$ are
absolutely continuous with respect to $\mathscr{L}^{1}$. Moreover,
$\mathcal{PC}^{N}(x)$ charges every interval $[x_{i},x_{i+1}]$ with equal mass
$\frac{1}{N}$. We shall use the shorthand notation
$\mathcal{PC}^{N}:=\mathcal{PC}^{N}(\mathcal{K}_{N})\subset\mathscr{P}(\mathbb{R})$.

Embedding via empirical measures. For any $N\in\mathbb{N}$, we consider the
map

$\displaystyle\mathcal{E}^{N}:\mathcal{K}_{N}\to\mathscr{P}(\mathbb{R})\,,\quad\mathcal{E}^{N}(x)=\frac{1}{N+1}\sum_{i=0}^{N}\delta_{x_{i}}\,.$
(2.11)

Note that the image of $\mathcal{E}^{N}$ consists of atomic measures over the
real line. In particular, $\mathcal{E}^{N}(x)$ is singular with respect to the
Lebesgue measure. We adopt the shorthand notation
$\mathcal{E}^{N}:=\mathcal{E}^{N}(\mathcal{K}_{N})\subset\mathscr{P}(\mathbb{R})$.

It’s easy to see that $\mathcal{E}^{N}$ and $\mathcal{PC}^{N}$ are injective.
Both embeddings maps play an important role in linking the discrete and
continuous framework. As we are going to see in Lemma 5, the two maps turn out
to be equivalent at the level of the convergence, as $N\to\infty$, with
respect to a broad class of weak topologies on $\mathscr{P}(\mathbb{R})$.

On one hand, the $\mathcal{PC}^{N}$-embedding appears to be more natural in
the discretisation of the continuous distances $\mathbb{W}_{p,m}$, for two
reasons: firstly, the image measure is always absolutely continuous, as every
measure in the domain of $\mathbb{W}_{p,m}$. Secondly, we note that, for
$x\in\mathcal{K}_{N}$ and $v\in T_{x}\mathcal{K}_{N}$, one has that

$\displaystyle\Phi_{p,m}(\mu^{N},\nu^{N})\leq\frac{N+1}{N}\Phi_{p,m}^{N}(x,v)$
(2.12)

for $\mu^{N}=\mathcal{PC}^{N}(x)$ and
$\nu^{N}:=\sum_{i=0}^{N-1}v_{i}R_{i}^{N}(x)\mathbbm{1}_{\left[x_{i},x_{i+1}\right]}(\cdot)\mathscr{L}^{1}\in\mathcal{M}(\mathbb{R})$.

On the other hand, the $\mathcal{E}^{N}$-embedding enjoys good properties at
the level of the $p$-moments. Indeed, by constructions, the $p$-th moments of
$\mathcal{E}^{N}(x)$ coincides with $\|x\|_{\ell_{p,N+1}}^{p}$, thus
properties in the $p$-Wasserstein space can be recast in terms of the behavior
in the space $(\mathcal{K}_{N},\|\cdot\|_{\ell_{p,N}})$.

Depending on the choice of the embedding, we can define the corresponding
embedded distance at the continuous level.

######  Definition  (Discrete distances - embedding).

Fix $N\in\mathbb{N}$, and let $\phi_{p,m}$ be an energy density as in
Definition 2.1. Define the pseudo-distances $\mathbb{W}_{p,m}^{N}$ and
$\widehat{\mathbb{W}}_{p,m}^{N}$ on $\mathscr{P}(\mathbb{R})$ as

$\displaystyle\mathbb{W}_{p,m}^{N}(\mu_{0},\mu_{1})$
$\displaystyle:=\begin{cases}d_{p,m}^{N}(x,y),&\text{if
}\mu_{0}=\mathcal{PC}^{N}(x),\;\mu_{1}=\mathcal{PC}^{N}(y),\;x,y\in\mathcal{K}_{N}\,,\\\
+\infty,&\text{otherwise}\,,\end{cases}$
$\displaystyle\widehat{\mathbb{W}}_{p,m}^{N}(\mu_{0},\mu_{1})$
$\displaystyle:=\begin{cases}d_{p,m}^{N}(x,y),&\text{if
}\mu_{0}=\mathcal{E}^{N}(x),\;\mu_{1}=\mathcal{E}^{N}(y),\;x,y\in\mathcal{K}_{N}\,,\\\
+\infty,&\text{otherwise}\,,\end{cases}$

where $d_{p,m}^{N}$ is the discrete pseudo-distance defined in (2.8).

###  2.3. The minimising movements scheme

Let $\phi_{2,m}$ be a density function as in Definition 2.1. Note that in this
quadratic case, $\mathbb{W}_{2,m}^{N}$ is induced by a Riemannian metric (2.9)
on $\mathcal{K}_{N}$, denoted by $g_{N}$. In this section, we introduce the
discrete-in-time schemes (both in $\mathcal{K}_{N}$ and $\mathbb{R}$) for the
gradient flow of an energy functional with respect to $\mathbb{W}_{2,m}$,
$\mathbb{W}_{2,m}^{N}$, in the spirit of [JKO98].

######  Definition  (Continuous JKO-scheme).

For a given $F:\mathscr{P}(\mathbb{R})\to(-\infty,+\infty]$ and $\tau>0$, we
iteratively define for $n\in\mathbb{N}$

$\displaystyle\mu^{(0)}\in\mathscr{P}(\mathbb{R}),\quad\mu_{\tau}^{(n)}\in\text{argmin}_{\mu}\left\\{F(\mu)+\frac{1}{2\tau}\mathbb{W}_{2,m}\big{(}\mu,\mu_{\tau}^{(n-1)}\big{)}^{2}\right\\}.$
(JKOτ)

Formally, in the limit as $\tau\to 0$, one formally obtains the PDE

$\displaystyle\partial_{t}\mu_{t}-\partial_{x}\left(m(\mu_{t})\partial_{x}\left(\text{D}F(\mu_{t})\right)\right)=0,\quad\mu_{0}:=\mu^{(0)}\in\mathscr{P}(\mathbb{R}),$

where D$F$ denotes the first variation of $F$ (i.e.
$($D$F(\rho),\omega)=\lim_{\varepsilon\to
0}\varepsilon^{-1}(F(\rho+\varepsilon w)-F(\rho))$). This equation is the
(formal) gradient flow of $F$ with respect to the generalised Wasserstein
distance $\mathbb{W}_{2,m}$. The corresponding discrete counterpart is given
by the following definition.

######  Definition  (Discrete JKO-schemes).

For a given $N\in\mathbb{N}$,
$F_{N}:\mathscr{P}(\mathbb{R})\to(-\infty,+\infty]$, $\tau>0$, we iteratively
define for $n\in\mathbb{N}$

$\displaystyle\mu_{N}^{(0)}\in\mathcal{E}^{N},\quad\mu_{\tau,N}^{(n)}\in\operatorname*{arg\,min}_{\mu\in\mathscr{P}(\mathbb{R})}\left\\{F_{N}(\mu)+\frac{1}{2\tau}\mathbb{W}_{2,m}^{N}\big{(}\mu,\mu_{\tau,N}^{(n-1)}\big{)}^{2}\right\\}\,,$
(JKOτ,N)

where $\mathcal{E}^{N}$ denotes the set of empirical measures as given in
(2.11).

Assumptions. In order to ensure well-posedness, we need the following
assumptions.

  1. (A)

There exist $C,D\in\mathbb{R}^{+}$, $s\in(0,2)$, such that

$\displaystyle
F(\mu)\geq-C\int|x|^{s}\,\mathrm{d}\mu(x)-D,\quad\forall\mu\in\mathscr{P}(\mathbb{R})\,.$
(2.13)

  2. (B)

$F:(\mathscr{P}(\mathbb{R}),\mathbb{W}_{q})\to(-\infty,+\infty]$ is proper and
lsc, for some $q<2$.

######  Example .

Fix $f:\mathbb{R}\to\mathbb{R}\cup\\{+\infty\\}$. Then the functional

$\displaystyle F(\mu)=\int f(x)\,\mathrm{d}\mu(x),\quad[f(x)]_{\\_}\leq
C|x|^{s}+D$ (2.14)

satisfies (2.13). If $f$ is lower semicontinuous, also the assumption (B) is
satisfied [AGS08, Lemma 5.1.7]. A simple example is given by the linear drift
$f(x)=\pm x$.

A consequence of the above assumptions is the well-posedness of the
variational problems in (JKOτ,N), (JKOτ). Fix $N\in\mathbb{N}$ and two
functionals $F_{N}$, $F$ as above, and define the discrete JKO functionals as

$\displaystyle\mathcal{J}_{\tau,N}:\mathscr{P}(\mathbb{R})\to[0,+\infty],\quad\mathcal{J}_{\tau,N}(\mu):=F_{N}(\mu)+\frac{1}{2\tau}\mathbb{W}_{2,m}^{N}(\mu,\mu_{N}^{(0)})^{2}\;,$
(2.15)

whereas the continuous JKO functional are denoted by

$\displaystyle\mathcal{J}_{\tau}:\mathscr{P}(\mathbb{R})\to[0,+\infty],\quad\mathcal{J}_{\tau}(\mu):=F(\mu)+\frac{1}{2\tau}\mathbb{W}_{2,m}(\mu,\mu^{(0)})^{2}\;.$
(2.16)

Recall the definition of $\theta$ asociated to a nonlinear mobility $m$ as in
Definition 2.1.

######  Proposition  (Well-posedness).

Fix $N\in\mathbb{N}$ and assume that $\theta\in\emph{L}^{\infty}(0,M)$. Let
$F$, $F_{N}$ be two functionals satisfying (A),(B).

  1. (i)

If the initial measure $\mu_{N}^{(0)}$ belongs to
$\mathscr{P}_{2}(\mathbb{R})$, then the minimisation problems (JKOτ,N) admit
solutions, for every $n\in\mathbb{N}$.

  2. (ii)

The property $(i)$ holds as well if $\mathbb{W}_{2,m}^{N}$ is replaced with
$\widehat{\mathbb{W}}_{2,m}^{N}$ in (JKOτ,N).

  3. (iii)

If $\mu^{(0)}\in\mathscr{P}_{2}(\mathbb{R})$, then the continuous minimisation
problems (JKOτ) admit solutions, for every $n\in\mathbb{N}$.

We first need the following preliminary result, concerning the relation
between the weighted distance $\mathbb{W}_{p,m}$ and the classical Wasserstein
$\mathbb{W}_{p}$.

######  Lemma  (Comparison with Wasserstein distances).

Assume $\theta\in\emph{L}^{\infty}(0,M)$. Then it holds

$\displaystyle\mathbb{W}_{p,m}(\mu_{0},\mu_{1})^{p}\geq\frac{1}{\|\theta\|_{\infty}^{p-1}}\mathbb{W}_{p}(\mu_{0},\mu_{1})^{p},\quad\forall\mu_{0},\mu_{1}\in\mathscr{P}(\mathbb{R}).$
(2.17)

Similarly, for every $N\in\mathbb{N}$, we have that

$\displaystyle\widehat{\mathbb{W}}_{p,m}^{N}(\mu_{0},\mu_{1})^{p}\geq\frac{1}{\|\theta\|_{\infty}^{p-1}}\mathbb{W}_{p}(\mu_{0},\mu_{1})^{p},\quad\forall\mu_{0},\mu_{1}\in\mathcal{E}^{N}.$
(2.18)

###### Proof.

The lower bound 2.17 is trivial, whereas 2.18 follows from (4.1). ∎

###### Proof of Proposition 2.3.

(i), (ii): We prove the claim for (JKOτ,N) and $n=1$, the proof for a general
$n$ follows by iteration. In the discrete setting, we work with the cone
$\mathcal{K}_{N}$ which is a subset of the finite dimensional euclidean space
$\mathbb{R}^{N+1}$.

The lower semicontinuity with respect to $\mathbb{W}_{q}$ of
$\mathcal{J}_{\tau,N}$ follows from the one of $F$ and the continuity of
$d_{2,m}^{N}$, together with the closedness of $\mathcal{K}_{N}$. Thus, to
prove the existence of minimisers, it is enough to show that the family
$\left(\mathcal{J}_{\tau,N}\right)_{N}$ is suitably coercive in
$(\mathscr{P}_{q}(\mathbb{R}),\mathbb{W}_{q})$. From (A) and Lemma 2.3, by
means of a triangle inequality and the monotonicity of the $\ell$-norms on
$\mathcal{K}_{N}$ with $s<2$, we obtain for every
$\mu\in\mathscr{P}_{2}(\mathbb{R})$

$\displaystyle\begin{aligned}
\mathcal{J}_{\tau,N}(\mu)&\geq\frac{1}{\|\theta\|_{\infty}}\mathbb{W}_{2}(\mu,\mu_{N}^{(0)})^{2}-C\mathbb{E}_{\mu}|x|^{s}-D\\\
&\geq\frac{1}{2\|\theta\|_{\infty}}\mathbb{E}_{\mu}|x|^{2}-\frac{1}{\|\theta\|_{\infty}}\mathbb{E}_{\mu_{N}^{(0)}}|x|^{2}-C\mathbb{E}_{\mu}|x|^{s}-D\\\
&\geq\frac{1}{2\|\theta\|_{\infty}}\mathbb{E}_{\mu}|x|^{2}-C\Big{(}\mathbb{E}_{\mu}|x|^{2}\Big{)}^{s/2}-\frac{1}{\|\theta\|_{\infty}}\mathbb{E}_{\mu_{N}^{(0)}}|x|^{2}-D\\\
&\geq
c\mathbb{E}_{\mu}|x|^{2}-\frac{1}{\|\theta\|_{\infty}}\mathbb{E}_{\mu_{N}^{(0)}}|x|^{2}-D^{\prime}\,,\end{aligned}$
(2.19)

where $c$, $D^{\prime}\in\mathbb{R}^{+}$ only depends on $\theta$, $C$, and
$D$. Under our assumptions, this shows that the sublevels of
$\mathcal{J}_{\tau,N}$ are bounded in $\mathbb{W}_{2}$. In particular, for
every $t\in\mathbb{R}$, we have

$\displaystyle\Big{\\{}\mu\in\mathscr{P}(\mathbb{R})\ :\
\mathcal{J}_{\tau,N}(\mu)\leq
t\Big{\\}}\subset\Big{\\{}\mu=\mathcal{E}^{N}(x)\ :\ \|x\|_{2,N}\leq
C\;\text{and}\;N(\Delta x)_{i}\geq 1/M\Big{\\}}\,,$

for some $C\in\mathbb{R}_{+}$ depending on $t$. This easily implies coercivity
of $\mathcal{J}_{\tau,N}$ and thus the existence of a minimiser. (ii) is
trivial.

(iii): By similar computations as in (2.19), one can show the
$\mathbb{W}_{q}$-coercivity of $\mathcal{J}_{\tau}$ using (2.17). The lower
semicontinuity with respect to $\mathbb{W}_{q}$ follows from (B). ∎

######  Remark  (Uniqueness).

In general, no uniqueness is guaranteed. Whenever $F$, $F_{N}$ satisfy a
suitable strict convexity assumption, then the scheme is uniquely defined.
Nevertheless, we choose to work without such assumption, due to some
interesting, non-strictly convex examples of interest (e.g. traffic flow
evolutions).

######  Proposition  (Equicoercivity).

Let $\theta\in\emph{L}^{\infty}(0,M)$. Let $\big{(}\mu_{N}^{(0)}\big{)}_{N}$
be a sequence of initial measures with equibounded second moments, i.e.

$\displaystyle\sup_{N\in\mathbb{N}}\mathbb{E}_{\mu_{N}^{(0)}}|x|^{2}<\infty.$
(2.20)

Assume that the family $\left(F_{N}\right)_{N}$ satisfies the assumption in
(A) with some uniform (in $N\in\mathbb{N}$) constants $C,D\in\mathbb{R}_{+}$.
Then the functionals $\left(\mathcal{J}_{\tau,N}\right)_{N}$ are equicoercive
with respect to the $\mathbb{W}_{q}$ distance, for every $q\in[1,2)$: thus,
for every $\lambda>0$, the set
$\\{\mu\in\mathscr{P}(\mathbb{R})\,:\,\mathcal{J}_{\tau,N}(\mu)\leq\lambda\\}$
is compact in $(\mathscr{P}_{q}(\mathbb{R}),\mathbb{W}_{q})$.

###### Proof.

It follows directly from (2.19) and from the fact that the balls of
$\mathbb{W}_{2}$ are compact in $\mathbb{W}_{q}$, for every $q<2$, see e.g.
[AGS08, Lemma 5.1.7]. ∎

######  Remark  (Alternative coercivity).

The conclusion of Proposition 2.3 remains valid if we replaced the assumption
in (A) with $\left(F_{N}\right)_{N}$ being equicoercive w.r.t. some
$\mathbb{W}_{s}$, $s\in[1,+\infty)$ (and it holds with $q=s$).

######  Remark  (Lack of $\mathbb{W}_{2}$-coercivity).

[MANU-SIMO: controllate se vi piace e cancellate questo] In general, without
any additional assumption of $\left(F_{N}\right)_{N}$ (as discussed in the
previous remark), we are not able to show the $\mathbb{W}_{2}$-convergence of
the JKO schemes. At least for certain energies, e.g. the one in (2.14) with a
Lipschitz $f$, one may expect to be able to obtain such stronger convergence.
Unfortunately, new ideas seem to be required to obtain coercivity in this
topology of the energies involved in the JKO schemes, including a more
detailed analysis of the discrete geodesics on $\mathcal{K}_{N}$. We leave
this to future analysis.

##  3\. Statements of the main results

We are ready to present the main contributions of this work. The first one is
the convergence, in the sense of $\Gamma$-convergence as $N\to\infty$, of the
discrete trasport distances $\mathbb{W}_{p,m}^{N}$,
$\widehat{\mathbb{W}}_{p,m}^{N}$ to the continuous one $\mathbb{W}_{p,m}$,
with respect to a broad class of weak topologies on $\mathscr{P}(\mathbb{R})$.

######  Theorem 3.1 ($\Gamma$-convergence of $\mathbb{W}_{p,m}^{N}$ to
$\mathbb{W}_{p,m}$).

Fix $1<p<\infty$, $M\in\mathbb{R}_{+}$, and let $\phi_{p,m}$ be as in
Definition 2.1. Let $\theta$ be as in Definition 2.1 and assume that
$\theta\in\emph{L}^{\infty}(0,M)$. Then we have $\Gamma$-convergence of
$\mathbb{W}_{p,m}^{N}$ towards $\mathbb{W}_{p,m}$ with respect to the vague
and the $\mathbb{W}_{q}$-topology, for any $1<q<+\infty$. More precisely, we
have the following bounds:

  1. (_LI_)

For every $\mu_{0}^{N},\mu_{1}^{N}\to\mu_{0},\mu_{1}$ vaguely in
$\mathscr{P}(\mathbb{R})$ as $N\rightarrow+\infty$, we have that

$\displaystyle\liminf_{N\rightarrow+\infty}\mathbb{W}_{p,m}^{N}(\mu_{0}^{N},\mu_{1}^{N})\geq\mathbb{W}_{p,m}(\mu_{0},\mu_{1}).$

  2. (_LS_)

For any given $1<q<+\infty$, for every
$\mu_{0},\mu_{1}\in\mathscr{P}_{q}(\mathbb{R})$, there exist
$\tilde{\mu}_{0}^{N},\tilde{\mu}_{1}^{N}\rightarrow\mu_{0},\mu_{1}$ in
$(\mathscr{P}_{q}(\mathbb{R}),\mathbb{W}_{q})$ as $N\rightarrow+\infty$ and
such that

$\displaystyle\limsup_{N\rightarrow+\infty}\mathbb{W}_{p,m}^{N}(\tilde{\mu}_{0}^{N},\tilde{\mu}_{1}^{N})\leq\mathbb{W}_{p,m}(\mu_{0},\mu_{1}).$

The very same conclusions hold if we replace $\mathbb{W}_{p,m}^{N}$ with
$\widehat{\mathbb{W}}_{p,m}^{N}$.

The second main result concerns the corresponding gradient-flow structures: we
have that, at every time-scale $\tau>0$, the sequences
$\left(\mu_{\tau,N}^{(n)}\right)_{n}$ defined via (JKOτ,N) converge as
$N\to\infty$ to the corresponding continuous (in space) steps of (JKOτ).

######  Theorem 3.2 (JKO-convergence in $\mathbb{W}_{q}$, for $q<2$).

Consider the same assumptions of Theorem 3.1. Fix $\tau\in\mathbb{R}_{+}$,
$q\in[1,2)$, and let $\left(F_{N}\right)_{N}$, $F$ be functionals satisfying
(A),(B), and

$\displaystyle\lim_{N\to\infty}F_{N}(\mu_{N})=F(\mu)\,,\quad\text{for every}\
\ \mu_{N}\to\mu\ \ \text{in}\ (\mathscr{P}_{q}(\mathbb{R}),\mathbb{W}_{q})\,.$
(3.1)

Let then $\left(\mu_{N}^{(0)}\right)_{N}$ be a sequence satisfying (2.20), and
suppose that $\mu_{N}^{(0)}\to\mu^{(0)}$ in
$(\mathscr{P}_{q}(\mathbb{R}),\mathbb{W}_{q})$. Then we have that

  1. (i)

$\mathcal{J}_{\tau,N}$ $\Gamma$-converge to $\mathcal{J}_{\tau}$ with respect
to the $\mathbb{W}_{q}$-topology, as $N\to\infty$.

  2. (ii)

Up to subsequences, $\mu_{\tau,N}^{(n)}\to\mu_{\tau}^{(n)}$ in
$(\mathscr{P}_{q}(\mathbb{R}),\mathbb{W}_{q})$ as $N\to\infty$, for all
$n\in\mathbb{N}$.

######  Remark  (The case $F_{N}\equiv F$).

The easiest example of a family satisfying (3.1) is when $F_{N}:=F$, for some
$F$ which is $\mathbb{W}_{q}$-continuous (thus (B) holds as well).

##  4\. Approximation of finite-energy solutions of $\mathsf{CE}$

In this chapter we provide some approximation results (Proposition 4.2) for
solutions to the continuity equation with bounded action. These techniques
allow us to obtain "good" recovery sequences for $\mathbb{W}_{p,m}$, which
turn out to be crucial in the proof of Theorem 3.1 and Theorem 3.2. We start
recalling some basic notions.

######  Definition  (Pseudo-inverse).

Fix $\mu\in\mathscr{P}(\mathbb{R})$. The pseudo-inverse or quantile function
$X_{\mu}:(0,1)\to\mathbb{R}$ of $\mu$ is the non-decreasing, right-continuous
map given by

$\displaystyle X_{\mu}(z)=\inf\left\\{a\in\mathbb{R}\ :\
\mu(-\infty,a)>z\right\\}\,.$

######  Remark  (Law of the pseudo-inverse).

It is easy to check that, by construction, the law of $X_{\mu}$ coincides with
$\mu$ itself, i.e. $(X_{\mu})_{\\#}\mathscr{L}^{1}=\mu$. In fact, it is the
unique right continuous monotone increasing map from $(0,1)$ to $\mathbb{R}$
with such property.

######  Remark  (Wasserstein distance on the real line).

A remarkable property of the one-dimensional optimal transport problem is the
possibility of computing the $p$-Wasserstein distance between two probability
measures via the $\text{L}^{p}$-distance of the corresponding pseudo-inverses.
Precisely, for $p\geq 1$, and every $\mu,\nu\in\mathscr{P}_{p}(\mathbb{R})$,
it holds

$\displaystyle\mathbb{W}_{p}(\mu,\nu)=\|X_{\mu}-X_{\nu}\|_{\text{L}^{p}(0,1)}\,.$
(4.1)

######  Remark  (Monotonocity and continuity of the pseudo-inverse).

Whenever $\mu$ does not charge single points, $X_{\mu}$ is an increasing map
(and in particular, injective). In general, even for absolute continuous
measures (with respect to $\mathscr{L}^{1}$) with smooth densities, the
corresponding pseudo-inverse might fail to be continuous. If the support of
$\mu$ is connected, then $X_{\mu}$ is continuous.

###  4.1. Reduction to compact support

In this section, we discuss a procedure to reduce measures to compactly
supported ones, and how this effects the value of the associated
$\phi_{p,m}$-energies.

######  Definition  (Compactification of a measure).

Fix $\mu\in\mathscr{P}(\mathbb{R})$ an atomless probability measure,
$\nu\in\mathcal{M}(\mathbb{R})$. For every $\varepsilon>0$, we define the
measure $\mathcal{C}_{\varepsilon}(\nu|\mu)\in\mathscr{P}(\mathbb{R})$ as

$\displaystyle\mathcal{C}_{\varepsilon}(\nu|\mu):=\frac{1}{1-2\varepsilon}\nu\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits I_{\varepsilon}[\mu]\,,\quad
I_{\varepsilon}[\mu]:=\Big{[}X_{\mu}(\varepsilon),X_{\mu}(1-\varepsilon)\Big{]}\,.$
(4.2)

We use the shorthand notation
$\mathcal{C}_{\varepsilon}\mu:=\mathcal{C}_{\varepsilon}(\mu|\mu)\in\mathscr{P}(\mathbb{R})$.

The next lemma provides a control of the error in applying the operator
$\mathcal{C}_{\varepsilon}$.

######  Lemma  (Compactification error).

Fix $\mu\in\mathscr{P}_{p}(\mathbb{R})$ an atomless probability measure,
$\varepsilon>0$. Then for every $q\leq p$,

$\displaystyle\emph{diam}(I_{\varepsilon}[\mu])^{p-q}\mathbb{W}_{q}(\mu,\mathcal{C}_{\varepsilon}\mu)^{q}\leq
C\int_{I_{\varepsilon}[\mu]^{c}}|x-x_{\mu}|^{p}\,\mathrm{d}\mu(x)\,,$ (4.3)

for some constant $C>0$ depending only on $p$, where
$x_{\mu}:=X_{\mu}\big{(}\frac{1}{2}\big{)}$ is the median of $\mu$. If $q=p$,
one can take $C=1$.

###### Proof.

Without loss of generality, we can assume $x_{\mu}=0$, otherwise we change
variables $x\mapsto x-x_{\mu}$. By density arguments, we can further assume
that $\mu$ is absolutely continuous with respect to $\mathscr{L}^{1}$. We
split the proof in two steps.

Step 1: the case $q=p$. Pick any admissible map $\tilde{T}$ between
$\mu\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits I_{\varepsilon}[\mu]^{c}$ and
$2\varepsilon\mathcal{C}_{\varepsilon}\mu$ (i.e.
$\tilde{T}_{\\#}(\mu\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits
I_{\varepsilon}[\mu]^{c})=2\varepsilon\mathcal{C}_{\varepsilon}\mu$).
Moreover, thanks to $x_{\mu}=0$, we can also assume that $\tilde{T}(x)x\geq
0$, for every $x\in\mathbb{R}$. Then, we define the map

$\displaystyle T:\mathbb{R}\to I_{\varepsilon}[\mu]\,,\quad
T(x)=\begin{cases}x,&\text{if }x\in I_{\varepsilon}[\mu]\,,\\\
\tilde{T}(x),&\text{otherwise}\,.\end{cases}$

Note that $T$ is an admissible map between $\mu$ and
$\mathcal{C}_{\varepsilon}\mu$. Indeed for every measurable map
$\varphi:\mathbb{R}\to\mathbb{R}$, we have that

$\displaystyle\int\varphi(T(x))\,\mathrm{d}\mu(x)$
$\displaystyle=\int_{I_{\varepsilon}[\mu]}\varphi\,\mathrm{d}\mu+\int_{I_{\varepsilon}[\mu]^{c}}\varphi(\tilde{T}(x))\,\mathrm{d}\mu(x)$
$\displaystyle=(1-2\varepsilon)\int\varphi\,\mathrm{d}\mathcal{C}_{\varepsilon}\mu+2\varepsilon\int\varphi\,\mathrm{d}\mathcal{C}_{\varepsilon}\mu=\int\varphi\,\mathrm{d}\mathcal{C}_{\varepsilon}\mu\,.$

Computing the transport cost with respect to $|\cdot|^{p}$, we obtain that

$\displaystyle
W_{p}^{p}(\mu,\mathcal{C}_{\varepsilon}\mu)\leq\int|x-T(x)|^{p}\,\mathrm{d}\mu(x)=\int_{I_{\varepsilon}[\mu]^{c}}|x-\tilde{T}(x)|^{p}\,\mathrm{d}\mu(x)\leq\int_{I_{\varepsilon}[\mu]^{c}}|x|^{p}\,\mathrm{d}\mu(x)\,,$

where at last we used $\tilde{T}(x)x\geq 0$, which concludes the proof of
(4.3).

Step 2: $q<p$. Fix $\mu\in\mathscr{P}_{p}(\mathbb{R})$ and $q<p$. Then
applying the first step of the proof and using that
$\mu\in\mathscr{P}_{q}(\mathbb{R})$ we deduce

$\displaystyle\mathbb{W}_{q}(\mu,\mathcal{C}_{\varepsilon}\mu)^{q}\leq\int_{I_{\varepsilon}[\mu]^{c}}|x|^{q}\,\mathrm{d}\mu(x)\,.$
(4.4)

Consider any transport map $S:I_{\varepsilon}[\mu]^{c}\to
I_{\varepsilon}[\mu]^{c}$ satisfying

$\displaystyle S_{\\#}(\mu\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits
I_{\varepsilon}[\mu]^{c})=\mu\mathop{\hbox{\vrule
height=7.0pt,width=0.5pt,depth=0.0pt\vrule
height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits
I_{\varepsilon}[\mu]^{c}\quad\text{and}\quad S(x)x\leq 0\,,\quad x\in
I_{\varepsilon}[\mu]^{c}\,.$ (4.5)

One can for example consider any map that sends the $\varepsilon$ mass of
$\mu$ which lives in $I_{\varepsilon}[\mu]^{c}\cap\\{x\leq 0\\}$ into
$I_{\varepsilon}[\mu]^{c}\cap\\{x\geq 0\\}$ and viceversa. In particular, we
observe that $|x|\leq|x-S(x)|$, for $x\in I_{\varepsilon}[\mu]^{c}$. Together
with (4.4), this yields

$\displaystyle\text{diam}(I_{\varepsilon}[\mu])^{p-q}\mathbb{W}_{q}(\mu,\mathcal{C}_{\varepsilon}\mu)^{q}$
$\displaystyle\leq\text{diam}(I_{\varepsilon}[\mu])^{p-q}\int_{I_{\varepsilon}[\mu]^{c}}|x-S(x)|^{q}\,\mathrm{d}\mu(x)$
$\displaystyle=\int_{I_{\varepsilon}[\mu]^{c}}\bigg{(}\frac{\text{diam}(I_{\varepsilon}[\mu])}{|x-S(x)|}\bigg{)}^{p-q}|x-S(x)|^{p}\,\mathrm{d}\mu(x)$
$\displaystyle\leq\int_{I_{\varepsilon}[\mu]^{c}}|x-S(x)|^{p}\,\mathrm{d}\mu(x)\,,$

where at last we used that for $x\in I_{\varepsilon}[\mu]^{c}$ we have
$|x-S(x)|\geq\text{diam}(I_{\varepsilon}[\mu])$. By (4.5)

$\displaystyle\int_{I_{\varepsilon}[\mu]^{c}}|x-S(x)|^{p}\,\mathrm{d}\mu(x)$
$\displaystyle\leq
2^{p-1}\int_{I_{\varepsilon}[\mu]^{c}}\big{(}|x|^{p}+|S(x)|^{p}\big{)}\,\mathrm{d}\mu(x)=2^{p}\int_{I_{\varepsilon}[\mu]^{c}}|x|^{p}\,\mathrm{d}\mu(x)\,,$

we conclude the proof. ∎

The operator $\mathcal{C}_{\varepsilon}$ is constructed in such a way to
preserve solutions to $\mathsf{CE}$.

######  Lemma  (Compactification of solutions to $\mathsf{CE}$).

Let $(\mu_{t},\nu_{t})\in\mathsf{CE}$ with $C^{1}$ (space-time) densities
$(\rho_{t},j_{t})_{t}$ with respect to $\mathscr{L}^{1}$. Assume also that
$\rho_{t}>0$ for every $t\in[0,1]$. For every $\varepsilon>0$, consider the
operator $\mathcal{C}_{\varepsilon}$ as defined in Definition 4.1. Then

$\big{(}\mathcal{C}_{\varepsilon}\mu_{t},\mathcal{C}_{\varepsilon}(\nu_{t}|\mu_{t})\big{)}\in\mathsf{CE}\,.$

###### Proof.

Fix any $\varphi\in C_{c}(\mathbb{R})$. For the sake of simplicity, set
$a_{t}:=X_{\mu_{t}}(\varepsilon)$, $b_{t}:=X_{\mu_{t}}(1-\varepsilon)$.
Firstly, for $t\in(0,1)$, set $F_{\mu}(x):=\mu(-\infty,x)$ and note that by
definition of quantile function one has that

$\displaystyle F_{\mu_{t}}\big{(}X_{\mu_{t}}(z)\big{)}=z\,,\quad\forall
z\in(0,1)\,.$

Using that $(\mu_{t},\nu_{t})\in\mathsf{CE}$, it is readily verified that
$\frac{\,\mathrm{d}}{\,\mathrm{d}t}F_{\mu_{t}}(x)=-j_{t}(x)$, for every
$t\in[0,1]$, $x\in\mathbb{R}$. Note that $(t,x)\mapsto F_{\mu_{t}}(x)$ is
$C^{1}$ in space and time. By applying the implicit function theorem, we infer
that $t\mapsto X_{\mu_{t}}(z)$ is differentiable for every $z\in(0,1)$ and it
holds

$\displaystyle-
j_{t}\big{(}X_{\mu_{t}}(z)\big{)}+\rho_{t}\big{(}X_{\mu_{t}}(z)\big{)}\frac{\,\mathrm{d}}{\,\mathrm{d}t}X_{\mu_{t}}(z)=0\,.$
(4.6)

Hence, $a_{t}$, $b_{t}$ both solves the differential equation
$\rho_{t}(c_{t})\dot{c}_{t}=j_{t}(c_{t})$. Therefore, we have that

$\displaystyle(1-2\varepsilon)\frac{\,\mathrm{d}}{\,\mathrm{d}t}\int\varphi\,\mathrm{d}\mathcal{C}_{\varepsilon}\mu_{t}$
$\displaystyle=\frac{\,\mathrm{d}}{\,\mathrm{d}t}\int_{I_{\varepsilon}[\mu_{t}]}\varphi\,\mathrm{d}\mu_{t}$
$\displaystyle=\int_{I_{\varepsilon}[\mu_{t}]}\varphi\dot{\rho}_{t}\,\mathrm{d}x+\dot{b}_{t}\varphi(b_{t})\rho_{t}(b_{t})-\dot{a}_{t}\varphi(a_{t})\rho_{t}(a_{t})$
$\displaystyle=-\int_{I_{\varepsilon}[\mu_{t}]}\varphi\nabla\cdot
j_{t}\,\mathrm{d}x+\varphi j_{t}\big{|}_{\partial I_{\varepsilon}[\mu_{t}]}$
$\displaystyle=\int_{I_{\varepsilon}[\mu_{t}]}\nabla\varphi\cdot
j_{t}=(1-2\varepsilon)\int\nabla\varphi\,\mathrm{d}\mathcal{C}_{\varepsilon}(\nu_{t}|\mu_{t})\,.\qed$

In view of the error estimate of Lemma 4.1, we also prove the following
result.

######  Lemma  (Uniform bounds and stability properties).

Fix $\varepsilon>0$, $\left(\mu_{n}\right)_{n}\subset\mathscr{P}(\mathbb{R})$
a sequence of atomless measures.

  1. (i)

If $\left(\mu_{n}\right)_{n}$ is tight in $\mathscr{P}(\mathbb{R})$, then for
$s\in(0,1)$

$\displaystyle\sup_{n\in\mathbb{N}}\emph{\text{diam}}(I_{\varepsilon}[\mu_{n}])<\infty\quad\text{and}\quad\sup_{n\in\mathbb{N}}\big{|}X_{\mu_{n}}(s)\big{|}<\infty\,.$
(4.7)

  2. (ii)

For every $\mu,\nu\in\mathscr{P}_{p}(\mathbb{R})$ atomless measures, it holds

$\displaystyle\left|\int_{I_{\varepsilon}[\mu]^{c}}|x|^{p}\,\mathrm{d}\mu(x)-\int_{I_{\varepsilon}[\nu]^{c}}|x|^{p}\,\mathrm{d}\nu(x)\right|^{\frac{1}{p}}\leq\Big{\|}X_{\mu}-X_{\nu}\Big{\|}_{\emph{L}^{p}([\varepsilon,1-\varepsilon]^{c})}.$

In particular, if $\mu_{n}\to\mu$ in $\mathbb{W}_{p}$, then

$\displaystyle\lim_{n\to\infty}\int_{I_{\varepsilon}[\mu_{n}]^{c}}|x|^{p}\,\mathrm{d}\mu_{n}(x)=\int_{I_{\varepsilon}[\mu]^{c}}|x|^{p}\,\mathrm{d}\mu(x)\,.$
(4.8)

###### Proof.

(i): We proceed by contradiction: assume that (up to subsequence)

$\displaystyle\lim_{n\to\infty}\text{diam}(I_{\varepsilon}[\mu_{n}])=+\infty\,.$
(4.9)

By tightness, we know that there exists a compact set
$K_{\varepsilon}\Subset\mathbb{R}$ such that
$\sup_{n}\mu_{n}(K_{\varepsilon}^{c})<\varepsilon$. On the other hand, by
definition of $I_{\varepsilon}$, we know that $I_{\varepsilon}[\mu_{n}]^{c}$
is a disjoint union of unbounded intervals $I_{n}^{-}$ and $I_{n}^{+}$, each
of $\mu_{n}$-mass equals to $\varepsilon$. From (4.9), we deduce that, for $n$
sufficiently big, either $I_{n}^{-}$ or $I_{n}^{+}$ must be subsets of
$K_{\varepsilon}^{c}$, which gives us the contradiction. A similar argument
shwos the uniform bound on the quantiles.

(ii): Fix $\mu\in\mathscr{P}_{p}(\mathbb{R})$. By definition of pseudo-inverse
we have

$\displaystyle\int_{I_{\varepsilon}[\mu]^{c}}|x|^{p}\,\mathrm{d}\mu(x)=\int_{0}^{\varepsilon}\Big{|}X_{\mu}(z)\Big{|}^{p}\,\mathrm{d}z+\int_{1-\varepsilon}^{1}\Big{|}X_{\mu}(z)\Big{|}^{p}\,\mathrm{d}z\,.$

The assumption on the $\mathbb{W}_{p}$ convergence on $\mu_{n}$ is equivalent
to the fact that $X_{\mu_{n}}\to X_{\mu}$ in $\text{L}^{p}(0,1)$ (see Remark
4), and thus $(ii)$ follows. ∎

###  4.2. The approximation procedure

We introduce the following

######  Definition  (Regular curves).

Let $(\mu_{t},\nu_{t})\in\mathsf{CE}$ be a solution of the continuity equation
in $\mathbb{R}^{d}$. We say that $(\mu_{t},\nu_{t})$ is a _regular curve_ if
the following properties hold

  1. i)

for all $t\in[0,1]$, $\mu_{t}$ and $\nu_{t}$ have compact and connected
support, $\mu_{t},\nu_{t}\ll\mathscr{L}^{1}$ and their respective densities
densities $\rho_{t}$, $j_{t}$ are $C^{\infty}$ in their support

  2. ii)

there exist a compact set $K$ and two constants $0<l\leq\tilde{M}<M$ such that

$\operatorname{supp}(\rho_{t}),\,\operatorname{supp}(j_{t})\subset
K\quad\text{and}\quad l<\rho_{t}(x)<\tilde{M}\quad\text{for all
$x\in\operatorname{supp}(\rho_{t})$ and $t\in[0,1]$.}$

We are finally ready to state and prove the main result of this section.

######  Proposition  (Approximation with regular curves).

Let $p\in(1,\infty)$, $M\in\mathbb{R}_{+}$, and $\phi_{p,m}$ be as in
Definition 2.1. Let $(\mu_{t},\nu_{t})\in\mathsf{CE}$ satisfying

$\displaystyle
E:=\int_{0}^{1}\Phi_{p,m}(\mu_{t},\nu_{t})\,\mathrm{d}t<\infty\,,\quad\sup_{t\in[0,1]}\|\rho_{t}\|_{\emph{L}^{\infty}(\mathbb{R})}=:\bar{M}<M\,\,.$
(4.10)

Then, for every $\eta>0$, there exists
$(\tilde{\mu}_{t},\tilde{\nu}_{t})\in\mathsf{CE}$ such that

  1. (i)

$\displaystyle\int_{0}^{1}\Phi_{p,m}(\tilde{\mu}_{t},\tilde{\nu}_{t})\,\mathrm{d}t\leq\int_{0}^{1}\Phi_{p,m}(\mu_{t},\nu_{t})\,\mathrm{d}t+\eta\,.$

  2. (ii)

$(\tilde{\mu}_{t},\tilde{\nu}_{t})$ are regular curves in the sense of
Definition 4.2, for some compact set $K\Subset\mathbb{R}$ and $l>0$ depending
only on $\eta$ and $(\mu_{t})_{t\in[0,1]}$, while $\tilde{M}$ only depends on
$\bar{M}$.

  3. (iii)

For $q\leq p$ and $t\in[0,1]$, we have that

$\displaystyle\max\left\\{\Big{(}\emph{diam}(\operatorname{supp}(\tilde{\mu}_{t}))^{p-q}\vee
1\Big{)}\mathbb{W}_{q}(\mu_{t},\tilde{\mu}_{t})^{q}\,,\int_{\operatorname{supp}(\tilde{\mu}_{t})^{c}}(|x|^{p}+1)\,\mathrm{d}\mu_{t}(x)\right\\}\leq\eta\,.$

###### Proof.

We split the proof into several steps.

Step 1: Space-time regularisation. We extend $\mu_{t},\nu_{t}$ as
$\mu_{t}=\mu_{0}$ for $t<0$, $\mu_{t}=\mu_{1}$ for $t>1$, and $\nu_{t}=0$ for
$t\in(-\infty,0)\cup(1,+\infty)$. Given two smooth functions
$f,g:\mathbb{R}\to\mathbb{R}_{+}$ such that

$\displaystyle\int
f(x)\,\mathrm{d}x=1\,,\quad\int|x|^{p}f(x)\,\mathrm{d}x<\infty\,,\quad f>0\
\text{on}\ \mathbb{R}\,,$ (4.11)

and the same for $g$. We define for $\tau>0$ the rescaled kernel on
$\mathbb{R}\times\mathbb{R}$ by setting
$h_{\tau}(t,x):=\tau^{-2}f(\tau^{-1}t)g(\tau^{-1}x)$, for
$(t,x)\in\mathbb{R}\times\mathbb{R}$. Note that $h_{\tau}$ has integral $1$
and full support, for every $\tau>0$. We then consider
$({\bm{\mu}},{\bm{\nu}})$ as in (2.2) and define

$\displaystyle\hat{\bm{\mu}}:=h_{\tau}*{\bm{\mu}}\,,\quad\hat{\bm{\nu}}:=h_{\tau}*{\bm{\nu}}\,,$
(4.12)

where the convolution has to be intended as in Section 2.1.1(II). In
particular,

$\displaystyle\int_{0}^{1}\Phi_{p,m}(\hat{\mu}_{t},\hat{\nu}_{t})\,\mathrm{d}t\leq\int_{0}^{1}\Phi_{p,m}(\mu_{t},\nu_{t})\,\mathrm{d}t\,.$
(4.13)

Moreover, $\hat{\bm{\mu}}$, $\hat{\bm{\nu}}$ have densities
$\hat{\rho}_{\cdot}(\cdot),\hat{j}_{\cdot}(\cdot)\in
C^{\infty}([0,1]\times\mathbb{R})$. Thanks to the positivity of $h_{\tau}$ and
$\int h_{\tau}=1$, we have that $\hat{\rho}_{t}(x)\in(0,\bar{M})$, for every
$x\in\mathbb{R}$, $t\in[0,1]$.

Step 2: Reduction to compact supports. Pick
$\bar{\varepsilon}=\varepsilon(\bar{M})\in(0,1)$ so that

$\displaystyle\tilde{M}:=(1-2\bar{\varepsilon})^{-1}\bar{M}<M\,.$ (4.14)

For $\varepsilon\in(0,\bar{\varepsilon})$ to be suitably chosen and
$t\in[0,1]$, we define

$\displaystyle\tilde{\mu}_{t}:=\mathcal{C}_{\varepsilon}\hat{\mu}_{t}\,,\quad\tilde{\nu}_{t}:=\mathcal{C}_{\varepsilon}(\hat{\nu}_{t}|\hat{\mu}_{t})\,,$

where $\mathcal{C}_{\varepsilon}$ is defined in Definition 4.1. We denote by
$\tilde{\rho}_{t}$, $\tilde{j}_{t}$ the corresponding densities. Step 3:
Energy estimates. Thanks to Lemma 4.1 and the fact that
$(\hat{\mu}_{t},\hat{\nu}_{t})\in\mathsf{CE}$, we know that
$(\tilde{\mu}_{t},\tilde{\nu}_{t})\in\mathsf{CE}$. By the convexity and
homogeneity of $\phi_{p,m}$, we have that

$\displaystyle\phi_{p,m}(\tilde{\rho}_{t},\tilde{j}_{t})=\frac{1}{(1-2\varepsilon)^{p}}\phi_{p,m}\Big{(}\frac{1}{1-2\varepsilon}\hat{\rho}_{t},\hat{j}_{t}\Big{)}\leq\frac{1+\varepsilon\bar{C}}{(1-2\varepsilon)^{p}}\phi_{p,m}(\hat{\rho}_{t},\hat{j}_{t})\,,$

for some $\bar{C}=\bar{C}(\bar{M},\phi_{p,m})<\infty$ depending only on
$\bar{M}$ and $\phi_{p,m}$, where at last we used Lemma 7, thanks to (4.14).
In particular, by (4.10), (4.13), and the latter bound,

$\displaystyle\int_{0}^{1}\Phi_{p,m}(\tilde{\mu}_{t},\tilde{\nu}_{t})\,\mathrm{d}t\leq\frac{1+\varepsilon\bar{C}}{(1-2\varepsilon)^{p}}\int_{0}^{1}\Phi_{p,m}(\mu_{t},\nu_{t})\,\mathrm{d}t\leq\int_{0}^{1}\Phi_{p,m}(\mu_{t},\nu_{t})\,\mathrm{d}t+\bar{C}E\varepsilon\,.$
(4.15)

The upper bound (i) then follows directly by choosing $\varepsilon>0$ small
enough.

Step 4: Regularity of the approximation. By construction and (4.14), we have
that

$\displaystyle\begin{gathered}\big{\\{}t\mapsto\tilde{\rho}_{t},\,\tilde{j}_{t}\in
C^{\infty}(\operatorname{supp}(\tilde{\mu}_{t}))\big{\\}}\in
C^{\infty}([0,1])\,,\quad\operatorname{supp}\tilde{\mu}_{t}=I_{\varepsilon}[\hat{\mu}_{t}]\subset\mathbb{R}\,,\\\
0<l_{\tau}\leq\tilde{\rho}_{s}(x)\leq\tilde{M}<M\,,\quad\forall
x\in\operatorname{supp}(\tilde{\mu}_{s})\,,\end{gathered}$ (4.18)

for every $s\in[0,1]$, for some $l_{\tau}>0$ depending only on $\tau$, $f$,
and $(\mu_{t})_{t\in[0,1]}$ (but not on $\varepsilon$, since
$(1-2\varepsilon)^{-1}>1$). Recall the definition of $I_{\varepsilon}[\cdot]$
as given in (4.2). In particular, (ii) follows with a compact set
$K=K_{\tau,\varepsilon}\subset\mathbb{R}$ only depending on $\varepsilon$,
$\tau$, and $(\mu_{t})_{t\in[0,1]}$.

Step 5: Estimate of the distance. We are left with the proof of (iii). By
triangle inequality, we have that
$\mathbb{W}_{q}(\mu_{t},\tilde{\mu}_{t})\leq\mathbb{W}_{q}(\mu_{t},\hat{\mu}_{t})+\mathbb{W}_{q}(\hat{\mu}_{t},\tilde{\mu}_{t})\,.$
Set $f_{\tau}:=\tau^{-1}f(\tau^{-1}\cdot)$. By Remark 2.1 with $q\leq p$ and
(4.11), for every $s,t\in[0,1]$ we have that

$\displaystyle\mathbb{W}_{q}(\mu_{t},f_{\tau}*\mu_{s})\leq\mathbb{W}_{q}(\mu_{t},\mu_{s})+\mathbb{W}_{q}(\mu_{s},f_{\tau}*\mu_{s})\leq
E^{\prime}|t-s|+\tau(\mathbb{E}_{f}|x|^{q})^{\frac{1}{q}}\,,$

for some $E^{\prime}=E^{\prime}(E,{\bm{\mu}},\theta)$, where the second
inequality follows using the admissible transport plan
$\,\mathrm{d}\pi(x,y):=f_{\tau}(y-x)\,\mathrm{d}y\,\mathrm{d}\mu_{s}(x)$.
Using the convexity of $\mathbb{W}_{q}$ and the latter bound, by (4.11)

$\displaystyle\mathbb{W}_{q}(\mu_{t},\hat{\mu}_{t})\leq\int
g_{\tau}(t-s)\mathbb{W}_{q}(\mu_{t},f_{\tau}*\mu_{s})\,\mathrm{d}s\leq
C\tau\,,$

for some $C\in\mathbb{R}_{+}$ depending on $f$, $g$, and $E^{\prime}$. By
Lemma 4.1, we get that

$\displaystyle\text{diam}(I_{\varepsilon}[\hat{\mu}_{t}])^{p-q}\mathbb{W}_{q}(\hat{\mu}_{t},\mathcal{C}_{\varepsilon}\hat{\mu}_{t})^{q}\leq\int_{I_{\varepsilon}[\hat{\mu}_{t}]^{c}}|x-x_{\hat{\mu}_{t}}|^{p}\,\mathrm{d}\hat{\mu}_{t}(x)\,.$

Collecting the various estimates, we end up with

$\displaystyle\text{diam}(I_{\varepsilon}[\hat{\mu}_{t}])^{p-q}\mathbb{W}_{q}(\mu_{t},\tilde{\mu}_{t})^{q}\leq\bigg{(}\text{diam}(I_{\varepsilon}[\hat{\mu}_{t}])^{\frac{p}{q}-1}C\tau+\left(\int_{I_{\varepsilon}[\hat{\mu}_{t}]^{c}}|x-x_{\hat{\mu}_{t}}|^{p}\,\mathrm{d}\hat{\mu}_{t}(x)\right)^{\frac{1}{q}}\bigg{)}^{q}\,.$

Employing Lemma 4.1, together with Remark 2.1, and taking the limit as
$\tau\to 0$, for every fixed $\varepsilon>0$ and $t\in[0,1]$ we find that

$\displaystyle\limsup_{\tau\to
0}\text{diam}(I_{\varepsilon}[\hat{\mu}_{t}])^{p-q}\mathbb{W}_{q}(\mu_{t},\tilde{\mu}_{t})^{q}$
$\displaystyle\leq
C^{\prime}\left(\int_{I_{\varepsilon}[\mu_{t}]^{c}}|x|^{p}\,\mathrm{d}\mu_{t}(x)+\varepsilon\right)$
$\displaystyle=C^{\prime}\left(\int_{0}^{\varepsilon}\Big{|}X_{\mu_{t}}(z)\Big{|}^{p}\,\mathrm{d}z+\int_{1-\varepsilon}^{1}\Big{|}X_{\mu_{t}}(z)\Big{|}^{p}\,\mathrm{d}z+\varepsilon\right)\,,$

for some $C^{\prime}\in\mathbb{R}_{+}$ independent of $\varepsilon$ and $t$.
The latter term is infinitesimal as $\varepsilon\to 0$ uniformly in time,
thanks to Remark 2.1 and (4.1). Therefore, we reach any threshold $\eta>0$ if
we suitably choose first $\tau$ and then $\varepsilon$ small enough.

Step 5: Estimate of the tail. We are left to prove we can choose
$\tau,\varepsilon>0$ such that

$\displaystyle\sup_{t\in[0,1]}\int_{\operatorname{supp}(\tilde{\mu}_{t})^{c}}|x|^{p}\,\mathrm{d}\mu_{t}(x)\leq\eta\,,$

for any given threshold $\eta>0$. We claim that
$I_{2\varepsilon}[\mu_{t}]\subset I_{\varepsilon}[\hat{\mu}_{t}]$, for $\tau$
sufficiently small. This would allow us to conclude, as in the proof of Step
4, that

$\displaystyle\sup_{t\in[0,1]}\int_{I_{\varepsilon}[\hat{\mu}_{t}]^{c}}|x|^{p}\,\mathrm{d}\mu_{t}(x)\leq\sup_{t\in[0,1]}\int_{I_{2\varepsilon}[\mu_{t}]^{c}}|x|^{p}\,\mathrm{d}\mu_{t}(x)\to
0$

as $\varepsilon\to 0$. More precisely, we shall prove that

$\displaystyle I_{\varepsilon+\delta_{\tau}}[\mu_{t}]\subset
I_{\varepsilon}[\hat{\mu}_{t}],\quad\forall t\in[0,1],$ (4.19)

where $\delta_{\tau}$ can be explicitly chosen depending only on $\tau$, $p$,
$M$, and the curve ${\bm{\mu}}$ as

$\displaystyle\delta_{\tau}:=\Big{(}\sup_{t\in[0,1]}\mathbb{W}_{p}(\mu_{t},\hat{\mu}_{t})^{\frac{p}{p+1}}\Big{)}\Big{(}(p+1)M^{p}\Big{)}^{\frac{1}{p+1}}.$

Using the fact that $\delta_{\tau}\to 0$ as $\tau\to 0$, the claim is proved.
To prove (4.19), suppose that $\delta>0$ is such that
$X_{\mu_{t}}(\varepsilon+\delta)<X_{\hat{\mu}_{t}}(\varepsilon)$, for some
$t\in[0,1]$. Using the fact that $\rho_{t}\leq M$, we obtain

$\displaystyle
X_{\mu_{t}}(\varepsilon+\delta)>X_{\mu_{t}}(\varepsilon+z)+M^{-1}(\delta-z),\quad\forall
z\in[0,\delta].$

Hence, by assumption on $\delta$, for $z\in[0,\delta]$ we have that

$X_{\hat{\mu}_{t}}(\varepsilon+z)-X_{\mu_{t}}(\varepsilon+z)>X_{\mu_{t}}(\varepsilon+\delta)-X_{\mu_{t}}(\varepsilon+z)>M^{-1}(\delta-z)\,.$

By Remark 4, we conclude that

$\displaystyle\mathbb{W}_{p}(\mu_{t},\hat{\mu}_{t})^{p}\geq\int_{0}^{\delta}\Big{|}X_{\mu_{t}}(\varepsilon+z)-X_{\hat{\mu}_{t}}(\varepsilon+z)\Big{|}^{p}\,\mathrm{d}z>\delta^{p+1}(p+1)^{-1}M^{-p},$

which shows that $\delta<\delta_{\tau}$. This shows that
$X_{\mu_{t}}(\varepsilon+\delta_{\tau})\geq X_{\hat{\mu}_{t}}(\varepsilon)$.
In a similar way, one can prove that
$X_{\mu_{t}}(1-\varepsilon-\delta_{\tau})\leq
X_{\hat{\mu}_{t}}(1-\varepsilon)$, and thus proving (4.19). ∎

##  5\. Proof of the main result: $\Gamma$-convergence.

In this section we prove Theorem 3.1. We start showing that for any sequence
$\left(x^{N}\in\mathcal{K}^{N}\right)_{N}$, the weak limits of the associated
empirical measures coincide with the weak limits of the corresponding
piecewise-constant measures.

######  Lemma  (Weak limits of empirical and piecewise-constant measures).

Let $p\in[1,\infty)$ and $\left(x^{N}\in\mathcal{K}_{N}\right)_{N}$. Set
$\hat{\mu}^{N}:=\mathcal{E}^{N}(x^{N})$, $\mu^{N}:=\mathcal{PC}^{N}(x^{N})$,
and fix $\mu\in\mathscr{P}(R)$.

  1. (i)

$\hat{\mu}^{N}\to\mu$ vaguely if and only if $\mu^{N}\to\mu$ vaguely.

  2. (ii)

The $p$th-moments of $\hat{\mu}^{N}$ are equi-integrable if and only if the
$p$th-moments of $\mu^{N}$ are equi-integrable.

  3. (iii)

$\mathbb{W}_{p}(\hat{\mu}^{N},\mu)\to 0$ if and only if
$\mathbb{W}_{p}(\mu^{N},\mu)\to 0$.

###### Proof.

Note that (iii) is a direct consequence of (i) and (ii). For the sake of
simplicity, we omit the dependence on $N$ and write $x_{i}=x_{i}^{N}$.

(i): It suffices to prove that $\hat{\mu}^{N}-\mu^{N}\to 0$ vaguely. In fact,
we claim that

$\hat{\mu}^{N}-\frac{1}{N}\sum_{i=0}^{N-1}\delta_{x_{i}}\to 0\quad\mbox{ and
}\quad\frac{1}{N}\sum_{i=0}^{N-1}\delta_{x_{i}}-\mu^{N}\to
0\quad\text{vaguely}\,.$

For simplicity of notation, we denote
$\bar{\mu}^{N}:=\frac{1}{N}\sum_{i=0}^{N-1}\delta_{x_{i}}$ (note that in
general $\bar{\mu}^{N}$ does not coincide with $\hat{\mu}^{N-1}$ because the
set of particles may differ). For any $\varphi\in C^{\infty}_{c}(\mathbb{R})$,
we denote by $\operatorname{supp}\varphi$ its (compact) support. On one hand

$\displaystyle\left|\int_{\mathbb{R}}\varphi\,\mathrm{d}(\hat{\mu}^{N}-\bar{\mu}^{N})\right|$
$\displaystyle=\left|\frac{1}{N+1}\sum_{i=0}^{N}\varphi(x_{i})-\frac{1}{N}\sum_{i=0}^{N-1}\varphi(x_{i})\right|$
$\displaystyle\leq\frac{1}{N(N+1)}\sum_{i=0}^{N-1}|\varphi(x_{i})|+\frac{\|\varphi(x_{N})\|}{N+1}\leq\frac{2\|\varphi\|_{\infty}}{N+1}\to
0\,.$

On the other side,

$\displaystyle\left|\int_{\mathbb{R}}\varphi\,\mathrm{d}(\bar{\mu}^{N}-\mu^{N})\right|$
$\displaystyle=\frac{1}{N}\sum_{i=0}^{N-1}\left|\varphi(x_{i})-\fint_{x_{i}}^{x_{i+1}}\varphi(x)\,\mathrm{d}x\right|\leq\frac{\|\varphi^{\prime}\|_{\infty}}{N}\text{diam(spt}\varphi)\to
0\,.$

(ii): Let us first assume that the $p$th-moments of $\hat{\mu}^{N}$ are equi-
integrable, namely for every $\varepsilon>0$ there is $L_{\varepsilon}>0$ such
that

$\limsup_{N\to\infty}\int_{\mathbb{R}\setminus[-L_{\varepsilon},L_{\varepsilon}]}|x|^{p}\,\mathrm{d}\hat{\mu}^{N}(x)<\varepsilon.$

We set $i_{\ell}:=\min\\{i:x_{i}\in[-L_{\varepsilon},L_{\varepsilon}]\\}$,
$i_{r}:=\max\\{i:x_{i}\in[-L_{\varepsilon},L_{\varepsilon}]\\}.$ In particular

$\limsup_{N\to\infty}\frac{1}{N+1}\left(\sum_{i=0}^{i_{\ell}-1}|x_{i}|^{p}+\sum_{i=i_{r}+1}^{N}|x_{i}|^{p}\right)<\varepsilon,$

from which, by monotonicity of the function $|\cdot|^{p}$ separately in
$(\infty,-L_{\varepsilon})$ and $(L_{\varepsilon},\infty)$, for $N$ big enough
we deduce

$\displaystyle\int_{\mathbb{R}\setminus[-L_{\varepsilon},L_{\varepsilon}]}|x|^{p}\,\mathrm{d}\mu^{N}$
$\displaystyle\leq\frac{1}{N}\left(\sum_{i=0}^{i_{\ell}-1}\fint_{x_{i}}^{x_{i+1}}|x|^{p}\,\mathrm{d}x+\sum_{i=i_{r}}^{N-1}\fint_{x_{i}}^{x_{i+1}}|x|^{p}\,\mathrm{d}x\right)$
$\displaystyle\leq\frac{1}{N}\left(\sum_{i=0}^{i_{\ell}-1}|x_{i}|^{p}+\sum_{i=i_{r}}^{N-1}|x_{i+1}|^{p}\right)\leq
2\varepsilon\,,$

thus providing the equi-integrability of the $p$th-moments of $\mu^{N}$.

Viceversa: assume that for $\varepsilon>0$, there exists $L_{\varepsilon}>0$
such that

$\limsup_{N\to\infty}\int_{\mathbb{R}\setminus[-L_{\varepsilon},L_{\varepsilon}]}|x|^{p}\,\mathrm{d}\mu^{N}<\varepsilon.$

Observe that, calling again $i_{\ell},i_{r}$ the first and the last particle
inside $[-L_{\varepsilon},L_{\varepsilon}]$ respectively, for $N$ big enough
we get

$\displaystyle\int_{\mathbb{R}\setminus[x_{i_{\ell}},x_{i_{r}}]}|x|^{p}\,\mathrm{d}\mu^{N}$
$\displaystyle=\int_{\mathbb{R}\setminus[-L_{\varepsilon},L_{\varepsilon}]}|x|^{p}\,\mathrm{d}\mu^{N}+\int_{[-L_{\varepsilon},L_{\varepsilon}]\setminus[x_{i_{\ell}},x_{i_{r}}]}|x|^{p}\,\mathrm{d}\mu^{N}$
$\displaystyle\leq\varepsilon+\frac{1}{N}\left(\frac{1}{\Delta
x_{i_{\ell}-1}}\int_{-L_{\varepsilon}}^{x_{i_{\ell}}}|x|^{p}\,\mathrm{d}x\frac{1}{\Delta
x_{i_{r}}}\int^{L_{\varepsilon}}_{x_{i_{r}}}|x|^{p}\,\mathrm{d}x\right)\leq\varepsilon+\frac{2|L_{\varepsilon}|^{p}}{N}\leq
2\varepsilon\,.$

Therefore

$\limsup_{N}\frac{1}{N}\left(\sum_{i=0}^{i_{\ell}-1}\fint_{x_{i}}^{x_{i+1}}(-x)^{p}\,\mathrm{d}x+\sum_{i=i_{r}}^{N-1}\fint_{x_{i}}^{x_{i+1}}x^{p}\,\mathrm{d}x\right)<2\varepsilon.$
(5.1)

Since $|x_{i_{\ell}-1}|\leq|x_{i_{\ell}-2}|$ and
$a^{p+1}-b^{p+1}\geq(a-b)a^{p}$ for $0<b<a$, we have that

$\displaystyle\int_{\mathbb{R}\setminus[-L_{\varepsilon},L_{\varepsilon}]}|x|^{p}\,\mathrm{d}\hat{\mu}^{N}$
$\displaystyle\leq
2\left(\sum_{i=0}^{i_{\ell}-2}|x_{i}|^{p}+\sum_{i=i_{r}}^{N-1}|x_{i+1}|^{p}\right)$
$\displaystyle\leq
2(p+1)\left(\sum_{i=0}^{i_{\ell}-1}\fint_{x_{i}}^{x_{i+1}}(-x)^{p}\,\mathrm{d}x+\sum_{i=i_{r}}^{N-1}\fint_{x_{i}}^{x_{i+1}}x^{p}\,\mathrm{d}x\right)\,.$

Hence, by (5.1) we conclude the equi-integrability of the $p$-moments of
$\hat{\mu}^{N}$. ∎

###  5.1. Proof of the $\Gamma$-limsup inequality

In the first result, we show the $\Gamma$-$\limsup$ inequality for regular
curves, in the sense of Definition 4.2, and we conclude using the
approximation result provided by Proposition 4.2. We emphasise that, for
regular curves, we are able to construct a recovery sequence satisfying a
discrete version of Definition 4.2. In principle, this is not necessary to
prove Theorem 3.1, nonetheless it plays a crucial role in Section 6 for the
convergence of the JKO-schemes. Throughout the whole section, we fix
$\phi_{p,m}$ as in Definition 2.1, with $1<p<\infty$. Moreover, we denote by
$\mathbb{W}_{\infty}$ the distance given by
$\mathbb{W}_{\infty}(\mu_{0},\mu_{1}):=\|X_{\mu_{0}}-X_{\mu_{1}}\|_{\infty}$,
for $\mu_{0},\mu_{1}\in\mathscr{P}(\mathbb{R})$ compactly supported.

######  Proposition  (Existence of (regular) recovery sequences for regular
curves).

Let $(\tilde{\bm{\mu}},\tilde{\bm{\nu}})\in\mathsf{CE}$ be regular curves in
the sense of Definition 4.2 for some compact set $K$ and constants
$0<l\leq\tilde{M}<M$. Then there exist discrete curves $\left(\tilde{x}^{N}\in
C^{\infty}([0,1];\mathcal{K}_{N})\right)_{N}$ such that

  1. (i)

the measures $\displaystyle\mathcal{PC}^{N}(\tilde{x}^{N}(k))$ (and hence
$\mathcal{E}^{N}(\tilde{x}^{N}(k))$ as well, by Lemma 5) converge to
$\tilde{\mu}_{k}$ in $\mathbb{W}_{q}$ as $N\to\infty$, for $k=0,1$, for every
$q\in(1,+\infty]$.

  2. (ii)

$\displaystyle\limsup_{N\to\infty}\int_{0}^{1}\Phi_{p,m}^{N}\big{(}\tilde{x}^{N}(t),\dot{\tilde{x}}^{N}(t)\big{)}\,\mathrm{d}t\leq\mathbb{W}_{p,m}(\tilde{\mu}_{0},\tilde{\mu}_{1})^{p}$
.

  3. (iii)

The following regularity properties holds: for $i=0,\dots,N$, $t\in[0,1]$,
$N\in\mathbb{N}$

$\displaystyle
l\leq\tilde{R}_{i}^{N}(\tilde{x}^{N}(t))\leq\tilde{M},\quad\operatorname{supp}\mathcal{PC}^{N}\big{(}\tilde{x}^{N}(t)\big{)}=\operatorname{supp}(\tilde{\mu}_{t})\subset
K\,.$

###### Proof.

The main idea is to consider the pseudo-inverse associated with the regular
curve $\tilde{\bm{\mu}}$ and discretise them in space on a scale $1/N$.

Step 1: Discretisation of $(\tilde{\mu}_{t},\tilde{\nu}_{t})$. Denote by
$\tilde{v}_{t}=\tilde{j}_{t}/\tilde{\rho}_{t}$ the associated velocity field,
and by $I_{t}$ its compact and connected support. By construction, we have
that $\big{\\{}t\mapsto\tilde{v}_{t}\in C^{\infty}(I_{t})\big{\\}}\in
C^{\infty}([0,1])$. In particular, the curve
$\tilde{X}_{t}:=X_{\tilde{\mu}_{t}}$ solves the differential equation (see
(4.6))

$\displaystyle\dot{\tilde{X}}_{t}(z)=\tilde{v}_{t}(\tilde{X}_{t}(z))\,,\quad\tilde{X}_{0}:=X_{\tilde{\mu}_{0}},\quad\tilde{X}_{0}\in
C^{1}([0,1])\,,$

where the latter regularity follows from the continuity of (the density of)
$\mu_{0}$ and the fact that $I_{t}$ is compact and connected. Moreover, by the
homogeneity of $\phi_{p,m}$, we have that for $t\in[0,1]$

$\displaystyle\int_{0}^{1}(\tilde{\rho}_{t}\circ\tilde{X}_{t})^{-1}\phi_{p,m}(\tilde{\rho}_{t}\circ\tilde{X}_{t},\tilde{j}_{t}\circ\tilde{X}_{t})\,\mathrm{d}z$
$\displaystyle=\int_{\mathbb{R}}\phi_{p,m}(\tilde{\rho}_{t},\tilde{j}_{t})\,\mathrm{d}x$
(5.2)

We define for every $N\in\mathbb{N}$ the discrete curve $\tilde{x}^{N}(t)$ as

$\displaystyle\tilde{x}_{i}^{N}(t):=\tilde{X}_{t}\left(\frac{i}{N}\right),\quad
t\in[0,1],\;i\in\\{0,...,N\\}\,,$

and consider the corresponding measures
$\tilde{\mu}_{t}^{N}=\mathcal{PC}^{N}(\tilde{x}^{N}(t))\in\mathscr{P}(\mathbb{R})$
with densities

$\displaystyle\tilde{\rho}_{t}^{N}(x)=\sum_{i=0}^{N-1}\tilde{R}_{i}^{N}\mathbbm{1}_{\left[\tilde{x}_{i}^{N},\tilde{x}_{i+1}^{N}\right)}(x)\,,\quad\tilde{R}_{j}^{N}:=R_{j}^{N}(\tilde{x}^{N})\,,$
(5.3)

for $x\in\mathbb{R}$ and $j=0,\dots,N-1$. Note that the pseudo-inverse of
$\tilde{\mu}_{t}^{N}$ is nothing but the piecewise affine interpolation of the
$\tilde{X}_{t}$ on $[0,1]$. Thus for $i=0,\dots,N-1$

$\tilde{X}_{\tilde{\mu}_{t}^{N}}(z)=(Nz-i)\tilde{X}_{t}\left(\frac{i+1}{N}\right)+(i+1-Nz)\tilde{X}_{t}\left(\frac{i}{N}\right)\qquad\text{for
}z\in\left[\frac{i}{N},\frac{i+1}{N}\right].$

From the continuity of $\tilde{X}_{t}$, it follows that
$\sup_{t\in[0,1]}\big{\|}\tilde{X}_{t}-\tilde{X}_{\tilde{\mu}_{t}^{N}}\big{\|}_{\text{L}^{\infty}(0,1)}\to
0$ as $N\to\infty$, which shows (i) (see also Remark 4). By construction, for
any $t\in[0,1]$ the
$\operatorname{supp}\tilde{\mu}_{t}^{N}=\operatorname{supp}\tilde{\mu}_{t}$
for every $N\in\mathbb{N}$, in particular
$\operatorname{supp}\tilde{\mu}_{t}^{N}\subset K$ and also
$\tilde{x}_{i}^{N}(t)\in K$ for every $i=0,\dots,N$. Also, being
$l\leq\tilde{\rho}_{t}\leq\tilde{M}$, it is immediate to see that
$l\leq\tilde{R}_{i}^{N}\leq\tilde{M}$, thus proving (iii).

Step 2: Energy estimates. Using that
$(\tilde{X}_{t})_{\\#}\,\mathrm{d}z=\tilde{\mu}_{t}$, we deduce that

$\displaystyle\fint_{\frac{i}{N}}^{\frac{i+1}{N}}\tilde{R}_{i}^{N}(\tilde{\rho}_{t}\circ\tilde{X}_{t})^{-1}(z)\,\mathrm{d}z=N\tilde{R}_{i}^{N}\int_{\tilde{X}_{t}(\frac{i}{N})}^{\tilde{X}_{t}(\frac{i+1}{N})}\tilde{\rho}_{t}^{-1}(x)\,\mathrm{d}\tilde{\mu}_{t}(x)=1\,,$

by the very definition of $\tilde{R}_{i}^{N}$. This means that the measures
$\,\mathrm{d}\xi_{t}^{N,i}:=N\tilde{R}_{i}^{N}(\tilde{\rho}_{t}\circ\tilde{X}_{t})^{-1}(z)\,\mathrm{d}z\in\mathscr{P}\big{(}\big{[}\frac{i}{N},\frac{i+1}{N}\big{]}\big{)}$.
By Jensen’s inequality and (5.2)

$\displaystyle\int_{\mathbb{R}}\phi_{p,m}(\tilde{\rho}_{t},\tilde{j}_{t})\,\mathrm{d}x$
$\displaystyle\geq\frac{1}{N}\sum_{i=0}^{N-1}(\tilde{R}_{i}^{N})^{-1}\phi_{p,m}\bigg{(}\int_{\frac{i}{N}}^{\frac{i+1}{N}}(\tilde{\rho}_{t}\circ\tilde{X}_{t},\tilde{j}_{t}\circ\tilde{X}_{t})\,\mathrm{d}\xi_{t}^{N,i}(z)\bigg{)}$
$\displaystyle=\frac{1}{N}\sum_{i=0}^{N-1}(\tilde{R}_{i}^{N})^{-1}\phi_{p,m}\bigg{(}\tilde{R}_{i}^{N},\tilde{R}_{i}^{N}\fint_{\frac{i}{N}}^{\frac{i+1}{N}}v_{t}\circ\tilde{X}_{t}(z)\,\mathrm{d}z\bigg{)}$
$\displaystyle=\frac{1}{N}\sum_{i=0}^{N-1}(\tilde{R}_{i}^{N})^{p-1}\phi_{p,m}\bigg{(}\tilde{R}_{i}^{N},\fint_{\frac{i}{N}}^{\frac{i+1}{N}}\partial_{t}\tilde{X}_{t}(z)\,\mathrm{d}z\bigg{)}\,,$
(5.4)

where at last we used the $p$-homogeneity of $\phi_{p,m}$ with respect to the
second variable. Moreover, by $\tilde{X}\in C^{1}([0,1]\times[0,1])$,
$\text{Im}(\tilde{X}_{t})\subset\operatorname{supp}\tilde{\rho}_{t}$, and the
boundedness of $\tilde{v}$,

$\displaystyle\left\\{(\tilde{R}_{i}^{N}\big{(}\tilde{x}_{t}^{N}\big{)},\dot{\tilde{X}}_{t}(z))\
:\
t>0\,,\;z\in[0,1]\,,\;N\in\mathbb{N}\right\\}\subset[l,\tilde{M}]\times[0,\|\tilde{v}\|_{\infty}]\,=K$

It is clear that $\phi_{p,m}$ is Lipschitz in $K$, therefore, using (2.7),
(5.1) and (2.6), we get that there exists a constant $C\in\mathbb{R}_{+}$
which only depends on $l$, $\tilde{M}$, $p$, $\theta$, and $\rho^{*}$ such
that

$\displaystyle\int_{0}^{1}\Phi_{p,m}(\tilde{\rho}_{t},\tilde{j}_{t})\,\mathrm{d}t\geq\int_{0}^{1}\Phi_{p,m}^{N}(\tilde{x}^{N},\dot{\tilde{x}}^{N})\,\mathrm{d}t-\frac{C}{N}\Big{(}\|\tilde{X}\|_{C^{2}}+\|\tilde{X}\|_{C^{1}}^{p}\Big{)}\,.$

Letting $N\to+\infty$ we obtain (ii). ∎

We are finally ready to prove the $\Gamma$-$\limsup$ inequality in Theorem
3.1.

###### Proof of Theorem 3.1, $\Gamma$-$\limsup$.

Fix $\mu_{0},\mu_{1}\in\mathscr{P}_{q}(\mathbb{R})$ be such that
$\mathbb{W}_{p,m}(\mu_{0},\mu_{1})<+\infty$. Let
$({\bm{\mu}},{\bm{\nu}})\in\mathsf{CE}(\mu_{0},\mu_{1})$ be a
$\mathbb{W}_{p,m}$-geodesic between $\mu_{0}$ and $\mu_{1}$ (see (III)). For
every $\eta>0$, we apply Proposition 4.2 and Proposition 5.1, and find regular
curves $\tilde{\mu}_{0}^{\eta}$, $\tilde{\mu}_{1}^{\eta}$ (thus belonging to
$\mathscr{P}_{q}(\mathbb{R})$) and discrete curves $\tilde{x}^{N,\eta}\in
W^{1,p}(0,1;\mathcal{K}_{N})$ such that
$\mathbb{W}_{q}(\mu_{k},\tilde{\mu}_{k}^{\eta})^{q}\leq\eta$,
$\mathcal{E}^{N}(\tilde{x}_{k}^{N,\eta})\xrightarrow{\mathbb{W}_{q}}\tilde{\mu}_{k}^{\eta}$,
and

$\displaystyle\limsup_{N\to\infty}\int_{0}^{1}\Phi_{p,m}^{N}\big{(}\tilde{x}^{N,\eta}(t),\dot{\tilde{x}}^{N,\eta}(t)\big{)}\,\mathrm{d}t\leq\mathbb{W}_{p,m}(\tilde{\mu}_{0}^{\eta},\tilde{\mu}_{1}^{\eta})^{p}\leq\mathbb{W}_{p,m}(\mu_{0},\mu_{1})^{p}+\eta\,.$

We conclude the proof by choosing a suitable diagonal sequence
$\tilde{x}^{N}:=\tilde{x}^{N,\eta(N)}$. ∎

###  5.2. Proof of the $\Gamma$-liminf inequality

In this section we prove the $\Gamma$-$\liminf$ in Theorem 3.1. A crucial tool
is a compactness result for "almost" solutions to the continuity equation.
This is the content of the next Proposition, which is inspired by [DNS12,
Lemma 4.5] (see also [LM10, Proposition 2.5]) and follows similar ideas. For
the sake of clarity, we provide a proof.

######  Definition  (Asymptotic solutions to $\mathsf{CE}$).

We say that a sequence $\left(\mu_{t}^{N},\nu_{t}^{N}\right)_{N}$ is
asymptotically a solution to $\mathsf{CE}$ on $[0,1]$ if the same assumptions
of Definition 2.1 are satisfied for every $N\in\mathbb{N}$, but with $3.$
replaced by

  * $3^{\prime}$.

For every $\varphi\in C_{c}^{1}(\mathbb{R})$, the measurable maps
$\mathcal{R}^{N}\varphi:=t\mapsto(\partial_{t}\mu_{t}^{N}+\partial_{x}\nu_{t}^{N},\varphi)$
converge to $0$ weakly in $\text{L}^{1}(0,1)$ as $N\to\infty$.

We use the notation $\mathcal{R}_{t}^{N}\varphi:=(\mathcal{R}^{N}\varphi)(t)$,
for $t\in(0,1)$.

######  Proposition  (Compactness).

Let $\left(\mu_{t}^{N},\nu_{t}^{N}\right)_{N}$ be asymptotically a solution to
$\mathsf{CE}$ satisfying

$\displaystyle
A:=\sup_{N\in\mathbb{N}}\int_{0}^{1}\Phi_{p,m}(\mu_{t}^{N},\nu_{t}^{N})\,\mathrm{d}t<\infty\,.$

Suppose that $\mu_{0}^{N}\to\mu_{0}\in\mathscr{P}(\mathbb{R})$ vaguely. Then
there exists a (non-relabeled) subsequence and a solution
$(\mu_{t},\nu_{t})\in\mathsf{CE}$ such that $\mu_{t}^{N}\to\mu_{t}$ vaguely in
$\mathscr{P}(\mathbb{R})$ for all $t\in[0,1]$, ${\bm{\nu}}^{N}\to{\bm{\nu}}$
vaguely in $\mathcal{M}((0,1)\times\mathbb{R})$, and

$\displaystyle\liminf_{N\rightarrow+\infty}\int_{0}^{1}\Phi_{p,m}(\mu_{t}^{N},\nu_{t}^{N})\,\mathrm{d}t\geq\int_{0}^{1}\Phi_{p,m}(\mu_{t},\nu_{t})\,\mathrm{d}t.$

###### Proof.

We denote by $\rho_{t}^{N}$, $j_{t}^{N}$ the corresponding densities of
$\mu_{t}^{N}$ and $\nu_{t}^{N}$ with respect to the Lebesgue measure on
$[0,1]\times\mathbb{R}$. Then an application of the Hölder inequality yields
for every Borel set $B\subset\mathbb{R}$ and $t_{1}<t_{2}\in[0,1]$ the bound

$\displaystyle\begin{aligned}
\int_{t_{1}}^{t_{2}}\int_{B}|j_{t}^{N}|\,\mathrm{d}x\,\mathrm{d}t&\leq\left(\int_{t_{1}}^{t_{2}}\int_{B}m(\rho_{t}^{N})\,\mathrm{d}x\,\mathrm{d}t\right)^{\frac{1}{p^{\prime}}}\left(\int_{0}^{1}\Phi_{p,m}(\mu_{t}^{N},\nu_{t}^{N})\,\mathrm{d}t\right)^{\frac{1}{p}}\\\
&\leq\left(\|m\|_{\infty}\mathscr{L}^{1}(B)(t_{2}-t_{1})\right)^{\frac{1}{p^{\prime}}}A^{\frac{1}{p}}\end{aligned}$
(5.5)

for any $N\in\mathbb{N}$, for $p^{\prime}$ conjugate of $p$ (note that
$m\in\text{L}^{\infty}(0,M)$ by concavity and $m(0)=0=m(M)$). Consequently
there exists ${\bm{\nu}}\in\mathcal{M}([0,1]\times\mathbb{R})$ and a
subsequence (not relabeled) such that ${\bm{\nu}}^{N}\to{\bm{\nu}}$ vaguely as
$N\rightarrow+\infty$.

Moreover, (5.5) shows that the map $t\mapsto|\nu_{t}^{N}|(B)$ is equi-
integrable in $[0,1]$ and uniformly bounded in $L^{1}(0,1)$, for every
$B\subset\mathbb{R}$. By Dunford–Pettis [Bre11, Theorem 4.30], this in
particular implies that ${\bm{\nu}}$ satisfies for every $R>0$

$\displaystyle|{\bm{\nu}}|(B_{R}\times
I)=\int_{I}m_{R}(t)\,\mathrm{d}t,\quad\forall
I\subset([0,1),\,B_{R}\subset\mathbb{R},\quad\text{with
}m_{R}\in\text{L}^{1}(0,1),$ (5.6)

so that we can write ${\bm{\nu}}=\int_{0}^{1}\nu_{t}\,\mathrm{d}t$, intended
as in (2.2), for a Borel family of (signed) finite measures
$\left(\nu_{t}\right)_{t\in(0,1)}$ (see [DNS12, Lemma 4.5] for a similar
argument).

By assumption we know that $\mu_{0}^{N}\to\mu_{0}\in\mathscr{P}(\mathbb{R})$
vaguely (in fact narrowly, by Remark 2.1). We now claim that for every time
$t\in[0,1]$, the sequence $\mu_{t}^{N}$ vaguely converges to some
$\mu_{t}\in\mathscr{P}(\mathbb{R})$, as a consequence of
$(\mu_{t}^{N},\nu_{t}^{N})$ being asymptotically a solution to $\mathsf{CE}$.

Indeed, consider a function $\varphi\in C_{c}^{1}(\mathbb{R})$ and define
$\phi(\tau,x):=\mathbbm{1}_{I}(\tau)\partial_{x}\varphi(x)$, $I:=[0,t]$. Note
that the discontinuity set of $\phi$ is concentrated on
$D:=\\{0,t\\}\times\mathbb{R}$ and from (5.6) one has $|{\bm{\nu}}|(D)=0$. It
follows [AGS08, Proposition 5.1.10] that

$\displaystyle\lim_{N\rightarrow+\infty}\int_{I}\int_{\mathbb{R}}\partial_{x}\varphi\,\mathrm{d}\nu_{\tau}^{N}\,\mathrm{d}\tau=\lim_{N\rightarrow+\infty}\int_{I\times\mathbb{R}}\phi\,\mathrm{d}{\bm{\nu}}^{N}=\int_{I\times\mathbb{R}}\phi\,\mathrm{d}{\bm{\nu}}=\int_{I}\int_{\mathbb{R}}\partial_{x}\varphi\,\mathrm{d}\nu_{\tau}\,\mathrm{d}\tau.$

Moreover, by definition of $\mathcal{R}_{\tau}^{N}$ and integrating over $I$
we obtain

$\displaystyle\begin{aligned}
\lim_{N\rightarrow+\infty}\int_{\mathbb{R}}\varphi\,\mathrm{d}\mu_{t}^{N}&=\lim_{N\rightarrow+\infty}\left(\int_{\mathbb{R}}\varphi\,\mathrm{d}\mu_{0}^{N}+\int_{0}^{t}\int_{\mathbb{R}}\partial_{x}\varphi\,\mathrm{d}\nu_{\tau}^{N}\,\mathrm{d}\tau+\int_{0}^{t}\mathcal{R}_{\tau}^{N}\varphi\,\mathrm{d}\tau\right)\\\
&=\int_{\mathbb{R}}\varphi\,\mathrm{d}\mu_{0}+\int_{I}\int_{\mathbb{R}}\partial_{x}\varphi\,\mathrm{d}\nu_{\tau}\,\mathrm{d}\tau\,,\end{aligned}$
(5.7)

where we used that $\mu_{0}^{N}\to\mu_{0}$ vaguely and that
$\mathcal{R}^{N}\varphi\to 0$ weakly in $\text{L}^{1}(0,1)$. Hence, (5.7)
shows that $\mu_{t}^{N}$ is vaguely converging to some
$\mu_{t}\in\mathscr{P}(\mathbb{R})$ as $N\rightarrow+\infty$ for every
$t\in[0,1]$, with $(\mu_{t},\nu_{t})\in\mathsf{CE}$. Finally, the lower
semicontinuity estimate follows from the lower semicontinuity of $\Phi_{p,m}$,
see (I). ∎

We are ready to prove the $\Gamma$-$\liminf$ inequality in Theorem 3.1.

###### Proof of Theorem 3.1, $\Gamma$-$\liminf$.

Let $\mu_{0}^{N}=\mathcal{PC}^{N}(x_{0}^{N})$,
$\mu_{1}^{N}=\mathcal{PC}^{N}(x_{1}^{N})$ be two sequences in
$\mathscr{P}(\mathbb{R})$ converging vaguely to $\mu_{0},\mu_{1}$,
respectively. We claim that

$\displaystyle\liminf_{N\rightarrow+\infty}\mathbb{W}_{p,m}^{N}(\mu_{0}^{N},\mu_{1}^{N})\geq\mathbb{W}_{p,m}(\mu_{0},\mu_{1}).$
(5.8)

Applying Lemma 2.2, with no loss of generality we can write

$\displaystyle\begin{aligned}
\mathbb{W}_{p,m}^{N}(\mu_{0}^{N},\mu_{1}^{N})^{p}=\int_{0}^{1}\Phi_{p,m}^{N}(x^{N}(t),\dot{x}^{N}(t))\,\mathrm{d}t\
,\ \
E:=\sup_{N\in\mathbb{N}}\sup_{t\in[0,1]}\Phi_{p,m}^{N}(x^{N}(t),\dot{x}^{N}(t))<\infty\,,\end{aligned}$
(5.9)

where $x^{N}\in W^{1,p}(0,1;\mathcal{K}_{N})$ is the discrete geodesic between
$x_{0}^{N}$ and $x_{1}^{N}$ with respect to $d_{p,m}^{N}$. We define
$\mu_{t}^{N}:=\mathcal{PC}^{N}(x_{t}^{N})$ with density (w.r.t the Lebesgue
measure on $\mathbb{R}$) $\rho_{t}^{N}$ given by the piecewise-constant map

$\displaystyle\rho_{t}^{N}(x)=\sum_{i=0}^{N-1}R_{i}^{N}(t)\mathbbm{1}_{\left[x_{i}^{N}(t),x_{i+1}^{N}(t)\right)}(x)\,,$
(5.10)

and define the momentum fields as

$\displaystyle
j_{t}^{N}(x):=\sum_{i=0}^{N-1}\dot{x}_{i}^{N}(t)R_{i}^{N}(t)\mathbbm{1}_{\left[x_{i}^{N}(t),x_{i+1}^{N}(t)\right)}(x),\quad\nu_{t}^{N}:=j_{t}^{N}\mathscr{L}^{1}.$
(5.11)

Step 1: Uniform error estimate. Let us pick any $\varphi\in
C_{c}^{\infty}(\mathbb{R})$. We claim that

$\displaystyle\sup_{t\in(0,1)}\left|\left(\partial_{t}\mu_{t}^{N}+\partial_{x}\nu_{t}^{N},\varphi\right)\right|\leq
cN^{-\frac{1}{p^{\prime}}}\|\varphi\|_{C^{2}}\left(\text{diam}(\operatorname{supp}\varphi)+2\right),$
(5.12)

for some $c=c(\theta,E,p)\in\mathbb{R}^{+}$, where $p^{\prime}$ is the
conjugate of $p$. For simplicity, in this proof we use the shorthand notation

$\displaystyle\Delta x_{i}:=x_{i+1}^{N}(t)-x_{i}^{N}(t),\
\quad\Delta\dot{x}_{i}:=\dot{x}_{i+1}^{N}(t)-\dot{x}_{i}^{N}(t),\quad
R_{i}:=R_{i}^{N}(t),\quad I_{i}:=[x_{i},x_{i+1}].$

In particular, we have that $\dot{R_{i}}=-NR_{i}^{2}\Delta\dot{x}_{i}\,$, from
which we infer

$\displaystyle\frac{\,\mathrm{d}}{\,\mathrm{d}t}\int\varphi\,\mathrm{d}\mu_{t}^{N}=-\sum_{i=0}^{N-1}R_{i}\Delta\dot{x}_{i}\fint_{I_{i}}\varphi\,\mathrm{d}x-R_{i}\big{(}\varphi(x_{i})\dot{x}_{i}-\varphi(x_{i+1})\dot{x}_{i+1}\big{)}.$

Using a second order Taylor expansion for $\varphi$, we know there exist a
family of points $\left(c_{x}\right)_{x}$ such that for every $x\in I_{i}$, we
have $c_{x}\in(x,x_{i+1})$ and

$\displaystyle\varphi(x)=\varphi(x_{i+1})+\varphi^{\prime}(x_{i+1})(x-x_{i+1})+\frac{1}{2}\varphi^{\prime\prime}(c_{x})(x-x_{i+1})^{2}\,,$

for $i=0,\dots,N-1$. It follows that

$\displaystyle\frac{\,\mathrm{d}}{\,\mathrm{d}t}\int\varphi\,\mathrm{d}\mu_{t}^{N}=\sum_{i=0}^{N-1}R_{i}\dot{x}_{i}\left(\varphi(x_{i+1})-\varphi(x_{i})\right)+\left(\mathcal{R}_{t}^{N},\varphi\right)=\left(\nu_{t}^{N},\partial_{x}\varphi\right)+\left(\mathcal{R}_{t}^{N},\varphi\right),$

where the error $\mathcal{R}_{t}^{N}\in\mathcal{D}^{\prime}(\mathbb{R})$ is
given by

$\displaystyle\left(\mathcal{R}_{t}^{N},\varphi\right)=\sum_{i=0}^{N-1}\frac{1}{2N}\Delta\dot{x}_{i}\varphi^{\prime}(x_{i+1})-\frac{1}{2}R_{i}\Delta\dot{x}_{i}\fint_{x_{i}}^{x_{i+1}}\varphi^{\prime\prime}(c_{x})(x-x_{i+1})^{2}\,\mathrm{d}x.$

By $\theta\in\text{L}^{\infty}(0,M)$ and (5.9), we deduce the uniform bound

$\displaystyle\frac{1}{N+1}\sum_{i=0}^{N}|\dot{x}_{i}|^{p}\leq\|\theta\|_{\infty}^{p-1}\Phi_{p,m}^{N}(x,\dot{x})\leq\|\theta\|_{\infty}^{p-1}E\quad\Longrightarrow\quad\sup_{i}\left|\dot{x}_{i}\right|\leq
cN^{\frac{1}{p}},$ (5.13)

for some $c=c(\|\theta\|_{\infty},E,p)\in\mathbb{R}^{+}$. Consequently, on one
hand by $x_{i}<x_{i+1}$ we get

$\displaystyle\left|\frac{1}{N}\sum_{i=0}^{N-1}\Delta\dot{x}_{i}\varphi^{\prime}(x_{i+1})\right|\leq\frac{1}{N}\sum_{i=0}^{N-1}|\dot{x}_{i}||\varphi^{\prime}(x_{i})-\varphi^{\prime}(x_{i+1})|+\frac{1}{N}\Big{(}|\dot{x}_{0}\varphi^{\prime}(x_{0})|+|\dot{x}_{N}\varphi^{\prime}(x_{N})|\Big{)}$
$\displaystyle\quad\leq
cN^{-\frac{1}{p^{\prime}}}\left(\operatorname{Lip}\varphi^{\prime}\sum_{i\in\mathcal{S}_{\varphi}^{N}}\Delta
x_{i}+2\operatorname{Lip}\varphi\right)\leq
cN^{-\frac{1}{p^{\prime}}}(\text{diam}(\operatorname{supp}\varphi)+2)\|\varphi\|_{C^{2}},$

where we set $\mathcal{S}_{\varphi}^{N}:=\\{i\in\\{0,\dots,N-1\\}\ :\
I_{i}\cap\operatorname{supp}\varphi\neq\emptyset\\}$. On the other hand, using
once more that $x_{i}<x_{i+1}$ and (5.13) we obtain

$\displaystyle\left|\sum_{i=0}^{N-1}R_{i}\Delta\dot{x}_{i}\fint_{x_{i}}^{x_{i+1}}\varphi^{\prime\prime}(c_{x})(x-x_{i+1})^{2}\,\mathrm{d}x\right|\leq\frac{\left\|\varphi\right\|_{C^{2}}}{N}\sum_{i\in\mathcal{S}_{\varphi}^{N}}|\Delta\dot{x}_{i}|\Delta
x_{i}$ $\displaystyle\qquad\leq
2cN^{-\frac{1}{p^{\prime}}}\left\|\varphi\right\|_{C^{2}}\sum_{i\in\mathcal{S}_{\varphi}^{N}}\Delta
x_{i}\leq
2cN^{-\frac{1}{p^{\prime}}}\left\|\varphi\right\|_{C^{2}}\text{diam}(\operatorname{supp}\varphi),$

which concludes the proof of (5.12).

Step 2: Compactness. Consider $\mu_{t}^{N},\nu_{t}^{N}$, defined in (5.10),
(5.11) and ${\bm{\nu}}$ as in (2.2). We claim that up to subsequence, we have
that

$\displaystyle{\bm{\nu}}^{N}\to{\bm{\nu}}=\int_{0}^{1}\nu_{t}\,\mathrm{d}t\in\mathcal{M}((0,1)\times\mathbb{R})\quad\text{and}\quad\mu_{s}^{N}\to\mu_{s}\in\mathscr{P}(\mathbb{R})\quad\text{vaguely},$

for all $s\in[0,1]$, with $(\mu_{t},\nu_{t})\in\mathsf{CE}$. To show this, we
prove that the assumptions of Proposition 5.2 are satisfied. The fact that
$(\mu_{t}^{N},\nu_{t}^{N})_{N}$ is asymptotically a solution to $\mathsf{CE}$
directly follows from (5.12), the convergence $\mu_{0}^{N}\to\mu_{0}$ is
already guaranteed by construction whereas the action bound follows from (5.9)
and (2.12). We can then apply Proposition 5.2 and conclude.

Step 3: Action estimate. We conclude the proof of (5.8) applying the lower
semicontinuous estimate provided by Proposition 5.2. Precisely we have

$\displaystyle\liminf_{N\rightarrow+\infty}\mathbb{W}_{p,m}^{N}(\mu_{0}^{N},\mu_{1}^{N})^{p}$
$\displaystyle=\liminf_{N\rightarrow+\infty}\int_{0}^{1}\Phi_{p,m}^{N}(x^{N},\dot{x}^{N})\,\mathrm{d}t\geq\liminf_{N\rightarrow+\infty}\int_{0}^{1}\Phi_{p,m}(\mu_{t}^{N},\nu_{t}^{N})\,\mathrm{d}t$
$\displaystyle\geq\int_{0}^{1}\Phi_{p,m}(\mu_{t},\nu_{t})\,\mathrm{d}t\geq\mathbb{W}_{p,m}(\mu_{0},\mu_{1})^{p}\,,$

where in the first inequality we used (2.12), and at last
$(\mu_{t},\nu_{t})\in\mathsf{CE}$. ∎

##  6\. Proof of the main result: convergence of the JKO schemes

In this last section of the paper, we prove the discrete-to-continuum
convergence of the JKO-schemes, i.e. Theorem 3.2.

Fix $\phi_{2,m}$ as in Definition 2.1 and $q\in[1,2)$. For a given initial
sequence $\mu_{N}^{(0)}\in\mathcal{E}^{N}$ and
$\mu^{(0)}\in\mathscr{P}_{2}(\mathbb{R})$, recall the definition of the
associated operators $\mathcal{J}_{\tau,N}$, $\mathcal{J}_{\tau}$ as given in
(2.15), (2.16). In order to prove the convergence of the minimisers, our goal
is to show a $\Gamma$-convergence result and an equi-coercivity property of
the families $\left(\mathcal{J}_{\tau,N}\right)_{N}$ in the $\mathbb{W}_{q}$
topology.

Assuming that $\mu_{N}^{(0)}\to\mu^{(0)}$ in $\mathbb{W}_{q}$, one of the
major difficulties is to show that $\mathcal{J}_{\tau,N}$ $\Gamma$-converge to
$\mathcal{J}_{\tau}$ as $N\to\infty$ despite the dependence on the quadratic
(generalised) Wasserstein distance. The extra assumption (2.20) of uniformly
bounded second moments on $\left(\mu_{N}^{(0)}\right)_{N}$ is crucial to this
purpose. Also important is the fact that such assumption can be "iterated",
namely from (2.20) we can deduce a similar bound $\mu_{N}^{(1)}$, and
iteratively for $\mu_{N}^{(n)}$, for every $n\in\mathbb{N}$.

The proof of the $\Gamma$-convergence relies on an improved version of the
$\Gamma$-$\limsup$ in Theorem 3.1, which is the main technical difficulty of
this last section.

###  6.1. An improved limsup inequality

The proof is based on the regular approximation result shown in Proposition
4.2 and Proposition 5.1.

######  Proposition  (Limsup inequality, JKO).

Fix $\mu_{0}$, $\mu_{1}\in\mathscr{P}_{2}(\mathbb{R})$ and an approximating
sequence $\mathcal{E}^{N}\ni\mu_{0}^{N}\to\mu_{0}$ in $\mathbb{W}_{q}$ for
some $q>1$, satisfying (2.20). Then, for every $\varepsilon>0$, there exists
$\bar{\mu}_{1}\in\mathscr{P}_{2}(\mathbb{R})$ and a sequence
$\left(\mu_{1}^{N}\in\mathcal{E}^{N}\right)_{N}$ such that

  1. (i)

$\displaystyle\mu_{1}^{N}\to\bar{\mu}_{1}\quad\text{in
}\big{(}\mathscr{P}_{q}(\mathbb{R}),\mathbb{W}_{q}\big{)}$.

  2. (ii)

$\displaystyle\mathbb{W}_{q}(\mu_{1},\bar{\mu}_{1})\leq\varepsilon$.

  3. (iii)

$\displaystyle\limsup_{N\to\infty}\mathbb{W}_{2,m}^{N}\big{(}\mu_{0}^{N},\mu_{1}^{N}\big{)}\leq\mathbb{W}_{2,m}(\mu_{0},\mu_{1})+\varepsilon$.

######  Remark  (Relation to Theorem 3.1, $\Gamma$-limsup).

Note that Proposition 6.1 is not a trivial consequence of the $\Gamma$-limsup
in Theorem 3.1, due to the fact $\mu_{0}^{N}$ does not necessarily coincide
with the recovery sequence for $\mu_{0}$ obtained in Theorem 3.1.

###### Proof.

Without loss of generality, we assume
$\mathbb{W}_{2,m}(\mu_{0},\mu_{1})<\infty$. Let $w^{N}\in\mathcal{K}^{N}$ such
that $\mu_{0}^{N}=\mathcal{E}^{N}(w^{N})$. Let
$({\bm{\mu}},{\bm{\nu}})\in\mathsf{CE}$ be the $\mathbb{W}_{2,m}$-geodesic
(III) between $\mu_{0}$ and $\mu_{1}$.

Case 1: We first assume that

$\displaystyle\max\left\\{\sup_{N\in\mathbb{N}}\sup_{i\leq
N}R_{i}^{N}(w^{N}),\,\sup_{t\in[0,1]}\left\|\rho_{t}\right\|_{L^{\infty}(\mathbb{R})}\right\\}=\bar{M}<M\,,$
(6.1)

where $\rho_{t}$ denotes the density of $\mu_{t}$ w.r.t. $\mathscr{L}^{1}$. In
this case, for every $\eta>0$, we apply Proposition 4.2 and find regular
curves $(\tilde{\bm{\mu}},\tilde{\bm{\nu}})\in\mathsf{CE}$ in the sense of
Definition 4.2 with densities $\tilde{\rho}_{t},\tilde{j}_{t}$ with

$\displaystyle\operatorname{supp}(\tilde{\rho}_{t})\cup\operatorname{supp}(\tilde{j}_{t})\subset
K\quad\text{and}\quad 0<l\leq\tilde{\rho}_{t}(x)\leq\tilde{M}<M\,,$

for a compact set $K\Subset\mathbb{R}$, $l>0$ (both depending on $\bar{M}$ and
$\eta$), and $\tilde{M}<M$ (only depending on $\bar{M}$), as well as
satisfying the bounds for all $s\in[0,1]$

$\displaystyle\begin{gathered}\int_{0}^{1}\Phi_{2,m}(\tilde{\mu}_{t},\tilde{\nu}_{t})\,\mathrm{d}t\leq\int_{0}^{1}\Phi_{2,m}(\mu_{t},\nu_{t})\,\mathrm{d}t+\eta,\\\
\max\left\\{\Big{(}\text{diam}(\operatorname{supp}(\tilde{\mu}_{s}))^{2-q}\vee
1\Big{)}\mathbb{W}_{q}(\mu_{s},\tilde{\mu}_{s})^{q}\,,\int_{\operatorname{supp}(\tilde{\mu}_{s})^{c}}(|x|^{2}+1)\,\mathrm{d}\mu_{s}(x)\right\\}\leq\eta\,.\end{gathered}$
(6.4)

Consequently, we apply Proposition 5.1 to
$(\tilde{\bm{\mu}},\tilde{\bm{\nu}})$ and find curves $\tilde{x}^{N}\in
C^{\infty}([0,1];\mathcal{K}_{N})$ such that

$\displaystyle\begin{gathered}\tilde{\mu}_{0}^{N}:=\mathcal{E}^{N}(\tilde{w}^{N})\to\tilde{\mu}_{0}\,,\quad\tilde{\mu}_{1}^{N}:=\mathcal{E}^{N}(\tilde{y}^{N})\to\tilde{\mu}_{1}\quad\text{in
}\mathbb{W}_{\infty}\,,\\\
\limsup_{N\to\infty}\int_{0}^{1}\Phi_{2,m}^{N}(\tilde{x}^{N}(t),\dot{\tilde{x}}^{N}(t))\,\mathrm{d}t\leq\int_{0}^{1}\Phi_{2,m}(\tilde{\mu}_{t},\tilde{\nu}_{t})\,\mathrm{d}t\,,\end{gathered}$
(6.7)

together with the regularity properties for $i=0,\dots,N$, $t\in[0,1]$,
$N\in\mathbb{N}$

$\displaystyle\begin{gathered}\tilde{x}_{i}^{N}(t)\in K\quad\text{and}\quad
l\leq R_{i}^{N}(\tilde{x}^{N}(t))\leq\tilde{M}\,.\end{gathered}$ (6.9)

We split the analysis of the problem into three different areas: one is inside
$\tilde{S}:=\operatorname{supp}(\tilde{\mu}_{0})=\operatorname{supp}(\tilde{w}^{N})$
(by Proposition 5.1), a second one far away from the compact set $K$ (and thus
from $\tilde{S}$), and the remaining area in between. Precisely, we define

$\displaystyle i_{1}$ $\displaystyle:=\inf\left\\{i\ :\
w_{i}^{N}>\inf(B_{1}(K))\right\\}\,,\quad$ $\displaystyle
i_{2}:=\sup\left\\{i\ :\ w_{i}^{N}<\sup(B_{1}(K))\right\\}\,,$
$\displaystyle\tilde{i}_{1}$ $\displaystyle:=\inf\left\\{i\ :\
w_{i}^{N}\geq\inf(\tilde{S})\right\\}\,,\quad$
$\displaystyle\tilde{i}_{2}:=\sup\left\\{i\ :\
w_{i}^{N}\leq\sup(\tilde{S})\right\\}\,,$

where $B_{1}(K)=\\{x\in\mathbb{R}\ :\ d(x,K)<1\\}$. See Figure 2. Step 1:
Definition of $y^{N}$. We define for every $N\in\mathbb{N}$ the particles
$y^{N}\in\mathcal{K}_{N}$ as follows: for $i\in\\{i_{1},\dots,i_{2}\\}$, we
choose the particles associated with $\tilde{y}^{N}$, otherwise we select the
particles associated with $w^{N}$. Precisely, we define

$\displaystyle y_{i}^{N}:=\begin{cases}\tilde{y}_{i}^{N}\,,&\text{if
}i_{1}\leq i\leq i_{2}\,,\\\ w_{i}^{N}\,,&\text{otherwise}\,,\end{cases}$

see Figure 2. We then set $\mu_{1}^{N}:=\mathcal{E}^{N}(y^{N})$ and claim it
to be the sought recovery sequence.

Figure 2. The definition of the recovery sequence $y^{N}$.

Our first goal is to estimate
$d_{2,m}^{N}(w^{N},y^{N})=\mathbb{W}_{2,m}^{N}(\mu_{0}^{N},\mu_{1}^{N})$ and
to show (iii). To do so, we shall introduce an intermediate step $z^{N}$,
defined as

$\displaystyle z_{i}^{N}:=\begin{cases}\tilde{w}_{i}^{N}\,,&\text{if
}i_{1}\leq i\leq i_{2}\,,\\\ w_{i}^{N}\,,&\text{otherwise}\,,\\\
\end{cases}\;=\begin{cases}\tilde{w}_{i}^{N}\,,&\text{if }i_{1}\leq i\leq
i_{2}\,,\\\ y_{i}^{N}\,,&\text{otherwise}\,.\end{cases}$

For simplicity, we denote by $I_{N}$ the set of indexes corresponding to
$B_{1}(K)\setminus\tilde{S}$, i.e.

$\displaystyle I_{N}^{-}:=\\{i\ :\ i_{1}\leq i<\tilde{i}_{1}\\}\,,\quad
I_{N}^{+}:=\\{i\ :\ \tilde{i}_{2}<i\leq i_{2}\\}\,,\quad I_{N}:=I_{N}^{-}\cup
I_{N}^{+}\,.$

Step 3: estimate of $d_{2,m}^{N}(w^{N},z^{N})$. The cost of transporting
$w^{N}$ onto $z^{N}$ consists in three parts. A trivial one, outside
$B_{1}(K)$, where the particles do not move. The second one is the cost of
connecting particles which live inside $\tilde{S}$, where the $\mathbb{W}_{q}$
and the $\mathbb{W}_{2}$ are equivalent (up to a factor depending on
$\text{diam}(\tilde{S})$). The last one is the cost of transporting the
particles $w^{N}$ from $B_{1}(K)\setminus\tilde{S}$ to the corresponding
particles $z^{N}$ (which live in $\tilde{S}$ and coincide with
$\tilde{w}^{N}$). See Figure 2. We estimate that in terms of the tails of the
second moments of $\mu_{0}$, see (6.13).

Note that $w_{i_{1}-1}^{N},w_{i_{2}+1}^{N}\in\mathbb{R}\setminus B_{1}(K)$, it
is immediate to see that $R_{i_{1}-1}^{N}(z^{N})\vee R_{i_{2}}^{N}(z^{N})\leq
1/N$ for $N$ big enough. By (6.1), (6.9), we have

$\displaystyle\sup_{N\in\mathbb{N}}\sup_{i\leq N}R_{i}^{N}(z^{N})\leq
M^{\prime}\,,\quad\text{where}\quad M^{\prime}=\max\\{\tilde{M},\bar{M}\\}<M$

does not depend on $\eta$. In particular, we obtain that

$\displaystyle\begin{aligned} d_{2,m}^{N}(w^{N},z^{N})^{2}\leq
c_{\theta}\left\|w^{N}-z^{N}\right\|_{\ell_{2,N}}^{2}=\frac{c_{\theta}}{N+1}\sum_{i=i_{1}}^{i_{2}}\left|w_{i}^{N}-z_{i}^{N}\right|^{2}\,,\end{aligned}$
(6.10)

where $c_{\theta}:=\theta(M^{\prime})^{-1}<\infty$. By construction, on one
side we have that (green lines in Figure 2)

$\displaystyle\begin{aligned}
\frac{1}{N+1}\sum_{i=\tilde{i}_{1}}^{\tilde{i}_{2}}\left|w_{i}^{N}-z_{i}^{N}\right|^{2}&\leq\frac{1}{N+1}\text{diam}(\tilde{S})^{2-q}\sum_{i=\tilde{i}_{1}}^{\tilde{i}_{2}}\left|w_{i}^{N}-\tilde{w}_{i}^{N}\right|^{q}\\\
&\leq\text{diam}(\tilde{S})^{2-q}\mathbb{W}_{q}(\mu_{0}^{N},\tilde{\mu}_{0}^{N})^{q}\,.\end{aligned}$
(6.11)

We are left to estimate the remaining terms of the sum in (6.10), i.e.
corresponding to $i\in I_{N}$. By (6.4), we have that
$\mu_{0}((\tilde{S})^{c})\leq\eta$, whereas by assumption we know that
$\mu_{0}^{N}\to\mu_{0}$ vaguely. Therefore, using that $\mu_{0}$ is absolutely
continuous with respect to $\mathscr{L}^{1}$ (in particular,
$\mu_{0}(\partial\tilde{S})=0$), we get $\mu_{0}^{N}(\tilde{S}^{c})\leq 2\eta$
for $N$ large enough. As a consequence, we infer that, for $\eta<\frac{1}{4}$,

$\displaystyle\forall i\in I_{N}^{-},\quad\tilde{w}_{i}^{N}\leq
x_{\tilde{\mu}_{0}^{N}},\quad\forall i\in I_{N}^{+},\quad\tilde{w}_{i}^{N}\geq
x_{\tilde{\mu}_{0}^{N}}\,,$ (6.12)

where $x_{\tilde{\mu}_{0}^{N}}=X_{\tilde{\mu}_{0}^{N}}(\frac{1}{2})$ is the
median of $\tilde{\mu}_{0}^{N}$. Note that $w_{i}^{N}\in
B_{1}(K)\setminus\tilde{S}$ if and only if $i\in I_{N}$ (corresponding to the
violet lines in Figure 2). Together with (6.12), this implies

$\displaystyle\frac{1}{N+1}\sum_{i\in
I_{N}}\left|w_{i}^{N}-z_{i}^{N}\right|^{2}$
$\displaystyle=\frac{1}{N+1}\sum_{i\in
I_{N}}\left|w_{i}^{N}-\tilde{w}_{i}^{N}\right|^{2}\leq\frac{1}{N}\sum_{i\in
I_{N}}\left|w_{i}^{N}-x_{\tilde{\mu}_{0}^{N}}\right|^{2}$
$\displaystyle=\int_{B_{1}(K)\setminus\tilde{S}}\left|x-x_{\tilde{\mu}_{0}^{N}}\right|^{2}\,\mathrm{d}\mu_{0}^{N}(x)\,.$
(6.13)

Thanks to the fact that
$\|X_{\tilde{\mu}_{0}^{N}}-X_{\tilde{\mu}_{0}}\|_{L^{\infty}(0,1)}\to 0$ by
(6.7) and $\mu_{0}^{N}\to\mu_{0}$ in $\mathbb{W}_{q}$, by (6.13) we obtain
that

$\displaystyle\begin{aligned} \limsup_{N\to\infty}\frac{1}{N+1}\sum_{i\in
I_{N}}\left|w_{i}^{N}-z_{i}^{N}\right|^{2}&\leq\int_{B_{1}(K)\setminus\tilde{S}}\left|x-x_{\tilde{\mu}_{0}}\right|^{2}\,\mathrm{d}\mu_{0}(x)\\\
&\leq
2\int_{\tilde{S}^{c}}|x|^{2}\,\mathrm{d}\mu_{0}(x)+2\eta\left|x_{\tilde{\mu}_{0}}\right|^{2}\end{aligned}$
(6.14)

where we used $\mu_{0}(\partial\tilde{S})=0$ and
$\mu_{0}(\tilde{S}^{c})\leq\eta$. Altogether, using (6.10), (6.11), (6.14),
and the fact that $\mu_{0}^{N}\to\mu_{0}$ and
$\tilde{\mu}_{0}^{N}\to\tilde{\mu}_{0}$ in $\mathbb{W}_{q}$, taking the limsup
in $N\to\infty$ we obtain

$\displaystyle\limsup_{N\to\infty}d_{2,m}^{N}(w^{N},z^{N})^{2}$
$\displaystyle\leq
2c_{\theta}\left(\text{diam}(\tilde{S})^{2-q}\mathbb{W}_{q}(\mu_{0},\tilde{\mu}_{0})^{q}+\int_{\tilde{S}^{c}}|x|^{2}\,\mathrm{d}\mu_{0}(x)+\eta\left|x_{\tilde{\mu}_{0}}\right|^{2}\right)$
$\displaystyle\leq
2c_{\theta}\eta\left(2+\left|x_{\tilde{\mu}_{0}}\right|^{2}\right)\,,$ (6.15)

where at last we used (6.4).

Step 4: estimate of $d_{2,m}^{N}(z^{N},y^{N})$. We bound from above the
discrete distance from $z^{N}$ to $y^{N}$ using the competitor

$\displaystyle x_{i}^{N}(t):=\begin{cases}\tilde{x}_{i}^{N}(t)\,,&\text{if
}i_{1}\leq i\leq i_{2}\,,\\\ w_{i}^{N}\,,&\text{otherwise}\,,\end{cases}$

which is admissible by construction, i.e. $x^{N}(0)=z^{N}$, $x^{N}(1)=y^{N}$,
and $x^{N}\in C^{\infty}([0,1];\mathcal{K}_{N})$, thanks to the fact that
$(\tilde{x}_{i_{1}}^{N}(\cdot)-w_{i_{1}-1}^{N})\wedge(w_{i_{2}+1}^{N}-\tilde{x}_{i_{2}}^{N}(\cdot))\geq
1$. Since $\phi\geq 0$ and $R_{i_{2}}^{N}(x^{N}(\cdot))\leq
R_{i_{2}}^{N}(\tilde{x}^{N}(\cdot))$, we obtain the estimate

$\displaystyle
d_{2,m}^{N}(z^{N},y^{N})^{2}\leq\int_{0}^{1}\Phi_{2,m}^{N}(\tilde{x}^{N}(t),\dot{\tilde{x}}^{N}(t))\,\mathrm{d}t\,.$

Using first (6.7) and then (6.4), we conclude

$\displaystyle\limsup_{N\to\infty}d_{2,m}^{N}(z^{N},y^{N})^{2}\leq\int_{0}^{1}\Phi_{2,m}(\tilde{\mu}_{t},\tilde{\nu}_{t})\,\mathrm{d}t\leq\mathbb{W}_{2,m}(\mu_{0},\mu_{1})^{2}+\eta\,.$
(6.16)

Step 5: proof of (iii). Putting together (6.15) and (6.16) we conclude for
every $\eta>0$

$\displaystyle\limsup_{N\to\infty}\mathbb{W}_{2,m}^{N}(\mu_{0}^{N},\mu_{1}^{N})$
$\displaystyle\leq\limsup_{N\to\infty}d_{2,m}^{N}(w^{N},z^{N})+d_{2,m}^{N}(z^{N},y^{N})$
$\displaystyle\leq\eta^{\frac{1}{2}}(2c_{\theta})^{\frac{1}{2}}\left(2+\left|x_{\tilde{\mu}_{0}}\right|^{2}\right)^{\frac{1}{2}}+\Big{(}\mathbb{W}_{2,m}(\mu_{0},\mu_{1})^{2}+\eta\Big{)}^{\frac{1}{2}}\,.$

Finally, by (6.4) for $s=0$ we know that $\tilde{\mu}_{0}\to\mu_{0}$ in
$\mathbb{W}_{q}$ as $\eta\to 0$. As a consequence of Lemma 4.1(i), the sought
bound (iii) then follows from choosing $\eta$ sufficiently small.

Step 6: conclusion. We note that, by the assumption (2.20) on $\mu_{0}^{N}$
and (6.7),

$\displaystyle\sup_{N\in\mathbb{N}}\mathbb{E}_{\mu_{1}^{N}}|x|^{2}\leq\sup_{N\in\mathbb{N}}\big{(}\mathbb{E}_{\mu_{0}^{N}}|x|^{2}+\mathbb{E}_{\tilde{\mu}_{1}^{N}}|x|^{2}\big{)}<\infty\,,$

therefore there exists $\bar{\mu}_{1}\in\mathscr{P}_{2}(\mathbb{R})$ such
that, up to a (non-relabeled) subsequence, we get that
$\mu_{1}^{N}\to\bar{\mu}_{1}$ in $\mathbb{W}_{q}$. We are left with the proof
of (ii). By (6.4), we have that

$\displaystyle\mathbb{W}_{q}(\mu_{1},\bar{\mu}_{1})\leq\mathbb{W}_{q}(\mu_{1},\tilde{\mu}_{1})+\mathbb{W}_{q}(\tilde{\mu}_{1},\bar{\mu}_{1})\leq\eta^{\frac{1}{q}}+\mathbb{W}_{q}(\tilde{\mu}_{1},\bar{\mu}_{1})\,.$
(6.17)

On the other hand, by $\tilde{\mu}_{1}^{N}\to\tilde{\mu}_{1}$ in
$\mathbb{W}_{\infty}$ and arguing as in (6.12)–(6.14), we find that

$\displaystyle\mathbb{W}_{q}$
$\displaystyle(\tilde{\mu}_{1},\bar{\mu}_{1})=\lim_{N\to\infty}\mathbb{W}_{q}(\tilde{\mu}_{1}^{N},\mu_{1}^{N})^{q}=\lim_{N\to\infty}\frac{1}{N+1}\sum_{i\notin\\{i_{1},\dots,i_{2}\\}}\left|w_{i}^{N}-\tilde{y}_{i}^{N}\right|^{q}$
$\displaystyle\leq\int_{B_{1}(K)^{c}}\left|x-x_{\tilde{\mu}_{1}}\right|^{q}\,\mathrm{d}\mu_{0}(x)\leq
2\int_{\tilde{S}^{c}}|x|^{q}\,\mathrm{d}\mu_{0}(x)+2\eta\left|x_{\tilde{\mu}_{1}}\right|^{q}\leq
2\eta\left(1+\left|x_{\tilde{\mu}_{1}}\right|^{q}\right)\,,$

where at last we used (6.4) once more. Together with (6.17), we deduce that

$\displaystyle\mathbb{W}_{q}(\mu_{1},\bar{\mu}_{1})\leq\eta^{\frac{1}{q}}+(2\eta)^{\frac{1}{q}}\left(1+\left|x_{\tilde{\mu}_{1}}\right|^{q}\right)^{\frac{1}{q}}\,.$

Finally, by $\tilde{\mu}_{1}\to\mu_{1}$ as $\eta\to 0$ and Lemma 4.1(i), for
every given $\varepsilon>0$ we obtain the sought bound (ii) choosing $\eta$
sufficiently small.

Case 2: We now turn to the general case, meaning we do not assume that (6.1)
is satisfied. For $\lambda>0$, we consider the regularised measures
$(\mu_{t}^{\lambda},\nu_{t}^{\lambda})\in\mathsf{CE}$ given by

$\displaystyle\mu_{t}^{\lambda}:=\big{(}(1+\lambda)\text{id}\big{)}_{\\#}\mu_{t}\,,\quad\nu_{t}^{\lambda}:=(1+\lambda)\big{(}(1+\lambda)\text{id}\big{)}_{\\#}\nu_{t}\,.$

Note that by construction, the corresponding densities $\rho_{t}^{\lambda}$,
$j_{t}^{\lambda}$ are given by

$\displaystyle\rho_{t}^{\lambda}(x)=\frac{1}{1+\lambda}\rho_{t}\Big{(}\frac{x}{1+\lambda}\Big{)}\,,\quad
j_{t}^{\lambda}(x)=j_{t}\Big{(}\frac{x}{1+\lambda}\Big{)}\,,\quad\forall
x\in\mathbb{R}\,,t\in[0,1]\,.$ (6.18)

Furthermore, we have that
$\mu_{0}^{N,\lambda}:=\big{(}(1+\lambda)\text{id}\big{)}_{\\#}\mu_{0}^{N}=\mathcal{E}^{N}(w^{N,\lambda})$,
where $w^{N,\lambda}=(1+\lambda)w^{N}$. In particular, we note that

$\displaystyle\max\left\\{\sup_{N\in\mathbb{N}}\sup_{i\leq
N}R_{i}^{N}(w^{N,\lambda}),\,\sup_{t}\|\rho_{t}^{\lambda}\|_{\infty}\right\\}\leq(1+\lambda)^{-1}M<M\,.$
(6.19)

We can then apply the first part of the proof (Case 1) to
$\mu_{0}^{N,\lambda}$, $\mu_{0}^{\lambda}$, and $\mu_{1}^{\lambda}$: for any
fixed $\varepsilon>0$, we can find $\mu_{1}^{N,\lambda}\in\mathcal{E}^{N}$,
$\bar{\mu}_{1}^{\lambda}\in\mathscr{P}_{2}(\mathbb{R})$ such that
$\mu_{1}^{N,\lambda}\to\bar{\mu}_{1}^{\lambda}$ in $\mathbb{W}_{q}$ and

$\displaystyle\mathbb{W}_{q}(\mu_{1}^{\lambda},\bar{\mu}_{1}^{\lambda})\leq\frac{\varepsilon}{2}\quad\text{and}\quad\limsup_{N\to\infty}\mathbb{W}_{2,m}^{N}(\mu_{0}^{N,\lambda},\mu_{1}^{N,\lambda})\leq\mathbb{W}_{2,m}(\mu_{0}^{\lambda},\mu_{1}^{\lambda})+\frac{\varepsilon}{2}\,.$
(6.20)

Using similar arguments as in [LM10, Theorem 4.1], it is not hard to see that

$\displaystyle\sup_{N\in\mathbb{N}}d_{2,m}^{N}(w^{N},w^{N,\lambda})^{2}\leq
C\,\lambda\,\sup_{N}\|w^{N}\|_{\ell_{2,N}}^{2}=:\bar{C}^{2}\lambda\,,$ (6.21)

where $C\in\mathbb{R}_{+}$ only depends on $\phi$ and $\bar{C}<\infty$ since
$\left(\mu_{0}^{N}\right)_{N}$ satisfies (2.20). Moreover, by (6.18), we also
have that

$\displaystyle\int_{0}^{1}\Phi_{2,m}(\mu_{t}^{\lambda},\nu_{t}^{\lambda})\,\mathrm{d}t\leq(1+\lambda)\int_{0}^{1}\Phi_{2,m}((1+\lambda)^{-1}\mu_{t},\nu_{t})\,\mathrm{d}t\leq(1+\lambda)\mathbb{W}_{2,m}(\mu_{0},\mu_{1})^{2}\,,$
(6.22)

where at last we used that $\theta$ is non-increasing (see Remark 2.1).
Putting (6.20), (6.21), and (6.22) together, we obtain that

$\displaystyle\limsup_{N\to\infty}\mathbb{W}_{2,m}^{N}(\mu_{0},\mu_{1}^{N,\lambda})$
$\displaystyle\leq\limsup_{N\to\infty}\mathbb{W}_{2,m}^{N}(\mu_{0},\mu_{0}^{N,\lambda})+\mathbb{W}_{2,m}^{N}(\mu_{0}^{N,\lambda},\mu_{1}^{N,\lambda})$
$\displaystyle\leq\bar{C}\lambda^{\frac{1}{2}}+\sqrt{(1+\lambda)}\mathbb{W}_{2,m}(\mu_{0},\mu_{1})+\frac{\varepsilon}{2}\,.$

This shows that (iii) is satisfied with $\mu_{1}^{N}:=\mu_{1}^{N,\lambda}$ if
$\lambda>0$ is sufficiently small. Finally, by (6.20) and the scaling
properties of $\mathbb{W}_{q}$, we have that

$\displaystyle\mathbb{W}_{q}(\mu_{1},\bar{\mu}_{1}^{\lambda})\leq\mathbb{W}_{q}(\mu_{1},\mu_{1}^{\lambda})+\mathbb{W}_{q}(\mu_{1}^{\lambda},\bar{\mu}_{1}^{\lambda})\leq\lambda\big{(}\mathbb{E}_{\mu_{1}}|x|^{q}\big{)}^{\frac{1}{q}}+\frac{\varepsilon}{2}\,.$

This shows that (ii) is satisfied with
$\bar{\mu}_{1}:=\bar{\mu}_{1}^{\lambda}\in\mathscr{P}_{2}(\mathbb{R})$ if
$\lambda>0$ is sufficiently small. Given that $\mu_{1}^{N}\to\bar{\mu}_{1}$ in
$\mathbb{W}_{q}$ by construction, we also deduce (i), which ends the proof. ∎

###  6.2. Proof of the convergence of the JKO schemes (Theorem 3.2)

We are finally ready to prove Theorem 3.2.

###### Proof of Theorem 3.2.

We start showing (i). With no loss of generality, we assume $\tau=\frac{1}{2}$
and $F_{N}\equiv 0$, the general case following trivially by the assumption in
(3.1). Thus,
$\mathcal{J}_{\tau,N}(\cdot)=\mathbb{W}_{2,m}^{N}(\mu_{N}^{(0)},\cdot)^{2}$
and $\mathcal{J}_{\tau}(\cdot)=\mathbb{W}_{2,m}(\mu^{(0)},\cdot)^{2}$. In this
case, the $\Gamma$-$\liminf$ inequality easily follows by Theorem 3.1(LI),
hence we shall prove the $\Gamma$-$\limsup$.

Let $\mu\in\mathscr{P}_{2}(\mathbb{R})$ such that
$\mathbb{W}_{2,m}(\mu^{(0)},\mu)<\infty$. For every $\varepsilon>0$, we apply
Proposition 6.1 and find a sequence
$\left(\mu^{N,\varepsilon}\in\mathcal{E}^{N}\right)_{N}$ and
$\bar{\mu}^{\varepsilon}\in\mathscr{P}_{2}(\mathbb{R})$ such that
$\mu^{N,\varepsilon}\to\bar{\mu}^{\varepsilon}$ in $\mathbb{W}_{q}$, with
$\mathbb{W}_{q}(\mu,\bar{\mu}^{\varepsilon})\leq\varepsilon$, and
$\limsup_{N}\mathbb{W}_{2,m}^{N}\big{(}\mu_{N}^{(0)},\mu^{N,\varepsilon}\big{)}\leq\mathbb{W}_{2,m}(\mu^{(0)},\mu)+\varepsilon$.
Hence, we can choose a diagonal sequence $\mu^{N}:=\mu^{N,\varepsilon_{N}}$
such that $\mu^{N}\to\mu$ in $\mathbb{W}_{q}$ and

$\displaystyle\limsup_{N\to\infty}\mathbb{W}_{2,m}^{N}\big{(}\mu_{N}^{(0)},\mu^{N}\big{)}\leq\mathbb{W}_{2,m}(\mu^{(0)},\mu)\,,$

which concludes the proof of (i).

Finally, the proof of (ii) follows from the $\Gamma$-convergence shown in (i)
and the equicoercivity property proved in Proposition 2.3, see e.g. [Bra02,
Theorem 1.21].

### Acknowledgments

This work started when L.P. and S.dM visited E.R. at the University of
l’Aquila and continued first at the Erwin Schrödinger Institute in Vienna,
then at the École Polytechnique Fédérale de Lausanne and also in the Simons
institute in Berkeley, CA. The authors thank the hospitality of these three
institutions and, especially, Marco di Francesco for suggesting the topic in
first place and for fruitful discussions and literature suggestions. S.dM
acknowledges support by GNAMPA (INdAM). L.P. acknowledges support by the
Austrian Science Fund (FWF), grants No W1245 and No F65, and by the Hausdorff
Center for Mathematics in Bonn. E.R. acknowledges support by GNAMPA (INdAM). ∎

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