# Dissipative probability vector fields and generation of evolution semigroups
in Wasserstein spaces

Giulia Cavagnari  Giulia Cavagnari: Politecnico di Milano, Dipartimento di
Matematica, Piazza Leonardo Da Vinci 32, 20133 Milano (Italy)
<EMAIL_ADDRESS>,  Giuseppe Savaré  Giuseppe Savaré: Bocconi
University, Department of Decision Sciences and BIDSA, Via Roentgen 1, 20136
Milano (Italy<EMAIL_ADDRESS>and  Giacomo Enrico Sodini
Giacomo Enrico Sodini: TUM Fakultät für Mathematik, Boltzmannstrasse 3, 85748
Garching bei München (Germany<EMAIL_ADDRESS>
###### Abstract.

We introduce and investigate a notion of multivalued $\lambda$-dissipative
probability vector field (MPVF) in the Wasserstein space
$\mathcal{P}_{2}(\mathsf{X})$ of Borel probability measures on a Hilbert space
$\mathsf{X}$. Taking inspiration from the theory of dissipative operators in
Hilbert spaces and of Wasserstein gradient flows of geodesically convex
functionals, we study local and global well posedness of evolution equations
driven by dissipative MPVFs. Our approach is based on a measure-theoretic
version of the Explicit Euler scheme, for which we prove novel convergence
results with optimal error estimates under an abstract CFL stability
condition, which do not rely on compactness arguments and also hold when
$\mathsf{X}$ has infinite dimension.

We characterize the limit solutions by a suitable Evolution Variational
Inequality (EVI), inspired by the Bénilan notion of integral solutions to
dissipative evolutions in Banach spaces. Existence, uniqueness and stability
of EVI solutions are then obtained under quite general assumptions, leading to
the generation of a semigroup of nonlinear contractions.

###### Key words and phrases:

Measure differential equations/inclusions in Wasserstein spaces, probability
vector fields, dissipative operators, evolution variational inequality,
explicit Euler scheme.

###### 1991 Mathematics Subject Classification:

Primary: 34A06, 34A45; Secondary: 34A12, 34A34, 34A60, 28A50

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Wasserstein distance in Hilbert spaces
    2. 2.2 A strong-weak topology on measures in product spaces
  3. 3 Directional derivatives of the Wasserstein distance
    1. 3.1 Directional derivatives of the Wasserstein distance and duality pairings
    2. 3.2 Right and left derivatives of the Wasserstein distance along a.c. curves
    3. 3.3 Convexity and semicontinuity of duality parings
    4. 3.4 Behaviour of duality pairings along geodesics
  4. 4 Dissipative probability vector fields: the metric viewpoint
    1. 4.1 Multivalued Probability Vector Fields and $\lambda$-dissipativity
    2. 4.2 Behaviour of $\lambda$-dissipative MPVF along geodesics
    3. 4.3 Extensions of dissipative MPVF
  5. 5 Examples of $\lambda$-dissipative MPVFs
    1. 5.1 Subdifferentials of $\lambda$-convex functionals
    2. 5.2 MPVF concentrated on the graph of a multifunction
    3. 5.3 Interaction field induced by a dissipative map
  6. 6 Solutions to Measure Differential Inclusions
    1. 6.1 Metric characterization and EVI
    2. 6.2 Local existence of $\lambda$-EVI solutions by the Explicit Euler Scheme
    3. 6.3 Stability and uniqueness
    4. 6.4 Global existence and generation of $\lambda$-flows
    5. 6.5 Barycentric property
    6. 6.6 A few borderline examples
  7. 7 Explicit Euler Scheme
    1. 7.1 The Explicit Euler Scheme: preliminary estimates
    2. 7.2 Error estimates for the Explicit Euler scheme
    3. 7.3 Error estimates between discrete and EVI solutions
  8. A Comparison with [Pic19]
  9. B Wasserstein differentiability along curves
  10. C Support function and Dini derivatives

##  1\. Introduction

The aim of this paper is to study the local and global well posedness of
evolution equations for Borel probability measures driven by a suitable notion
of probability vector fields in an Eulerian framework.

For the sake of simplicity, let us consider here a finite dimensional
Euclidean space $\mathsf{X}$ with scalar product $\langle\cdot,\cdot\rangle$
and norm $|\cdot|$ (our analysis however will not be confined to finite
dimension and will be carried out in a separable Hilbert space) and the space
$\mathcal{P}(\mathsf{X})$ (resp. $\mathcal{P}_{b}(\mathsf{X})$) of Borel
probability measures in $\mathsf{X}$ (resp. with bounded support).

#### A Cauchy-Lipschitz approach, via vector fields

A first notion of vector field can be described by maps
$\boldsymbol{b}:\mathcal{P}_{b}(\mathsf{X})\to\mathrm{C}(\mathsf{X};\mathsf{X})$,
typically taking values in some subset of continuous vector fields in
$\mathsf{X}$ (as the locally Lipschitz ones of
$\mathrm{Lip}_{loc}(\mathsf{X};\mathsf{X})$), and satisfying suitable growth-
continuity conditions. In this respect, the evolution driven by
$\boldsymbol{b}$ can be described by a continuous curve
$t\mapsto\mu_{t}\in\mathcal{P}_{b}(\mathsf{X})$, $t\in[0,T]$, starting from an
initial measure $\mu_{0}\in\mathcal{P}_{b}(\mathsf{X})$ and satisfying the
continuity equation

$\displaystyle\partial_{t}\mu_{t}+\nabla\cdot(\boldsymbol{v}_{t}\mu_{t})$
$\displaystyle=0$ $\displaystyle\text{in }(0,T)\times\mathsf{X},$ (1.1a)
$\displaystyle\boldsymbol{v}_{t}$ $\displaystyle=\boldsymbol{b}[\mu_{t}]$
$\displaystyle\text{$\mu_{t}$-a.e.\leavevmode\nobreak\ for every
$t\in(0,T)$},$ (1.1b)

in the distributional sense, i.e.

$\int_{0}^{T}\int_{\mathsf{X}}\Big{(}\partial_{t}\zeta+\langle\nabla\zeta,\boldsymbol{v}_{t}\rangle\Big{)}\,\mathrm{d}\mu_{t}\,\mathrm{d}t=0,\quad\boldsymbol{v}_{t}=\boldsymbol{b}[\mu_{t}],\quad\text{for
every }\zeta\in\mathrm{C}^{1}_{c}((0,T)\times\mathsf{X}).$ (1.2)

If $\boldsymbol{b}$ is sufficiently smooth, solutions to (1.1a,b) can be
obtained by many techniques. Recent contributions in this direction are given
by the papers [Pic19, Pic18, BF21, Cav+20], we also mention [PR14, PR19] for
the analysis in presence of sources. In particular, in [BF21] the aim of the
authors is to develop a suitable Cauchy-Lipschitz theory in Wasserstein spaces
for differential inclusions which generalizes (1.1b) to multivalued maps
$\boldsymbol{b}:\mathcal{P}_{b}(\mathsf{X})\rightrightarrows\mathrm{Lip}_{loc}(\mathsf{X};\mathsf{X})$
and requires (1.1b), (1.2) to hold for a suitable measurable selection of
$\boldsymbol{b}$. As it occurs in the classical finite-dimensional case, the
differential-inclusion approach is suitable to describe the dynamics of
control systems, when the velocity vector field involved in the continuity
equation depends on a control parameter.

#### The Explicit Euler method

A natural approach, that is suitable for a great generalization, is to
approximate (1.1a,b) by a measure-theoretic version of the Explicit Euler
scheme. Choosing a step size $\tau>0$ and a partition
$\\{0,\tau,\cdots,n\tau,\cdots,N\tau\\}$ of the interval $[0,T]$,
$N={\mathrm{N}(T,\tau)}=\left\lceil T/\tau\right\rceil$, we construct a
sequence $M^{n}_{\tau}\in\mathcal{P}_{b}(\mathsf{X})$, $n=0,\cdots,N,$ by the
algorithm

$M^{0}_{\tau}:=\mu_{0},\quad
M^{n+1}_{\tau}:=(\boldsymbol{i}_{\mathsf{X}}+\tau\boldsymbol{b}^{n}_{\tau})_{\sharp}M^{n}_{\tau},\quad\boldsymbol{b}^{n}_{\tau}\in\boldsymbol{b}[M^{n}_{\tau}],$
(1.3)

where $\boldsymbol{i}_{\mathsf{X}}(x):=x$ is the identity map and
$\boldsymbol{r}_{\sharp}\mu$ denotes the push forward of
$\mu\in\mathcal{P}(\mathsf{X})$ induced by a Borel map
$\boldsymbol{r}:\mathsf{X}\to\mathsf{X}$ and defined by
$\boldsymbol{r}_{\sharp}\mu(B):=\mu(\boldsymbol{r}^{-1}(B))$ for every Borel
set $B\subset\mathsf{X}$. If $\bar{M}_{\tau}$ is the piecewise constant
interpolation of the discrete values $(M^{n}_{\tau})_{n=0}^{N}$, one can then
study the convergence of $\bar{M}_{\tau}$ as $\tau\downarrow 0$, hoping to
obtain a solution to (1.1a,b) in the limit.

It is then natural to investigate a few relevant questions:

  1. $\langle$E.1$\rangle$

what is the most general framework where the Explicit Euler scheme can be
implemented,

  2. $\langle$E.2$\rangle$

what are the structural conditions ensuring its convergence,

  3. $\langle$E.3$\rangle$

how to characterize the limit solutions and their properties.

Concerning the first question $\langle$E.1$\rangle$, one immediately realizes
that each iteration of (1.3) actually depends on the probability distribution
on the tangent bundle $\mathsf{T\kern-1.5ptX}$ (which we may identify with
$\mathsf{X}\times\mathsf{X}$, where the second component plays the role of
velocity)

$\Phi^{n}_{\tau}:=(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{b}^{n}_{\tau})_{\sharp}M^{n}_{\tau},\in\mathcal{P}(\mathsf{T\kern-1.5ptX})$

whose first marginal is $M^{n}_{\tau}$. If we denote by
$\mathsf{x},\mathsf{v}:\mathsf{T\kern-1.5ptX}\to\mathsf{X}$ the projections
$\mathsf{x}(x,v)=x,\ \mathsf{v}(x,v)=v$, and by
$\operatorname{\mathsf{exp}}^{\tau}:\mathsf{T\kern-1.5ptX}\to\mathsf{X}$ the
exponential map in the flat space $\mathsf{X}$
$\operatorname{\mathsf{exp}}^{\tau}(x,v):=x+\tau v$, we recover
$M^{n+1}_{\tau}$ by a single step of “free motion” driven by $\Phi^{n}_{\tau}$
and given by

$M^{n+1}_{\tau}=\operatorname{\mathsf{exp}}^{\tau}_{\sharp}\Phi^{n}_{\tau}=(\mathsf{x}+\tau\mathsf{v})_{\sharp}\Phi^{n}_{\tau}.$

This operation does not depend on the fact that $\Phi^{n}_{\tau}$ is
concentrated on the graph of a map (in this case
$\boldsymbol{b}^{n}_{\tau}\in\boldsymbol{b}[M^{n}_{\tau}]$): one can more
generally assign a multivalued map
${\boldsymbol{\mathrm{F}}}:\mathcal{P}_{b}(\mathsf{X})\rightrightarrows\mathcal{P}_{b}(\mathsf{T\kern-1.5ptX})$
such that for every $\mu\in\mathcal{P}_{b}(\mathsf{X})$ every measure
$\Phi\in{\boldsymbol{\mathrm{F}}}[\mu]\in\mathcal{P}_{b}(\mathsf{T\kern-1.5ptX})$
has first marginal $\mu=\mathsf{x}_{\sharp}\Phi$. We call
${\boldsymbol{\mathrm{F}}}$ a _multivalued probability vector field_ (MPVF in
the following), which is in good analogy with a Riemannian interpretation of
$\mathcal{P}_{b}(\mathsf{T\kern-1.5ptX})$. The disintegration
$\Phi_{x}\in\mathcal{P}_{b}(\mathsf{X})$ of $\Phi$ with respect to $\mu$
provides a (unique up to $\mu$-negligible sets) Borel family of probability
measures on the space of velocities such that
$\Phi=\int_{\mathsf{X}}\Phi_{x}\,\mathrm{d}\mu(x)$. $\Phi$ is induced by a
vector field $\boldsymbol{b}$ only if $\Phi_{x}=\delta_{\boldsymbol{b}(x)}$ is
a Dirac mass for $\mu$-a.e.$x$. (1.3) now reads as

$M^{0}_{\tau}:=\mu_{0},\quad
M^{n+1}_{\tau}:=\operatorname{\mathsf{exp}}^{\tau}_{\sharp}\Phi^{n}_{\tau}=(\mathsf{x}+\tau\mathsf{v})_{\sharp}\Phi^{n}_{\tau},\quad\Phi^{n}_{\tau}\in{\boldsymbol{\mathrm{F}}}[M^{n}_{\tau}].$
(1.4)

Besides greater generality, this point of view has other advantages: working
with the joint distribution ${\boldsymbol{\mathrm{F}}}[\mu]$ instead of the
disintegrated vector field $\boldsymbol{b}[\mu]$ potentially allows for the
weakening of the continuity assumption with respect to $\mu$. This relaxation
corresponds to the introduction of Young’s measures to study the limit
behaviour of weakly converging maps [CRV04]. Adopting this viewpoint, the
classical discontinuous example in $\mathbb{R}$ (see [Fil88]), where
$\boldsymbol{b}(x)=-\mathrm{sign}(x)$, admits a natural closed realization as
MPVF given by

$\Phi\in{\boldsymbol{\mathrm{F}}}[\mu]\quad\Leftrightarrow\quad\Phi_{x}=\begin{cases}\delta_{\boldsymbol{b}(x)}&\text{if
}x\neq 0\\\ (1-\theta)\delta_{-1}+\theta\delta_{1}&\text{if
}x=0\end{cases}\quad\text{for some }\theta\in[0,1].$

In particular,
${\boldsymbol{\mathrm{F}}}[\delta_{0}]=\big{\\{}\delta_{0}\otimes\left((1-\theta)\delta_{-1}+\theta\delta_{1}\right)\mid\theta\in[0,1]\big{\\}}$
(see also [Cam+21, Example 6.2]).

The study of measure-driven differential equations/inclusions is not new in
the literature DalMasoRampazzo,SilvaVinter. However, these studies, devoted to
the description of impulsive control systems [Bre96] and mainly motivated by
applications in rational mechanics and engineering, have been used to describe
evolutions in $\mathbb{R}^{d}$ rather than in the space of measures.

A second advantage in considering a MPVF is the consistency with the theory of
Wasserstein gradient flows generated by geodesically convex functionals
introduced in [AGS08] (Wasserstein subdifferentials are particular examples of
MPVFs) and with the multivalued version of the notion of probability vector
fields introduced in [Pic19, Pic18], whose originating idea was indeed to
describe the uncertainty affecting not only the state of the system, but
possibly also the distribution of the vector field itself.

A third advantage is to allow for a more intrinsic geometric view, inspired by
Otto’s non-smooth Riemannian interpretation of the Wasserstein space:
probability vector fields provide an appropriate description of infinitesimal
deformations of probability measures, which should be measured by, e.g., the
$L^{2}$-Kantorovich-Rubinstein-Wasserstein distance

$W_{2}^{2}(\mu,\nu):=\min\Big{\\{}\int_{\mathsf{X}\times\mathsf{X}}|x-y|^{2}\,\mathrm{d}\boldsymbol{\gamma}(x,y):\boldsymbol{\gamma}\in\Gamma(\mu,\nu)\Big{\\}},$
(1.5)

where $\Gamma(\mu,\nu)$ is the set of couplings with marginals $\mu$ and $\nu$
respectively. It is well known [AGS08, Vil09, San15] that if $\mu,\nu$ belong
to the space $\mathcal{P}_{2}(\mathsf{X})$ of Borel probability measures with
finite second moment

$\mathsf{m}_{2}^{2}(\mu):=\int_{\mathsf{X}}|x|^{2}\,\mathrm{d}\mu(x)<\infty,$

then the minimum in (1.5) is attained in a compact convex set
$\Gamma_{o}(\mu,\nu)$ and $(\mathcal{P}_{2}(\mathsf{X}),W_{2})$ is a complete
and separable metric space. Adopting this viewpoint and proceeding by analogy
with the theory of dissipative operators in Hilbert spaces, a natural class of
MPVFs for evolutionary problems should at least satisfy a
$\lambda$-dissipativity condition, $\lambda\in\mathbb{R}$, as

$\forall\,\Phi\in{\boldsymbol{\mathrm{F}}}[\mu],\
\Psi\in{\boldsymbol{\mathrm{F}}}[\nu],\ \mu\neq\nu:\quad
W_{2}(\operatorname{\mathsf{exp}}^{\tau}_{\sharp}\Phi,\operatorname{\mathsf{exp}}^{\tau}_{\sharp}\Psi)\leq(1+\lambda\tau)W_{2}(\mu,\nu)+o(\tau)\quad\text{as
}\tau\downarrow 0.$ (1.6)

#### Metric dissipativity

Condition (1.6) in the simple case $\lambda=0$ has a clear interpretation in
terms of one step of the Explicit Euler method: it is an asymptotic
contraction as the time step goes to $0$. By using the properties of the
Wasserstein distance, we will first compute the right derivative of the
(squared) Wasserstein distance along the deformation
$\operatorname{\mathsf{exp}}^{\tau}$

$\displaystyle\left[\Phi,\Psi\right]_{r}$
$\displaystyle:=\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}\tau}W_{2}^{2}(\operatorname{\mathsf{exp}}^{\tau}_{\sharp}\Phi,\operatorname{\mathsf{exp}}^{\tau}_{\sharp}\Psi)\Big{|}_{\tau=0+}$
(1.7) $\displaystyle={}$
$\displaystyle\min\Big{\\{}\int_{\mathsf{T\kern-1.5ptX}\times\mathsf{T\kern-1.5ptX}}\langle
w-v,y-x\rangle\,\mathrm{d}\boldsymbol{\Theta}(x,v;y,w):\boldsymbol{\Theta}\in\Gamma(\Phi,\Psi),\
(\mathsf{x},\mathsf{y})_{\sharp}\boldsymbol{\Theta}\in\Gamma_{o}(\mu,\nu)\Big{\\}}$

and we will show that (1.6) admits the equivalent characterization

$\left[\Phi,\Psi\right]_{r}\leq\lambda W_{2}^{2}(\mu,\nu)\quad\text{for every
$\Phi\in{\boldsymbol{\mathrm{F}}}[\mu],\
\Psi\in{\boldsymbol{\mathrm{F}}}[\nu]$}.$ (1.8)

If we interpret the left hand side of (1.8) as a sort of Wasserstein pseudo-
scalar product of $\Phi$ and $\Psi$ along the direction of an optimal coupling
between $\mu$ and $\nu$, (1.8) is in perfect analogy with the canonical
definition of $\lambda$-dissipativity (also called one-sided Lipschitz
condition) for a multivalued map
${\boldsymbol{F}}:\mathsf{X}\rightrightarrows\mathsf{X}$:

$\langle w-v,y-x\rangle\leq\lambda|x-y|^{2}\quad\text{for every
$v\in{\boldsymbol{F}}[x],\ w\in{\boldsymbol{F}}[y]$}.$ (1.9)

It turns out that the (opposite of the) Wasserstein subdifferential
$\boldsymbol{\partial}\mathcal{F}$ [AGS08, Section 10.3] of a geodesically
$(-\lambda)$-convex functional
$\mathcal{F}:\mathcal{P}_{2}(\mathsf{X})\to(-\infty,+\infty]$ is a MPVF and
satisfies a condition equivalent to (1.6) and (1.8). We also notice that (1.8)
reduces to (1.9) in the particular case when
$\Phi=\delta_{(x,v)},\Psi=\delta_{(y,w)}$ are Dirac masses in
$\mathsf{T\kern-1.5ptX}$.

#### Conditional convergence of the Explicit Euler method

Differently from the Implicit Euler method, however, even if a MPVF satisfies
(1.8), every step of the Explicit Euler scheme (1.4) affects the distance by a
further quadratic correction according to the formula

$W_{2}^{2}(\operatorname{\mathsf{exp}}^{\tau}_{\sharp}\Phi,\operatorname{\mathsf{exp}}^{\tau}_{\sharp}\Psi)\leq
W_{2}^{2}(\mu,\nu)+2\tau\left[\Phi,\Psi\right]_{r}+\tau^{2}\Big{(}|\Phi|_{2}^{2}+|\Psi|_{2}^{2}\Big{)},\quad|\Phi|_{2}^{2}:=\int_{\mathsf{T\kern-1.5ptX}}|v|^{2}\,\mathrm{d}\Phi(x,v),$

which depends on the order of magnitude of $\Phi$ and $\Psi$, and thus of
${\boldsymbol{\mathrm{F}}}$, at $\mu$ and $\nu$.

Our first main result (Theorems 7.5,7.7), which provides an answer to question
$\langle$E.2$\rangle$, states that if ${\boldsymbol{\mathrm{F}}}$ is a
$\lambda$-dissipative MPVF according to (1.8) then every family of discrete
solutions $(\bar{M}_{\tau})_{\tau>0}$ of (1.4) in an interval $[0,T]$
satisfying the abstract CFL condition

$|\Phi^{n}_{\tau}|_{2}\leq L\quad\text{if }0\leq n\leq
N={\mathrm{N}(T,\tau)},$ (1.10)

is uniformly converging to a limit curve
$\mu\in\mathrm{Lip}([0,T];\mathcal{P}_{2}(\mathsf{X}))$ starting from
$\mu_{0}$, with a uniform error estimate

$W_{2}(\mu_{t},\bar{M}_{\tau}(t))\leq
CL\sqrt{\tau(t+\tau)}\mathrm{e}^{\lambda_{+}t}\quad\text{for every }t\in[0,T]$
(1.11)

and a universal constant $C\leq 14$. Apart from the precise value of $C$, the
estimate (1.11) is sharp [Rul96] and reproduces in the measure-theoretic
framework the celebrated Crandall-Liggett [CL71] estimate for the generation
of dissipative semigroups in Banach spaces. We derive it by adapting to the
metric-Wasserstein setting the relaxation and doubling variable techniques of
[NS06], strongly inspired by the ideas of Kružkov [Kru70] and Crandall-Evans
[CE75].

This crucial result does not require any bound on the support of the measures
neither local compactness of the underlying space $\mathsf{X}$, so that we
will prove it in a general Hilbert space, possibly with infinite dimension.
Moreover, if $\mu,\nu$ are two limit solutions starting from $\mu_{0},\nu_{0}$
we show that

$W_{2}(\mu_{t},\nu_{t})\leq W_{2}(\mu_{0},\nu_{0})\mathrm{e}^{\lambda t}\quad
t\in[0,T],$

as it happens in the case of gradient flows of $(-\lambda)$-convex functions.
Once one has these building blocks, it is not too difficult to construct a
local and global existence theory, mimicking the standard arguments for ODEs.

#### Metric characterization of the limit solution

As we stated in question $\langle$E.3$\rangle$, a further important point is
to get an effective characterization of the solution $\mu$ obtained as limit
of the approximation scheme.

As a first property, considered in [Pic19, Pic18] in the case of a single-
valued PVF, one could hope that $\mu$ satisfies the continuity equation (1.1a)
coupled with the barycentric condition replacing (1.1b)

$\boldsymbol{v}_{t}(x)=\int_{\mathsf{T\kern-1.5ptX}}v\,\mathrm{d}\Phi_{t}(x,v),\quad\Phi_{t}={\boldsymbol{\mathrm{F}}}[\mu_{t}].$
(1.12)

This is in fact true, as shown in [Pic19, Pic18] in the finite dimensional
case, if ${\boldsymbol{\mathrm{F}}}$ is single valued and satisfies a stronger
Lipschitz dependence w.r.t. $\mu$ (see (H1) in Appendix A).

In the framework of dissipative MPVFs, we will replace (1.12) with its
relaxation _à la Filippov_ (see e.g. [Vin10, Chapter 2] and [AF09, Chapter
10])

$\boldsymbol{v}_{t}(x)=\int_{\mathsf{T\kern-1.5ptX}}v\,\mathrm{d}\Phi_{t}(x,v)\quad\text{for
some}\quad\Phi_{t}\in\overline{\operatorname{co}}(\operatorname{cl}({\boldsymbol{\mathrm{F}}})[\mu_{t}]),$

where $\operatorname{cl}({\boldsymbol{\mathrm{F}}})$ is the sequential closure
of the graph of ${\boldsymbol{\mathrm{F}}}$ in the strong-weak topology of
$\mathcal{P}_{2}^{sw}(\mathsf{T\kern-1.5ptX})$ (see [NS21] and Section 2.2 for
more details; in fact, a more restrictive “directional” closure could be
considered, see (6.34)) and
$\overline{\operatorname{co}}(\operatorname{cl}({\boldsymbol{\mathrm{F}}})[\mu])$
denotes the closed convex hull of the given section
$\operatorname{cl}({\boldsymbol{\mathrm{F}}})[\mu]$.

However, even in the case of a single valued map, (1.12) is not enough to
characterize the limit solution, as it has been shown by an interesting
example in [Pic19, Cam+21] (see also the gradient flow of Example 6.34).

Here we follow the metric viewpoint adopted in [AGS08] for gradient flows and
we will characterize the limit solutions by a suitable Evolution Variational
Inequality satisfied by the squared distance function from given test
measures. This approach is also strongly influenced by the Bénilan notion of
integral solutions to dissipative evolutions in Banach spaces [Bén72]. The
main idea is that any differentiable solution to
$\dot{x}(t)\in{\boldsymbol{F}}[x(t)]$ driven by a $\lambda$-dissipative
operator in a Hilbert space as in (1.9) satisfies

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}|x(t)-y|^{2}$
$\displaystyle=\langle\dot{x}(t),x(t)-y\rangle=\langle\dot{x}(t)-w,x(t)-y\rangle+\langle
w,x(t)-y\rangle$ $\displaystyle\leq\lambda|x(t)-y|^{2}-\langle
w,y-x(t)\rangle\quad\text{for every }w\in{\boldsymbol{\mathrm{F}}}[y].$

In the framework of $\mathcal{P}_{2}(\mathsf{X})$, we replace
$w\in{\boldsymbol{F}}[y]$ with $\Psi\in{\boldsymbol{\mathrm{F}}}[\nu]$ and the
scalar product $\langle w,y-x(t)\rangle$ with

$\left[\Psi,\mu_{t}\right]_{r}:=\min\Big{\\{}\int_{\mathsf{T\kern-1.5ptX}\times\mathsf{X}}\langle
w,y-x\rangle\,\mathrm{d}\boldsymbol{\Theta}(y,w;x):\boldsymbol{\Theta}\in\Gamma(\Psi,\mu_{t}),\
(\mathsf{y},\mathsf{x})_{\sharp}\boldsymbol{\Theta}\in\Gamma_{o}(\nu,\mu_{t})\Big{\\}},$

as in (1.7). According to this formal heuristic, we obtain the $\lambda$-EVI
characterization of a limit curve $\mu$ as

$\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}W_{2}^{2}(\mu_{t},\nu)\leq\lambda
W_{2}^{2}(\mu_{t},\nu)-\left[\Psi,\mu_{t}\right]_{r}\quad\text{for every
}\Psi\in{\boldsymbol{\mathrm{F}}}[\nu].$ ($\lambda$-EVI)

As for Bénilan integral solutions, we can considerably relax the apriori
smoothness assumptions on $\mu$, just imposing that $\mu$ is continuous and
($\lambda$-EVI) holds in the sense of distributions in $(0,T)$. In this way,
we obtain a robust characterization, which is stable under uniform convergence
and also allows for solutions taking values in the closure of the domain of
${\boldsymbol{\mathrm{F}}}$. This is particularly important when
${\boldsymbol{\mathrm{F}}}$ involves drift terms with superlinear growth (see
Example 6.32).

The crucial point of this approach relies on a general error estimate, which
extends the validity of (1.11) to a general $\lambda$-EVI solution $\mu$ and
therefore guarantees its uniqueness, whenever the Explicit Euler method is
solvable, at least locally in time.

Combining local in time existence with suitable global confinement conditions
(see e.g. Theorem 6.31) we can eventually obtain a robust theory for the
generation of a $\lambda$-flow, i.e. a semigroup $(\mathrm{S}_{t})_{t\geq 0}$
in a suitable subset $D$ of $\mathcal{P}_{2}(\mathsf{X})$ such that
$\mathrm{S}_{t}[\mu_{0}]$ is the unique $\lambda$-EVI solution starting from
$\mu_{0}$ and for every $\mu_{0},\mu_{1}\in D$

$W_{2}(\mathrm{S}_{t}[\mu_{0}],\mathrm{S}_{t}[\mu_{1}])\leq
W_{2}(\mu_{0},\mu_{1})\mathrm{e}^{\lambda t},\quad t\geq 0,$

as in the case of Wasserstein gradient flows of geodesically
$(-\lambda)$-convex functionals.

#### Explicit vs Implicit Euler method

In the framework of contraction semigroups generated by $\lambda$-dissipative
operators in Hilbert or Banach spaces, a crucial role is played by the
Implicit Euler scheme, which has the advantage to be unconditionally stable,
and thus avoids any apriori restriction on the local bound of the operator, as
we did in (1.10). In Hilbert spaces, it is well known that the solvability of
the Implicit Euler scheme is equivalent to the maximality of the graph of the
operator.

In the case of a Wasserstein gradient flow of a geodesically convex
$\mathcal{F}:\mathcal{P}_{2}(\mathsf{X})\to(-\infty,+\infty]$, every step of
the Implicit Euler method (also called JKO/Minimizing Movement scheme [JKO98,
AGS08]) can be solved by a variational approach: $M^{n+1}_{\tau}$ has to be
selected among the solutions of

$\min_{M\in\mathcal{P}_{2}(\mathsf{X})}\frac{1}{2\tau}W_{2}^{2}(M,M^{n}_{\tau})+\mathcal{F}(M).$
(1.13)

Notice, however, that in this case the MPVF $\boldsymbol{\partial}\mathcal{F}$
is defined implicitely in terms of $\mathcal{F}$ and each step of (1.13)
provides a suitable variational selection in
$\boldsymbol{\partial}\mathcal{F}$, leading in the limit to the minimal
selection principle.

In the case of more general dissipative evolutions, it is not at all clear how
to solve the Implicit Euler scheme, in particular when
${\boldsymbol{\mathrm{F}}}[\mu]$ is not concentrated on a map, and to
characterize the maximal extension of ${\boldsymbol{\mathrm{F}}}$ (in the
Hilbertian case the maximal extension of a dissipative operator
${\boldsymbol{F}}$ is explicitly computable at least when the domain of
${\boldsymbol{F}}$ has not empty interior, see the Theorems of Robert and
Bénilan in [Qi83]). Indeed, the analogy with the Hilbertian theory does not
extend to some properties which play a crucial role. In particular, a
dissipative MPVF ${\boldsymbol{\mathrm{F}}}$ in $\mathcal{P}_{2}(\mathsf{X})$
is not locally bounded in the interior of its domain (see Example 5.2) and
maximality may fail also for single-valued continuous PVFs (see Example 5.3).
Even more remarkably, in the Hilbertian case a crucial equivalent
characterization of dissipativity reads as

$v\in{\boldsymbol{F}}[x],\
w\in{\boldsymbol{F}}[y]\quad\Rightarrow\quad|x-y|\leq|(x-\tau v)-(y-\tau w)|$

which implies that the resolvent operators $(I-\tau{\boldsymbol{F}})^{-1}$
(and every single step of the Implicit Euler scheme) are contractions in
$\mathsf{X}$. On the contrary, if we assume the forward characterizations
(1.6) and (1.8) of dissipativity in $\mathcal{P}_{2}(\mathsf{X})$ (with
$\lambda=0$) we _cannot_ conclude in general that

$\Phi\in{\boldsymbol{\mathrm{F}}}[\mu],\
\Psi\in{\boldsymbol{\mathrm{F}}}[\nu]\quad\Rightarrow\quad W_{2}(\mu,\nu)\leq
W_{2}(\operatorname{\mathsf{exp}}^{-\tau}_{\sharp}\Phi,\operatorname{\mathsf{exp}}^{-\tau}_{\sharp}\Psi),$
(1.14)

since the squared distance map
$f(t):=W^{2}_{2}(\operatorname{\mathsf{exp}}^{t}_{\sharp}\Phi,\operatorname{\mathsf{exp}}^{t}_{\sharp}\Psi)$,
$t\in\mathbb{R}$, is not convex in general (see e.g. [AGS08, Example 9.1.5])
and the fact that its right derivative at $t=0$ (corresponding to
$\left[\Phi,\Psi\right]_{r}$) is $\leq 0$ according to (1.8) does not imply
that $f(0)\leq f(t)$ for $t<0$ (corresponding to (1.14) for $t=-\tau$).

For these reasons, we decided to approach the investigation of dissipative
evolutions in $\mathcal{P}_{2}(\mathsf{X})$ by the Explicit Euler method, and
we defer the study of the implicit one to a forthcoming paper.

#### Plan of the paper

As we already mentioned, our theory works in a general separable Hilbert space
$\mathsf{X}$: we collect some preliminary material concerning the Wasserstein
distance in Hilbert spaces and the properties of strong-weak topology for
$\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})$ in Section 2.

In Section 3, we will study the semi-concavity properties of $W_{2}$ along
general deformations induced by the exponential map
$\operatorname{\mathsf{exp}}^{\tau}$ and we introduce and study the pairings
$\left[\cdot,\cdot\right]_{r}$, $\left[\cdot,\cdot\right]_{l}$. We will apply
such tools to derive the precise expressions of the left and right derivatives
of $W_{2}$ along absolutely continuous curves in $\mathcal{P}_{2}(\mathsf{X})$
in Section 3.2.

In Section 4, we will introduce and study the notion of $\lambda$-dissipative
MPVF, in particular its behaviour along geodesics (Section 4.2) and its
extension properties (Section 4.3). A few examples are collected in Section 5.

The last two sections contain the core of our results. Section 6 is devoted to
the notion of $\lambda$-EVI solutions and to their properties: local
uniqueness, stability and regularity in Section 6.3, global existence in
Section 6.4 and barycentric characterizations in Section 6.5. Section 7
contains the main estimates for the Explicit Euler scheme: the Cauchy
estimates between two discrete solutions corresponding to different step sizes
in Section 7.2 and the uniform error estimates between a discrete and a
$\lambda$-EVI solution in Section 7.3.

#### Acknowledgments.

G.S. and G.E.S. gratefully acknowledge the support of the Institute of
Advanced Study of the Technical University of Munich. The authors thank the
Department of Mathematics of the University of Pavia where this project was
partially carried out. G.S. also thanks IMATI-CNR, Pavia. G.C. and G.S. have
been supported by the MIUR-PRIN 2017 project _Gradient flows, Optimal
Transport and Metric Measure Structures_. G.C. also acknowledges the partial
support of the funds FARB 2016 Politecnico di Milano Prog. TDG6ATEN04.

##  2\. Preliminaries

In this section, we introduce the main concepts and results of Optimal
Transport theory that will be extensively used in the rest of the paper. We
start by listing the adopted notation.

$\boldsymbol{b}_{\Phi}$ |  the barycenter of $\Phi\in\mathcal{P}(\mathsf{T\kern-1.5ptX})$ as in Definition 3.1;
---|---
$\mathrm{B}_{X}(x,r)$ |  the open ball with radius $r>0$ centered at $x\in X$;
$\mathrm{C}(X;Y)$ |  the set of continuous functions from $X$ to $Y$;
$\mathrm{C}_{b}(X)$ |  the set of bounded continuous real valued functions defined in $X$;
$\mathrm{C}_{c}(X)$ | the set of continuous real valued functions with compact support;
$\operatorname{Cyl}(\mathsf{X})$ |  the space of cylindrical functions on $\mathsf{X}$, see Definition 2.9;
$\operatorname{cl}({\boldsymbol{\mathrm{F}}}),\operatorname{co}({\boldsymbol{\mathrm{F}}})[\mu]$ |  the sequential closure and convexification of ${\boldsymbol{\mathrm{F}}}$, see Section 4.3;
$\overline{\operatorname{co}}({\boldsymbol{\mathrm{F}}})[\mu],\hat{{\boldsymbol{\mathrm{F}}}}$ |  sequential closure of convexification and extension of ${\boldsymbol{\mathrm{F}}}$, see Section 4.3;
${\frac{\mathrm{d}}{\mathrm{d}t}}^{\kern-3.0pt+}\zeta,{\frac{\mathrm{d}}{\mathrm{d}t}}_{\kern-1.0pt+}\zeta$ |  the right upper/lower Dini derivatives of $\zeta$, see (6.3);
$\mathrm{D}({\boldsymbol{\mathrm{F}}})$ |  the proper domain of a set-valued function as in Definition 4.1
$f_{\sharp}\nu$ |  the push-forward of $\nu\in\mathcal{P}(X)$ through the map $f:X\to Y$;
$\Gamma(\mu,\nu)$ |  the set of admissible couplings between $\mu,\nu$, see (2.1);
$\Gamma_{o}(\mu,\nu)$ |  the set of optimal couplings between $\mu,\nu$, see Definition 2.5;
$\Gamma_{o}^{i}({\mu_{0}},{\mu_{1}}|{\boldsymbol{\mathrm{F}}}),\,i=0,1$ |  the set of optimal couplings conditioned to ${\boldsymbol{\mathrm{F}}}$, see Definition 4.8;
$\mathcal{I}$ |  an interval of $\mathbb{R}$;
$\boldsymbol{i}_{X}(\cdot)$ |  the identity function on a set $X$;
$\mathrm{I}(\boldsymbol{\mu}|{\boldsymbol{\mathrm{F}}})$ |  the set of time instants $t$ s.t. $\mathsf{x}^{t}_{\sharp}\boldsymbol{\mu}$ belongs to $\mathrm{D}({\boldsymbol{\mathrm{F}}})$, see Definition 4.8;
$\Lambda,\Lambda_{o}$ |  the sets of couplings as in Definition 3.8 and Theorem 3.9;
$\mathsf{m}_{2}(\nu)$ |  the $2$-nd moment of $\nu\in\mathcal{P}(X)$ as in Definition 2.5;
$|\Phi|_{2}$ |  the $2$-nd moment of $\Phi\in\mathcal{P}(\mathsf{T\kern-1.5ptX})$ as in (3.2);
$|{\boldsymbol{\mathrm{F}}}|_{2}(\mu)$ |  the $2$-nd moment of ${\boldsymbol{\mathrm{F}}}$ at $\mu$ as in (6.17);
$|\dot{\mu}|(t)$ |  the metric derivative of a locally absolutely continuous curve $\mu$;
$\mathcal{P}(X)$ |  the set of Borel probability measures on the topological space $X$;
$\mathcal{P}_{b}(X)$ | the set of Borel probability measures with bounded support;
$\mathcal{P}_{2}(X)$ |  the subset of measures in $\mathcal{P}(X)$ with finite quadratic moments;
$\mathcal{P}_{2}^{sw}(\mathsf{X}\times\mathsf{Y})$ |  the space $\mathcal{P}_{2}(\mathsf{X}\times\mathsf{Y})$ endowed with a weaker topology as in Definition 2.14;
$\mathcal{P}(\mathsf{T\kern-1.5ptX}|\mu)$ |  the subset of $\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})$ with fixed first marginal $\mu$ as in (3.3);
$\left[\cdot,\cdot\right]_{r}$, $\left[\cdot,\cdot\right]_{l}$ |  the pseudo scalar products as in Definition 3.5;
$[\Phi,\boldsymbol{\vartheta}]_{r,t}$, $[\Phi,\boldsymbol{\vartheta}]_{l,t}$ |  the duality pairings as in Definition 3.17;
$[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{r,t}$, $[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{l,t}$ |  the duality pairings as in Definition 4.9;
$[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{0+},[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{1-}$ |  the limiting duality pairings as in Definition 4.11;
$\operatorname{supp}(\nu)$ |  the support of $\nu\in\mathcal{P}(X)$;
$\operatorname{Tan}_{\mu}\mathcal{P}_{2}(X)$ |  the tangent space defined in Theorem 2.10;
$W_{2}(\mu,\nu)$ |  the $L^{2}$-Wasserstein distance between $\mu$ and $\nu$, see Definition 2.5;
$\mathsf{X}$ | a separable Hilbert space;
$\mathsf{T\kern-1.5ptX}$ |  the tangent bundle to $\mathsf{X}$, usually endowed with the strong-weak topology;
$\mathsf{x},\mathsf{v},\operatorname{\mathsf{exp}}^{t}$ |  the projection and exponential maps defined in (3.1);
$\mathsf{x}^{t}$ |  the evaluation map defined in (3.4).

In the present paper we will mostly deal with Borel probability measures
defined in (subsets of) some separable Hilbert space endowed with the strong
or a weaker topology. The convenient setting is therefore provided by
Polish/Lusin and completely regular topological spaces.

Recall that a topological space $X$ is Polish (resp. Lusin) if its topology is
induced by a complete and separable metric (resp. is coarser than a Polish
topology). We will denote by $\mathcal{P}(X)$ the set of Borel probability
measures on $X$. If $X$ is Lusin, every measure $\mu\in\mathcal{P}(X)$ is also
a Radon measure, i.e. it satisfies

$\forall\,B\subset X\text{ Borel,
}\forall\,\varepsilon>0\quad\exists\,K\subset B\text{ compact s.t.
}\mu(B\setminus K)<\varepsilon.$

$X$ is _completely regular_ if it is Hausdorff and for every closed set $C$
and point $x\in X\setminus C$ there exists a continuous function $f:X\to[0,1]$
s.t. $f(x)=0$ and $f(C)=\\{1\\}$.

Given $X$ and $Y$ Lusin spaces, $\mu\in\mathcal{P}(X)$ and a Borel function
$f:X\to Y$, there is a canonical way to transfer the measure $\mu$ from $X$ to
$Y$ through $f$. This is called the _push forward_ of $\mu$ through $f$,
denoted by $f_{\sharp}\mu$ and defined by $(f_{\sharp}\mu)(B):=\mu(f^{-1}(B))$
for every Borel set $B$ in $Y$, or equivalently

$\int_{Y}\varphi\,\mathrm{d}(f_{\sharp}\mu)=\int_{X}\varphi\circ
f\,\mathrm{d}\mu$

for every $\varphi$ bounded (or nonnegative) real valued Borel function on
$Y$. A particular case occurs if $X=X_{1}\times X_{2}$, $Y=X_{i}$ and
$f=\pi^{i}$ is the projection on the $i$-th component, $i=1,2$. In this case,
$f$ is usually denoted with $\pi^{i}$ or $\pi^{X_{i}}$, and
$\pi^{X_{i}}_{\sharp}\mu$ is called the $i$-th marginal of $\mu$.
This notation is particularly useful when dealing with transport plans: given
$X_{1}$ and $X_{2}$ completely regular spaces and $\mu\in\mathcal{P}(X_{1})$,
$\nu\in\mathcal{P}(X_{2})$, we define

$\Gamma(\mu,\nu):=\left\\{\boldsymbol{\gamma}\in\mathcal{P}(X_{1}\times
X_{2})\mid\pi^{1}_{\sharp}\boldsymbol{\gamma}=\mu\,,\,\pi^{2}_{\sharp}\boldsymbol{\gamma}=\nu\right\\},$
(2.1)

i.e. the set of probability measures on the product space having $\mu$ and
$\nu$ as marginals.
On $\mathcal{P}(X)$ we consider the so called narrow topology which is the
coarsest topology on $\mathcal{P}(X)$ s.t. the maps
$\mu\mapsto\int_{X}\varphi\,\mathrm{d}\mu$ are continuous for every
$\varphi\in\rm\mathrm{C}_{b}(X)$, the space of real valued and bounded
continuous functions on $X$. In this way a net
$(\mu_{\alpha})_{\alpha\in\mathbb{A}}\subset\mathcal{P}(X)$ indexed by a
directed set $\mathbb{A}$ is said to converge narrowly to
$\mu\in\mathcal{P}(X)$, and we write $\mu_{\alpha}\to\mu$ in $\mathcal{P}(X)$,
if

$\lim_{\alpha}\int_{X}\varphi\,\mathrm{d}\mu_{\alpha}=\int_{X}\varphi\,\mathrm{d}\mu\quad\forall\varphi\in\rm\mathrm{C}_{b}(X).$

We recall the well known Prokhorov’s theorem in the context of completely
regular topological spaces (see [Sch73, Appendix]).

######  Theorem 2.1 (Prokhorov).

Let $X$ be a completely regular topological space and let
$\mathcal{F}\subset\mathcal{P}(X)$ be a tight subset i.e.

$\text{for every }\varepsilon>0\text{ there exists }K_{\varepsilon}\subset
X\text{ compact s.t. }\mu(X\setminus
K_{\varepsilon})<\varepsilon\,\;\forall\,\mu\in\mathcal{F}.$

Then $\mathcal{F}$ is relatively compact in $\mathcal{P}(X)$ w.r.t. the narrow
topology.

It is then relevant to know when a given $\mathcal{F}\subset\mathcal{P}(X)$ is
tight. If $X$ is a Lusin completely regular topological space, then the set
$\mathcal{F}=\\{\mu\\}\subset\mathcal{P}(X)$ is tight. Another trivial
criterion for tightness is the following: if
$\mathcal{F}\subset\mathcal{P}(X_{1}\times X_{2})$ is s.t.
$\mathcal{F}_{i}:=\\{\pi_{\sharp}^{i}\boldsymbol{\gamma}\mid\boldsymbol{\gamma}\in\mathcal{F}\\}\subset\mathcal{P}(X_{i})$
are tight for $i=1,2$, then also $\mathcal{F}$ is tight. We also recall the
following useful proposition (see [AGS08, Remark 5.1.5]).

######  Proposition 2.2.

Let $X$ be a Lusin completely regular topological space and let
$\mathcal{F}\subset\mathcal{P}(X)$. Then $\mathcal{F}$ is tight if and only if
there exists $\varphi:X\to[0,+\infty]$ with compact sublevels s.t.

$\sup_{\mu\in\mathcal{F}}\int_{X}\varphi\,\mathrm{d}\mu<+\infty.$

We recall the so-called _disintegration theorem_ (see e.g. [AGS08, Theorem
5.3.1]).

######  Theorem 2.3.

Let $\mathbb{X},X$ be Lusin completely regular topological spaces,
$\boldsymbol{\mu}\in\mathcal{P}(\mathbb{X})$ and $r:\mathbb{X}\to X$ a Borel
map. Denote with $\mu=r_{\sharp}\boldsymbol{\mu}\in\mathcal{P}(X)$. Then there
exists a $\mu$-a.e. uniquely determined Borel family of probability measures
$\\{\boldsymbol{\mu}_{x}\\}_{x\in X}\subset\mathcal{P}(\mathbb{X})$ such that
$\boldsymbol{\mu}_{x}(\mathbb{X}\setminus r^{-1}(x))=0$ for $\mu$-a.e. $x\in
X$, and

$\int_{\mathbb{X}}\varphi(\boldsymbol{x})\,\mathrm{d}\boldsymbol{\mu}(\boldsymbol{x})=\int_{X}\left(\int_{r^{-1}(x)}\varphi(\boldsymbol{x})\,\mathrm{d}\boldsymbol{\mu}_{x}(\boldsymbol{x})\right)\,\mathrm{d}\mu(x)$

for every bounded Borel map $\varphi:\mathbb{X}\to\mathbb{R}$.

######  Remark 2.4.

When $\mathbb{X}=X_{1}\times X_{2}$ and $r=\pi^{1}$, we can canonically
identify the disintegration $\\{\boldsymbol{\mu}_{x}\\}_{x\in
X_{1}}\subset\mathcal{P}(\mathbb{X})$ of
$\boldsymbol{\mu}\in\mathcal{P}(X_{1}\times X_{2})$ w.r.t.
$\mu=\pi^{1}_{\sharp}\boldsymbol{\mu}$ with a family of probability measures
$\\{\mu_{x_{1}}\\}_{x_{1}\in X_{1}}\subset\mathcal{P}(X_{2})$. We write
$\boldsymbol{\mu}=\displaystyle\int_{X_{1}}\mu_{x_{1}}\,\mathrm{d}\mu(x_{1})$.

###  2.1. Wasserstein distance in Hilbert spaces

Let $X$ be a separable (possibly infinite dimensional) Hilbert space. We will
denote by $X^{s}$ (respt. $X^{w}$) the Hilbert space endowed with its strong
(resp. weak) topology. Notice that $X^{w}$ is a Lusin completely regular
space. $X^{s}$ and $X^{w}$ share the same class of Borel sets and therefore of
Borel probability measures, which we will simply denote by $\mathcal{P}(X)$,
using $\mathcal{P}(X^{s})$ and $\mathcal{P}(X^{w})$ only when we will refer to
the correspondent topology. Finally, if $X$ has finite dimension then the two
topologies coincide.

We now list some properties of Wasserstein spaces and we refer to [AGS08, §7]
for a complete account of this matter.

######  Definition 2.5.

Given $\mu\in\mathcal{P}(X)$ we define

$\mathsf{m}_{2}^{2}(\mu):=\int_{X}|x|^{2}\,\mathrm{d}\mu(x),\qquad\mathcal{P}_{2}(X):=\\{\mu\in\mathcal{P}(X)\mid\mathsf{m}_{2}(\mu)<+\infty\\}.$

The $L^{2}$-Wasserstein distance between
$\mu,\mu^{\prime}\in\mathcal{P}_{2}(X)$ is defined as

$\displaystyle W_{2}^{2}(\mu,\mu^{\prime})$
$\displaystyle:=\inf\left\\{\int_{X\times
X}|x-y|^{2}\,\mathrm{d}\boldsymbol{\gamma}(x,y)\mid\boldsymbol{\gamma}\in\Gamma(\mu,\mu^{\prime})\right\\}.$
(2.2)

The set of elements of $\Gamma(\mu,\mu^{\prime})$ realizing the infimum in
(2.2) is denoted with $\Gamma_{o}(\mu,\mu^{\prime})$. We say that a measure
$\boldsymbol{\gamma}\in\mathcal{P}_{2}(X\times X)$ is optimal if
$\boldsymbol{\gamma}\in\Gamma_{o}(\pi^{1}_{\sharp}\boldsymbol{\gamma},\pi^{2}_{\sharp}\boldsymbol{\gamma})$.

We will denote by $\mathrm{B}(\mu,\varrho)$ the open ball centered at $\mu$
with radius $\varrho$ in $\mathcal{P}_{2}(X)$. The metric space
$(\mathcal{P}_{2}(X),W_{2})$ enjoys many interesting properties: here we only
recall that it is a complete and separable metric space and that
$W_{2}$-convergence (sometimes denoted with
$\overset{W_{2}}{\longrightarrow}$) is stronger than the narrow convergence.
In particular, given $(\mu_{n})_{n\in\mathbb{N}}\subset\mathcal{P}_{2}(X)$ and
$\mu\in\mathcal{P}_{2}(X)$, we have [AGS08, Remark 7.1.11] that

$\mu_{n}\overset{W_{2}}{\to}\mu,\text{ as
}n\to+\infty\quad\Longleftrightarrow\quad\begin{cases}\mu_{n}\to\mu\text{ in
}\mathcal{P}(X^{s}),\\\
\mathsf{m}_{2}(\mu_{n})\to\mathsf{m}_{2}(\mu),\end{cases}\text{ as
}n\to+\infty.\\\ $ (2.3)

Finally, we recall that sequences converging in $(\mathcal{P}_{2}(X),W_{2})$
are tight. More precisely we have the following characterization of
compactness in $\mathcal{P}_{2}(X)$.

######  Lemma 2.6 (Relative compactness in $\mathcal{P}_{2}(X)$).

A subset $\mathcal{K}\subset\mathcal{P}_{2}(X)$ is relatively compact w.r.t.
the $W_{2}$-topology if and only if

  1. (1)

$\mathcal{K}$ is tight w.r.t. $X^{s}$,

  2. (2)

$\mathcal{K}$ is uniformly $2$-integrable, i.e.

$\lim_{k\to\infty}\sup_{\mu\in\mathcal{K}}\int_{|x|\geq
k}|x|^{2}\,\mathrm{d}\mu=0.$ (2.4)

###### Proof.

Tightness is clearly a necessary condition; concerning (2.4) let us notice
that the maps $F_{k}:\mathcal{P}_{2}(X)\to[0,\infty)$,
$F_{k}(\mu):=\int_{|x|\geq k}|x|^{2}\,\mathrm{d}\mu$ are upper semicontinuous,
are decreasing w.r.t. $k$, and converge pointwise to $0$ for every
$\mu\in\mathcal{P}_{2}(X)$. Therefore, if $\mathcal{K}$ is relatively compact,
they converge uniformly to $0$ thanks to Dini’s Theorem.

In order to prove that (1) and (2) are also sufficient for relative
compactness, it is sufficient to check that every sequence
$(\mu_{n})_{n\in\mathbb{N}}$ in $\mathcal{K}$ has a convergent subsequence.
Applying Prokhorov Theorem 2.1 we can find $\mu\in\mathcal{P}(X)$ and a
convergent subsequence $k\mapsto\mu_{n(k)}$ such that $\mu_{n(k)}\to\mu$ in
$\mathcal{P}(X^{s})$. Since $\mathsf{m}_{2}(\mu_{n})$ is uniformly bounded,
then $\mu\in\mathcal{P}_{2}(X)$. Applying [AGS08, Lemma 5.1.7], we also get
$\lim_{k\to\infty}\mathsf{m}_{2}(\mu_{n(k)})=\mathsf{m}_{2}(\mu)$ so that
$\lim_{k\to\infty}W_{2}(\mu_{n(k)},\mu)=0$ by (2.3). ∎

######  Definition 2.7 (Geodesics).

A curve $(\mu_{t})_{t\in[0,1]}\subset\mathcal{P}_{2}(X)$ is said to be a
_(constant speed) geodesic_ if for all $0\leq s\leq t\leq 1$ we have

$W_{2}(\mu_{s},\mu_{t})=(t-s)W_{2}(\mu_{0},\mu_{1}).$

We also say that $(\mu_{t})_{t\in[0,1]}$ is a _geodesic from $\mu_{0}$ to
$\mu_{1}$_. We say that $A\subset\mathcal{P}_{2}(X)$ is a _geodesically
convex_ set if for any pair $\mu_{0},\mu_{1}\in A$ there exists a geodesic
$(\mu_{t})_{t\in[0,1]}$ from $\mu_{0}$ to $\mu_{1}$ such that
$(\mu_{t})_{t\in[0,1]}\subset A$.

We recall also the following useful properties of geodesics (see [AGS08,
Theorem 7.2.1, Theorem 7.2.2]).

######  Theorem 2.8 (Properties of geodesics).

Let $\mu_{0},\mu_{1}\in\mathcal{P}_{2}(X)$ and
$\boldsymbol{\mu}\in\Gamma_{o}(\mu_{0},\mu_{1})$. Then $(\mu_{t})_{t\in[0,1]}$
defined by

$\mu_{t}:=(\mathsf{x}^{t})_{\sharp}\boldsymbol{\mu},\quad t\in[0,1],$ (2.5)

is a (constant speed) geodesic from $\mu_{0}$ to $\mu_{1}$, where
$\mathsf{x}^{t}:X^{2}\to X$ is given by,
$\mathsf{x}^{t}(x_{0},x_{1}):=(1-t)x_{0}+tx_{1}$. Conversely, any (constant
speed) geodesic $(\mu_{t})_{t\in[0,1]}$ from $\mu_{0}$ to $\mu_{1}$ admits the
representation (2.5) for a suitable plan
$\boldsymbol{\mu}\in\Gamma_{o}(\mu_{0},\mu_{1})$.
Finally, if $(\mu_{t})_{t\in[0,1]}$ is a geodesic connecting $\mu_{0}$ to
$\mu_{1}$, then for every $t\in(0,1)$ there exists a unique optimal plan
between $\mu_{0}$ and $\mu_{t}$ (resp. between $\mu_{t}$ and $\mu_{1}$) and it
is concentrated on a map.

We define moreover the analogous of $\mathrm{C}^{\infty}_{c}(\mathbb{R}^{d})$
when we have $X$ in place of $\mathbb{R}^{d}$.

######  Definition 2.9 ($\operatorname{Cyl}(X)$).

We denote by $\Pi_{d}(X)$ the space of linear maps $\pi:X\to\mathbb{R}^{d}$ of
the form $\pi(x)=(\langle x,e_{1}\rangle,\cdots,\langle x,e_{d}\rangle)$ for
an orthonormal set $\\{e_{1},\cdots,e_{d}\\}$ of $X$. A function
$\varphi:X\to\mathbb{R}$ belongs to the space of cylindrical functions on $X$,
$\operatorname{Cyl}(X)$, if it is of the form

$\varphi=\psi\circ\pi$

where $\pi\in\Pi_{d}(X)$ and $\psi\in\mathrm{C}^{\infty}_{c}(\mathbb{R}^{d})$.

We recall the following result (see [AGS08, Theorem 8.3.1, Proposition 8.4.5
and Proposition 8.4.6]) characterizing locally absolutely continuous curves in
$\mathcal{P}_{2}(X)$ defined in a (bounded or unbounded) open interval
$\mathcal{I}\subset\mathbb{R}$. We use the equivalent notation
$\mu(t)\equiv\mu_{t}$ for the evaluation at time $t\in\mathcal{I}$ of a map
$\mu:\mathcal{I}\to\mathcal{P}_{2}(X)$.

######  Theorem 2.10 (Wasserstein velocity field).

Let $\mu:\mathcal{I}\to\mathcal{P}_{2}(X)$ be a locally absolutely continuous
curve defined in an open interval $\mathcal{I}\subset\mathbb{R}$. There exists
a Borel vector field $\boldsymbol{v}:\mathcal{I}\times X\to X$ and a set
$A(\mu)\subset\mathcal{I}$ with $\mathcal{L}(\mathcal{I}\setminus A(\mu))=0$
such that for every $t\in A(\mu)$

$\begin{gathered}\boldsymbol{v}_{t}\in\operatorname{Tan}_{\mu_{t}}\mathcal{P}_{2}(X):={}\overline{\\{\nabla\varphi\mid\varphi\in\operatorname{Cyl}(X)\\}}^{L^{2}_{\mu_{t}}(X;X)},\\\
\int_{X}|\boldsymbol{v}_{t}|^{2}\,\mathrm{d}\mu_{t}=|\dot{\mu}_{t}|^{2}=\lim_{h\to
0}\frac{W_{2}^{2}(\mu_{t+h},\mu_{t})}{h^{2}},\end{gathered}$

and the continuity equation

$\partial_{t}\mu_{t}+\nabla\cdot(\boldsymbol{v}_{t}\mu_{t})=0$

holds in the sense of distributions in $\mathcal{I}\times X$. Moreover,
$\boldsymbol{v}_{t}$ is uniquely determined in $L^{2}_{\mu_{t}}(X;X)$ for
$t\in A(\mu)$ and

$\lim_{h\to
0}\frac{W_{2}((\boldsymbol{i}_{X}+h\boldsymbol{v}_{t})_{\sharp}\mu_{t},\mu_{t+h})}{|h|}=0\quad\text{for
every }t\in A(\mu).$ (2.6)

We conclude this section with a useful property concerning the upper
derivative of the Wasserstein distance, which in fact holds in every metric
space.

######  Lemma 2.11.

Let $\mu:\mathcal{I}\to\mathcal{P}_{2}(\mathsf{X})$,
$\nu\in\mathcal{P}_{2}(\mathsf{X})$, $t\in\mathcal{I}$,
$\boldsymbol{\sigma}\in\Gamma_{o}(\mu_{t},\nu)$, and consider the constant
speed geodesic $(\nu_{t,s})_{s\in[0,1]}$ defined by
$\nu_{t,s}:=(\mathsf{x}^{s})_{\sharp}\boldsymbol{\sigma}$ for every
$s\in[0,1]$. The upper right and left Dini derivatives
$b^{\pm}:(0,1]\to\mathbb{R}$ defined by

$\displaystyle b^{+}(s):={}$ $\displaystyle\frac{1}{2s}\limsup_{h\downarrow
0}\frac{W_{2}^{2}(\mu_{t+h},\nu_{t,s})-W_{2}^{2}(\mu_{t},\nu_{t,s})}{h},$
$\displaystyle b^{-}(s):={}$ $\displaystyle\frac{1}{2s}\limsup_{h\downarrow
0}\frac{W_{2}^{2}(\mu_{t},\nu_{t,s})-W_{2}^{2}(\mu_{t-h},\nu_{t,s})}{h}$

are respectively decreasing and increasing in $(0,1]$.

###### Proof.

Take $0\leq s^{\prime}<s\leq 1$. Since $(\nu_{t,s})_{s\in[0,1]}$ is a constant
speed geodesic from $\mu_{t}$ to $\nu$, we have

$W_{2}(\mu_{t},\nu_{t,s})=W_{2}(\mu_{t},\nu_{t,s^{\prime}})+W_{2}(\nu_{t,s^{\prime}},\nu_{t,s}),$

then, by triangular inequality

$\displaystyle W_{2}(\mu_{t+h},\nu_{t,s})-W_{2}(\mu_{t},\nu_{t,s})$
$\displaystyle\leq
W_{2}(\mu_{t+h},\nu_{t,s^{\prime}})+W_{2}(\nu_{t,s^{\prime}},\nu_{t,s})-W_{2}(\mu_{t},\nu_{t,s})$
$\displaystyle=W_{2}(\mu_{t+h},\nu_{t,s^{\prime}})-W_{2}(\mu_{t},\nu_{t,s^{\prime}}).$

Dividing by $h>0$ and passing to the limit as $h\downarrow 0$ we obtain that
the function $a:[0,1]\to\mathbb{R}$ defined by

$a^{+}(s):=\limsup_{h\downarrow
0}\frac{W_{2}(\mu_{t+h},\nu_{t,s})-W_{2}(\mu_{t},\nu_{t,s})}{h}$

is decreasing. It is then sufficient to observe that for $s>0$

$b^{+}(s)=a^{+}(s)\frac{W_{2}(\mu_{t},\nu_{t,s})}{s}=a^{+}(s)W_{2}(\mu_{t},\nu).$

The monotonicity property of $b^{-}$ follows by the same argument. ∎

###  2.2. A strong-weak topology on measures in product spaces

Let us consider the case when $X=\mathsf{X}\times\mathsf{Y}$ where
$\mathsf{X},\mathsf{Y}$ are separable Hilbert spaces. $X$ is naturally endowed
with the product Hilbert norm and $\mathcal{P}_{2}(X)$ with the corresponding
topology induced by the $L^{2}$-Wasserstein distance. However, it will be
extremely useful to endow $\mathcal{P}_{2}(X)$ with a weaker topology which is
related to the strong-weak topology on $X$, i.e. the product topology of
$\mathsf{X}^{s}\times\mathsf{Y}^{w}$. We follow the approach of [NS21], to
which we refer for the proofs of the results presented in this section.

In order to define the topology, we consider the space
$\mathrm{C}^{sw}_{2}(\mathsf{X}\times\mathsf{Y})$ of test functions
$\zeta:\mathsf{X}\times\mathsf{Y}\to\mathbb{R}$ such that

$\displaystyle\zeta\text{ is sequentially continuous in
$\mathsf{X}^{s}\times\mathsf{Y}^{w}$,}$ $\displaystyle\forall\,\varepsilon>0\
\exists\,A_{\varepsilon}\geq 0:|\zeta(x,y)|\leq
A_{\varepsilon}(1+|x|_{\mathsf{X}}^{2})+\varepsilon|y|_{\mathsf{Y}}^{2}\quad\text{for
every }(x,y)\in\mathsf{X}\times\mathsf{Y}.$

Notice in particular that functions in
$\mathrm{C}^{sw}_{2}(\mathsf{X}\times\mathsf{Y})$ have quadratic growth. We
endow $\mathrm{C}^{sw}_{2}(X)$ with the norm

$\|\zeta\|_{\mathrm{C}^{sw}_{2}(X)}:=\sup_{(x,y)\in
X}\frac{|\zeta(x,y)|}{1+|x|_{\mathsf{X}}^{2}+|y|_{\mathsf{Y}}^{2}}.$

######  Remark 2.12.

When $\mathsf{Y}$ is finite dimensional, (2.2) is equivalent to the continuity
of $\zeta$.

######  Lemma 2.13.

$(\mathrm{C}^{sw}_{2}(\mathsf{X}\times\mathsf{Y}),\|\cdot\|_{\mathrm{C}^{sw}_{2}(\mathsf{X}\times\mathsf{Y})})$
is a Banach space.

######  Definition 2.14 (Topology of
$\mathcal{P}_{2}^{sw}(\mathsf{X}\times\mathsf{Y})$, [NS21]).

We denote by $\mathcal{P}_{2}^{sw}(\mathsf{X}\times\mathsf{Y})$ the space
$\mathcal{P}_{2}(\mathsf{X}\times\mathsf{Y})$ endowed with the coarsest
topology which makes the following functions continuous

$\boldsymbol{\mu}\mapsto\int\zeta(x,y)\,\mathrm{d}\boldsymbol{\mu}(x,y),\quad\zeta\in\mathrm{C}^{sw}_{2}(\mathsf{X}\times\mathsf{Y}).$

It is obvious that the topology of
$\mathcal{P}_{2}(\mathsf{X}\times\mathsf{Y})$ is finer than the topology of
$\mathcal{P}_{2}^{sw}(\mathsf{X}\times\mathsf{Y})$ and the latter is finer
than the topology of $\mathcal{P}(\mathsf{X}^{s}\times\mathsf{Y}^{w})$. It is
worth noticing that

$\text{any bounded bilinear form $B:\mathsf{X}\times\mathsf{Y}\to\mathbb{R}$
belongs to }\mathrm{C}^{sw}_{2}(\mathsf{X}\times\mathsf{Y}),$

so that for every net
$(\boldsymbol{\mu}_{\alpha})_{\alpha\in\mathbb{A}}\subset\mathcal{P}(\mathsf{X}\times\mathsf{Y})$
indexed by a directed set $\mathbb{A}$, we have

$\lim_{\alpha\in\mathbb{A}}\boldsymbol{\mu}_{\alpha}=\boldsymbol{\mu}\quad\text{in
}\mathcal{P}_{2}^{sw}(\mathsf{X}\times\mathsf{Y})\quad\Rightarrow\quad\lim_{\alpha\in\mathbb{A}}\int
B\,\mathrm{d}\boldsymbol{\mu}_{\alpha}=\int B\,\mathrm{d}\boldsymbol{\mu}.$
(2.7)

The following proposition justifies the interest in the
$\mathcal{P}_{2}^{sw}(\mathsf{X}\times\mathsf{Y})$-topology.

######  Proposition 2.15.

  1. (1)

If
$(\boldsymbol{\mu}_{\alpha})_{\alpha\in\mathbb{A}}\subset\mathcal{P}_{2}(\mathsf{X}\times\mathsf{Y})$
is a net indexed by the directed set $\mathbb{A}$ and
$\boldsymbol{\mu}\in\mathcal{P}_{2}(\mathsf{X}\times\mathsf{Y})$ satisfies

    1. (a)

$\boldsymbol{\mu}_{\alpha}\to\boldsymbol{\mu}$ in
$\mathcal{P}(\mathsf{X}^{s}\times\mathsf{Y}^{w})$,

    2. (b)

$\displaystyle\lim_{\alpha\in\mathbb{A}}\int|x|_{\mathsf{X}}^{2}\,\mathrm{d}\boldsymbol{\mu}_{\alpha}(x,y)=\int|x|_{\mathsf{X}}^{2}\,\mathrm{d}\boldsymbol{\mu}(x,y)$,

    3. (c)

$\displaystyle\sup_{\alpha\in\mathbb{A}}\int|y|_{\mathsf{Y}}^{2}\,\mathrm{d}\boldsymbol{\mu}_{\alpha}(x,y)<\infty$,

then $\boldsymbol{\mu}_{\alpha}\to\boldsymbol{\mu}$ in
$\mathcal{P}_{2}^{sw}(\mathsf{X}\times\mathsf{Y})$. The converse property
holds for sequences: if $\mathbb{A}=\mathbb{N}$ and
$\boldsymbol{\mu}_{n}\to\boldsymbol{\mu}$ in
$\mathcal{P}_{2}^{sw}(\mathsf{X}\times\mathsf{Y})$ as $n\to\infty$ then
properties (a), (b), (c) hold.

  2. (2)

For every compact set $\mathcal{K}\subset\mathcal{P}_{2}(\mathsf{X}^{s})$ and
every constant $c<\infty$ the sets

$\mathcal{K}_{c}:=\Big{\\{}\boldsymbol{\mu}\in\mathcal{P}_{2}(\mathsf{X}\times\mathsf{Y}):\pi^{\mathsf{X}}_{\sharp}\boldsymbol{\mu}\in\mathcal{K},\quad\int|y|_{\mathsf{Y}}^{2}\,\mathrm{d}\boldsymbol{\mu}(x,y)\leq
c\Big{\\}}$

are compact and metrizable in
$\mathcal{P}_{2}^{sw}(\mathsf{X}\times\mathsf{Y})$ (in particular they are
sequentially compact).

It is worth noticing that the topology
$\mathcal{P}_{2}^{ws}(\mathsf{X}\times\mathsf{Y})$ is strictly weaker than
$\mathcal{P}_{2}(\mathsf{X}\times\mathsf{Y})$ even when $\mathsf{Y}$ is finite
dimensional. In fact, $\mathrm{C}^{sw}_{2}(\mathsf{X}\times\mathsf{Y})$ does
not contain the quadratic function $(x,y)\mapsto|y|_{\mathsf{Y}}^{2}$, so that
convergence of the quadratic moment w.r.t. $y$ is not guaranteed.

##  3\. Directional derivatives and probability measures on the tangent bundle

From now on, we will denote by $\mathsf{X}$ a separable Hilbert space with
norm $|\cdot|$ and scalar product $\langle\cdot,\cdot\rangle$. We denote by
$\mathsf{T\kern-1.5ptX}$ the tangent bundle to $\mathsf{X}$, which is
identified with the set $\mathsf{X}\times\mathsf{X}$ with the induced norm
$|(x,v)|:=\big{(}|x|^{2}+|v|^{2}\big{)}^{1/2}$ and the strong-weak topology of
$\mathsf{X}^{s}\times\mathsf{X}^{w}$(i.e. the product of the strong topology
on the first component and the weak topology on the second one). We will
denote by $\mathsf{x},\mathsf{v}:\mathsf{T\kern-1.5ptX}\to\mathsf{X}$ the
projection maps and by
$\operatorname{\mathsf{exp}}^{t}:\mathsf{T\kern-1.5ptX}\to\mathsf{X}$ the
exponential map defined by

$\mathsf{x}(x,v):=x,\quad\mathsf{v}(x,v)=v,\quad\operatorname{\mathsf{exp}}^{t}(x,v):=x+tv.$
(3.1)

The set $\mathcal{P}(\mathsf{T\kern-1.5ptX})$ is defined thanks to the
identification of $\mathsf{T\kern-1.5ptX}$ with $\mathsf{X}\times\mathsf{X}$
and it is endowed with the narrow topology induced by the strong-weak topology
in $\mathsf{T\kern-1.5ptX}$. For $\Phi\in\mathcal{P}(\mathsf{T\kern-1.5ptX})$
we define

$|\Phi|_{2}^{2}:=\int_{\mathsf{T\kern-1.5ptX}}|v|^{2}\,\mathrm{d}\Phi(x,v).$
(3.2)

We denote by $\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})$ the subset of
$\mathcal{P}(\mathsf{T\kern-1.5ptX})$ of measures for which
$\int\big{(}|x|^{2}+|v|^{2}\big{)}\,\mathrm{d}\Phi<\infty$ endowed with the
topology of $\mathcal{P}_{2}^{sw}(\mathsf{T\kern-1.5ptX})$ as in Section 2.2.
If $\mu\in\mathcal{P}(\mathsf{X})$ we will also consider

$\mathcal{P}(\mathsf{T\kern-1.5ptX}|\mu):=\Big{\\{}\Phi\in\mathcal{P}(\mathsf{T\kern-1.5ptX})\mid\mathsf{x}_{\sharp}\Phi=\mu\Big{\\}},\quad\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}|\mu):=\Big{\\{}\Phi\in\mathcal{P}(\mathsf{T\kern-1.5ptX}|\mu):|\Phi|_{2}<\infty\Big{\\}}.$
(3.3)

We will also deal with the product space $\mathsf{X}^{2}$: we will use the
notation

$\mathsf{x}^{t}:\mathsf{X}^{2}\to\mathsf{X},\quad\mathsf{x}^{t}(x_{0},x_{1}):=(1-t)x_{0}+tx_{1},\quad
t\in[0,1].$ (3.4)

If $\boldsymbol{v}\in L^{2}_{\mu}(\mathsf{X};\mathsf{X})$ we can consider the
probability

$\Phi_{\mu,\boldsymbol{v}}:=(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v})_{\sharp}\mu\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}|\mu).$
(3.5)

In this case we will say that $\Phi$ is concentrated on the graph of the map
$\boldsymbol{v}$. More generally, given a Borel family of probability measures
$(\Phi_{x})_{x\in\mathsf{X}}\subset\mathcal{P}_{2}(\mathsf{X})$ satisfying

$\int\Big{(}\int|v|^{2}\,\mathrm{d}\Phi_{x}(v)\Big{)}\,\mathrm{d}\mu(x)<\infty$
(3.6)

we can consider the probability

$\Phi=\int_{\mathsf{X}}\Phi_{x}\,\mathrm{d}\mu(x)\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}|\mu).$
(3.7)

Conversely, every $\Phi\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}|\mu)$ can be
disintegrated by a Borel family
$(\Phi_{x})_{x\in\mathsf{X}}\subset\mathcal{P}_{2}(\mathsf{X})$ satisfying
(3.6) and (3.7). $\Phi$ can be associated to a vector field $\boldsymbol{v}\in
L^{2}_{\mu}(\mathsf{X};\mathsf{X})$ if and only if for $\mu$-a.e.
$x\in\mathsf{X}$ $\Phi_{x}=\delta_{\boldsymbol{v}(x)}$. Recalling the
disintegration Theorem 2.3 and Remark 2.4, we give the following definition.

######  Definition 3.1.

Given $\Phi\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}|\mu)$, the _barycenter of
$\Phi$_ is the function $\boldsymbol{b}_{\Phi}\in
L^{2}_{\mu}(\mathsf{X};\mathsf{X})$ defined by

$\boldsymbol{b}_{\Phi}(x):=\int_{\mathsf{X}}v\,\mathrm{d}\Phi_{x}(v)\quad\text{for
}\mu\text{-a.e. }x\in\mathsf{X},$

where $\\{\Phi_{x}\\}_{x\in\mathsf{X}}\subset\mathcal{P}_{2}(\mathsf{X})$ is
the disintegration of $\Phi$ w.r.t. $\mu$.

######  Remark 3.2.

Notice that, by the linearity of the scalar product, we get the following
identity which will be useful later

$\int_{\mathsf{X}}\langle\boldsymbol{\zeta}(x),\boldsymbol{b}_{\Phi}(x)\rangle\,\mathrm{d}\mu(x)=\int_{\mathsf{T\kern-1.5ptX}}\langle\boldsymbol{\zeta}(x),v\rangle\,\mathrm{d}\Phi(x,v)\quad\forall\,\boldsymbol{\zeta}\in
L^{2}_{\mu}(\mathsf{X};\mathsf{X}).$ (3.8)

###  3.1. Directional derivatives of the Wasserstein distance and duality
pairings

Our starting point is a relevant semi-concavity property of the function

$f(s,t):=\frac{1}{2}W_{2}^{2}(\operatorname{\mathsf{exp}}^{s}_{\sharp}\Phi_{0},\operatorname{\mathsf{exp}}^{t}_{\sharp}\Phi_{1}),\quad
s,t\in\mathbb{R},$ (3.9)

with $\Phi_{0},\Phi_{1}\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})$. We first
state an auxiliary result, whose proof is based on [AGS08, Proposition 7.3.1].

######  Lemma 3.3.

Let $\Phi_{0},\Phi_{1}\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})$,
$s,t\in\mathbb{R}$, and let
$\boldsymbol{\vartheta}^{s,t}\in\Gamma(\operatorname{\mathsf{exp}}^{s}_{\sharp}\Phi_{0},\operatorname{\mathsf{exp}}^{t}_{\sharp}\Phi_{1})$.
Then there exists $\boldsymbol{\Theta}^{s,t}\in\Gamma(\Phi_{0},\Phi_{1})$ such
that
$(\operatorname{\mathsf{exp}}^{s},\operatorname{\mathsf{exp}}^{t})_{\sharp}\boldsymbol{\Theta}^{s,t}=\boldsymbol{\vartheta}^{s,t}$.

###### Proof.

Define, for every $r,s,t\in\mathbb{R}$,

$\Sigma^{r}:\mathsf{T\kern-1.5ptX}\to\mathsf{T\kern-1.5ptX},\quad\Sigma^{r}(x,v)=(\operatorname{\mathsf{exp}}^{r}(x,v),v);\quad\Lambda^{s,t}:\mathsf{T\kern-1.5ptX}\times\mathsf{T\kern-1.5ptX}\to\mathsf{T\kern-1.5ptX}\times\mathsf{T\kern-1.5ptX},\quad\Lambda^{s,t}:=(\Sigma^{s},\Sigma^{t}).$

Consider the probabilities
$(\Sigma^{s})_{\sharp}\Phi_{0},(\Sigma^{t})_{\sharp}\Phi_{1}$ and
$\boldsymbol{\vartheta}^{s,t}$. They are constructed in such a way that there
exists
$\boldsymbol{\Psi}^{s,t}\in\mathcal{P}(\mathsf{T\kern-1.5ptX}\times\mathsf{T\kern-1.5ptX})$
s.t.

$(\mathsf{x}^{0},\mathsf{v}^{0})_{\sharp}\boldsymbol{\Psi}^{s,t}=(\Sigma^{s})_{\sharp}\Phi_{0},\quad(\mathsf{x}^{1},\mathsf{v}^{1})_{\sharp}\boldsymbol{\Psi}^{s,t}=(\Sigma^{t})_{\sharp}\Phi_{1},\quad(\mathsf{x}^{0},\mathsf{x}^{1})_{\sharp}\boldsymbol{\Psi}^{s,t}=\boldsymbol{\vartheta}^{s,t},$

where we adopted the notation $\mathsf{x}^{i}(x_{0},v_{0},x_{1},v_{1}):=x_{i}$
and $\mathsf{v}^{i}(x_{0},v_{0},x_{1},v_{1}):=v_{i}$, $i=0,1$. We conclude by
taking
$\boldsymbol{\Theta}^{s,t}:=(\Lambda^{-s,-t})_{\sharp}\boldsymbol{\Psi}^{s,t}$.
∎

######  Proposition 3.4.

Let $\Phi_{0},\Phi_{1}\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})$ with
$\mu_{1}=\mathsf{x}_{\sharp}\Phi_{1}$ and
$\varphi^{2}:=|\Phi_{0}|_{2}^{2}+|\Phi_{1}|_{2}^{2}$, let
$f:\mathbb{R}^{2}\to\mathbb{R}$ be the function defined by (3.9) and let
$h,g:\mathbb{R}\to\mathbb{R}$ be defined by

$h(s):=f(s,s)=\frac{1}{2}W_{2}^{2}(\operatorname{\mathsf{exp}}^{s}_{\sharp}\Phi_{0},\operatorname{\mathsf{exp}}^{s}_{\sharp}\Phi_{1}),\quad
g(s):=f(s,0)=\frac{1}{2}W_{2}^{2}(\operatorname{\mathsf{exp}}^{s}_{\sharp}\Phi_{0},\mu_{1}),\quad
s\in\mathbb{R}.$ (3.10)

  1. (1)

The function $(s,t)\mapsto f(s,t)-\frac{1}{2}\varphi^{2}(s^{2}+t^{2})$ is
concave, i.e. it holds

$\displaystyle f((1-\alpha)s_{0}+\alpha s_{1},(1-\alpha)t_{0}+\alpha t_{1})$
$\displaystyle\geq(1-\alpha)f(s_{0},t_{0})+\alpha f(s_{1},t_{1})$ (3.11)
$\displaystyle\quad-\frac{1}{2}\alpha(1-\alpha)\Big{[}(s_{1}-s_{0})^{2}+(t_{1}-t_{0})^{2}\Big{]}\varphi^{2}$

for every $s_{0},s_{1},t_{0},t_{1}\in\mathbb{R}$ and every $\alpha\in[0,1]$.

  2. (2)

The function $s\mapsto h(s)-\varphi^{2}s^{2}$ is concave.

  3. (3)

the function $s\mapsto g(s)-\frac{1}{2}s^{2}|\Phi_{0}|_{2}^{2}$ is concave.

###### Proof.

Let us first prove (3.11). We set $s:=(1-\alpha)s_{0}+\alpha s_{1}$,
$t:=(1-\alpha)t_{0}+\alpha t_{1}$ and we apply Lemma 3.3 to find
$\boldsymbol{\Theta}\in\Gamma(\Phi_{0},\Phi_{1})$ such that
$(\operatorname{\mathsf{exp}}^{s},\operatorname{\mathsf{exp}}^{t})_{\sharp}\boldsymbol{\Theta}\in\Gamma_{o}(\operatorname{\mathsf{exp}}^{s}_{\sharp}\Phi_{0},\operatorname{\mathsf{exp}}^{t}_{\sharp}\Phi_{1})$.
Then, recalling the Hilbertian identity

$|(1-\alpha)a+\alpha
b|^{2}=(1-\alpha)|a|^{2}+\alpha|b|^{2}-\alpha(1-\alpha)|a-b|^{2},\quad
a,b\in\mathsf{X},$

we have

$\displaystyle W_{2}^{2}$
$\displaystyle(\operatorname{\mathsf{exp}}^{s}_{\sharp}\Phi_{0},\operatorname{\mathsf{exp}}^{t}_{\sharp}\Phi_{1})=\int|x_{0}+sv_{0}-(x_{1}+tv_{1})|^{2}\,\mathrm{d}\boldsymbol{\Theta}=$
$\displaystyle=\int|(1-\alpha)(x_{0}+s_{0}v_{0})+\alpha(x_{0}+s_{1}v_{0})-(1-\alpha)(x_{1}+t_{0}v_{1})-\alpha(x_{1}+t_{1}v_{1})|^{2}\,\mathrm{d}\boldsymbol{\Theta}$
$\displaystyle=(1-\alpha)\int|x_{0}+s_{0}v_{0}-(x_{1}+t_{0}v_{1})|^{2}\,\mathrm{d}\boldsymbol{\Theta}+\alpha\int|x_{0}+s_{1}v_{0}-(x_{1}+t_{1}v_{1})|^{2}\,\mathrm{d}\boldsymbol{\Theta}$
$\displaystyle\quad-\alpha(1-\alpha)\int|(s_{1}-s_{0})v_{0}+(t_{1}-t_{0})v_{1}|^{2}\,\mathrm{d}\boldsymbol{\Theta}$
$\displaystyle\geq(1-\alpha)W_{2}^{2}(\operatorname{\mathsf{exp}}_{\sharp}^{s_{0}}\Phi_{0},\operatorname{\mathsf{exp}}^{t_{0}}_{\sharp}\Phi_{1})+\alpha
W_{2}^{2}(\operatorname{\mathsf{exp}}_{\sharp}^{s_{1}}\Phi_{0},\operatorname{\mathsf{exp}}^{t_{1}}_{\sharp}\Phi_{1})$
$\displaystyle\quad-\alpha(1-\alpha)\Big{(}(s_{1}-s_{0})^{2}+(t_{1}-t_{0})^{2}\Big{)}\Big{(}\int|v_{0}|^{2}\,\mathrm{d}\Phi_{0}+\int|v_{1}|^{2}\,\mathrm{d}\Phi_{1}\Big{)}.$

which is the thesis. Claims (2) and (3) follow as particular cases when $t=s$
or $t=0$. ∎

Semi-concavity is a useful tool to guarantee the existence of one-sided
partial derivatives at $(0,0)$: for every $\alpha,\beta\in\mathbb{R}$ we have
(see e.g. [HL93, Ch. VI, Prop. 1.1.2]) that

$\displaystyle f_{r}^{\prime}(\alpha,\beta)$
$\displaystyle=\lim_{\varrho\downarrow
0}\frac{f(\alpha\varrho,\beta\varrho)-f(0,0)}{\varrho}=\sup_{\varrho>0}\frac{f(\alpha\varrho,\beta\varrho)-f(0,0)}{\varrho}-\frac{\varrho\varphi^{2}}{2}(\alpha^{2}+\beta^{2}),$
$\displaystyle f_{l}^{\prime}(\alpha,\beta)$
$\displaystyle=\lim_{\varrho\downarrow
0}\frac{f(0,0)-f(-\alpha\varrho,-\beta\varrho)}{\varrho}=\inf_{\varrho>0}\frac{f(0,0)-f(-\alpha\varrho,-\beta\varrho)}{\varrho}+\frac{\varrho\varphi^{2}}{2}(\alpha^{2}+\beta^{2}).$

$f_{r}^{\prime}$ (resp. $f^{\prime}_{l}$) is a concave (resp. convex) and
positively $1$-homogeneous function, i.e. a superlinear (resp. sublinear)
function. They satisfy

$\displaystyle f_{r}^{\prime}(-\alpha,-\beta)$
$\displaystyle=-f^{\prime}_{l}(\alpha,\beta),\quad
f_{l}^{\prime}(\alpha,\beta)\geq f_{r}^{\prime}(\alpha,\beta)\quad\text{for
every }\alpha,\beta\in\mathbb{R},$ (3.12) $\displaystyle
f_{r}^{\prime}(\alpha,\beta)$ $\displaystyle\geq\alpha
f_{r}^{\prime}(1,0)+\beta f_{r}^{\prime}(0,1)\quad\text{for every
}\alpha,\beta\geq 0,$ (3.13) $\displaystyle f(s,t)$ $\displaystyle\leq
f(0,0)+f^{\prime}_{r}(s,t)-\frac{\varphi^{2}}{2}(s^{2}+t^{2})\quad\text{for
every }s,t\in\mathbb{R}.$

Notice moreover that

$f_{r}^{\prime}(1,0)=g^{\prime}_{r}(0)=\lim_{\varrho\downarrow
0}\frac{g(\varrho)-g(0)}{\varrho}$

where $g$ is the function defined in (3.10); a similar representation holds
for $f_{l}^{\prime}(1,0)$. We introduce the following notation for
$f^{\prime}_{r}$, $f^{\prime}_{l}$, $g^{\prime}_{r}$ and $g^{\prime}_{l}$.

######  Definition 3.5.

Let $\mu_{0},\mu_{1}\in\mathcal{P}_{2}(\mathsf{X})$,
$\Phi_{0}\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}|\mu_{0})$ and
$\Phi_{1}\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}|\mu_{1})$. Recalling the
definitions of $f$ and $g$ given by (3.9) and (3.10), we define

$\displaystyle\left[\Phi_{0},\mu_{1}\right]_{r}$
$\displaystyle:=g^{\prime}_{r}(0)=f_{r}^{\prime}(1,0)=\lim_{s\downarrow
0}\frac{W_{2}^{2}(\operatorname{\mathsf{exp}}^{s}_{\sharp}\Phi_{0},\mu_{1})-W_{2}^{2}(\mu_{0},\mu_{1})}{2s},$
$\displaystyle\left[\Phi_{0},\mu_{1}\right]_{l}$
$\displaystyle:=g^{\prime}_{l}(0)=f_{l}^{\prime}(1,0)=\lim_{s\downarrow
0}\frac{W_{2}^{2}(\mu_{0},\mu_{1})-W_{2}^{2}(\operatorname{\mathsf{exp}}^{-s}_{\sharp}\Phi_{0},\mu_{1})}{2s},$
and analogously $\displaystyle\left[\Phi_{0},\Phi_{1}\right]_{r}$
$\displaystyle:=f^{\prime}_{r}(1,1)=\lim_{t\downarrow
0}\frac{W_{2}^{2}(\operatorname{\mathsf{exp}}^{t}_{\sharp}\Phi_{0},\operatorname{\mathsf{exp}}^{t}_{\sharp}\Phi_{1})-W_{2}^{2}(\mu_{0},\mu_{1})}{2t},$
$\displaystyle\left[\Phi_{0},\Phi_{1}\right]_{l}$
$\displaystyle:=f_{l}^{\prime}(1,1)=\lim_{t\downarrow
0}\frac{W_{2}^{2}(\mu_{0},\mu_{1})-W_{2}^{2}(\operatorname{\mathsf{exp}}^{-t}_{\sharp}\Phi_{0},\operatorname{\mathsf{exp}}^{-t}_{\sharp}\Phi_{1})}{2t}.$

######  Remark 3.6.

Notice that
$\left[\Phi_{0},\mu_{1}\right]_{r}=\left[\Phi_{0},\Phi_{\mu_{1}}\right]_{r}$
and
$\left[\Phi_{0},\mu_{1}\right]_{l}=\left[\Phi_{0},\Phi_{\mu_{1}}\right]_{l}$,
where

$\Phi_{\mu_{1}}=(\boldsymbol{i}_{\mathsf{X}},0)_{\sharp}\mu_{1}\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}).$

Moreover, using the notation

$-\Phi:=J_{\sharp}\Phi,\quad\Phi\in\mathcal{P}(\mathsf{T\kern-1.5ptX}),\
J(x,v):=(x,-v),$ (3.14)

we have

$\left[-\Phi_{0},-\Phi_{1}\right]_{r}=-\left[\Phi_{0},\Phi_{1}\right]_{l},\quad\text{
and
}\quad\left[-\Phi_{0},\mu_{1}\right]_{r}=-\left[\Phi_{0},\mu_{1}\right]_{l}.$

In particular, the properties of $\left[\cdot,\cdot\right]_{l}$ (in
$\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})\times\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})$
or $\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})\times\mathcal{P}_{2}(\mathsf{X})$)
and the ones of $\left[\cdot,\cdot\right]_{r}$ in
$\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})\times\mathcal{P}_{2}(\mathsf{X})$ can
be easily derived by the corresponding ones of $\left[\cdot,\cdot\right]_{r}$
in
$\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})\times\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})$.

Recalling (3.13) and (3.12) we obtain the following result.

######  Corollary 3.7.

For every $\mu_{0},\mu_{1}\in\mathcal{P}_{2}(\mathsf{X})$ and for every
$\Phi_{0}\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}|\mu_{0})$,
$\Phi_{1}\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}|\mu_{1})$, it holds

$\left[\Phi_{0},\mu_{1}\right]_{r}+\left[\Phi_{1},\mu_{0}\right]_{r}\leq\left[\Phi_{0},\Phi_{1}\right]_{r}\quad\text{
and
}\quad\left[\Phi_{0},\mu_{1}\right]_{l}+\left[\Phi_{1},\mu_{0}\right]_{l}\geq\left[\Phi_{0},\Phi_{1}\right]_{l}.$

Let us now show an important equivalent characterization of the quantities we
have just introduced. As usual we will denote by
$\mathsf{x}^{0},\mathsf{v}^{0},\mathsf{x}^{1}:\mathsf{T\kern-1.5ptX}\times\mathsf{X}\to\mathsf{X}$
the projection maps of a point $(x_{0},v_{0},x_{1})$ in
$\mathsf{T\kern-1.5ptX}\times\mathsf{X}$ (and similarly for
$\mathsf{T\kern-1.5ptX}\times\mathsf{T\kern-1.5ptX}$ with
$\mathsf{x}^{0},\mathsf{v}^{0},\mathsf{x}^{1},\mathsf{v}^{1}$).

First of all we introduce the following sets.

######  Definition 3.8.

For every $\Phi_{0}\in\mathcal{P}(\mathsf{T\kern-1.5ptX})$ with
$\mu_{0}=\mathsf{x}_{\sharp}\Phi_{0}$ and
$\mu_{1}\in\mathcal{P}_{2}(\mathsf{X})$ we set

$\Lambda(\Phi_{0},\mu_{1}):=\left\\{\boldsymbol{\sigma}\in\Gamma(\Phi_{0},\mu_{1})\mid(\mathsf{x}^{0},\mathsf{x}^{1})_{\sharp}\boldsymbol{\sigma}\in\Gamma_{o}(\mu_{0},\mu_{1})\right\\}.$

Analogously, for every
$\Phi_{0},\Phi_{1}\in\mathcal{P}(\mathsf{T\kern-1.5ptX})$ with
$\mu_{0}=\mathsf{x}_{\sharp}\Phi_{0}$ and
$\mu_{1}=\mathsf{x}_{\sharp}\Phi_{1}$ in $\mathcal{P}_{2}(\mathsf{X})$ we set

$\Lambda(\Phi_{0},\Phi_{1}):=\left\\{\boldsymbol{\Theta}\in\Gamma(\Phi_{0},\Phi_{1})\mid(\mathsf{x}^{0},\mathsf{x}^{1})_{\sharp}\boldsymbol{\Theta}\in\Gamma_{o}(\mu_{0},\mu_{1})\right\\}.$

In the following proposition and subsequent corollary, we provide a useful
characterization of the pairings $\left[\cdot,\cdot\right]_{r}$ and
$\left[\cdot,\cdot\right]_{l}$.

######  Theorem 3.9.

For every $\Phi_{0},\Phi_{1}\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})$ and
$\mu_{1}\in\mathcal{P}_{2}(\mathsf{X})$ we have

$\displaystyle\left[\Phi_{0},\mu_{1}\right]_{r}$
$\displaystyle=\min\left\\{\int_{\mathsf{T\kern-1.5ptX}\times\mathsf{X}}\langle
x_{0}-x_{1},v_{0}\rangle\,\mathrm{d}\boldsymbol{\sigma}\mid\boldsymbol{\sigma}\in\Lambda(\Phi_{0},\mu_{1})\right\\},$
(3.15) $\displaystyle\left[\Phi_{0},\Phi_{1}\right]_{r}$
$\displaystyle=\min\left\\{\int_{\mathsf{T\kern-1.5ptX}\times\mathsf{T\kern-1.5ptX}}\langle
x_{0}-x_{1},v_{0}-v_{1}\rangle\,\mathrm{d}\boldsymbol{\Theta}\mid\boldsymbol{\Theta}\in\Lambda(\Phi_{0},\Phi_{1})\right\\}.$
(3.16)

We denote by $\Lambda_{o}(\Phi_{0},\mu_{1})$ (resp.
$\Lambda_{o}(\Phi_{0},\Phi_{1})$) the subset of $\Lambda(\Phi_{0},\mu_{1})$
(resp. $\Lambda(\Phi_{0},\Phi_{1})$) where the minimum in (3.15) (resp.
(3.16)) is attained.

###### Proof.

First, we recall that the minima in the right hand side are attained since
$\Lambda(\Phi_{0},\mu_{1})$ and $\Lambda(\Phi_{0},\Phi_{1})$ are compact
subsets of $\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}\times\mathsf{X})$ and
$\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}\times\mathsf{T\kern-1.5ptX})$
respectively by Lemma 2.6 and the integrands are continuous functions with
quadratic growth. Thanks to Remark 3.6, we only need to prove the second
equality. For every $\boldsymbol{\Theta}\in\Lambda(\Phi_{0},\Phi_{1})$ and
setting $\mu_{0}=\mathsf{x}_{\sharp}\Phi_{0}$,
$\mu_{1}=\mathsf{x}_{\sharp}\Phi_{1}$, we have

$\displaystyle W_{2}^{2}$
$\displaystyle(\operatorname{\mathsf{exp}}^{t}_{\sharp}(\Phi_{0}),\operatorname{\mathsf{exp}}^{t}_{\sharp}(\Phi_{1}))\leq\int_{\mathsf{T\kern-1.5ptX}\times\mathsf{T\kern-1.5ptX}}|(x_{0}-x_{1})+t(v_{0}-v_{1})|^{2}\,\mathrm{d}\boldsymbol{\Theta}$
$\displaystyle=\int_{\mathsf{X}^{2}}|x_{0}-x_{1}|^{2}\,\mathrm{d}(\mathsf{x}^{0},\mathsf{x}^{1})_{\sharp}\boldsymbol{\Theta}+2t\int_{\mathsf{T\kern-1.5ptX}\times\mathsf{T\kern-1.5ptX}}\langle
x_{0}-x_{1},v_{0}-v_{1}\rangle\,\mathrm{d}\boldsymbol{\Theta}+t^{2}\int_{\mathsf{X}^{2}}|v_{0}-v_{1}|^{2}\,\mathrm{d}\boldsymbol{\Theta}$
$\displaystyle=W_{2}^{2}(\mu_{0},\mu_{1})+2t\int_{\mathsf{T\kern-1.5ptX}\times\mathsf{T\kern-1.5ptX}}\langle
x_{0}-x_{1},v_{0}-v_{1}\rangle\,\mathrm{d}\boldsymbol{\Theta}+t^{2}\int_{\mathsf{X}^{2}}|v_{0}-v_{1}|^{2}\,\mathrm{d}\boldsymbol{\Theta}.$

and this immediately implies

$\displaystyle\left[\Phi_{0},\Phi_{1}\right]_{r}\leq\min\left\\{\int_{\mathsf{T\kern-1.5ptX}\times\mathsf{T\kern-1.5ptX}}\langle
x_{0}-x_{1},v_{0}-v_{1}\rangle\,\mathrm{d}\boldsymbol{\Theta}\mid\boldsymbol{\Theta}\in\Lambda(\Phi_{0},\Phi_{1})\right\\}.$

In order to prove the converse inequality, thanks to Lemma 3.3, for every
$t>0$ we can find $\boldsymbol{\Theta}_{t}\in\Gamma(\Phi_{0},\Phi_{1})$ s.t.

$(\operatorname{\mathsf{exp}}^{t},\operatorname{\mathsf{exp}}^{t})_{\sharp}\boldsymbol{\Theta}_{t}\in\Gamma_{o}(\operatorname{\mathsf{exp}}^{t}_{\sharp}\Phi_{0},\operatorname{\mathsf{exp}}^{t}_{\sharp}\Phi_{1}).$

Then

$\begin{split}\frac{W_{2}^{2}(\operatorname{\mathsf{exp}}^{t}_{\sharp}\Phi_{0},\operatorname{\mathsf{exp}}^{t}_{\sharp}\Phi_{1})-W_{2}^{2}(\mu_{0},\mu_{1})}{2t}&\geq\frac{1}{2t}\int_{\mathsf{T\kern-1.5ptX}\times\mathsf{T\kern-1.5ptX}}|(x_{0}-x_{1})+t(v_{0}-v_{1})|^{2}\,\mathrm{d}\boldsymbol{\Theta}_{t}\\\
&\quad-\frac{1}{2t}\int_{\mathsf{T\kern-1.5ptX}\times\mathsf{T\kern-1.5ptX}}|x_{0}-x_{1}|^{2}\,\mathrm{d}\boldsymbol{\Theta}_{t}\\\
&\geq\int_{\mathsf{T\kern-1.5ptX}\times\mathsf{T\kern-1.5ptX}}\langle
x_{0}-x_{1},v_{0}-v_{1}\rangle\,\mathrm{d}\boldsymbol{\Theta}_{t}.\end{split}$
(3.17)

Since $\Gamma(\Phi_{0},\Phi_{1})$ is compact in
$\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}\times\mathsf{T\kern-1.5ptX})$, there
exists a vanishing sequence $k\mapsto t(k)$ and
$\boldsymbol{\Theta}\in\Gamma(\Phi_{0},\Phi_{1})$ s.t.
$\boldsymbol{\Theta}_{t(k)}\to\boldsymbol{\Theta}$ in
$\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}\times\mathsf{T\kern-1.5ptX})$.
Moreover it holds
$(\operatorname{\mathsf{exp}}^{t(k)},\operatorname{\mathsf{exp}}^{t(k)})_{\sharp}\boldsymbol{\Theta}_{t(k)}\to(\mathsf{x}^{0},\mathsf{x}^{1})_{\sharp}\boldsymbol{\Theta}$
in $\mathcal{P}(\mathsf{T\kern-1.5ptX}\times\mathsf{T\kern-1.5ptX})$ so that
$(\mathsf{x}^{0},\mathsf{x}^{1})_{\sharp}\boldsymbol{\Theta}\in\Gamma_{o}(\mu_{0},\mu_{1})$,
and therefore $\boldsymbol{\Theta}\in\Lambda(\Phi_{0},\Phi_{1})$. The
convergence in
$\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}\times\mathsf{T\kern-1.5ptX})$ yields

$\lim_{k}\int_{\mathsf{T\kern-1.5ptX}\times\mathsf{T\kern-1.5ptX}}\langle
x_{0}-x_{1},v_{0}-v_{1}\rangle\,\mathrm{d}\boldsymbol{\Theta}_{t(k)}=\int_{\mathsf{T\kern-1.5ptX}\times\mathsf{T\kern-1.5ptX}}\langle
x_{0}-x_{1},v_{0}-v_{1}\rangle\,\mathrm{d}\boldsymbol{\Theta},$

so that, passing to the limit in (3.17) along the sequence $t(k)$, we obtain

$\displaystyle\left[\Phi_{0},\Phi_{1}\right]_{r}\geq\int_{\mathsf{T\kern-1.5ptX}\times\mathsf{T\kern-1.5ptX}}\langle
x_{0}-x_{1},v_{0}-v_{1}\rangle\,\mathrm{d}\boldsymbol{\Theta}\quad\text{for
some }\boldsymbol{\Theta}\in\Lambda(\Phi_{0},\Phi_{1}).\qed$

######  Corollary 3.10.

Let $\Phi_{0},\Phi_{1}\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})$ and
$\mu_{1}\in\mathcal{P}_{2}(\mathsf{X})$, then

$\displaystyle\left[\Phi,\mu_{1}\right]_{l}$
$\displaystyle=\max\left\\{\int_{\mathsf{T\kern-1.5ptX}\times\mathsf{X}}\langle
x_{0}-x_{1},v_{0}\rangle\,\mathrm{d}\boldsymbol{\sigma}\mid\boldsymbol{\sigma}\in\Lambda(\Phi_{0},\mu_{1})\right\\},$
(3.18) $\displaystyle\left[\Phi_{0},\Phi_{1}\right]_{l}$
$\displaystyle=\max\left\\{\int_{\mathsf{T\kern-1.5ptX}\times\mathsf{T\kern-1.5ptX}}\langle
x_{0}-x_{1},v_{0}-v_{1}\rangle\,\mathrm{d}\boldsymbol{\Theta}\mid\boldsymbol{\Theta}\in\Lambda(\Phi_{0},\Phi_{1})\right\\}.$

###  3.2. Right and left derivatives of the Wasserstein distance along a.c.
curves

Let us now discuss the differentiability of the map $\mathcal{I}\ni
t\mapsto\frac{1}{2}W_{2}^{2}(\mu(t),\nu)$ along a locally absolutely
continuous curve $\mu:\mathcal{I}\to\mathcal{P}_{2}(\mathsf{X})$, with
$\mathcal{I}$ an open interval of $\mathbb{R}$ and
$\nu\in\mathcal{P}_{2}(\mathsf{X})$.

######  Theorem 3.11.

Let $\mu:\mathcal{I}\to\mathcal{P}_{2}(\mathsf{X})$ be a locally absolutely
continuous curve and let
$\boldsymbol{v}:\mathcal{I}\times\mathsf{X}\to\mathsf{X}$ and $A(\mu)$ be as
in Theorem 2.10. Then, for every $\nu\in\mathcal{P}_{2}(\mathsf{X})$ and every
$t\in A(\mu)$, it holds

$\displaystyle\lim_{h\downarrow
0}\frac{W_{2}^{2}(\mu_{t+h},\nu)-W_{2}^{2}(\mu_{t},\nu)}{2h}$
$\displaystyle=\left[(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}_{t})_{\sharp}\mu_{t},\nu\right]_{r},$
(3.19) $\displaystyle\lim_{h\uparrow
0}\frac{W_{2}^{2}(\mu_{t+h},\nu)-W_{2}^{2}(\mu_{t},\nu)}{2h}$
$\displaystyle=\left[(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}_{t})_{\sharp}\mu_{t},\nu\right]_{l},$

so that the map $s\mapsto W_{2}^{2}(\mu_{s},\nu)$ is left and right
differentiable at every $t\in A(\mu)$. In particular,

  1. (1)

if $t\in A(\mu)$ and $\nu\in\mathcal{P}_{2}(\mathsf{X})$ are s.t. there exists
a unique optimal transport plan between $\mu_{t}$ and $\nu$, then the map
$s\mapsto W_{2}^{2}(\mu_{s},\nu)$ is differentiable at $t$;

  2. (2)

there exists a subset $A({\mu,\nu})\subset A(\mu)$ of full Lebesgue measure
such that $s\mapsto W_{2}^{2}(\mu_{s},\nu)$ is differentiable in
$A({\mu,\nu})$ and

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}W_{2}^{2}(\mu_{t},\nu)$
$\displaystyle=\left[(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}_{t})_{\sharp}\mu_{t},\nu\right]_{r}=\left[(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}_{t})_{\sharp}\mu_{t},\nu\right]_{l}$
$\displaystyle=\int\langle\boldsymbol{v}_{t}(x_{1}),x_{1}-x_{2}\rangle\,\mathrm{d}\boldsymbol{\mu}(x_{1}.x_{2})\quad\text{for
every }\boldsymbol{\mu}\in\Gamma_{o}(\mu_{t},\nu),\ t\in A({\mu,\nu}).$

###### Proof.

Let $\nu\in\mathcal{P}_{2}(\mathsf{X})$ and for every $t\in\mathcal{I}$ we set
$\Phi_{t}:=(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}_{t})_{\sharp}\mu_{t}\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})$.
By Theorem 3.9, we have

$\displaystyle\lim_{h\downarrow
0}\frac{W_{2}^{2}(\operatorname{\mathsf{exp}}_{\sharp}^{h}\Phi_{t},\nu)-W_{2}^{2}(\mu_{t},\nu)}{2h}$
$\displaystyle=\left[(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}_{t})_{\sharp}\mu_{t},\nu\right]_{r},$
$\displaystyle\lim_{h\uparrow
0}\frac{W_{2}^{2}(\operatorname{\mathsf{exp}}_{\sharp}^{h}\Phi_{t},\nu)-W_{2}^{2}(\mu_{t},\nu)}{2h}$
$\displaystyle=\left[(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}_{t})_{\sharp}\mu_{t},\nu\right]_{l}.$

Since
$\operatorname{\mathsf{exp}}_{\sharp}^{h}\Phi_{t}=(\boldsymbol{i}_{\mathsf{X}}+h\boldsymbol{v}_{t})_{\sharp}\mu_{t}$,
then thanks to Theorem 2.10 we have that the above limits coincide
respectively with the limits in the statement, for all $t\in A(\mu)$.

Claim (1) comes by the characterizations given in Theorem 3.9 and Corollary
3.10. Indeed, if there exists a unique optimal transport plan between
$\mu_{t}$ and $\nu$, then
$\left[(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}_{t})_{\sharp}\mu_{t},\nu\right]_{r}=\left[(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}_{t})_{\sharp}\mu_{t},\nu\right]_{l}$.

Claim (2) is a simple consequence of the fact that $s\mapsto
W_{2}^{2}(\mu_{s},\nu)$ is differentiable a.e. in $\mathcal{I}$. ∎

######  Remark 3.12.

Thanks to [AGS08, Proposition 8.5.4], in Theorem 3.11 we can actually replace
$\boldsymbol{v}$ with any Borel velocity field $\boldsymbol{w}$ solving the
continuity equation for $\mu$ and s.t.
$\|\boldsymbol{w}_{t}\|_{L^{2}_{\mu_{t}}}\in L^{1}_{loc}(\mathcal{I})$.
Indeed, we notice that by [AGS08, Lemma 5.3.2],

$\Lambda((\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}_{t})_{\sharp}\mu_{t},\nu)=\\{(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}_{t},\boldsymbol{i}_{\mathsf{X}})_{\sharp}\boldsymbol{\gamma}\mid\boldsymbol{\gamma}\in\Gamma_{o}(\mu_{t},\nu)\\}.$

See Appendix B for a further discussion about Theorem 3.11.

######  Theorem 3.13.

Let $\mu^{1},\mu^{2}:\mathcal{I}\to\mathcal{P}_{2}(\mathsf{X})$ be locally
absolutely continuous curves and let
$\boldsymbol{v}^{1},\boldsymbol{v}^{2}:\mathcal{I}\times\mathsf{X}\to\mathsf{X}$
be the corresponding Wasserstein velocity fields satisfying (2.6) in
$A({\mu^{1}})$ and $A({\mu^{2}})$ respectively. Then, for every $t\in
A({\mu^{1}})\cap A({\mu^{2}})$, it holds

$\displaystyle\lim_{h\downarrow
0}\frac{W_{2}^{2}(\mu^{1}_{t+h},\mu^{2}_{t+h})-W_{2}^{2}(\mu^{1}_{t},\mu^{2}_{t})}{2h}$
$\displaystyle=\left[(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}^{1}_{t})_{\sharp}\mu^{1}_{t},(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}^{2}_{t})_{\sharp}\mu^{2}_{t}\right]_{r},$
$\displaystyle\lim_{h\uparrow
0}\frac{W_{2}^{2}(\mu^{1}_{t+h},\mu^{2}_{t+h})-W_{2}^{2}(\mu^{1}_{t},\mu^{2}_{t})}{2h}$
$\displaystyle=\left[(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}^{1}_{t})_{\sharp}\mu^{1}_{t},(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}^{2}_{t})_{\sharp}\mu^{2}_{t}\right]_{l}.$

In particular, there exists a subset $A\subset A({\mu^{1}})\cap A({\mu^{2}})$
of full Lebesgue measure such that $s\mapsto
W_{2}^{2}(\mu^{1}_{s},\mu^{2}_{s})$ is differentiable in $A$ and

$\displaystyle\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}W_{2}^{2}(\mu^{1}_{t},\mu^{2}_{t})$
$\displaystyle=\left[(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}^{1}_{t})_{\sharp}\mu^{1}_{t},(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}^{2}_{t})_{\sharp}\mu^{2}_{t}\right]_{r}=\left[(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}^{1}_{t})_{\sharp}\mu^{1}_{t},(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}^{2}_{t})_{\sharp}\mu^{2}_{t}\right]_{l}$
(3.20)
$\displaystyle=\int\langle\boldsymbol{v}^{1}_{t}-\boldsymbol{v}^{2}_{t},x_{1}-x_{2}\rangle\,\mathrm{d}\boldsymbol{\mu}(x_{1},x_{2})\quad\text{for
every }\boldsymbol{\mu}\in\Gamma_{o}(\mu^{1}_{t},\mu^{2}_{t}),\ t\in A.$

The proof of Theorem 3.13 follows by the same argument of the proof of Theorem
3.11.

###  3.3. Convexity and semicontinuity of duality parings

We want now to investigate the semicontinuity and convexity properties of the
functionals $\left[\cdot,\cdot\right]_{r}$ and $\left[\cdot,\cdot\right]_{l}$.

######  Lemma 3.14.

Let
$(\Phi_{n})_{n\in\mathbb{N}}\subset\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})$ be
converging to $\Phi$ in $\mathcal{P}_{2}^{sw}(\mathsf{T\kern-1.5ptX})$, and
let $(\nu_{n})_{n\in\mathbb{N}}\subset\mathcal{P}_{2}(\mathsf{X})$ be
converging to $\nu$ in $\mathcal{P}_{2}(\mathsf{X})$. Then

$\liminf_{n}\left[\Phi_{n},\nu_{n}\right]_{r}\geq\left[\Phi,\nu\right]_{r}\quad\text{
and
}\quad\limsup_{n}\left[\Phi_{n},\nu_{n}\right]_{l}\leq\left[\Phi,\nu\right]_{l}.$
(3.21)

Finally, if $(\Phi_{n}^{i})_{n\in\mathbb{N}}$, $i=0,1$, are sequences
converging to $\Phi^{i}$ in $\mathcal{P}_{2}^{sw}(\mathsf{T\kern-1.5ptX})$
then

$\liminf_{n\to\infty}\left[\Phi^{0}_{n},\Phi^{1}_{n}\right]_{r}\geq\left[\Phi^{0},\Phi^{1}\right]_{r},\qquad\limsup_{n\to\infty}\left[\Phi^{0}_{n},\Phi^{1}_{n}\right]_{l}\geq\left[\Phi^{0},\Phi^{1}\right]_{l}.$
(3.22)

###### Proof.

We just consider the proof of the first inequality (3.21); the other
statements follow by similar arguments and by Remark 3.6.

We can extract a subsequence of $(\Phi_{n})_{n\in\mathbb{N}}$ (not relabeled)
s.t. the $\liminf$ is achieved as a limit. We have to prove that

$\lim_{n}\left[\Phi_{n},\nu_{n}\right]_{r}\geq\left[\Phi,\nu\right]_{r}.$
(3.23)

For every $n\in\mathbb{N}$ take
$\boldsymbol{\sigma}_{n}\in\Lambda_{o}(\Phi_{n},\nu_{n})$ with
$\bar{\boldsymbol{\vartheta}}_{n}=(\mathsf{x}^{0},\mathsf{x}^{1})_{\sharp}\boldsymbol{\sigma}_{n}$,
and observe that the family
$(\bar{\boldsymbol{\vartheta}}_{n})_{n\in\mathbb{N}}$ is relatively compact in
$\mathcal{P}_{2}(\mathsf{X}^{2})$ (since the marginals of
$\bar{\boldsymbol{\vartheta}}_{n}$ are converging w.r.t. $W_{2}$) so that
$(\boldsymbol{\sigma}_{n})_{n\in\mathbb{N}}$ is relatively compact in
$\mathcal{P}_{2}^{sws}(\mathsf{T\kern-1.5ptX}\times\mathsf{X})$ by Proposition
2.15 since the moments
$\int|v_{0}|^{2}\,\mathrm{d}\boldsymbol{\sigma}_{n}(x_{0},v_{0},x_{1})=|\Phi_{n}|^{2}_{2}$
are uniformly bounded by assumption. Thus, possibly passing to a further
subsequence, we have that $(\boldsymbol{\sigma}_{n})_{n\in\mathbb{N}}$
converges to some $\boldsymbol{\sigma}$ in
$\mathcal{P}_{2}^{sws}(\mathsf{T\kern-1.5ptX}\times\mathsf{X})$. In particular
$\boldsymbol{\sigma}\in\Lambda(\Phi,\nu)$ since optimality of the
$\mathsf{X}^{2}$ marginals is preserved by narrow convergence.

(2.7) then yields

$\lim_{n\to\infty}\left[\Phi_{n},\nu_{n}\right]_{r}=\lim_{n\to\infty}\int\langle
v_{0},x_{0}-x_{1}\rangle\,\mathrm{d}\boldsymbol{\sigma}_{n}=\int\langle
v_{0},x_{0}-x_{1}\rangle\,\mathrm{d}\boldsymbol{\sigma}$

which yields (3.23) since the RHS is larger than $\left[\Phi,\nu\right]_{r}$
by Theorem 3.9. ∎

######  Remark 3.15.

Notice that in the special case in which $\Lambda(\Phi,\nu)$ is a singleton,
then the limit exists and it holds

$\lim_{n\to\infty}\left[\Phi_{n},\nu_{n}\right]_{r}=\left[\Phi,\nu\right]_{r},\qquad\lim_{n\to\infty}\left[\Phi_{n},\nu_{n}\right]_{l}=\left[\Phi,\nu\right]_{l}.$

######  Lemma 3.16.

For every $\mu,\nu\in\mathcal{P}_{2}(\mathsf{X})$ the maps
$\Phi\mapsto\left[\Phi,\nu\right]_{r}$ and
$(\Phi,\Psi)\mapsto\left[\Phi,\Psi\right]_{r}$ (resp.
$\Phi\mapsto\left[\Phi,\nu\right]_{l}$ and
$(\Phi,\Psi)\mapsto\left[\Phi,\Psi\right]_{l}$) are convex (resp. concave) in
$\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}|\mu)$ and
$\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}|\mu)\times\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}|\nu)$.

###### Proof.

We prove the convexity of $(\Phi,\Psi)\mapsto\left[\Phi,\Psi\right]_{r}$ in
$\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}|\mu)\times\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}|\nu)$;
the argument of the proofs of the other statements are completely analogous.

Let $\Phi_{k}\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}|\mu)$,
$\Psi_{k}\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}|\nu)$, and let
$\beta_{k}\geq 0$, with $\sum_{k}\beta_{k}=1$, $k=1,\cdots,K$. We set
$\Phi=\sum_{k=1}^{K}\beta_{k}\Phi_{k}$,
$\Psi=\sum_{k=1}^{K}\beta_{k}\Psi_{k}$, For every $k$ let us select
$\boldsymbol{\Theta}_{k}\in\Lambda(\Phi_{k},\Psi_{k})$ such that

$\left[\Phi_{k},\Psi_{k}\right]_{r}=\int\langle
v_{1}-v_{0},x_{1}-x_{0}\rangle\,\mathrm{d}\boldsymbol{\Theta}_{k}.$

It is not difficult to check that
$\boldsymbol{\Theta}:=\sum_{k}\beta_{k}\boldsymbol{\Theta}_{k}\in\Lambda(\Phi,\Psi)$
so that

$\left[\Phi,\Psi\right]_{r}\leq\int\langle
v_{1}-v_{0},x_{1}-x_{0}\rangle\,\mathrm{d}\boldsymbol{\Theta}=\sum_{k}\beta_{k}\int\langle
v_{1}-v_{0},x_{1}-x_{0}\rangle\,\mathrm{d}\boldsymbol{\Theta}_{k}=\sum_{k}\beta_{k}\left[\Phi_{k},\Psi_{k}\right]_{r}.\qed$

###  3.4. Behaviour of duality pairings along geodesics

We have seen that the duality pairings $\left[\cdot,\cdot\right]_{r}$ and
$\left[\cdot,\cdot\right]_{l}$ may differ when the collection of optimal plans
$\Gamma_{o}(\mu_{0},\mu_{1})$ contains more than one element. It is natural to
expect a simpler behaviour along geodesics. We will introduce the following
definition, where we use the notation

$\mathsf{x}^{t}(x_{0},x_{1}):=(1-t)x_{0}+tx_{1},\quad\mathsf{v}^{0}(x_{0},v_{0},x_{1}):=v_{0}\quad\text{for
every
}(x_{0},v_{0},x_{1})\in\mathsf{T\kern-1.5ptX}\times\mathsf{X},\,t\in[0,1].$

######  Definition 3.17.

For $\boldsymbol{\vartheta}\in\mathcal{P}_{2}(\mathsf{X}\times\mathsf{X})$,
$t\in[0,1]$, $\vartheta_{t}=\mathsf{x}^{t}_{\sharp}\boldsymbol{\vartheta}$ and
$\Phi\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}|\vartheta_{t})$, we set

$\Gamma_{t}(\Phi,\boldsymbol{\vartheta}):=\left\\{\boldsymbol{\sigma}\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}\times\mathsf{X})\mid(\mathsf{x}^{0},\mathsf{x}^{1})_{\sharp}\boldsymbol{\sigma}=\boldsymbol{\vartheta},\quad(\mathsf{x}^{t}\circ(\mathsf{x}^{0},\mathsf{x}^{1}),\mathsf{v}^{0})_{\sharp}\boldsymbol{\sigma}=\Phi\right\\},$

which is not empty since
$\vartheta_{t}=\mathsf{x}^{t}_{\sharp}\boldsymbol{\vartheta}=\mathsf{x}_{\sharp}\Phi$.
We set

$\displaystyle\left[\Phi,\boldsymbol{\vartheta}\right]_{b,t}$
$\displaystyle:=\int\langle
x_{0}-x_{1},\boldsymbol{b}_{\Phi}(\mathsf{x}^{t}(x_{0},x_{1}))\rangle\,\mathrm{d}\boldsymbol{\vartheta}(x_{0},x_{1}),$
$\displaystyle[\Phi,\boldsymbol{\vartheta}]_{r,t}$
$\displaystyle:=\min\left\\{\int\langle
x_{0}-x_{1},v_{0}\rangle\,\mathrm{d}\boldsymbol{\sigma}(x_{0},v_{0},x_{1})\mid\boldsymbol{\sigma}\in\Gamma_{t}(\Phi,\boldsymbol{\vartheta})\right\\},$
$\displaystyle[\Phi,\boldsymbol{\vartheta}]_{l,t}$
$\displaystyle:=\max\left\\{\int\langle
x_{0}-x_{1},v_{0}\rangle\,\mathrm{d}\boldsymbol{\sigma}(x_{0},v_{0},x_{1})\mid\boldsymbol{\sigma}\in\Gamma_{t}(\Phi,\boldsymbol{\vartheta})\right\\}.$
If moreover
$\Phi_{0}\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}|\vartheta_{0})$,
$\Phi_{1}\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}|\vartheta_{1})$,
$\boldsymbol{\vartheta}\in\Gamma(\vartheta_{0},\vartheta_{1})$, we define
$\displaystyle[\Phi_{0},\Phi_{1}]_{r,\boldsymbol{\vartheta}}$
$\displaystyle:=[\Phi_{0},\boldsymbol{\vartheta}]_{r,0}-[\Phi_{1},\boldsymbol{\vartheta}]_{l,1},$
$\displaystyle[\Phi_{0},\Phi_{1}]_{l,\boldsymbol{\vartheta}}$
$\displaystyle:=[\Phi_{0},\boldsymbol{\vartheta}]_{l,0}-[\Phi_{1},\boldsymbol{\vartheta}]_{r,1}.$

Notice that, if $\Phi_{x}$ is the disintegration of $\Phi$ with respect to
$\vartheta_{t}=\mathsf{x}_{\sharp}\Phi$, we can consider the barycentric
coupling
$\boldsymbol{\sigma}_{t}:=\int_{\mathsf{X}\times\mathsf{X}}\Phi_{\mathsf{x}^{t}}\,\mathrm{d}\boldsymbol{\vartheta}\in\Gamma_{t}(\Phi,\boldsymbol{\vartheta})$,
i.e.

$\int\psi(x_{0},v_{0},x_{1})\,\mathrm{d}\boldsymbol{\sigma}_{t}=\int\Big{[}\int\psi(x_{0},v_{0},x_{1})\,\mathrm{d}\Phi_{(1-t)x_{0}+tx_{1}}(v_{0})\Big{]}\,\mathrm{d}\boldsymbol{\vartheta}(x_{0},x_{1})$

so that
$\left[\Phi,\boldsymbol{\vartheta}\right]_{b,t}=\int\langle{v_{0}},x_{0}-x_{1}\rangle\,\mathrm{d}\boldsymbol{\sigma}_{t}$
and

$[\Phi,\boldsymbol{\vartheta}]_{r,t}\leq\left[\Phi,\boldsymbol{\vartheta}\right]_{b,t}\leq[\Phi,\boldsymbol{\vartheta}]_{l,t}.$

If we define by $\mathsf{s}:\mathsf{X}^{2}\to\mathsf{X}^{2}$ the map
$\mathsf{s}(x_{0},x_{1}):=(x_{1},x_{0})$ (with a similar definition for
$\mathsf{T\kern-1.5ptX}\times\mathsf{X}$:
$\mathsf{s}(x_{0},v_{0},x_{1}):=(x_{1},v_{0},x_{0})$) it is easy to check that

$\boldsymbol{\sigma}\in\Gamma_{t}(\Phi,\boldsymbol{\vartheta})\quad\Leftrightarrow\quad\mathsf{s}_{\sharp}\boldsymbol{\sigma}\in\Gamma_{1-t}(\Phi,\mathsf{s}_{\sharp}\boldsymbol{\vartheta})$

so that

$[\Phi,\boldsymbol{\vartheta}]_{r,t}=-[\Phi,\mathsf{s}_{\sharp}\boldsymbol{\vartheta}]_{l,1-t},\quad[\Phi,\boldsymbol{\vartheta}]_{l,t}=-[\Phi,\mathsf{s}_{\sharp}\boldsymbol{\vartheta}]_{r,1-t}.$
(3.24)

(3.15) and (3.18) have simpler versions in two particular cases, which will be
explained in the next remark.

######  Remark 3.18 (Particular cases).

Suppose that $\boldsymbol{\vartheta}\in\mathcal{P}_{2}(\mathsf{X}^{2})$,
$t\in[0,1]$, $\vartheta_{t}=\mathsf{x}^{t}_{\sharp}\boldsymbol{\vartheta}$,
$\Phi\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}|\vartheta_{t})$ and
$\mathsf{x}^{t}:\mathsf{X}^{2}\to\mathsf{X}$ is
$\boldsymbol{\vartheta}$-essentially injective so that
$\boldsymbol{\vartheta}$ is concentrated on a Borel map
$(X_{0},X_{1}):\mathsf{X}\to\mathsf{X}\times\mathsf{X}$, i.e.
$\boldsymbol{\vartheta}=(X_{0},X_{1})_{\sharp}\vartheta_{t}$. In this case
$\Gamma_{t}(\Phi,\boldsymbol{\vartheta})$ contains a unique element given by
$(X_{0}\circ\mathsf{x},\mathsf{v},X_{1}\circ\mathsf{x})_{\sharp}\Phi$ and

$[\Phi,\boldsymbol{\vartheta}]_{r,t}=[\Phi,\boldsymbol{\vartheta}]_{l,t}=\left[\Phi,\boldsymbol{\vartheta}\right]_{b,t}=\int\langle
v,X_{0}(x)-X_{1}(x)\rangle\,\mathrm{d}\Phi(x,v)=\int\langle\boldsymbol{b}_{\Phi},X_{0}-X_{1}\rangle\,\mathrm{d}\vartheta_{t},$
(3.25)

where in the last formula we have applied the barycentric reduction (3.8).
When $t=0$ and $\boldsymbol{\vartheta}$ is the unique element of
$\Gamma_{o}(\vartheta_{0},\vartheta_{1})$ then $X_{0}(x)=x$ and we obtain

$\left[\Phi,\vartheta_{1}\right]_{r}=\left[\Phi,\vartheta_{1}\right]_{l}=[\Phi,\boldsymbol{\vartheta}]_{r,0}=[\Phi,\boldsymbol{\vartheta}]_{l,0}=\int\langle
v,x-X_{1}(x)\rangle\,\mathrm{d}\Phi(x,v)=\int\langle\boldsymbol{b}_{\Phi},x-X_{1}(x)\rangle\,\mathrm{d}\vartheta_{0}(x).$

Another simple case is when $\Phi=\Phi_{\vartheta_{t},\boldsymbol{w}}$ for
some vector field $\boldsymbol{w}\in
L^{2}_{\vartheta_{t}}(\mathsf{X};\mathsf{X})$ as in (3.5) (i.e. its
disintegration $\Phi_{x}$ w.r.t. $\vartheta_{t}$ takes the form
$\delta_{\boldsymbol{w}(x)}$ and $\boldsymbol{w}=\boldsymbol{b}_{\Phi}.$). We
have

$[\Phi,\boldsymbol{\vartheta}]_{r,t}=[\Phi,\boldsymbol{\vartheta}]_{l,t}=\int\langle\boldsymbol{w}((1-t)x_{0}+tx_{1}),x_{0}-x_{1}\rangle\,\mathrm{d}\boldsymbol{\vartheta}(x_{0},x_{1}).$

In particular we get

$\left[\Phi,\vartheta_{1}\right]_{r}=\min\Big{\\{}\int\langle\boldsymbol{w}(x),x_{0}-x_{1}\rangle\,\mathrm{d}\boldsymbol{\vartheta}(x_{0},x_{1})\mid\boldsymbol{\vartheta}\in\Gamma_{o}(\vartheta_{0},\vartheta_{1})\Big{\\}}.$

An important case in which the previous Remark 3.18 applies is that of
geodesics in $\mathcal{P}_{2}(\mathsf{X})$.

######  Lemma 3.19.

Let $\mu_{0},\mu_{1}\in\mathcal{P}_{2}(\mathsf{X})$, $(\mu_{t})_{t\in[0,1]}$
be a constant speed geodesic induced by an optimal plan
$\boldsymbol{\mu}\in\Gamma_{o}(\mu_{0},\mu_{1})$ by the relation

$\mu_{t}=\mathsf{x}^{t}_{\sharp}\boldsymbol{\mu},\quad
t\in[0,1],\quad\mathsf{x}^{t}(x_{0},x_{1})=(1-t)x_{0}+tx_{1}.$

If $t\in(0,1)$, $\Phi_{t}\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}|\mu_{t})$,
$\hat{\boldsymbol{\mu}}=\mathsf{s}_{\sharp}\boldsymbol{\mu}\in\Gamma_{o}(\mu_{1},\mu_{0})$,
then

$\displaystyle\frac{1}{1-t}\left[\Phi_{t},\mu_{1}\right]_{r}$ $\displaystyle=$
$\displaystyle\frac{1}{1-t}\left[\Phi_{t},\mu_{1}\right]_{l}$ $\displaystyle=$
$\displaystyle[\Phi_{t},\boldsymbol{\mu}]_{r,t}$ $\displaystyle=$
$\displaystyle[\Phi_{t},\boldsymbol{\mu}]_{l,t}$ (3.26) $\displaystyle=$
$\displaystyle-\frac{1}{t}\left[\Phi_{t},\mu_{0}\right]_{r}$ $\displaystyle=$
$\displaystyle-\frac{1}{t}\left[\Phi_{t},\mu_{0}\right]_{l}$ $\displaystyle=$
$\displaystyle-[\Phi_{t},\hat{\boldsymbol{\mu}}]_{r,1-t}$ $\displaystyle=$
$\displaystyle-[\Phi_{t},\hat{\boldsymbol{\mu}}]_{l,1-t}.$

###### Proof.

The crucial fact is that $\mathsf{x}^{t}:\mathsf{X}^{2}\to\mathsf{X}$ is
injective on $\operatorname{supp}(\boldsymbol{\mu})$ and thus a bijection on
its image $\operatorname{supp}(\mu_{t})$. Indeed, take
$(x_{0},x_{1}),(x_{0}^{\prime},x_{1}^{\prime})\in\operatorname{supp}(\boldsymbol{\mu})$,
then

$\displaystyle\left|\mathsf{x}^{t}(x_{0},x_{1})-\mathsf{x}^{t}(x_{0}^{\prime},x_{1}^{\prime})\right|^{2}$
$\displaystyle=(1-t)^{2}|x_{0}-x_{0}^{\prime}|^{2}+t^{2}|x_{1}-x_{1}^{\prime}|^{2}+2t(1-t)\langle
x_{0}-x_{0}^{\prime},x_{1}-x_{1}^{\prime}\rangle$
$\displaystyle\geq(1-t)^{2}|x_{0}-x_{0}^{\prime}|^{2}+t^{2}|x_{1}-x_{1}^{\prime}|^{2}$

thanks to the cyclical monotonicity of $\operatorname{supp}(\boldsymbol{\mu})$
(see [AGS08, Remark 7.1.2]).

Then, for every $x\in\operatorname{supp}(\mu_{t})$, there exists a unique
couple
$(x_{0},x_{1})=(X_{0}(x),X_{1}(x))\in\operatorname{supp}(\boldsymbol{\mu})$
s.t. $x=(1-t)x_{0}+tx_{1}$, where we refer to Remark 3.18 for the definitions
of $X_{0},X_{1}$ (cf. also [San15, Theorem 5.29]). Hence, in the following
diagram all maps are bijections:

${\operatorname{supp}(\boldsymbol{\mu}_{t0})}$${\operatorname{supp}(\boldsymbol{\mu})}$${\operatorname{supp}(\boldsymbol{\mu}_{t1})}$
${\operatorname{supp}(\mu_{t})}$
$\scriptstyle{(\mathsf{x}^{t}\text{,}\,\mathsf{x}^{0})}$$\scriptstyle{(\mathsf{x}^{t}\text{,}\,\mathsf{x}^{1})}$$\scriptstyle{\mathsf{x}^{t}}$$\scriptstyle{(\boldsymbol{i}_{\mathsf{X}}\text{,}\,X_{1})}$$\scriptstyle{(\boldsymbol{i}_{\mathsf{X}}\text{,}\,X_{0})}$

where
$\boldsymbol{\mu}_{t1}=(\mathsf{x}^{t},\mathsf{x}^{1})_{\sharp}\boldsymbol{\mu}=(\boldsymbol{i}_{\mathsf{X}},X_{1})_{\sharp}\mu_{t}$
is the unique element of $\Gamma_{o}(\mu_{t},\mu_{1})$ and
$\boldsymbol{\mu}_{t0}=(\mathsf{x}^{t},\mathsf{x}^{0})_{\sharp}\boldsymbol{\mu}=(\boldsymbol{i}_{\mathsf{X}},X_{0})_{\sharp}\mu_{t}=(\mathsf{x}^{1-t},\mathsf{x}^{1})_{\sharp}\hat{\boldsymbol{\mu}}$
is the unique element of $\Gamma_{o}(\mu_{t},\mu_{0})$ (see Theorem 2.8).
Since

$\frac{x-X_{1}(x)}{1-t}=\frac{x-x_{1}}{1-t}=x_{0}-x_{1}=-\frac{x-x_{0}}{t}=-\frac{x-X_{0}(x)}{t},$

and
$\Lambda(\Phi_{t},\mu_{1})=\\{(\boldsymbol{i}_{\mathsf{T\kern-1.5ptX}},X_{1}\circ\mathsf{x})_{\sharp}\Phi_{t}\\}$
thanks to Theorem 2.8, by Theorem 3.9 and Corollary 3.10 we have

$\left[\Phi_{t},\mu_{1}\right]_{r}=\left[\Phi_{t},\mu_{1}\right]_{l}=\int_{\mathsf{T\kern-1.5ptX}}\langle
v,x-X_{1}(x)\rangle\,\mathrm{d}\Phi_{t}(x,v).$

Analogously,
$\Lambda(\Phi_{t},\mu_{0})=\\{(\boldsymbol{i}_{\mathsf{T\kern-1.5ptX}},X_{0}\circ\mathsf{x})_{\sharp}\Phi_{t}\\}$.
Hence

$\left[\Phi_{t},\mu_{0}\right]_{r}=\left[\Phi_{t},\mu_{0}\right]_{l}=\int_{\mathsf{T\kern-1.5ptX}}\langle
v,x-X_{0}(x)\rangle\,\mathrm{d}\Phi_{t}(x,v).$

Also recalling (3.24) and (3.25) we conclude. ∎

##  4\. Dissipative probability vector fields: the metric viewpoint

###  4.1. Multivalued Probability Vector Fields and $\lambda$-dissipativity

######  Definition 4.1 (Multivalued Probability Vector Field - MPVF).

A _multivalued probability vector field_ ${\boldsymbol{\mathrm{F}}}$ is a
nonempty subset of $\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})$ with domain
$\mathrm{D}({\boldsymbol{\mathrm{F}}}):=\mathsf{x}_{\sharp}({\boldsymbol{\mathrm{F}}})=\\{\mathsf{x}_{\sharp}\Phi:\Phi\in{\boldsymbol{\mathrm{F}}}\\}$.
Given $\mu\in\mathcal{P}_{2}(\mathsf{X})$, we define the _section_
${\boldsymbol{\mathrm{F}}}[\mu]$ of ${\boldsymbol{\mathrm{F}}}$ as

${\boldsymbol{\mathrm{F}}}[\mu]:=(\mathsf{x}_{\sharp})^{-1}(\mu)\cap{\boldsymbol{\mathrm{F}}}=\left\\{\Phi\in{\boldsymbol{\mathrm{F}}}\mid\mathsf{x}_{\sharp}\Phi=\mu\right\\}.$

A _selection_ ${\boldsymbol{\mathrm{F}}}^{\prime}$ of
${\boldsymbol{\mathrm{F}}}$ is a subset of ${\boldsymbol{\mathrm{F}}}$ such
that
$\mathrm{D}({\boldsymbol{\mathrm{F}}}^{\prime})=\mathrm{D}({\boldsymbol{\mathrm{F}}})$.
We call ${\boldsymbol{\mathrm{F}}}$ a _probability vector field_ (PVF) if
$\mathsf{x}_{\sharp}$ is injective in ${\boldsymbol{\mathrm{F}}}$, i.e.
${\boldsymbol{\mathrm{F}}}[\mu]$ contains a unique element for every
$\mu\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$. ${\boldsymbol{\mathrm{F}}}$ is
a vector field if for every $\mu\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$
${\boldsymbol{\mathrm{F}}}[\mu]$ contains a unique element $\Phi$ concentrated
on a map, i.e.
$\Phi=(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{b}_{\Phi})_{\sharp}\mu$.

######  Remark 4.2.

We can equivalently formulate Definition 4.1 by considering
${\boldsymbol{\mathrm{F}}}$ as a multifunction, as in the case, e.g., of the
Wasserstein subdifferential $\boldsymbol{\partial}\mathcal{F}$ of a function
$\mathcal{F}:\mathcal{P}_{2}(\mathsf{X})\to(-\infty,+\infty]$, see [AGS08, Ch.
10] and the next Section 5.1. According to this viewpoint, a MPVF is a set-
valued map
${\boldsymbol{\mathrm{F}}}:\mathcal{P}_{2}(\mathsf{X})\supset\mathrm{D}({\boldsymbol{\mathrm{F}}})\rightrightarrows\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})$
such that $\mathsf{x}_{\sharp}\Phi=\mu$ for all
$\Phi\in{\boldsymbol{\mathrm{F}}}[\mu]$. In this way, each section
${\boldsymbol{\mathrm{F}}}[\mu]$ is nothing but the image of
$\mu\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$ through
${\boldsymbol{\mathrm{F}}}$. In this case, _probability vector fields_
correspond to single valued maps: this notion has been used in [Pic19] with
the aim of describing a sort of velocity field on $\mathcal{P}(\mathsf{X})$,
and later in [Pic18] dealing with Multivalued Probability Vector Fields
(called Probability Multifunctions).

######  Definition 4.3 (Metrically $\lambda$-dissipative MPVF).

A MPVF
${\boldsymbol{\mathrm{F}}}\subset\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})$ is
(metrically) _$\lambda$ -dissipative_, $\lambda\in\mathbb{R}$, if

$\left[\Phi_{0},\Phi_{1}\right]_{r}\leq\lambda
W_{2}^{2}(\mu_{0},\mu_{1})\quad\text{for every
}\Phi_{0},\Phi_{1}\in{\boldsymbol{\mathrm{F}}},\
\mu_{i}=\mathsf{x}_{\sharp}\Phi_{i}.$ (4.1)

We say that ${\boldsymbol{\mathrm{F}}}$ is (metrically) _$\lambda$
-accretive_, if
$-{\boldsymbol{\mathrm{F}}}=\\{-\Phi:\Phi\in{\boldsymbol{\mathrm{F}}}\\}$
(recall (3.14)) is $-\lambda$-dissipative, i.e.

$\left[\Phi_{0},\Phi_{1}\right]_{l}\geq\lambda
W_{2}^{2}(\mu_{0},\mu_{1})\quad\text{for every
}\Phi_{0},\Phi_{1}\in{\boldsymbol{\mathrm{F}}},\
\mu_{i}=\mathsf{x}_{\sharp}\Phi_{i}.$

######  Remark 4.4.

Notice that (4.1) is equivalent to ask for the existence of a coupling
$\boldsymbol{\Theta}\in\Lambda(\Phi_{0},\Phi_{1})$ (thus
$(\mathsf{x}^{0},\mathsf{x}^{1})_{\sharp}\boldsymbol{\Theta}$ is optimal
between $\mu_{0}=\mathsf{x}_{\sharp}\Phi_{0}$ and
$\mu_{1}=\mathsf{x}_{\sharp}\Phi_{1}$) such that

$\int\langle
v_{1}-v_{0},x_{1}-x_{0}\rangle\,\mathrm{d}\boldsymbol{\Theta}\leq\lambda
W_{2}^{2}(\mu_{0},\mu_{1})=\lambda\int|x_{1}-x_{0}|^{2}\,\mathrm{d}\boldsymbol{\Theta}.$

Recalling the discussion of the previous section, $\lambda$-dissipativity has
a natural metric interpretation: for every
$\Phi_{0},\Phi_{1}\in{\boldsymbol{\mathrm{F}}}$ with
$\mu_{0}=\mathsf{x}_{\sharp}\Phi_{0}$, $\mu_{1}=\mathsf{x}_{\sharp}\Phi_{1}$
we have the asymptotic expansion

$W_{2}^{2}(\operatorname{\mathsf{exp}}^{t}\Phi_{0},\operatorname{\mathsf{exp}}^{t}\Phi_{1})\leq(1+2\lambda
t)W_{2}^{2}(\mu_{0},\mu_{1})+o(t)\quad\text{as }t\downarrow 0.$

######  Remark 4.5.

Thanks to Corollary 3.7, (4.1) implies the weaker condition

$\left[\Phi_{0},\mu_{1}\right]_{r}+\left[\Phi_{1},\mu_{0}\right]_{r}\leq\lambda
W_{2}^{2}(\mu_{0},\mu_{1}),\quad\forall\,\Phi_{0},\Phi_{1}\in{\boldsymbol{\mathrm{F}}},\
\mu_{0}=\mathsf{x}_{\sharp}\Phi_{0},\ \mu_{1}=\mathsf{x}_{\sharp}\Phi_{1}.$
(4.2)

It is clear that the inequality of (4.2) implies the inequality of (4.1)
whenever $\Gamma_{o}(\mu_{0},\mu_{1})$ contains only one element. More
generally, we will see in Corollary 4.13 that (4.2) is in fact equivalent to
(4.1) when $\mathrm{D}({\boldsymbol{\mathrm{F}}})$ is geodesically convex
(according to Definition 2.7).

As in the standard Hilbert case, $\lambda$-dissipativity can be reduced to
dissipativity (meaning $0$-dissipativity) by a simple transformation. Let us
introduce the map

$L^{\lambda}:\mathsf{T\kern-1.5ptX}\to\mathsf{T\kern-1.5ptX},\quad
L^{\lambda}(x,v):=(x,v-\lambda x),$

observing that for every
$\boldsymbol{\sigma}\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}\times\mathsf{X})$
with $(\mathsf{x}^{i})_{\sharp}\boldsymbol{\sigma}=\mu_{i}$, $i=0,1$, the
transformed plan
$\boldsymbol{\sigma}^{\lambda}:=(L^{\lambda},\boldsymbol{i}_{\mathsf{X}})_{\sharp}\boldsymbol{\sigma}$
satisfies

$\displaystyle\int\langle
v_{0},x_{0}-x_{1}\rangle\,\mathrm{d}\boldsymbol{\sigma}^{\lambda}$
$\displaystyle=\int\langle v_{0}-\lambda
x_{0},x_{0}-x_{1}\rangle\,\mathrm{d}\boldsymbol{\sigma}$
$\displaystyle=\int\langle
v_{0},x_{0}-x_{1}\rangle\,\mathrm{d}\boldsymbol{\sigma}-\frac{\lambda}{2}\int|x_{0}-x_{1}|^{2}\,\mathrm{d}\boldsymbol{\sigma}+\frac{\lambda}{2}\Big{(}\mathsf{m}_{2}^{2}(\mu_{1})-\mathsf{m}_{2}^{2}(\mu_{0})\Big{)}.$
(4.3)

Similarly, if
$\Theta\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}\times\mathsf{T\kern-1.5ptX})$
with $\mathsf{x}^{i}_{\sharp}\Theta=\mu_{i}$, the plan
$\Theta^{\lambda}:=(L^{\lambda},L^{\lambda})_{\sharp}\Theta$ satisfies

$\displaystyle\int\langle
v_{0}-v_{1},x_{0}-x_{1}\rangle\,\mathrm{d}\Theta^{\lambda}$
$\displaystyle=\int\langle
v_{0}-v_{1}-\lambda(x_{0}-x_{1}),x_{0}-x_{1}\rangle\,\mathrm{d}\Theta$
$\displaystyle=\int\langle
v_{0}-v_{1},x_{0}-x_{1}\rangle\,\mathrm{d}\Theta-\lambda\int|x_{0}-x_{1}|^{2}\,\mathrm{d}\Theta.$
(4.4)

######  Lemma 4.6.

${\boldsymbol{\mathrm{F}}}$ is a $\lambda$-dissipative MPVF (resp. satisfies
(4.2)) if and only if
${\boldsymbol{\mathrm{F}}}^{\lambda}:=L^{\lambda}_{\sharp}({\boldsymbol{\mathrm{F}}})=\\{L^{\lambda}_{\sharp}\Phi\mid\Phi\in{\boldsymbol{\mathrm{F}}}\\}$
is dissipative (resp. satisfies (4.2) with $\lambda=0$).

###### Proof.

Let us first check the case of (4.2). Since
$\boldsymbol{\sigma}\in\Lambda_{o}(\Phi_{0},\mu_{1})$ if and only if
$\boldsymbol{\sigma}^{\lambda}\in\Lambda_{o}(L^{\lambda}_{\sharp}\Phi_{0},\mu_{1})$,
(4.3) yields

$\displaystyle\int\langle
v_{0},x_{0}-x_{1}\rangle\,\mathrm{d}\boldsymbol{\sigma}^{\lambda}=\int\langle
v_{0},x_{0}-x_{1}\rangle\,\mathrm{d}\boldsymbol{\sigma}-\frac{\lambda}{2}\Big{(}\mathsf{m}_{2}^{2}(\mu_{0})-\mathsf{m}_{2}^{2}(\mu_{1})+W_{2}^{2}(\mu_{0},\mu_{1})\Big{)}$

and therefore

$\left[L^{\lambda}_{\sharp}\Phi_{0},\mu_{1}\right]_{r}=\left[\Phi_{0},\mu_{1}\right]_{r}-\frac{\lambda}{2}\Big{(}\mathsf{m}_{2}^{2}(\mu_{0})-\mathsf{m}_{2}^{2}(\mu_{1})+W_{2}^{2}(\mu_{0},\mu_{1})\Big{)}.$
(4.5)

Using the corresponding identity for
$\left[L^{\lambda}_{\sharp}\Phi_{1},\mu_{0}\right]_{r}$ we obtain that
${\boldsymbol{\mathrm{F}}}^{\lambda}$ is dissipative.

A similar argument, using the identity (4.4), shows the equivalence between
the $\lambda$-dissipativity of ${\boldsymbol{\mathrm{F}}}$ and the
dissipativity of ${\boldsymbol{\mathrm{F}}}^{\lambda}$. ∎

Let us conclude this section by showing that $\lambda$-dissipativity can be
deduced from a Lipschitz like condition similar to the one considered in
[Pic19] (see Appendix A).

######  Lemma 4.7.

Suppose that the MPVF ${\boldsymbol{\mathrm{F}}}$ satisfies

$\mathcal{W}_{2}({\boldsymbol{\mathrm{F}}}[\nu],{\boldsymbol{\mathrm{F}}}[\nu^{\prime}])\leq
LW_{2}(\nu,\nu^{\prime}),\quad\forall\,\nu,\nu^{\prime}\in\mathrm{D}({\boldsymbol{\mathrm{F}}}),$

where
$\mathcal{W}_{2}:\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})\times\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})\to[0,+\infty)$
is defined by

$\mathcal{W}_{2}^{2}(\Phi_{0},\Phi_{1})=\inf\left\\{\int_{\mathsf{T\kern-1.5ptX}\times\mathsf{T\kern-1.5ptX}}|v_{0}-v_{1}|^{2}\,\mathrm{d}\boldsymbol{\Theta}(x_{0},v_{0},x_{1},v_{1}):\boldsymbol{\Theta}\in\Lambda(\Phi_{0},\Phi_{1})\right\\},$

with $\Lambda(\cdot,\cdot)$ as in Definition 3.8. Then
${\boldsymbol{\mathrm{F}}}$ is $\lambda$-dissipative, for
$\lambda:=\frac{1}{2}(1+L^{2})$

###### Proof.

Let $\nu^{\prime},\nu^{\prime\prime}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$,
then by Theorem 3.9 and Young’s inequality, we have

$\begin{split}\left[{\boldsymbol{\mathrm{F}}}[\nu^{\prime}],{\boldsymbol{\mathrm{F}}}[\nu^{\prime\prime}]\right]_{r}&=\min\left\\{\int_{\mathsf{T\kern-1.5ptX}\times\mathsf{T\kern-1.5ptX}}\langle
x^{\prime}-x^{\prime\prime},v^{\prime}-v^{\prime\prime}\rangle\,\mathrm{d}\boldsymbol{\Theta}\,:\,\boldsymbol{\Theta}\in\Lambda({\boldsymbol{\mathrm{F}}}[\nu^{\prime}],{\boldsymbol{\mathrm{F}}}[\nu^{\prime\prime}])\right\\}\\\
&\leq\frac{1}{2}\left(W_{2}^{2}(\nu^{\prime},\nu^{\prime\prime})+\mathcal{W}_{2}^{2}({\boldsymbol{\mathrm{F}}}[\nu^{\prime}],{\boldsymbol{\mathrm{F}}}[\nu^{\prime\prime}])\right)\\\
&\leq\frac{L^{2}+1}{2}\,W_{2}^{2}(\nu^{\prime},\nu^{\prime\prime}).\qed\end{split}$

###  4.2. Behaviour of $\lambda$-dissipative MPVF along geodesics

Let us now study the behaviour of a MPVF ${\boldsymbol{\mathrm{F}}}$ along
geodesics. Recall that in the case of a dissipative map
${\boldsymbol{F}}:\mathsf{H}\to\mathsf{H}$ in a Hilbert space $\mathsf{H}$, it
is quite immediate to prove that the real function

$f(t):=\langle F(x_{t}),x_{0}-x_{1}\rangle,\quad x_{t}=(1-t)x_{0}+tx_{1},\quad
t\in[0,1]$

is monotone increasing. This property has a natural counterpart in the case of
measures.

######  Definition 4.8.

Let ${\boldsymbol{\mathrm{F}}}\subset\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})$,
$\mu_{0},\mu_{1}\in\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$,
$\boldsymbol{\mu}\in\Gamma_{o}(\mu_{0},\mu_{1})$. We define the sets

$\displaystyle\mathrm{I}(\boldsymbol{\mu}|{\boldsymbol{\mathrm{F}}}):={}$
$\displaystyle\Big{\\{}t\in[0,1]:\mathsf{x}^{t}_{\sharp}\boldsymbol{\mu}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})\Big{\\}},$
$\displaystyle\Gamma_{o}^{i}({\mu_{0}},{\mu_{1}}|{\boldsymbol{\mathrm{F}}}):={}$
$\displaystyle\Big{\\{}\boldsymbol{\mu}\in\Gamma_{o}(\mu_{0},\mu_{1}):i\text{
is an accumulation point of
}\mathrm{I}(\boldsymbol{\mu}|{\boldsymbol{\mathrm{F}}})\Big{\\}},i=0,1$ (4.6)
$\displaystyle\Gamma_{o}^{01}({\mu_{0}},{\mu_{1}}|{\boldsymbol{\mathrm{F}}}):={}$
$\displaystyle\Gamma_{o}^{0}({\mu_{0}},{\mu_{1}}|{\boldsymbol{\mathrm{F}}})\cap\Gamma_{o}^{1}({\mu_{0}},{\mu_{1}}|{\boldsymbol{\mathrm{F}}}).$

Notice that these sets depend on ${\boldsymbol{\mathrm{F}}}$ just through
$\mathrm{D}({\boldsymbol{\mathrm{F}}})$. In particular, if
$\mu_{0},\mu_{1}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$ and
$\mathrm{D}({\boldsymbol{\mathrm{F}}})$ is open or geodesically convex
according to Definition 2.7 then
$\Gamma_{o}^{01}({\mu_{0}},{\mu_{1}}|{\boldsymbol{\mathrm{F}}})\neq\emptyset$.

######  Definition 4.9.

Let ${\boldsymbol{\mathrm{F}}}\subset\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})$
be a MPVF. Let
$\mu_{0},\mu_{1}\in\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$,
$\boldsymbol{\mu}\in\Gamma_{o}(\mu_{0},\mu_{1})$ and let
$\mu_{t}:=\mathsf{x}^{t}_{\sharp}\boldsymbol{\mu}$, $t\in[0,1]$. For every
$t\in\mathrm{I}(\boldsymbol{\mu}|{\boldsymbol{\mathrm{F}}})$ we define

$\displaystyle[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{r,t}:=\sup\left\\{[\Phi,\boldsymbol{\mu}]_{r,t}\mid\Phi\in{\boldsymbol{\mathrm{F}}}[\mu_{t}]\right\\},\qquad[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{l,t}:=\inf\left\\{[\Phi,\boldsymbol{\mu}]_{l,t}\mid\Phi\in{\boldsymbol{\mathrm{F}}}[\mu_{t}]\right\\}.$

######  Theorem 4.10.

Let us suppose that the MPVF ${\boldsymbol{\mathrm{F}}}$ satisfies (4.2), let
$\mu_{0},\mu_{1}\in\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$, and let
$\boldsymbol{\mu}\in\Gamma_{o}(\mu_{0},\mu_{1})$ with
$W^{2}:=W_{2}^{2}(\mu_{0},\mu_{1})$. Then the following properties hold

  1. (1)

$[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{l,t}\leq[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{r,t}$
for every
$t\in(0,1)\cap\mathrm{I}(\boldsymbol{\mu}|{\boldsymbol{\mathrm{F}}})$;

  2. (2)

$[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{r,s}\leq[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{l,t}+\lambda
W^{2}(t-s)$ for every
$s,t\in\mathrm{I}(\boldsymbol{\mu}|{\boldsymbol{\mathrm{F}}})$, $s<t$;

  3. (3)

$t\mapsto[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{r,t}+\lambda W^{2}t$
and $t\mapsto[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{l,t}+\lambda
W^{2}t$ are increasing respectively in
$\mathrm{I}(\boldsymbol{\mu}|{\boldsymbol{\mathrm{F}}})\setminus\\{1\\}$ and
in $\mathrm{I}(\boldsymbol{\mu}|{\boldsymbol{\mathrm{F}}})\setminus\\{0\\}$.

  4. (4)

the right (resp. left) limits of
$t\mapsto[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{r,t}$ and
$t\mapsto[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{l,t}$ exist at every
right (resp. left) accumulation point of
$\mathrm{I}(\boldsymbol{\mu}|{\boldsymbol{\mathrm{F}}})$, and in those points
the right (resp. left) limits of
$[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{r,t}$ coincide with the right
(resp. left) limits of $[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{l,t}$.

  5. (5)

$[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{l,t}=[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{r,t}$
at every interior point $t$ of
$\mathrm{I}(\boldsymbol{\mu}|{\boldsymbol{\mathrm{F}}})$ where one of them is
continuous.

###### Proof.

Throughout all the proof we set
$f_{r}(t):=[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{r,t}$ and
$f_{l}(t):=[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{l,t}$. Thanks to
Lemma 4.6 and in particular to (4.5), it is easy to check that it is
sufficient to consider the dissipative case $\lambda=0$.

  1. (1)

It is a direct consequence of Lemma 3.19 and the definitions of $f_{r}$ and
$f_{l}$.

  2. (2)

We prove that for every $\Phi\in{\boldsymbol{\mathrm{F}}}[\mu_{s}]$ and
$\Phi^{\prime}\in{\boldsymbol{\mathrm{F}}}[\mu_{t}]$ it holds

$[\Phi,\boldsymbol{\mu}]_{r,s}\leq[\Phi^{\prime},\boldsymbol{\mu}]_{l,t}.$
(4.7)

The thesis will follow immediately passing to the $\sup$ over
$\Phi\in{\boldsymbol{\mathrm{F}}}[\mu_{s}]$ in the LHS and to the $\inf$ over
$\Phi^{\prime}\in{\boldsymbol{\mathrm{F}}}[\mu_{t}]$ in the RHS. It is enough
to prove (4.7) in case at least one between $s,t$ belongs to $(0,1)$. Let us
define the map
$L:\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}\times\mathsf{X})\to\mathbb{R}$ as

$L(\gamma):=\int_{\mathsf{T\kern-1.5ptX}\times\mathsf{X}}\langle
v_{0},x_{0}-x_{1}\rangle\,\mathrm{d}\gamma(x_{0},v_{0},x_{1})\quad\gamma\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}\times\mathsf{X}).$

Observe that, since it never happens that $s=0$ and $t=1$ at the same time,
the map $T:\Gamma_{s}(\Phi,\boldsymbol{\mu})\to\Lambda(\Phi,\mu_{t})$ defined
as

$T(\boldsymbol{\sigma}):=(\mathsf{x}^{s}\circ(\mathsf{x}^{0},\mathsf{x}^{1}),\mathsf{v}^{0},\mathsf{x}^{t}\circ(\mathsf{x}^{0},\mathsf{x}^{1}))_{\sharp}\boldsymbol{\sigma}$

is a bijection s.t. $(t-s)L(\boldsymbol{\sigma})=L(T(\boldsymbol{\sigma}))$
for every $\boldsymbol{\sigma}\in\Gamma_{s}(\Phi,\boldsymbol{\mu})$. This
immediately gives that

$(t-s)[\Phi,\boldsymbol{\mu}]_{r,s}=\left[\Phi,\mu_{t}\right]_{r}.$

In the same way we can deduce that

$(s-t)[\Phi^{\prime},\boldsymbol{\mu}]_{l,t}=\left[\Phi^{\prime},\mu_{s}\right]_{r}.$

Thanks to the dissipativity of ${\boldsymbol{\mathrm{F}}}$ we get

$(t-s)[\Phi,\boldsymbol{\mu}]_{r,s}-(t-s)[\Phi^{\prime},\boldsymbol{\mu}]_{l,t}=\left[\Phi,\mu_{t}\right]_{r}+\left[\Phi^{\prime},\mu_{s}\right]_{r}\leq
0.$

  3. (3)

Combining (1) and (2) we have that for every
$s,t\in\mathrm{I}(\boldsymbol{\mu}|{\boldsymbol{\mathrm{F}}})$ with $0<s<t<1$
it holds

$f_{l}(s)\leq f_{r}(s)\leq f_{l}(t)\leq f_{r}(t).$ (4.8)

This implies that both $f_{l}$ and $f_{r}$ are increasing in
$\mathrm{I}(\boldsymbol{\mu}|{\boldsymbol{\mathrm{F}}})\cap(0,1)$. Observe
that, again combining (1) and (2), it also holds

$\displaystyle f_{r}(0)$ $\displaystyle\leq f_{l}(t)\leq f_{r}(t),$
$\displaystyle f_{l}(t)$ $\displaystyle\leq f_{r}(t)\leq f_{l}(1)$

for every
$t\in\mathrm{I}(\boldsymbol{\mu}|{\boldsymbol{\mathrm{F}}})\setminus\\{0,1\\}$,
and then $f_{r}$ is increasing in
$\mathrm{I}(\boldsymbol{\mu}|{\boldsymbol{\mathrm{F}}})\setminus\\{1\\}$ and
$f_{l}$ is increasing in
$\mathrm{I}(\boldsymbol{\mu}|{\boldsymbol{\mathrm{F}}})\setminus\\{0\\}$.

  4. (4)

It is an immediate consequence of (4.8).

  5. (5)

It is a straightforward consequence of (4). ∎

Thanks to the previous Theorem 4.10 the next definition is well posed.

######  Definition 4.11.

Let us suppose that the MPVF ${\boldsymbol{\mathrm{F}}}$ satisfies (4.2), let
$\mu_{0},\mu_{1}\in\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$.

$\displaystyle\text{If
$\boldsymbol{\mu}\in\Gamma_{o}^{0}({\mu_{0}},{\mu_{1}}|{\boldsymbol{\mathrm{F}}})$
we set}\quad[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{0+}$
$\displaystyle:=\lim_{t\downarrow
0}[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{r,t}=\lim_{t\downarrow
0}[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{l,t}$ $\displaystyle\text{If
$\boldsymbol{\mu}\in\Gamma_{o}^{1}({\mu_{0}},{\mu_{1}}|{\boldsymbol{\mathrm{F}}})$
we set}\quad[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{1-}$
$\displaystyle:=\lim_{t\uparrow
1}[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{r,t}=\lim_{t\uparrow
1}[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{l,t}.$

######  Corollary 4.12.

Let us keep the same notation of Theorem 4.10 and let
$s\in\mathrm{I}(\boldsymbol{\mu}|{\boldsymbol{\mathrm{F}}})\cap(0,1)$ with
$\Phi\in{\boldsymbol{\mathrm{F}}}[\mu_{s}]$.

  1. (1)

If
$\boldsymbol{\mu}\in\Gamma_{o}^{0}({\mu_{0}},{\mu_{1}}|{\boldsymbol{\mathrm{F}}})$,
we have that

$\displaystyle[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{0+}$
$\displaystyle\leq[\Phi,\boldsymbol{\mu}]_{l,s}+\lambda
sW^{2}=[\Phi,\boldsymbol{\mu}]_{r,s}+\lambda sW^{2};$ (4.9)

if moreover $\Phi_{0}\in{\boldsymbol{\mathrm{F}}}[\mu_{0}]$ then

$\left[\Phi_{0},\mu_{1}\right]_{r}\leq[\Phi_{0},\boldsymbol{\mu}]_{r,0}\leq[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{0+}.$
(4.10)

  2. (2)

If
$\boldsymbol{\mu}\in\Gamma_{o}^{1}({\mu_{0}},{\mu_{1}}|{\boldsymbol{\mathrm{F}}})$,
we have that

$\displaystyle[\Phi,\boldsymbol{\mu}]_{l,s}-\lambda(1-s)W^{2}=[\Phi,\boldsymbol{\mu}]_{r,s}-\lambda(1-s)W^{2}\leq[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{1-};$

if moreover $\Phi_{1}\in{\boldsymbol{\mathrm{F}}}[\mu_{1}]$ then

$[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{1-}\leq[\Phi_{1},\boldsymbol{\mu}]_{l,1}\leq-\left[\Phi_{1},\mu_{0}\right]_{r}$
(4.11)

  3. (3)

In particular, for every $\Phi_{0}\in{\boldsymbol{\mathrm{F}}}[\mu_{0}]$,
$\Phi_{1}\in{\boldsymbol{\mathrm{F}}}[\mu_{1}]$ and
$\boldsymbol{\mu}\in\Gamma_{o}^{01}({\mu_{0}},{\mu_{1}}|{\boldsymbol{\mathrm{F}}})$
we obtain

$[\Phi_{0},\Phi_{1}]_{r,\boldsymbol{\mu}}\leq[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{0+}-[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{1-}\leq\lambda
W^{2}_{2}(\mu_{0},\mu_{1}).$ (4.12)

(4.12) immediately yields the following property.

######  Corollary 4.13.

Suppose that a MPVF ${\boldsymbol{\mathrm{F}}}$ satisfies

for every $\mu_{0},\mu_{1}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$ the set
$\Gamma_{o}^{01}({\mu_{0}},{\mu_{1}}|{\boldsymbol{\mathrm{F}}})$ of (4.6) is
not empty (4.13)

(e.g. if $\mathrm{D}({\boldsymbol{\mathrm{F}}})$ is open or geodesically
convex), then ${\boldsymbol{\mathrm{F}}}$ is $\lambda$-dissipative if and only
if it satisfies (4.2).

######  Proposition 4.14.

Let ${\boldsymbol{\mathrm{F}}}\subset\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})$
be a MPVF satisfying (4.2), let
$\mu_{0}\in\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$ and let
$\Phi\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}|\mu_{0})$. Consider the
following statements

  1. $(\mathrm{P1})$

$\left[\Phi,\mu\right]_{r}+\left[\Psi,\mu_{0}\right]_{r}\leq\lambda
W_{2}^{2}(\mu_{0},\mu)$ for every $\Psi\in{\boldsymbol{\mathrm{F}}}$ with
$\mu=\mathsf{x}_{\sharp}\Psi$;

  2. $(\mathrm{P2})$

for every $\mu\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$ there exists
$\Psi\in{\boldsymbol{\mathrm{F}}}[\mu]$ s.t.
$\left[\Phi,\mu\right]_{r}+\left[\Psi,\mu_{0}\right]_{r}\leq\lambda
W_{2}^{2}(\mu_{0},\mu)$;

  3. $(\mathrm{P3})$

$[\Phi,\boldsymbol{\mu}]_{r,0}\leq[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{0+}$
for every $\mu_{1}\in\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$,
$\boldsymbol{\mu}\in\Gamma_{o}^{0}({\mu_{0}},{\mu_{1}}|{\boldsymbol{\mathrm{F}}})$;

  4. $(\mathrm{P4})$

$[\Phi,\boldsymbol{\mu}]_{r,0}\leq[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{0+}$
for every $\mu_{1}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$,
$\boldsymbol{\mu}\in\Gamma_{o}^{0}({\mu_{0}},{\mu_{1}}|{\boldsymbol{\mathrm{F}}})$;

  5. $(\mathrm{P5})$

$[\Phi,\boldsymbol{\mu}]_{r,0}\leq\lambda
W_{2}^{2}(\mu_{0},\mu_{1})+[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{1-}$
for every $\mu_{1}\in\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$,
$\boldsymbol{\mu}\in\Gamma_{o}^{1}({\mu_{0}},{\mu_{1}}|{\boldsymbol{\mathrm{F}}})$;

  6. $(\mathrm{P6})$

$[\Phi,\boldsymbol{\mu}]_{r,0}\leq\lambda
W_{2}^{2}(\mu_{0},\mu_{1})+[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{1-}$
for every $\mu_{1}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$,
$\boldsymbol{\mu}\in\Gamma_{o}^{1}({\mu_{0}},{\mu_{1}}|{\boldsymbol{\mathrm{F}}})$.

Then the following hold

  1. (1)

$(\mathrm{P1})$ $\Rightarrow$ $(\mathrm{P2})$ $\Rightarrow$ $(\mathrm{P3})$
$\Rightarrow$ $(\mathrm{P4})$;

  2. (2)

$(\mathrm{P1})$ $\Rightarrow$ $(\mathrm{P2})$ $\Rightarrow$ $(\mathrm{P5})$
$\Rightarrow$ $(\mathrm{P6})$;

  3. (3)

if for every $\mu_{1}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$
$\Gamma_{o}^{0}({\mu_{0}},{\mu_{1}}|{\boldsymbol{\mathrm{F}}})\neq\emptyset$,
then $(\mathrm{P4})$ $\Rightarrow$ $(\mathrm{P1})$ (in particular,
$(\mathrm{P1})$, $(\mathrm{P2})$, $(\mathrm{P3})$, $(\mathrm{P4})$ are
equivalent);

  4. (4)

if for every $\mu_{1}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$
$\Gamma_{o}^{1}({\mu_{0}},{\mu_{1}}|{\boldsymbol{\mathrm{F}}})\neq\emptyset$,
then $(\mathrm{P6})$ $\Rightarrow$ $(\mathrm{P1})$ (in particular,
$(\mathrm{P1})$, $(\mathrm{P2})$, $(\mathrm{P5})$, $(\mathrm{P6})$ are
equivalent).

###### Proof.

We first prove that $(\mathrm{P2})$ $\Rightarrow$
$(\mathrm{P3})$,$(\mathrm{P5})$. Let us choose an arbitrary
$\mu_{1}\in\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$; by the
definition of $[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{r,t}$ and arguing
as in the proof of Theorem 4.10(2), for all
$\boldsymbol{\mu}\in\Gamma_{o}(\mu_{0},\mu_{1})$ and
$t\in\mathrm{I}(\boldsymbol{\mu}|{\boldsymbol{\mathrm{F}}})$ there exists
$\Psi\in{\boldsymbol{\mathrm{F}}}[\mu_{t}]$ such that

$\displaystyle[\Phi,\boldsymbol{\mu}]_{r,0}$
$\displaystyle=\frac{1}{t}\left[\Phi,\mu_{t}\right]_{r}\leq-\frac{1}{t}\left[\Psi,\mu_{0}\right]_{r}+t\lambda
W_{2}^{2}(\mu_{0},\mu_{1})=[\Psi,\boldsymbol{\mu}]_{r,t}+t\lambda
W_{2}^{2}(\mu_{0},\mu_{1})$
$\displaystyle\leq[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{r,t}+t\lambda
W_{2}^{2}(\mu_{0},\mu_{1})$

where we also used (3.26). If
$\boldsymbol{\mu}\in\Gamma_{o}^{0}({\mu_{0}},{\mu_{1}}|{\boldsymbol{\mathrm{F}}})$,
by passing to the limit as $t\downarrow 0$ we get $(\mathrm{P3})$.

In the second case, assuming that
$\boldsymbol{\mu}\in\Gamma_{o}^{1}({\mu_{0}},{\mu_{1}}|{\boldsymbol{\mathrm{F}}})$,
we can pass to the limit as $t\uparrow 1$ and we get $(\mathrm{P5})$.

We now prove item (3). Let $\mu_{1}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$,
$\Psi\in{\boldsymbol{\mathrm{F}}}[\mu_{1}]$,
$\boldsymbol{\mu}\in\Gamma_{o}^{0}({\mu_{0}},{\mu_{1}}|{\boldsymbol{\mathrm{F}}})$,
$s\in\mathrm{I}(\boldsymbol{\mu}|{\boldsymbol{\mathrm{F}}})\cap(0,1)$,
$\Phi_{s}\in{\boldsymbol{\mathrm{F}}}[\mu_{s}]$, with
$\mu_{s}=\mathsf{x}^{s}_{\sharp}\boldsymbol{\mu}$. Assuming $(\mathrm{P4})$
and using (4.10), (4.9), (3.26) and (4.2), we have

$\displaystyle\left[\Phi,\mu_{1}\right]_{r}$
$\displaystyle\leq[\Phi,\boldsymbol{\mu}]_{r,0}\leq[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{0+}\leq[\Phi_{s},\boldsymbol{\mu}]_{r,s}+\lambda
sW_{2}^{2}(\mu_{0},\mu_{1})$
$\displaystyle=\frac{1}{1-s}\left[\Phi_{s},\mu_{1}\right]_{r}+\lambda
sW_{2}^{2}(\mu_{0},\mu_{1})\leq-\frac{1}{1-s}\left[\Psi,\mu_{s}\right]_{r}+\lambda(1+s)W_{2}^{2}(\mu_{0},\mu_{1}).$

By Lemma 3.14, letting $s\downarrow 0$ we get $(\mathrm{P1})$. Item (4)
follows by (4.10), (4.11). ∎

###  4.3. Extensions of dissipative MPVF

Let us briefly study a few simple properties about extensions of
$\lambda$-dissipative MPVFs. The first one concerns the sequential closure in
$\mathcal{P}_{2}^{sw}(\mathsf{T\kern-1.5ptX})$ (the sequential closure may be
smaller than the topological closure, but see Proposition 2.15): given
$A\subset\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})$, we will denote by
$\operatorname{cl}(A)$ its sequential closure defined by

$\operatorname{cl}(A):=\Big{\\{}\Phi\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}):\exists\,\Phi_{n}\in
A:\Phi_{n}\to\Phi\ \text{in
}\mathcal{P}_{2}^{sw}(\mathsf{T\kern-1.5ptX})\Big{\\}}.$

######  Proposition 4.15.

If ${\boldsymbol{\mathrm{F}}}$ is a $\lambda$-dissipative MPVF then its
sequential closure $\operatorname{cl}({\boldsymbol{\mathrm{F}}})$ is
$\lambda$-dissipative as well.

###### Proof.

If $\Phi^{i}$, $i=0,1$, belong to
$\operatorname{cl}({\boldsymbol{\mathrm{F}}})$, we can find sequences
$\Phi^{i}_{n}\in{\boldsymbol{\mathrm{F}}}$ such that $\Phi^{i}_{n}\to\Phi^{i}$
in $\mathcal{P}_{2}^{sw}(\mathsf{T\kern-1.5ptX})$ as $n\to\infty$, $i=0,1$. It
is then sufficient to pass to the limit in the inequality

$\left[\Phi^{0}_{n},\Phi^{1}_{n}\right]_{r}\leq\lambda
W_{2}^{2}(\mu^{0}_{n},\mu^{1}_{n}),\quad\mu^{i}_{n}=\mathsf{x}_{\sharp}\Phi^{i}_{n}$

using the lower semicontinuity property (3.22) and the fact that convergence
in $\mathcal{P}_{2}^{sw}(\mathsf{T\kern-1.5ptX})$ yields
$\mu^{i}_{n}\to\mathsf{x}_{\sharp}\Phi^{i}$ in $\mathcal{P}_{2}(\mathsf{X})$
as $n\to\infty$. ∎

A second result concerns the convexification of the sections of
${\boldsymbol{\mathrm{F}}}$. For every
$\mu\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$ we set

$\displaystyle\operatorname{co}({\boldsymbol{\mathrm{F}}})[\mu]:={}$
$\displaystyle\text{the convex hull of
}{\boldsymbol{\mathrm{F}}}[\mu]=\Big{\\{}\sum_{k}\alpha_{k}\Phi_{k}:\Phi_{k}\in{\boldsymbol{\mathrm{F}}}[\mu],\alpha_{k}\geq
0,\sum_{k}\alpha_{k}=1\Big{\\}},$
$\displaystyle\overline{\operatorname{co}}({\boldsymbol{\mathrm{F}}})[\mu]:={}$
$\displaystyle\operatorname{cl}(\operatorname{co}({\boldsymbol{\mathrm{F}}})[\mu]).$

Notice that if ${\boldsymbol{\mathrm{F}}}[\mu]$ is bounded in
$\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})$ then
$\overline{\operatorname{co}}({\boldsymbol{\mathrm{F}}})[\mu]$ coincides with
the closed convex hull of ${\boldsymbol{\mathrm{F}}}[\mu]$.

######  Proposition 4.16.

If ${\boldsymbol{\mathrm{F}}}$ is $\lambda$-dissipative, then
$\operatorname{co}({\boldsymbol{\mathrm{F}}})$ and
$\overline{\operatorname{co}}({\boldsymbol{\mathrm{F}}})$ are
$\lambda$-dissipative as well.

###### Proof.

By Proposition 4.15 and noting that
$\overline{\operatorname{co}}({\boldsymbol{\mathrm{F}}})\subset\operatorname{cl}(\operatorname{co}({\boldsymbol{\mathrm{F}}}))$,
it is sufficient to prove that $\operatorname{co}({\boldsymbol{\mathrm{F}}})$
is $\lambda$-dissipative. By Lemma 4.6 it is not restrictive to assume
$\lambda=0$. Let
$\Phi^{i}\in\operatorname{co}({\boldsymbol{\mathrm{F}}})[\mu_{i}]$, $i=0,1$;
there exist positive coefficients $\alpha^{i}_{k}$, $k=1,\cdots,K$, with
$\sum_{k}\alpha^{i}_{k}=1$, and elements
$\Phi^{i}_{k}\in{\boldsymbol{\mathrm{F}}}[\mu^{i}]$, $i=0,1$, such that
$\Phi^{i}=\sum_{k=1}^{K}\alpha^{i}_{k}\Phi^{i}_{k}$. Setting
$\beta_{h,k}:=\alpha^{0}_{h}\alpha^{1}_{k}$, we can apply Lemma 3.16 and we
obtain

$\left[\Phi^{0},\Phi^{1}\right]_{r}=\Big{[}{\sum_{h,k}\beta_{h,k}\Phi^{0}_{h}},{\sum_{h,k}\beta_{h,k}\Phi^{1}_{k}}\Big{]}_{r}\leq\sum_{h,k}\beta_{h,k}\left[\Phi^{0}_{h},\Phi^{1}_{k}\right]_{r}\leq
0.\qed$

As a last step, we want to study the properties of the extended MPVF

$\displaystyle\hat{{\boldsymbol{\mathrm{F}}}}:=\Big{\\{}$
$\displaystyle\Phi\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}):\mu=\mathsf{x}_{\sharp}\Phi\in\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})},$
(4.14)
$\displaystyle\left[\Phi,\nu\right]_{r}+\left[\Psi,\mu\right]_{r}\leq\lambda
W_{2}^{2}(\mu,\nu)\quad\forall\,\Psi\in{\boldsymbol{\mathrm{F}}},\
\nu=\mathsf{x}_{\sharp}\Psi\Big{\\}}.$

It is obvious that
${\boldsymbol{\mathrm{F}}}\subset\hat{\boldsymbol{\mathrm{F}}}$; if the domain
of ${\boldsymbol{\mathrm{F}}}$ satisfies the geometric condition (4.16), the
following result shows that $\hat{\boldsymbol{\mathrm{F}}}$ provides the
maximal $\lambda$-dissipative extension of ${\boldsymbol{\mathrm{F}}}$.

######  Proposition 4.17.

Let ${\boldsymbol{\mathrm{F}}}$ be a $\lambda$-dissipative MPVF.

  1. (a)

If ${\boldsymbol{\mathrm{F}}}^{\prime}\supset{\boldsymbol{\mathrm{F}}}$ is
$\lambda$-dissipative with
$\mathrm{D}({\boldsymbol{\mathrm{F}}}^{\prime})\subset\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$,
then ${\boldsymbol{\mathrm{F}}}^{\prime}\subset\hat{\boldsymbol{\mathrm{F}}}$.
In particular
$\overline{\operatorname{co}}(\operatorname{cl}({\boldsymbol{\mathrm{F}}}))\subset\hat{\boldsymbol{\mathrm{F}}}$.

  2. (b)

$\widehat{{\operatorname{cl}({\boldsymbol{\mathrm{F}}})}}=\hat{\boldsymbol{\mathrm{F}}}$
and
$\widehat{{\operatorname{co}({\boldsymbol{\mathrm{F}}})}}=\hat{\boldsymbol{\mathrm{F}}}$.

  3. (c)

$\hat{\boldsymbol{\mathrm{F}}}$ is sequentially closed and
$\hat{\boldsymbol{\mathrm{F}}}[\mu]$ is convex for every
$\mu\in\mathrm{D}(\hat{\boldsymbol{\mathrm{F}}})$.

  4. (d)

If $\mathrm{D}({\boldsymbol{\mathrm{F}}})$ satisfies (4.13), then the
restriction of $\hat{\boldsymbol{\mathrm{F}}}$ to
$\mathrm{D}({\boldsymbol{\mathrm{F}}})$ is $\lambda$-dissipative and for every
$\mu_{0},\mu_{1}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$

$[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{0+}=[\hat{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{0+},\quad[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{1-}=[\hat{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{1-}\quad\text{for
every
}\boldsymbol{\mu}\in\Gamma_{o}^{01}({\mu_{0}},{\mu_{1}}|{\boldsymbol{\mathrm{F}}}).$
(4.15)

  5. (e)

If $\mu_{0}\in\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})},\
\mu_{1}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$ and
$\Gamma_{o}^{1}({\mu_{0}},{\mu_{1}}|{\boldsymbol{\mathrm{F}}})\neq\emptyset$
then

$\Phi_{i}\in\hat{\boldsymbol{\mathrm{F}}}[\mu_{i}]\quad\Rightarrow\quad\left[\Phi_{0},\Phi_{1}\right]_{r}\leq\lambda
W_{2}^{2}(\mu_{0},\mu_{1}).$

  6. (f)

If

for every $\mu_{0},\mu_{1}\in\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$