the set $\Gamma_{o}^{01}({\mu_{0}},{\mu_{1}}|{\boldsymbol{\mathrm{F}}})$ is
not empty, (4.16)

then $\hat{\boldsymbol{\mathrm{F}}}$ is $\lambda$-dissipative as well and for
every $\mu_{0},\mu_{1}\in\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$
(4.15) holds.

###### Proof.

Claim (a) is obvious since every $\lambda$-dissipative extension
${\boldsymbol{\mathrm{F}}}^{\prime}$ of ${\boldsymbol{\mathrm{F}}}$ in
$\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$ satisfies
${\boldsymbol{\mathrm{F}}}^{\prime}\subset\hat{\boldsymbol{\mathrm{F}}}$.

(b) Let us prove that if $\Phi\in\hat{\boldsymbol{\mathrm{F}}}$ then
$\Phi\in\widehat{\operatorname{cl}({\boldsymbol{\mathrm{F}}})}$. If
$\Psi\in\operatorname{cl}({\boldsymbol{\mathrm{F}}})$ we can find a sequence
$\Psi_{n}\in{\boldsymbol{\mathrm{F}}}$ converging to $\Psi$ in
$\mathcal{P}_{2}^{sw}(\mathsf{T\kern-1.5ptX})$ as $n\to\infty$. We can then
pass to the limit in the inequalities

$\left[\Phi,\nu_{n}\right]_{r}+\left[\Phi_{n},\mu\right]_{r}\leq\lambda
W_{2}^{2}(\mu,\nu_{n}),\quad\mu=\mathsf{x}_{\sharp}\Phi,\
\nu_{n}=\mathsf{x}_{\sharp}\Psi_{n},$

using the lower semicontinuity results of Lemma 3.14. We conclude since
$\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}=\overline{\mathrm{D}(\operatorname{cl}({\boldsymbol{\mathrm{F}}}))}$.

In order to prove that $\Phi\in\hat{\boldsymbol{\mathrm{F}}}\ \Rightarrow\
\Phi\in\widehat{{\operatorname{co}({\boldsymbol{\mathrm{F}}})}}$ we take
$\Psi=\sum\alpha_{k}\Psi_{k}\in\operatorname{co}({\boldsymbol{\mathrm{F}}})$;
for some $\Psi_{k}\in{\boldsymbol{\mathrm{F}}}[\nu]$,
$\nu=\mathsf{x}_{\sharp}\Psi\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$, and
positive coefficients $\alpha_{k}$, $k=1,\cdots,K$, with
$\sum_{k}\alpha_{k}=1$. Taking a convex combination of the inequalities

$\left[\Phi,\nu\right]_{r}+\left[\Psi_{k},\mu\right]_{r}\leq\lambda
W_{2}^{2}(\mu,\nu),\quad\text{for every }k=1,\cdots,K,$

and using Lemma 3.16 we obtain

$\left[\Phi,\nu\right]_{r}+\left[\Psi,\mu\right]_{r}\leq\sum_{k}\alpha_{k}\Big{(}\left[\Phi,\nu\right]_{r}+\left[\Psi_{k},\mu\right]_{r}\Big{)}\leq\lambda
W_{2}^{2}(\mu,\nu).$

The proof of claim (c) follows by a similar argument.

(d) Let $\mu_{i}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$,
$\Phi_{i}\in\hat{\boldsymbol{\mathrm{F}}}[\mu_{i}]$, $i=0,1$, and
$\boldsymbol{\mu}\in\Gamma_{o}^{01}({\mu_{0}},{\mu_{1}}|{\boldsymbol{\mathrm{F}}})$.
The implication $(\mathrm{P1})$$\Rightarrow$$(\mathrm{P4})$ of Proposition
4.14 applied to $\boldsymbol{\mu}$ and to
$\mathsf{s}_{\sharp}\boldsymbol{\mu}$ yields

$[\Phi_{0},\boldsymbol{\mu}]_{r,0}\leq[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{0+},\quad[\Phi_{1},\mathsf{s}_{\sharp}\boldsymbol{\mu}]_{r,0}\leq[{\boldsymbol{\mathrm{F}}},\mathsf{s}_{\sharp}\boldsymbol{\mu}]_{0+}=-[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{1-}$

so that (4.12) yields

$\left[\Phi_{0},\Phi_{1}\right]_{r}\leq[\Phi_{0},\boldsymbol{\mu}]_{r,0}+[\Phi_{1},\mathsf{s}_{\sharp}\boldsymbol{\mu}]_{r,0}\leq[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{0+}-[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{1-}\leq\lambda
W_{2}^{2}(\mu_{0},\mu_{1}).$

In order to prove (4.15) we observe that
${\boldsymbol{\mathrm{F}}}\subset\hat{\boldsymbol{\mathrm{F}}}$ so that, for
every
$\boldsymbol{\mu}\in\Gamma_{o}^{01}({\mu_{0}},{\mu_{1}}|{\boldsymbol{\mathrm{F}}})$
and every $t\in\mathrm{I}(\boldsymbol{\mu}|{\boldsymbol{\mathrm{F}}})$, we
have
$[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{r,t}\leq[\hat{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{r,t}$
and
$[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{l,t}\geq[\hat{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{l,t}$,
hence (4.15) is a consequence of Definition 4.11 and Theorem 4.10.

The proof of claim (f) follows by the same argument.

In the case of claim (e), we use the implication
$(\mathrm{P1})$$\Rightarrow$$(\mathrm{P6})$ of Proposition 4.14 applied to
$\boldsymbol{\mu}$ and the implication
$(\mathrm{P1})$$\Rightarrow$$(\mathrm{P3})$ applied to
$\mathsf{s}_{\sharp}\boldsymbol{\mu}$, obtaining

$[\Phi_{0},\boldsymbol{\mu}]_{r,0}\leq\lambda
W_{2}^{2}(\mu_{0},\mu_{1})+[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{1-},\quad[\Phi_{1},\mathsf{s}_{\sharp}\boldsymbol{\mu}]_{r,0}\leq[{\boldsymbol{\mathrm{F}}},\mathsf{s}_{\sharp}\boldsymbol{\mu}]_{0+}=-[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{1-}$

and then

$\left[\Phi_{0},\Phi_{1}\right]_{r}\leq[\Phi_{0},\boldsymbol{\mu}]_{r,0}+[\Phi_{1},\mathsf{s}_{\sharp}\boldsymbol{\mu}]_{r,0}\leq\lambda
W_{2}^{2}(\mu_{0},\mu_{1}).\qed$

##  5\. Examples of $\lambda$-dissipative MPVFs

In this section we present significant examples of $\lambda$-dissipative MPVFs
which are interesting for applications.

###  5.1. Subdifferentials of $\lambda$-convex functionals

Recall that a functional
$\mathcal{F}:\mathcal{P}_{2}(\mathsf{X})\to(-\infty,+\infty]$ is
$\lambda$-(geodesically) convex on $\mathcal{P}_{2}(\mathsf{X})$ (see [AGS08,
Definition 9.1.1]) if for any $\mu_{0},\mu_{1}$ in the proper domain
$D(\mathcal{F}):=\\{\mu\in\mathcal{P}_{2}(\mathsf{X})\mid\mathcal{F}(\mu)<+\infty\\}$
there exists $\boldsymbol{\mu}\in\Gamma_{o}(\mu_{0},\mu_{1})$ such that

$\mathcal{F}(\mu_{t})\leq(1-t)\mathcal{F}(\mu_{0})+t\mathcal{F}(\mu_{1})-\frac{\lambda}{2}t(1-t)W_{2}^{2}(\mu_{0},\mu_{1})\qquad\forall\,t\in[0,1],$

where $(\mu_{t})_{t\in[0,1]}$ is the constant speed geodesic induced by
$\boldsymbol{\mu}$, i.e. $\mu_{t}=\mathsf{x}^{t}_{\sharp}\boldsymbol{\mu}$.

The _Fréchet subdifferential $\boldsymbol{\partial}\mathcal{F}$_ of
$\mathcal{F}$ [AGS08, Definition 10.3.1] is a MPVF which can be characterized
[AGS08, Theorem 10.3.6] by

$\Phi\in\boldsymbol{\partial}\mathcal{F}[\mu]\quad\Leftrightarrow\quad\mu\in
D(\mathcal{F}),\
\mathcal{F}(\nu)-\mathcal{F}(\mu)\geq-\left[\Phi,\nu\right]_{l}+\frac{\lambda}{2}W_{2}^{2}(\mu,\nu)\quad\text{for
every }\nu\in D(\mathcal{F}).$

According to the notation introduced in (3.14), we set

$-\boldsymbol{\partial}\mathcal{F}[\mu]=J_{\sharp}\boldsymbol{\partial}\mathcal{F}[\mu],\quad
J(x,v)=(x,-v),$ (5.1)

and we have the following result.

######  Theorem 5.1.

If $\mathcal{F}:\mathcal{P}_{2}(\mathsf{X})\to(-\infty,+\infty]$ is a proper,
lower semicontinuous and $\lambda$-convex functional, then
$-\boldsymbol{\partial}\mathcal{F}$ is a _$(-\lambda)$ -dissipative_ MPVF.

Referring to [AGS08], here we list interesting and explicit examples of
$(-\lambda)$-dissipative MPVFs induced by proper, lower semicontinuous and
$\lambda$-convex functionals, focusing on the cases when
$\mathrm{D}(\boldsymbol{\partial}\mathcal{F})=\mathcal{P}_{2}(\mathsf{X}).$

  1. (1)

_Potential energy._ Let $P:\mathsf{X}\to\mathbb{R}$ be a l.s.c. and
$\lambda$-convex functional satisfying

$|\partial^{o}P(x)|\leq C(1+|x|)\quad\text{for every }x\in\mathsf{X},$

for some constant $C>0$, where $\partial^{o}P(x)$ is the element of minimal
norm in $\partial P(x)$. By [AGS08, Proposition 10.4.2] the PVF

${\boldsymbol{\mathrm{F}}}[\mu]:=(\boldsymbol{i}_{\mathsf{X}},-\partial^{o}P)_{\sharp}\mu,\quad\mu\in\mathcal{P}_{2}(\mathsf{X}),$

is a $(-\lambda)$-dissipative selection of
$-\boldsymbol{\partial}\mathcal{F}_{P}$ for the potential energy functional

$\mathcal{F}_{P}(\mu):=\int_{\mathsf{X}}P\,\mathrm{d}\mu,\quad\mu\in\mathcal{P}_{2}(\mathsf{X}).$

  2. (2)

_Interaction energy._ If $W:\mathsf{X}\to[0,+\infty)$ is an even,
differentiable, and $\lambda$-convex function for some $\lambda\in\mathbb{R}$,
whose differential has a linear growth, then, by [AGS08, Theorem 10.4.11], the
PVF

${\boldsymbol{\mathrm{F}}}[\mu]:=(\boldsymbol{i}_{\mathsf{X}},(-\nabla
W\ast\mu))_{\sharp}\mu,\quad\mu\in\mathcal{P}_{2}(\mathsf{X}),$

is a $(-\lambda)$-dissipative selection of
$-\boldsymbol{\partial}\mathcal{F}_{W}$, the opposite of the Wasserstein
subdifferential of the interaction energy functional

$\mathcal{F}_{W}(\mu):=\frac{1}{2}\int_{\mathsf{X}^{2}}W(x-y)\,\mathrm{d}(\mu\otimes\mu)(x,y),\quad\mu\in\mathcal{P}_{2}(\mathsf{X}).$

  3. (3)

_Opposite Wasserstein distance._ Let $\bar{\mu}\in\mathcal{P}_{2}(\mathsf{X})$
be fixed and consider the functional
$\mathcal{F}_{\text{Wass}}:\mathcal{P}_{2}(\mathsf{X})\to\mathbb{R}$ defined
as

$\mathcal{F}_{\text{Wass}}(\mu):=-\frac{1}{2}W_{2}^{2}(\mu,\bar{\mu}),\quad\mu\in\mathcal{P}_{2}(\mathsf{X}),$

which is geodesically $(-1)$-convex [AGS08, Proposition 9.3.12]. Setting

$\boldsymbol{b}(\mu):=\operatorname*{arg\,min}\left\\{\int_{\mathsf{X}}|\boldsymbol{b}(x)-x|^{2}\,\mathrm{d}\mu:\boldsymbol{b}=\boldsymbol{b}_{\boldsymbol{\gamma}}\in
L^{2}_{\mu}(\mathsf{X};\mathsf{X}),\
\boldsymbol{\gamma}\in\Gamma_{o}(\mu,\bar{\mu})\right\\},$

the PVF

${\boldsymbol{\mathrm{F}}}[\mu]:=(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{i}_{\mathsf{X}}-\boldsymbol{b}(\mu))_{\\#}\mu,\quad\mu\in\mathcal{P}_{2}(\mathsf{X})$

is a selection of $-\boldsymbol{\partial}\mathcal{F}_{\text{Wass}}(\mu)$ and
it is therefore $1$-dissipative.

###  5.2. MPVF concentrated on the graph of a multifunction

The previous example of Section 5.1 has a natural generalization in terms of
dissipative graphs in $\mathsf{X}\times\mathsf{X}$ [AC84, AF09, Bré73]. We
consider a (not empty) $\lambda$-dissipative set
$F\subset\mathsf{X}\times\mathsf{X}$, i.e. satisfying

$\langle
v_{0}-v_{1},x_{0}-x_{1}\rangle\leq\lambda|x_{0}-x_{1}|^{2}\quad\text{for any
}(x_{0},v_{0}),\ (x_{1},v_{1})\in F.$

The corresponding MPVF defined as

${\boldsymbol{\mathrm{F}}}:=\left\\{\Phi\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})\mid\Phi\text{
is concentrated on }F\right\\}$

is $\lambda$-dissipative as well. In fact, if
$\Phi_{0},\Phi_{1}\in{\boldsymbol{\mathrm{F}}}$ with
$\nu_{i}=\mathsf{x}_{\sharp}\Phi_{i}$, $i=0,1$, and
$\boldsymbol{\Theta}\in\Lambda(\Phi_{0},\Phi_{1})$ then
$(x_{0},v_{0},x_{1},v_{1})\in F\times F$ $\boldsymbol{\Theta}$-a.e., so that

$\displaystyle\int_{\mathsf{T\kern-1.5ptX}\times\mathsf{T\kern-1.5ptX}}\langle
v_{0}-v_{1},x_{0}-x_{1}\rangle\,\mathrm{d}\boldsymbol{\Theta}(x_{0},v_{0},x_{1},v_{1})\leq\lambda\int_{\mathsf{T\kern-1.5ptX}\times\mathsf{T\kern-1.5ptX}}|x_{0}-x_{1}|^{2}\,\mathrm{d}\boldsymbol{\Theta}=\lambda
W_{2}^{2}(\nu_{0},\nu_{1}).$

since
$(\mathsf{x}^{0},\mathsf{x}^{1})_{\sharp}\boldsymbol{\Theta}\in\Gamma_{o}(\nu_{0},\nu_{1})$.
Taking the supremum w.r.t. $\boldsymbol{\Theta}\in\Lambda(\Phi_{0},\Phi_{1})$
we obtain $\left[\Phi_{0},\Phi_{1}\right]_{l}\leq\lambda
W_{2}^{2}(\nu_{0},\nu_{1})$ which is even stronger than
$\lambda$-dissipativity. If $\mathrm{D}(F)=\mathsf{X}$ then
$\mathrm{D}({\boldsymbol{\mathrm{F}}})$ contains $\mathcal{P}_{\rm
c}(\mathsf{X})$, the set of Borel probability measures with compact support.
If $F$ has also a linear growth, then it is easy to check that
$\mathrm{D}({\boldsymbol{\mathrm{F}}})=\mathcal{P}_{2}(\mathsf{X})$ as well.

Despite the analogy just shown with dissipative operators in Hilbert spaces,
there are important differences with the Wasserstein framework, as highlighted
in the following examples. The main point here is that the dissipativity
property of Definition 4.3 does not force the sections
$\mathsf{v}_{\sharp}{\boldsymbol{\mathrm{F}}}[\mu]$ to belong to the tangent
space $\operatorname{Tan}_{\mu}\mathcal{P}_{2}(\mathsf{X})$.

######  Example 5.2.

Let $\mathsf{X}=\mathbb{R}^{2}$, let $B:=\\{x\in\mathbb{R}^{2}\mid|x|\leq
1\\}$ be the closed unit ball, let ${\mathcal{L}_{B}}$ be the (normalized)
Lebesgue measure on $B$, and let
${\boldsymbol{r}}:\mathbb{R}^{2}\to\mathbb{R}^{2}$,
${\boldsymbol{r}}(x_{1},x_{2})=(x_{2},-x_{1})$ be the anti-clockwise rotation
of $\pi/2$ degrees. We define the MPVF

${\boldsymbol{\mathrm{F}}}[\nu]=\begin{cases}(\boldsymbol{i}_{\mathbb{R}^{2}},0)_{\sharp}\nu,\quad&\text{
if }\nu\in\mathcal{P}_{2}(\mathbb{R}^{2})\setminus\\{{\mathcal{L}_{B}}\\},\\\
\left\\{(\boldsymbol{i}_{\mathbb{R}^{2}},a{\boldsymbol{r}})_{\sharp}{\mathcal{L}_{B}}\mid
a\in\mathbb{R}\right\\},\quad&\text{ if }\nu={\mathcal{L}_{B}}.\end{cases}$

Observe that
$\mathrm{D}({\boldsymbol{\mathrm{F}}})=\mathcal{P}_{2}(\mathbb{R}^{2})$ and
${\boldsymbol{\mathrm{F}}}$ is obviously unbounded at $\nu={\mathcal{L}_{B}}$.
${\boldsymbol{\mathrm{F}}}$ also satisfies (4.2) with $\lambda=0$ (hence it is
dissipative): it is enough to check that

$\left[(\boldsymbol{i}_{\mathbb{R}^{2}},a{\boldsymbol{r}})_{\sharp}{\mathcal{L}_{B}},\nu\right]_{r}=0\quad\text{
for every }\nu\in\mathcal{P}_{2}(\mathbb{R}^{2}),\,a\in\mathbb{R}.$ (5.2)

To prove (5.2), we notice that the optimal transport plan from
${\mathcal{L}_{B}}$ to $\nu$ is concentrated on a map and optimal maps belong
to the tangent space
$\operatorname{Tan}_{{\mathcal{L}_{B}}}\mathcal{P}_{2}(\mathbb{R}^{2})$
[AGS08, Prop. 8.5.2]; by Remark 3.18 we have just to check that

$\int_{\mathbb{R}^{2}}\langle{\boldsymbol{r}}(x),\nabla\varphi(x)\rangle\,\mathrm{d}{\mathcal{L}_{B}}(x)=0\quad\forall\varphi\in\mathrm{C}^{\infty}_{c}(\mathbb{R}^{2}),$

that is a consequence of the Divergence Theorem on $B$. This example is in
contrast with the Hilbertian theory of dissipative operators according to
which an everywhere defined dissipative operator is locally bounded (see
[Bré73, Proposition 2.9]).

######  Example 5.3.

In the same setting of the previous example, let us define the MPVF

${\boldsymbol{\mathrm{F}}}[\nu]=(\boldsymbol{i}_{\mathbb{R}^{2}},{\boldsymbol{r}})_{\sharp}\nu,\quad{\boldsymbol{r}}(x_{1},x_{2})=(x_{2},-x_{1}),\quad\nu\in\mathcal{P}_{2}(\mathbb{R}^{2}).$

It is easy to check that ${\boldsymbol{\mathrm{F}}}$ is dissipative and
Lipschitz continuous (as a map from $\mathcal{P}_{2}(\mathbb{R}^{2})$ to
$\mathcal{P}_{2}(\mathrm{T}\mathbb{R}^{2})$). Moreover, arguing as in Example
5.2, we can show that
$(\boldsymbol{i}_{\mathbb{R}^{d}},0)_{\sharp}{\mathcal{L}_{B}}\in\hat{\boldsymbol{\mathrm{F}}}[{\mathcal{L}_{B}}]$,
where $\hat{\boldsymbol{\mathrm{F}}}$ is defined in (4.14). This is again in
contrast with the Hilbertian theory of dissipative operators, stating that a
single valued, everywhere defined, and continuous dissipative operator
coincides with its maximal extension (see [Bré73, Proposition 2.4]).

###  5.3. Interaction field induced by a dissipative map

Let us consider the Hilbert space $\mathsf{Y}=\mathsf{X}^{n}$,
$n\in\mathbb{N}$, endowed with the scalar product
$\langle\boldsymbol{x},\boldsymbol{y}\rangle:=\frac{1}{n}\sum_{i=1}^{n}\langle
x_{i},y_{i}\rangle$, for every $\boldsymbol{x}=(x_{i})_{i=1}^{n},\
\boldsymbol{y}=(y_{i})_{i=1}^{n}\in\mathsf{X}^{n}$. We identify
$\mathsf{T\kern-1.5ptY}$ with $(\mathsf{T\kern-1.5ptX})^{n}$ and we denote by
$\mathsf{x}^{i},\mathsf{v}^{i}$ the $i$-th coordinate maps. Every permutation
$\sigma:\\{1,\cdots,n\\}\to\\{1,\cdots,n\\}$ in $\mathrm{Sym}(n)$ operates on
$\mathsf{Y}$ by the obvious formula
$\sigma(\boldsymbol{x})_{i}=x_{\sigma(i)}$, $i=1,\cdots,n$,
$\boldsymbol{x}\in\mathsf{Y}$.

Let $G:\mathsf{Y}\to\mathsf{Y}$ be a Borel $\lambda$-dissipative map bounded
on bounded sets (this property is always true if $\mathsf{Y}$ has finite
dimension) and satisfying

${\boldsymbol{x}}\in\mathrm{D}(G)\quad\Rightarrow\quad\sigma({\boldsymbol{x}})\in\mathrm{D}(G),\
G(\sigma({\boldsymbol{x}}))=\sigma(G({\boldsymbol{x}}))\quad\text{for every
permutation $\sigma$}.$ (5.3)

Denoting by $(G^{1},\cdots,G^{n})$ the components of $G$, by $\mathsf{x}^{i}$
the projections from $\mathsf{Y}$ to $\mathsf{X}$ and by $\mu^{\otimes
n}=\bigotimes_{i=1}^{n}\mu$, the MPVF

${\boldsymbol{\mathrm{F}}}[\mu]:=(\mathsf{x}^{1},G^{1})_{\sharp}\mu^{\otimes
n}\quad\text{with domain
}\mathrm{D}({\boldsymbol{\mathrm{F}}}):=\mathcal{P}_{b}(\mathsf{X})$

is $\lambda$-dissipative as well. In fact, if
$\mu,\nu\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$,
$\Phi=(\mathsf{x}^{1},G^{1})_{\sharp}\mu^{\otimes n}$ and
$\Psi=(\mathsf{x}^{1},G^{1})_{\sharp}\nu^{\otimes n}$, and
$\boldsymbol{\gamma}\in\Gamma_{o}(\mu,\nu)$, we can consider the plan
$\boldsymbol{\beta}:=P_{\sharp}\boldsymbol{\gamma}^{\otimes
n}\in\Gamma(\mu^{\otimes n},\nu^{\otimes n})$, where
$P((x_{1},y_{1}),\cdots,(x_{n},y_{n})):=((x_{1},\cdots,x_{n}),(y_{1},\cdots,y_{n}))$.
Considering the map
$H^{1}({\boldsymbol{x}},{\boldsymbol{y}}):=(x_{1},G^{1}({\boldsymbol{x}}),y_{1},G^{1}({\boldsymbol{y}}))$
we have
$\boldsymbol{\Theta}:=H^{1}_{\sharp}\boldsymbol{\beta}\in\Lambda(\Phi,\Psi)$,
so that

$\displaystyle\left[\Phi,\Psi\right]_{r}$ $\displaystyle\leq\int\langle
v_{1}-w_{1},x_{1}-y_{1}\rangle\,\mathrm{d}\boldsymbol{\Theta}(x_{1},v_{1},y_{1},w_{1})=\int\langle
G^{1}({\boldsymbol{x}})-G^{1}({\boldsymbol{y}}),x_{1}-y_{1}\rangle\,\mathrm{d}\boldsymbol{\beta}({\boldsymbol{x}},{\boldsymbol{y}})$
$\displaystyle=\frac{1}{n}\sum_{k=1}^{n}\int\langle
G^{k}({\boldsymbol{x}})-G^{k}({\boldsymbol{y}}),x_{k}-y_{k}\rangle\,\mathrm{d}\boldsymbol{\beta}({\boldsymbol{x}},{\boldsymbol{y}})=\int\langle
G({\boldsymbol{x}})-G({\boldsymbol{y}}),{\boldsymbol{x}}-{\boldsymbol{y}}\rangle\,\mathrm{d}\boldsymbol{\beta}({\boldsymbol{x}},{\boldsymbol{y}})$

where we used (5.3) and the invariance of $\boldsymbol{\beta}$ with respect to
permutations. The $\lambda$-dissipativity of $G$ then yields

$\displaystyle\int\langle
G({\boldsymbol{x}})-G({\boldsymbol{y}}),{\boldsymbol{x}}-{\boldsymbol{y}}\rangle\,\mathrm{d}\boldsymbol{\beta}({\boldsymbol{x}},{\boldsymbol{y}})$
$\displaystyle\leq\lambda\int|{\boldsymbol{x}}-{\boldsymbol{y}}|^{2}_{\mathsf{Y}}\,\mathrm{d}\boldsymbol{\beta}({\boldsymbol{x}},{\boldsymbol{y}})=\lambda\frac{1}{n}\sum_{k=1}^{n}\int|x_{k}-y_{k}|^{2}_{\mathsf{Y}}\,\mathrm{d}\boldsymbol{\beta}({\boldsymbol{x}},{\boldsymbol{y}})$
$\displaystyle=\lambda\frac{1}{n}\sum_{k=1}^{n}\int|x_{k}-y_{k}|^{2}_{\mathsf{Y}}\,\mathrm{d}\boldsymbol{\boldsymbol{\gamma}}(x_{k},y_{k})=\lambda
W_{2}^{2}(\mu,\nu).$

A typical example when $n=2$ is provided by

$G(x_{1},x_{2}):=(A(x_{1}-x_{2}),A(x_{2}-x_{1}))$

where $A:\mathsf{X}\to\mathsf{X}$ is a Borel, locally bounded, dissipative and
antisymmetric map satisfying $A(-z)=-A(z)$. We easily get

$\displaystyle\langle$ $\displaystyle
G({\boldsymbol{x}})-G({\boldsymbol{y}}),{\boldsymbol{x}}-{\boldsymbol{y}}\rangle$
$\displaystyle=\frac{1}{2}\Big{(}\langle
A(x_{1}-x_{2})-A(y_{1}-y_{2}),x_{1}-y_{1}\rangle-\langle
A(x_{1}-x_{2})-A(y_{1}-y_{2}),x_{2}-y_{2}\rangle\Big{)}$
$\displaystyle=\frac{1}{2}\langle
A(x_{1}-x_{2})-A(y_{1}-y_{2}),x_{1}-x_{2}-(y_{1}-y_{2})\rangle\leq 0.$

In this case

${\boldsymbol{\mathrm{F}}}[\mu]=(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{a}[\mu])_{\sharp}\mu,\quad\boldsymbol{a}[\mu](x)=\int_{\mathsf{X}}A(x-y)\,\mathrm{d}\mu(y)\quad\text{for
every }x\in\mathsf{X}.$

##  6\. Solutions to Measure Differential Inclusions

###  6.1. Metric characterization and EVI

Let $\mathcal{I}$ denote an arbitrary (bounded or unbounded) interval in
$\mathbb{R}$.

The aim of this section is to study a suitable notion of solution to the
following differential inclusion in the $L^{2}$-Wasserstein space of
probability measures

$\dot{\mu}(t)\in{\boldsymbol{\mathrm{F}}}[\mu(t)],\qquad t\in\mathcal{I},$
(6.1)

driven by a MPVF ${\boldsymbol{\mathrm{F}}}$ as in Definition 4.1. In
particular, we will address the usual Cauchy problem when (6.1) is
supplemented by a given initial condition.

Measure Differential Inclusions have been introduced in [Pic18] extending to
the multi-valued framework the theory of Measure Differential Equations
developed in [Pic19]. In these papers, the author aims to describe the
evolution of curves in the space of probability measures under the action of a
so called _probability vector field_ ${\boldsymbol{\mathrm{F}}}$ (see
Definition 4.1 and Remark 4.2). However, as exploited also in [Cam+21], the
definition of solution to (6.1) given in Piccoli_2019,Piccoli_MDI,Camilli_MDE
is too weak and it does not enjoy uniqueness property which is recovered only
at the level of the semigroup through an approximation procedure.

From the Wasserstein viewpoint, the simplest way to interpret (6.1) is to ask
for a locally absolutely continuous curve
$\mu:\mathcal{I}\to\mathcal{P}_{2}(\mathsf{X})$ to satisfy

$(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}_{t})_{\sharp}\mu_{t}\in{{\boldsymbol{\mathrm{F}}}}[\mu_{t}]\quad\text{for
a.e. }t\in\mathcal{I},$ (6.2)

where $\boldsymbol{v}$ is the Wasserstein metric velocity vector associated to
$\mu$ (see Theorem 2.10). Even in the case of a regular PVF, however, (6.2) is
too strong, since there is no reason why a given
${\boldsymbol{\mathrm{F}}}[\mu_{t}]$ should be associated to a vector field of
the tangent space $\operatorname{Tan}_{\mu_{t}}\mathcal{P}_{2}(\mathsf{X})$.
Starting from (6.2), we thus introduce a weaker definition of solution to
(6.1), modeled on the so-called EVI formulation for gradient flows, which will
eventually suggest, as a natural formulation of (6.1), the relaxed version of
(6.2) as a differential inclusion with respect to the extension
$\hat{\boldsymbol{\mathrm{F}}}$ of ${\boldsymbol{\mathrm{F}}}$ introduced in
(4.14).

We start from this simple remark: whenever ${\boldsymbol{\mathrm{F}}}$ is
$\lambda$-dissipative, recalling Theorem 3.11 and Remark 4.5, one easily sees
that every locally absolutely continuous solution according to the above
definition (6.2) also satisfies the Evolution Variational Inequality
($\lambda$-EVI)

$\frac{1}{2}\frac{\,\mathrm{d}}{\,\mathrm{d}t}W_{2}^{2}(\mu_{t},\nu)\leq\lambda
W_{2}^{2}(\mu_{t},\nu)-\left[\Phi,\mu_{t}\right]_{r}\quad\text{in
}\mathscr{D}^{\prime}\big{(}\operatorname{int}\left(\mathcal{I}\right)\big{)},$
($\lambda$-EVI)

for every $\nu\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$ and every
$\Phi\in{\boldsymbol{\mathrm{F}}}[\nu]$, where $\left[\cdot,\cdot\right]_{r}$
is the functional pairing in Definition 3.5 (in fact, ($\lambda$-EVI) holds
a.e. in $\mathcal{I}$). This provides a heuristic motivation for the following
definition.

######  Definition 6.1 ($\lambda$-Evolution Variational Inequality).

Let ${\boldsymbol{\mathrm{F}}}$ be a MPVF and let $\lambda\in\mathbb{R}$. We
say that a continuous curve
$\mu:\mathcal{I}\to\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$ is a
_$\lambda$ -EVI solution_ to (6.1) for the MPVF ${\boldsymbol{\mathrm{F}}}$ if
($\lambda$-EVI) holds for every $\nu\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$
and every $\Phi\in{\boldsymbol{\mathrm{F}}}[\nu]$.
A $\lambda$-EVI solution $\mu$ is said to be a _strict solution_ if
$\mu_{t}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$ for every $t\in\mathcal{I}$,
$t>\inf\mathcal{I}$.
A $\lambda$-EVI solution $\mu$ is said to be a _global solution_ if
$\sup\mathcal{I}=+\infty$.

In Example 6.32 we will clarify the interest in imposing no more than
continuity in the above definition.

Recall that the right upper and lower Dini derivatives of a function
$\zeta:\mathcal{I}\to\mathbb{R}$ are defined for every $t\in\mathcal{I}$,
$t<\sup\mathcal{I}$ by

${\frac{\mathrm{d}}{\mathrm{d}t}}^{\kern-3.0pt+}\zeta(t)\limsup_{h\downarrow
0}\frac{\zeta(t+h)-\zeta(t)}{h},\qquad{\frac{\mathrm{d}}{\mathrm{d}t}}_{\kern-1.0pt+}\zeta(t)\liminf_{h\downarrow
0}\frac{\zeta(t+h)-\zeta(t)}{h}.$ (6.3)

######  Remark 6.2.

Arguing as in [MS20, Lemma A.1] and using the lower semicontinuity of the map
$t\mapsto\left[\Phi,\mu_{t}\right]_{r}$, the distributional inequality of
($\lambda$-EVI) can be equivalently reformulated in terms of the right upper
or lower Dini derivatives of the squared distance function and requiring the
condition to hold for every $t\in\operatorname{int}\left(\mathcal{I}\right)$:

$\displaystyle\frac{1}{2}{\frac{\mathrm{d}}{\mathrm{d}t}}^{\kern-3.0pt+}W_{2}^{2}(\mu_{t},\nu)$
$\displaystyle\leq\lambda
W_{2}^{2}(\mu_{t},\nu)-\left[\Phi,\mu_{t}\right]_{r}$ $\displaystyle\text{for
every }t\in\operatorname{int}\left(\mathcal{I}\right),\
\Phi\in{\boldsymbol{\mathrm{F}}},\ \nu=\mathsf{x}_{\sharp}\Phi,$
($\lambda$-EVI1)
$\displaystyle\frac{1}{2}{\frac{\mathrm{d}}{\mathrm{d}t}}_{\kern-1.0pt+}W_{2}^{2}(\mu_{t},\nu)$
$\displaystyle\leq\lambda
W_{2}^{2}(\mu_{t},\nu)-\left[\Phi,\mu_{t}\right]_{r}$ $\displaystyle\text{for
every }t\in\operatorname{int}\left(\mathcal{I}\right),\
\Phi\in{\boldsymbol{\mathrm{F}}},\ \nu=\mathsf{x}_{\sharp}\Phi.$
($\lambda$-EVI2)

A further equivalent formulation [MS20, Theorem 3.3] involves the difference
quotients: for every $s,t\in\mathcal{I}$, $s<t$

${\mathrm{e}^{-2\lambda(t-s)}}\,W_{2}^{2}(\mu_{t},\nu)-W_{2}^{2}(\mu_{s},\nu)\leq-2\int_{s}^{t}\mathrm{e}^{-2\lambda(r-s)}\left[\Phi,\mu_{r}\right]_{r}\,\mathrm{d}r\quad\text{for
every }\Phi\in{\boldsymbol{\mathrm{F}}},\ \nu=\mathsf{x}_{\sharp}\Phi.$
($\lambda$-EVI3)

Finally, if $\mu$ is also locally absolutely continuous, then ($\lambda$-EVI1)
and ($\lambda$-EVI2) are also equivalent to

$\begin{aligned}
\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}W_{2}^{2}(\mu_{t},\nu)&\leq\lambda
W_{2}^{2}(\mu_{t},\nu)-\left[\Phi,\mu_{t}\right]_{r}\end{aligned}\quad\text{for
a.e.\leavevmode\nobreak\ }t\in\mathcal{I}\text{ and every}\
\Phi\in{\boldsymbol{\mathrm{F}}},\ \nu=\mathsf{x}_{\sharp}\Phi.$

The following Lemma provides a further insight.

######  Lemma 6.3.

Let ${\boldsymbol{\mathrm{F}}}$ be a $\lambda$-dissipative MPVF and let
$\mu:\mathcal{I}\to\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$ be a
continuous $\lambda$-EVI solution to (6.1). We have

$\displaystyle\frac{1}{2}{\frac{\mathrm{d}}{\mathrm{d}t}}^{\kern-3.0pt+}W_{2}^{2}(\mu_{t},\nu)\leq[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{0+}\quad$
for every $\nu\in\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$,
$t\in\operatorname{int}\left(\mathcal{I}\right)$,
$\boldsymbol{\mu}\in\Gamma_{o}^{0}({\mu_{t}},{\nu}|{\boldsymbol{\mathrm{F}}})$,
(6.4a)
$\displaystyle\frac{1}{2}{\frac{\mathrm{d}}{\mathrm{d}t}}^{\kern-3.0pt+}W_{2}^{2}(\mu_{t},\nu)\leq\lambda
W_{2}^{2}(\mu_{t},\nu)+[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{1-}\quad$
for every $\nu\in\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$,
$t\in\operatorname{int}\left(\mathcal{I}\right),$
$\boldsymbol{\mu}\in\Gamma_{o}^{1}({\mu_{t}},{\nu}|{\boldsymbol{\mathrm{F}}})$.
(6.4b)

If moreover $\mu$ is locally absolutely continuous with Wasserstein velocity
field $\boldsymbol{v}$ satisfying (2.6) for every $t$ in the subset
$A(\mu)\subset\mathcal{I}$ of full Lebesgue measure, then

$\displaystyle\left[(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}_{t})_{\sharp}\mu_{t},\nu\right]_{r}$
$\displaystyle\leq\lambda
W_{2}^{2}(\mu_{t},\nu)-\left[\Phi,\mu_{t}\right]_{r}$ $\displaystyle\text{if
$t\in A(\mu)$,\ $\Phi\in{\boldsymbol{\mathrm{F}}},\
\nu=\mathsf{x}_{\sharp}\Phi$},$ (6.5a)
$\displaystyle[(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}_{t})_{\sharp}\mu_{t},\boldsymbol{\mu}]_{r,0}$
$\displaystyle\leq[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{0+}$
$\displaystyle\text{if $t\in A(\mu)$,
}\nu\in\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})},\
\boldsymbol{\mu}\in\Gamma_{o}^{0}({\mu_{t}},{\nu}|{\boldsymbol{\mathrm{F}}}),$
(6.5b)
$\displaystyle[(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}_{t})_{\sharp}\mu_{t},\boldsymbol{\mu}]_{r,0}$
$\displaystyle\leq\lambda
W_{2}^{2}(\mu_{t},\nu)+[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{1-}$
$\displaystyle\text{if $t\in A(\mu)$,
}\nu\in\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})},\
\boldsymbol{\mu}\in\Gamma_{o}^{1}({\mu_{t}},{\nu}|{\boldsymbol{\mathrm{F}}}).$
(6.5c)

###### Proof.

In order to check (6.5a) it is sufficient to combine (3.19) of Theorem 3.11
with ($\lambda$-EVI1). (6.5b) and (6.5c) then follow applying Proposition
4.14. Let us now prove (6.4a): let us fix
$\nu\in\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$ and
$t\in\operatorname{int}\left(\mathcal{I}\right)$. Take
$\boldsymbol{\mu}\in\Gamma_{o}(\mu_{t},\nu)$ and define the constant speed
geodesic $(\nu_{s})_{s\in[0,1]}$ by
$\nu_{s}:=(\mathsf{x}^{s})_{\sharp}\boldsymbol{\mu}$, thus in particular
$\nu_{0}=\mu_{t}$ and $\nu_{1}=\nu$. Then by Lemma 2.11, for every
$s\in\mathrm{I}(\boldsymbol{\mu}|{\boldsymbol{\mathrm{F}}})\cap(0,1)$ and
$\Phi_{s}\in{\boldsymbol{\mathrm{F}}}(\nu_{s})$ we have

$\begin{split}\frac{1}{2}{\frac{\mathrm{d}}{\mathrm{d}t}}^{\kern-3.0pt+}W_{2}^{2}(\mu_{t},\nu)&\leq\frac{1}{2s}{\frac{\mathrm{d}}{\mathrm{d}t}}^{\kern-3.0pt+}W_{2}^{2}(\mu_{t},\nu_{s})\leq-\frac{1}{s}\left[\Phi_{s},\mu_{t}\right]_{r}+\frac{\lambda}{s}W_{2}^{2}(\mu_{t},\nu_{s})\\\
&\leq[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}]_{r,s}+\lambda
sW_{2}^{2}(\mu_{t},\nu),\end{split}$

where the second inequality comes from ($\lambda$-EVI1). Taking
$\boldsymbol{\mu}\in\Gamma_{o}^{0}({\mu_{t}},{\nu}|{\boldsymbol{\mathrm{F}}})$
and passing to the limit as $s\downarrow 0$ we get (6.4a). Analogously for
(6.4b). ∎

We can now give an interpretation of absolutely continuous $\lambda$-EVI
solutions in terms of differential inclusions.

######  Theorem 6.4.

Let ${\boldsymbol{\mathrm{F}}}$ be a $\lambda$-dissipative MPVF and let
$\mu:\mathcal{I}\to\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$ be a
locally absolutely continuous curve.

  1. (1)

If $\mu$ satisfies the differential inclusion (6.2) driven by any
$\lambda$-dissipative extension of ${\boldsymbol{\mathrm{F}}}$ in
${\mathrm{D}({\boldsymbol{\mathrm{F}}})}$, then $\mu$ is also a $\lambda$-EVI
solution to (6.1) for ${\boldsymbol{\mathrm{F}}}$.

  2. (2)

$\mu$ is a $\lambda$-EVI solution of (6.1) for ${\boldsymbol{\mathrm{F}}}$ if
and only if

$(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}_{t})_{\sharp}\mu_{t}\in{\hat{\boldsymbol{\mathrm{F}}}}[\mu_{t}]\quad\text{for
a.e. }t\in\mathcal{I}.$ (6.6)

  3. (3)

If $\mathrm{D}({\boldsymbol{\mathrm{F}}})$ satisfies (4.13) and
$\mu_{t}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$ for a.e. $t\in\mathcal{I}$,
then the following properties are equivalent:

     * -

$\mu$ is a $\lambda$-EVI solution to (6.1) for ${\boldsymbol{\mathrm{F}}}$.

     * -

$\mu$ satisfies (6.5b).

     * -

$\mu$ is a $\lambda$-EVI solution to (6.1) for the restriction of
$\hat{\boldsymbol{\mathrm{F}}}$ to $\mathrm{D}({\boldsymbol{\mathrm{F}}})$.

  4. (4)

If ${\boldsymbol{\mathrm{F}}}$ satisfies (4.16) then $\mu$ is a $\lambda$-EVI
solution to (6.1) for ${\boldsymbol{\mathrm{F}}}$ if and only if it is a
$\lambda$-EVI solution to (6.1) for $\hat{\boldsymbol{\mathrm{F}}}$.

###### Proof.

(1) It is sufficient to apply Theorem 3.11 and the definition of
$\lambda$-dissipativity.

The left-to-right implication $\Rightarrow$ of (2) follows by (6.5a) of Lemma
6.3 and the definition of $\hat{\boldsymbol{\mathrm{F}}}$.

Conversely, if $\mu$ satisfies (6.6),
$\nu\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$,
$\Phi\in{\boldsymbol{\mathrm{F}}}[\nu]$, then Theorem 3.11 and the definition
of $\hat{\boldsymbol{\mathrm{F}}}$ yield

$\displaystyle\frac{1}{2}\frac{\,\mathrm{d}}{\,\mathrm{d}t}W_{2}^{2}(\mu_{t},\nu)$
$\displaystyle=\left[(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}_{t})_{\sharp}\mu_{t},\nu\right]_{r}\leq\lambda
W_{2}^{2}(\mu_{t},\nu)-\left[\Phi,\mu_{t}\right]_{r}\quad\text{a.e.\leavevmode\nobreak\
in }\mathcal{I}.$

Claim (3) is an immediate consequence of Lemma 6.3, Proposition 4.17(d) and
Proposition 4.14.

Claim (4) is a consequence of Proposition 4.17(f) and the
$\lambda$-dissipativity of $\hat{\boldsymbol{\mathrm{F}}}$. ∎

######  Proposition 6.5.

Let $\mathcal{F}:\mathcal{P}_{2}(\mathsf{X})\to(-\infty,+\infty]$ be a proper,
lower semicontinuous and $\lambda$-convex functional and let
$\mu\in\mathrm{C}(\mathcal{I};\mathrm{D}(\boldsymbol{\partial}\mathcal{F}))$
be a locally absolutely continuous curve. Then

  1. (1)

if $\mu$ is a Gradient Flow for $\mathcal{F}$ i.e.

$(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}_{t})_{\sharp}\mu_{t}\in-\boldsymbol{\partial}\mathcal{F}(\mu_{t})\quad\text{
a.e. }t\in\mathcal{I},$

then $\mu$ is a $(-\lambda)$-EVI solution of (6.1) for the MPVF
$-\boldsymbol{\partial}\mathcal{F}$ as in (5.1);

  2. (2)

if $\mu$ is a $(-\lambda)$-EVI solution of (6.1) for the MPVF
$-\boldsymbol{\partial}\mathcal{F}$ and the domain of
$\boldsymbol{\partial}\mathcal{F}$ satisfies

$\text{ for a.e.
}t\in\mathcal{I},\,\Gamma_{o}^{0}({\mu_{t}},{\nu}|\boldsymbol{\partial}\mathcal{F})\neq\emptyset\quad\forall\nu\in\mathrm{D}(\boldsymbol{\partial}\mathcal{F}),$

then $\mu$ is a Gradient Flow for $\mathcal{F}$.

###### Proof.

The first assertion is a consequence Theorem 6.4(1). We prove the second
claim; by (6.5b) we have that for a.e. $t\in\mathcal{I}$ it holds

$\left[(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}_{t})_{\sharp}\mu_{t},\nu\right]_{r}\leq[(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}_{t})_{\sharp}\mu_{t},\boldsymbol{\mu}]_{r,0}\leq[-\boldsymbol{\partial}\mathcal{F},\boldsymbol{\mu}]_{0+}\quad\forall\,\nu\in\mathrm{D}(\mathcal{F})\,\,\forall\,\boldsymbol{\mu}\in\Gamma_{o}^{0}({\mu_{t}},{\nu}|\boldsymbol{\partial}\mathcal{F}).$

We show that for every
$\nu_{0},\nu_{1}\in\mathrm{D}(\boldsymbol{\partial}\mathcal{F})$ and every
$\boldsymbol{\nu}\in\Gamma_{o}^{0}({\nu_{0}},{\nu_{1}}|{\boldsymbol{\mathrm{F}}})$

$[-\boldsymbol{\partial}\mathcal{F},\boldsymbol{\nu}]_{0+}\leq\mathcal{F}(\nu_{1})-\mathcal{F}(\nu_{0})-\frac{\lambda}{2}W_{2}^{2}(\nu_{0},\nu_{1}).$
(6.7)

To prove that, we take
$s\in\mathrm{I}(\boldsymbol{\nu}|\boldsymbol{\partial}\mathcal{F})\cap(0,1)$
and $\Phi_{s}\in-\boldsymbol{\partial}\mathcal{F}(\nu_{s})$. By definition of
subdifferential we have

$\left[\Phi_{s},\nu_{1}\right]_{r}\leq\mathcal{F}(\nu_{1})-\mathcal{F}(\nu_{s})-\frac{\lambda}{2}W_{2}^{2}(\nu_{s},\nu_{1})$

where $\nu_{s}=\mathsf{x}^{s}_{\sharp}\boldsymbol{\nu}$. Dividing by $(1-s)$,
using (3.26) and passing to the infimum w.r.t.
$\Phi_{s}\in-\boldsymbol{\partial}\mathcal{F}(\nu_{s})$ we obtain

$[-\boldsymbol{\partial}\mathcal{F},\boldsymbol{\nu}]_{r,s}\leq\frac{1}{1-s}\left(\mathcal{F}(\nu_{1})-\mathcal{F}(\nu_{s})\right)-\frac{\lambda(1-s)}{2}W_{2}^{2}(\nu_{0},\nu_{1}).$

Passing to the limit as $s\downarrow 0$ and using the lower semicontinuity of
$\mathcal{F}$ lead to the result. Once that (6.7) is established we have that
for a.e. $t\in\mathcal{I}$ it holds

$\left[(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}_{t})_{\sharp}\mu_{t},\nu\right]_{r}\leq\mathcal{F}(\nu)-\mathcal{F}(\mu_{t})-\frac{\lambda}{2}W_{2}^{2}(\mu_{t},\nu)\text{
for every }\nu\in\mathrm{D}(\boldsymbol{\partial}\mathcal{F}).$ (6.8)

To conclude it is enough to use the lower semicontinuity of the LHS (see Lemma
3.14) and the fact that $\mathrm{D}(\boldsymbol{\partial}\mathcal{F})$ is
dense in $\mathrm{D}(\mathcal{F})$ in energy: indeed we can apply [NS21,
Corollary 4.5] and [AGS08, Lemma 3.1.2] to the proper, lower semicontinuous
and convex functional
$\mathcal{F}^{\lambda}:\mathcal{P}_{2}(\mathsf{X})\to(-\infty,+\infty]$
defined as

$\mathcal{F}^{\lambda}(\nu)=\mathcal{F}(\nu)-\frac{\lambda}{2}\mathsf{m}_{2}^{2}(\nu)$

to get the existence, for every $\nu\in\mathrm{D}(\mathcal{F})$, of a family
$(\nu_{\tau})_{\tau>0}\subset\mathrm{D}(\mathcal{F}^{\lambda})=\mathrm{D}(\mathcal{F})$
s.t.

$\nu_{\tau}\to\nu,\quad\mathcal{F}^{\lambda}(\nu_{\tau})\to\mathcal{F}^{\lambda}(\nu)\quad\text{
as }\tau\downarrow 0.$

Of course $\mathcal{F}(\nu_{\tau})\to\mathcal{F}(\nu)$ as $\tau\downarrow 0$
and, applying [AGS08, Lemma 10.3.4], we see that
$\nu_{\tau}\in\mathrm{D}(\boldsymbol{\partial}\mathcal{F}^{\lambda})$. However
$\boldsymbol{\partial}\mathcal{F}^{\lambda}=L^{\lambda}_{\sharp}\boldsymbol{\partial}\mathcal{F}$
(see (4.5)) so that
$\nu_{\tau}\in\mathrm{D}(\boldsymbol{\partial}\mathcal{F})$. We can thus write
(6.8) for $\nu_{\tau}$ in place of $\nu$ and pass to the limit as
$\tau\downarrow 0$, obtaining that, by definition of subdifferential,
$(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}_{t})_{\sharp}\mu_{t}\in-\boldsymbol{\partial}\mathcal{F}(\mu_{t})$
for a.e. $t\in\mathcal{I}$. ∎

We derive a further useful a priori bound for $\lambda$-EVI solutions.

######  Proposition 6.6.

Let ${\boldsymbol{\mathrm{F}}}$ be a $\lambda$-dissipative MPVF and let
$T\in(0,+\infty]$. Every $\lambda$-EVI solution
$\mu:[0,T)\to\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$ with initial
datum $\mu_{0}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$ satisfies the a priori
bound

$W_{2}(\mu_{t},\mu_{0})\leq
2|{\boldsymbol{\mathrm{F}}}|_{2}(\mu_{0})\int_{0}^{t}\mathrm{e}^{\lambda
s}\,\mathrm{d}s,\quad\text{for all }t\in[0,T),$ (6.9)

where

$|{\boldsymbol{\mathrm{F}}}|_{2}(\mu):={}\inf\Big{\\{}|\Phi|_{2}:\Phi\in{\boldsymbol{\mathrm{F}}}[\mu]\Big{\\}}\quad\text{for
every }\mu\in\mathrm{D}({\boldsymbol{\mathrm{F}}}).$

###### Proof.

Let $\Phi\in{\boldsymbol{\mathrm{F}}}(\mu_{0})$. ($\lambda$-EVI) with
$\nu:=\mu_{0}$ then yields

${\frac{\mathrm{d}}{\mathrm{d}t}}^{\kern-3.0pt+}W_{2}^{2}(\mu_{t},\mu_{0})-2\lambda
W_{2}^{2}(\mu_{t},\mu_{0})\leq-2\left[\Phi,\mu_{t}\right]_{r}\leq
2|\Phi|_{2}\,W_{2}(\mu_{t},\mu_{0}),\quad\text{for every }t\in[0,T).$

We can then apply the estimate of [AGS08, Lemma 4.1.8] to obtain

$\mathrm{e}^{-\lambda t}W_{2}(\mu_{t},\mu_{0})\leq
2|\Phi|_{2}\int_{0}^{t}\mathrm{e}^{-\lambda s}\,\mathrm{d}s,\quad\text{for all
}t\in[0,T),$

which in turn yields (6.9). ∎

We conclude this section with a result showing the robustness of the notion of
$\lambda$-EVI solution.

######  Proposition 6.7.

If $\mu_{n}:\mathcal{I}\to\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$ is
a sequence of $\lambda$-EVI solutions locally uniformly converging to $\mu$ as
$n\to\infty$, then $\mu$ is a $\lambda$-EVI solution.

###### Proof.

$\mu$ is a continuous curve defined in $\mathcal{I}$ with values in
$\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$. Using pointwise
convergence, the lower semicontinuity of $\mu\mapsto\left[\Phi,\mu\right]_{r}$
of Lemma 3.14, and Fatou’s Lemma, it is easy to pass to the limit in the
equivalent characterization ($\lambda$-EVI3) of $\lambda$-EVI solutions,
written for $\mu_{n}$. ∎

###  6.2. Local existence of $\lambda$-EVI solutions by the Explicit Euler
Scheme

In order to prove the existence of a $\lambda$-EVI solution to (6.1), our
strategy is to employ an approximation argument through an Explicit Euler
scheme as it occurs for ODEs.
In the following $\left\lfloor\cdot\right\rfloor$ and
$\left\lceil\cdot\right\rceil$ denote the floor and the ceiling functions
respectively.

######  Definition 6.8 (Explicit Euler Scheme).

Let ${\boldsymbol{\mathrm{F}}}$ be a MPVF and suppose we are given a step size
$\tau>0$, an initial datum
$\mu^{0}_{\tau}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$, a bounded interval
$[0,T]$, corresponding to the final step ${\mathrm{N}(T,\tau)}:=\left\lceil
T/\tau\right\rceil,$ and a stability bound $L>0$. A sequence
$(M^{n}_{\tau},{\boldsymbol{F}}_{\tau}^{n})_{0\leq
n\leq{\mathrm{N}(T,\tau)}}\subset\mathrm{D}({\boldsymbol{\mathrm{F}}})\times{\boldsymbol{\mathrm{F}}}$
is a _$L$ -stable solution to the Explicit Euler Scheme in $[0,T]$ starting
from $\mu^{0}_{\tau}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$_ if

$\left\\{\begin{aligned} M_{\tau}^{0}&=\mu^{0}_{\tau},\\\
{\boldsymbol{F}}_{\tau}^{n}&\in{\boldsymbol{\mathrm{F}}}[M_{\tau}^{n}],\
|{\boldsymbol{F}}_{\tau}^{n}|_{2}\leq L&&0\leq n<{\mathrm{N}(T,\tau)},\\\
M_{\tau}^{n}&=(\operatorname{\mathsf{exp}}^{\tau})_{\sharp}{\boldsymbol{F}}_{\tau}^{n-1}&&1\leq
n\leq{\mathrm{N}(T,\tau)}.\end{aligned}\right.$ (EE)

We define the following two different interpolations of the sequence
$(M^{n}_{\tau},{\boldsymbol{F}}_{\tau}^{n})$:

  * •

the affine interpolation:

$M_{\tau}(t):=(\operatorname{\mathsf{exp}}^{t-n\tau})_{\sharp}{\boldsymbol{F}}_{\tau}^{n}\text{\quad
if }t\in[n\tau,(n+1)\tau]\text{ for some }n\in\mathbb{N},\ 0\leq
n<{\mathrm{N}(T,\tau)},$ (6.10)

  * •

the piecewise constant interpolation:

$\begin{split}\bar{M}_{\tau}(t)&:=M^{\left\lfloor
t/\tau\right\rfloor}_{\tau},\quad t\in[0,T],\\\
{\boldsymbol{F}}_{\tau}(t)&:={\boldsymbol{\mathrm{F}}}^{\left\lfloor
t/\tau\right\rfloor}_{\tau},\quad t\in[0,T].\end{split}$

We will call $\mathscr{E}(\mu^{0}_{\tau},\tau,T,L)$ (resp.
$\mathscr{M}(\mu^{0}_{\tau},\tau,T,L)$) the (possibly empty) set of all the
curves $(M_{\tau},{\boldsymbol{F}}_{\tau})$ (resp. $M_{\tau}$) arising from
the solution of (EE).

The affine interpolation can be trivially written as

$M_{\tau}(t)=\left(\operatorname{\mathsf{exp}}^{t-\left\lfloor
t/\tau\right\rfloor\tau}\right)_{\sharp}\left({\boldsymbol{F}}_{\tau}(t)\right),\quad
t\in[0,T],$

and $M_{\tau}$ satisfies the uniform Lipschitz bound

$W_{2}(M_{\tau}(t),M_{\tau}(s))\leq L|t-s|\quad 0\leq s\leq t\leq T,\quad
M_{\tau}\in\mathscr{E}(\mu_{0},\tau,T,L).$ (6.11)

Notice that, since in general ${\boldsymbol{\mathrm{F}}}[\mu]$ is not reduced
to a singleton, the sets $\mathscr{E}(\mu_{0},\tau,T,L)$ and
$\mathscr{M}(\mu_{0},\tau,T,L)$ may contain more than one element (or may be
empty). Stable solutions to the Explicit Euler scheme generated by a
$\lambda$-dissipative MPVF exhibit a nice behaviour, which is clarified by the
following important result, which will be proved in Section 7 (see Proposition
7.3 and Theorems 7.4, 7.5, 7.7), with a more accurate estimate of the error
constants $A(\vartheta)$. We stress that in the next statement $A(\vartheta)$
solely depend on $\vartheta$ (in particular, it is independent of
$\lambda,L,T,\tau,\eta,M_{\tau},M_{\eta}$).

######  Theorem 6.9.

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

  1. (1)

For every $\mu_{0},\mu_{0}^{\prime}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$,
every $M_{\tau}\in\mathscr{M}(\mu_{0},\tau,T,L)$,
$M_{\tau}^{\prime}\in\mathscr{M}(\mu_{0}^{\prime},\tau,T,L)$ with
$\tau\lambda_{+}\leq 2$ we have

$W_{2}(M_{\tau}(t),M_{\tau}^{\prime}(t))\leq\mathrm{e}^{\lambda
t}W_{2}(\mu_{0},\mu_{0}^{\prime})+8L\sqrt{t\tau}\Big{(}1+|\lambda|\sqrt{t\tau}\Big{)}\mathrm{e}^{\lambda_{+}t}\quad\text{for
every }t\in[0,T].$ (6.12)

  2. (2)

For every $\vartheta>1$ there exists a constant $A(\vartheta)$ such that if
$M_{\tau}\in\mathscr{M}(M^{0}_{\tau},\tau,T,L)$ and
$M_{\eta}\in\mathscr{M}(M^{0}_{\eta},\eta,T,L)$ with
$\lambda_{+}(\tau+\eta)\leq 1$ then

$W_{2}(M_{\tau}(t),M_{\eta}(t))\leq\Big{(}\vartheta
W_{2}(M^{0}_{\tau},M^{0}_{\eta})+A(\vartheta)L\sqrt{(\tau+\eta)(t+\tau+\eta)}\Big{)}\mathrm{e}^{\lambda_{+}\,t},\quad
t\in[0,T].$

  3. (3)

For every $\vartheta>1$ there exists a constant $A(\vartheta)$ such that if
$\mu\in\mathrm{C}([0,T];\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})})$ is
a $\lambda$-EVI solution and $M_{\tau}\in\mathscr{M}(M^{0}_{\tau},\tau,T,L)$
then

$W_{2}(\mu(t),M_{\tau}(t))\leq\Big{(}\vartheta\,W_{2}(\mu_{0},M^{0}_{\tau})+A(\vartheta)L\sqrt{\tau(t+\tau)}\Big{)}\mathrm{e}^{\lambda_{+}t}\quad\text{for
every }t\in[0,T].$ (6.13)

  4. (4)

If $n\mapsto\tau(n)$ is a vanishing sequence of time steps,
$(\mu_{0,n})_{n\in\mathbb{N}}$ is a sequence in
$\mathrm{D}({\boldsymbol{\mathrm{F}}})$ converging to
$\mu_{0}\in\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$ in
$\mathcal{P}_{2}(\mathsf{X})$ and
$M_{n}\in\mathscr{M}(\mu_{0,n},\tau(n),T,L)$, then $M_{n}$ is uniformly
converging to a limit curve
$\mu\in\mathrm{Lip}([0,T];\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})})$
which is a $\lambda$-EVI solution starting from $\mu_{0}$.

If we assume that the Explicit Euler scheme is locally solvable, Theorem 6.9
provides a crucial tool to obtain local existence and uniqueness of
$\lambda$-EVI solutions.

######  Definition 6.10 (Local and global solvability of (EE)).

We say that the Explicit Euler Scheme (EE) associated to a MPVF
${\boldsymbol{\mathrm{F}}}$ is _locally solvable_ at
$\mu_{0}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$ if there exist strictly
positive constants $\boldsymbol{\tau},T,L$ such that
$\mathscr{E}(\mu_{0},\tau,T,L)$ is not empty for every
$\tau\in(0,\boldsymbol{\tau})$.
We say that (EE) is _globally solvable_ at
$\mu_{0}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$ if for every $T>0$ there
exist strictly positive constants $\boldsymbol{\tau},L$ such that
$\mathscr{E}(\mu_{0},\tau,T,L)$ is not empty for every
$\tau\in(0,\boldsymbol{\tau})$.

Let us now state the main existence result for $\lambda$-EVI solutions. Given
$T\in(0,+\infty]$ and $\mu:[0,T)\to\mathcal{P}_{2}(\mathsf{X})$ we denote by
$|\dot{\mu}|_{+}(t)$ the right upper metric derivative

$|\dot{\mu}|_{+}(t):=\limsup_{h\downarrow
0}\frac{W_{2}(\mu_{t+h},\mu_{t})}{h}.$

######  Theorem 6.11 (Local existence and uniqueness).

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

  1. (a)

If the Explicit Euler Scheme is locally solvable at
$\mu_{0}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$, then there exists $T>0$ and
a unique $\lambda$-EVI solution
$\mu\in\mathrm{Lip}([0,T];\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})})$
starting from $\mu_{0}$, satisfying

$t\mapsto\mathrm{e}^{-\lambda t}|\dot{\mu}|_{+}(t)\quad\text{is decreasing in
}[0,T).$ (6.14)

If
$\mu^{\prime}:[0,T^{\prime}]\to\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$
is any other $\lambda$-EVI solution starting from $\mu_{0}$ then
$\mu(t)=\mu^{\prime}(t)$ if $0\leq t\leq T\land T^{\prime}$.

  2. (b)

If the Explicit Euler Scheme is locally solvable in
$\mathrm{D}({\boldsymbol{\mathrm{F}}})$ and

$\begin{gathered}\text{for any local $\lambda$-{\rm EVI} solution $\mu$
starting from $\mu_{0}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$}\\\
\text{there exists }\delta>0:\quad
t\in[0,\delta]\quad\Rightarrow\quad\mu(t)\in\mathrm{D}({\boldsymbol{\mathrm{F}}}),\end{gathered}$
(6.15)

then for every $\mu_{0}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$ there exist a
unique maximal time $T\in(0,\infty]$ and a unique strict $\lambda$-EVI
solution $\mu\in\mathrm{Lip}_{\rm
loc}([0,T);\mathrm{D}({\boldsymbol{\mathrm{F}}}))$ starting from $\mu_{0}$,
which satisfies (6.14) and

$T<\infty\quad\Rightarrow\quad\lim_{t\uparrow
T}\mu_{t}\not\in\mathrm{D}({\boldsymbol{\mathrm{F}}}).$ (6.16)

Any other $\lambda$-EVI solution
$\mu^{\prime}:[0,T^{\prime})\to\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$
starting from $\mu_{0}$ coincides with $\mu$ in $[0,T\land T^{\prime})$.

###### Proof.

(a) Let $\boldsymbol{\tau},T,L$ positive constants such that
$\mathscr{E}(\mu_{0},\tau,T,L)$ is not empty for every
$\tau\in(0,\boldsymbol{\tau})$. Thanks to Theorem 6.9(2), the family
$M_{\tau}\in\mathscr{E}(\mu_{0},\tau,T,L)$ satisfies the Cauchy condition in
$\mathrm{C}([0,T];\mathcal{P}_{2}(\mathsf{X}))$ so that there exists a unique
limit curve $\mu=\lim_{\tau\downarrow 0}M_{\tau}$ which is also Lipschitz in
time, thanks to the a-priori bound (6.11). Theorem 6.9(4) shows that $\mu$ is
a $\lambda$-EVI solution starting from $\mu_{0}$ and the estimate (6.13) of
Theorem 6.9(3) shows that any other $\lambda$-EVI solution in an interval
$[0,T^{\prime}]$ starting from $\mu_{0}$ should coincide with $\mu$ in
$[0,T^{\prime}\land T]$.

Let us now check (6.14): we fix $s,t$ such that $0\leq s<t<T$ and
$h\in(0,T-t)$, and we set $s_{\tau}:=\tau\left\lfloor s/\tau\right\rfloor$,
$h_{\tau}:=\tau\left\lfloor h/\tau\right\rfloor$. The curves $r\mapsto
M_{\tau}(s_{\tau}+r)$, $r\mapsto M_{\tau}(s_{\tau}+h_{\tau}+r)$ belong to
$\mathscr{M}(M_{\tau}(s_{\tau}),\tau,t-s,L)$ and
$\mathscr{M}(M_{\tau}(s_{\tau}+h_{\tau}),\tau,t-s,L)$, so that (6.12) yields

$W_{2}(M_{\tau}(s_{\tau}+t-s),M_{\tau}(s_{\tau}+h_{\tau}+(t-s)))\leq\mathrm{e}^{\lambda(t-s)}W_{2}(M_{\tau}(s_{\tau}),M_{\tau}(s_{\tau}+h_{\tau}))+B\sqrt{\tau},$

for $B=B(\lambda,L,\boldsymbol{\tau},T)$. Passing to the limit as
$\tau\downarrow 0$ we get

$W_{2}(\mu(t),\mu(t+h))\leq\mathrm{e}^{\lambda(t-s)}W_{2}(\mu(s),\mu(s+h)).$

Dividing by $h$ and passing to the limit as $h\downarrow 0$ we get (6.14).

(b) Let us call $\mathcal{S}$ the collection of $\lambda$-EVI solutions
$\mu:[0,S)\to\mathrm{D}({\boldsymbol{\mathrm{F}}})$ starting from $\mu_{0}$
with values in $\mathrm{D}({\boldsymbol{\mathrm{F}}})$ and defined in some
interval $[0,S)$, $S=S(\mu)$. Thanks to (6.15) and the previous claim the set
$\mathcal{S}$ is not empty.

It is also easy to check that two curves
$\mu^{\prime},\mu^{\prime\prime}\in\mathcal{S}$ coincide in the common domain
$[0,S)$ with $S:=S(\mu^{\prime})\land S(\mu^{\prime\prime})$: in fact the set
$\\{t\in[0,S):\mu^{\prime}(r)=\mu^{\prime\prime}(r)\text{ if }0\leq r\leq
t\\}$ contains $t=0$, is closed since $\mu^{\prime},\mu^{\prime\prime}$ are
continuous, and it is also open since if $\mu^{\prime}=\mu^{\prime\prime}$ in
$[0,t]$ then the previous claim and the fact that
$\mu^{\prime}(t)=\mu^{\prime\prime}(t)\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$
show that $\mu^{\prime}=\mu^{\prime\prime}$ also in a right neighborhood of
$t$. Since $[0,S)$ is connected, we conclude that
$\mu^{\prime}=\mu^{\prime\prime}$ in $[0,S)$.

We can thus define $T:=\sup\\{S(\mu):\mu\in\mathcal{S}\\}$ obtaining that
there exists a unique $\lambda$-EVI solution $\mu$ starting from $\mu_{0}$ and
defined in $[0,T)$ with values in $\mathrm{D}({\boldsymbol{\mathrm{F}}})$.

If $T<\infty$, since $\mu$ is Lipschitz in $[0,T)$ thanks to (6.14), we know
that there exists the limit $\bar{\mu}:=\lim_{t\uparrow T}\mu(t)$ in
$\mathcal{P}_{2}(\mathsf{X})$. If
$\bar{\mu}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$ we can extend $\mu$ to a
$\lambda$-EVI solution with values in $\mathrm{D}({\boldsymbol{\mathrm{F}}})$
and defined in an interval $[0,T^{\prime})$ with $T^{\prime}>T$, which
contradicts the maximality of $T$. ∎

Recall that a set $A$ in a metric space $X$ is locally closed if every point
of $A$ has a neighborhood $U$ such that $A\cap U=\bar{A}\cap U$. Equivalently,
$A$ is the intersection of an open and a closed subset of $X$. In particular,
open or closed sets are locally closed.

######  Corollary 6.12.

Let ${\boldsymbol{\mathrm{F}}}$ be a $\lambda$-dissipative MPVF for which the
Explicit Euler Scheme is locally solvable in
$\mathrm{D}({\boldsymbol{\mathrm{F}}})$. If
$\mathrm{D}({\boldsymbol{\mathrm{F}}})$ is locally closed then for every
$\mu_{0}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$ there exists a unique
maximal strict $\lambda$-EVI solution $\mu\in\mathrm{Lip}_{\rm
loc}([0,T);\mathrm{D}({\boldsymbol{\mathrm{F}}}))$, $T\in(0,+\infty]$,
satisfying (6.16).

Let us briefly discuss the question of local solvability of the Explicit Euler
scheme. The main constraints of the Explicit Euler construction relies on the
a priori stability bound and in the condition
$M_{\tau}^{n}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$ for every step $0\leq
n\leq{\mathrm{N}(T,\tau)}$. This constraint is feasible if at each measure
$M^{n}_{\tau}$, $0\leq n<{\mathrm{N}(T,\tau)}$, the set
$\operatorname{Adm}_{\tau,L}(M^{n}_{\tau})$ defined by

$\operatorname{Adm}_{\tau,L}(\mu):=\Big{\\{}\Phi\in{\boldsymbol{\mathrm{F}}}[\mu]:|\Phi|_{2}\leq
L\quad\text{and}\quad\operatorname{\mathsf{exp}}_{\sharp}^{\tau}\Phi\in\mathrm{D}({\boldsymbol{\mathrm{F}}})\Big{\\}}$

is not empty. If $\mathrm{D}({\boldsymbol{\mathrm{F}}})$ is open and
${\boldsymbol{\mathrm{F}}}$ is locally bounded, then it is easy to check that
the Explicit Euler scheme is locally solvable (see Lemma 6.13). We will adopt
the following notation:

$\displaystyle|{\boldsymbol{\mathrm{F}}}|_{2}(\mu):={}$
$\displaystyle\inf\Big{\\{}|\Phi|_{2}:\Phi\in{\boldsymbol{\mathrm{F}}}[\mu]\Big{\\}}\quad\text{for
every }\mu\in\mathrm{D}({\boldsymbol{\mathrm{F}}}),$ (6.17)

and we will also introduce the upper semicontinuous envelope
$|{\boldsymbol{\mathrm{F}}}|_{2\star}$ of the function
$|{\boldsymbol{\mathrm{F}}}|_{2}$: i.e.

$\displaystyle|{\boldsymbol{\mathrm{F}}}|_{2\star}(\mu):={}$
$\displaystyle\inf_{\delta>0}\sup\Big{\\{}|{\boldsymbol{\mathrm{F}}}|_{2}(\nu):\nu\in\mathrm{D}({\boldsymbol{\mathrm{F}}}),\
W_{2}(\nu,\mu)\leq\delta\Big{\\}}$ $\displaystyle={}$
$\displaystyle\sup\Big{\\{}\limsup_{k\to\infty}|{\boldsymbol{\mathrm{F}}}|_{2}(\mu_{k}):\mu_{k}\in\mathrm{D}({\boldsymbol{\mathrm{F}}}),\
\mu_{k}\to\mu\text{ in }\mathcal{P}_{2}(\mathsf{X})\Big{\\}}.$

######  Lemma 6.13.

If ${\boldsymbol{\mathrm{F}}}$ is a $\lambda$-dissipative MPVF,
$\mu_{0}\in\mathrm{Int}(\mathrm{D}({\boldsymbol{\mathrm{F}}}))$ and
${\boldsymbol{\mathrm{F}}}$ is bounded in a neighborhood of $\mu_{0}$, i.e.
there exists $\varrho>0$ such that $|{\boldsymbol{\mathrm{F}}}|_{2}$ is
bounded in $\mathrm{B}(\mu_{0},\varrho)$, then the Explicit Euler scheme is
locally solvable at $\mu_{0}$ and the locally Lipschitz solution $\mu$ given
by Theorem 6.11(a) satisfies

$|\dot{\mu}|_{+}(t)\leq e^{\lambda
t}|{\boldsymbol{\mathrm{F}}}|_{2\star}(\mu_{0})\quad\forall\,t\in[0,T).$
(6.18)

In particular, if $\mathrm{D}({\boldsymbol{\mathrm{F}}})$ is open and
${\boldsymbol{\mathrm{F}}}$ is locally bounded, for every
$\mu_{0}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$ there exists a unique
maximal $\lambda$-EVI solution $\mu\in\mathrm{Lip}_{\rm
loc}([0,T);\mathcal{P}_{2}(\mathsf{X}))$ satisfying (6.16) and (6.18).

###### Proof.

Let $\mu_{0}\in\mathrm{Int}(\mathrm{D}({\boldsymbol{\mathrm{F}}}))$ and let
$\varrho,L>0$ so that $|{\boldsymbol{\mathrm{F}}}|_{2}(\mu)<L$ for every
$\mu\in\mathrm{B}(\mu_{0},\varrho)$. We set $T:=\varrho/(2L)$,
$\boldsymbol{\tau}=T\land 1$ and we perform a simple induction argument to
prove that $W_{2}(M^{n}_{\tau},\mu_{0})\leq Ln\tau<\varrho$ if
$n\leq{\mathrm{N}(T,\tau)}$ so that we can always find an element
${\boldsymbol{F}}^{n}_{\tau}\in{\boldsymbol{\mathrm{F}}}_{\tau,L}$. In fact,
if $W_{2}(M^{n}_{\tau},\mu_{0})<Ln\tau$ and $n<{\mathrm{N}(T,\tau)}$ then
$W_{2}(M^{n+1}_{\tau},\mu_{0})\leq
W_{2}(M^{n+1}_{\tau},M^{n}_{\tau})+W_{2}(M^{n}_{\tau},\mu_{0})\leq
L(n+1)\tau$. (6.14) shows that $|\dot{\mu}_{t}|_{+}\leq L\mathrm{e}^{\lambda
t}$ for every $L>|{\boldsymbol{\mathrm{F}}}|_{2\star}(\mu_{0})$, so that we
obtain (6.18). ∎

More refined estimates will be discussed in the next sections. Here we will
show another example, tailored to the case of measures with bounded support.

######  Proposition 6.14.

Let ${\boldsymbol{\mathrm{F}}}$ be a $\lambda$-dissipative MPVF such that
$\mathrm{D}({\boldsymbol{\mathrm{F}}})\subset\mathcal{P}_{\rm b}(\mathsf{X})$
and for every $\mu_{0}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$ there exist
$\varrho>0$, $L>0$ such that for every $\mu\in\mathcal{P}_{\rm b}(\mathsf{X})$

$\operatorname{supp}(\mu)\subset\operatorname{supp}(\mu_{0})+\mathrm{B}_{\mathsf{X}}(\varrho)\quad\Rightarrow\quad\exists\Phi\in{\boldsymbol{\mathrm{F}}}[\mu]:\operatorname{supp}(\mathsf{v}_{\sharp}\Phi)\subset\mathrm{B}_{\mathsf{X}}(L).$

Then for every $\mu_{0}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$ there exists
$T\in(0,+\infty]$ and a unique maximal strict $\lambda$-EVI solution
$\mu\in\mathrm{Lip}_{\rm{loc}}([0,T);\mathrm{D}({\boldsymbol{\mathrm{F}}}))$
satisfying (6.16).

###### Proof.

Arguing as in the proof of Lemma 6.13, it is easy to check that setting
$T:=\varrho/4L$, $\boldsymbol{\tau}=T\wedge 1$ we can find a discrete solution
$(M_{\tau},{\boldsymbol{F}}_{\tau})\in\mathscr{E}(\mu_{0},\tau,T,L)$
satisfying the more restrictive condition
$\operatorname{supp}(M^{n}_{\tau})\subset\operatorname{supp}(\mu_{0})+\mathrm{B}_{\mathsf{X}}(Ln\tau)\subset\operatorname{supp}(\mu_{0})+\mathrm{B}_{\mathsf{X}}(\varrho/2)$,
$\operatorname{supp}(\mathsf{v}_{\sharp}{\boldsymbol{F}}^{n}_{\tau})\subset\mathrm{B}_{\mathsf{X}}(L)$
so that the Explicit Euler scheme is locally solvable and $M_{\tau}$ satisfies
the uniform bound

$\operatorname{supp}(M_{\tau}(t))\subset\operatorname{supp}(\mu_{0})+\mathrm{B}_{\mathsf{X}}(\varrho/2)\quad\text{for
every }t\in[0,T].$ (6.19)

Theorem 6.11 then yields the existence of a local solution, and Theorem 6.9(3)
shows that the local solution satisfies the same bound (6.19) on the support,
so that (6.15) holds. ∎

###  6.3. Stability and uniqueness

In the following theorem we prove a stability result for $\lambda$-EVI
solutions of (6.1), as it occurs in the classical Hilbertian case scenario. We
distinguish three cases: the first one assumes that the Explicit Euler scheme
is locally solvable in $\mathrm{D}({\boldsymbol{\mathrm{F}}})$.

######  Theorem 6.15 (Uniqueness and Stability).

Let ${\boldsymbol{\mathrm{F}}}$ be a $\lambda$-dissipative MPVF such that the
Explicit Euler scheme is locally solvable in
$\mathrm{D}({\boldsymbol{\mathrm{F}}})$, and let
$\mu^{1},\mu^{2}:[0,T)\to\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$,
$T\in(0,+\infty]$, be $\lambda$-EVI solutions to (6.1). If $\mu^{1}$ is
strict, then

$W_{2}(\mu^{1}_{t},\mu^{2}_{t})\leq
W_{2}(\mu^{1}_{0},\mu^{2}_{0})\mathrm{e}^{\lambda_{+}\,t}\quad\text{for every
}t\in[0,T).$ (6.20)

In particular, if $\mu^{1}_{0}=\mu^{2}_{0}$ then $\mu^{1}\equiv\mu^{2}$ in
$[0,T)$.
If $\mu^{1},\mu^{2}$ are both strict, then

$W_{2}(\mu_{t}^{1},\mu_{t}^{2})\leq
W_{2}(\mu^{1}_{0},\mu^{2}_{0})\mathrm{e}^{\lambda t}\quad\text{for all
}t\in[0,T).$ (6.21)

###### Proof.

In order to prove (6.20), let us fix $t\in(0,T)$. Since the Explicit Euler
scheme is locally solvable and
$\mu^{1}_{t}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$, there exist
$\boldsymbol{\tau},\delta,L$ such that
$\mathscr{M}(\mu^{1}_{t},\tau,\delta,L)$ is not empty for every
$\tau\in(0,\boldsymbol{\tau})$. If
$M^{1}_{\tau}\in\mathscr{M}(\mu^{1}_{t},\tau,\delta,L)$, then (6.13) yields

$\displaystyle W_{2}(\mu^{1}_{t+h},\mu^{2}_{t+h})$ $\displaystyle\leq
W_{2}(M^{1}_{\tau}(h),\mu^{2}_{t+h})+W_{2}(M^{1}_{\tau}(h),\mu^{1}_{t+h})$
$\displaystyle\leq\vartheta
W_{2}(\mu^{1}_{t},\mu^{2}_{t})\mathrm{e}^{\lambda_{+}h}+B\sqrt{\tau}\quad\text{if
}0\leq h\leq\delta,$

for $B=B(\lambda,L,\boldsymbol{\tau},\delta)$ Passing to the limit as
$\tau\downarrow 0$ we obtain

$W_{2}(\mu^{1}_{t+h},\mu^{2}_{t+h})\leq\vartheta
W_{2}(\mu^{1}_{t},\mu^{2}_{t})\mathrm{e}^{\lambda_{+}h}$

and a further limit as $\vartheta\downarrow 1$ yields

$W_{2}(\mu^{1}_{t+h},\mu^{2}_{t+h})\leq
W_{2}(\mu^{1}_{t},\mu^{2}_{t})\mathrm{e}^{\lambda_{+}h}\quad\text{for every
}h\in[0,\delta],$

which implies that the map
$t\mapsto\mathrm{e}^{-\lambda_{+}t}W_{2}(\mu^{1}_{t},\mu^{2}_{t})$ is
decreasing in $[t,t+\delta]$. Since $t$ is arbitrary, we obtain (6.20).

In order to prove the estimate (6.21) (which is better than (6.20) when
$\lambda<0$), we argue in a similar way, using (6.12).

As before, for a given $t\in(0,T)$, since the Explicit Euler scheme is locally
solvable and
$\mu^{1}_{t},\mu^{2}_{t}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$, there exist
$\boldsymbol{\tau},\delta,L$ such that
$\mathscr{M}(\mu^{1}_{t},\tau,\delta,L)$ and
$\mathscr{M}(\mu^{2}_{t},\tau,\delta,L)$ are not empty for every
$\tau\in(0,\boldsymbol{\tau})$. If
$M^{i}_{\tau}\in\mathscr{M}(\mu^{i}_{t},\tau,\delta,L)$, for $i=1,2$, (6.12)
and (6.13) then yield

$\displaystyle W_{2}(\mu^{1}_{t+h},\mu^{2}_{t+h})$ $\displaystyle\leq
W_{2}(\mu^{1}_{t+h},M^{1}_{\tau}(h))+W_{2}(M^{1}_{\tau}(h),M^{2}_{\tau}(h))+W_{2}(\mu^{2}_{t+h},M^{2}_{\tau}(h))$
$\displaystyle\leq\mathrm{e}^{\lambda
h}W_{2}(\mu^{1}_{t},\mu^{2}_{t})+B\sqrt{\tau}\quad\text{if }0\leq
h\leq\delta,$

for $B=B(\lambda,L,\boldsymbol{\tau},\delta)$. Passing to the limit as
$\tau\downarrow 0$ we obtain

$W_{2}(\mu^{1}_{t+h},\mu^{2}_{t+h})\leq\mathrm{e}^{\lambda
h}W_{2}(\mu^{1}_{t},\mu^{2}_{t})$

which implies that the map $t\mapsto\mathrm{e}^{-\lambda
t}W_{2}(\mu^{1}_{t},\mu^{2}_{t})$ is decreasing in $(0,T)$. ∎

It is possible to prove (6.21) by a direct argument depending on the
definition of $\lambda$-EVI solution and a geometric condition on
$\mathrm{D}({\boldsymbol{\mathrm{F}}})$. The simplest situation deals with
absolutely continuous curves.

######  Theorem 6.16 (Stability for absolutely continuous solutions).

Let ${\boldsymbol{\mathrm{F}}}$ be a $\lambda$-dissipative MPVF and let
$\mu^{1},\mu^{2}:[0,T)\to\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$,
$T\in(0,+\infty]$, be _locally absolutely continuous_ $\lambda$-EVI solutions
to (6.1). If
$\Gamma_{o}^{0}({\mu^{1}_{t}},{\mu^{2}_{t}}|{\boldsymbol{\mathrm{F}}})\neq\emptyset$
for a.e. $t\in(0,T)$, then (6.21) holds. In particular, if
$\mu^{1}_{0}=\mu^{2}_{0}$ then $\mu^{1}\equiv\mu^{2}$ in $[0,T)$.

###### Proof.

Since $\mu^{1},\mu^{2}$ are locally absolutely continuous curves, we can apply
Theorem 3.13 and find a subset $A\subset A({\mu^{1}})\cap A({\mu^{2}})$ of
full Lebesgue measure such that (3.20) holds and
$\Gamma_{o}^{0}({\mu^{1}_{t}},{\mu^{2}_{t}}|{\boldsymbol{\mathrm{F}}})\neq\emptyset$
for every $t\in A$. Selecting
$\boldsymbol{\mu}_{t}\in\Gamma_{o}^{0}({\mu^{1}_{t}},{\mu^{2}_{t}}|{\boldsymbol{\mathrm{F}}})$,
we have

$\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}W_{2}^{2}(\mu^{1}_{t},\mu^{2}_{t})=\int\langle\boldsymbol{v}_{t}^{1}(x_{1}),x_{1}-x_{2}\rangle\,\mathrm{d}\boldsymbol{\mu}_{t}(x_{1},x_{2})+\int\langle\boldsymbol{v}_{t}^{2}(x_{2}),x_{2}-x_{1}\rangle\,\mathrm{d}\boldsymbol{\mu}_{t}(x_{1},x_{2}).$

Using (6.5b), (6.5c), for every $t\in A$ we get

$\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}W_{2}^{2}(\mu^{1}_{t},\mu^{2}_{t})=\left[(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}_{t})_{\sharp}\mu_{t}^{1},\mu_{t}^{2}\right]_{r}\leq[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}_{t}]_{0+}+\lambda
W_{2}^{2}(\mu^{1}_{t},\mu^{2}_{t})+[{\boldsymbol{\mathrm{F}}},\mathsf{s}_{\sharp}\boldsymbol{\mu}_{t}]_{1-}=\lambda
W_{2}^{2}(\mu^{1}_{t},\mu^{2}_{t}),$

where we also used the property

$[{\boldsymbol{\mathrm{F}}},\mathsf{s}_{\sharp}\boldsymbol{\mu}_{t}]_{1-}=-[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}_{t}]_{0+}.\qed$

The last situation deals with comparison between an absolutely continuous and
a merely continuous $\lambda$-EVI solution. The argument is technically more
involved and takes inspiration from the proof of [NS06, Theorem 1.1]: we refer
to the Introduction of [NS06] for an explanation of the heuristic idea. Since
it is also at the core of the discrete estimates of Theorem 6.9, we present it
here in the easier continuous setting.

######  Theorem 6.17 (Refined stability).

Let $T>0$ and let
$\mu^{1}\in\mathrm{AC}([0,T];\mathrm{D}({\boldsymbol{\mathrm{F}}}))$ and
$\mu^{2}\in\mathrm{C}([0,T];\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})})$
be $\lambda$-EVI solutions for the $\lambda$-dissipative MPVF
${\boldsymbol{\mathrm{F}}}$. If at least one of the following properties hold:

  1. (1)

$\Gamma_{o}^{0}({\mu^{1}_{r}},{\mu^{2}_{s}}|{\boldsymbol{\mathrm{F}}})\neq\emptyset\text{
for every }s\in(0,T)\text{ and }r\in[0,T)\setminus N$ with $N\subset(0,T),\
\mathcal{L}^{1}(N)=0$;

  2. (2)

$\mu^{1}$ satisfies (6.2),

then

$W_{2}(\mu^{1}_{t},\mu^{2}_{t})\leq e^{\lambda
t}W_{2}(\mu^{1}_{0},\mu^{2}_{0})\quad\text{ for every }t\in[0,T].$

###### Proof.

We extend $\mu^{1}$ in $(-\infty,0)$ with the constant value $\mu^{1}_{0}$, we
denote by $\boldsymbol{v}$ the Wasserstein velocity field associated to
$\mu^{1}$ (and extended to $0$ outside $A(\mu^{1})$) and we define the
functions $w,f,h:(-\infty,T]\times[0,T]\to\mathbb{R}$ by

$w(r,s):=W_{2}(\mu^{1}_{r},\mu^{2}_{s})$
$f(r,s):=\begin{cases}2|{\boldsymbol{\mathrm{F}}}|_{2}(\mu_{0}^{1})w(0,s)\quad&\text{
if }r<0,\\\ 0&\text{ if }r\geq 0,\end{cases}\qquad
h(r,s):=\begin{cases}0\quad&\text{ if }r<0,\\\
2\left[(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}_{r})_{\sharp}\mu^{1}_{r},\mu^{2}_{s}\right]_{r}&\text{
if }r\geq 0.\end{cases}$

Theorem 3.11 yields

$\displaystyle\frac{\partial}{\partial r}w^{2}(r,s)=h(r,s)\quad$
$\displaystyle\text{ in }\mathcal{D}^{\prime}(-\infty,T)\text{ for every
}s\in[0,T].$ (6.22)

In case (1) holds, writing (6.4b) for $\mu^{2}$ with $\nu=\mu^{1}_{r}$ with
$r\in(-\infty,T]\setminus N$, then for every
$\boldsymbol{\mu}_{rs}\in\Gamma_{o}^{0}({\mu^{1}_{r}},{\mu^{2}_{s}}|{\boldsymbol{\mathrm{F}}})$
we obtain

$\displaystyle{\frac{\mathrm{d}}{\mathrm{d}s}}^{\kern-3.0pt+}w^{2}(r,s)\leq
2\lambda
w^{2}(r,s)-2[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}_{rs}]_{0+}\quad$
$\displaystyle\text{ for }s\in(0,T)\text{ and }r\in(-\infty,T)\setminus N.$
(6.23)

On the other hand (6.5b) yields

$\displaystyle-2[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}_{rs}]_{0+}$
$\displaystyle\leq-2[(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}_{r})_{\sharp}\mu^{1}_{r},\boldsymbol{\mu}_{rs}]_{r,0}\leq-2\left[(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}_{r})_{\sharp}\mu^{1}_{r},\mu^{2}_{s}\right]_{r}\quad\text{for
every }r\in A(\mu^{1})\setminus N,$ (6.24)
$\displaystyle-2[{\boldsymbol{\mathrm{F}}},\boldsymbol{\mu}_{rs}]_{0+}$
$\displaystyle\leq
2|{\boldsymbol{\mathrm{F}}}|_{2}(\mu_{0}^{1})w(0,s)=f(r,s)\quad\text{for every
}r<0.$

Combining (6.3) and (6.24) we obtain

${\frac{\mathrm{d}}{\mathrm{d}s}}^{\kern-3.0pt+}w^{2}(r,s)\leq 2\lambda
w^{2}(r,s)+f(r,s)-h(r,s)\quad\text{for }s\in(0,T),\ r\in(-\infty,0]\cup
A(\mu^{1})\setminus N.$

Since $|h(r,s)|\leq 2|\dot{\mu}^{1}_{r}|\,w(r,s)$, applying Lemma C.2 we get

$\frac{\partial}{\partial s}w^{2}(r,s)\leq 2\lambda
w^{2}(r,s)+f(r,s)-h(r,s)\quad\text{in }\mathcal{D}^{\prime}(0,T)\text{ for
a.e.\leavevmode\nobreak\ }r\in(-\infty,T].$ (6.25)

(6.25) can also be deduced in case (2) using (6.2).

By multiplying both inequalities (6.22) and (6.25) by $e^{-2\lambda s}$ we get

$\displaystyle\frac{\partial}{\partial r}\Big{(}e^{-2\lambda
s}w^{2}(r,s)\Big{)}=e^{-2\lambda s}h(r,s)\quad$ $\displaystyle\text{ in
}\mathcal{D}^{\prime}(-\infty,T)\text{ and every }s\in[0,T],$
$\displaystyle\frac{\partial}{\partial s}\Big{(}e^{-2\lambda
s}w^{2}(r,s)\Big{)}\leq e^{-2\lambda s}\big{(}f(r,s)-h(r,s)\big{)}\quad$
$\displaystyle\text{ in }\mathcal{D}^{\prime}(0,T)\text{ and
a.e.\leavevmode\nobreak\ }r\in(-\infty,T].$

We fix $t\in[0,T]$ and $\varepsilon>0$ and we apply the Divergence theorem in
[NS06, Lemma 6.15] on the two-dimensional strip $Q_{0,t}^{\varepsilon}$ as in
Figure 1,

$Q_{0,t}^{\varepsilon}:=\\{(r,s)\in\mathbb{R}^{2}\mid 0\leq s\leq
t\,,\,s-\varepsilon\leq r\leq s\\},$

$\varepsilon$$\varepsilon$$r=t$$s=t$$Q^{\varepsilon}_{0,t}$$r$$s$$r=s$$r=s-\varepsilon$
Figure 1. Strip $Q^{\varepsilon}_{0,t}$ corresponding to penalization about
the diagonal $\\{r=s\\}$.

and we get

$\int_{t-\varepsilon}^{t}e^{-2\lambda
t}w^{2}(r,t)\,\mathrm{d}r\leq\int_{-\varepsilon}^{0}w^{2}(r,0)\,\mathrm{d}r+\iint_{Q^{\varepsilon}_{0,t}}e^{-2\lambda
s}f(r,s)\,\mathrm{d}r\,\mathrm{d}s.$

Using

$w(t,t)\leq\int_{r}^{t}|\dot{\mu}^{1}|(u)\,\mathrm{d}u+w(r,t)\leq\int_{t-\varepsilon}^{t}|\dot{\mu}^{1}|(u)\,\mathrm{d}u+w(r,t)\quad\text{
if }t-\varepsilon\leq r\leq t,$

then, for every $\vartheta,\vartheta_{\star}>1$ conjugate coefficients
($\vartheta_{\star}=\vartheta/(\vartheta-1)$), we get

$w^{2}(t,t)\leq\vartheta
w^{2}(r,t)+\vartheta_{\star}\left(\int_{t-\varepsilon}^{t}|\dot{\mu}^{1}|(u)\,\mathrm{d}u\right)^{2}.$
(6.26)

Integrating (6.26) w.r.t. $r$ in the interval $(t-\varepsilon,t)$, we obtain

$e^{-2\lambda
t}w^{2}(t,t)\leq\frac{\vartheta}{\varepsilon}\int_{t-\varepsilon}^{t}e^{-2\lambda
t}w^{2}(r,t)\,\mathrm{d}r+\vartheta_{\star}\left(\int_{t-\varepsilon}^{t}|\dot{\mu}^{1}|(u)\,\mathrm{d}u\right)^{2}\max\\{1,e^{2|\lambda|T}\\}.$
(6.27)

Finally, we have the following inequality

$\varepsilon^{-1}\iint_{Q^{\varepsilon}_{0,t}}e^{-2\lambda
s}f(r,s)\,\mathrm{d}r\,\mathrm{d}s\leq
2|{\boldsymbol{\mathrm{F}}}|_{2}(\mu_{0})\int_{0}^{\varepsilon}e^{-2\lambda
s}w(0,s)\,\mathrm{d}s.$ (6.28)

Summing up (6.27) and (6.28) we obtain

$e^{-2\lambda
t}w^{2}(t)\leq\vartheta\left(w^{2}(0)+2|{\boldsymbol{\mathrm{F}}}|_{2}(\mu_{0})\int_{0}^{\varepsilon}e^{-2\lambda
s}w(0,s)\,\mathrm{d}s\right)+\vartheta_{\star}\left(\int_{t-\varepsilon}^{t}|\dot{\mu}^{1}|(u)\,\mathrm{d}u\right)^{2}\max\\{1,e^{2|\lambda|T}\\}.$

where we have used the notation $w(s)=w(s,s)$. Taking the limit as
$\varepsilon\downarrow 0$ and $\vartheta\downarrow 1$, we obtain the thesis. ∎

######  Corollary 6.18 (Local Lipschitz estimate).

Let ${\boldsymbol{\mathrm{F}}}$ be a $\lambda$-dissipative MPVF and let
$\mu:(0,T)\to\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$,
$T\in(0,+\infty]$, be a $\lambda$-EVI solution to (6.1). If at least one of
the following two conditions holds

  1. (a)

$\mu$ is strict and (EE) is locally solvable in
$\mathrm{D}({\boldsymbol{\mathrm{F}}})$,

  2. (b)

$\mu$ is locally absolutely continuous and (4.16) holds,

then $\mu$ is locally Lipschitz and

$t\mapsto\mathrm{e}^{-\lambda t}|\dot{\mu}|_{+}(t)\quad\text{is decreasing in
}(0,T).$ (6.29)

###### Proof.

Since for every $h>0$ the curve $t\mapsto\mu_{t+h}$ is a $\lambda$-EVI
solution, (6.21) yields

$\mathrm{e}^{-\lambda(t-s)}W_{2}(\mu_{t+h},\mu_{t})\leq
W_{2}(\mu_{s+h},\mu_{s})\quad\text{for every $0<s<t$.}$

Dividing by $h$ and taking the limsup as $h\downarrow 0$, we get (6.29), which
in turn shows the local Lipschitz character of $\mu$. ∎

###  6.4. Global existence and generation of $\lambda$-flows

We collect here a few simple results on the existence of global solutions and
the generation of a $\lambda$-flow. A first result can be deduced from the
global solvability of the Explicit Euler scheme.

######  Theorem 6.19 (Global existence).

Let ${\boldsymbol{\mathrm{F}}}$ be a $\lambda$-dissipative MPVF. If the
Explicit Euler Scheme is globally solvable at
$\mu_{0}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$, then there exists a unique
global $\lambda$-EVI solution $\mu\in\mathrm{Lip}_{\rm
loc}([0,\infty);\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})})$ starting
from $\mu_{0}$.

###### Proof.

We can argue as in the proof of Theorem 6.11(a), observing that the global
solvability of (EE) allows for the construction of a limit solution on every
interval $[0,T]$, $T>0$. ∎

Let us provide a simple condition ensuring global solvability, whose proof is
deferred to Section 7.

######  Proposition 6.20.

Let ${\boldsymbol{\mathrm{F}}}$ be a $\lambda$-dissipative MPVF such that for
every $R>0$ there exist $M=\mathrm{M}(R)>0$ and $\bar{\tau}=\bar{\tau}(R)>0$
such that

$\mu\in\mathrm{D}({\boldsymbol{\mathrm{F}}}),\ \mathsf{m}_{2}(\mu)\leq R,\
0<\tau\leq\bar{\tau}\quad\Rightarrow\quad\exists\,\Phi\in{\boldsymbol{\mathrm{F}}}[\mu]:|\Phi|_{2}\leq\mathrm{M}(R),\
\operatorname{\mathsf{exp}}^{\tau}_{\sharp}\Phi\in\mathrm{D}({\boldsymbol{\mathrm{F}}}).$
(6.30)

Then the Explicit Euler scheme is globally solvable in
$\mathrm{D}({\boldsymbol{\mathrm{F}}})$.

Global existence of $\lambda$-EVI solution is also related to the existence of
a $\lambda$-flow.

######  Definition 6.21.

We say that the $\lambda$-dissipative MPVF ${\boldsymbol{\mathrm{F}}}$
_generates a $\lambda$-flow_ if for every
$\mu_{0}\in\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$ there exists a
unique $\lambda$-EVI solution $\mu=\mathrm{S}[\mu_{0}]$ starting from
$\mu_{0}$ and the maps
$\mu_{0}\mapsto\mathrm{S}_{t}[\mu_{0}]=(\mathrm{S}[\mu_{0}])_{t}$ induce a
semigroup of Lipschitz transformations $(\mathrm{S}_{t})_{t\geq 0}$ of
$\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$ satisfying

$W_{2}(\mathrm{S}_{t}[\mu_{0}],\mathrm{S}_{t}[\mu_{1}])\leq\mathrm{e}^{\lambda
t}W_{2}(\mu_{0},\mu_{1})\quad\text{for every }t\geq 0.$ (6.31)

######  Theorem 6.22 (Generation of a $\lambda$-flow).

Let ${\boldsymbol{\mathrm{F}}}$ be a $\lambda$-dissipative MPVF. If at least
one of the following properties is satisfied:

  1. (a)

the Explicit Euler Scheme is globally solvable for every $\mu_{0}$ in a dense
subset of $\mathrm{D}({\boldsymbol{\mathrm{F}}})$;

  2. (b)

the Explicit Euler Scheme is locally solvable in
$\mathrm{D}({\boldsymbol{\mathrm{F}}})$ and, for every $\mu_{0}$ in a dense
subset of $\mathrm{D}({\boldsymbol{\mathrm{F}}})$, there exists a strict
global $\lambda$-EVI solution starting from $\mu_{0}$;

  3. (c)

the Explicit Euler Scheme is locally solvable in
$\mathrm{D}({\boldsymbol{\mathrm{F}}})$ and
$\mathrm{D}({\boldsymbol{\mathrm{F}}})$ is closed;

  4. (d)

for every $\mu_{0}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$,
$\mu_{1}\in\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$
$\Gamma_{o}^{0}({\mu_{0}},{\mu_{1}}|{\boldsymbol{\mathrm{F}}})\neq\emptyset$
and, for every $\mu_{0}$ in a dense subset of
$\mathrm{D}({\boldsymbol{\mathrm{F}}})$, there exists a locally absolutely
continuous strict global $\lambda$-EVI solution starting from $\mu_{0}$;

  5. (e)

for every $\mu_{0}$ in a dense subset of
$\mathrm{D}({\boldsymbol{\mathrm{F}}})$, there exists a locally absolutely
continuous solution of (6.2) starting from $\mu_{0}$,

then ${\boldsymbol{\mathrm{F}}}$ generates a $\lambda$-flow.

###### Proof.

(a) Let $D$ be the dense subset of $\mathrm{D}({\boldsymbol{\mathrm{F}}})$ for
which (EE) is globally solvable. For every $\mu_{0}\in D$ we define
$\mathrm{S}_{t}[\mu_{0}]$, $t\geq 0$, as the value at time $t$ of the unique
$\lambda$-EVI solution starting from $\mu_{0}$, whose existence is guaranteed
by Theorem 6.19.

If $\mu_{0},\mu_{1}\in D$, $T>0$, we can find $\boldsymbol{\tau},L$ such that
$\mathscr{M}(\mu_{0},\tau,T,L)$ and $\mathscr{M}(\mu_{1},\tau,T,L)$ are not
empty for every $\tau\in(0,\boldsymbol{\tau})$. We can then pass to the limit
in the uniform estimate (6.12) for every choice of
$M^{i}_{\tau}\in\mathscr{M}(\mu_{i},\tau,T,L)$, $i=0,1$, obtaining (6.31) for
every $\mu_{0},\mu_{1}\in D$.

We can then extend the map $\mathrm{S}_{t}$ to
$\overline{D}=\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$ still
preserving the same property. Proposition 6.7 shows that for every
$\mu_{0}\in\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$ the continuous
curve $t\mapsto\mathrm{S}_{t}[\mu_{0}]$ is a $\lambda$-EVI solution starting
from $\mu_{0}$.

Finally, if
$\mu\in\mathrm{C}([0,T^{\prime});\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})})$
is any $\lambda$-EVI solution starting from $\mu_{0}$, we can apply (6.13) to
get

$W_{2}(\mu_{t},M^{1}_{\tau}(t))\leq\Big{(}2W_{2}(\mu_{0},\mu_{1})+C(\boldsymbol{\tau},L,T)\sqrt{\tau}\Big{)}\mathrm{e}^{\lambda_{+}t}\quad\text{for
every }t\in[0,T],$ (6.32)

for every $T<T^{\prime}$, $\tau<\boldsymbol{\tau}$, where
$C(\boldsymbol{\tau},L,T)>0$ is a suitable constant. Passing to the limit as
$\tau\downarrow 0$ in (6.32) we obtain

$W_{2}(\mu_{t},\mathrm{S}_{t}[\mu_{1}])\leq
2W_{2}(\mu_{0},\mu_{1})\mathrm{e}^{\lambda_{+}t}\quad\text{for every
}t\in[0,T].$ (6.33)

Choosing now a sequence $\mu_{1,n}$ in $D$ converging to $\mu_{0}$ and
observing that we can choose arbitrary $T<T^{\prime}$, we eventually get
$\mu_{t}=\mathrm{S}_{t}[\mu_{0}]$ for every $t\in[0,T^{\prime})$.

(b) Let $D$ be the dense subset of $\mathrm{D}({\boldsymbol{\mathrm{F}}})$
such that there exists a global strict $\lambda$-EVI solution starting from
$D$. By Theorem 6.15 such a solution is unique and the corresponding family of
solution maps $\mathrm{S}_{t}:D\to\mathrm{D}({\boldsymbol{\mathrm{F}}})$
satisfy (6.31). Arguing as in the previous claim, we can extend
$\mathrm{S}_{t}$ to $\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$ still
preserving (6.31) and the fact that $t\mapsto\mathrm{S}_{t}[\mu_{0}]$ is a
$\lambda$-EVI solution.

If $\mu$ is $\lambda$-EVI solution starting from $\mu_{0}$, Theorem 6.15 shows
that (6.33) holds for every $\mu_{1}\in D$. By approximation we conclude that
$\mu_{t}=\mathrm{S}_{t}[\mu_{0}]$.

(c) Corollary 6.12 shows that for every initial datum
$\mu_{0}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$ there exists a global
$\lambda$-EVI solution. We can then apply Claim (b).

(d) Let $D$ be the dense subset of $\mathrm{D}({\boldsymbol{\mathrm{F}}})$
such that there exists a locally absolutely continuous strict global
$\lambda$-EVI solution starting from $D$. By Theorem 6.16 such a solution is
the unique locally absolutely continuous solution starting from $\mu_{0}$ and
the corresponding family of solution maps
$\mathrm{S}_{t}:D\to\mathrm{D}({\boldsymbol{\mathrm{F}}})$ satisfy (6.31).
Arguing as in the previous claim (b), we can extend $\mathrm{S}_{t}$ to
$\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$ still preserving (6.31)
(again thanks to Theorem 6.16) and the fact that
$t\mapsto\mathrm{S}_{t}[\mu_{0}]$ is a $\lambda$-EVI solution.

If $\mu$ is a $\lambda$-EVI solution starting from
$\mu_{0}\in\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$ and
$(\mu_{0}^{n})_{n\in\mathbb{N}}\subset D$ is a sequence converging to
$\mu_{0}$, we can apply Theorem 6.17(1) and conclude that
$\mu_{t}=\mathrm{S}_{t}[\mu_{0}]$.

(e) The proof follows by the same argument of the previous claim, eventually
applying Theorem 6.17(2). ∎

By Lemma 6.13 we immediately get the following result.

######  Corollary 6.23.

If ${\boldsymbol{\mathrm{F}}}$ is locally bounded $\lambda$-dissipative MPVF
with $\mathrm{D}({\boldsymbol{\mathrm{F}}})=\mathcal{P}_{2}(\mathsf{X})$ then
for every $\mu_{0}\in\mathcal{P}_{2}(\mathsf{X})$ there exists a unique global
$\lambda$-EVI solution starting from $\mu_{0}$.

We conclude this section by showing a consistency result with the Hilbertian
theory, related to the example of Section 5.2.

######  Corollary 6.24 (Consistency with the theory of contraction semigroups
in Hilbert spaces).

Let $F\subset\mathsf{X}\times\mathsf{X}$ be a dissipative maximal subset
generating the semigroup $(R_{t})_{t\geq 0}$ of nonlinear contractions [Bré73,
Theorem 3.1]. Let ${\boldsymbol{\mathrm{F}}}$ be the dissipative MPVF

${\boldsymbol{\mathrm{F}}}:=\\{\Phi\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})\mid\Phi\text{
is concentrated on }F\\}.$

The semigroup
$\mu_{0}\mapsto\mathrm{S}_{t}[\mu_{0}]:=(R_{t})_{\sharp}\mu_{0}$, $t\geq 0$,
is the $0$-flow generated by ${\boldsymbol{\mathrm{F}}}$ in
$\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$.

###### Proof.

Let $D$ be the set of discrete measures
$\frac{1}{n}\sum_{j=1}^{n}\delta_{x_{j}}$ with $x_{j}\in\mathrm{D}(F)$. Since
every $\mu_{0}\in\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$ is
supported in $\overline{\mathrm{D}(F)}$, $D$ is dense in
$\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$. Our thesis follows by
applying Theorem 6.22(e) if we show that for every
$\mu_{0}=\frac{1}{n}\sum_{j=1}^{n}\delta_{x_{j,0}}\in D$ there exists a
locally absolutely continuous solution $\mu:[0,\infty)\to D$ of (6.2) starting
from $\mu_{0}$.

It can be directly checked that

$\mu_{t}:=(R_{t})_{\sharp}\mu_{0}=\frac{1}{n}\sum_{j=1}^{n}\delta_{x_{j,t}},\quad
x_{j,t}:=R_{t}(x_{j,0})$

satisfies the continuity equation with Wasserstein velocity vector
$\boldsymbol{v}_{t}$ (defined on the finite support of $\mu_{t}$) satisfying

$\boldsymbol{v}_{t}(x_{j,t})=\dot{x}_{j,t}=F^{\circ}(x_{j,t}),\quad|\boldsymbol{v}_{t}(x_{j,t})|\leq|F^{\circ}(x_{j,0})|\quad\text{for
every }j=1,\cdots,n,\ \text{and a.e.\leavevmode\nobreak\ }t>0,$

where $F^{\circ}$ is the minimal selection of $F$. It follows that

$(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}_{t})_{\sharp}\mu_{t}\in{\boldsymbol{\mathrm{F}}}[\mu_{t}]\quad\text{
for a.e. }t>0,$

so that $\mu$ is a Lipschitz EVI solution for ${\boldsymbol{\mathrm{F}}}$
starting from $\mu_{0}$. We can thus conclude observing that the map
$\mu_{0}\mapsto(R_{t})_{\sharp}\mu_{0}$ are contractions in
$\mathcal{P}_{2}(\mathsf{X})$ and the curve $\mu_{t}=(R_{t})_{\sharp}\mu_{0}$
is continuous with values in
$\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}$. ∎

###  6.5. Barycentric property

If we assume that the MPVF ${\boldsymbol{\mathrm{F}}}$ is a sequentially
closed subset of $\mathcal{P}_{2}^{sw}(\mathsf{T\kern-1.5ptX})$ with convex
sections, we are able to provide a stronger result showing a particular
property satisfied by the solutions of (6.1) (see Theorem 6.27). This is
called _barycentric property_ and it is strictly connected with the weaker
definition of solution discussed in Piccoli_2019, Piccoli_MDI, Camilli_MDE.

We first introduce a directional closure of ${\boldsymbol{\mathrm{F}}}$ along
smooth cylindrical deformations. We set

$\mathrm{exp}^{\varphi}(x):=x+\nabla\varphi(x)\quad\text{for every
}\varphi\in\operatorname{Cyl}(\mathsf{X})$

and

$\displaystyle{\overline{{\boldsymbol{\mathrm{F}}}}}[\mu]:={}\Big{\\{}\Phi\in\mathcal{P}_{2}(\mathsf{X})\mid{}$
$\displaystyle\exists\,\varphi\in\operatorname{Cyl}(\mathsf{X}),\
(r_{n})_{n\in\mathbb{N}}\subset[0,+\infty),\ r_{n}\downarrow 0,\
\Phi_{n}\in{\boldsymbol{\mathrm{F}}}[\mathrm{exp}^{r_{n}\varphi}_{\sharp}\mu]:$
(6.34) $\displaystyle\Phi_{n}\to\Phi\text{ in
}\mathcal{P}_{2}^{sw}(\mathsf{T\kern-1.5ptX})\Big{\\}}.$

######  Definition 6.25 (Barycentric property).

Let ${\boldsymbol{\mathrm{F}}}$ be a MPVF. We say that a locally absolutely
continuous curve $\mu:\mathcal{I}\to\mathrm{D}({\boldsymbol{\mathrm{F}}})$
satisfies the _barycentric property_ (resp. the _relaxed barycentric
property_) if for a.e. $t\in\mathcal{I}$ there exists
$\Phi_{t}\in{\boldsymbol{\mathrm{F}}}[\mu_{t}]$ (resp.
$\Phi_{t}\in\overline{\operatorname{co}}({\overline{{\boldsymbol{\mathrm{F}}}}}[\mu_{t}])$)
s.t.

$\frac{\,\mathrm{d}}{\,\mathrm{d}t}\int_{\mathsf{X}}\varphi(x)\,\mathrm{d}\mu_{t}(x)=\int_{\mathsf{T\kern-1.5ptX}}\langle\nabla\varphi(x),v\rangle\,\mathrm{d}\Phi_{t}(x,v)\quad\forall\,\varphi\in\operatorname{Cyl}(\mathsf{X}).$
(6.35)

Notice that
${\boldsymbol{\mathrm{F}}}\subset{\overline{{\boldsymbol{\mathrm{F}}}}}\subset\operatorname{cl}({\boldsymbol{\mathrm{F}}})$
and ${\overline{{\boldsymbol{\mathrm{F}}}}}={\boldsymbol{\mathrm{F}}}$ if
${\boldsymbol{\mathrm{F}}}$ is sequentially closed in
$\mathcal{P}_{2}^{sw}(\mathsf{T\kern-1.5ptX})$. Recalling Proposition 4.17(a)
we also get

$\overline{\operatorname{co}}({\overline{{\boldsymbol{\mathrm{F}}}}})\subset\hat{\boldsymbol{\mathrm{F}}},$

so that the relaxed barycentric property implies the corresponding property
for the extended MPVF $\hat{\boldsymbol{\mathrm{F}}}$.

######  Remark 6.26.

If $\mathsf{X}=\mathbb{R}^{d}$, the property stated in Definition 6.25
coincides with the weak definition of solution to (6.1) given in [Pic18].

The aim is to prove that the $\lambda$-EVI solution of (6.1) enjoys the
barycentric property of Definition 6.25, under suitable mild conditions on
${\boldsymbol{\mathrm{F}}}$. This is strictly related to the behaviour of
${\boldsymbol{\mathrm{F}}}$ along the family of smooth deformations induced by
cylindrical functions. Let us denote by $\mathbf{pr}_{\mu}$ the orthogonal
projection in $L^{2}_{\mu}(\mathsf{X};\mathsf{X})$ onto the tangent space
$\operatorname{Tan}_{\mu}\mathcal{P}_{2}(\mathsf{X})$ and by
$\boldsymbol{b}_{\Phi}$ the barycenter of $\Phi$ as in Definition 3.1.

######  Theorem 6.27.

Let ${\boldsymbol{\mathrm{F}}}$ be a $\lambda$-dissipative MPVF such that for
every $\mu\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$ there exist constants
$M,\varepsilon>0$ such that

$\forall\varphi\in\operatorname{Cyl}(\mathsf{X}):\
\sup_{\mathsf{X}}|\nabla\varphi|\leq\varepsilon\quad\Rightarrow\quad\mathrm{exp}^{\varphi}_{\sharp}\mu\in\mathrm{D}({\boldsymbol{\mathrm{F}}}),\
|{\boldsymbol{\mathrm{F}}}|_{2}(\mathrm{exp}^{\varphi}_{\sharp}\mu)<M.$ (6.36)

If $\mu:\mathcal{I}\to\mathrm{D}({\boldsymbol{\mathrm{F}}})$ is a locally
absolutely continuous $\lambda$-EVI solution of (6.1) with Wasserstein
velocity field $\boldsymbol{v}$ satisfying (2.6) for every $t$ in the subset
$A(\mu)\subset\mathcal{I}$ of full Lebesgue measure, then

$\text{for every $t\in A(\mu)$ there exists
$\Phi_{t}\in\overline{\operatorname{co}}({\overline{{\boldsymbol{\mathrm{F}}}}})[\mu_{t}]$
such
that}\quad\boldsymbol{v}_{t}=\mathbf{pr}_{\mu_{t}}\circ\boldsymbol{b}_{\Phi_{t}}.$
(6.37)

In particular, $\mu$ satisfies the relaxed barycentric property.

If moreover ${\overline{{\boldsymbol{\mathrm{F}}}}}={\boldsymbol{\mathrm{F}}}$
and for every $\nu\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$
${\boldsymbol{\mathrm{F}}}[\nu]$ is a convex subset of
$\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})$, then $\mu$ satisfies (6.35).

###### Proof.

In the following $t$ is a fixed element of $A(\mu)$ and $M$ is the constant
associated to the measure $\mu_{t}$ in (6.36). For every
$\zeta\in\operatorname{Cyl}(\mathsf{X})$ there exists $\delta=\delta(\zeta)>0$
such that
$\nu^{\zeta}:=\mathrm{exp}^{-\delta\zeta}_{\sharp}\mu_{t}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$
and
$\boldsymbol{\sigma}^{\zeta}:=(\boldsymbol{i}_{\mathsf{X}},\mathrm{exp}^{-\delta\zeta})_{\sharp}\mu_{t}\in\Gamma_{o}^{01}({\mu_{t}},{\nu^{\zeta}}|{\boldsymbol{\mathrm{F}}})$
is the unique optimal transport plan between $\mu_{t}$ and $\nu^{\zeta}$.
Thanks to Theorem 3.11, the map $s\mapsto W_{2}^{2}(\mu_{s},\nu^{\zeta})$ is
differentiable at $s=t$, moreover by employing also (6.5b), it holds

$\delta\int_{\mathsf{X}}\langle\boldsymbol{v}_{t}(x),\nabla\zeta(x)\rangle\,\mathrm{d}\mu_{t}(x)=\frac{\,\mathrm{d}}{\,\mathrm{d}t}\frac{1}{2}W_{2}^{2}(\mu_{t},\nu^{\zeta})\leq[{\boldsymbol{\mathrm{F}}},\boldsymbol{\sigma}^{\zeta}]_{0+}=\lim_{s\downarrow
0}\,[{\boldsymbol{\mathrm{F}}},\boldsymbol{\sigma}^{\zeta}]_{l,s}.$ (6.38)

We can choose a decreasing vanishing sequence
$(s_{k})_{k\in\mathbb{N}}\subset(0,1)$, measures
$\nu_{k}^{\zeta}:=\mathsf{x}^{s_{k}}_{\sharp}\boldsymbol{\sigma}^{\zeta}$ and
$\Phi_{k}^{\zeta}\in{\boldsymbol{\mathrm{F}}}[\nu_{k}^{\zeta}]$ such that
$\sup_{k}|\Phi_{k}^{\zeta}|_{2}\leq M$ and $\Phi_{k}^{\zeta}\to\Phi^{\zeta}$
in $\mathcal{P}_{2}^{sw}(\mathsf{T\kern-1.5ptX})$. Then, by (6.13), we get
$\Phi^{\zeta}\in{\overline{{\boldsymbol{\mathrm{F}}}}}[\mu_{t}]$ with
$|\Phi^{\zeta}|_{2}\leq M$ and by (6.38) and the upper semicontinuity of
$\left[\cdot,\cdot\right]_{l}$ (see Lemma 3.14) we get

$\delta\int_{\mathsf{X}}\langle\boldsymbol{v}_{t}(x),\nabla\zeta(x)\rangle\,\mathrm{d}\mu_{t}(x)\leq\left[\Phi^{\zeta},\nu^{\zeta}\right]_{l}=\delta\int_{\mathsf{T\kern-1.5ptX}}\langle
v,\nabla\zeta(x)\rangle\,\mathrm{d}\Phi^{\zeta}(x,v).$ (6.39)

Indeed, notice that, by [AGS08, Lemma 5.3.2], we have
$\Lambda(\Phi^{\zeta},\nu^{\zeta})=\\{\Phi^{\zeta}\otimes\nu^{\zeta}\\}$ with
$(\mathsf{x}^{0},\mathsf{x}^{1})_{\sharp}(\Phi^{\zeta}\otimes\nu^{\zeta})=\boldsymbol{\sigma}^{\zeta}$.

By means of the identity highlighted in Remark 3.2, the expression in (6.39)
can be written as follows

$\langle\boldsymbol{v}_{t},\nabla\zeta\rangle_{L^{2}_{\mu_{t}}(\mathsf{X};\mathsf{X})}\leq\langle\boldsymbol{b}_{\Phi^{\zeta}},\nabla\zeta\rangle_{L^{2}_{\mu_{t}}(\mathsf{X};\mathsf{X})}=\langle\mathbf{pr}_{\mu_{t}}(\boldsymbol{b}_{\Phi^{\zeta}}),\nabla\zeta\rangle_{L^{2}_{\mu_{t}}(\mathsf{X};\mathsf{X})}$

so that

$\langle\boldsymbol{v}_{t},\nabla\zeta\rangle_{L^{2}_{\mu_{t}}(\mathsf{X};\mathsf{X})}\leq\sup_{\boldsymbol{b}\in
K}\,\langle\boldsymbol{b},\nabla\zeta\rangle_{L^{2}_{\mu_{t}}(\mathsf{X};\mathsf{X})}\quad\text{for
all }\,\zeta\in\operatorname{Cyl}(\mathsf{X})$

where

$K:=\left\\{\mathbf{pr}_{\mu_{t}}(\boldsymbol{b}_{\Phi})\,:\,\Phi\in{\overline{{\boldsymbol{\mathrm{F}}}}}[\mu_{t}],\,|\Phi|_{2}\leq
M\right\\}\subset\operatorname{Tan}_{\mu_{t}}\mathcal{P}_{2}(\mathsf{X}).$
(6.40)

Applying Lemma C.1 in
$\operatorname{Tan}_{\mu_{t}}\mathcal{P}_{2}(\mathsf{X})\subset
L^{2}_{\mu_{t}}(\mathsf{X};\mathsf{X})$ we obtain that
$\boldsymbol{v}_{t}\in\overline{\operatorname{co}}(K)$. In order to obtain
(6.37) it is sufficient to prove that $\boldsymbol{v}_{t}$ is the
$L^{2}$-projection of the barycenter of an element of
$\overline{\operatorname{co}}({\overline{{\boldsymbol{\mathrm{F}}}}}[\mu_{t}])$.

Notice that an element
$\boldsymbol{v}\in\operatorname{Tan}_{\mu}\mathcal{P}_{2}(\mathsf{X})$
coincides with $\mathbf{pr}_{\mu}(\boldsymbol{b}_{\Phi})$ for
$\Phi\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX}|\mu)$ if and only if

$\int\langle\boldsymbol{v},\nabla\zeta\rangle\,\mathrm{d}\mu=\int\langle
v,\nabla\zeta\rangle\,\mathrm{d}\Phi(x,v)\quad\text{for every
}\zeta\in\mathrm{Cyl}(\mathsf{X}).$ (6.41)

It is easy to check that any element $\boldsymbol{v}\in\operatorname{co}(K)$
can be represented as $\mathbf{pr}_{\mu_{t}}(\boldsymbol{b}_{\Phi})$ (and thus
as in (6.41)) for some
$\Phi\in\operatorname{co}({\overline{{\boldsymbol{\mathrm{F}}}}}[\mu_{t}])$.
If $\boldsymbol{v}\in\overline{\operatorname{co}}(K)$ we can find a sequence
$\Phi_{n}\in\operatorname{co}({\overline{{\boldsymbol{\mathrm{F}}}}}[\mu_{t}])$
such that $|\Phi_{n}|_{2}\leq M$ and
$\boldsymbol{v}_{n}=\mathbf{pr}_{\mu_{t}}(\boldsymbol{b}_{\Phi_{n}})\to\boldsymbol{v}$
in $L^{2}_{\mu_{t}}(\mathsf{X};\mathsf{X})$. Since the sequence
$(\Phi_{n})_{n\in\mathbb{N}}$ is relatively compact in
$\mathcal{P}_{2}^{sw}(\mathsf{T\kern-1.5ptX})$ by Proposition 2.15(2), we can
extract a (not relabeled) subsequence converging to a limit $\Phi$ in
$\mathcal{P}_{2}^{sw}(\mathsf{T\kern-1.5ptX})$, as $n\to+\infty$. By
definition
$\Phi\in\overline{\operatorname{co}}({\overline{{\boldsymbol{\mathrm{F}}}}}[\mu_{t}])$
with $|\Phi|_{2}\leq M$. We can eventually pass to the limit in (6.41) written
for $\boldsymbol{v}_{n}$ and $\Phi_{n}$ thanks to
$\mathcal{P}_{2}^{sw}(\mathsf{T\kern-1.5ptX})$ convergence, obtaining the
corresponding identity for $\boldsymbol{v}$ and $\Phi$ in the limit.
Finally, being $\mu$ locally absolutely continuous, it satisfies the
continuity equation driven by $\boldsymbol{v}$ in the sense of distributions
(see Theorem 2.10), so that

$\frac{\,\mathrm{d}}{\,\mathrm{d}t}\int_{\mathsf{X}}\zeta(x)\,\mathrm{d}\mu_{t}(x)=\int_{\mathsf{X}}\langle\nabla\zeta(x),\boldsymbol{v}_{t}(x)\rangle\,\mathrm{d}\mu_{t}(x)=\int_{\mathsf{T\kern-1.5ptX}}\langle\nabla\zeta(x),v\rangle\,\mathrm{d}\Phi_{t}(x,v)\quad\forall\zeta\in\text{Cyl}(\mathsf{X}).\qed$

######  Remark 6.28.

We notice that it is always possible to estimate the value of $M$ in (6.40) by
$|{\boldsymbol{\mathrm{F}}}|_{2\star}(\mu_{t})$.

######  Remark 6.29.

Using a standard approximation argument (see for example the proof of Lemma
5.1.12(f) in [AGS08]) it is possible to show that actually the barycentric
property (6.35) holds for every
$\varphi\in\mathrm{C}^{1,1}(\mathsf{X};\mathbb{R})$ (indeed, in this case,
$\nabla\varphi\in\operatorname{Tan}_{\mu}\mathcal{P}_{2}(\mathsf{X})$ for
every $\mu\in\mathcal{P}_{2}(\mathsf{X})$).

As a complement to the studies investigated in this section, we prove the
converse characterization of Theorem 6.27 in the particular case of _regular
measures_ or _regular vector fields_. We refer to [AGS08, Definitions 6.2.1,
6.2.2] for the definition of $\mathcal{P}_{2}^{r}(\mathsf{X})$, that is the
space of regular measures on $\mathsf{X}$. When $\mathsf{X}=\mathbb{R}^{d}$
has finite dimension, $\mathcal{P}_{2}^{r}(\mathsf{X})$ is just the subset of
measures in $\mathcal{P}_{2}(\mathsf{X})$ which are absolutely continuous
w.r.t. the Lebesgue measure $\mathcal{L}^{d}$.

######  Theorem 6.30.

Let ${\boldsymbol{\mathrm{F}}}$ be a $\lambda$-dissipative MPVF. Let
$\mu:\mathcal{I}\to\mathrm{D}({\boldsymbol{\mathrm{F}}})$ be a locally
absolutely continuous curve satisfying the relaxed barycentric property of
Definition 6.25. If for a.e. $t\in\mathcal{I}$ at least one of the following
properties holds:

  1. (1)

$\mu_{t}\in\mathcal{P}_{2}^{r}(\mathsf{X})$,

  2. (2)

${\overline{{\boldsymbol{\mathrm{F}}}}}[\mu_{t}]$ contains a unique element
$\Phi_{t}$ concentrated on a map, i.e.
$\Phi_{t}=(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{b}_{\Phi_{t}})_{\sharp}\mu_{t}$

then $\mu$ is $\lambda$-EVI solution of (6.1).

###### Proof.

Take $\varphi\in\operatorname{Cyl}(\mathsf{X})$ and observe that, since $\mu$
has the relaxed barycentric property, then for a.e. $t\in\mathcal{I}$ (recall
Theorem 3.11) there exists
$\Phi_{t}\in\overline{\operatorname{co}}({\overline{{\boldsymbol{\mathrm{F}}}}}[\mu_{t}])$
such that

$\frac{\,\mathrm{d}}{\,\mathrm{d}t}\int_{\mathsf{X}}\varphi(x)\,\mathrm{d}\mu_{t}(x)=\int_{\mathsf{T\kern-1.5ptX}}\langle\nabla\varphi(x),v\rangle\,\mathrm{d}\Phi_{t}=\int_{\mathsf{X}}\langle\nabla\varphi,\mathbf{pr}_{\mu_{t}}\circ\boldsymbol{b}_{\Phi_{t}}\rangle\,\mathrm{d}\mu_{t}=\int_{\mathsf{X}}\langle\boldsymbol{v}_{t},\nabla\varphi\rangle\,\mathrm{d}\mu_{t},$

hence $\mu$ solves the continuity equation
$\partial_{t}\mu_{t}+\text{div}(\boldsymbol{v}_{t}\mu_{t})=0$, with
$\boldsymbol{v}_{t}=\mathbf{pr}_{\mu_{t}}\circ\boldsymbol{b}_{\Phi_{t}}\in\operatorname{Tan}_{\mu_{t}}\mathcal{P}_{2}(\mathsf{X})$.
By Theorem 3.11, we also know that

$\frac{\,\mathrm{d}}{\,\mathrm{d}t}\frac{1}{2}W_{2}^{2}(\mu_{t},\nu)=\int_{\mathsf{X}^{2}}\langle\boldsymbol{v}_{t}(x_{0}),x_{0}-x_{1}\rangle\,\mathrm{d}\boldsymbol{\gamma}_{t}(x_{0},x_{1}),\quad
t\in A(\mu,\nu),\
\boldsymbol{\gamma}_{t}\in\Gamma_{o}(\mu_{t},\nu),\nu\in\mathcal{P}_{2}(\mathsf{X}).$
(6.42)

Possibly disregarding a Lebesgue negligible set, we can decompose the set
$A(\mu,\nu)$ in the union $A_{1}\cup A_{2}$, where $A_{1},A_{2}$ correspond to
the times $t$ for which the properties (1) and (2) hold.

If $t\in A_{1}$ and $\nu\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$, then by
[AGS08, Theorem 6.2.10], since $\mu_{t}\in\mathcal{P}_{2}^{r}(\mathsf{X})$,
there exists a unique $\boldsymbol{\gamma}_{t}\in\Gamma_{o}(\mu_{t},\nu)$ and
$\boldsymbol{\gamma}_{t}=(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{r}_{t})_{\sharp}\mu_{t}$
for some map $\boldsymbol{r}_{t}$ s.t.
$\boldsymbol{i}_{\mathsf{X}}-\boldsymbol{r}_{t}\in\operatorname{Tan}_{\mu_{t}}\mathcal{P}_{2}(\mathsf{X})\subset
L^{2}_{\mu_{t}}(\mathsf{X};\mathsf{X})$ (recall [AGS08, Proposition 8.5.2]),
so that

$\displaystyle\int_{\mathsf{X}^{2}}\langle\boldsymbol{v}_{t}(x_{0}),x_{0}-x_{1}\rangle\,\mathrm{d}\boldsymbol{\gamma}_{t}(x_{0},x_{1})=\int_{\mathsf{X}}\langle\boldsymbol{v}_{t}(x_{0}),x_{0}-\boldsymbol{r}_{t}(x_{0})\rangle\,\mathrm{d}\mu_{t}(x_{0})$
$\displaystyle=\int_{\mathsf{X}}\langle\boldsymbol{b}_{\Phi_{t}},x_{0}-\boldsymbol{r}_{t}(x_{0})\rangle\,\mathrm{d}\mu_{t}(x_{0})=\int_{\mathsf{T\kern-1.5ptX}}\langle
v,x-\boldsymbol{r}_{t}(x)\rangle\,\mathrm{d}\Phi_{t}(x,v)=\left[\Phi_{t},\nu\right]_{r},$
(6.43)

where we also applied Theorem 3.9 and Remark 3.18, recalling that in this case
$\Lambda(\Phi_{t},\nu)$ is a singleton.

If $t\in A_{2}$ we can select the optimal plan
$\boldsymbol{\gamma}_{t}\in\Gamma_{o}(\mu_{t},\nu)$ along which

$\left[\Phi_{t},\nu\right]_{r}=[\Phi_{t},\boldsymbol{\gamma}_{t}]_{r,0}=\int_{\mathsf{X}}\langle\boldsymbol{b}_{\Phi_{t}}(x_{0}),x_{0}-x_{1}\rangle\,\mathrm{d}\boldsymbol{\gamma}_{t}(x_{0},x_{1}).$

If $\boldsymbol{r}_{t}$ is the barycenter of $\boldsymbol{\gamma}_{t}$ with
respect to its first marginal $\mu_{t}$, recalling that
$\boldsymbol{i}_{\mathsf{X}}-\boldsymbol{r}_{t}\in\operatorname{Tan}_{\mu_{t}}\mathcal{P}_{2}(\mathsf{X})$
(see also the proof of [AGS08, Thm. 12.4.4]) we also get

$\displaystyle\int_{\mathsf{X}^{2}}$
$\displaystyle\langle\boldsymbol{v}_{t}(x_{0}),x_{0}-x_{1}\rangle\,\mathrm{d}\boldsymbol{\gamma}_{t}(x_{0},x_{1})=\int_{\mathsf{X}}\langle\boldsymbol{v}_{t}(x_{0}),x_{0}-\boldsymbol{r}_{t}(x_{0})\rangle\,\mathrm{d}\mu_{t}(x_{0})$
$\displaystyle=\int_{\mathsf{X}}\langle\boldsymbol{b}_{\Phi_{t}}(x_{0}),x_{0}-\boldsymbol{r}_{t}(x_{0})\rangle\,\mathrm{d}\mu_{t}(x_{0})=\int_{\mathsf{X}}\langle\boldsymbol{b}_{\Phi_{t}}(x_{0}),x_{0}-x_{1}\rangle\,\mathrm{d}\boldsymbol{\gamma}_{t}(x_{0},x_{1})=\left[\Phi_{t},\nu\right]_{r}$
(6.44)

where we still applied Theorem 3.9 and Remark 3.18.

Combining (6.42) with (6.43) and (6.44) we eventually get

$\displaystyle\frac{\,\mathrm{d}}{\,\mathrm{d}t}\frac{1}{2}W_{2}^{2}(\mu_{t},\nu)$
$\displaystyle=\left[\Phi_{t},\nu\right]_{r}\leq-\left[\Psi,\mu_{t}\right]_{r}+\lambda
W_{2}^{2}(\mu_{t},\nu),\quad\forall\,\Psi\in{\boldsymbol{\mathrm{F}}}[\nu],$

by definition of $\hat{\boldsymbol{\mathrm{F}}}$ and the fact that
$\overline{\operatorname{co}}(\overline{{\boldsymbol{\mathrm{F}}}})[\mu_{t}]\subset\hat{\boldsymbol{\mathrm{F}}}[\mu_{t}]$.
∎

Thanks to Theorem 6.30, we can apply to barycentric solutions the uniqueness
and approximation results of the previous Sections. We conclude this section
with a general result on the existence of a $\lambda$-flow for
$\lambda$-dissipative MPVFs, which is the natural refinement of Proposition
6.14

######  Theorem 6.31 (Generation of $\lambda$-flow).

Let ${\boldsymbol{\mathrm{F}}}$ be a $\lambda$-dissipative MPVF such that
$\mathcal{P}_{b}(\mathsf{X})\subset\mathrm{D}({\boldsymbol{\mathrm{F}}})$ and
for every $\mu_{0}\in\mathcal{P}_{b}(\mathsf{X})$ there exist $\varrho>0$ and
$L>0$ such that

$\operatorname{supp}(\mu)\subset\operatorname{supp}(\mu_{0})+\mathrm{B}_{\mathsf{X}}(\varrho)\quad\Rightarrow\quad\exists\Phi\in{\boldsymbol{\mathrm{F}}}[\mu]:\operatorname{supp}(\mathsf{v}_{\sharp}\Phi)\subset\mathrm{B}_{\mathsf{X}}(L).$
(6.45)

Let
${\boldsymbol{\mathrm{F}}}_{b}:={\boldsymbol{\mathrm{F}}}\cap\mathcal{P}_{b}(\mathsf{T\kern-1.5ptX})$.
If there exists $a\geq 0$ such that for every
$\Phi\in{\boldsymbol{\mathrm{F}}}_{b}$

$\operatorname{supp}(\Phi)\subset\Big{\\{}(x,v)\in\mathsf{T\kern-1.5ptX}:\langle
v,x\rangle\leq a(1+|x|^{2})\Big{\\}},$ (6.46)

then ${\boldsymbol{\mathrm{F}}}$ generates a $\lambda$-flow.

###### Proof.

It is enough to prove that ${\boldsymbol{\mathrm{F}}}_{b}$ generates a
$\lambda$-flow. Applying Proposition 6.14 to the MPVF
${\boldsymbol{\mathrm{F}}}_{b}$, we know that for every
$\mu_{0}\in\mathrm{D}({\boldsymbol{\mathrm{F}}}_{b})$ there exists a unique
maximal strict $\lambda$-EVI solution $\mu\in\mathrm{Lip}_{\rm
loc}([0,T);\mathcal{P}_{b}(\mathsf{X}))$ driven by
${\boldsymbol{\mathrm{F}}}_{b}$ and satisfying (6.16). We argue by
contradiction, and we assume that $T<+\infty$. Notice that by (6.45)
${\boldsymbol{\mathrm{F}}}$ satisfies (6.36), so that $\mu$ is a relaxed
barycentric solution for ${\boldsymbol{\mathrm{F}}}_{b}$. Since
$\mu_{0}\in\mathcal{P}_{b}(\mathsf{X})$, we know that
$\operatorname{supp}(\mu_{0})\subset\mathrm{B}_{\mathsf{X}}(r_{0})$ for some
$r_{0}>1$.

It is easy to check that (6.46) holds also for every
$\Phi\in\overline{\operatorname{co}}({\overline{{\boldsymbol{\mathrm{F}}}}}_{b})$.
Moreover, setting $b:=2a$, condition (6.46) yields

$\langle v,x\rangle\leq b|x|^{2}\quad\text{for
every}\quad(x,v)\in\operatorname{supp}\Phi\in{\boldsymbol{\mathrm{F}}}_{b},\
|x|\geq 1.$ (6.47)

Let $\phi(r):\mathbb{R}\to\mathbb{R}$ be any smooth increasing function such
that $\phi(r)=0$ if $r\leq r_{0}$ and $\phi(r)=1$ if $r\geq r_{0}+1$, and let
$\varphi(t,x):=\phi(|x|\mathrm{e}^{-bt})$. Clearly
$\varphi\in\mathrm{C}^{1,1}(\mathsf{X}\times[0,+\infty))$, with
$\nabla\varphi(t,x)=\frac{x}{|x|}\phi^{\prime}(|x|\mathrm{e}^{-bt})\mathrm{e}^{-bt}$
if $x\neq 0$, $\nabla\varphi(t,0)=0$, and
$\partial_{t}\varphi(t,x)=-b\phi^{\prime}(|x|\mathrm{e}^{-bt})|x|\mathrm{e}^{-bt}.$
We thus have for a.e. $t\in[0,T)$

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\int_{\mathsf{X}}\varphi(t,x)\,\mathrm{d}\mu_{t}$
$\displaystyle=\mathrm{e}^{-bt}\int_{\mathsf{T\kern-1.5ptX}}\Big{(}-b\phi^{\prime}(|x|\mathrm{e}^{-bt})|x|+\langle
v,x\rangle|x|^{-1}\phi^{\prime}(|x|\mathrm{e}^{-bt})\Big{)}\mathrm{d}\Phi_{t}(v,x)$
$\displaystyle\leq\mathrm{e}^{-bt}\int_{\mathsf{T\kern-1.5ptX}}\Big{(}-b\phi^{\prime}(|x|\mathrm{e}^{-bt})|x|+b|x|\phi^{\prime}(|x|\mathrm{e}^{-bt})\Big{)}\mathrm{d}\Phi_{t}(v,x)=0$

where in the last inequality we used (6.47) and the fact that the integrand
vanishes if $|x|\leq 1$. We get

$\int_{\mathsf{X}}\varphi(t,x)\,\mathrm{d}\mu_{t}=0\quad\text{in }[0,T);$

this implies that
$\operatorname{supp}(\mu_{t})\subset\mathrm{B}_{\mathsf{X}}((r_{0}+1)\mathrm{e}^{bt})$
so that the limit measure $\mu_{T}$ belongs to $\mathcal{P}_{b}(\mathsf{X})$
as well, leading to a contradiction with (6.16) for
${\boldsymbol{\mathrm{F}}}_{b}$.

We deduce that $\mu$ is a global strict $\lambda$-EVI solution for
${\boldsymbol{\mathrm{F}}}_{b}$. We can then apply Theorem 6.22(b) to
${\boldsymbol{\mathrm{F}}}_{b}$. ∎

###  6.6. A few borderline examples

We conclude this section with a few examples which reveal the importance of
some of the technical tools we developed so far. First of all we exhibit an
example of dissipative MPVF generating a $0$-flow, for which solutions
starting from initial data are merely continuous (in particular the nice
regularizing effect of gradient flows does not hold for general dissipative
evolutions). This clarifies the interest in a definition of continuous, not
necessarily absolutely continuous, solution.

######  Example 6.32 (Lifting of dissipative evolutions and lack of
regularizing effect).

Let us consider the situation of Corollary 6.24, choosing the Hilbert space
$\mathsf{X}=\ell^{2}(\mathbb{N})$. Following [Rul96, Example 3] we can easily
find a maximal linear dissipative operator
$A:\mathrm{D}(A)\subset\ell^{2}(\mathbb{N})\to\ell^{2}(\mathbb{N})$ whose
semigroup does not provide a regularizing effect.

The domain of $A$ is
$\mathrm{D}(A):=\\{x\in\ell^{2}(\mathbb{N}):\sum_{k=1}^{\infty}k^{2}|x_{k}|^{2}<\infty\\}$
and $A$ is defined as

$A(x_{1},x_{2},\dots,x_{2k-1},x_{2k},\dots)=(-x_{2},x_{1},\dots,-kx_{2k},kx_{2k-1},\dots),\quad
x\in\mathrm{D}(A),$

so that there is no regularizing effect for the semigroup $(R_{t})_{t\geq 0}$
generated by (the graph of) $A$: evolutions starting outside the domain
$\mathrm{D}(A)$ stay outside the domain and do not give raise to locally
Lipschitz or a.e. differentiable curves. Corollary 6.24 shows that the
$0$-flow $(S_{t})_{t\geq 0}$ generated by ${\boldsymbol{\mathrm{F}}}$ on
$\mathcal{P}_{2}(X)$ is given by

$S_{t}[\mu_{0}]=(R_{t})_{\sharp}\mu_{0}\quad\text{ for every
}\mu_{0}\in\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})}=\mathcal{P}_{2}(\mathsf{X})$

so that there is the same lack of regularizing effect on probability measures.

In the next example we show that a constant MPVF generates a barycentric
solution.

######  Example 6.33 (Constant PVF and barycentric evolutions).

Given $\theta\in\mathcal{P}_{2}(\mathsf{X})$, we consider the constant PVF

${\boldsymbol{\mathrm{F}}}[\mu]:=\mu\otimes\theta.$

${\boldsymbol{\mathrm{F}}}$ is dissipative: in fact, if
$\Phi_{i}=\mu_{i}\otimes\theta$, $i=0,1$,
$\boldsymbol{\mu}\in\Gamma_{o}(\mu_{0},\mu_{1})$, and
$\boldsymbol{r}:\mathsf{X}\times\mathsf{X}\times
X\to\mathsf{T\kern-1.5ptX}\times\mathsf{T\kern-1.5ptX}$ is defined by
$\boldsymbol{r}(x_{0},x_{1},v):=(x_{0},v;x_{1},v)$, then

$\Theta=\boldsymbol{r}_{\sharp}(\boldsymbol{\mu}\otimes\theta)\in\Lambda(\Phi_{0},\Phi_{1})$

so that (3.16) yields

$\left[\Phi_{0},\Phi_{1}\right]_{r}\leq\int\langle
x_{0}-x_{1},v-v\rangle\,\mathrm{d}(\boldsymbol{\mu}\otimes\theta)(x_{0},x_{1},v)=0.$

Applying Proposition 6.20 and Theorem 6.19 we immediately see that
${\boldsymbol{\mathrm{F}}}$ generates a $0$-flow $(\mathrm{S}_{t})_{t\geq 0}$
in $\mathcal{P}_{2}(\mathsf{X})$, obtained as a limit of the Explicit Euler
scheme. It is also straightforward to notice that we can apply Theorem 6.27 to
${\boldsymbol{\mathrm{F}}}$ so that for every
$\mu_{0}\in\mathcal{P}_{2}(\mathsf{X})$ the unique EVI solution
$\mu_{t}=\mathrm{S}_{t}\mu_{0}$ satisfies the continuity equation

$\partial_{t}\mu_{t}+\nabla\cdot(\boldsymbol{b}\mu_{t})=0,\quad\boldsymbol{b}=\int_{\mathsf{X}}x\,\mathrm{d}\theta(x).$

Since $\boldsymbol{b}$ is constant, we deduce that $\mathrm{S}_{t}$ acts as a
translation with constant velocity $\boldsymbol{b}$, i.e.

$\mu_{t}=(\boldsymbol{i}_{\mathsf{X}}+t\boldsymbol{b})_{\sharp}\mu_{0},$

so that $\mathrm{S}_{t}$ coincides with the semigroup generated by the PVF
${\boldsymbol{\mathrm{F}}}^{\prime}[\mu]:=(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{b})_{\sharp}\mu$.

We conclude this section with a $1$-dimensional example of a curve which
satisfies the barycentric property but it is not an EVI solution.

######  Example 6.34.

Let $\mathsf{X}=\mathbb{R}$. It is well known (see e.g. [NS09]) that
$\mathcal{P}_{2}(\mathbb{R})$ is isometric to the closed convex subset
$\mathcal{K}\subset L^{2}(0,1)$ of the (essentially) increasing maps and the
isometry $\mathcal{J}:\mathcal{P}_{2}(\mathbb{R})\to\mathcal{K}$ maps each
measure $\mu\in\mathcal{P}_{2}(\mathbb{R})$ into the pseudo inverse of its
cumulative distribution function.

It follows that for every $\bar{\nu}\in\mathcal{P}_{2}(\mathbb{R})$ the
functional $\mathcal{F}:\mathcal{P}_{2}(\mathbb{R})\to\mathbb{R}$ defined as

$\mathcal{F}(\mu):=\frac{1}{2}W_{2}^{2}(\mu,\bar{\nu})$

is $1$-convex, since it satisfies
$\mathcal{F}(\mu)=\mathcal{G}(\mathcal{J}(\mu))$ where
$\mathcal{G}:L^{2}(0,1)\to\mathbb{R}$ is defined as

$\mathcal{G}(u):=\frac{1}{2}\|u-\mathcal{J}(\bar{\nu})\|^{2}\quad\text{for
every }u\in L^{2}(0,1).$

Thus $\mathcal{F}$ generates a gradient flow $(\mathrm{S}_{t})_{t\geq 0}$
which is a semigroup of contractions in $\mathcal{P}_{2}(\mathbb{R})$; for
every $\mu_{0}\in\mathcal{P}_{2}(\mathbb{R})$ $\mathrm{S}_{t}[\mu_{0}]$ is the
unique $(-1)$-EVI solution for the MPVF $-\boldsymbol{\partial}\mathcal{F}$
starting from $\mu_{0}\in\mathcal{P}_{2}(\mathsf{X})$ (see Proposition 6.5).
Since the notion of gradient flow is purely metric, the gradient flow of
$\mathcal{G}$ starting from $\mathcal{J}(\mu_{0})$ is just the image through
$\mathcal{J}$ of the gradient flow of $\mathcal{F}$ starting from
$\mu_{0}\in\mathcal{P}_{2}(\mathsf{X})$. It is easy to check that

$u(t):=\mathrm{e}^{-t}\mathcal{J}(\mu_{0})+(1-\mathrm{e}^{-t})\mathcal{J}(\bar{\nu})$

is the gradient flow of $\mathcal{G}$ starting from
$u_{0}=\mathcal{J}(\mu_{0})$. Note that $u(t)$ is the $L^{2}(0,1)$ geodesic
from $\mathcal{J}(\bar{\nu})$ to $\mathcal{J}(\mu_{0})$ evaluated at the
rescaled time $e^{-t}$, so that $\mathrm{S}_{t}[\mu_{0}]$ must coincide with
the evaluation at time $\mathrm{e}^{-t}$ of the (unique) geodesic connecting
$\bar{\nu}$ to $\mu_{0}$ i.e.

$\mathrm{S}_{t}[\mu_{0}]=\mathsf{x}^{s}_{\sharp}\boldsymbol{\gamma},\quad
s=\mathrm{e}^{-t}\in(0,1],$

where $\boldsymbol{\gamma}\in\Gamma_{o}(\bar{\nu},\mu_{0})$.

Let us now consider the particular case
$\bar{\nu}=\frac{1}{2}\delta_{-a}+\frac{1}{2}\delta_{a}$, where $a>0$ is a
fixed parameter and $\mu_{0}=\delta_{0}$. It is straightforward to see that

$\mu_{t}=\mathrm{S}_{t}[\delta_{0}]=\frac{1}{2}\delta_{a(1-\mathrm{e}^{-t})}+\frac{1}{2}\delta_{a(\mathrm{e}^{-t}-1)},\quad
t\geq 0$

so that

$(\boldsymbol{i}_{\mathsf{X}},\boldsymbol{v}_{t})_{\sharp}\mu_{t}=\frac{1}{2}\delta_{((1-e^{-t})a,e^{-t}a)}+\frac{1}{2}\delta_{((e^{-t}-1)a,-e^{-t}a)}\in-\boldsymbol{\partial}\mathcal{F}(\mu_{t}),\quad\text{
a.e. }t>0,$

where $\boldsymbol{v}$ is the Wasserstein velocity field of $\mu_{t}$. On the
other hand, [AGS08, Lemma 10.3.8] shows that

$\delta_{0}\otimes\left(\frac{1}{2}\delta_{-a}+\frac{1}{2}\delta_{a}\right)\in-\boldsymbol{\partial}\mathcal{F}(\delta_{0})$

so that the constant curve $\bar{\mu}_{t}:=\delta_{0}$ for $t\geq 0$ has the
barycentric property for the MPVF $-\boldsymbol{\partial}\mathcal{F}$ but it
is not a EVI solution for $-\boldsymbol{\partial}\mathcal{F}$, being different
from $\mu_{t}=\mathrm{S}_{t}[\delta_{0}]$.

##  7\. Explicit Euler Scheme

In this section, we collect all the main estimates concerning the Explicit
Euler scheme (EE).

###  7.1. The Explicit Euler Scheme: preliminary estimates

Our first step is to prove simple a priori estimates and a discrete version of
($\lambda$-EVI) as a consequence of Proposition 3.4.

######  Proposition 7.1.

Every solution
$(M_{\tau},{\boldsymbol{F}}_{\tau})\in\mathscr{E}(\mu_{0},\tau,T,L)$ of (EE)
satisfies

$W_{2}(M_{\tau}(t),\mu_{0})\leq Lt,\quad|{\boldsymbol{F}}_{\tau}(t)|_{2}\leq
L\quad\text{for every $t\in[0,T]$,}$ (7.1) $W_{2}(M_{\tau}(t),M_{\tau}(s))\leq
L|t-s|\quad\text{for every }s,t\in[0,T],$ (7.2)
$\frac{\,\mathrm{d}}{\,\mathrm{d}t}\frac{1}{2}W_{2}^{2}(M_{\tau}(t),\nu)\leq\left[{\boldsymbol{F}}_{\tau}(t),\nu\right]_{r}+\tau|{\boldsymbol{F}}_{\tau}(t)|_{2}^{2}\leq\left[{\boldsymbol{F}}_{\tau}(t),\nu\right]_{r}+\tau
L^{2}\quad\text{in $[0,T]$, }\forall\nu\in\mathcal{P}_{2}(\mathsf{X}),$ (IEVI)

with possibly countable exceptions. In particular

$\frac{1}{2}W_{2}^{2}(M_{\tau}^{n+1},\nu)-\frac{1}{2}W_{2}^{2}(M_{\tau}^{n},\nu)\leq\tau\left[{\boldsymbol{F}}_{\tau}^{n},\nu\right]_{r}+\frac{1}{2}{\tau^{2}}L^{2}\quad\text{for
every }0\leq n<{\mathrm{N}(T,\tau)},\forall\nu\in\mathcal{P}_{2}(\mathsf{X}).$
(7.3)

###### Proof.

The second inequality of (7.1) is a trivial consequence of the definition of
$\mathscr{E}(\mu_{0},\tau,T,L)$, the first inequality is a particular case of
(7.2). The estimate (7.2) is immediate if $n\tau\leq s<t\leq(n+1)\tau$ since

$\displaystyle W_{2}(M_{\tau}(s),M_{\tau}(t))$
$\displaystyle=W_{2}((\operatorname{\mathsf{exp}}^{s-n\tau})_{\sharp}{\boldsymbol{F}}_{\tau}^{n},(\operatorname{\mathsf{exp}}^{t-n\tau})_{\sharp}{\boldsymbol{F}}_{\tau}^{n})\leq\sqrt{\int_{\mathsf{T\kern-1.5ptX}}|(t-s)v)|^{2}\,\mathrm{d}{\boldsymbol{F}}_{\tau}^{n}}$
$\displaystyle=(t-s)\sqrt{\int_{\mathsf{T\kern-1.5ptX}}|v|^{2}\,\mathrm{d}{\boldsymbol{F}}_{\tau}^{n}}\leq(t-s)L.$

This implies that the metric velocity of $M_{\tau}$ is bounded by $L$ in
$[0,T]$ and therefore $M_{\tau}$ is $L$-Lipschitz.

Let us recall that for every $\nu\in\mathcal{P}_{2}(\mathsf{X})$ and
$\Phi\in\mathcal{P}_{2}(\mathsf{T\kern-1.5ptX})$ the function
$g(t):=\frac{1}{2}W_{2}^{2}(\operatorname{\mathsf{exp}}^{t}_{\sharp}\Phi,\nu)$
satisfies

$t\mapsto g(t)-\frac{1}{2}t^{2}|\Phi|_{2}^{2}\text{ is concave},\quad
g^{\prime}_{r}(0)=\left[\Phi,\nu\right]_{r},\quad
g^{\prime}(t)\leq\left[\Phi,\nu\right]_{r}+t|\Phi|_{2}^{2}\quad t\geq 0,$
(7.4)

by Definition 3.5 and Proposition 3.4. In particular, the concavity yields the
differentiability of $g$ with at most countable exceptions. Thus, taking any
$n\in\mathbb{N}$, $0\leq n<{\mathrm{N}(T,\tau)}$, $t\in[n\tau,(n+1)\tau)$ and
$\Phi={\boldsymbol{F}}_{\tau}^{n}$ so that
$\operatorname{\mathsf{exp}}^{t}_{\sharp}\Phi=M_{\tau}(t)$, (7.4) yields
(IEVI). (7.3) follows by integration in each interval $[n\tau,(n+1)\tau]$. ∎

In the following, we prove a uniform bound on curves
$M_{\tau}\in\mathscr{M}(\mu_{0},\tau,T,L)$ which is useful to prove global
solvability of the Explicit Euler scheme, as stated in Proposition 6.20. We
will use the following discrete Gronwall estimate: if a sequence
$(x_{n})_{n\in\mathbb{N}}$ of positive real numbers satisfies

$x_{n+1}-x_{n}\leq\tau y+\tau\alpha x_{n},\quad 1\leq n\leq N,\,\alpha\geq
0,\,y\geq 0,\,\tau>0,$

then

$x_{n}\leq(x_{0}+\tau ny)\mathrm{e}^{\alpha n\tau}\quad 0\leq n\leq N+1.$
(7.5)

######  Proposition 7.2.

Let ${\boldsymbol{\mathrm{F}}}$ be a $\lambda$-dissipative MPVF such that for
every $R>0$ there exist $M=\mathrm{M}(R)>0$ and $\bar{\tau}=\bar{\tau}(R)>0$
such that

$\mu\in\mathrm{D}({\boldsymbol{\mathrm{F}}}),\ \mathsf{m}_{2}(\mu)\leq R,\
0<\tau\leq\bar{\tau}\quad\Rightarrow\quad\exists\,\Phi\in{\boldsymbol{\mathrm{F}}}[\mu]:|\Phi|_{2}\leq\mathrm{M}(R),\
\operatorname{\mathsf{exp}}^{\tau}_{\sharp}\Phi\in\mathrm{D}({\boldsymbol{\mathrm{F}}}),$
(7.6)

then the Explicit Euler scheme is globally solvable in
$\mathrm{D}({\boldsymbol{\mathrm{F}}})$. More precisely, if for a given
$\mu_{0}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$ with
$\Psi_{0}\in{\boldsymbol{\mathrm{F}}}[\mu_{0}]$,
$\mathsf{m}_{0}:=\mathsf{m}_{2}(\mu_{0}),$ and we set

$R:=\mathsf{m}_{0}+\Big{(}|\Psi_{0}|_{2}+1\Big{)}\sqrt{2T}\mathrm{e}^{(1+2\lambda_{+})T},\quad
L:=\mathrm{M}(R),\quad\boldsymbol{\tau}=\frac{1}{L^{2}}\land\bar{\tau}(R)\wedge
T,$ (7.7)

then for every $\tau\in(0,\boldsymbol{\tau}]$ the set
$\mathscr{E}(\mu_{0},\tau,T,L)$ is not empty.

###### Proof.

We want to prove by induction that for every integer
$N\leq{\mathrm{N}(T,\tau)}$, (EE) has a solution up to the index $N$
satisfying the upper bound

$\mathsf{m}_{2}(M^{N}_{\tau})\leq R,$ (7.8)

corresponding to the constants $R,L$ given by (7.7). For $N=0$ the statement
is trivially satisfied. Assuming that $0\leq N<{\mathrm{N}(T,\tau)}$ and
elements $(M^{n}_{\tau},{\boldsymbol{\mathrm{F}}}^{n}_{\tau})$, $0\leq n<N$,
$M^{N}_{\tau}$, are given satisfying (EE) and (7.8), we want to show that we
can perform a further step of the Euler Scheme so that (EE) is solvable up to
the index $N+1$ and $\mathsf{m}_{2}(M^{N+1}_{\tau})\leq R$.

Notice that by the induction hypothesis, for $n=0,\dots,N-1$, we have
$|{\boldsymbol{F}}^{n}_{\tau}|_{2}\leq L$; since
$\mathsf{m}_{2}(M^{N}_{\tau})\leq R$, by (7.6) we can select
${\boldsymbol{F}}^{N}_{\tau}\in{\boldsymbol{\mathrm{F}}}[M^{N}_{\tau}]$ with
$|{\boldsymbol{F}}^{N}_{\tau}|_{2}\leq L$ such that
$M^{N+1}_{\tau}=\operatorname{\mathsf{exp}}^{\tau}_{\sharp}{\boldsymbol{F}}^{N}_{\tau}\in\mathrm{D}({\boldsymbol{\mathrm{F}}})$.
Using (7.3) with $\nu=\mu_{0}$, the $\lambda$-dissipativity with
$\Psi_{0}\in{\boldsymbol{\mathrm{F}}}[\mu_{0}]$

$\left[{\boldsymbol{F}}_{\tau}^{n},\mu_{0}\right]_{r}\leq\lambda
W_{2}^{2}(M_{\tau}^{n},\mu_{0})-\left[\Psi_{0},M_{\tau}^{n}\right]_{r},$

and the bound

$-\left[\Psi_{0},M_{\tau}^{n}\right]_{r}\leq\frac{1}{2}W_{2}^{2}(M_{\tau}^{n},\mu_{0})+\frac{1}{2}|\Psi_{0}|_{2}^{2},$

we end up with

$\frac{1}{2}W_{2}^{2}(M_{\tau}^{n+1},\mu_{0})-\frac{1}{2}W_{2}^{2}(M_{\tau}^{n},\mu_{0})\leq\frac{\tau^{2}}{2}L^{2}+\tau\left(\frac{1}{2}+\lambda_{+}\right)\,W_{2}^{2}(M_{\tau}^{n},\mu_{0})+\frac{\tau}{2}|\Psi_{0}|_{2}^{2},$

for every $n\leq N$. Using the Gronwall estimate (7.5) we get

$\displaystyle W_{2}(M_{\tau}^{n},\mu_{0})$
$\displaystyle\leq\sqrt{T+\tau}\Big{(}|\Psi_{0}|_{2}+\sqrt{\tau}L\Big{)}\mathrm{e}^{(\frac{1}{2}+\lambda_{+})\,(T+\tau)}\leq\sqrt{2T}\Big{(}|\Psi_{0}|_{2}+1\Big{)}\mathrm{e}^{(1+2\lambda_{+})T}$

for every $n\leq N+1$, so that

$\mathsf{m}_{2}(M^{N+1}_{\tau})\leq\mathsf{m}_{0}+\sqrt{2T}\Big{(}|\Psi_{0}|_{2}+1\Big{)}\mathrm{e}^{(1+2\lambda_{+})T}\leq
R.\qed$

We conclude this section by proving the stability estimate (6.12) of Theorem
6.9. We introduce the notation

$I_{\kappa}(t):=\int_{0}^{t}\mathrm{e}^{\kappa
r}\,\mathrm{d}r=\frac{1}{\kappa}(\mathrm{e}^{\kappa t}-1)\quad\text{if
}\kappa\neq 0;\quad I_{0}(t):=t.$

Notice that for every $t\geq 0$

$I_{\kappa}(t)\leq t\mathrm{e}^{\kappa t}\quad\text{if }\kappa\geq 0;\qquad$
(7.9)

######  Proposition 7.3.

Let $M_{\tau}\in\mathscr{M}(\mu_{0},\tau,T,L)$ and
$M_{\tau}^{\prime}\in\mathscr{M}(\mu_{0}^{\prime},\tau,T,L)$. If
$\lambda_{+}\tau\leq 2$ then

$W_{2}(M_{\tau}(t),M_{\tau}^{\prime}(t))\leq
W_{2}(\mu_{0},\mu_{0}^{\prime})\mathrm{e}^{\lambda
t}+8L\sqrt{t\tau}\Big{(}1+|\lambda|\sqrt{t\tau}\Big{)}\mathrm{e}^{\lambda_{+}t}$

for every $t\in[0,T]$.

###### Proof.

Let us set $w(t):=W_{2}(M_{\tau}(t),M_{\tau}^{\prime}(t))$. Since by
Proposition 3.4(2), in every interval $[n\tau,(n+1)\tau]$ the function
$t\mapsto w^{2}(t)-4L^{2}(t-n\tau)^{2}$ is concave, with

$\frac{\mathrm{d}}{\mathrm{d}t}w^{2}(t)\bigg{|}_{t=n\tau+}=2\left[{\boldsymbol{F}}_{\tau}(t),{\boldsymbol{F}}_{\tau}^{\prime}(t)\right]_{r}\leq
2\lambda W_{2}^{2}(\bar{M}_{\tau}(t),\bar{M}_{\tau}^{\prime}(t)),$

we obtain

$\frac{\mathrm{d}}{\mathrm{d}t}w^{2}(t)\leq 2\lambda
W_{2}^{2}(\bar{M}_{\tau}(t),\bar{M}_{\tau}^{\prime}(t))+8L^{2}\tau\quad
t\in[0,T],$

with possibly countable exceptions. Using the identity
$a^{2}-b^{2}=2b(a-b)+|a-b|^{2}$ with
$a=W_{2}(\bar{M}_{\tau}(t),\bar{M}_{\tau}^{\prime}(t))$ and
$b=W_{2}(M_{\tau}(t),M_{\tau}^{\prime}(t))$ and observing that $|a-b|\leq
W_{2}(\bar{M}_{\tau}(t),M_{\tau}(t))+W_{2}(\bar{M}_{\tau}^{\prime}(t),M_{\tau}^{\prime}(t))\leq
2L\tau$, we eventually get

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}w^{2}(t)$ $\displaystyle\leq
2\lambda w^{2}(t)+8L^{2}\tau+8|\lambda|L\tau w(t)+\lambda_{+}8L^{2}\tau^{2}$
$\displaystyle\leq 2\lambda w^{2}(t)+8|\lambda|L\tau w(t)+24L^{2}\tau,$

since $\lambda_{+}\tau\leq 2$ by assumption. The Gronwall lemma [AGS08, Lemma
4.1.8] and (7.9) yield

$\displaystyle w(t)$ $\displaystyle\leq\Big{(}w^{2}(0)\mathrm{e}^{2\lambda
t}+24L^{2}\tau\mathrm{I}_{2\lambda}(t)\Big{)}^{1/2}+8|\lambda|L\tau\mathrm{I}_{\lambda}(t)$
$\displaystyle\leq w(0)\mathrm{e}^{\lambda
t}+8L\sqrt{t\tau}\Big{(}1+|\lambda|\sqrt{t\tau}\Big{)}\mathrm{e}^{\lambda_{+}t}.\qed$

###  7.2. Error estimates for the Explicit Euler scheme

######  Theorem 7.4.

Let ${\boldsymbol{\mathrm{F}}}$ be a $\lambda$-dissipative MPVF. If
$M_{\tau}\in\mathscr{M}(M^{0}_{\tau},\tau,T,L)$,
$M_{\eta}\in\mathscr{M}(M^{0}_{\eta},\eta,T,L)$ with
$\lambda\sqrt{T(\tau+\eta)}\leq 1$, then for every $\vartheta>1$ there exists
a constant $C(\vartheta)$ such that

$W_{2}(M_{\tau}(t),M_{\eta}(t))\leq\Big{(}\sqrt{\vartheta}W_{2}(M^{0}_{\tau},M^{0}_{\eta})+C(\vartheta)L\sqrt{(\tau+\eta)(t+\tau+\eta)}\Big{)}\mathrm{e}^{\lambda_{+}\,t}$

for every $t\in[0,T]$.

###### Proof.

We argue as in the proof of Theorem 6.17 with the aim to gain a convenient
order of convergence. Since $\lambda$-dissipativity implies
$\lambda^{\prime}$-dissipativity for $\lambda^{\prime}\geq\lambda$, it is not
restrictive to assume $\lambda>0$. We set $\sigma:=\tau+\eta$. We will
extensively use the a priori bounds (7.1) and (7.2); in particular,

$W_{2}(M_{\tau}(t),\bar{M}_{\tau}(t))\leq L\tau,\quad
W_{2}(M_{\eta}(t),\bar{M}_{\eta}(t))\leq L\eta.$

We will also extend $M_{\tau}$ and $\bar{M}_{\tau}$ for negative times by
setting

$M_{\tau}(t)=\bar{M}_{\tau}(t)=M^{0}_{\tau},\quad{\boldsymbol{\mathrm{F}}}_{\tau}(t)=M^{0}_{\tau}\otimes\delta_{0}\quad\text{if
}t<0.$ (7.10)

The proof is divided into several steps.

_1\. Doubling variables._

We fix a final time $t\in[0,T]$ and two variables $r,s\in[0,t]$ together with
the functions

$\displaystyle w(r,s):={}$ $\displaystyle
W_{2}(M_{\tau}(r),M_{\eta}(s)),\quad$ $\displaystyle w_{\tau}(r,s):={}$
$\displaystyle W_{2}(\bar{M}_{\tau}(r),M_{\eta}(s)),$ (7.11) $\displaystyle
w_{\eta}(r,s):={}$ $\displaystyle W_{2}(M_{\tau}(r),\bar{M}_{\eta}(s)),\quad$
$\displaystyle w_{\tau,\eta}(r,s):={}$ $\displaystyle
W_{2}(\bar{M}_{\tau}(r),\bar{M}_{\eta}(s)),$

observing that

$|w-w_{\tau}|\lor|w_{\eta}-w_{\tau,\eta}|\leq
L\tau,\quad|w-w_{\eta}|\lor|w_{\tau}-w_{\tau,\eta}|\leq L\eta.$ (7.12)

By Proposition 7.1, we can write (IEVI) both for $M_{\tau}$ and $M_{\eta}$ and
we obtain

$\displaystyle\frac{\partial}{\partial
r}\frac{1}{2}W_{2}^{2}(M_{\tau}(r),\nu_{1})$
$\displaystyle\leq\tau|{\boldsymbol{F}}_{\tau}(r)|_{2}^{2}+\left[{\boldsymbol{F}}_{\tau}(r),\nu_{1}\right]_{r}\quad\forall\nu_{1}\in\mathcal{P}_{2}(\mathsf{X})$
($\text{IEVI}_{\tau}$) $\displaystyle\frac{\partial}{\partial
s}\frac{1}{2}W_{2}^{2}(M_{\eta}(s),\nu_{2})$
$\displaystyle\leq\eta|{\boldsymbol{F}}_{\eta}(s)|_{2}^{2}+\left[{\boldsymbol{F}}_{\eta}(s),\nu_{2}\right]_{r}$
$\displaystyle\leq\eta|{\boldsymbol{F}}_{\eta}(s)|_{2}^{2}+\lambda
W_{2}^{2}(\bar{M}_{\eta}(s),\nu_{2})-\left[\Phi,\bar{M}_{\eta}(s)\right]_{r},\quad\forall\nu_{2}\in\mathrm{D}({\boldsymbol{\mathrm{F}}}),\,\Phi\in{\boldsymbol{\mathrm{F}}}[\nu_{2}].$
($\text{IEVI}_{\eta}$)

Apart from possible countable exceptions, ($\text{IEVI}_{\tau}$) holds for
$r\in(-\infty,t]$ and ($\text{IEVI}_{\eta}$) for $s\in[0,t]$. Taking
$\nu_{1}=\bar{M}_{\eta}(s)$, $\nu_{2}=\bar{M}_{\tau}(r)$,
$\Phi={\boldsymbol{F}}_{\tau}(r\lor
0)\in{\boldsymbol{\mathrm{F}}}[\bar{M}_{\tau}(r)]$, summing the two
inequalities $(\text{IEVI}_{\tau,\eta})$, setting

$f(r,s):=\begin{cases}2LW_{2}(\bar{M}_{\eta}(s),M_{\tau}(0))=2Lw_{\eta}(0,s)&\text{if
}r<0,\\\ 0&\text{if }r\geq 0,\end{cases}$

using (7.1) and the $\lambda$-dissipativity of ${\boldsymbol{\mathrm{F}}}$, we
obtain

$\frac{\partial}{\partial r}w_{\eta}^{2}(r,s)+\frac{\partial}{\partial
s}w_{\tau}^{2}(r,s)\leq 2\lambda w_{\tau,\eta}^{2}(r,s)+2L^{2}\sigma+f(r,s)$

in $(-\infty,t]\times[0,t]$ (see also [NS06, Lemma 6.15]). By multiplying both
sides by $e^{-2\lambda s}$, we have

$\begin{split}&\frac{\partial}{\partial r}\mathrm{e}^{-2\lambda
s}w_{\eta}^{2}+\frac{\partial}{\partial s}\mathrm{e}^{-2\lambda
s}w_{\tau}^{2}\leq\Big{(}2\lambda\left(w_{\tau,\eta}^{2}-w_{\tau}^{2}\right)+f+2L^{2}\sigma\Big{)}\mathrm{e}^{-2\lambda
s}.\end{split}$ (7.13)

Using (7.12), the inequality

$w_{\tau,\eta}+w_{\tau}=w_{\tau,\eta}-w_{\tau}+2(w_{\tau}-w)+2w\leq
2L\sigma+2w,\quad|w(r,s)-w(s,s)|\leq L|r-s|$

and the elementary inequality $a^{2}-b^{2}\leq|a-b||a+b|,$ we get

$2\big{(}w_{\tau,\eta}^{2}(r,s)-w_{\tau}^{2}(r,s)\big{)}\leq R,\quad
R:=4L^{2}\sigma(\sigma+|r-s|)+4L\sigma w(s,s)\quad\text{if }r,s\leq t.$

Thus (7.13) becomes

$\frac{\partial}{\partial r}\mathrm{e}^{-2\lambda
s}w_{\eta}^{2}+\frac{\partial}{\partial s}\mathrm{e}^{-2\lambda
s}w_{\tau}^{2}\leq Z,\quad
Z:=\Big{(}R\lambda+f+2L^{2}\sigma\Big{)}\mathrm{e}^{-2\lambda s}.$ (7.14)

_2\. Penalization._

We fix any $\varepsilon>0$ and apply the Divergence Theorem to the inequality
(7.14) in the two-dimensional strip $Q^{\varepsilon}_{0,t}$ as in Figure 1 and
we get

$\displaystyle\begin{split}\int_{t-\varepsilon}^{t}&\mathrm{e}^{-2\lambda
t}w_{\tau}^{2}(r,t)\,\mathrm{d}r\leq\int_{-\varepsilon}^{0}w^{2}_{\tau}(r,0)\,\mathrm{d}r+\\\
&+\int_{0}^{t}\mathrm{e}^{-2\lambda
s}\left(w_{\tau}^{2}(s,s)-w_{\eta}^{2}(s,s)\right)\,\mathrm{d}s+\int_{0}^{t}\mathrm{e}^{-2\lambda
s}\left(w_{\eta}^{2}(s-\varepsilon,s)-w_{\tau}^{2}(s-\varepsilon,s)\right)\,\mathrm{d}s\\\
&+\iint_{Q_{0,t}^{\varepsilon}}Z\,\mathrm{d}r\mathrm{d}s.\end{split}$ (7.15)

_3\. Estimates of the RHS._

We want to estimate the integrals (say $I_{0},I_{1},I_{2},I_{3})$ of the right
hand side of (7.15) in terms of

$w(s):=w(s,s)\quad\text{and}\quad W(t):=\sup_{0\leq s\leq
t}\mathrm{e}^{-\lambda s}w(s).$

We easily get

$I_{0}=\int_{-\varepsilon}^{0}w_{\tau}^{2}(r,0)\,\mathrm{d}r=\varepsilon
w^{2}(0).$

(7.12) yields

$\displaystyle|w_{\tau}(s,s)-w_{\eta}(s,s)|\leq L(\tau+\eta)=L\sigma$

and

$|w^{2}_{\tau}(s,s)-w^{2}_{\eta}(s,s)|\leq
L\sigma\Big{(}L\sigma+2w(s)\Big{)};$

after an integration,

$\displaystyle I_{1}$
$\displaystyle\leq{L^{2}\sigma^{2}t}+2L\sigma\int_{0}^{t}e^{-2\lambda
s}w(s)\,\mathrm{d}s\leq L^{2}\sigma^{2}t+2L\sigma tW(t).$

Performing the same computations for the third integral term at the RHS of
(7.15) we end up with

$\begin{split}I_{2}&=\int_{0}^{t}e^{-2\lambda
s}\left(w_{\eta}^{2}(s-\varepsilon,s)-w_{\tau}^{2}(s-\varepsilon,s)\right)\,\mathrm{d}s\leq{L^{2}t\sigma^{2}}+2L\sigma\int_{0}^{t}e^{-2\lambda
s}w(s-\varepsilon,s)\,\mathrm{d}s\\\ &\leq
L^{2}\sigma^{2}t+2L^{2}\sigma\varepsilon t+2L\sigma\int_{0}^{t}e^{-2\lambda
s}w(s)\,\mathrm{d}s\leq L^{2}\sigma^{2}t+2L^{2}\sigma\varepsilon t+2L\sigma
tW(t).\end{split}$

Eventually, using the elementary inequalities,

$\iint_{Q^{\varepsilon}_{0,t}}\lambda\mathrm{e}^{-2\lambda
s}\,\mathrm{d}r\,\mathrm{d}s\leq\frac{\varepsilon}{2},\quad\iint_{Q^{\varepsilon}_{0,t}}\mathrm{e}^{-2\lambda
s}w(s,s)\,\mathrm{d}r\,\mathrm{d}s=\varepsilon\int_{0}^{t}\mathrm{e}^{-2\lambda
s}w(s)\,\mathrm{d}s,$

and $f(r,s)\leq 2L^{2}(\eta+s)+2Lw(s)$ for $r<0$ and $f(r,s)=0$ for $r\geq 0$,
we get

$\displaystyle I_{3}$
$\displaystyle=\iint_{Q^{\varepsilon}_{0,t}}Z\,\mathrm{d}r\mathrm{d}s\leq
2L^{2}\sigma\varepsilon(\sigma+\varepsilon)+4L\lambda\sigma\varepsilon\int_{0}^{t}\mathrm{e}^{-2\lambda
s}w(s)\,\mathrm{d}s+2L^{2}\sigma\varepsilon t$
$\displaystyle\qquad+2\iint_{Q^{\varepsilon}_{0,\varepsilon\wedge
t}}(L^{2}(\eta+s)+Lw(s))\mathrm{e}^{-2\lambda s}\,\mathrm{d}r\mathrm{d}s$
$\displaystyle\leq
2L^{2}\sigma\varepsilon(\sigma+\varepsilon)+2L^{2}\varepsilon^{2}(\sigma+\varepsilon)+2L^{2}\sigma\varepsilon
t+4L\lambda\sigma\varepsilon tW(t)+2L\varepsilon^{2}W(t\land\varepsilon).$

We eventually get

$\sum_{k=0}^{3}I_{k}\leq\varepsilon
w^{2}(0)+2L^{2}\sigma^{2}t+4L^{2}\sigma\varepsilon
t+2L^{2}\varepsilon(\sigma+\varepsilon)^{2}+4L\sigma(1+\lambda\varepsilon)tW(t)+2L\varepsilon^{2}W(t\land\varepsilon).$
(7.16)

_4\. LHS and penalization_

We want to use the first integral term in (7.15) to derive a pointwise
estimate for $w(t)$;

(7.2) and (7.11) yield

$\displaystyle w(t)=w(t,t)\leq L(t-r)+w(r,t)\leq L(\tau+|t-r|)+w_{\tau}(r,t)$

so that we get for every $\vartheta,\vartheta_{\star}>1$ conjugate
coefficients

$\mathrm{e}^{-2\lambda
t}w^{2}(t)\leq\frac{\vartheta}{\varepsilon}\int_{t-\varepsilon}^{t}e^{-2\lambda
t}w_{\tau}^{2}(r,t)\,\mathrm{d}r+\vartheta_{\star}L^{2}(\tau+\varepsilon)^{2}\leq\frac{\vartheta}{\varepsilon}(I_{0}+I_{1}+I_{2}+I_{3})+\vartheta_{\star}L^{2}(\tau+\varepsilon)^{2}.$

(7.16) yields

$\displaystyle\begin{split}\mathrm{e}^{-2\lambda
t}w^{2}(t)\leq&(2\vartheta+\vartheta_{\star})L^{2}(\sigma+\varepsilon)^{2}+\vartheta\Big{(}w^{2}(0)+2L^{2}\sigma^{2}t/\varepsilon+4L^{2}\sigma
t\Big{)}\\\
&+\frac{4L(1+\lambda\varepsilon)\sigma\vartheta}{\varepsilon}tW(t)+2L\varepsilon\vartheta
W(t\land\varepsilon).\end{split}$

_5\. Conclusion._

Choosing $\varepsilon:=\sqrt{\sigma(\sigma\lor t)}$ and assuming
$\lambda\sqrt{T\sigma}\leq 1$, we obtain

$\displaystyle\mathrm{e}^{-2\lambda t}w^{2}(t)$ $\displaystyle\leq\vartheta
w^{2}(0)+(14\vartheta+4\vartheta_{\star})L^{2}\sigma(\sigma\lor t)+10\vartheta
L\sqrt{\sigma(\sigma\lor t)}W(t).$ (7.17)

Since the right hand side of (7.17) is an increasing function of $t$, (7.17)
holds even if we substitute the left hand side with $\mathrm{e}^{-2\lambda
s}w^{2}(s)$ for every $s\in[0,t]$; we thus obtain the inequality

$W^{2}(t)\leq\vartheta
w^{2}(0)+(14\vartheta+4\vartheta_{\star})L^{2}\sigma(\sigma\lor t)+10\vartheta
L\sqrt{\sigma(\sigma\lor t)}W(t).$

Using the elementary property for positive $a,b$

$W^{2}\leq a+2bW\quad\Rightarrow\quad W\leq b+\sqrt{b^{2}+a}\leq 2b+\sqrt{a},$
(7.18)

we eventually obtain

$\displaystyle\mathrm{e}^{-\lambda t}w(t)$ $\displaystyle\leq\Big{(}\vartheta
w^{2}(0)+(14\vartheta+4\vartheta_{\star})L^{2}\sigma(\sigma\lor
t)\Big{)}^{1/2}+10\vartheta L\sqrt{\sigma(\sigma\lor t)}$
$\displaystyle\leq\sqrt{\vartheta}w(0)+C(\vartheta)L\sqrt{\sigma(\sigma\lor
t)},\quad
C(\vartheta):=(14\vartheta+4\vartheta_{\star})^{1/2}+10\vartheta.\qed$

###  7.3. Error estimates between discrete and EVI solutions

######  Theorem 7.5.

Let ${\boldsymbol{\mathrm{F}}}$ be a $\lambda$-dissipative MPVF. If
$\mu\in\mathrm{C}([0,T];\overline{\mathrm{D}({\boldsymbol{\mathrm{F}}})})$ is
a $\lambda$-EVI solution and $M_{\tau}\in\mathscr{M}(M^{0}_{\tau},\tau,T,L)$,
then for every $\vartheta>1$ there exists a constant $C(\vartheta)$ such that

$W_{2}(\mu(t),M_{\tau}(t))\leq\Big{(}\sqrt{\vartheta}\,W_{2}(\mu_{0},M^{0}_{\tau})+C(\vartheta)L\sqrt{\tau(t+\tau)}\Big{)}\mathrm{e}^{\lambda_{+}t}\quad\text{for
every }t\in[0,T].$

######  Remark 7.6.

When $\mu_{0}=M^{0}_{\tau}$ and $\lambda\leq 0$ we obtain the optimal error
estimate

$W_{2}(\mu(t),M_{\tau}(t))\leq 13L\sqrt{\tau(t+\tau)}.$

###### Proof.

We repeat the same argument of the previous proof, still assuming $\lambda>0$,
extending $M_{\tau},\bar{M}_{\tau},\bar{\boldsymbol{F}}_{\tau}$ as in (7.10)
and setting

$w(r,s):=W_{2}(M_{\tau}(r),\mu(s)),\quad
w_{\tau}(r,s):=W_{2}(\bar{M}_{\tau}(r),\mu(s)).$

We use ($\lambda$-EVI) for $\mu(s)$ with $\nu=\bar{M}_{\tau}(r)$ and
$\Phi={\boldsymbol{\mathrm{F}}}_{\tau}(r\lor 0)$ and (IEVI) for $M_{\tau}(r)$
with $\nu=\mu(s)$ obtaining

$\displaystyle\frac{\partial}{\partial r}\frac{\mathrm{e}^{-2\lambda
s}}{2}W_{2}^{2}(M_{\tau}(r),\mu(s))$ $\displaystyle\leq\mathrm{e}^{-2\lambda
s}\Big{(}\tau|{\boldsymbol{F}}_{\tau}(r)|_{2}^{2}+\left[{\boldsymbol{F}}_{\tau}(r),\mu(s)\right]_{r}\Big{)}$
$\displaystyle s\in[0,T],r\in(-\infty,T)$
$\displaystyle\frac{\partial}{\partial s}\frac{\mathrm{e}^{-2\lambda
s}}{2}W_{2}^{2}(\mu(s),\bar{M}_{\tau}(r))$
$\displaystyle\leq-\mathrm{e}^{-2\lambda s}\left[{\boldsymbol{F}}_{\tau}(r\lor
0),\mu(s)\right]_{r}$ $\displaystyle\text{in $\mathscr{D}^{\prime}(0,T)$,
}r\in(-\infty,T).$

Using [NS06, Lemma 6.15] we can sum the two contributions obtaining

$\frac{\partial}{\partial r}\mathrm{e}^{-2\lambda
s}w^{2}(r,s)+\frac{\partial}{\partial s}\mathrm{e}^{-2\lambda
s}w_{\tau}^{2}(r,s)\leq Z,\quad Z:=(2L^{2}\tau+2f(r,s))\mathrm{e}^{-2\lambda
s},$

where

$f(r,s):=\begin{cases}LW_{2}(M_{\tau}(0),\mu(s))=Lw(0,s)&\text{if }r<0,\\\
0&\text{if }r\geq 0.\end{cases}$

Let $t\in[0,T]$ and $\varepsilon>0$. Applying the Divergence Theorem in
$Q_{0,t}^{\varepsilon}$ (see Figure 1) we get

$\displaystyle\begin{split}\int_{t-\varepsilon}^{t}&\mathrm{e}^{-2\lambda
t}w_{\tau}^{2}(r,t)\,\mathrm{d}r\leq\int_{-\varepsilon}^{0}w_{\tau}^{2}(r,0)\,\mathrm{d}r\\\
&+\int_{0}^{t}\mathrm{e}^{-2\lambda
s}\left(w_{\tau}^{2}(s,s)-w^{2}(s,s)\right)\,\mathrm{d}s+\int_{0}^{t}\mathrm{e}^{-2\lambda
s}\left(w^{2}(s-\varepsilon,s)-w_{\tau}^{2}(s-\varepsilon,s)\right)\,\mathrm{d}s\\\
&+\iint_{Q_{0,t}^{\varepsilon}}Z\,\mathrm{d}r\mathrm{d}s.\end{split}$ (7.19)

Using

$w(t,t)\leq w(r,t)+L(t-r)\leq w_{\tau}(r,t)+L(\tau+\varepsilon)\quad\text{if
}t-\varepsilon\leq r\leq t,$

we get for every $\vartheta,\vartheta_{\star}>1$ conjugate coefficients
($\vartheta_{\star}=\vartheta/(\vartheta-1)$)

$\mathrm{e}^{-2\lambda
t}w^{2}(t)\leq\frac{\vartheta}{\varepsilon}\int_{t-\varepsilon}^{t}e^{-2\lambda
t}w_{\tau}^{2}(r,t)\,\mathrm{d}r+\vartheta_{\star}L^{2}(\tau+\varepsilon)^{2}.$
(7.20)

Similarly to (7.12) we have

$\displaystyle|w_{\tau}(s,s)-w(s,s)|\leq
L\tau,\quad|w^{2}_{\tau}(s,s)-w^{2}(s,s)|\leq L\tau\Big{(}L\tau+2w(s)\Big{)}$

and, after an integration,

$\displaystyle\int_{0}^{t}e^{-2\lambda
s}\left(w^{2}_{\tau}(s,s)-w^{2}(s,s)\right)\,\mathrm{d}s\leq{L^{2}t\tau^{2}}+2L\tau\int_{0}^{t}e^{-2\lambda
s}w(s)\,\mathrm{d}s.$ (7.21)

Performing the same computations for the third integral term at the RHS of
(7.19) we end up with

$\begin{split}\int_{0}^{t}&e^{-2\lambda
s}\left(w^{2}(s-\varepsilon,s)-w_{\tau}^{2}(s-\varepsilon,s)\right)\,\mathrm{d}s\leq{L^{2}t\tau^{2}}+2L\tau\int_{0}^{t}e^{-2\lambda