${\mathcal{L}}_{T}$-a.e. $t\in[0,T]$ such that

(A.12) $\begin{cases}\dot{z}_{t}=F(t,z_{t}),\qquad\text{for
${\mathcal{L}}_{T}$-a.e. }t\in[0,T]\\\ z(0)=z_{0}.\end{cases}$

Moreover, the following estimates hold:

(A.13) $\sup_{t\in[0,T]}\|z_{t}\|\leq
e^{\tilde{L}T}\left(\|z_{0}\|+C_{0}T\right),$ (A.14)
$\|z_{t}-z_{s}\|\leq|t-s|\,\big{(}C_{0}+\tilde{L}e^{\tilde{L}T}\left(\|z_{0}\|+C_{0}T\right)\big{)}\qquad\forall\,s,t\in[0,T].$

###### Proof.

We provide only a sketch of the proof.

Assumptions (A.10) and (A.11) yield the following growth property

(A.15) $\|F(t,z)\|\leq C_{0}+\tilde{L}\|z\|,\qquad\forall\,(t,z)\in[0,T]\times
E.$

We define the Banach space $(\mathscr{S},\|\cdot\|_{\mathscr{S}})$ as follows

$\SS:=\left\\{z\in
C([0,T];E)\,:\,\|z\|_{\SS}:=\sup_{t\in[0,T]}e^{-\tilde{L}t}\|z_{t}\|<+\infty\right\\},$

and the operator $S:\SS\to\SS$ by

$S(y)_{t}=z_{0}+\int_{0}^{t}F(s,y_{s})\,{\mathrm{d}}s.$

By (A.10) and (A.15), using (A.1), it is classical to prove that $S$ is well
defined and it is a contraction. Then, by Banach fixed point Theorem we get
the existence and uniqueness of $z\in\mathscr{S}$ such that (A.9) holds. The
estimates (A.13) and (A.14) follow from (A.9), (A.15), and Gronwall
inequality. Finally, $z$ belongs to $\mathrm{AC}^{\infty}([0,T];E)$ thanks to
(A.14). ∎

######  Proposition A.6.

Let $F,F^{n}:[0,T]\times E\to E$, $n\in\mathbb{N}$, be Carathéodory functions
satisfying (A.10),(A.11) with the same constant $\tilde{L}$ and $C_{0}$. Let
$z_{0},z^{n}_{0}\in E$, $z\in\mathrm{AC}^{\infty}([0,T];E)$ the solution of
(A.12) and $z^{n}\in\mathrm{AC}^{\infty}([0,T];E)$ the solutions of

$\begin{cases}\dot{z}^{n}_{t}=F^{n}(t,z^{n}_{t}),\qquad\text{for
${\mathcal{L}}_{T}$-a.e. }t\in[0,T]\\\ z^{n}(0)=z^{n}_{0}.\end{cases}$

Then

(A.16) $\sup_{t\in[0,T]}\|z^{n}_{t}-z_{t}\|\leq
e^{\tilde{L}T}\left(\|z^{n}_{0}-z_{0}\|+\int_{0}^{T}\|F^{n}(t,z_{t})-F(t,z_{t})\|\,{\mathrm{d}}t\right).$

In particular, if $\lim_{n\to+\infty}\|z_{0}^{n}-z_{0}\|=0$ and

(A.17)
$\lim_{n\to+\infty}\int_{0}^{T}\|F^{n}(t,z_{t})-F(t,z_{t})\|\,{\mathrm{d}}t=0,$

then

$\lim_{n\to+\infty}\sup_{t\in[0,T]}\|z^{n}_{t}-z_{t}\|=0.$

###### Proof.

We have

$\displaystyle\|z^{n}_{t}-z_{t}\|$
$\displaystyle=\left\|z^{n}_{0}-z_{0}+\int_{0}^{t}(F^{n}(s,z^{n}_{s})-F(s,z_{s}))\,{\mathrm{d}}s\right\|$
$\displaystyle\leq\|z^{n}_{0}-z_{0}\|+\int_{0}^{t}\|F^{n}(s,z^{n}_{s})-F(s,z_{s}))\|\,{\mathrm{d}}s.$

Using (A.10) it holds

$\displaystyle\|F^{n}(s,z^{n}_{s})-F(s,z_{s}))\|$
$\displaystyle\leq\|F^{n}(s,z^{n}_{s})-F^{n}(s,z_{s}))\|+\|F^{n}(s,z_{s})-F(s,z_{s}))\|$
$\displaystyle\leq\tilde{L}\|z^{n}_{s}-z_{s}\|+\|F^{n}(s,z_{s})-F(s,z_{s}))\|.$

The last two inequalities yield

$\|z^{n}_{t}-z_{t}\|\leq\|z^{n}_{0}-z_{0}\|+\int_{0}^{T}\|F^{n}(s,z_{s})-F(s,z_{s}))\|\,{\mathrm{d}}s+\tilde{L}\int_{0}^{t}\|z^{n}_{s}-z_{s}\|\,{\mathrm{d}}s.$

By Gronwall lemma we obtain (A.16). ∎

######  Proposition A.7.

Let $F:[0,T]\times L^{p}(\Omega;\mathbb{R}^{d})\to
L^{p}(\Omega;\mathbb{R}^{d})$ be a Carathéodory function satisfying (A.11) Let
$Y_{0}\in L^{p}(\Omega;\mathbb{R}^{d})$. Then the following assertions are
equivalent:

  1. (1)

$Y\in C([0,T];L^{p}(\Omega;\mathbb{R}^{d}))$ and

(A.18)
$Y_{t}=Y_{0}+\int_{0}^{t}F(s,Y_{s})\,{\mathrm{d}}s,\qquad\forall\,t\in[0,T];$

  2. (2)

$Y\in\mathrm{AC}^{p}([0,T];L^{p}(\Omega;\mathbb{R}^{d}))$ (if $p=1$ it is also
differentiable for a.e. $t\in[0,T]$) and

(A.19) $\begin{cases}\dot{Y}_{t}=F(t,Y_{t}),\qquad\text{for
${\mathcal{L}}_{T}$-a.e. }t\in[0,T]\\\ Y_{t=0}=Y_{0};\end{cases}$

  3. (3)

$Y\in L^{p}(\Omega;\mathrm{AC}^{p}([0,T];\mathbb{R}^{d})))$ and

(A.20)
$Y_{t}(\omega)=Y_{0}(\omega)+\int_{0}^{t}F(s,Y_{s})(\omega)\,{\mathrm{d}}s,\qquad\forall\,t\in[0,T],\quad\text{for
$\mathbb{P}$-a.e. }\omega\in\Omega,$

where $Y_{t}:\Omega\to\mathbb{R}^{d}$ is defined by
$Y_{t}(\omega):=Y(t,\omega)$ for $\mathbb{P}$-a.e. $\omega\in\Omega$.

  4. (4)

$Y\in L^{p}(\Omega;\mathrm{AC}^{p}([0,T];\mathbb{R}^{d})))$ and for
$\mathbb{P}$-a.e. $\omega\in\Omega$ it holds

(A.21) $\begin{cases}\dot{Y}_{t}(\omega)=F(t,Y_{t})(\omega),\qquad\text{for
${\mathcal{L}}_{T}$-a.e. }t\in[0,T]\\\
Y_{t=0}(\omega)=Y_{0}(\omega),\end{cases}$

where $Y_{t}:\Omega\to\mathbb{R}^{d}$ is defined by
$Y_{t}(\omega):=Y(t,\omega)$ for $\mathbb{P}$-a.e. $\omega\in\Omega$.

###### Proof.

The assertions (1)-(2) and (3)-(4) are equivalent by Proposition A.4. The
equivalence (1)-(3) is a consequence of the equivalences (3)-(5) in
Proposition A.3 and items (i)-(ii)-(iii). ∎

##  Appendix B A convergence result for solutions of Cauchy problems

We state and prove the following well known result, for sake of completeness.

######  Lemma B.1.

Let $U$ be a Polish space. Let $u^{n}:[0,T]\to U$ be a sequence of
${\mathcal{L}}_{T}$-measurable functions such that
$(i_{[0,T]},u^{n})_{\sharp}{\mathcal{L}}_{T}\xrightarrow{\mathcal{Y}}\nu_{t}\otimes{\mathcal{L}}_{T}\in\mathscr{P}([0,T]\times
U)$.
Let $g:[0,T]\times(\mathbb{R}^{d}\times U)\to\mathbb{R}^{d}$ a Carathéodory
function such that

(B.1) $|g(t,x_{1},u)-g(t,x_{2},u)|\leq
L|x_{1}-x_{2}|,\qquad\forall\,(t,x_{1},u),(t,x_{2},u)\in[0,T]\times\mathbb{R}^{d}\times
U,$

for some $L>0$, and

(B.2) $C_{0}:=\sup_{(t,u)\in[0,T]\times U}|g(t,0,u)|<+\infty.$

Given $X_{0}\in\mathbb{R}^{d}$ and $n\in\mathbb{N}$, we denote by
$X^{n}\in\mathrm{AC}^{1}([0,T];\mathbb{R}^{d})$ the unique solution of the
Cauchy problem

(B.3) $\begin{cases}\dot{X}^{n}_{t}=g(t,X^{n}_{t},u^{n}_{t}),&\textrm{for a.e.
}t\in(0,T)\\\ X^{n}(0)=X_{0},\end{cases}$

and by $X\in\mathrm{AC}^{1}([0,T];\mathbb{R}^{d})$ the unique solution of the
Cauchy problem

(B.4)
$\begin{cases}\dot{X}_{t}=\displaystyle\int_{U}g(t,X_{t},u)\,{\mathrm{d}}\nu_{t}(u),&\textrm{for
a.e. }t\in(0,T)\\\ X(0)=X_{0}.\end{cases}$

Then

(B.5) $\lim_{n\to+\infty}\sup_{t\in[0,T]}|X^{n}_{t}-X_{t}|=0.$

###### Proof.

Observe that existence and uniqueness of solutions of Cauchy problems (B.3)
and (B.4) is consequence of the fact that
$G^{n},G:[0,T]\times\mathbb{R}^{d}\to\mathbb{R}^{d}$, defined by
$G^{n}(t,x):=g(t,x,u^{n}_{t})$ and
$G(t,x):=\displaystyle\int_{U}g(t,x,u)\,{\mathrm{d}}\nu_{t}(u)$, are
Charathéodory and $L$-Lipschitz continuous w.r.t. $x\in\mathbb{R}^{d}$.
Moreover, by (B.1) and (B.2) it holds

(B.6) $|g(t,x,u)|\leq
C_{0}+L|x|,\qquad\forall\,(t,x,u)\in[0,T]\times\mathbb{R}^{d}\times U.$

We define $Y^{n}\in C([0,T];\mathbb{R}^{d})$ by

(B.7) $Y^{n}_{t}:=X_{0}+\int_{0}^{t}g(s,X_{s},u^{n}_{s})\,{\mathrm{d}}s.$

From the convergence
$(i_{[0,T]},u^{n})_{\sharp}{\mathcal{L}}_{T}\xrightarrow{\mathcal{Y}}\nu\in\mathscr{P}([0,T]\times
U)$ it follows that

(B.8)
$\lim_{n\to+\infty}Y^{n}_{t}=X_{0}+\int_{0}^{t}\int_{U}g(s,X_{s},u)\,{\mathrm{d}}\nu_{s}(u)\,{\mathrm{d}}s=X_{t},\qquad\forall\,t\in[0,T].$

Denoting by $C:=\sup_{s\in[0,T]}|X_{s}|$, from (B.2) and (B.1) it is simple to
prove that

(B.9) $|Y^{n}_{t}|\leq(|X_{0}|+(C_{0}+LC)T),\qquad\forall t\in[0,T]$

and

(B.10) $|Y^{n}_{t}-Y^{n}_{s}|\leq|t-s|(C_{0}+LC)T,\qquad\forall t,s\in[0,T].$

By (B.9) and (B.10), Ascoli-Arzelà theorem and (B.8) imply that

(B.11) $\lim_{n\to+\infty}\sup_{t\in[0,T]}|Y^{n}_{t}-X_{t}|=0.$

Since

$\begin{split}|X^{n}_{t}-X_{t}|&\leq|X^{n}_{t}-Y^{n}_{t}|+|Y^{n}_{t}-X_{t}|\leq\int_{0}^{t}\left|g(s,X^{n}_{s},u^{n}_{s})-g(s,X_{s},u^{n}_{s})\right|\,{\mathrm{d}}s+|Y^{n}_{t}-X_{t}|\\\
&\leq
L\int_{0}^{t}|X^{n}_{s}-X_{s}|\,{\mathrm{d}}s+\sup_{s\in[0,T]}|Y^{n}_{s}-X_{s}|,\end{split}$

by Gronwall inequality we have that

(B.12) $|X^{n}_{t}-X_{t}|\leq e^{Lt}\sup_{s\in[0,T]}|Y^{n}_{s}-X_{s}|.$

The convergence (B.5) follows from (B.12) and (B.11). ∎

##  Appendix C An empirical Superposition Principle

In this appendix, we give a refined version of the Superposition Principle
(see Theorem 2.5 for the classical result) in the case of trajectories of the
form $\mu_{t}\in\mathscr{P}^{N}(\mathbb{R}^{d})$ for any $t\in[0,T]$, where
$\mathscr{P}^{N}(\mathbb{R}^{d})$ is the space of empirical probability
measures

$\mathscr{P}^{N}(\mathbb{R}^{d}):=\left\\{\mu=\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}}\;\text{
for some }x_{i}\in\mathbb{R}^{d}\right\\}.$

The novelty consists in proving that if
$\mu_{t}\in\mathscr{P}^{N}(\mathbb{R}^{d})$ for every $t\in[0,T]$, then there
exists representative $\eta\in\mathscr{P}^{N}(\Gamma_{T})$. This result has
been used to prove Proposition 9.8 and Corollary 9.3.

######  Theorem C.1.

Let $N\in\mathbb{N}$ and
$\mu\in\mathrm{AC}([0,T];\mathscr{P}_{1}(\mathbb{R}^{d}))$ such that
$\mu_{t}\in\mathscr{P}^{N}(\mathbb{R}^{d})$ for every $t\in[0,T]$.

  1. (1)

There exists a unique (up to ${\mathcal{L}}_{T}\otimes\mu_{t}$-negligible
sets) Borel vector field $v:[0,T]\times\mathbb{R}^{d}\to\mathbb{R}^{d}$
satisfying

(C.1)
$\int_{0}^{T}\int_{\mathbb{R}^{d}}|v_{t}(x)|\,{\mathrm{d}}\mu_{t}(x)\,{\mathrm{d}}t<+\infty,$

such that $\mu$ is a distributional solution of the continuity equation

(C.2) $\partial_{t}\mu_{t}+\mathrm{div}(v_{t}\mu_{t})=0,\quad\text{in
}[0,T]\times\mathbb{R}^{d}.$

  2. (2)

There exists $\eta\in\mathscr{P}^{N}(\Gamma_{T})$ of the form

(C.3) $\eta=\frac{1}{N}\sum_{i=1}^{N}\delta_{\gamma_{i}},$

such that $(e_{t})_{\sharp}\eta=\mu_{t}$ for every $t\in[0,T]$ and for any
$i=1,\dots,N$, $\gamma_{i}\in\mathrm{AC}([0,T];\mathbb{R}^{d})$ solves the
differential equation

(C.4) $\dot{\gamma}_{i}(t)=v_{t}(\gamma_{i}(t))\quad\text{for
}{\mathcal{L}}_{T}\text{-a.e. }t\in[0,T].$

###### Proof.

Let us recall that the metric derivative $|\mu^{\prime}|$ of the absolutely
continuous curve $\mu$, given by

$|\mu^{\prime}|(t):=\lim_{s\to
t}\frac{W_{1}(\mu_{t},\mu_{s})}{|t-s|},\quad\text{for a.e. }t\in[0,T],$

belongs to $L^{1}(0,T)$ and satisfies

(C.5)
$W_{1}(\mu_{t_{1}},\mu_{t_{2}})\leq\displaystyle\int_{t_{1}}^{t_{2}}|\mu^{\prime}|(t)\,{\mathrm{d}}t,\quad\text{
for any }0\leq t_{1}\leq t_{2}\leq T.$

in particular there exists $\psi:[0,+\infty)\to[0,+\infty)$ increasing, convex
and superlinear at $+\infty$ such that

(C.6)
$\int_{0}^{T}\psi\big{(}|\mu^{\prime}|(t)\big{)}\,{\mathrm{d}}t<+\infty.$

First of all we prove the existence of $\eta\in\mathscr{P}(\Gamma_{T})$ such
that $(e_{t})_{\sharp}\eta=\mu_{t}$ for all $t\in[0,T]$ and $\eta$ is of the
form (C.3) for some $\gamma_{i}\in\Gamma_{T}$, $i=1,\dots,N$.

Let $M\in\mathbb{N}$ and consider the diadic discretization of the interval
$[0,T]$, with time step $\tau_{M}=T\,2^{-M}$. Since
$\mu_{t}\in\mathscr{P}^{N}(\mathbb{R}^{d})$ for every $t\in[0,T]$, there
exists $x_{i}(t)\in\mathbb{R}^{d}$, $i=1,\ldots,N$, such that

$\mu_{t}=\frac{1}{N}\sum_{i=1}^{N}\delta_{x_{i}(t)},\qquad\forall\,t\in[0,T].$

For $n=0,\dots,2^{M}$ and $i=1,\dots,N$, we set
$x^{n}_{M,i}:=x_{i}(n\tau_{M})$, and $\mu^{n}_{M}:=\mu_{n\tau_{M}}$. For
$n=1,\dots,2^{M}$, let
$\varrho^{n-1,n}_{M}\in\Gamma_{o}(\mu^{n-1}_{M},\mu^{n}_{M})$ be an optimal
plan for the $1$-Wasserstein distance. Since $\mu^{n-1}_{M}$ and $\mu^{n}_{M}$
belong to $\mathscr{P}^{N}(\mathbb{R}^{d})$, then $\varrho^{n-1,n}_{M}$ is of
the form

(C.7)
$\varrho_{M}^{n-1,n}=\frac{1}{N}\sum_{i=1}^{N}\delta_{x^{n-1}_{M,i}}\otimes\delta_{x^{n}_{M,\sigma^{n}_{M}(i)}},$

for some permutation $\sigma^{n}_{M}$ of $\\{1,\dots,N\\}$. Let us define
$\sigma_{M}^{0,n}:=\sigma_{M}^{n}\circ\sigma_{M}^{n-1}\circ\dots\circ\sigma_{M}^{1}$
and $\sigma_{M}^{0,0}(i)=i$ for $i=1,\ldots,N$.
For $i=1,\dots,N$ we define the curves $\gamma_{M,i}\in\Gamma_{T}$ by linear
time interpolation as

$\gamma_{M,i}(t):=\frac{n\tau_{M}-t}{\tau_{M}}\,x^{n-1}_{M,\sigma_{M}^{0,n-1}(i)}+\frac{t-(n-1)\tau_{M}}{\tau_{M}}\,x^{n}_{M,\sigma_{M}^{0,n}(i)},\quad\text{for
}t\in[(n-1)\tau_{M},n\tau_{M}],$

$n=1,\dots,2^{M}$.

We claim that, for any $k=1,\ldots,N$, the sequence
$\\{\gamma_{M,k}\\}_{M\in\mathbb{N}}$ uniformly converges to a curve
$\gamma_{k}\in\mathrm{AC}([0,T];\mathbb{R}^{d})$. Indeed,

(C.8)
$\begin{split}\int_{0}^{T}\psi\Big{(}\frac{1}{N}|\dot{\gamma}_{M,k}(t)|\Big{)}\,{\mathrm{d}}t&\leq\int_{0}^{T}\psi\Big{(}\frac{1}{N}\sum_{i=1}^{N}|\dot{\gamma}_{M,i}(t)|\Big{)}\,{\mathrm{d}}t\\\
&=\sum_{n=1}^{2^{M}}\tau_{M}\,\psi\Big{(}\frac{1}{N}\sum_{i=1}^{N}\frac{\left|x^{n}_{M,\sigma^{0,n}_{M}(i)}-x^{n-1}_{M,\sigma^{0,n-1}_{M}(i)}\right|}{\tau_{M}}\Big{)}\\\
&=\sum_{n=1}^{2^{M}}\tau_{M}\,\psi\Big{(}\frac{1}{\tau_{M}}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}|x-y|\,{\mathrm{d}}\varrho_{M}^{n-1,n}(x,y)\Big{)}\\\
&=\sum_{n=1}^{2^{M}}\tau_{M}\,\psi\Big{(}\frac{1}{\tau_{M}}W_{1}(\mu_{M}^{n-1},\mu_{M}^{n})\Big{)}\\\
&\leq\sum_{n=1}^{2^{M}}\tau_{M}\,\psi\Big{(}\frac{1}{\tau_{M}}\int_{(n-1)\tau_{M}}^{n\tau_{M}}|\mu^{\prime}|(t)\,{\mathrm{d}}t\Big{)}\\\
&\leq\sum_{n=1}^{2^{M}}\int_{(n-1)\tau_{M}}^{n\tau_{M}}\psi(|\mu^{\prime}|(t))\,{\mathrm{d}}t=\int_{0}^{T}\psi(|\mu^{\prime}|(t))\,{\mathrm{d}}t,\end{split}$

where we employed the definition of the optimal plan in (C.7), (C.5), Jensen’s
inequality, and (C.6).
Since $\gamma_{M,k}(0)=x_{k}(0)$ for any $M\in\mathbb{N}$, and (C.8) and (C.6)
hold, by Ascoli-Arzelá Theorem the sequence
$\\{\gamma_{M,k}\\}_{M\in\mathbb{N}}$ is compact in $C([0,T];\mathbb{R}^{d})$.
Furthermore, by (C.8) and the lower semicontinuity of the functional

$\gamma\mapsto\int_{0}^{T}\psi\left(\frac{1}{N}|\dot{\gamma}(t)|\right){\mathrm{d}}t$

w.r.t. weak convergence in $\mathrm{AC}([0,T];\mathbb{R}^{d})$, we get
$\gamma_{k}\in\mathrm{AC}([0,T];\mathbb{R}^{d})$. Moreover, if
$t=n_{0}\tau_{M_{0}}$ for some $M_{0}\in\mathbb{N}$ and
$n_{0}\in\\{0,1,\ldots,2^{M_{0}}\\}$, then $\gamma_{M,k}(t)$ is constant for
any $M\in\mathbb{N}$, $M>M_{0}$ and the claim is proved.

Defining

(C.9)
$\eta_{M}:=\frac{1}{N}\sum_{i=1}^{N}\delta_{\gamma_{M,i}},\qquad\eta:=\frac{1}{N}\sum_{i=1}^{N}\delta_{\gamma_{i}},$

from the convergence of $\gamma_{M,i}$ to $\gamma_{i}$ it follows that
$\eta_{M}$ weakly converges to $\eta$ as $M\to+\infty$. Moreover, if
$t=n_{0}\tau_{M_{0}}$ for some $M_{0}\in\mathbb{N}$ and
$n_{0}\in\\{0,1,\ldots,2^{M_{0}}\\}$, then $(e_{t})_{\sharp}\eta_{M}=\mu_{t}$
for any $M\in\mathbb{N}$, $M>M_{0}$. Then, by the continuity of
$t\mapsto\mu_{t}$ and of $t\mapsto(e_{t})_{\sharp}\eta$, we conclude that
$(e_{t})_{\sharp}\eta=\mu_{t}$ and
$\mu_{t}=\frac{1}{N}\sum_{i=1}^{N}\delta_{\gamma_{i}(t)}$ for all $t\in[0,T]$.

It remains to define a vector field $v$ such that (C.4) and (C.2) hold, also
showing that $v$ is uniquely characterized by (C.2).

Since $\gamma_{i}\in\mathrm{AC}([0,T];\mathbb{R}^{d})$ for any $i=1,\ldots,N$,
the Borel set $A:=\\{t\in[0,T]:\exists\,k\in\\{1,\ldots,N\\}\text{ such that
}\gamma_{k}\text{ is not differentiable at }t\\}$ is
${\mathcal{L}}_{T}$-negligible. Moreover, the sets

(C.10) $N_{i,k}:=\\{t\in[0,T]\setminus
A:\gamma_{i}(t)=\gamma_{k}(t),\dot{\gamma}_{i}(t)\neq\dot{\gamma}_{k}(t)\\}$

satisfy ${\mathcal{L}}_{T}(N_{i,k})=0$ for any $i,k\in\\{1,\ldots,N\\}$. We
define
$\tilde{N}:=\Big{(}\bigcup_{i,k\in\\{1,\ldots,N\\}}N_{i,k}\Big{)}\bigcup A$
noticing that ${\mathcal{L}}_{T}(\tilde{N})=0$ so that
${\mathcal{L}}_{T}\otimes\mu_{t}(\tilde{N}\times\mathbb{R}^{d})=0$, and
$S:=\\{(t,\gamma_{i}(t)):t\in[0,T],\ i\in\\{1,\ldots,N\\}\\}=\mathop{\rm
supp}\nolimits(\mu_{t}\otimes{\mathcal{L}}_{T})$. We can thus define a Borel
vector field $v:[0,T]\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ by

$v(t,x):=\begin{cases}0&\text{if }t\in\tilde{N}\text{ or
}(t,x)\in\big{(}[0,T]\times\mathbb{R}^{d}\big{)}\setminus S,\\\
\dot{\gamma}_{i}(t)&\text{if }x=\gamma_{i}(t)\ \text{for
$t\in[0,T]\setminus\tilde{N}$ and some }i\in\\{1,\ldots,N\\},\end{cases}$

so that $\dot{\gamma}_{i}(t)=v_{t}(\gamma_{i}(t))$ for every
$t\in[0,T]\setminus\tilde{N}$. It is then easy to check that (C.1) and (C.2)
hold.

Let us eventually check that (C.2) uniquely characterizes $v(t_{0},x_{0})$ for
every $(t_{0},x_{0})\in S\setminus(\tilde{N}\times\mathbb{R}^{d})$. Notice
that for every $\varphi\in C_{c}^{\infty}(\mathbb{R}^{d})$ we have

(C.11)
$\frac{{\mathrm{d}}}{{\mathrm{d}}t}\int_{\mathbb{R}^{d}}\varphi\,{\mathrm{d}}\mu_{t}\Big{|}_{t=t_{0}}=\frac{1}{N}\sum_{i=1}^{N}\nabla\varphi(\gamma_{i}(t_{0}))\cdot\dot{\gamma}_{i}(t_{0})=\frac{1}{N}\sum_{i=1}^{N}\nabla\varphi(\gamma_{i}(t_{0}))\cdot
v({t_{0}},\gamma_{i}(t_{0})).$

Setting $K:=\\{k\in\\{1,\ldots,N\\}:\gamma_{k}(t_{0})\neq x_{0}\\}$ and
$r_{0}:=\min\\{|\gamma_{k}(t_{0})-x_{0}|:k\in K\\}>0$, for every
$\xi\in\mathbb{R}^{d}$ we can find a test function $\varphi\in
C^{\infty}_{c}(\mathbb{R}^{d})$ such that $\mathop{\rm
supp}\nolimits\varphi\subset B_{r_{0}}(x_{0})$ and $\nabla\varphi(x_{0})=\xi$:
(C.11) then yields

(C.12)
$\frac{{\mathrm{d}}}{{\mathrm{d}}t}\int_{\mathbb{R}^{d}}\varphi\,{\mathrm{d}}\mu_{t}\Big{|}_{t=t_{0}}=\frac{n}{N}\xi\cdot
v({t_{0}},x_{0})\quad\text{where}\quad n:=N-\\#K.$

Since $\xi$ is arbitrary, (C.12) uniquely characterizes $v(t_{0},x_{0})$ in
terms of $\mu$. ∎

##  Appendix D Finite Partitions

In this section we provide a proof of Proposition 4.16. For sake of clarity,
we divide the statement of Proposition 4.16 in three separate lemmas of
independent interest.

Given a standard Borel space $(\Omega,\mathfrak{B},\mathbb{P})$, in Lemma D.2
we construct a family of algebras $\mathfrak{B}^{n}$, $n\in\mathbb{N}$,
satisfying the finite approximation property of Definition 4.15. Then we fix
$(\Omega,\mathfrak{B},\mathbb{P})=([0,1],{\mathcal{B}},{\mathcal{L}}_{1})$,
where ${\mathcal{B}}$ is the Borel $\sigma$-algebra and ${\mathcal{L}}_{1}$
the Lebesgue measure restricted to the interval $[0,1]$. With this choice of
parametrization space, in Lemma D.3 we show that the family of algebras
${\mathcal{B}}^{N}$ associated to the uniform partition of $[0,1]$ with
elements’ size $1/N$ satisfies the finite approximation property. Finally, we
combine the previous results in Lemma D.4, where we consider a general
standard Borel space $(\Omega,\mathfrak{B},\mathbb{P})$ and $\mathbb{P}$ is
without atoms. This is possible thanks to the following fundamental result on
Borel equivalence of Probability spaces (see e.g. [Royden, Chapter 15, Theorem
9]).

######  Proposition D.1.

Let $\Omega$ be a Polish space and $\mathbb{P}\in\mathscr{P}(\Omega)$ without
atoms. Then there exist a Borel set $\Omega_{0}\subset\Omega$ such that
$\mathbb{P}(\Omega_{0})=0$, a Borel set $I_{0}\subset[0,1]$ such that
${\mathcal{L}}_{1}(I_{0})=0$ and a bijective function
$\psi:\Omega\setminus\Omega_{0}\to[0,1]\setminus I_{0}$ such that $\psi$ and
$\psi^{-1}$ are Borel, $\psi_{\sharp}\mathbb{P}={\mathcal{L}}_{1}$ and
$(\psi^{-1})_{\sharp}{\mathcal{L}}_{1}=\mathbb{P}$.

The first part of Proposition 4.16 is restated in the following Lemma.

######  Lemma D.2 (Proposition 4.16, part 1).

Let $(\Omega,\mathfrak{B},\mathbb{P})$ be a standard Borel space. Then there
exists a family of finite algebras $\mathfrak{B}^{n}\subset\mathfrak{B}$,
$n\in\mathbb{N}$, satisfying the finite approximation property of Definition
4.15.

###### Proof.

Since $(\Omega,\mathfrak{B},\mathbb{P})$ is standard Borel we can choose a
Polish topology $\tau$ such that $\mathfrak{B}={\mathcal{B}}_{(\Omega,\tau)}$,
then there exists a countable basis ${\mathcal{A}}=\\{B^{i}:i\in\mathbb{N}\\}$
of its topology. Then
${\mathcal{B}}_{(\Omega,\tau)}=\sigma(\\{B^{i}:i\in\mathbb{N}\\})$. We define
$\mathfrak{B}^{1}:=\sigma(B^{1})$ and
$\mathfrak{B}^{n}:=\sigma(\\{B^{n}\\}\cup\mathfrak{B}^{n-1})$. It follows from
the definition that $\mathfrak{B}^{n}\subset\mathfrak{B}^{n+1}$ for any
$n\in\mathbb{N}$ and
${\mathcal{B}}_{(\Omega,\tau)}=\sigma\left(\bigcup_{n=1}^{+\infty}\mathfrak{B}^{n}\right)$.

For any $n\in\mathbb{N}$, the finite algebra $\mathfrak{B}^{n}$ induces a
minimal (with respect to the inclusion) partition of $\Omega$, denoted by
${\mathcal{P}}^{n}=\\{A_{k}^{n}:k=1,\ldots,k(n)\\}\subset\mathfrak{B}^{n}$.
Then for any $A_{k}^{n+1}\in{\mathcal{P}}^{n+1}$ there exists $h\in\mathbb{N}$
such that $A_{h}^{n}\in{\mathcal{P}}^{n}$ and $A_{k}^{n+1}\subset A_{h}^{n}$.

We define the sequence of linear operators $P_{n}:L^{1}(\Omega;E)\to
L^{1}(\Omega;E)$ defined by

$P_{n}g:=\sum_{k=1}^{k(n)}\mathds{1}_{A^{n}_{k}}\,\fint_{A^{n}_{k}}g(\omega)\,{\mathrm{d}}\mathbb{P}(\omega),$

with the convention that
$\fint_{A^{n}_{k}}g(\omega)\,{\mathrm{d}}\mathbb{P}(\omega)=0$ if
$\mathbb{P}(A^{n}_{k})=0$.

It is simple to prove that

(D.1)
$\|P_{n}g\|_{L_{\mathbb{P}}^{1}(\Omega;E)}\leq\|g\|_{L_{\mathbb{P}}^{1}(\Omega;E)},\qquad\forall\,g\in{L_{\mathbb{P}}^{1}(\Omega;E)}.$

Given a Borel function $g:\Omega\to E$ such that $g\in
L_{\mathbb{P}}^{1}(\Omega;E)$, we define $g^{n}:=P_{n}g$ and we prove that the
properties of Definition 4.15 hold.

Property (i) is obvious since $g^{n}$ is constant on the elements of the
partition ${\mathcal{P}}^{n}\subset\mathfrak{B}^{n}$.

Property (ii) follows from the fact that (see for instance [diestel1977vector,
Corollary 8, p. 48])

$\fint_{A}g(\omega)\,{\mathrm{d}}\mathbb{P}(\omega)\in\mathrm{\overline{co}}\left(g(A)\right),\quad\forall\,A\in{\mathcal{B}}_{(\Omega,\tau)}:\mathbb{P}(A)>0.$

In order to prove property (iii) we start with the particular case
$g=a\mathds{1}_{A}$ for a given Borel set $A$ and a given $a\in E$. Since
$(\Omega,\tau)$ is Polish, for any $\varepsilon>0$ there exists an open set
$A_{\varepsilon}$ and a compact set $K_{\varepsilon}$ such that
$K_{\varepsilon}\subset A\subset A_{\varepsilon}$ and
$\mathbb{P}(A_{\varepsilon}\setminus K_{\varepsilon})<\varepsilon$. Since
$A_{\varepsilon}$ is union of elements of the basis ${\mathcal{A}}$, there
exists a finite covering of $K_{\varepsilon}$ of the form $\\{B^{j}:j\in
J\\}\subset{\mathcal{A}}$, for a suitable finite $J\subset\mathbb{N}$, such
that $\cup_{j\in J}B^{j}\subset A_{\varepsilon}$. Since

$P_{n}g-g=a\sum_{k=1}^{k(n)}\frac{\mathbb{P}(A_{k}^{n}\cap
A)}{\mathbb{P}(A_{k}^{n})}\mathds{1}_{A_{k}^{n}}-a\mathds{1}_{A},$

by setting $n_{\varepsilon}:=\max J$, it holds that

$\|P_{n}g-g\|_{L_{\mathbb{P}}^{1}(\Omega;E)}\leq\|a\mathds{1}_{A_{\varepsilon}}-a\mathds{1}_{A}\|_{L_{\mathbb{P}}^{1}(\Omega;E)}\leq\|a\|_{E}\
\mathbb{P}(A_{\varepsilon}\setminus A)<\|a\|_{E}\,\varepsilon\ ,\quad\forall\
n\geq n_{\varepsilon}\ .$

Since $P_{n}$ is linear, then (iii) holds for any $g$ simple function. In the
general case, take $g\in L_{\mathbb{P}}^{1}(\Omega;E)$ and $\varepsilon>0$,
and let $g_{\varepsilon}:\Omega\to E$ be a simple function such that
$\|g-g_{\varepsilon}\|_{L_{\mathbb{P}}^{1}(\Omega;E)}<\varepsilon$. Observing
that

$\|P_{n}g-g\|_{L_{\mathbb{P}}^{1}(\Omega;E)}\leq\|P_{n}g-P_{n}g_{\varepsilon}\|_{L_{\mathbb{P}}^{1}(\Omega;E)}+\|P_{n}g_{\varepsilon}-g_{\varepsilon}\|_{L_{\mathbb{P}}^{1}(\Omega;E)}+\|g_{\varepsilon}-g\|_{L_{\mathbb{P}}^{1}(\Omega;E)},$

by (D.1) and property (iii) applied to $g_{\varepsilon}$ it holds that
$\limsup_{n\to+\infty}\|P_{n}g-g\|_{L_{\mathbb{P}}^{1}(\Omega;E)}\leq
2\varepsilon$ and we conclude.

Finally, property (iv) follows from the measurability of $G$, Fubini Theorem
and the definition of $P_{n}$. ∎

Consider now the Polish space $([0,1],{\mathcal{B}},{\mathcal{L}}_{1})$.

######  Lemma D.3 (Proposition 4.16, part 2).

For any $N\in\mathbb{N}$ we define $I_{k}^{N}:=[(k-1)/N,k/N)$,
$k=1,\ldots,N-1$, $I_{N}^{N}:=[(N-1)/N,1]$. If $E$ is a Banach space, $g\in
L^{1}([0,1];E)$ and

$g^{N}:=\sum_{k=1}^{N}\mathds{1}_{I^{N}_{k}}\,\fint_{I^{N}_{k}}g(s)\,{\mathrm{d}}s,$

then

(D.2) $\lim_{N\to+\infty}\|g^{N}-g\|_{L^{1}([0,1];E)}=0.$

Moreover, the family of finite algebras
${\mathcal{B}}^{N}:=\sigma(\\{I^{N}_{k}:k=1,\ldots,N\\})$, $N\in\mathbb{N}$,
satisfies the finite approximation property of Definition 4.15.
Finally, if $g\in L^{p}([0,1];E)$, for some $p\in(1,+\infty)$, then
$\lim_{N\to+\infty}\|g^{N}-g\|_{L^{p}([0,1];E)}=0$.

###### Proof.

For any $x\in[0,1]$ and $N\in\mathbb{N}$, there exists a unique $k(x,N)$ such
that $x\in I_{k(x,N)}^{N}$. From the definition of $I^{N}_{k}$ it follows that
$I_{k(x,N)}^{N}\subset B_{1/N}(x)$. Since

$g^{N}(x)-g(x)=\sum_{k=1}^{N}\mathds{1}_{I^{N}_{k}}(x)\,\fint_{I^{N}_{k}}(g(s)-g(x))\,{\mathrm{d}}s,$

then

$\|g^{N}(x)-g(x)\|_{E}\leq
2\fint_{B_{1/N}(x)}\|g(s)-g(x)\|_{E}\,{\mathrm{d}}s.$

By the Bochner version of the Lebesgue differentiation Theorem (see for
instance [diestel1977vector, Theorem 9, p. 49]) we obtain that
$\lim_{N\to+\infty}\|g^{N}(x)-g(x)\|_{E}=0$ for ${\mathcal{L}}_{1}$-a.e.
$x\in[0,1]$.

Since $\int_{0}^{1}\|g(x)\|_{E}\,{\mathrm{d}}x<+\infty$ there exists a convex,
increasing, superlinear function $\psi:[0,+\infty)\to[0,+\infty)$ such that
$\int_{0}^{1}\psi\left(\|g(x)\|_{E}\right)\,{\mathrm{d}}x<+\infty$. Since

$\|g^{N}(x)\|_{E}\leq 2\fint_{B_{1/N}(x)}\|g(s)\|_{E}\,{\mathrm{d}}s,$

by Jensen’s inequality,

$\begin{split}\int_{0}^{1}\psi\left(\|g^{N}(x)\|_{E}\right)\,{\mathrm{d}}x&\leq
2\int_{0}^{1}\psi\left(\fint_{B_{1/N}(x)}\|g(s)\|_{E}\,{\mathrm{d}}s\right)\,{\mathrm{d}}x\\\
&\leq
2\int_{0}^{1}\fint_{B_{1/N}(x)}\psi\left(\|g(s)\|_{E}\right)\,{\mathrm{d}}s\,{\mathrm{d}}x\\\
&=N\int_{0}^{1}\int_{0}^{1}\mathds{1}_{(x-1/N,x+1/N)}(s)\psi\left(\|g(s)\|_{E}\right)\,{\mathrm{d}}s\,{\mathrm{d}}x\\\
&=N\int_{0}^{1}\int_{0}^{1}\mathds{1}_{(s-1/N,s+1/N)}(x)\psi\left(\|g(s)\|_{E}\right)\,{\mathrm{d}}s\,{\mathrm{d}}x\\\
&\leq
2\int_{0}^{1}\psi\left(\|g(s)\|_{E}\right)\,{\mathrm{d}}s<+\infty,\end{split}$

which implies the equi-integrability of the sequence $\|g^{N}\|_{E}$. Then
(D.2) holds.

The finite approximation property for ${\mathcal{B}}^{N}$ follows as in the
proof of Lemma D.2. The final assertion is a consequence of the equi-
integrability of the sequence $\|g^{N}\|^{p}_{E}$. ∎

######  Lemma D.4 (Proposition 4.16, part 3).

Let $(\Omega,\mathfrak{B},\mathbb{P})$ be a standard Borel space and
$\mathbb{P}$ without atoms. Then there exists a family
$\mathfrak{B}^{N}\subset\mathfrak{B}$, $N\in\mathbb{N}$, satisfying the finite
approximation property of Definition 4.15 such that the associated minimal
partition ${\mathcal{P}}^{N}=\\{A_{k}^{N}:k=1,\ldots,N\\}$ contains exactly
$N$ elements and $\mathbb{P}(A^{N}_{k})=\frac{1}{N}$, for $k=1,\ldots,N$.

###### Proof.

Let $\tau$ be a Polish topology on $\Omega$ such that
$\mathfrak{B}={\mathcal{B}}_{(\Omega,\tau)}$. Let also $\Omega_{0}$, $I_{0}$,
$\psi$, $\psi^{-1}$ be given by Proposition D.1. Using the notation of Lemma
D.3 we define the sets $A^{N}_{1}:=\psi^{-1}(I^{N}_{1}\setminus
I_{0})\cup\Omega_{0}$ and $A^{N}_{k}:=\psi^{-1}(I^{N}_{k}\setminus I_{0})$ for
$k=2,\ldots,N$.

It is immediate to prove that $\mathbb{P}(A^{N}_{j})=\frac{1}{N}$ for
$j=1,\ldots,N$ and $\\{A^{N}_{1},\ldots,A^{N}_{N}\\}$ is a partition of
$\Omega$. Moreover, given a Banach space $E$ and $g\in
L^{1}_{\mathbb{P}}(\Omega;E)$, we denote by $\tilde{g}:=g\circ\psi^{-1}\in
L^{1}([0,1];E)$. Denoting by $\tilde{g}^{N}$ the sequence given by Lemma D.3
applied to $\tilde{g}$, we define $g^{N}:=\tilde{g}^{N}\circ\psi$ and the
finite approximation property for
$(\Omega,{\mathcal{B}}_{(\Omega,\tau)},\mathbb{P})$ follows by Lemma D.3.

∎

## Acknowledgements

The authors acknowledge the support of MIUR-PRIN 2017 project _Gradient flows,
Optimal Transport and Metric Measure Structures_. G. Cavagnari, S. Lisini and
C. Orrieri acknowledge the support of the INDAM-GNAMPA project 2019 “Trasporto
ottimo per dinamiche con interazione”. C. Orrieri has also been supported by
the project Fondazione Cariplo-Regione Lombardia MEGAsTAR “Matematica
d’Eccellenza in biologia ed ingegneria come acceleratore di una nuova
strateGia per l’ATtRattività dell’ateneo pavese”. G. Savaré gratefully
acknowledges the support of the Institute of Advanced Study or the Technical
University of Munich and of IMATI-CNR, Pavia. G. Cavagnari and G. Savaré are
also grateful to the Department of Mathematics of the University of Pavia
where this project has been developed.

## References