us to define the first moment $m_1(\Psi)$ for a probability measure $\Psi \in {\mathcal P}(Y)$ as $$m_1(\Psi)\coloneqq\int_Y \lVert y\rVert_{\overline Y} \,\mathrm{d}\Psi$$ so that the space ${\mathcal P}_1(Y)$ can be equivalently characterized as $${\mathcal P}_1(Y)=\{\Psi\in{\mathcal P}(Y): m_1(\Psi)<+\infty\}.$$ We will sometimes use the notation ${\mathcal P}(K)$ to denote probability measures with support contained on a given compact subset $K\subset Y$. Notice that trivially we have ${\mathcal P}(K)\subset {\mathcal P}_1(Y)$.

Well-posedness of ODE’s in Banach spaces
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We recall here a theorem by Brezis [@Brezis Sect. I.3, Thm. 1.4, Cor. 1.1] on the well-posedness of ODE’s in Banach spaces.

\[thm:Brezis\] Let $(E,\|\cdot\|_E)$ be a Banach space, $C$ a closed convex subset of $E$, and let $A(t,\cdot)\colon C\to E$, $t\in [0,T]$, be a family of operators satisfying the following properties:

1.  there exists a constant $L{\geqslant}0$ such that for every $c_1,\,c_2\in C$ and $t\in [0,T]$ $$\label{eq:15}
    \|A(t,c_1)-A(t,c_2)\|_E\le L\|c_1-c_2\|_E;$$

2.  for every $c\in C$ the map $t\mapsto A(t,c)$ is continuous in $[0,T]$;

3.  for every $R>0$ there exists $\theta>0$ such that $$\label{eq:16}
    c\in C,\ \|c\|_E\le R\quad \Rightarrow\quad c+\theta A(t,c)\in C.$$

Then for every $\bar c\in C$ there exists a unique curve $c\colon[0,T]\to C$ of class $C^1$ satisfying $c_t\in C$ for all $t\in [0,T]$ and $$\label{eq:17}
