where the infimum is taken over all admissible pairs $(\mu,v)$ with the following properties:
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1. $\mu:[0,1]\to\mathcal P_2(\mathbb R^n)$, $t\mapsto\mu_t$, is narrowly continuous.
2. $\mu_{t=0}=\mu_0$ and $\mu_{t=1}=\mu_1$.
3. $v:[0,1]\times\mathbb R^n\to\mathbb R^n$ is Borel measurable.
4. The kinetic action is finite:
5. The continuity equation $\partial_t\mu_t+\operatorname{div}(v_t\mu_t)=0$ holds in the sense that, for every $\phi\in C_c^\infty((0,1)\times\mathbb R^n)$,