from simply by letting ${{\mathcal R}}_0
\colon C_\mathrm{be}([0,T]; {{\mathcal P}}({{\mathbb T}}^d\times {{\mathcal V}}))\to [0,+\infty]$ be the functional defined by ${{\mathcal R}}_0(P) = \inf_{\Theta} {{\mathcal R}}(P,\Theta)$ where the infimum is carried out over all $\Theta\in {{\mathcal M}}_\mathrm{a}([0,T]\times {{\mathbb T}}^d\times {{\mathcal V}}\times {{\mathcal V}})$ such that the pair $(P,\Theta)$ satisfies the balance equation . It is however unclear to us whether ${{\mathcal R}}_0(P)$ could be represented by the metric derivative of $t\mapsto P_t$ with respect to a suitable distance on ${{\mathcal P}}({{\mathbb T}}^d\times {{\mathcal V}})$.

We now state an existence and uniqueness result for the above formulation, together with a continuous dependence on the initial condition and the coefficients. In particular uniqueness implies that solutions to with bounded entropy are characterized by the gradient flow formulation in Definition \[t:dlbe\].

\[t:eu\] For each $Q\in{{\mathcal P}}({{\mathbb T}}^d\times {{\mathcal V}})$, with ${{\mathcal H}}(Q)< +\infty$, there exists a unique solution $(P,\Theta)$ to the linear Boltzmann equation with initial condition $Q$. Furthermore

-   Set $\Theta^P({\mathop{}\!\mathrm{d}}t,{\mathop{}\!\mathrm{d}}x, {\mathop{}\!\mathrm{d}}v, {\mathop{}\!\mathrm{d}}v'):={\mathop{}\!\mathrm{d}}t\,\sigma
        (v, v')\big[P_t({\mathop{}\!\mathrm{d}}x, {\mathop{}\!\mathrm{d}}v)\pi ({\mathop{}\!\mathrm{d}}v')-\pi ({\mathop{}\!\mathrm{d}}v) P_t({\mathop{}\!\mathrm{d}}x,
        {\mathop{}\!\mathrm{d}}v') \big]$. Then $\Theta=\Theta^P$.

-   Let $\{Q^n\}\subset {{\mathcal P}}({{\mathbb T}}^d\times {{\mathcal V}})$ be such that $Q^n\to Q$ and ${{\mathcal H}}(Q^n)\to {{\mathcal H}}(Q)$ and denote by $(P^n,\Theta^n)\in {{\mathcal S}}_\mathrm{be}$ the solution to the linear Boltzmann 