the coercive properties of the relative entropy and Prohorov theorem, that there exist a compact ${{\mathcal K}}
  \subset\subset {{\mathcal P}}({{\mathbb T}}^d\times {{\mathcal V}})$ such that $P^n_t\in {{\mathcal K}}$ for any $n$ and $t\in[0,T]$. Hence, by Ascoli-Arzelà theorem, to prove the precompactness of $\{P^n\}$ it is enough to show that for each continuous $g\colon {{\mathbb T}}^d\times {{\mathcal V}}\to {{\mathbb R}}$ with compact support and continuously differentiable with respect to $x$ we have $$\label{equic} 
    \lim_{\delta\downarrow 0} \, \sup_n\, \sup_{|t-s|<\delta} 
    \big| P^n_t(g) - P^n_s(g) \big| =0.$$ From the balance equation we deduce $$\begin{split}
    & P^n_t(g) - P^n_s(g) = 
    - \int_s^t\!{\mathop{}\!\mathrm{d}}\tau\, P^n_\tau\big( b \cdot \nabla_x g \big) 
    \\
    &\qquad \qquad 
    -\frac 12 \int_{[s,t]\times {{\mathbb T}}^d\times {{\mathcal V}}\times {{\mathcal V}}} 
    \Theta^n({\mathop{}\!\mathrm{d}}\tau, {\mathop{}\!\mathrm{d}}x, {\mathop{}\!\mathrm{d}}v, {\mathop{}\!\mathrm{d}}v')
    \big[ g(x,v) - g(x,v')\big].
    \end{split}$$ By Assumption \[t:asb\], , and the basic entropy inequality, the first term on the right hand side vanishes as $|t-s|\to 0$ uniformly in $n$. On the other hand, by choosing $\alpha=1$ in the variational representation , $$\begin{split}
     & \Big| \int_{[s,t]\times {{\mathbb T}}^d\times {{\mathcal V}}\times {{\mathcal V}}} 
    \Theta^n({\mathop{}\!\mathrm{d}}\tau, {\mathop{}\!\mathrm{d}}x, {\mathop{}\!\mathrm{d}}v, {\mathop{}\!\mathrm{d}}v')
    \big[ g(x,v) - g(x,v')\big] \Big| 
    \le  {{\mathcal R}} (P^n,\Theta^n)
    \\ 
    & + \quad 
    2 \int_s^t\!{\mathop{}\!\mathrm{d}}\tau \iiint \! P_\tau({\mathop{}\!\mathrm{d}}x,{\mathop{}\!\mathrm{d}}v)\pi({\mathop{}\!\mathrm{d}}v') \, 
    \sigma(v,v') 
    \big\{ {\mathop{\rm ch}\nolimits}\big[ g(x,v) - g(x,v')  