\big] -1 \big\}. 
  \end{split}$$ Replacing $g$ by $\gamma g$ with $\gamma>0$, using , Assumption \[t:asb\], the basic entropy inequality, and ${{\mathcal R}} (P^n,\Theta^n) \le {{\mathcal H}}(Q^n)$, we obtain that there exists a constant $C$ independent on $n$, $t,s$ such that $$\begin{split}
     & \Big| \int_{[s,t]\times {{\mathbb T}}^d\times {{\mathcal V}}\times {{\mathcal V}}} 
    \Theta^n({\mathop{}\!\mathrm{d}}\tau, {\mathop{}\!\mathrm{d}}x, {\mathop{}\!\mathrm{d}}v, {\mathop{}\!\mathrm{d}}v')
    \big[ g(x,v) - g(x,v')\big] \Big| 
    \\ & \qquad 
    \le \frac 1\gamma \, \sup_n {{\mathcal H}}(Q^n)  
    + \frac C\gamma \, |t-s| \, \exp\{ 2 \gamma \| g\|_\infty\}. 
  \end{split}$$ By choosing $\gamma = (2 \|g\|_\infty)^{-1} \log
  (1/|t-s|)$ when $|t-s|\le 1$ the bound follows.

In view of the second statement in Lemma \[t:lem\], the proof of item (iii) is achieved by the same arguments.

We finally prove the existence result. Consider first the case in which $b$ and $\sigma$ are continuous and bounded. If $Q({\mathop{}\!\mathrm{d}}x, {\mathop{}\!\mathrm{d}}v) =f_0(x,v) {\mathop{}\!\mathrm{d}}x \pi({\mathop{}\!\mathrm{d}}v)$ for some continuous density $f_0$ uniformly bounded away from zero, by classical results, the linear Boltzmann equation has a continuous solution $f(t,x,v)$ uniformly bounded away from zero. Set $P_t({\mathop{}\!\mathrm{d}}x, {\mathop{}\!\mathrm{d}}v) := f(t,x,v) {\mathop{}\!\mathrm{d}}x \pi({\mathop{}\!\mathrm{d}}v)$ and $\Theta({\mathop{}\!\mathrm{d}}t,
  {\mathop{}\!\mathrm{d}}x, {\mathop{}\!\mathrm{d}}v, {\mathop{}\!\mathrm{d}}v') := {\mathop{}\!\mathrm{d}}t {\mathop{}\!\mathrm{d}}x \pi({\mathop{}\!\mathrm{d}}v) \pi ({\mathop{}\!\mathrm{d}}v') \sigma
  (v,v') \big[ f(t,x,v)- f(t,x,v')\big]$. It is then straightforward to justify the informal computations presented 