before and deduce that $(P, \Theta)$ solves the linear Boltzmann equation according to Definition \[t:dlbe\]. Existence in the general case of $Q$ satisfying the relative entropy bound ${{\mathcal H}} (Q) <+\infty$ and for coefficients $b$ and $\sigma$ satisfying Assumption \[t:asb\] is then achieved by items [(ii)]{} and [(iii)]{}.

Diffusive limit
===============

In this Section we discuss the asymptotic behavior of linear Boltzmann equation, showing that in the diffusive scaling limit the marginal distribution of the position evolves according to the heat flow. This is a classical topic and has been much investigated in the literature, see [@LK; @BLP; @BSS; @EP] and [@DGP] for a more general setting. We observe that this issue has natural counterpart in probabilistic terms namely, the central limit for additive functional of Markov chains. Indeed, the linear Boltzmann equation is the Fokker-Planck equation for the Markov process $(V_t,X_t)$ where $V_t$ is the continuous time Markov chain on ${{\mathcal V}}$ with transition rates $\sigma(v',v)\pi({\mathop{}\!\mathrm{d}}v')$ while $X_t$ is the ${{\mathbb R}}^d$-valued additive functional $X_t=\int_0^t\!{\mathop{}\!\mathrm{d}}s\,
b(V_s)$. We refer to [@KLO] for a recent monograph on this topic.

The gradient flow formulation of linear Boltzmann equations discussed before allows a novel approach to the analysis of the diffusive limit. According 