of the heat equation with initial condition $\nu$.

Linear Boltzmann equations {#linear-boltzmann-equations .unnumbered}
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We do not need any particular hypotheses on the velocity space ${{\mathcal V}}$ that is assumed to be a Polish space, i.e. a metrizable complete and separable topological space. We denote by ${{\mathcal P}}({{\mathbb T}}^d\times {{\mathcal V}})$ the set of probabilities on ${{\mathbb T}}^d\times {{\mathcal V}}$, that we consider endowed with the topology of weak convergence. We suppose given a Borel probability measure $\pi$ on ${{\mathcal V}}$, a symmetric scattering kernel $\sigma$, i.e. a Borel function $\sigma\colon {{\mathcal V}}\times {{\mathcal V}} \to [0,+\infty)$ satisfying $\sigma(v,v')=\sigma(v',v)$, $v,v'\in
{{\mathcal V}}$, and a *drift* $b\colon {{\mathcal V}}\to {{\mathbb R}}^d$. Given $P\in {{\mathcal P}}({{\mathbb T}}^d\times{{\mathcal V}})$, we denote by ${{\mathcal H}}(P)$ the relative entropy of $P$ with respect to the probability ${\mathop{}\!\mathrm{d}}x\,\pi({\mathop{}\!\mathrm{d}}v)$ namely, ${{\mathcal H}}(P)
= \iint \!{\mathop{}\!\mathrm{d}}x \pi({\mathop{}\!\mathrm{d}}v)\, f \log f $ if ${\mathop{}\!\mathrm{d}}P=f \,{\mathop{}\!\mathrm{d}}x\,\pi({\mathop{}\!\mathrm{d}}v)$ and ${{\mathcal H}}(P)=+\infty$ otherwise.

Also in this case we start by an informal discussion. Fix $T>0$. Given a path $(P_t)_{t\in[0,T]}$ on ${{\mathcal P}}({{\mathbb T}}^d\times {{\mathcal V}})$ with ${\mathop{}\!\mathrm{d}}P_t = f(t,x,v) \, {\mathop{}\!\mathrm{d}}x
\,\pi({\mathop{}\!\mathrm{d}}v)$, we use the shorthand notation $f=f(t,x,v)$, $f'=f(t,x,v')$ and set $$\label{etaf}
  \eta^f =\eta^f(t,x,v,v') 
  := \sigma (f-f') = \sigma(v,v') \big[ f(t,x,v)-f(t,x,v')\big].$$ We then 