V}}$ defined by $$\label{tpi}
\tilde \pi({\mathop{}\!\mathrm{d}}v) :=\frac{\lambda(v)}{\pi(\lambda)}\pi({\mathop{}\!\mathrm{d}}v).$$

$\phantom{i}$\[assumpt1\]

-   The drift $b:{{\mathcal V}}\to\mathbb R^d$ is centered with respect to the measure $\pi$, namely $\pi(b)=0$.

-   The scattering rate $\lambda$ satisfies $\pi[\lambda=0]=0$.

-   $|b|^2/\lambda$ has all exponential moments, i.e. $\pi[\exp \{\gamma |b|^2/\lambda\}]< +\infty$ for any $\gamma>0$.

-   There exists a constant $C_0>0$ such that for any $g\in L^2(\tilde \pi)$ $$\label{pin}
      \int {\mathop{}\!\mathrm{d}}\tilde\pi \big[ g-\tilde\pi(g)\big]^2\leq C_0\iint\tilde\pi({\mathop{}\!\mathrm{d}}v)\tilde\pi({\mathop{}\!\mathrm{d}}v')\frac {\sigma(v,v')}
      {\lambda(v)\lambda(v')}\big[g(v)-g(v') \big]^2.$$

We remark that, as in the case of phonon Boltzmann equation, in item (ii) we allow the case in which $\lambda(v)=0$ for some $v\in {{\mathcal V}}$. Item (iv) corresponds to the assumption that the continuous time Markov chain with transition rates $\frac
{\sigma(v,v')} {\lambda(v)\lambda(v')}\tilde\pi({\mathop{}\!\mathrm{d}}v')$ has spectral gap. The generator of this Markov chain is $(K-{{1 \mskip -5mu {\rm I}}})$, where $K$ is given by $$\label{def:K}
\big(K g\big)(v)=\pi(\lambda)\int  \tilde\pi({\mathop{}\!\mathrm{d}}v')\frac{\sigma(v,v')}{ \lambda(v)\lambda(v')}g(v').$$ Observe that $$- \big({{\mathcal L}} f\big)(v)=\lambda(v) \big[\big({{1 \mskip -5mu {\rm I}}}-K\big) f\big](v),$$ where ${{\mathcal L}}$ is the generator of the original Markov chain as defined in . We emphasize that we do not assume the spectral gap of the generator ${{\mathcal L}}$, in fact the linear phonon Boltzmann equation, that will be discussed in the next section, meets the requirements in Assumption \[assumpt1\] but 