  The object of this paper is to provide a new and systematic tauberian
approach to quantitative long time behaviour of $C_{0}$-semigroups
$\left(\mathcal{V}(t)\right)_{t \geq0}$ in $L^{1}(\mathbb{T}^{d}\times
\mathbb{R}^{d})$ governing conservative linear kinetic equations on the torus
with general scattering kernel $ {k}(v,v')$ and degenerate (i.e. not bounded
away from zero) collision frequency $\sigma(v)=\int_{\mathbb{R}^{d}}
{k}(v',v)m(\mathrm{d} v')$, (with $m(\mathrm{d} v)$ being absolutely continuous
with respect to the Lebesgue measure). We show in particular that if $N_{0}$ is
the maximal integer $s \geq0$ such that
$$\frac{1}{\sigma(\cdot)}\int_{\mathbb{R}^{d}}k(\cdot,v)\sigma^{-s}(v)m(\mathrm{d}
v) \in L^{\infty}(\mathbb{R}^{d})$$ then, for initial datum $f$ such that
$\mathrm{d} s\int_{\mathbb{T}^{d}\times
\mathbb{R}^{d}}|f(x,v)|\sigma^{-N_{0}}(v)\mathrm{d} x m(\mathrm{d} v) <\infty$
it holds
$$\left\|\mathcal{V}(t)f-\varrho_{f}\Psi\right\|_{L^{1}}=\dfrac{{\epsilon}_{f}(t)}{(1+t)^{N_{0}-1}},
\qquad \varrho_{f}:= \int_{\mathbb{R}^{d}}f(x,v)m(\mathrm{d} v)$$ where $\Psi$
is the unique invariant density of $\left(\mathcal{V}(t)\right)_{t \geq0}$ and
$\lim_{t\to\infty}{\epsilon}_{f}(t)=0$. We in particular provide a new criteria
of the existence of invariant density. The proof relies on the explicit
computation of the time decay of each term of the Dyson-Phillips expansion of
$\mathcal{V}(t)$ and on suitable smoothness and integrability properties of the
trace on the imaginary axis of Laplace transform of remainders of large order
of this Dyson-Phillips expansion. Our construction resorts also on collective
compactness arguments and provides various technical results of independent
interest. Finally, as a by-product of our analysis, we derive essentially sharp
``subgeometric'' convergence rate for Markov semigroups associated to general
transition kernels.
