The object of this paper is to provide a new and systematic tauberian approach to quantitative long time behaviour of
C
0
C_{0}
-semigroups
(
V
(
t
)
)
t
⩾
0
\left (\mathcal {V}(t)\right )_{t \geqslant 0}
in
L
1
(
T
d
×
R
d
)
L^{1}(\mathbb {T}^{d}\times \mathbb {R}^{d})
governing conservative linear kinetic equations on the torus with general scattering kernel
k
(
v
,
v
′
)
\boldsymbol {k}(v,v’)
and degenerate (i.e. not bounded away from zero) collision frequency
σ
(
v
)
=
∫
R
d
k
(
v
′
,
v
)
m
(
d
v
′
)
\sigma (v)=\int _{\mathbb {R}^{d}}\boldsymbol {k}(v’,v)\boldsymbol {m}(\mathrm {d}v’)
, (with
m
(
d
v
)
\boldsymbol {m}(\mathrm {d}v)
being absolutely continuous with respect to the Lebesgue measure). We show in particular that if
N
0
N_{0}
is the maximal integer
s
⩾
0
s \geqslant 0
such that
1
σ
(
⋅
)
∫
R
d
k
(
⋅
,
v
)
σ
−
s
(
v
)
m
(
d
v
)
∈
L
∞
(
R
d
)
,
\begin{equation*} \frac {1}{\sigma (\cdot )}\int _{\mathbb {R}^{d}}\boldsymbol {k}(\cdot ,v)\sigma ^{-s}(v)\boldsymbol {m}(\mathrm {d}v) \in L^{\infty }(\mathbb {R}^{d}), \end{equation*}
then, for initial datum
f
f
such that
∫
T
d
×
R
d
|
f
(
x
,
v
)
|
σ
−
N
0
(
v
)
d
x
m
(
d
v
)
>
∞
\displaystyle \int _{\mathbb {T}^{d}\times \mathbb {R}^{d}}|f(x,v)|\sigma ^{-N_{0}}(v)\mathrm {d}x\boldsymbol {m}(\mathrm {d}v) >\infty
it holds
‖
V
(
t
)
f
−
ϱ
f
Ψ
‖
L
1
(
T
d
×
R
d
)
=
ε
f
(
t
)
(
1
+
t
)
N
0
−
1
,
ϱ
f
≔
∫
R
d
f
(
x
,
v
)
d
x
m
(
d
v
)
,
\begin{equation*} \left \|\mathcal {V}(t)f-\varrho _{f}\Psi \right \|_{L^{1}(\mathbb {T}^{d}\times \mathbb {R}^{d})}=\dfrac {{\varepsilon }_{f}(t)}{(1+t)^{N_{0}-1}}, \qquad \varrho _{f}≔\int _{\mathbb {R}^{d}}f(x,v)\mathrm {d}x\boldsymbol {m}(\mathrm {d}v), \end{equation*}
where
Ψ
\Psi
is the unique invariant density of
(
V
(
t
)
)
t
⩾
0
\left (\mathcal {V}(t)\right )_{t \geqslant 0}
and
lim
t
→
∞
ε
f
(
t
)
=
0
\lim _{t\to \infty }{\varepsilon }_{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
(
V
(
t
)
)
t
⩾
0
\left (\mathcal {V}(t)\right )_{t \geqslant 0}
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.
