In both cases, the fact that $a \in \text{Lip}$ ensures that $\varpi_q := \varpi + (1 - 1/q)a'$ enjoys $(\varpi_q)_- \in L^\infty$ for any $q \in [1, \infty)$. On the other hand, (9.4) guarantees that $K \lesssim \langle \varpi_+ \rangle$ and $a/x \lesssim \langle \varpi_+ \rangle$, and finally (8.35) is verified. The condition (8.61) is equivalent to the Lyapunov type condition

$$ (9.11) \qquad \mathcal{H}^*[m] \le (\alpha_{\mathcal{H}} \varpi_+ + M_{\mathcal{H}})m, $$

where $\mathcal{H}^*[m](x) = 2K(x)m(x/2)$. For $m(x) = x^r$ with $r > 1$, we compute

$$ \mathcal{H}^*[m]/m = 2^{1-r}K, $$

and for $m(x) = \exp(\eta \int_{x_0}^x K/a)$ with $0 < \eta < 1$,

$$ \frac{\mathcal{H}^*[m](x)}{m(x)} = 2 \exp \left( - \eta \int_{x/2}^x \frac{K}{a} \right) K(x). $$

Using (9.4), we obtain that (9.11) is satisfied, for any $\alpha_{\mathcal{H}} \in (2^{1-r}, 1)$ in the first case, and for any $\alpha_{\mathcal{H}} \in (0, 1)$ in the second case, by choosing $M_{\mathcal{H}}$ large enough.

We can then apply Proposition 8.23-(1) for associating to equation (9.1) a strongly continuous semigroup $S$ in $X = L_m^1(x_0, \infty)$, and (H1) then follows from Lemma 2.2-(i). Moreover, we readily have that $\kappa_1 \le \kappa + M$ for any couple $(\kappa, M)$ such that (9.10) is verified.

**Condition (H2).** We aim at verifying (H2) for some $\kappa_0 > 0$. Recalling assumption (9.5), we pick up $\ell > k$ and we consider the function

$$ \phi_0(x) = xe^{-x^\ell/n} $$

with $n$ large enough to be chosen later. We compute

$$ \frac{\mathcal{L}^*\phi_0(x)}{\phi_0(x)} = \frac{a(x)}{x} \left(1 - \frac{\ell}{n}x^\ell\right) + K(x) \left(e^{\frac{1-2^{-\ell}}{n}x^\ell} - 1\right). $$

Choosing $R > x_0$ such that $xK(x)/a(x) \ge \frac{2\ell}{1-2^{-\ell}}$ and $K(x) \ge e^{-x^k}$ for all $x \ge R$, we get that

$$ \begin{aligned} \frac{\mathcal{L}^*\phi_0(x)}{\phi_0(x)} &\ge \frac{a(x)}{x} + K(x) \left( e^{\frac{1-2^{-\ell}}{n}x^\ell} - 1 - \frac{1-2^{-\ell}}{2n}x^\ell \right) \\ &\ge e^{-x^k} \left( e^{\frac{1-2^{-\ell}}{n}x^\ell} - e^{\frac{1-2^{-\ell}}{2n}x^\ell} \right) \ge e^{-\frac{1-2^{-\ell}}{2n}x^\ell} - e^{-\frac{1-2^{-\ell}}{2n}x^\ell} \left( e^{\frac{1-2^{-\ell}}{2n}R^\ell} - 1 \right) \end{aligned} $$

on $[R, \infty)$. Choosing then $n \ge \frac{\ell}{2}R^\ell$, we have

$$ \frac{\mathcal{L}^*\phi_0(x)}{\phi_0(x)} \ge \frac{a(x)}{2x} $$

on $(x_0, R)$. Gathering the two above estimates, we deduce the existence of an explicit $\kappa_0 > 0$ such that $\mathcal{L}^*\phi_0 \ge \kappa_0\phi_0$. We conclude by invoking Lemma 2.4-(i).

**Condition (H3).** We consider the weight function $m(x) = x^r$ for some $r > 1$ or $m(x) = \exp(\eta \int_{x_0}^x K/a)$ with $0 < \eta < 1$ and we define the stronger weight function $m_1(x) = \exp(\eta_1 \int_{x_0}^x K/a)$ for some $\eta_1 \in (\eta, 1)$. We fix $\kappa_B \in [0, \kappa_0)$, $M > 0$, and $R > x_0$ such that (9.10) is verified by $m_1$ with $\kappa = \kappa_B$. Using the splitting $\mathcal{L} = \mathcal{A} + \mathcal{B}$ with $\mathcal{A}f = M1_{(x_0,R)}f$, the inequality (9.10) for $m_1$ reads $\mathcal{B}^*m_1 \le \kappa_B m_1$ and this ensures (see the proof of Corollary 2.20) that $\kappa - \mathcal{B}$ is invertible in $L_{m_1}^1$ for any $\kappa > \kappa_B$, with positive inverse, and

$$ \|(\kappa - \mathcal{B})^{-1}\|_{\mathcal{B}(L_{m_1}^1)} \le \frac{1}{\kappa - \kappa_B}. $$

The operator $\mathcal{A}$ maps $L_m^1$ into $L_{m_1}^1$ with

$$ \|\mathcal{A}\|_{\mathcal{B}(L_m^1, L_{m_1}^1)} \le M \frac{m_1(R)}{m(x_0)}. $$

Besides, due to the derivative part $\partial_x(a \cdot)$ in the operator $\mathcal{B}$, we also have that $\mathcal{R}_B(\kappa)$ maps $L_{m_1}^1$ into $W_{\text{loc}}^{1,1}$. Finally, we have $\mathcal{R}_B(\kappa)\mathcal{A} : L_m^1 \to L_{m_1}^1 \cap W_{\text{loc}}^{1,1}$, and thus $\mathcal{R}_B(\kappa)\mathcal{A} \in \mathcal{H}(L_m^1)$, for any $\kappa \ge \kappa_0 > \kappa_B$. We deduce from Lemma 2.8-(2) that the condition (H3) holds for both the primal and the dual problems.
**Proof of Proposition 9.5. Step 1.** The rescaled semigroup $\tilde{S}$ is a contraction semigroup in $M_{\phi_1}^1 = (C_{0,\phi_1})'$. This ensures in particular that for all $f \in M_{\phi_1}^1$ the trajectory $(\tilde{S}_t f)_{t \ge 0}$ is bounded in $M_{\phi_1}^1$. We can thus use Theorem 5.23-(2) to infer the non-triviality of the boundary spectrum, by proving that the conclusion cannot hold, see Remark 5.26-(ii). We start from the fact that for any $\phi \in C_{0,\phi_1}(x_0, \infty)$, the solution $S^*\phi$ to the dual mitosis equation is the unique fixed point of $\Gamma$ defined in (9.20), and that it belongs to any closed invariant subset of $C([0, \infty) \times (x_0, \infty))$. For $y > x_0$ we define the set

$$ \mathcal{E}_y = \{x > x_0, x = 2^k y \text{ for some } k \in \mathbb{Z}\} $$

and we consider a function $\phi$ such that $\phi(x) = 0$ if $x \in \mathcal{E}_y$ and $\phi(x) > 0$ if $x \notin \mathcal{E}_y$. Then, since (9.19) ensures that $X_t(2x) = 2X_t(x)$ for all $t \ge 0$, the set

$$ \{\varphi \in C([0, \infty) \times (x_0, \infty)), \varphi(t, x) = 0 \text{ if } X_t(x) \in \mathcal{E}_y \text{ and } \varphi(t, x) > 0 \text{ if } X_t(x) \notin \mathcal{E}_y\} $$

is invariant under $\Gamma$. Consequently, the unique fixed point $S_t^*\phi$ belongs to this set, and we deduce that $S_t^*\phi(x) = 0$ if and only if $X_t(x) \in \mathcal{E}_y$. In other words, by duality, $\text{supp}(S_t\delta_x) \subset \mathcal{E}_y$ if and only if $X_t(x) \in \mathcal{E}_y$, and in particular $\text{supp}(S_t\delta_x) \subset \mathcal{E}_{X_t(x)}$ for all $x > x_0$ and $t \ge 0$. This prevents the convergence of $\tilde{S}_t\delta_x$ toward $\langle \delta_x, \phi_1 \rangle f_1$ and we infer from (the negation of) Theorem 5.23-(2) that the boundary point spectrum cannot be trivial.

**Step 2.** We next formulate the following simple but fundamental observation. If $(\lambda, f)$ is a solution to the eigenvalue problem in $X = M_m^1$, then $f \in D(\mathcal{L}) \subset BV_{\text{loc}} \subset L_{\text{loc}}^1$ and it is a solution to the eigenvalue problem in $X = L_m^1$. Symmetrically, if $(\lambda, \phi)$ is a solution to the eigenvalue problem in $Y = L_{m-1}^\infty$, then $\phi \in D(\mathcal{L}^*) \subset \text{Lip}_{\text{loc}} \cap L_{m-1}^\infty$, $m_1(x)/m_2(x) \to 0$ as $x \to \infty$, and it is a solution to the eigenvalue problem in $Y = C_{0,m_1}$. In other words, the point spectrum and the associated eigenelements are the same in the two frameworks $(L_m^1, L_{m-1}^\infty)$ and $(M_m^1, C_{0,m})$. Now, as a consequence of this observation and Step 1, we know that the boundary point spectrum $\Sigma_P^+(\mathcal{L})$ is not trivial. Because we have proved that $(\kappa - B)^{-1}\mathcal{A}$ is compact in $L_m^1$, $\kappa < \lambda_1$, we may apply Theorem 5.7 and we obtain that $\Sigma_P^+(\mathcal{L}) = \{\lambda_1\} + i\alpha\mathbb{Z}$ for some $\alpha > 0$, and each eigenvalue is algebraically simple. Using finally Theorem 5.25 in the situation (2), we get the weak-* convergence (9.21). $\square$

This result is proved by means of entropy techniques in [169] for a linear growth rate $a(x) = x$, by taking advantage of the explicit formulation of the eigenvectors $g_k$ and $\psi_k$ in terms of $f_1$ and $\phi_1$ in that case. Here we extend it to any $a$ satisfying $a(2x) = 2a(x)$. Note that arguing similarly as in [169], the convergence (9.21) may be strengthened into an exponential strong convergence in $M_m^1$ for $m(x) = x^r$, $r > 1$, or $m(x) = \exp(\eta \int_{x_0}^x K/a)$, $0 < \eta < 1$, meaning that there is a spectral gap between $\Sigma_+(\mathcal{L})$ and the rest of the spectrum in these spaces.

**Long-time convergence in $X_p$.** We prove the following result, the case $p = 1$ of which corresponds to the convergence result of Theorem 9.4.

**Proposition 9.6.** Under the same assumptions as in Proposition 9.5, the convergence (9.21) holds for the strong topology in $X_p$, $1 \le p < \infty$ for all $f \in X_p$, and the convergence of the Fejér sum in the definition of the projector $\Pi$ is also for this topology.

**Proof.** The case $p = 1$ is an immediate consequence of Theorem 5.25, case (4). The proof in the case $p > 1$ is a direct adaptation of the case $p = 1$. We aim at verifying that the trajectories $(\tilde{S}_t f)_{t \ge 0}$ are relatively compact in $X_p$. We have already seen that $X_\infty \subset X_p$ is dense. Besides, the domain $D(\mathcal{L})$ of the generator $\mathcal{L} - \lambda_1$ of $\tilde{S}$ in $X_p$ is also dense in $X_p$, so that it suffices to check the relative compactness of $(\tilde{S}_t f)_{t \ge 0}$ for $f \in X_\infty \cap D(\mathcal{L})$. For $f$ in $X_\infty \cap D(\mathcal{L})$ the bounds

$$ \|\tilde{S}_t f\|_{X_p} \le \|f\|_{X_p}, \quad \|\mathcal{L}\tilde{S}_t f\|_{X_p} = \|\tilde{S}_t \mathcal{L} f\|_{X_p} \le \|\mathcal{L} f\|_{X_p} \quad \text{and} \quad \|\tilde{S}_t f\|_{X_\infty} \le \|f\|_{X_\infty} $$

yield the relative compactness of $(\tilde{S}_t f)_{t \ge 0}$, the second bound guaranteeing uniform $W_{\text{loc}}^{1,1}$ estimates. We can thus apply the case (1) of Theorem 5.25 to deduce the convergence (9.21) in $X_p$ for the strong topology. $\square$

**Proposition 9.6** extends the result of [50] where it is proved in the case $p = 2$ for $a(x) = x$ by taking advantage of the Hilbert structure of $X_2$ and of the explicit formulation of the eigenvectors $g_k$ and
**Proof of the existence part of Theorem 9.1.** We deduce from Theorem 2.21 that the conclusion (S1) about the existence of a solution $(\lambda_1, f_1, \phi_1) \in \mathbb{R} \times X_+ \times X'_+$ to the first eigentriplet problem holds true. □

Moreover, we have $\lambda_1 \ge \kappa_0 > 0$ and $f_1 \in W_{\text{loc}}^{1,1} \cap L_m^1$ with $m(x) = \exp(\eta \int_{x_0}^x K/a)$ for any $\eta \in (0, 1)$.

**Additional estimates on the eigenfunction $\phi_1$.** We consider the weight function $m(x) = x^r$ for some $r > 1$ or $m(x) = \exp(\eta \int_{x_0}^x K/a)$ with $0 < \eta < 1$ and we define the weaker weight function $m_0(x) = x^{r_0}$ for some $r_0 \in (1, r)$. We fix $\kappa_B \in [0, \kappa_0)$, $M > 0$, and $R > x_0$ such that (9.10) is verified by $m_0$. Using again the splitting $\mathcal{L} = \mathcal{A} + \mathcal{B}$ with $\mathcal{A}f = M1_{(x_0, R)}f$, (9.10) means that $\mathcal{B}^*m_0 \le \kappa_B m_0$ and this ensures that for any $\kappa > \kappa_B$ the operator $\kappa - \mathcal{B}^*$ is invertible in $L_{m_0}^\infty$, with positive inverse, and

$$ \|(\kappa - \mathcal{B}^*)^{-1}\|_{\mathcal{B}(L_{m_0}^\infty)} \le \frac{1}{\kappa - \kappa_B}. $$

Because of the derivative part of $\mathcal{B}^*$, we also have that $\mathcal{R}_{\mathcal{B}^*}(\kappa) : L_{m_0}^\infty \to W_{\text{loc}}^{1,\infty}$. Besides, the operator $\mathcal{A}^* = \mathcal{A}$ maps $L_{m_0}^\infty$ into $L_{m_0}^\infty$ with

$$ \|\mathcal{A}^*\|_{\mathcal{B}(L_{m_0}^\infty, L_{m_0}^\infty)} \le M \frac{m(R)}{m_0(x_0)}. $$

Finally we have that $\mathcal{R}_{\mathcal{B}^*}(\kappa)\mathcal{A} : L_{m_0}^\infty \to L_{m_0}^\infty \cap W_{\text{loc}}^{1,\infty}$. Consequently $\phi_1 \in L_{m_0}^\infty \cap W_{\text{loc}}^{1,\infty}$ and

$$ (9.12) \qquad \|\phi_1\|_{L_{m_0}^\infty} = \|(\lambda_1 - \mathcal{B}^*)^{-1} \mathcal{A}^* \phi_1\|_{L_{m_0}^\infty} \le \frac{m(R)}{m_0(x_0)} \frac{M}{\kappa_0 - \kappa_B} \|\phi_1\|_{L_{m_0}^\infty}. $$

We also easily deduce quantitative estimates of $\phi_1$ in $W_{\text{loc}}^{1,\infty}$ from the identity

$$ \phi_1'(x) = \frac{1}{a(x)} \left[ \lambda_1 \phi_1(x) + K(x) \phi_1(x) - 2K(x) \phi_1(x/2) \right]. $$

**Condition (H4).** The operator $\mathcal{L}$ satisfies the strong maximum principle. Let $\lambda \in \mathbb{R}$ and $f \in X_+ \cap D(\mathcal{L}) \setminus \{0\}$ such that $(\lambda - \mathcal{L})f \ge 0$, i.e.

$$ \lambda f(x) + (af)'(x) + K(x)f(x) \ge 4K(2x)f(2x) \quad \forall x > x_0. $$

Denoting by $\Lambda_\lambda$ a function such that $\Lambda'_\lambda(x) = \frac{\lambda+K(x)}{a(x)}$, we get that

$$ (9.13) \qquad a(x)f(x) \ge 4 \int_{x_0}^{x} e^{\Lambda_{\lambda}(y)-\Lambda_{\lambda}(x)} K(2y)f(2y) dy. $$

Since $K(2y) > 0$ for all $y > x_0$, $f \in X_+ \setminus \{0\}$, and $a(x) > 0$ for all $x > x_0$, we deduce from (9.13) that the set $\{x > x_0, f(x) > 0\}$ is an interval of the form $(\underline{x}, +\infty)$. Using again (9.13) we remark that we must have $\underline{x} = \max(x_0, \underline{x}/2)$, which enforces $\underline{x} = x_0$ and finally $f > 0$.

**Proof of the uniqueness and positivity part of Theorem 9.1.** We deduce from Theorem 4.13 the validity of the conclusion (S2) about uniqueness and positivity of the solution $(\lambda_1, f_1, \phi_1)$ to the first eigentriplet problem. □

For deriving the exponential stability, we start by verifying a quantified irreducibility and aperiodicity condition on $S$, given in the next lemma, which then allows us to prove that the Doblin-Harris condition (6.9) is met.

**Lemma 9.2.** Assume that (9.6) is satisfied. Then for all $\varepsilon > 0$, $R_1 > x_0$, and $R_2 > x_0 + \varepsilon$, there exists $T_1 > 0$ such that for any $T > T_1$, there exists $c_T > 0$ such that

$$ S_T^* \phi \ge c_T \mathbf{1}_{(x_0, R_1)} \int_{x_0+\varepsilon}^{R_2} \phi \, dx, \quad \forall \phi \ge 0. $$

**Proof of Lemma 9.2.** Throughout the proof we denote by $c_t$ any positive constant that depends only on $t$. It is proved in [353, Prop. 5] the existence of $(x_2, x_3) \subset (x_0, \infty)$ such that for all $R_1 > x_0$ there exists $T_0 > 0$ such that for any $T > T_0$ and any $\phi \ge 0$

$$ (9.14) \qquad S_T^* \phi \ge c_T \mathbf{1}_{(x_0, R_1)} \int_{x_2}^{x_3} \phi(x) dx. $$
$\psi_k$ in terms of $f_1$ and $\phi_1$. In this Hilbert setting it is proved that the Fourier series $\sum_{k=-n}^n \langle f, \psi_k \rangle g_k$ converges as $n$ goes to infinity, and $\Pi f$ is then given by the limit.

**About the value of $\alpha$.** For ensuring that the boundary spectrum is discrete, we have used a compactness argument. The period $2\pi/\alpha$ of the periodic semigroup $\tilde{S}\Pi$ is thus not quantified. It is expected to be equal to the time needed for a particle to double its size by following the flow of $a$, namely

$$ (9.22) \qquad \frac{2\pi}{\alpha} = \int_x^{2x} \frac{dt}{a(t)}, $$

which is independent of the choice of $x \ge x_0$ due to the condition $a(2t) = 2a(t)$. This is known to be true in the case of a linear growth rate $a(x) = x$, see [134] or [50], and also for $a$ general when the size domain is $(x_0, 4x_0)$, see [188], where explicit computations can be carried out. In the general case, we have not been able to prove (9.22). Yet the fact that for any $x > x_0$ the support of $S_t\delta_x$ is a subset of $\mathcal{E}_{X_t(x)}$ guarantees that the period cannot be too small as shown now.

**Proposition 9.7.** We have the estimate

$$ (9.23) \qquad \frac{2\pi}{\alpha} \ge \ell_a := \int_x^{2x} \frac{dt}{a(t)}. $$

*Proof of Proposition 9.7.* Let $x > x_0$ such that $\psi_1(x) \ne 0$ (actually any $x > x_0$ is suitable). We have

$\tilde{S}_t\delta_x - \tilde{S}_t\Pi\delta_x \to 0,$
as $t \to +\infty$, and $\text{supp } \tilde{S}_t\delta_x \subset \mathcal{E}_{X_t(x)}$, so $\text{supp } \tilde{S}_t\Pi\delta_x \subset \mathcal{E}_{X_t(x)}$ since $\tilde{S}_t\Pi\delta_x$ is periodic. Besides,

$$ \tilde{S}_t \Pi \delta_x = \lim_{n \to \infty} \frac{1}{n} \sum_{\ell=0}^{n} \sum_{k=-\ell}^{\ell} \psi_k(x) e^{i\alpha k t} g_k $$

and, since $\psi_1(x) \ne 0$, the period of this periodic function of time is $\frac{2\pi}{\alpha}$. But since $\text{supp } \tilde{S}_t\Pi\delta_x \subset \mathcal{E}_{X_t(x)}$ and the period of the set $\mathcal{E}_{X_t(x)}$ is $\ell_a$, we have proved that (9.23) holds. $\square$

On the other hand, we can use Theorem 6.5 for deriving a quantified lower bound on $\alpha$, and thus an upper bound on the period. We work in the space $X = L_m^1$, recalling that $\Sigma_P^+(\mathcal{L})$ is the same in this space and in $M_{\phi_1}^1$.

**Proposition 9.8.** There exists a constructive constant $\alpha_1 > 0$ such that $\Sigma(\mathcal{L}) \cap B(\lambda_1, \alpha_1) = \{\lambda_1\}$. In particular, $\alpha \ge \alpha_1$.

To prove this result, we check that the conditions (6.7), (6.9) and (6.13) are verified, and we invoke Theorem 6.5. We start with a lemma which, together with Lemma 9.3, will guarantee the validity of (6.13).

**Lemma 9.9.** For all $\varepsilon > 0$, $R_1 > x_0$, and $R_2 > x_0 + \varepsilon$, there exist $T > 0$ and $c_T > 0$ such that

$$ \int_0^T S_t^* \phi \, dt \ge c_T \mathbf{1}_{(x_0, R_1)} \int_{x_0+\varepsilon}^{R_2} \phi \, dx, \quad \forall \phi \ge 0. $$

*Proof of Lemma 9.9.* Throughout the proof we denote by $c_t$ any positive constant that depend only on $t$. From the Duhamel formula (9.15), we get by positivity that for any $\phi \ge 0$

$$ \int_0^T S_t^* \phi(x) dt \ge \int_0^T \phi(X_t(x)) e^{-\int_0^t K(X_s(x)) ds} dt. $$

We deduce that for all $x \in (x_0, R_1)$ and for $T_0$ large enough so that $X_{T_0}(x_0) > R_2$, we have

$$ \int_0^{T_0} S_t^* \phi(x) dt \ge c_{T_0} \int_x^{X_T(x)} \phi(y) dy \ge c_{T_0} \int_{R_1}^{R_2} \phi(y) dy, $$

and the conclusion follows if $R_1 \le x_0 + \varepsilon$. If not, we have with the same argument the existence of $T$ such that for all $x \in (x_0, R_1)$

$$ \int_0^T S_t^* \phi(x) dt \ge c_T \int_{\max(R_1, 2x_0+\varepsilon)}^{2R_2} \phi(y) dy. $$
We may now extend the integral to $[x_0 + \varepsilon, R_2]$. The Duhamel formula

$$S_{\mathcal{L}}^{*} = S_{\mathcal{B}_0}^{*} + S_{\mathcal{B}_0}^{*} A_0 * S_{\mathcal{L}}^{*}$$

for the splitting $\mathcal{L}^* = A_0^* + B_0^*$ with $A_0^*\phi = \mathcal{H}^*[\phi]$ and $B_0^*\phi = b\phi' - K\phi$, also reads

$$(9.15) \quad S_t^* \phi(x) = \phi(X_t(x))e^{-\int_0^t K(X_s(x))ds} + 2 \int_0^t K(X_{t-s}(x))S_s^* \phi(X_{t-s}(x)/2)e^{-\int_0^{t-s} K(X_{s'}(x))ds'}ds,$$

where $X_t(x)$ is the solution to the characteristic equation

$$(9.16) \qquad \dot{X}_t(x) = a(X_t(x)) \quad \text{with} \quad X_0(x) = x.$$

Applying (9.14) to $S_t^*\phi$, that we bound from below by the first in Duhamel's formula (9.15), we obtain

$$S_{T+t}^{*} \phi \geq c_{T} \mathbf{1}_{(x_0, R_1)} \int_{x_2}^{x_3} \phi(X_t(x)) e^{-\int_{0}^{t} K(X_s(x)) ds} dx \geq c_{T} c_{t} \mathbf{1}_{(x_0, R_1)} \int_{X_t(x_2)}^{X_t(x_3)} \phi(y) dy.$$

Choosing $t_0$ such that $X_{t_0}(x_2) = x_3$, we get that for all $T > T_0 + t_0$

$$S_T^* \phi \geq c_T \mathbf{1}_{(x_0, R_1)} \int_{x_2}^{x_{t_0}(x_3)} \phi(x) dx.$$

Iterating this argument and using the strict positivity of $a$ we get for any $R_2 > x_2$ the existence of a time $t_1$ such that for all $T > T_0 + t_1$

$$(9.17) \qquad S_T^* \phi \geq c_T \mathbf{1}_{(x_0, R_1)} \int_{x_2}^{R_2} \phi(x) dx.$$

For decreasing the lower bound of the integral from $x_2$ to $x_0 + \varepsilon$, we iterate once Duhamel's formula (9.15) to get

$$S_t^* \phi(x) \geq 2 \int_0^t K(X_{t-s}(x)) \phi(X_s(X_{t-s}(x)/2)) e^{-\int_0^{t-s} K(X_{s'}(x))ds' - \int_0^s K(X_{s'}(x))ds'} ds$$

and then, using (9.17),

$$S_{T+t}^{*} \phi \geq c_{t} c_{T} \mathbf{1}_{(x_{0}, R_{1})} \int_{0}^{t} \int_{x_{2}}^{R_{2}} K(X_{t-s}(x)) \phi(X_{s}(X_{t-s}(x)/2)) dx \, ds.$$

We can assume that $x_2 > 2x_0$ and $R_2 > 2x_2$. The fact that $x_2 > 2x_0$ ensures, due to assumption (9.3), that $K$ is bounded from below by a positive constant on $[x_2, X_t(R_2)]$. We thus deduce, by using of a change of variables, that for any $t > 0$

$$S_{T+t}^{*} \phi \geq c_{t} c_{T} \mathbf{1}_{(x_{0}, R_{1})} \int_{X_{t}(X_{t}(x_{2})/2)}^{R_{2}/2} \phi(y) dy.$$

Since $X_t(x) \to x$ when $t \to 0$, we deduce for all $\zeta > 0$ the existence of $t > 0$ such that

$$S_{T+t}^{*} \phi \geq c_{t} c_{T} \mathbf{1}_{(x_{0}, R_{1})} \int_{x_{2}/2+\zeta}^{R_{2}/2} \phi(y) dy.$$

Since $R_2 > 2x_2$, we deduce by combining the above inequality with (9.17) that for all $T > T_0 + t_1 + t$

$$S_T^* \phi \geq c_T \mathbf{1}_{(x_0, R_1)} \int_{x_2/2+\zeta}^{R_2} \phi(x) dx.$$

Let us take $\zeta = x_0$. Since the sequence $(u_n)$ defined by $u_0 = x_2$ and $u_{n+1} = u_n/2 + x_0$ converges to $2x_0$, we obtain by an iteration argument the existence of a time $t_2$ such that for all $T > T_0 + t_2$

$$S_T^* \phi \geq c_T \mathbf{1}_{(x_0, R_1)} \int_{2x_0+\varepsilon}^{R_2} \phi(x) dx.$$

Using a last time the argument with $\zeta = \varepsilon/2$ yields the desired result. $\square$

We now prove another positivity result which allows making the time $T$ independent of $R_1$ in Lemma 9.2.
Iterating once Duhamel's formula and using that $X_t(x/2) = X_t(x)/2$ and (9.3) we get that for all $t \ge 0$ and all $y \in (2x_0 + \varepsilon, R_2)$

$$S_t^* \phi(y) \ge c_t \int_0^t K(X_{t-s}(y)) \phi(X_s(X_{t-s}(y)/2)) ds \ge c_t \phi(X_t(y/2)).$$

This yields for all $x \in (x_0, R_1)$

$$\begin{aligned} \int_0^{T+t} S_s^* \phi(x) ds &\ge \int_t^{T+t} S_s^* \phi(x) ds = \int_0^T S_s^* S_t^* \phi(x) ds \\ &\ge c_T \int_{\max(R_1, 2x_0+\varepsilon)}^{2R_2} S_t^* \phi(y) dy \\ &\ge c_T c_t \int_{\max(R_1, 2x_0+\varepsilon)}^{2R_2} \phi(X_t(y/2)) dy = c_T c_t \int_{\max(X_t(\frac{R_1}{2}), X_t(x_0+\frac{\varepsilon}{2}))}^{X_t(R_2)} \phi(z) dz. \end{aligned}$$

Choosing $t > 0$ small enough so that $\max(X_t(\frac{R_1}{2}), X_t(x_0 + \frac{\varepsilon}{2})) \le \max(\frac{R_1+\varepsilon}{2}, x_0 + \varepsilon)$, we get

$$\int_0^{T_1} S_s^* \phi(x) ds \ge c_{T_1} \int_{\max(\frac{R_1+\varepsilon}{2}, x_0+\varepsilon)}^{R_2} \phi(z) dz, \text{ for } T_1 := T+t.$$

Iterating the argument we can build an increasing sequence of times $T_n$ such that

$$\int_0^{T_n} S_t^* \phi(x) dt \ge c_{T_n} \int_{\max(u_n, x_0+\varepsilon)}^{R_2} \phi(z) dz, \quad \forall n \ge 0,$$

where $(u_n)$ is defined by $u_0 = R_1$ and $u_{n+1} = \frac{u_n+\varepsilon}{2}$. Since this sequence $(u_n)$ converges to $\varepsilon < x_0 + \varepsilon$, we get the conclusion by taking $n$ large enough. $\square$

We are now in position to prove Proposition 9.8.

*Proof of Proposition 9.8.* Arguing similarly as in the proof of Theorem 9.1 and using Lemma 9.9 instead of Lemma 9.2, we can prove that the conditions (6.7), (6.9) and (6.13) are verified. Applying Theorem 6.5 then gives the result. $\square$

**9.3. The model with variability.** In this last part, we consider the model with variability given by the equation (9.2). Compared to equation (9.1), the main consequence of introducing a variability in terms of the spectrum and the asymptotic behavior is that for equation (9.2) the boundary spectrum is trivial and the first eigenfunction is exponentially stable, no matter if a satisfies (9.6) or (9.19).

Additionally to the assumptions (9.3), (9.4) and (9.5), we ask that

$$(9.24) \qquad K(x) = O\left(\exp\left(\delta \int_{x_0}^{x/2} K/a\right)\right) \quad \text{as } x \to +\infty,$$

for some $\delta > 0$. About the variability kernel $\wp$ we suppose that

$$(9.25) \qquad \int_1^2 \wp(v, v_*) dv = 1, \forall v_* \in [1, 2], \quad \wp \in W^{1,\infty}([1, 2]^2) \quad \text{and} \quad \wp \ge \wp_*$$

for some $\wp_* > 0$. We still work in the space $X = L_m^1$ by considering the weight $m = m(x)$ as function of $(x, v)$ constant in $v$.

**Theorem 9.10.** Suppose that (9.3), (9.4), (9.5), (9.24) and (9.25) are satisfied. The first eigen-triplet problem for equation (9.2) admits a unique solution $(\lambda_1, f_1, \phi_1) \in \mathbb{R} \times X_+ \times X'_+$ with the normalization $\|\phi_1\| = \langle \phi_1, f_1 \rangle = 1$, and this triplet additionally satisfies $\lambda_1 > 0$, $f_1 > 0$ and $\phi_1 > 0$. Besides, there are some constructive constants $C \ge 1$, $\omega > 0$ such that

$$\|e^{-\lambda_1 t} S_L(t) f - \langle \phi_1, f \rangle f_1\|_X \le C e^{-\omega t} \|f - \langle \phi_1, f \rangle f_1\|_X$$

for any $f \in X$ and $t \ge 0$.
**Lemma 9.3.** Let $R_1 > 2x_0$. Then there exists $t_0 > 0$ such that for any $R > R_1$ we have

$$S_{t_0}^* \mathbf{1}_{(x_0, R_1)} \geq c_R \mathbf{1}_{(x_0, R)}$$

for some $c_R > 0$.

*Proof of Lemma 9.3.* Since $a$ is Lipschitz continuous, we can find $t_0 > 0$ small enough so that $X_{t_0}(x) \leq \alpha x$ for all $x > x_0$, with $\alpha > 1$ to be determined later. Then for any $t \in (0, t_0]$ and any $x \in (x_0, \frac{R_1}{\alpha})$, we have by using the first term in (9.15)

$$S_t^* \mathbf{1}_{(x_0, R_1)}(x) \geq c_{t_0} \mathbf{1}_{(x_0, R_1)}(X_t(x)) = c_{t_0} > 0.$$

Iterating once (9.15) and keeping only the second term, we get that for any $t \in (0, t_0]$ and any $x \in (2x_0, \frac{2R_1}{\alpha^2})$

$$S_t^* \mathbf{1}_{(x_0, R_1)}(x) \geq c_{t_0} \int_0^t \mathbf{1}_{(x_0, R_1)}(X_s(X_{t-s}(x)/2)) ds = c_{t_0} t.$$

Choosing $\alpha > 1$ such that $\frac{R_1}{\alpha} > 2x_0$ and $\frac{2R_1}{\alpha^2} > R_1$, we deduce that for any $t \in (0, t_0]$ there exists $c_t > 0$ such that

$$S_t^* \mathbf{1}_{(x_0, R_1)} \geq c_t \mathbf{1}_{(x_0, 2\alpha^{-2}R_1)}.$$

Dividing $[0, t_0]$ into $n$ sub-intervals $[\frac{k}{n}t_0, \frac{k+1}{n}t_0]$, $0 \leq k \leq n-1$, and iterating the above inequality with $t = t_0/n$, we deduce for all integer $n \geq 1$ the existence of $c_n > 0$ such that

$$S_{t_0}^* \mathbf{1}_{(x_0, R_1)} \geq c_n \mathbf{1}_{(x_0, (2\alpha^{-2})^n R_1)}$$

and the proof is complete since $2\alpha^{-2} > 1$. $\square$

With Lemmas 9.2 and 9.3, we are now in position to prove the convergence result in Theorem 9.1.

*Proof of the exponential stability part of Theorem 9.1.* We apply Theorem 6.4. We start by proving that (6.10) is verified, in a quantitative way, for the function $g_0 = \mathbf{1}_{(x_0+\varepsilon, R_2)}$ with a suitable choice of $R_2$ and $\varepsilon$. Choosing $r_0 \in (1, r)$ if $m(x) = x^r$ or any $r_0 > 1$ is $m(x) = \exp(\eta \int_{x_0}^x K/a)$ and defining $m_0(x) = x^{r_0}$ we have from (9.12), because of the normalization $\|\phi_1\|_{L_{m-1}^\infty} = 1$,

$$\|\phi_1\|_{L_{m_0-1}^\infty} \leq C_0$$

for some explicit constant $C_0 > 0$. Defining

$$R_2 := \inf\{R > 0; m_0(x)/m(x) \leq 1/2C_0, \forall x > R\},$$

we have

$$1 = \|\phi_1\|_{L_{m-1}^\infty} = \sup_{(x_0, \infty)} \frac{\phi_1}{m} = \sup_{(x_0, R_2)} \frac{\phi_1}{m},$$

because

$$\sup_{(R_2, \infty)} \frac{\phi_1}{m} \leq \sup_{(R_2, \infty)} \frac{\phi_1}{m_0} \frac{m_0}{m} \leq C_0 \frac{1}{2C_0} < 1.$$

Together with the fact that $\phi'_1 \in L_{\text{loc}}^\infty$, with a quantitative estimate on $\|\phi'_1\|_{L^\infty(x_0, R_2)}$, we see that $\phi_1$ has some quantifiable mass on $(x_0 + \varepsilon, R_2)$ for $\varepsilon > 0$ small enough, which exactly means that $\langle g_0, \phi_1 \rangle$ is quantified from below.

Now we prove that the Doblin-Harris condition (6.9) is verified. Choosing $R_1 > 2x_0$ and combining Lemma 9.2 and Lemma 9.3, we have for any $\varepsilon > 0$ and $R_2 > x_0 + \varepsilon$ the existence of $T > 0$ such that for any $R > R_1$

$$(9.18) \qquad S_T^* \phi \geq c_R \mathbf{1}_{(x_0, R)} \int_{x_0+\varepsilon}^{R_2} \phi \, dx, \quad \forall \phi \geq 0.$$

Defining $g_0 = \mathbf{1}_{(x_0+\varepsilon, R_2)}$, we deduce by duality that for all $f \geq 0$,

$$S_T f \geq c_R \langle f, \mathbf{1}_{(x_0, R)} \rangle g_0.$$
Yet expected, this result was known only in the case of a discrete set of variabilities [108, 328]. Theorem 9.10 is thus new in the literature.

Because of the assumption (9.25), we easily see that the construction of the semigroup and the proof of the conditions (H1), (H2) and (H4) given in Section 9.1 for the model without variability readily extend to the model (9.2). We thus only have to verify (H3) and some Harris type condition.

**Condition (H3).** Let $\delta \in (0,1)$ such that (9.24) is verified, and consider the weight function $m(x) = x^r$ with $r > 1$ or $m(x) = \exp(\eta \int_{x_0}^x K/a)$ with $\eta \in (0, 1 - \delta)$. We also use the two other weights

$$m_1(x) = \exp(\eta_1 \int_{x_0}^{x} K/a), \quad m_2(x) = \exp(\eta_2 \int_{x_0}^{x} K/a)$$

for some $\eta_1 \in (\eta, 1 - \delta)$ and $\eta_2 = \eta_1 + \delta$. We combine the two different splittings $\mathcal{L} = \mathcal{A} + \mathcal{B}$ and $\mathcal{L} = \mathcal{A}_0 + \mathcal{B}_0$, where

$$\mathcal{A}f(x, v) = M\mathbf{1}_{(x_0, R)}(x)f(x, v), \quad \mathcal{A}_0f(x, v) = 4 \int_1^2 K(2x)\wp(v, v_*)f(x, v_*) dv_*.$$

We prove that for any $\kappa > \kappa_B$ the operator

$$\mathcal{C} := (\kappa - \mathcal{B}_0)^{-1} \mathcal{A}_0 (\kappa - \mathcal{B})^{-1} \mathcal{A}$$

is well defined and maps continuously $L_m^1$ into $L_{m_1}^1 \cap W_{\text{loc}}^{1,1}$, in the sense that if $(f_n)$ is bounded in $L_m^1$ then the image is bounded in $L_{m_1}^1 \cap W^{1,1}((x_0, R) \times [1, 2])$ for all $R > 0$. In particular, $\mathcal{C} \in \mathcal{K}(L_m^1)$. More precisely, for any $\kappa > 0$, we prove

$$L_m^1 \xrightarrow{\mathcal{A}} L_{m_2}^1 \xrightarrow{(\kappa - \mathcal{B})^{-1}} L_{m_2}^1 \xrightarrow{\mathcal{A}_0} L_{m_1}^1 \cap W_{v, \text{loc}}^{1,1} \xrightarrow{(\kappa - \mathcal{B}_0)^{-1}} L_{m_1}^1 \cap W_{\text{loc}}^{1,1}$$

where $W_{v, \text{loc}}^{1,1} := \{f \in L_{\text{loc}}^1((x_0, \infty) \times [1, 2]), \partial_v f \in L_{\text{loc}}^1((x_0, \infty) \times [1, 2])\}$.

The results for $\mathcal{A}$ and $(\kappa - \mathcal{B})^{-1}$ are proved as in the case without variability. For the third one, the fact that $\mathcal{A}_0$ maps $L_{m_2}^1$ in $L_{m_1}^1$ follows from assumption (9.24), and the fact that the range is in $W_{v, \text{loc}}^{1,1}$ is a direct consequence of the assumption that $\wp \in W^{1,\infty}([1, 2]^2)$.

Finally we consider $\kappa - \mathcal{B}_0$ and we first verify that it is invertible in $L_{m_1}^1$ for any $\kappa > 0$. If $(\kappa - \mathcal{B}_0)g = f$, then necessarily

$$(9.26) \qquad g(x, v) = \frac{1}{va(x)} \int_{x_0}^{x} e^{(\Lambda_{\kappa}(y) - \Lambda_{\kappa}(x))/v} f(y, v) dy,$$

where $\Lambda_{\kappa}(x) = \int_{x_0}^{x} \frac{\kappa + K}{a}$, and consequently

$$g(x, v)m_1(x) = \frac{e^{(\eta_1 \Lambda(x) - \Lambda_\kappa(x))/v}}{va(x)} \int_{x_0}^{x} e^{(\Lambda_\kappa(y) - \eta_1 \Lambda_0(y))/v} f(y, v)m_1(y) dy.$$

Since

$$\Lambda_{\kappa}(x) - \eta_1 \Lambda_0(x) = (1 - \eta_1) \Lambda_{\kappa/(1-\eta_1)}(x)$$

we have for all $v \in [1, 2]$

$$\begin{align*}
& \int_{x_0}^{\infty} \left( \frac{\kappa}{1 - \eta_1} + K(x) \right) g(x, v) m_1(x) dx \\
&= \int_{x_0}^{\infty} \frac{1}{v} \Lambda'_{\kappa/(1-\eta_1)}(x) e^{-\frac{1-\eta_1}{v} \Lambda_{\kappa/(1-\eta_1)}(x)} \int_{x_0}^{x} e^{(1-\eta_1) \Lambda_{\kappa/(1-\eta_1)}(y, v)} f(y, v) m_1(y) dy dx \\
&= \int_{x_0}^{\infty} e^{\frac{1-\eta_1}{v} \Lambda_{\kappa/(1-\eta_1)}(y)} f(y, v) m_1(y) \int_{y}^{\infty} \frac{1}{v} \Lambda'_{\kappa/(1-\eta_1)}(x) e^{-\frac{1-\eta_1}{v} \Lambda_{\kappa/(1-\eta_1)}(x)} dx dy \\
&= \frac{1}{1 - \eta_1} \int_{x_0}^{\infty} f(y, v) m_1(y) dy.
\end{align*}$$

We deduce that the operator $\kappa - \mathcal{B}_0$ is invertible in $L_{m_1}^1$ with $\|(\kappa - \mathcal{B})^{-1}\| \le 1/\kappa$. We have also proved that $(\kappa - \mathcal{B}_0)^{-1}$ maps $L_{m_1}^1$ into $L_{Km_1}^1$ with $\|(\kappa - \mathcal{B}_0)^{-1}\|_{\mathcal{B}(L_{m_1}^1, L_{Km_1}^1)} \le \frac{1}{1-\eta_1}$. The fact that
Let us now consider $A > 0$ and $f \in X_+$ such that $\|f\| \le A[f]_{\phi_1}$. Since $m_0(x)/m(x) \to 0$ as $x \to +\infty$ and $\|\phi_1/m_0\|_{\infty} \le C_0$, we have

$$ 
\begin{aligned}
[f]_{\phi_1} &= \int_{x_0}^{R} f \frac{\phi_1}{m} m + \int_{R}^{\infty} fm \frac{\phi_1}{m_0} \frac{m_0}{m} \\
&\le \langle f, \mathbf{1}_{(x_0, R)} \rangle \sup_{(x_0, R)} m + \|f\| C_0 \sup_{(R, \infty)} \frac{m_0}{m} \\
&\le \langle f, \mathbf{1}_{(x_0, R)} \rangle m(R) + \frac{1}{2} [f]_{\phi_1},
\end{aligned}
$$

by choosing $R$ large enough. We deduce that $S_T f \ge \frac{c_R}{2m(R)} [f]_{\phi_1} g_0$, which is equivalent to (6.9) where we recall the definition $\tilde{S}_t := S_t e^{-\lambda_1 t}$.

Finally, we prove the Lyapunov condition (6.8). On the one hand, we get from (9.10) that

$$ \frac{d}{dt} \tilde{S}_t^* m = \tilde{S}_t^* (\mathcal{L}^* - \lambda_1) m \le (\kappa_B - \lambda_1) \tilde{S}_t^* m + M \tilde{S}_t^* (\mathbf{1}_{(x_0, R)} m). $$

On the other hand, arguing as in (6.6), we infer from (9.18) that

$$ \phi_1 = e^{-\lambda_1 T} S_T^* \phi_1 \ge c_R e^{-\lambda_1 T} \langle g_0, \phi_1 \rangle \mathbf{1}_{(x_0, R)}. $$

Combining both we deduce that

$$ \frac{d}{dt} \tilde{S}_t^* m \le (\kappa_B - \lambda_1) \tilde{S}_t^* m + \tilde{M} \phi_1 $$

with $\tilde{M} = \frac{m(R)}{c_R} \frac{e^{\lambda_1 T}}{\langle g_0, \phi_1 \rangle}$, and Grönwall's inequality then yields

$$ \tilde{S}_t^* m \le e^{(\kappa_B - \lambda_1)t} m + \tilde{M} t e^{(\kappa_B - \lambda_1)t} \phi_1. $$

This guarantees that (6.8) is verified with $\gamma_L = e^{(\kappa_B - \lambda_1)T} \in (0, 1)$ and $K = \tilde{M}T$.

We have proved that the conditions (6.15), (6.8) and (6.10) are verified. The conclusion of the proof then follows from Theorem 6.4. $\square$

## 9.2. The mitosis equation with non-mixing growth rate.
In this section, we investigate the case when the mixing condition (9.6) is not verified. In other words, we place ourselves under the singular condition

$$ (9.19) \qquad \forall x > x_0, \quad a(2x) = 2a(x). $$

In this case, we still have the existence of a unique eigentriplet $(\lambda_1, f_1, \phi_1)$ but the boundary point spectrum is not reduced to $\lambda_1$. As a consequence, the long time behavior of the semigroup does not stabilizes along $f_1$ but it exhibits periodic oscillations.

**Theorem 9.4.** Suppose that (9.3), (9.4), (9.5) and (9.19) are satisfied. The conclusions (S1), (S2) and (S31) with $\lambda_1 > 0$ hold in $X := L_m^1$. Besides, $\Sigma_P^+(\mathcal{L}) = \{\lambda_1 + i\kappa\alpha, k \in \mathbb{Z}\}$ for some quantifiable $\alpha > 0$, there exists a family $(g_k, \psi_k)_{k \in \mathbb{Z}}$ of corresponding primal and dual eigenvectors that verifies $\langle \psi_k, g_\ell \rangle = \delta_{k\ell}$, and for all $f \in L_{\phi_1}^1$, we have the convergence

$$ \|e^{-\lambda_1 t} S_{\mathcal{L}}(t) (f - \Pi f)\|_{L_{\phi_1}^1} \to 0 \quad \text{as } t \to +\infty, $$

where $\Pi f = \lim_{n \to \infty} \frac{1}{n} \sum_{\ell=0}^{n} \sum_{k=-\ell}^{\ell} \langle \psi_k, f \rangle g_k$.

This new result complements the scarce literature on the long time behavior of equation (9.1) in the singular case (9.19) which, to the best of our knowledge, is limited to the references [50, 168, 187]. We will actually prove that the convergence in Theorem 9.4 also holds in other spaces, and this will be the occasion to compare our method and results to the three above mentioned papers.

The proof of the conclusion (C2) in Section 9.1 does not use the mixing assumption (9.6). It thus also proves the existence, uniqueness and strict positivity of eigentriplet $(\lambda_1, f_1, \phi_1)$ under the assumptions of Theorem 9.4, as well as the fact that equation (9.1) is associated with a semigroup $S$ in $X$. For proving the long time convergence result, we start by verifying that this semigroup extends to other relevant Banach spaces.
it maps $W_{v,loc}^{1,1}$ into $W_{loc}^{1,1}$ readily follows from the formula (9.26). We conclude to the compactness of $C$ and then to the validity of (H3). Indeed, we can write (2.20) as

$$ \hat{f}_n = \mathcal{R}_B(\lambda_n)A\hat{f}_n + \mathcal{R}_B(\lambda_n)\varepsilon_n, $$

but also as

$$ \hat{f}_n = \mathcal{R}_{B_0}(\lambda_n)A_0\hat{f}_n + \mathcal{R}_{B_0}(\lambda_n)\varepsilon_n. $$

Combining both, we get

$$ \hat{f}_n = C\hat{f}_n + [\mathcal{R}_{B_0}(\lambda_n)A_0\mathcal{R}_B(\lambda_n) + \mathcal{R}_{B_0}(\lambda_n)]\varepsilon_n. $$

Since $C$ is compact, we conclude to (H3) with the same argument as in the proof of Lemma 2.8.

From (H1), (H2), (H3) and (H4) we infer the conclusion (C2) about existence and uniqueness of $(\lambda_1, f_1, \phi_1)$, which gives a part of Theorem 9.10. For the quantitative exponential stability, we start with a lemma.

**Lemma 9.11.** For all $\varepsilon > 0$, $R_1 > x_0$, and $R_2 > x_0 + \varepsilon$, there exist $T > 0$ and $c_T > 0$ such that

$$ (9.27) \qquad S_T^* \phi \ge c_T \mathbf{1}_{(x_0, R_1) \times [1, 2]} \int_1^2 \int_{x_0+\varepsilon}^{R_2} \phi \, dx \, dv, \quad \forall \phi \ge 0. $$

*Proof of Lemma 9.11.* Let us fix $\varepsilon > 0$, $R_1 > x_0$, and $R_2 > x_0 + \varepsilon$. Throughout the proof we denote by $c_t$ any positive constant that depend on $t$, and also possibly on the ingredients of the model $g$, $K$, $\wp$ and on $\varepsilon$, $R_1$, $R_2$, but is independent of $(x, v) \in (x_0, R_1) \times [1, 2]$.

*First step.* We prove first that there exists $T_1 > 0$ and $x_3 > x_2 > x_0$ such that $x_3 > \max(R_2, 2x_2)$ and

$$ (9.28) \qquad S_{T_1}^* \phi \ge c_{T_1} \mathbf{1}_{(x_0, R_1) \times [1, 2]} \int_1^2 \int_{x_2}^{x_3} \phi \, dxdv, $$

for all $\phi \ge 0$. We start from the Duhamel formula

$$ (9.29) \qquad S_t^* \phi(x, v) = \phi(X_t^v(x), v) e^{-\int_0^t K(X_s^v(x)) ds} \\ \qquad \qquad + 2 \int_0^t \int_1^2 K(X_s^v(x)) S_{t-s}^* \phi(X_s^v(x)/2, v_*) e^{-\int_0^s K(X_{s'}^v(x)) ds' \wp(v_*, v)} dv_* ds, $$

where $X_t^v(x)$ is the solution to the characteristic equation

$$ \dot{X}_t^v(x) = v a(X_t^v(x)) \quad \text{with} \quad X_0^v(x) = x. $$

Iterating twice (9.29), using positivity and the fact that $K$ and $a$ are locally bounded and $\wp$ is bounded from below, we deduce that

$$ S_t^* \phi(x, v) \ge c_t \int_0^t \int_1^2 \int_0^s \int_1^2 K(X_{s'}^v(x)) K(X_{s-s'}^v(X_{s'}^v(x)/2)) \phi(X_{t-s}^{v**}(X_{s-s'}^v(X_{s'}^v(x)/2)/2), v_{**}) dv_{**} ds' dv_* ds, $$

on $(x_0, R_1) \times [1, 2]$, for all $\phi \ge 0$. Let $t_0$ be such that $X_{t_0}^1(x_0) = 2x_0 + 1$. Then, for $t > 2t_0$, we deduce, from the fact that $K$ is locally bounded from below on $(2x_0, \infty)$, that

$$ S_t^* \phi(x, v) \ge c_t \int_{2t_0}^t \int_1^2 \int_{t_0}^s \int_1^2 \phi(X_{t-s}^{v**}(X_{s-s'}^v(X_{s'}^v(x)/2)/2), v_{**}) \, dv_{**} \, ds' \, dv_* \, ds. $$

For $t > 2t_0 + 2$, by using the Fubini-Tonelli theorem, we thus have

$$ S_t^* \phi(x, v) \ge c_t \int_1^2 \int_{t-1}^t \int_{t_0}^{t_0+1} \left( \int_1^2 \phi(X_{t-s}^{v**}(X_{s-s'}^v(X_{s'}^v(x)/2)/2), v_{**}) dv_* \right) ds' ds dv_{**}. $$

Using now a change of variables, we get

$$ S_t^* \phi(x, v) \ge c_t \int_1^2 \int_{t-1}^t \int_{t_0}^{t_0+1} \left( \int_{X_{t-s}^{v**}(X_{s-s'}^1(X_{s'}^v(x)/2)/2)}^{X_{t-s}^{v**}(X_{s-s'}^2(X_{s'}^v(x)/2)/2)} \phi(y, v_{**}) dy \right) ds' ds dv_{**} \\ \ge c_t \int_1^2 \int_{X_1^2(X_{t-t_0}^1(X_{t_0+1}^2(R_1)/2)/2)}^{X_{t-t_0-2}^2(x_0/2)/2} \phi(y, v_{**}) dy dv_{**}. $$
**Well-posedness in entropic $L^p$ and $M^1$ spaces.** The dual eigenfunction $\phi_1$ satisfies by definition $\mathcal{L}^*\phi_1 = \lambda_1\phi_1$ and the rescaled semigroup $\tilde{S}_t = S_t e^{-\lambda_1 t}$ is thus a contraction for the norm of $L_{\phi_1}^1$. In particular $S_t$ is a bounded operator for this norm and, since $L_m^1$ is dense in $L_{\phi_1}^1$, we can uniquely extend the semigroup $S$ into a strongly continuous semigroup in $L_{\phi_1}^1$. Similarly, due to the weak-* density of $L_{\phi_1}^1$ into $M_{\phi_1}^1$, this semigroup extends uniquely into a weakly-* continuous semigroup in $M_{\phi_1}^1$. We still denote by $S$ these extensions.

The General Relative Entropy principle, see [268, 49], ensures that the weighted $L^p$ sub-spaces of $L_{\phi_1}^1$ defined by

$$X_p := L_{f_1^{1-p}\phi_1}^p (x_0, \infty) \quad \text{for } p \in [1, \infty) \quad \text{and} \quad X_\infty := L_{f_1^{-1}}^\infty (x_0, \infty)$$

are invariant under the semigroup $S$ and the restriction to these spaces is a contraction. Besides, Jensen's inequality yields that it is a decreasing sequence for the inclusion

$$p > q \quad \implies \quad X_p \supset X_q.$$

Since $X_\infty \subset X_p$ is dense, we can infer the strong continuity of $S$ in $X_p$ from the strong continuity in $X_1$ by writing for any $f \in X_\infty$

$$\|\tilde{S}_t f - f\|_{X_p}^p \le \|\tilde{S}_t f - f\|_{X_\infty}^{p-1} \|\tilde{S}_t f - f\|_{X_1} \le 2^{p-1} \|f\|_{X_\infty}^{p-1} \|\tilde{S}_t f - f\|_{X_1} \to 0,$$

as $t \to 0$.

**Long-time convergence in $M_{\phi_1}^1$.** We start by giving some useful properties of the dual semigroup $S^*$ in $X' = L_{m-1}^\infty$. Splitting $\mathcal{L}^*$ as $\mathcal{L}^* = \mathcal{A}_0^* + \mathcal{B}_0^*$ with $\mathcal{A}_0^*\phi = \mathcal{H}^*[\phi]$, so that $\mathcal{B}_0^*\phi = a\phi' - K\phi$, the Duhamel formula

$$S_{\mathcal{L}}^* = S_{B_0}^* + S_{B_0}^* A_0 * S_{\mathcal{L}}^*$$

ensures that $\bar{\varphi}(t, x) := S_t^*\phi(x)$ is a fixed point of the operator $\Gamma$ defined by

$$(9.20) \quad \Gamma\varphi(t, x) := S_{B_0}^*(t)\phi(x) + [S_{B_0}^* A_0 * \varphi(\cdot, x)](t) \\ = \phi(X_t(x))e^{-\int_0^t K(X_s(x))ds} + 2\int_0^t K(X_{t-s}(x))\varphi(s, X_{t-s}(x)/2)e^{-\int_0^{t-s} K(X_{s'}(x))ds'}ds,$$

where we recall that $X_t(x)$ is the solution to the characteristic equation (9.16). It turns out that $\Gamma$ has a unique fixed point in $L_{\text{loc}}^\infty([0, \infty) \times (x_0, \infty))$, and that this fixed point also lies in any closed subset of $L_{\text{loc}}^\infty([0, \infty) \times (x_0, \infty))$ which is left invariant by $\Gamma$. This property is proved in [168] or in [35, Sec. 6.3], by building $\bar{\varphi}$ thanks to the Banach-Picard fixed point theorem. It has very useful consequences, as for instance the fact that if $\phi \in C(x_0, \infty)$, then $\bar{\varphi} \in C([0, \infty) \times (x_0, \infty))$. In particular, this implies that $C(x_0, \infty) \cap L_{m-1}^\infty$ is invariant under the semigroup $S^*$. Since $C(x_0, \infty) \cap L_{m-1}^\infty$ is a dense subspace of $C_{0,\phi_1}(x_0, \infty)$, this ensures that $C_{0,\phi_1}$ is invariant under $S^*$ and that the duality relation

$$\langle S_t f, \phi \rangle = \langle f, S_t^* \phi \rangle$$

is valid for any $f \in M_{\phi_1}^1$ and $\phi \in C_{0,\phi_1}$. The proof of the next result crucially relies on another application of the fact that the fixed point of $\Gamma$ belongs to any closed invariant subset.

**Proposition 9.5.** Suppose that (9.3), (9.4), (9.5) and (9.19) are satisfied. Then $\Sigma_P^+(\mathcal{L}) = \{\lambda_1 + ik\alpha, k \in \mathbb{Z}\}$ for some $\alpha > 0$, there exists a family $(g_k, \psi_k)_{k \in \mathbb{Z}}$ of corresponding primal and dual eigenvectors that verifies $\langle \psi_k, g_\ell \rangle = \delta_{k\ell}$, and for all $f \in M_{\phi_1}^1$ we have the convergence

$$(9.21) \quad \tilde{S}_t f - \tilde{S}_t \Pi f \to 0, \quad \text{as } t \to \infty, \quad \Pi f := \lim_{n \to \infty} \frac{1}{n} \sum_{\ell=0}^{n} \sum_{k=-\ell}^{\ell} \langle \psi_k, f \rangle g_k,$$

both convergences having to be understood in the sense of the weak-* topology.

Note that we did not specify the space in which we define the boundary point spectrum $\Sigma_P^+(\mathcal{L})$ in Proposition 9.5. It is because this set is the same in all the Banach lattices we consider. Indeed, any $g \in M_{\phi_1}^1$ such that $\mathcal{L}g = \lambda g$ for some $\lambda \in \mathbb{C}$ with $\Re e(\lambda) = \lambda_1$ satisfies $|g| \in \text{Span}(f_1)$, so that $g \in X = L_m^1$ for any weight $m$ as in (9.7).
Due to the strict positivity of $a$, we can choose $t = T_1$ large enough so that

$$X_{T_1-t_0-2}^2(x_0/2)/2 > \max(R_2, 2X_1^2(X_{T_1-t_0}^1(X_{t_0+1}^2(R_1)/2)/2)),$$

and we obtain (9.28) by setting $x_3 = X_{T_1-t_0-2}^2(x_0/2)/2$ and $x_2 = X_1^2(X_{T_1-t_0}^1(X_{t_0+1}^2(R_1)/2)/2)$, which concludes the first step of the proof.

**Second step.** We deduce (9.27) from (9.28) as follows. On the one hand, applying (9.28) to the function $S_t^*\phi$, we obtain

$$S_{T_1+t}^*\phi \geq c_{T_1} \mathbf{1}_{(x_0, R_1)\times[1,2]} \int_1^2 \int_{x_2}^{x_3} S_t^* \phi \,dx\,dv.$$

On the other hand, iterating once the Duhamel formula (9.29), we get by positivity that

$$S_t^* \phi(x, v) \geq c_t \left[ \phi(X_t^v(x), v) + \int_0^t \int_1^2 K(X_s^v(x)) \phi(X_{t-s}^{v_*}(X_s^v(x)/2), v_*) dv_* ds \right].$$

We first assume $x_2 > 2x_0$. In that case, the term $K(X_s^v(x))$ is bounded from below uniformly in $s \in [0, t]$, $v \in [1, 2]$ and $x \in [x_2, x_3]$, so that we infer from the two above inequalities that

$$S_{T_1+t}^* \phi \geq c_{T_1} c_t \mathbf{1}_{(x_0, R_1) \times [1,2]} \int_1^2 \int_{x_2}^{x_3} \left[ \phi(X_t^v(x), v) + \int_0^t \int_1^2 \phi(X_{t-s}^{v_*}(X_s^v(x)/2), v_*) dv_* ds \right] dx dv.$$

By a change of variable, we have

$$\int_{x_2}^{x_3} \phi(X_t^v(x), v) dx \geq c_t \int_{X_t^2(x_2)}^{x_3} \phi(y, v) dy$$

and

$$\int_0^t \int_{x_2}^{x_3} \phi(X_{t-s}^{v_*}(X_s^v(x)/2), v_*) dx ds \geq c_t \int_{X_t^2(X_t^2(x)/2)}^{x_3/2} \phi(y, v_*) dy.$$

Since $X_t^2(x) \to x$ as $t \to 0$, we deduce that we can find, for any $\zeta > 0$, a time $t > 0$ such that

$$S_{T_1+t}^* \phi \geq c_{T_1} c_t \mathbf{1}_{(x_0, R_1) \times [1,2]} \left[ \int_1^2 \int_{x_2+\zeta}^{x_3} \phi(y, v) dy dv + \int_1^2 \int_{x_2/2+\zeta}^{x_3/2} \phi(y, v_*) dy dv_* \right].$$

As $x_3 > 2x_2$, we can choose $\zeta$ small enough so that $x_3/2 > x_2 + \zeta$, and we get

$$S_{T_1+t}^* \phi \geq c_{T_1} c_t \mathbf{1}_{(x_0, R_1) \times [1,2]} \int_1^2 \int_{x_2/2+\zeta}^{x_3} \phi(y, v) dy dv.$$

Impose additionally that $\zeta \leq x_0$. Since the sequence $(u_n)$ defined by $u_0 = x_2$ and $u_{n+1} = u_n/2 + \zeta$ converges to $2\zeta \leq 2x_0$, we deduce by an iteration argument the existence of a time $T_2$ such that

$$(9.30) \qquad S_{T_2}^* \phi(x) \geq c_{T_2} \mathbf{1}_{(x_0, R_1)}(x) \int_1^2 \int_{2x_0+\varepsilon}^{x_3} \phi(y, v) dy dv.$$

Using a last time the argument with $\zeta \leq \varepsilon/2$ yields (9.27), since $x_3 > R_2$.

In the case where $x_2 \leq 2x_0$, (9.28) directly implies (9.30) with $T_2 = T_1$, and only one iteration of the extension argument is enough for concluding. $\square$

We are now in position to finish the proof of Theorem 9.10.

**Proof of Theorem 9.10.** The proof is exactly the same as for Theorem 9.1, Lemma 9.11 replacing Lemma 9.2. The only missing information is a quantitative $L_{\text{loc}}^{\infty}$ estimate on the derivatives $\partial_v\phi_1$ and $\partial_x\phi_1$, in order to use the same argument as in the proof of Proposition 9.8 for verifying (6.9). The estimate on $\partial_x\phi_1$ follows directly from the equation $\mathcal{L}^*\phi_1 = \lambda_1\phi_1$, which also reads

$$\partial_x \phi_1 = \frac{1}{va(x)} \left[ \lambda_1 \phi_1 + K \phi_1 - 2K(x) \int_1^2 \phi_1(x, v_*) \wp(v_*, v) dv_* \right].$$

For $\partial_v\phi_1$ we argue by duality, using that

$$\|\phi\|_{L^{\infty}} = \sup_{\|f\|_{L^1}=1} \langle \phi, f \rangle.$$

We start from

$$\phi_1 = (\lambda_1 - \mathcal{B}_0^*)^{-1} \mathcal{A}_0^* \phi_1,$$
Proof of Proposition 9.5. Step 1. The rescaled semigroup $\tilde{S}$ is a contraction semigroup in $M_{\phi_1}^1 = (C_{0,\phi_1})'$. This ensures in particular that for all $f \in M_{\phi_1}^1$ the trajectory $(\tilde{S}_t f)_{t \ge 0}$ is bounded in $M_{\phi_1}^1$. We can thus use Theorem 5.23-(2) to infer the non-triviality of the boundary spectrum, by proving that the conclusion cannot hold, see Remark 5.26-(ii). We start from the fact that for any $\phi \in C_{0,\phi_1}(x_0, \infty)$, the solution $S^*\phi$ to the dual mitosis equation is the unique fixed point of $\Gamma$ defined in (9.20), and that it belongs to any closed invariant subset of $C([0, \infty) \times (x_0, \infty))$. For $y > x_0$ we define the set

$$ \mathcal{E}_y = \{x > x_0, x = 2^k y \text{ for some } k \in \mathbb{Z}\} $$

and we consider a function $\phi$ such that $\phi(x) = 0$ if $x \in \mathcal{E}_y$ and $\phi(x) > 0$ if $x \notin \mathcal{E}_y$. Then, since (9.19) ensures that $X_t(2x) = 2X_t(x)$ for all $t \ge 0$, the set

$$ \{\varphi \in C([0, \infty) \times (x_0, \infty)), \varphi(t, x) = 0 \text{ if } X_t(x) \in \mathcal{E}_y \text{ and } \varphi(t, x) > 0 \text{ if } X_t(x) \notin \mathcal{E}_y\} $$

is invariant under $\Gamma$. Consequently, the unique fixed point $S_t^*\phi$ belongs to this set, and we deduce that $S_t^*\phi(x) = 0$ if and only if $X_t(x) \in \mathcal{E}_y$. In other words, by duality, $\text{supp}(S_t\delta_x) \subset \mathcal{E}_y$ if and only if $X_t(x) \in \mathcal{E}_y$, and in particular $\text{supp}(S_t\delta_x) \subset \mathcal{E}_{X_t(x)}$ for all $x > x_0$ and $t \ge 0$. This prevents the convergence of $\tilde{S}_t\delta_x$ toward $(\delta_x, \phi_1)f_1$ and we infer from (the negation of) Theorem 5.23-(2) that the boundary point spectrum cannot be trivial.

Step 2. We next formulate the following simple but fundamental observation. If $(\lambda, f)$ is a solution to the eigenvalue problem in $X = M_m^1$, then $f \in D(\mathcal{L}) \subset BV_{\text{loc}} \subset L_{\text{loc}}^1$ and it is a solution to the eigenvalue problem in $X = L_m^1$. Symmetrically, if $(\lambda, \phi)$ is a solution to the eigenvalue problem in $Y = L_{m^{-1}}^{\infty}$, then $\phi \in D(\mathcal{L}^*) \subset \text{Lip}_{\text{loc}} \cap L_{m^{-1}}^{\infty}$, $m_1(x)/m_2(x) \to 0$ as $x \to \infty$, and it is a solution to the eigenvalue problem in $Y = C_{0,m^{-1}}$. In other words, the point spectrum and the associated eigenelements are the same in the two frameworks $(L_m^1, L_{m^{-1}}^\infty)$ and $(M_m^1, C_{0,m^{-1}})$. Now, as a consequence of this observation and Step 1, we know that the boundary point spectrum $\Sigma_P^+(\mathcal{L})$ is not trivial. Because we have proved that $(\kappa - \mathcal{B})^{-1}\mathcal{A}$ is compact in $L_m^1$, $\kappa < \lambda_1$, we may apply Theorem 5.7 and we obtain that $\Sigma_P^+(\mathcal{L}) = \{\lambda_1\} + i\alpha\mathbb{Z}$ for some $\alpha > 0$, and each eigenvalue is algebraically simple. Using finally Theorem 5.25 in the situation (2), we get the weak-* convergence (9.21). $\square$

This result is proved by means of entropy techniques in [168] for a linear growth rate $a(x) = x$, by taking advantage of the explicit formulation of the eigenvectors $g_k$ and $\psi_k$ in terms of $f_1$ and $\phi_1$ in that case. Here we extend it to any $a$ satisfying $a(2x) = 2a(x)$. Note that arguing similarly as in [168], the convergence (9.21) may be strengthened into an exponential strong convergence in $M_m^1$ for $m(x) = x^r$, $r > 1$, or $m(x) = \exp(\eta \int_{x_0}^x K/a)$, $0 < \eta < 1$, meaning that there is a spectral gap between $\Sigma_+(\mathcal{L})$ and the rest of the spectrum in these spaces.

**Long-time convergence in $X_p$.** We prove the following result, the case $p = 1$ of which corresponds to the convergence result of Theorem 9.4.

**Proposition 9.6.** Under the same assumptions as in Proposition 9.5, the convergence (9.21) holds for the strong topology in $X_p$, $1 \le p < \infty$ for all $f \in X_p$, and the convergence of the Fejér sum in the definition of the projector $\Pi$ is also for this topology.

**Proof of Proposition 9.6.** The case $p = 1$ is an immediate consequence of Theorem 5.25, case (4). The proof in the case $p > 1$ is a direct adaptation of the case $p = 1$. We aim at verifying that the trajectories $(\tilde{S}_t f)_{t \ge 0}$ are relatively compact in $X_p$. We have already seen that $X_\infty \subset X_p$ is dense. Besides, the domain $D(\mathcal{L})$ of the generator $\mathcal{L} - \lambda_1$ of $\tilde{S}$ in $X_p$ is also dense in $X_p$, so that it suffices to check the relative compactness of $(\tilde{S}_t f)_{t \ge 0}$ for $f \in X_\infty \cap D(\mathcal{L})$. For $f$ in $X_\infty \cap D(\mathcal{L})$ the bounds

$$ \|\tilde{S}_t f\|_{X_p} \le \|f\|_{X_p}, \quad \|\mathcal{L}\tilde{S}_t f\|_{X_p} = \|\tilde{S}_t \mathcal{L} f\|_{X_p} \le \|\mathcal{L} f\|_{X_p} \quad \text{and} \quad \|\tilde{S}_t f\|_{X_\infty} \le \|f\|_{X_\infty} $$

yield the relative compactness of $(\tilde{S}_t f)_{t \ge 0}$, the second bound guaranteeing uniform $W_{\text{loc}}^{1,1}$ estimates. We can thus apply the case (1) of Theorem 5.25 to deduce the convergence (9.21) in $X_p$ for the strong topology. $\square$

Proposition 9.6 extends the result of [50] where it is proved in the case $p = 2$ for $a(x) = x$ by taking advantage of the Hilbert structure of $X_2$ and of the explicit formulation of the eigenvectors $g_k$ and
which yields

$$
\begin{align*}
\|\partial_v \phi_1\|_{L^\infty((x_0,R) \times [1,2])} &= \sup_{\|f\|_{L^1}=1} \langle \partial_v(\lambda_1 - \mathcal{B}_0^*)^{-1} \mathcal{A}_0^* \phi_1, f \rangle \\
&= \sup_{\|f\|_{L^1}=1} \langle \phi_1, \mathcal{A}_0(\lambda_1 - \mathcal{B}_0)^{-1} \partial_v f \rangle
\end{align*}
$$

where the supremum can be taken over the functions $f \in C_c^1((x_0, R) \times (1, 2))$. Using an integration by parts in $v_*$, we have

$$
\begin{align*}
\mathcal{A}_0(\lambda_1 - \mathcal{B}_0)^{-1} \partial_v f(x, v) &= 4 \int_1^2 \frac{K(2x)}{v_* a(x)} \int_{x_0}^x e^{(\Lambda_{\lambda_1}(y) - \Lambda_{\lambda_1}(x))/v_*} \partial_v f(y, v_*) dy \wp(v, v_*) dv_* \\
&= -4 \int_1^2 \int_{x_0}^x \partial_{v_*} \left( \frac{K(2x)}{v_* a(x)} e^{(\Lambda_{\lambda_1}(y) - \Lambda_{\lambda_1}(x))/v_*} \wp(v, v_*) \right) f(y, v_*) dy dv_*.
\end{align*}
$$

Since $\|\phi_1\|_{L^\infty((x_0,R)\times[1,2])} \le m(R)$, we deduce

$$
\|\partial_v \phi_1\|_{L^\infty((x_0,R)\times[1,2])} \le 4m(R) \sup_{(y,v_*) \in (x_0,R) \times [1,2]} \int_{x_0}^R \int_1^2 \left| \partial_{v_*} \left( \frac{K(2x)}{v_* a(x)} e^{(\Lambda_{\lambda_1}(y) - \Lambda_{\lambda_1}(x))/v_*} \wp(v, v_*) \right) \right| dv dx,
$$

and this last quantity is finite due to the assumptions made on the functions $a, K$ and $\wp$. $\square$

10. THE KINETIC LINEAR BOLTZMANN EQUATION

In this section, we consider the kinetic linear Boltzmann type equation

$$
(10.1) \qquad \partial_t f + v \cdot \nabla_x f - \nabla_x \Phi(x) \cdot \nabla_v f = \mathcal{H}[f] - Kf, \quad \text{in } (0, \infty) \times \mathcal{O}
$$

on the function $f = f(t, x, v)$, $t \ge 0$, $(x, v) \in \mathcal{O} := \Omega \times \mathbb{R}^d$. We assume that $K = K(x, v) \ge 0$ and
that the collision operator $\mathcal{H}$ is linear and defined by

$$
(10.2) \qquad \mathcal{H} = r\mathcal{H}_1, \quad (\mathcal{H}_1 g)(x, v) := \int_{\mathbb{R}^d} k g_* dv_*,
$$

for a real number $r > 0$ and some collision kernel $k : \Omega \times \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}_+$.
Here and below, we use
the common shorthands

$$
g_* := g(v_*), \quad k := k(x, v, v_*), \quad k_* := k(x, v_*, v).
$$

The most classical example for the collisional operator $\mathcal{C} = \mathcal{H} - K$ is the mass conservative operator

$$
(10.3) \qquad (\mathcal{C}g)(v) := \int_{\mathbb{R}^d} |v - v_*|^\gamma \{\mathcal{M}g_* - \mathcal{M}_*g\} \, dv_*,
$$

for some function $\mathcal{M} \in L_+^1(\mathbb{R}^d)$ and some exponent $\gamma \in \mathbb{R}$, which includes the relaxation operator

$$
(10.4) \qquad (\mathcal{C}g)(v) := \sigma(\mathcal{M}\rho_g - g), \quad \rho_g := \int_{\mathbb{R}^d} g_* \, dv_*.
$$

We make the following strong positivity and boundedness assumption on the collision kernel $k$ and
the function $K$. There exist $\gamma \ge 0$ and $K_i > 0$ such that

$$
(10.5) \qquad \forall (x, v) \in \Omega \times \mathbb{R}^d, \quad K_0 \le K(x, v) \langle v \rangle^{-\gamma} \le K_1.
$$

There exists a weight function $m : \mathbb{R}^d \to [1, \infty)$ such that

$$
(10.6) \qquad \forall p \in [1, \infty], \quad k m_*^{-1} m \in L_x^\infty L_v^p L_{v_*}^{p'}.
$$

For all $R > 0$, there exists $k_R > 0$ such that

$$
(10.7) \qquad \forall (x, v, v_*) \in \Omega \times B_R \times B_R, \quad k(x, v, v_*) \ge k_R.
$$

It is worth emphasizing that for $\mathcal{H}$ and $K$ defined in (10.3), the above assumptions are met when
$m := \mathcal{M}^{-1/2} : \mathbb{R}^d \to [1, \infty)$ (so that in particular $\mathcal{M} > 0$ a.e.) and $\mathcal{M}^{1/2} \langle v \rangle^\gamma \in L^1 \cap L^\infty$. We
$\psi_k$ in terms of $f_1$ and $\phi_1$. In this Hilbert setting it is proved that the Fourier series $\sum_{k=-n}^n \langle f, \psi_k \rangle g_k$ converges as $n$ goes to infinity, and $\Pi f$ is then given by the limit.

**About the value of $\alpha$.** For ensuring that the boundary spectrum is discrete, we have used a compactness argument. The period $2\pi/\alpha$ of the periodic semigroup $\tilde{S}\Pi$ is thus not quantified. It is expected to be equal to the time needed for a particle to double its size by following the flow of $a$, namely

$$ (9.22) \qquad \frac{2\pi}{\alpha} = \int_x^{2x} \frac{dt}{a(t)}, $$

which is independent of the choice of $x \ge x_0$ due to the condition $a(2t) = 2a(t)$. This is known to be true in the case of a linear growth rate $a(x) = x$, see [133] or [50], and also for $a$ general when the size domain is $(x_0, 4x_0)$, see [187], where explicit computations can be carried out. In the general case, we have not been able to prove (9.22). Yet the fact that for any $x > x_0$ the support of $S_t\delta_x$ is a subset of $\mathcal{E}_{X_t(x)}$ guarantees that the period cannot be too small as shown now.

**Proposition 9.7.** We have the estimate

$$ (9.23) \qquad \frac{2\pi}{\alpha} \ge \ell_a := \int_x^{2x} \frac{dt}{a(t)}. $$

*Proof of Proposition 9.7.* Let $x > x_0$ such that $\psi_1(x) \ne 0$ (actually any $x > x_0$ is suitable). We have

$$ \tilde{S}_t \delta_x - \tilde{S}_t \Pi \delta_x \to 0, $$

as $t \to +\infty$, and $\text{supp} \tilde{S}_t \delta_x \subset \mathcal{E}_{X_t(x)}$, so $\text{supp} \tilde{S}_t \Pi \delta_x \subset \mathcal{E}_{X_t(x)}$ since $\tilde{S}_t \Pi \delta_x$ is periodic. Besides,

$$ \tilde{S}_t \Pi \delta_x = \lim_{n \to \infty} \frac{1}{n} \sum_{\ell=0}^{n} \sum_{k=-\ell}^{\ell} \psi_k(x) e^{i\alpha k t} g_k $$

and, since $\psi_1(x) \ne 0$, the period of this periodic function of time is $\frac{2\pi}{\alpha}$. But since $\text{supp} \tilde{S}_t \Pi \delta_x \subset \mathcal{E}_{X_t(x)}$ and the period of the set $\mathcal{E}_{X_t(x)}$ is $\ell_a$, we have proved that (9.23) holds. $\square$

On the other hand, we can use Theorem 6.6 for deriving a quantified lower bound on $\alpha$, and thus an upper bound on the period. We work in the space $X = L_m^1$, recalling that $\Sigma_P^+(\mathcal{L})$ is the same in this space and in $M_{\phi_1}^1$.

**Proposition 9.8.** There exists a constructive constant $\alpha_1 > 0$ such that $\Sigma(\mathcal{L}) \cap B(\lambda_1, \alpha_1) = \{\lambda_1\}$. In particular, $\alpha \ge \alpha_1$.

To prove this result, we check that the conditions (6.8), (6.10) and (6.15) are verified, and we invoke Theorem 6.6. We start with a lemma which, together with Lemma 9.3, will guarantee the validity of (6.15).

**Lemma 9.9.** For all $\varepsilon > 0$, $R_1 > x_0$, and $R_2 > x_0 + \varepsilon$, there exist $T > 0$ and $c_T > 0$ such that

$$ \int_0^T S_t^* \phi \, dt \ge c_T \mathbf{1}_{(x_0, R_1)} \int_{x_0+\varepsilon}^{R_2} \phi \, dx, \quad \forall \phi \ge 0. $$

*Proof of Lemma 9.9.* Throughout the proof we denote by $c_t$ any positive constant that depends only on $t$. From the Duhamel formula (9.15), we get by positivity that for any $\phi \ge 0$

$$ \int_0^T S_t^* \phi(x) dt \ge \int_0^T \phi(X_t(x)) e^{-\int_0^t K(X_s(x)) ds} dt. $$

We deduce that for all $x \in (x_0, R_1)$ and for $T_0$ large enough so that $X_{T_0}(x_0) > R_2$, we have

$$ \int_0^{T_0} S_t^* \phi(x) dt \ge c_{T_0} \int_x^{X_T(x)} \phi(y) dy \ge c_{T_0} \int_{R_1}^{R_2} \phi(y) dy, $$

and the conclusion follows if $R_1 \le x_0 + \varepsilon$. If not, we have with the same argument the existence of $T$ such that for all $x \in (x_0, R_1)$

$$ \int_0^T S_t^* \phi(x) dt \ge c_T \int_{\max(R_1, 2x_0+\varepsilon)}^{2R_2} \phi(y) dy. $$
finally assume that for some weight function $m_1 : \mathbb{R}^d \to [1, \infty)$ such that $m_1/m \to \infty$ at infinity, we have

$$ (10.8) \qquad k m_*^{-1} m_1 \in L_x^\infty L_{vv_*}^2, $$

what holds true for the relaxation operator when $\mathcal{M} m_1 \in L^2(\mathbb{R}^d)$ and $m^{-1} \in L^2(\mathbb{R}^d)$, and that for some weight function $m_0 : \mathbb{R}^d \to [1, \infty)$ such that $m_0/m \to 0$ at infinity, we have

$$ (10.9) \qquad k m_{0*}^{-1} m \in L_x^\infty L_{vv_*}^1, $$

what holds true for the relaxation operator when $\mathcal{M} m \in L^1(\mathbb{R}^d)$ and $m_0^{-1} \in L^1(\mathbb{R}^d)$.

For the space domain $\Omega$, we consider the two following cases:

(1) $\Omega := \mathbb{T}^d$, the torus;

(2) $\Omega := \mathbb{R}^d$, the whole space.

In case (1), and for the sake of simplicity, we will always assume that $\Phi = 0$. In case (2), we will need a confinement mechanism which will be provided by the mean of the confinement force associated to the confinement potential $\Phi$. We do not consider here the case of a bounded domain with zero influx boundary condition because (1) our approach applies exactly as for the torus case and (2) this case has already been considered in the pioneering work by Vidav [351], where existence, uniqueness and exponential stability (with non constructive constants) have been established. We do not consider either the case of a bounded domain complemented with a reflection as we will consider in Section 11 for the kinetic Fokker-Planck evolution equation, because we have not been able to establish some crucial regularity estimates which seem to be necessary in our approach. We let this issue for a future work.

**10.1. The torus.** In this section, we are first concerned with the kinetic linear Boltzmann equation in the torus

$$ (10.10) \qquad \partial_t f + v \cdot \nabla_x f = \mathcal{K}[f] - Kf, \quad \text{in } (0, \infty) \times \mathbb{T}^d \times \mathbb{R}^d. $$

We make the boundedness and strong positivity assumptions listed above together with the additional assumption

$$ (10.11) \qquad k m_*^{-1} m_1, k m_{1*}^{-1/2} m \in L_{xv_*}^\infty, \quad \sum_{u \in \mathbb{Z}^d} \|m_1^{-1/2}(u+\cdot)\|_{L^\infty(\mathbb{T}^d)} < \infty, $$

for some $m_1$ such that $m/m_1 \in L^1 \cap L^2$.

**Theorem 10.1.** For the kinetic equation (10.10) in the torus and under conditions (10.5)-(10.6)-(10.7)-(10.8)-(10.9) and (10.11) for some weight functions $m$, $m_0$, $m_1$, there exists $r^* > 0$ such that for any $r \ge r^*$, the conclusions (C3) holds in $L_m^2$ and the conclusion (E31) holds in $L_m^1$.

Our result may be compared to [351] which establishes the same result without constructive rate and to [103] which establishes the same result using a probabilistic approach, both in the case of a bounded domain with zero influx boundary condition. It also extends to a non mass conservative situation the many results devoted to the conservative framework, see for instance the recent papers [285, 200, 79] and the references therein. When $\gamma > 0$, we may probably establish the same above result under the sole condition $r > 0$ (no need for $r$ to be large enough) by using some arguments developed in the next section.

We present now the proof of Theorem 10.1 by establishing that the conditions presented in the abstract part are satisfied.

**Condition (H1).** For an exponent $p \in [1, \infty)$ and a weight function $m$ satisfying (10.6), we set

$$ k_\infty := \|k m_*^{-1} m\|_{L_x^\infty L_v^p L_{v_*}^{p'}} < \infty. $$
Iterating once Duhamel's formula and using that $X_t(x/2) = X_t(x)/2$ and (9.3) we get that for all $t \ge 0$ and all $y \in (2x_0 + \varepsilon, R_2)$

$$S_t^* \phi(y) \ge c_t \int_0^t K(X_{t-s}(y)) \phi(X_s(X_{t-s}(y)/2)) ds \ge c_t \phi(X_t(y/2)).$$

This yields for all $x \in (x_0, R_1)$

$$\begin{aligned} \int_0^{T+t} S_s^* \phi(x) ds &\ge \int_t^{T+t} S_s^* \phi(x) ds = \int_0^T S_s^* S_t^* \phi(x) ds \\ &\ge c_T \int_{\max(R_1, 2x_0+\varepsilon)}^{2R_2} S_t^* \phi(y) dy \\ &\ge c_T c_t \int_{\max(R_1, 2x_0+\varepsilon)}^{2R_2} \phi(X_t(y/2)) dy = c_T c_t \int_{\max(X_t(\frac{R_1}{2}), X_t(x_0+\frac{\varepsilon}{2}))}^{X_t(R_2)} \phi(z) dz. \end{aligned}$$

Choosing $t > 0$ small enough so that $\max(X_t(\frac{R_1}{2}), X_t(x_0 + \frac{\varepsilon}{2})) \le \max(\frac{R_1+\varepsilon}{2}, x_0 + \varepsilon)$, we get

$$\int_0^{T_1} S_s^* \phi(x) ds \ge c_{T_1} \int_{\max(\frac{R_1+\varepsilon}{2}, x_0+\varepsilon)}^{R_2} \phi(z) dz, \text{ for } T_1 := T + t.$$

Iterating the argument we can build an increasing sequence of times $T_n$ such that

$$\int_0^{T_n} S_t^* \phi(x) dt \ge c_{T_n} \int_{\max(u_n, x_0+\varepsilon)}^{R_2} \phi(z) dz, \forall n \ge 0,$$

where $(u_n)$ is defined by $u_0 = R_1$ and $u_{n+1} = \frac{u_n+\varepsilon}{2}$. Since this sequence $(u_n)$ converges to $\varepsilon < x_0 + \varepsilon$, we get the conclusion by taking $n$ large enough. $\square$

We are now in position to prove Proposition 9.8.

*Proof of Proposition 9.8.* Arguing similarly as in the proof of Theorem 9.1 and using Lemma 9.9 instead of Lemma 9.2, we can prove that the conditions (6.8), (6.10) and (6.15) are verified. Applying Theorem 6.6 then gives the result. $\square$

### 9.3. The model with variability.
In this last part, we consider the model with variability given by the equation (9.2). Compared to equation (9.1), the main consequence of introducing a variability in terms of the spectrum and the asymptotic behavior is that for equation (9.2) the boundary spectrum is trivial and the first eigenfunction is exponentially stable, no matter if a satisfies (9.6) or (9.19).

Additionally to the assumptions (9.3), (9.4) and (9.5), we ask that

$$(9.24) \qquad K(x) = O\left(\exp\left(\delta \int_{x_0}^{x/2} K/a\right)\right) \quad \text{as } x \to +\infty,$$

for some $\delta > 0$. About the variability kernel $\wp$ we suppose that

$$(9.25) \qquad \int_1^2 \wp(v, v_*) dv = 1, \forall v_* \in [1, 2], \quad \wp \in W^{1,\infty}([1, 2]^2) \text{ and } \wp \ge \wp_*$$

for some $\wp_* > 0$. We work in the space $X := L_m^1(\mathcal{O})$ with $m = m(x)$ still given by (9.7) and $\mathcal{O} := (x_0, \infty) \times (1, 2)$.

**Theorem 9.10.** Suppose that (9.3), (9.4), (9.5), (9.24) and (9.25) are satisfied. Then the conclusions (S1), (S2), (S3) and (E3) hold with constructive constants in $L_m^1$.

Yet expected, this result was known only in the case of a discrete set of variabilities [107, 327]. Theorem 9.10 is thus new in the literature.

Because of the assumption (9.25), we easily see that the construction of the semigroup and the proof of the conditions (H1), (H2) and (H4) given in Section 9.1 for the model without variability readily extend to the model (9.2). We thus only have to verify (H3) and some Doblin-Harris type condition.
Considering then a solution $f$ to the evolution equation (10.10), we compute

$$ 
\begin{aligned}
\frac{1}{p} \frac{d}{dt} \int f^p m^p &= \int \mathcal{H}[f] f^{p-1} m^p - K(x, v) f^p m^p \\
&\le \|\mathcal{H}[f]m\|_{L^p} \left(\int f^p m^p\right)^{1-1/p} - \int K(x, v) f^p m^p \\
&\le rk_\infty \int f^p m^p - K_0 \int \langle v \rangle^\gamma f^p m^p,
\end{aligned}
$$

where we have used twice the Holder inequality. This differential inequality together with the Gronwall lemma provides an apriori estimates about the growth of the $L_m^p$ norm. As a consequence, the same arguments as in section 8.3 imply that $S_\mathcal{L}$ is a positive semigroup in $L_m^p$ with growth bound $\omega(S_\mathcal{L}) = rk_\infty - K_0$. In particular, condition (H1) holds thanks to Lemma 2.2.

**Condition (H2).** For $f_0 := 1_{T^d \times B_1}$, where $B_1$ denotes the unit ball in $\mathbb{R}_v^d$, we compute

$$ \mathcal{L}f_0 = \mathcal{H}[f_0] - Kf_0 \ge \inf_{v \in B_1} \{r\mathcal{H}_1[f_0] - K\}f_0. $$

Using (10.5) and the strong positivity condition (10.7), we get

$$ (10.12) \qquad \inf_{v \in B_1} \{\mathcal{H}[f_0] - K\} \ge rk_1 - 2^{\gamma/2}K_0 =: \kappa_0, $$

which provides a constructive lower bound of the set $\mathcal{I}$ defined in (2.15) thanks to Lemma 2.4-(ii). We have thus established that $\mathcal{L}$ satisfies (H2).

**Condition (H3).** We define the operator

$$ \mathcal{B}f := -v \cdot \nabla_x f - K(x, v)f, $$

and we assume $\kappa_B := -\inf K \le -K_0 < \kappa_0$, what holds whenever $r \ge r^*$, with $r^* > 0$ large enough thanks to (10.12). In the sequel, we assume $p = 2$ and we work in $X = L_m^2$. We immediately deduce that $\mathcal{B} - \kappa$ is dissipative for any $\kappa > \kappa_B$, and thus $\mathcal{R}_B(z)$ is bounded in $\mathcal{B}(L_m^2)$, uniformly in $z \in \Delta_\kappa$. For $\kappa > \kappa_B$ and $g \in L_m^2$, the function $f = \mathcal{R}_B(\kappa)g$ satisfies

$$ v \cdot \nabla_x f + (\kappa + K)f = g \quad \text{in } \mathcal{O}, $$

from what we deduce

$$ (\kappa - \kappa_B) \int_{\mathcal{O}} f^2 m^2 \le \int_{\mathcal{O}} (\kappa + K) f^2 m^2 = \int_{\mathcal{O}} fgm^2, $$

and finally

$$ \|f\|_{L_m^2}^2 \le \frac{1}{\kappa - \kappa_B} \|g\|_{L_m^2}^2. $$

Because of assumption (10.8), and defining $\mathcal{A} := \mathcal{H}$, we immediately deduce that

$$ (10.13) \qquad \mathcal{A}\mathcal{R}_B(\kappa) : L_m^2 \to L_{m_1}^2. $$

On the other hand, from the classical averaging lemma [182], we know that

$$ (10.14) \qquad \mathcal{A}_\varphi \mathcal{R}_B(\kappa) : L^2(\mathcal{O}) \to H^{1/2}(\mathcal{O}), $$

where for $\varphi = \varphi_1 \otimes \varphi_2 \in C_c^1(\mathcal{O}) \otimes C_c^1(\mathbb{R}^d)$, we have defined the mapping $\mathcal{A}_\varphi : L^2(\mathcal{O}) \to L^2(\mathcal{O})$ by

$$ \mathcal{A}_\varphi(f)(x) := \varphi_1(x, v) \int_{\mathbb{R}^d} f(x, v_*) \varphi_2(v_*) dv_*. $$

By classical approximation arguments, there exists a sequence $(\varphi_n)$ such that $\varphi_n \to k$ in the space $L^\infty(\mathbb{T}^d; L_{m_1 \otimes m^{-1}}^2(\mathbb{R}^d \times \mathbb{R}^d))$ and such that $\varphi_n$ is a linear combination of functions of $C_c^1(\mathcal{O}) \otimes C_c^1(\mathbb{R}^d)$. As a consequence of (10.13) and (10.14), we deduce that $\mathcal{A}\mathcal{R}_B(\kappa) \in \mathcal{H}(L_m^2)$ and next. $(\mathcal{R}_B(\kappa)\mathcal{A})^2 \in \mathcal{H}(L_m^2)$ for any $\kappa > \kappa_B$. We may use Lemma 2.8 (and Remark 2.9-(2)) with $N = 2$ in $X = L_m^2$, and deduce that (H3) holds.

**Condition (H4).** We start with a result of independent interest about strict positivity. Such an argument is reminiscent from [88, 327] in the study of the Boltzmann equation and has been used for instance in [293, 79].
**Condition (H3).** Let $\delta \in (0,1)$ such that (9.24) is verified, and consider the weight function $m(x) = x^r$ with $r > 1$ or $m(x) = \exp(\eta \int_{x_0}^x K/a)$ with $\eta \in (0, 1 - \delta)$. We also use the two other weights

$$m_1(x) = \exp(\eta_1 \int_{x_0}^{x} K/a), \quad m_2(x) = \exp(\eta_2 \int_{x_0}^{x} K/a)$$

for some $\eta_1 \in (\eta, 1 - \delta)$ and $\eta_2 = \eta_1 + \delta$. We combine the two different splittings $\mathcal{L} = \mathcal{A} + \mathcal{B}$ and $\mathcal{L} = \mathcal{A}_0 + \mathcal{B}_0$, where

$$\mathcal{A}f(x, v) = M\mathbf{1}_{(x_0, R)}(x)f(x, v), \quad \mathcal{A}_0f(x, v) = 4 \int_1^2 K(2x)\wp(v, v_*)f(x, v_*) dv_*.$$

We prove that for any $\kappa > \kappa_B$ the operator

$$\mathcal{C} := (\kappa - \mathcal{B}_0)^{-1} \mathcal{A}_0 (\kappa - \mathcal{B})^{-1} \mathcal{A}$$

is well defined and maps continuously $L_m^1$ into $L_{m_1}^1 \cap W_{\text{loc}}^{1,1}$, in the sense that if $(f_n)$ is bounded in $L_m^1$ then the image is bounded in $L_{m_1}^1 \cap W^{1,1}((x_0, R) \times [1, 2])$ for all $R > 0$. In particular, $\mathcal{C} \in \mathcal{K}(L_m^1)$. More precisely, for any $\kappa > 0$, we prove

$$L_m^1 \xrightarrow{\mathcal{A}} L_{m_2}^1 \xrightarrow{(\kappa - \mathcal{B})^{-1}} L_{m_2}^1 \xrightarrow{\mathcal{A}_0} L_{m_1}^1 \cap W_{v, \text{loc}}^{1,1} \xrightarrow{(\kappa - \mathcal{B}_0)^{-1}} L_{m_1}^1 \cap W_{\text{loc}}^{1,1}$$

where $W_{v,\text{loc}}^{1,1} := \{f \in L_{\text{loc}}^1((x_0, \infty) \times [1, 2]), \partial_v f \in L_{\text{loc}}^1((x_0, \infty) \times [1, 2])\}$.

The results for $\mathcal{A}$ and $(\kappa - \mathcal{B})^{-1}$ are proved as in the case without variability. For the third one, the fact that $\mathcal{A}_0$ maps $L_{m_2}^1$ in $L_{m_1}^1$ follows from assumption (9.24), and the fact that the range is in $W_{v,\text{loc}}^{1,1}$ is a direct consequence of the assumption that $\wp \in W^{1,\infty}([1, 2]^2)$.

Finally we consider $\kappa - \mathcal{B}_0$ and we first verify that it is invertible in $L_{m_1}^1$ for any $\kappa > 0$. If $(\kappa - \mathcal{B}_0)g = f$, then necessarily

$$(9.26) \qquad g(x, v) = \frac{1}{va(x)} \int_{x_0}^{x} e^{(\Lambda_{\kappa}(y) - \Lambda_{\kappa}(x))/v} f(y, v) dy,$$

where $\Lambda_{\kappa}(x) = \int_{x_0}^{x} \frac{\kappa + K}{a}$, and consequently

$$g(x, v)m_1(x) = \frac{e^{(\eta_1 \Lambda(x) - \Lambda_\kappa(x))/v}}{va(x)} \int_{x_0}^{x} e^{(\Lambda_\kappa(y) - \eta_1 \Lambda_0(y))/v} f(y, v)m_1(y) dy.$$

Since

$$\Lambda_{\kappa}(x) - \eta_1 \Lambda_0(x) = (1 - \eta_1) \Lambda_{\kappa/(1-\eta_1)}(x)$$

we have for all $v \in [1, 2]$

$$\begin{align*}
\int_{x_0}^{\infty} \left( \frac{\kappa}{1 - \eta_1} + K(x) \right) g(x, v) m_1(x) dx \\
&= \int_{x_0}^{\infty} \frac{1}{v} \Lambda'_{\kappa/(1-\eta_1)}(x) e^{-\frac{1-\eta_1}{v} \Lambda_{\kappa/(1-\eta_1)}(x)} \int_{x_0}^{x} e^{(1-\eta_1)\Lambda_{\kappa/(1-\eta_1)}(y,v)} f(y, v) m_1(y) dy dx \\
&= \int_{x_0}^{\infty} e^{\frac{1-\eta_1}{v} \Lambda_{\kappa/(1-\eta_1)}(y)} f(y, v) m_1(y) \int_{y}^{\infty} \frac{1}{v} \Lambda'_{\kappa/(1-\eta_1)}(x) e^{-\frac{1-\eta_1}{v} \Lambda_{\kappa/(1-\eta_1)}(x)} dx dy \\
&= \frac{1}{1 - \eta_1} \int_{x_0}^{\infty} f(y, v) m_1(y) dy.
\end{align*}$$

We deduce that the operator $\kappa - \mathcal{B}_0$ is invertible in $L_{m_1}^1$ with $\|(\kappa - \mathcal{B})^{-1}\| \le 1/\kappa$. We have also proved that $(\kappa - \mathcal{B}_0)^{-1}$ maps $L_{m_1}^1$ into $L_{Km_1}^1$ with $\|(\kappa - \mathcal{B}_0)^{-1}\|_{\mathcal{B}(L_{m_1}^1, L_{Km_1}^1)} \le \frac{1}{1-\eta_1}$. The fact that it maps $W_{v,\text{loc}}^{1,1}$ into $W_{\text{loc}}^{1,1}$ readily follows from the formula (9.26). We conclude to the compactness of $\mathcal{C}$ and then to the validity of (H3). Indeed, we can write (2.20) as

$$\hat{f}_n = \mathcal{R}_B(\lambda_n) \mathcal{A} \hat{f}_n + \mathcal{R}_B(\lambda_n) \varepsilon_n,$$

but also as

$$\hat{f}_n = \mathcal{R}_{\mathcal{B}_0}(\lambda_n) \mathcal{A}_0 \hat{f}_n + \mathcal{R}_{\mathcal{B}_0}(\lambda_n) \varepsilon_n.$$

Combining both, we get

$$\hat{f}_n = \mathcal{C} \hat{f}_n + [\mathcal{R}_{\mathcal{B}_0}(\lambda_n) \mathcal{A}_0 \mathcal{R}_B(\lambda_n) + \mathcal{R}_{\mathcal{B}_0}(\lambda_n)] \varepsilon_n.$$
**Lemma 10.2.** For any $\rho, \rho_*, t > 0$, there exists $c > 0$ such that

$$ (10.15) \qquad (S_{\mathcal{L}}(t)f_0)(x, v) \ge c \mathbf{1}_{B_{\rho}}(v) \int_{\mathbb{T}^d \times B_{\rho_*}} f_0 \, dv_* dx_*, $$

for all $f_0 \ge 0$ and $(x, v) \in \mathbb{T}^d \times \mathbb{R}^d$.

*Proof of Lemma 10.2.* We observe that the semigroup $S_B$ has explicit representation

$$ (S_B(t)f_0)(x, v) = f_0(x - vt, v)e^{-\int_0^t K(x - \tau v, v)d\tau}. $$

We next write the associated iterated Duhamel formula

$$ S_{\mathcal{L}} = S_B + S_B \mathcal{H} * S_B + S_B \mathcal{H} * S_B \mathcal{H} * S_B + S_B \mathcal{H} * S_B \mathcal{H} * S_B \mathcal{H} * S_{\mathcal{L}}. $$

Since all the terms are nonnegative, we may through away the first terms and the last one, and we get

$$ S_{\mathcal{L}} \ge S_B * \mathcal{H} S_B * \mathcal{H} S_B. $$

On the one hand, using the explicit expression of $S_B$ and (10.7), we have

$$ (\mathcal{H}_1 S_B(s)f_0)(y, w) \ge k_{\rho'} e^{-s K_{\rho_*}} \mathbf{1}_{B_{\rho'}}(w) \int_{B_{\rho_*}} f_0(y - w_* s, w_*) dw_* =: g(s, y, w), $$

for any $s > 0$ and any $\rho' \ge \rho_* > 0$, with $K_\rho := \sup_{z \in \mathbb{T}^d, |v| \le \rho} K(z, v)$. On the other hand, for the same reasons, we have

$$ \begin{aligned} (\mathcal{H}_1 S_B * g(t))(x, v) &= \int_0^t \int_{\mathbb{R}^d} k(x, v, v_*) g(s, x - v_*(t-s), v_*) e^{-\int_0^{t-s} K(x-\tau v_*, v_*) d\tau} dv_* ds \\ &\ge k_{\rho'} \mathbf{1}_{B_\rho}(v) \int_0^t \int_{B_{\rho'}} g(s, x - v_*(t-s), v_*) e^{-(t-s)K_{\rho'}} dv_* ds \\ &\ge k_{\rho'}^2 e^{-tK_{\rho'}} \mathbf{1}_{B_\rho}(v) \int_0^t \int_{B_{\rho'}} \int_{B_{(x+w_*s,(t-s)\rho')}} f_0(x - v_*(t-s) - w_*s, w_*) dw_* dv_* ds \\ &\ge k_{\rho'}^2 e^{-tK_{\rho'}} \mathbf{1}_{B_\rho}(v) \int_0^{t/2} \int_{B_{\rho_*}} \int_{\mathbb{T}^d} f_0(y_*, w_*) dy_* dw_* ds \\ &\ge k_{\rho'}^2 \frac{e^{-tK_{\rho'}}}{(t/2)^d} \mathbf{1}_{B_\rho}(v) \int_0^{t/2} \int_{B_{\rho_*}} \int_{\mathbb{T}^d} f_0(y_*, w_*) dy_* dw_* ds \\ &\ge k_{\rho'}^2 \frac{e^{-tK_{\rho'}}}{(t/2)^d} \mathbf{1}_{B_\rho}(v) \frac{t}{2} \int_{B_{\rho_*}} \int_{\mathbb{T}^d} f_0(y_*, w_*) dy_* dw_* \end{aligned} $$

for any $t > 0$ and $\rho' \ge \max(\rho, \rho_*)$ such that $t\rho' \ge 2$, in such a way that $B(z, (t/2)\rho') \supset \mathbb{T}^d$. We then have

$$ (\mathcal{H}_1 S_B * \mathcal{H}_1 S_B(t))(x, v) \ge k_{\rho'}^2 \frac{e^{-tK_{\rho'}}}{(t/2)^{d-1}} \mathbf{1}_{B_\rho}(v) \int_{\mathbb{T}^d} \int_{B_{\rho_*}} f_{0*} dw_* dx_* $$

for any $t \ge 2/\rho'$. We finally conclude

$$ S_{\mathcal{L}}(t)f_0(x, v) \ge r^2 k_{\rho'}^2 e^{-tK_{\rho'}} \int_{2/\rho'}^t \frac{ds}{(s/2)^{d-1}} \mathbf{1}_{B_\rho}(v) \int_{\mathbb{T}^d} \int_{B_{\rho_*}} f_{0*} dw_* dx_*, $$

from what we deduce (10.15) by choosing $\rho' = 8/t$. $\square$

<figure><img src="image_10.png" /></figure>

We now consider $\lambda \ge \lambda_1$ and $0 \le f \in L_m^2$, $f \ne 0$, such that

$$ \lambda f + v \cdot \nabla_x f + K f - \mathcal{H}[f] \ge 0 \quad \text{in} \quad \mathbb{T}^d \times \mathbb{R}^d. $$

We fix $\rho_* > 0$ such that $f \ne 0$ on $B_{\rho_*}$. From (2.13), we have

$$ f \ge \int_0^\infty e^{-(1+\lambda)t} S_{\mathcal{L}}(t) f dt, $$

and we conclude that $f > 0$ a.e. on any set $\mathbb{T}^d \times B_\rho$ thanks to Lemma 10.2. We have established that the strong maximum holds true, and thus (H4).
Since $\mathcal{C}$ is compact, we conclude to (H3) with the same argument as in the proof of Lemma 2.8.

From (H1), (H2), (H3) and (H4) we infer the conclusion (S1)-(S2) about existence and uniqueness of $(\lambda_1, f_1, \phi_1)$, which gives a part of Theorem 9.10. For the quantitative exponential stability, we start with a lemma.

**Lemma 9.11.** For all $\varepsilon > 0$, $R_1 > x_0$, and $R_2 > x_0 + \varepsilon$, there exist $T > 0$ and $c_T > 0$ such that

$$ (9.27) \qquad S_T^* \phi \ge c_T \mathbf{1}_{(x_0, R_1) \times [1, 2]} \int_1^2 \int_{x_0+\varepsilon}^{R_2} \phi \, dx \, dv, \quad \forall \phi \ge 0. $$

*Proof of Lemma 9.11.* Let us fix $\varepsilon > 0$, $R_1 > x_0$, and $R_2 > x_0 + \varepsilon$. Throughout the proof we denote by $c_t$ any positive constant that depend on $t$, and also possibly on the ingredients of the model $g$, $K$, $\wp$ and on $\varepsilon$, $R_1$, $R_2$, but is independent of $(x, v) \in (x_0, R_1) \times [1, 2]$.

*First step.* We prove first that there exists $T_1 > 0$ and $x_3 > x_2 > x_0$ such that $x_3 > \max(R_2, 2x_2)$ and

$$ (9.28) \qquad S_{T_1}^* \phi \ge c_{T_1} \mathbf{1}_{(x_0, R_1) \times [1, 2]} \int_1^2 \int_{x_2}^{x_3} \phi \, dx \, dv, $$

for all $\phi \ge 0$. We start from the Duhamel formula

$$ (9.29) \qquad S_t^* \phi(x, v) = \phi(X_t^v(x), v) e^{-\int_0^t K(X_s^v(x)) ds} \\ \qquad \qquad + 2 \int_0^t \int_1^2 K(X_s^v(x)) S_{t-s}^* \phi(X_s^v(x)/2, v_*) e^{-\int_0^s K(X_s^v(x)) ds'} \wp(v_*, v) dv_* ds, $$

where $X_t^v(x)$ is the solution to the characteristic equation

$$ \dot{X}_t^v(x) = v a(X_t^v(x)) \quad \text{with} \quad X_0^v(x) = x. $$

Iterating twice (9.29), using positivity and the fact that $K$ and $a$ are locally bounded and $\wp$ is bounded from below, we deduce that

$$ S_t^* \phi(x, v) \ge c_t \int_0^t \int_1^2 \int_0^s \int_1^2 K(X_{s'}^v(x)) K(X_{s-s'}^v(x^2)) \phi(X_{t-s}^{v**}(X_{s-s'}^v(x^2)/2), v_*) dv_** ds' dv_* ds, $$

on $(x_0, R_1) \times [1, 2]$, for all $\phi \ge 0$. Let $t_0$ be such that $X_{t_0}^1(x_0) = 2x_0 + 1$. Then, for $t > 2t_0$, we deduce, from the fact that $K$ is locally bounded from below on $(2x_0, \infty)$, that

$$ S_t^* \phi(x, v) \ge c_t \int_{2t_0}^t \int_1^2 \int_{t_0}^s \int_1^2 \phi(X_{t-s}^{v**}(X_{s-s'}^v(X_s^v(x)/2)/2), v_*) dv_* ds' dv_* ds. $$

For $t > 2t_0 + 2$, by using the Fubini-Tonelli theorem, we thus have

$$ S_t^* \phi(x, v) \ge c_t \int_1^2 \int_{t-1}^t \int_{t_0}^{t_0+1} \left( \int_1^2 \phi(X_{t-s}^{v**}(X_{s-s'}^v(X_s^v(x)/2)/2), v_*) dv_* \right) ds' ds dv_**. $$

Using now a change of variables, we get

$$ \begin{aligned} S_t^* \phi(x, v) &\ge c_t \int_1^2 \int_{t-1}^t \int_{t_0}^{t_0+1} \left( \int_{X_{t-s}^{v**}(X_{s-s'}^1(X_s^v(x)/2)/2)}^{X_{t-s}^{v**}(X_{s-s'}^2(X_s^v(x)/2)/2)} \phi(y, v_*) dy \right) ds' ds dv_** \\ &\ge c_t \int_1^2 \int_{X_1^1(X_{t-t_0}^1(X_{t_0+1}^2(R_1)/2)/2)}^{X_1^2(X_{t-t_0-2}^1(x_0/2)/2)} \phi(y, v_*) dy dv_**. \end{aligned} $$

Due to the strict positivity of $a$, we can choose $t = T_1$ large enough so that

$$ X_{T_1-t_0-2}^2(x_0/2)/2 > \max(R_2, 2X_1^2(X_{T_1-t_0}^1(X_{t_0+1}^2(R_1)/2)/2)), $$

and we obtain (9.28) by setting $x_3 = X_{T_1-t_0-2}^2(x_0/2)/2$ and $x_2 = X_1^2(X_{T_1-t_0}^1(X_{t_0+1}^2(R_1)/2)/2)$, which concludes the first step of the proof.

*Second step.* We deduce (9.27) from (9.28) as follows. On the one hand, applying (9.28) to the function $S_t^* \phi$, we obtain

$$ S_{T_1+t}^* \phi \ge c_{T_1} \mathbf{1}_{(x_0, R_1) \times [1, 2]} \int_1^2 \int_{x_2}^{x_3} S_t^* \phi \, dx \, dv. $$
**Condition (H5).** Assume that $(\lambda, f) \in \mathbb{C} \times X \setminus \{0\}$ satisfies

$$ \mathcal{L}f = \lambda f, \quad \mathcal{L}|f| = (\Re e\lambda)|f| = \Re e(\operatorname{sign}f)\mathcal{L}f. $$

From (H4) and the first identity satisfied by $|f|$, we know that $|f| > 0$ a.e. on $\mathbb{T}^d \times \mathbb{R}^d$. Using the second identity, we get

$$ \mathcal{H}[|f|] = \Re e(\operatorname{sign}f)\mathcal{H}[f]. $$

Writing $f = e^{i\alpha}|f|$, we deduce

$$ \int_{\mathbb{R}^d} k(x, v, v_*)|f(x, v_*)|(1 - \cos(\alpha - \alpha_*))dv_* = 0 \quad \text{a.e. on } \mathbb{T}^d \times \mathbb{R}^d, $$

and thus $\alpha = \alpha(x)$ thanks to (10.7). Next, coming back to the first equation, we have

$$ \begin{aligned} \lambda|f|e^{i\alpha} &= \mathcal{L}(|f|e^{i\alpha}) \\ &= e^{i\alpha}\mathcal{L}|f| - |f|e^{i\alpha}iv \cdot \nabla_x \alpha \\ &= e^{i\alpha}(\Re e\lambda)|f| - |f|e^{i\alpha}iv \cdot \nabla_x \alpha. \end{aligned} $$

The equation simplifies into

$$ v \cdot \nabla_x \alpha = \Im m\lambda, $$

so that $\alpha(x) = \alpha$ is a constant and the reverse Kato's inequality holds.

Alternatively to (H5), we readily infer from Lemma 10.2 that the variant condition (H5') is verified.

At this stage, because of Theorem 2.21, Theorem 4.13 and Theorem 5.16 (or Theorem 5.18), we deduce the conclusions (C1), (C2) and (C3) about the existence and uniqueness of the eigentriplet $(\lambda_1, f_1, \phi_1)$ which satisfies $f_1 > 0$, $\phi_1 > 0$, $\lambda_1$ is algebraically simple and on the triviality of the boundary punctual spectrum. We now establish the exponential asymptotic stability with constructive constants.

We start with a gain of uniform boundness estimate.

**Lemma 10.3.** There exists $N \ge 1$ such that $(\mathcal{A}\mathcal{R}_B(\kappa))^N : L_m^1 \to L_m^\infty$, for any $\kappa > \kappa_B$. As a consequence, $\phi_1 \in L_{m-1}^\infty$.

*Proof of Lemma 10.3. Step 1.* We argue similarly as in [279, Sec. 3.1]. On the one hand, denoting $\mathcal{A}_1 = \mathcal{H}_1$, so that $\mathcal{A} = r\mathcal{A}_1$, we have for any $f_0 \in L_m^1$

$$ (\mathcal{A}_1 S_B(t)f_0)(x, v) = \int_{\mathbb{R}^d} k(x, v, v_*)f_0(x - v_*t, v_*) e^{-\int_0^t K(x-v_*\tau, v_*)d\tau} dv_* $$

and, using estimate (10.11), we deduce that

$$ \begin{aligned} \|m_1 \mathcal{A}_1 S_B(t) f_0\|_{L_1^\infty L_v^\infty} &\le \|km_*^{-1}m_1\|_{L_{xvv_*}^\infty} \int_{\mathcal{O}} |f_0(x - v_*t, v_*)| m_* dv_* dx e^{t\kappa_B} \\ &\lesssim \|f_0\|_{L_m^1} e^{t\kappa_B}, \end{aligned} $$

for any $t \ge 0$. Now, we consider $f_0 \in L_x^1 L_{vm}^\infty$, we write

$$ (\mathcal{A}_1 S_B(t)f_0)(x, v) = \sum_{u \in \mathbb{Z}^d} \int_{\mathbb{T}^d} k(x, v, v_*) f_0(x - ut - v_*t, u + v_*) e^{-\int_0^t K(x-(u+v_*)\tau, u+v_*)d\tau} dv_* $$

and using estimate (10.11) again, we compute

$$ \begin{aligned} (\mathcal{A}_1 S_B(t)f_0)(x, v)m(v) &\le \|km_*^{-1/2}m\|_{L_{xvv_*}^\infty} \sum_{u \in \mathbb{Z}^d} \int_{\mathbb{T}^d} m_1^{1/2}(u+v_*)f_0(x-ut-v_*t, u+v_*) e^{t\kappa_B} dv_* \\ &\lesssim e^{t\kappa_B} \left( \sum_{u \in \mathbb{Z}^d} \|m_1^{-1/2}(u+\cdot)\|_{L^\infty(\mathbb{T}^d)} \right) \int_{\mathbb{T}^d} \|f_0(y, \cdot)\|_{L_m^\infty} \frac{dy}{t^d} \\ &\lesssim e^{t\kappa_B} \left( 1 + \frac{1}{t^d} \right) \|f_0\|_{L_x^1 L_{vm_1}^\infty}. \end{aligned} $$

Defining $\tilde{u}(t) := e^{-\kappa t} \mathcal{A}_1 S_B(t)$, $\kappa > \kappa_B$, we have first established $\tilde{u} : L_m^1 \to L_x^1 L_{vm_1}^\infty$ uniformly in time, and thus $\tilde{u} : L_x^1 L_{vm_1}^\infty \to L_x^1 L_{vm_1}^\infty$ uniformly in time because $L_x^1 L_{vm_1}^\infty \subset L_m^1$ (we use here the fact that $m/m_1 \in L^1$). On the other hand, we have established that $t^d \tilde{u} : L_x^1 L_{vm_1}^\infty \to L_m^\infty$ uniformly in
On the other hand, iterating once the Duhamel formula (9.29), we get by positivity that

$$S_t^* \phi(x, v) \geq c_t \left[ \phi(X_t^v(x), v) + \int_0^t \int_1^2 K(X_s^v(x)) \phi(X_{t-s}^{v*}(X_s^v(x)/2), v_*) dv_* ds \right].$$

We first assume $x_2 > 2x_0$. In that case, the term $K(X_s^v(x))$ is bounded from below uniformly in $s \in [0, t]$, $v \in [1, 2]$ and $x \in [x_2, x_3]$, so that we infer from the two above inequalities that

$$S_{T_1+t}^* \phi \geq c_{T_1} c_t \mathbf{1}_{(x_0, R_1) \times [1, 2]} \int_1^2 \int_{x_2}^{x_3} \left[ \phi(X_t^v(x), v) + \int_0^t \int_1^2 \phi(X_{t-s}^{v*}(X_s^v(x)/2), v_*) dv_* ds \right] dxdv.$$

By a change of variable, we have

$$\int_{x_2}^{x_3} \phi(X_t^v(x), v) dx \geq c_t \int_{X_t^2(x_2)}^{x_3} \phi(y, v) dy$$

and

$$\int_0^t \int_{x_2}^{x_3} \phi(X_{t-s}^{v*}(X_s^v(x)/2), v_*) dx ds \geq c_t \int_{X_t^2(X_t^2(x)/2)}^{x_3/2} \phi(y, v_*) dy.$$

Since $X_t^2(x) \to x$ as $t \to 0$, we deduce that we can find, for any $\zeta > 0$, a time $t > 0$ such that

$$S_{T_1+t}^* \phi \geq c_{T_1} c_t \mathbf{1}_{(x_0, R_1) \times [1, 2]} \left[ \int_1^2 \int_{x_2+\zeta}^{x_3} \phi(y, v) dy dv + \int_1^2 \int_{x_2/2+\zeta}^{x_3/2} \phi(y, v_*) dy dv_* \right].$$

As $x_3 > 2x_2$, we can choose $\zeta$ small enough so that $x_3/2 > x_2 + \zeta$, and we get

$$S_{T_1+t}^* \phi \geq c_{T_1} c_t \mathbf{1}_{(x_0, R_1) \times [1, 2]} \int_1^2 \int_{x_2/2+\zeta}^{x_3} \phi(y, v) dy dv.$$

Impose additionally that $\zeta \leq x_0$. Since the sequence $(u_n)$ defined by $u_0 = x_2$ and $u_{n+1} = u_n/2 + \zeta$ converges to $2\zeta \leq 2x_0$, we deduce by an iteration argument the existence of a time $T_2$ such that

$$(9.30) \qquad S_{T_2}^* \phi(x) \geq c_{T_2} \mathbf{1}_{(x_0, R_1)}(x) \int_1^2 \int_{2x_0+\epsilon}^{x_3} \phi(y, v) dy dv.$$

Using a last time the argument with $\zeta \leq \epsilon/2$ yields (9.27), since $x_3 > R_2$.

In the case where $x_2 \leq 2x_0$, (9.28) directly implies (9.30) with $T_2 = T_1$, and only one iteration of the extension argument is enough for concluding. $\square$

We are now in position to finish the proof of Theorem 9.10.

**Proof of Theorem 9.10.** The proof is exactly the same as for Theorem 9.1, Lemma 9.11 replacing Lemma 9.2. The only missing information is a quantitative $L_{\text{loc}}^{\infty}$ estimate on the derivatives $\partial_v \phi_1$ and $\partial_x \phi_1$, in order to use the same argument as in the proof of Proposition 9.8 for verifying (6.10).

The estimate on $\partial_x \phi_1$ follows directly from the equation $L^* \phi_1 = \lambda_1 \phi_1$, which also reads

$$\partial_x \phi_1 = \frac{1}{va(x)} \left[ \lambda_1 \phi_1 + K \phi_1 - 2K(x) \int_1^2 \phi_1(x, v_*) \varphi(v_*, v) dv_* \right].$$

For $\partial_v \phi_1$ we argue by duality, using that

$$\|\phi\|_{L^\infty} = \sup_{\|f\|_{L^1}=1} \langle \phi, f \rangle.$$

We start from

$$\phi_1 = (\lambda_1 - \mathcal{B}_0^*)^{-1} \mathcal{A}_0^* \phi_1,$$

which yields

$$\begin{align*}
\|\partial_v \phi_1\|_{L^\infty((x_0, R) \times [1, 2])} &= \sup_{\|f\|_{L^1}=1} \langle \partial_v (\lambda_1 - \mathcal{B}_0^*)^{-1} \mathcal{A}_0^* \phi_1, f \rangle \\
&= \sup_{\|f\|_{L^1}=1} \langle \phi_1, \mathcal{A}_0 (\lambda_1 - \mathcal{B}_0)^{-1} \partial_v f \rangle
\end{align*}$$
time. Using [276, Prop. 2.5] with $X := L_x^1 L_{vm_1}^\infty$ and $Y := L_m^\infty$, we deduce $\tilde{u}^{*(d+1)} : L_x^1 L_{vm_1}^\infty \to L_m^\infty$ uniformly in time, and we thus conclude that $\tilde{u}^{*N} : L_m^1 \to L_m^\infty$ uniformly in time, for $N := d+2$. Using formula (2.13), we deduce that $(AR_B)^N(z) : L_m^1 \to L_m^\infty$, uniformly for any $z \in \Delta_\kappa$.

Step 2. In particular, $(AR_B)^N(z) : L_m^1 \to L_m^2$ because $L_{m_1}^\infty \subset L_m^2$ (we use here the fact that $m/m_1 \in L^2$). By duality, we deduce that $(RB^*A^*)^N(z) : L_{m-1}^2 \to L_{m-1}^\infty$. Coming back to the eigenvalue equation

$$A^*\phi_1 + B^*\phi_1 = \lambda_1\phi_1,$$

we deduce

$$(10.16) \qquad \phi_1 = R_{B^*}(\lambda_1)A^*\phi = (R_{B^*}(\lambda_1)A^*)^N\phi_1.$$

By construction $\phi_1 \in L_{m-1}^2$ and we thus conclude that $\phi_1 \in L_{m-1}^\infty$.

□

From now on, we choose the normalization convention $\|\phi_1\|_{L_{m-1}^\infty} = 1$ and $\langle f_1, \phi_1 \rangle = 1$.

Because of (10.9) and proceeding similarly as during the proof of condition (H3), we have

$$AR_B(\kappa) : L_{m_0}^1 \to L_m^1, \quad \forall \kappa > \kappa_B,$$

so that

$$R_{B^*}(\kappa)A^* : L_{m-1}^\infty \to L_{m_0-1}^\infty, \quad \forall \kappa > \kappa_B.$$

From the first identity in (10.16), we deduce

$$\|\phi_1\|_{L_{m_0-1}^\infty} \le C_{01}\|\phi_1\|_{L_{m-1}^\infty},$$

with constructive constant $C_{01} \in (0, \infty)$. We may here proceed along an already used argument. Consider $0 \le f \in L_m^1$ and assume $\|f\|_{L_m^1} \le A[f]_{\phi_1}$. We then compute

$$\begin{aligned} \int f \phi_1 &= \int_{\mathcal{O}_\rho} f \frac{\phi_1}{m} m + \int_{\mathcal{O}_\rho^c} f m \frac{\phi_1}{m_0} \frac{m_0}{m} \\ &\le \langle f, \mathbf{1}_{\mathcal{O}_\rho} \rangle \sup_{\mathcal{O}_\rho} m + \|f\|_{L_m^1} C_{01} \sup_{\mathcal{O}_\rho^c} \frac{m_0}{m} \\ &\le \langle f, \mathbf{1}_{\mathcal{O}_\rho} \rangle \sup_{\mathcal{O}_\rho} m + \frac{1}{2} [f]_{\phi_1} \end{aligned}$$

by choosing $\rho = \rho(A)$ large enough, where we denote $\mathcal{O}_\rho := \mathbb{T}^d \times B_\rho$. Together with Lemma 10.2, we deduce that there exists $T > 0$ and $g_A \ge 0$, $g_A \ne 0$, such that

$$(10.17) \qquad S_{\mathcal{L}}(T)f \ge g_A[f]_{\phi_1},$$

what is nothing but the Harris condition (6.8). On the other hand, from the above regularization estimate, we have in the same time

$$1 = \|\phi_1\|_{L_{m-1}^\infty} \le C_0 \|\phi_1\|_{L_{m-1}^1}, \quad \|\phi_1\|_{L_{m_0-1}^1} \le C_1 \|\phi_1\|_{L_{m-1}^1},$$

for some constructive constants $C_i \in (0, \infty)$. We may thus compute

$$\begin{aligned} \int \phi_1 m^{-1} &\le \int_{B_\rho} \phi_1 m^{-1} + \sup_{B_\rho^c} \frac{m_0}{m} \int \phi_1 m_0^{-1} \\ &\le \int_{B_\rho} \phi_1 m^{-1} + \sup_{B_\rho^c} \frac{m_0}{m} C_1 \int \phi_1 m^{-1}, \end{aligned}$$

so that for $\rho > 0$ large enough, we deduce

$$(10.18) \qquad C_0^{-1} \le \|\phi_1\|_{L_{m-1}^1} \le 2 \int_{B_\rho} \phi_1 m^{-1}.$$

Together with the definition of $g_A$, we deduce that the positivity condition (6.3) holds.
where the supremum can be taken over the functions $f \in C_c^1((x_0, R) \times (1, 2))$. Using an integration by parts in $v_*$, we have

$$
\begin{align*}
\mathcal{A}_0(\lambda_1 - \mathcal{B}_0)^{-1} \partial_v f(x, v) &= 4 \int_1^2 \frac{K(2x)}{v_* a(x)} \int_{x_0}^x e^{(\Lambda_{\lambda_1}(y) - \Lambda_{\lambda_1}(x))/v_*} \partial_v f(y, v_*) dy \wp(v, v_*) dv_* \\
&= -4 \int_1^2 \int_{x_0}^x \partial_{v_*} \left( \frac{K(2x)}{v_* a(x)} e^{(\Lambda_{\lambda_1}(y) - \Lambda_{\lambda_1}(x))/v_*} \wp(v, v_*) \right) f(y, v_*) dy dv_*.
\end{align*}
$$

Since $\|\phi_1\|_{L^\infty((x_0,R)\times[1,2])} \le m(R)$, we deduce

$$
\begin{align*}
\|\partial_v \phi_1\|_{L^\infty((x_0,R)\times[1,2])} &\le 4m(R) \sup_{(y,v_*) \in (x_0,R) \times [1,2]} \int_{x_0}^R \int_1^2 \left| \partial_{v_*} \left( \frac{K(2x)}{v_* a(x)} e^{(\Lambda_{\lambda_1}(y) - \Lambda_{\lambda_1}(x))/v_*} \wp(v, v_*) \right) \right| dv dx,
\end{align*}
$$

and this last quantity is finite due to the assumptions made on the functions $a, K$ and $\wp$. $\square$

<figure><img src="image_11.png" /></figure>

10. THE KINETIC LINEAR BOLTZMANN EQUATION

In this section, we consider the kinetic linear Boltzmann type equation

$$
(10.1) \qquad \partial_t f + v \cdot \nabla_x f - \nabla_x \Phi(x) \cdot \nabla_v f = \mathcal{K}[f] - Kf, \quad \text{in } (0, \infty) \times \mathcal{O}
$$

on the function $f = f(t, x, v)$, $t \ge 0$, $(x, v) \in \mathcal{O} := \Omega \times \mathbb{R}^d$. We assume that $K = K(x, v) \ge 0$ and
that the collision operator $\mathcal{K}$ is linear and defined by

$$
(10.2) \qquad \mathcal{K} = r\mathcal{K}_1, \quad (\mathcal{K}_1 g)(x, v) := \int_{\mathbb{R}^d} k g_* dv_*,
$$

for a real number $r > 0$ and some collision kernel $k : \Omega \times \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}_+$.
Here and below, we use
the common shorthands

$$
g_* := g(v_*), \quad k := k(x, v, v_*), \quad k_* := k(x, v_*, v).
$$

The most classical example for the collisional operator $\mathcal{C} = \mathcal{K} - K$ is the mass conservative operator

$$
(10.3) \qquad (\mathcal{C}g)(v) := \int_{\mathbb{R}^d} |v - v_*|^{\gamma} \{\mathcal{M}g_* - \mathcal{M}_*g\} dv_*,
$$

for some function $\mathcal{M} \in L_+^1(\mathbb{R}^d)$ and some exponent $\gamma \in \mathbb{R}$, which includes the relaxation operator

$$
(10.4) \qquad (\mathcal{C}g)(v) := \sigma(\mathcal{M}\rho_g - g), \quad \rho_g := \int_{\mathbb{R}^d} g_* dv_*.
$$

We make the following strong positivity and boundedness assumption on the collision kernel $k$ and
the function $K$. There exist $\gamma \ge 0$ and $K_i > 0$ such that

$$
(10.5) \qquad \forall (x,v) \in \Omega \times \mathbb{R}^d, \quad K_0 \le K(x,v) \langle v \rangle^{-\gamma} \le K_1.
$$

There exists a weight function $m : \mathbb{R}^d \to [1, \infty)$ such that

$$
(10.6) \qquad \forall p \in [1, \infty], \quad k m_*^{-1} m \in L_x^\infty L_p^p L_{v_*}^{p'}.
$$

For all $R > 0$, there exists $k_R > 0$ such that

$$
(10.7) \qquad \forall (x, v, v_*) \in \Omega \times B_R \times B_R, \quad k(x, v, v_*) \ge k_R.
$$

It is worth emphasizing that for $\mathcal{K}$ and $K$ defined in (10.3), the above assumptions are met when
$m := \mathcal{M}^{-1/2} : \mathbb{R}^d \to [1, \infty)$ (so that in particular $\mathcal{M} > 0$ a.e.) and $\mathcal{M}^{1/2}\langle v \rangle^\gamma \in L^1 \cap L^\infty$. We
finally assume that for some weight function $m_1 : \mathbb{R}^d \to [1, \infty)$ such that $m_1/m \to \infty$ at infinity,
we have

$$
(10.8) \qquad k m_*^{-1} m_1 \in L_x^\infty L_{vv_*}^2,
$$

what holds true for the relaxation operator when $\mathcal{M} m_1 \in L^2(\mathbb{R}^d)$ and $m^{-1} \in L^2(\mathbb{R}^d)$, and that for
some weight function $m_0 : \mathbb{R}^d \to [1, \infty)$ such that $m_0/m \to 0$ at infinity, we have

$$
(10.9) \qquad k m_{0*}^{-1} m \in L_x^\infty L_{vv_*}^1,
$$
Finally, as during the proof of (H3) above, for $0 \le f_0 \in L_m^1$ and denoting $f := S_L(t)f_0$, we compute

$$ \begin{aligned} \frac{d}{dt} \int fmdvdx &= \int \mathcal{H}[f]mdvdx - \int Kfmdxdv \\ &\le C_0 \int fm_0dvdx + \kappa_B \int fmdxdv, \end{aligned} $$

with $C_0 := \|kmm_{0*}^{-1}\|_{L_{xv_*}^{\infty} L_v^1} < \infty$ and $m_0/m \to 0$ as $v \to \infty$. Observing that

$$ \int fm_0 \le \int_{B_\varrho} f\phi_1 \sup_{B_\varrho} \frac{m_0}{\phi_1} + \int_{B_\varrho^c} fm \sup_{B_\varrho^c} \frac{m_0}{m}, $$

for any $\kappa > \kappa_B$, we may choose $\varrho > 0$ large enough in such a way that $\sup_{B_\varrho^c} \frac{m_0}{m} \le (\kappa - \kappa_B)/C_0$ and we deduce that

$$ \frac{d}{dt} \int fmdvdx \le C_1 \int f\phi_1 dvdx + \kappa \int fmdxdv $$

with $C_1 = \sup_{B_\varrho} \frac{m_0}{\phi_1}$. From the Gronwall lemma, we obtain

$$ \|f_t\|_{L_m^1} \le e^{\kappa t} \|f_0\|_{L_m^1} + \frac{e^{\lambda_1 t} - e^{\kappa t}}{\lambda_1 - \kappa} C_1 \int f_0 \phi_1, $$

from what we immediately deduce that $S_L$ satisfies the Lyapunov condition (6.7) for any $t > 0$. It remains to quantify the constant $C_1$. The dual formulation of (10.15) applied to the dual eigenfunction $\phi_1$ with $t = 1$ and $\varrho_* = \varrho$ yields

$$ \phi_1 = e^{-\lambda_1} S_L^*(1)\phi_1 \ge e^{-\lambda_1 c} \mathbf{1}_{\mathbb{T}^d \times B_\varrho} \int_{\mathbb{T}^d \times B_\varrho} \phi_1 dv_* dx_* $$

Together with Equation (10.18), this provides the explicit bound $C_1 \le 2C_0e^{\lambda_1 c} c^{-1} \frac{\sup_{B_\varrho} m_0}{\inf_{B_\varrho} m}$.

We have established that the three conditions (6.8), (6.7) and (6.9) hold, so that we conclude the proof of Theorem 10.1 by just applying Theorem 6.3.

**10.2. The whole space case.** In this section, we assume that $\Omega := \mathbb{R}^d$ and we consider the kinetic equation (10.1) with an additional force field confinement $F = -\nabla_x \Phi$ associated to a potential $\Phi$. More precisely from now-on, we assume that

$$ (10.19) \qquad \Phi(x) = |x|^\beta, \quad \beta > 2, \quad K(v) = \langle v \rangle^\gamma, \quad \gamma > 0, $$

that (10.6) holds (for $p = 2$) and that there exist $\zeta, c_\zeta > 0$ such that

$$ (10.20) \qquad \mathcal{H}[\mathcal{M}^\zeta] \ge c_\zeta \langle v \rangle^\gamma \mathcal{M}^\zeta, \quad \mathcal{M} := e^{-|v|^2/2}. $$

Observe that condition (10.20) is satisfied when $\mathcal{H}$ is the positive part of the mass conservative operator (10.3). For further references, we write $\mathcal{L} := \mathcal{T} + \mathcal{C}$ with

$$ \mathcal{T} := -v \cdot \nabla_x f + \nabla_x \Phi \cdot \nabla_v, \quad \mathcal{C}f := \mathcal{H}[f] - Kf $$

and we define the Hamiltonian

$$ \mathcal{H} := \Phi(x) + \frac{1}{2}|v|^2. $$

In the sequel, we will only consider some weight functions $m = \omega(\mathcal{H})$ with $\omega(y) = y^\varrho$, $\varrho \ge 0$, or $\omega(y) = e^{\kappa y}$, $\kappa \in (0, 1)$, so that $\omega(\mathcal{H}) \sim \omega(|v|^2)\omega(\Phi)$. For $p \in [1, \infty)$, we further assume that $v \mapsto \omega^{-1}(|v|^2) \in L^{p'}$ (which imposes $\varrho > d/(2p')$ for a polynomial weight).

**Theorem 10.4.** For the kinetic equation (10.1) in the whole space with confinement force and under conditions (10.5)-(10.6)-(10.7)-(10.8)-(10.19)-(10.20) for some weight function $m = \omega(\mathcal{H})$ as discussed above, the conclusion (C3) about existence, uniqueness and positivity of the eigentriplet solution $(\lambda_1, f_1, \phi_1)$ holds as well as the ergodicity (E2) for the weak convergence in $L_{\phi_1}^1$.
what holds true for the relaxation operator when $\mathcal{M}m \in L^1(\mathbb{R}^d)$ and $m_0^{-1} \in L^1(\mathbb{R}^d)$.

For the space domain $\Omega$, we consider the two following cases:

(1) $\Omega := \mathbb{T}^d$, the torus;

(2) $\Omega := \mathbb{R}^d$, the whole space.

In case (1), and for the sake of simplicity, we will always assume that $\Phi = 0$. In case (2), we will need a confinement mechanism which will be provided by the mean of the confinement force associated to the confinement potential $\Phi$. We do not consider here the case of a bounded domain with zero influx boundary condition because (1) our approach applies exactly as for the torus case and (2) this case has already been considered in the pioneering work by Vidav [350], where existence, uniqueness and exponential stability (with non constructive constants) have been established. We do not consider either the case of a bounded domain complemented with a reflection as we will consider in Section 11 for the kinetic Fokker-Planck evolution equation, because we have not been able to establish some crucial regularity estimates which seem to be necessary in our approach.

**10.1. The torus.** In this section, we are first concerned with the kinetic linear Boltzmann equation in the torus

$$ (10.10) \qquad \partial_t f + v \cdot \nabla_x f = \mathcal{K}[f] - Kf, \quad \text{in } (0, \infty) \times \mathbb{T}^d \times \mathbb{R}^d. $$

We make the boundedness and strong positivity assumptions listed above together with the additional assumption

$$ (10.11) \qquad k m_*^{-1} m_1, k m_{1*}^{-1/2} m \in L_{xvv_*}^{\infty}, \quad \sum_{u \in \mathbb{Z}^d} \|m_1^{-1/2}(u+\cdot)\|_{L^{\infty}(\mathbb{T}^d)} < \infty, $$

for some $m_1$ such that $m/m_1 \in L^1 \cap L^2$.

**Theorem 10.1.** For the kinetic equation (10.10) in the torus and under conditions (10.5)-(10.6)-(10.7)-(10.8)-(10.9) and (10.11) for some weight functions $m$, $m_0$, $m_1$, there exists $r^* > 0$ such that for any $r \ge r^*$, the conclusions (S1), (S2), (S3$_2$) hold in $L_m^2$ and the conclusion (E3$_1$) holds in $L_m^1$.

Our result may be compared to [350] which establishes the same result without constructive rate and to [102] which establishes the same result using a probabilistic approach, both in the case of a bounded domain with zero influx boundary condition. It also extends to a non mass conservative situation the many results devoted to the conservative framework, see for instance the recent papers [284, 199, 78] and the references therein. When $\gamma > 0$, we may probably establish the same above result under the sole condition $r > 0$ (no need for $r$ to be large enough) by using some arguments developed in the next section.

We present now the proof of Theorem 10.1 by establishing that the conditions presented in the abstract part are satisfied.

**Condition (H1).** For an exponent $p \in [1, \infty)$ and a weight function $m$ satisfying (10.6), we set

$$ k_{\infty} := \|km_*^{-1}m\|_{L_{x}^{\infty}L_{v}^{p}L_{v_*}^{p'}} < \infty. $$

Considering then a solution $f$ to the evolution equation (10.10), we compute

$$ \begin{aligned} \frac{1}{p} \frac{d}{dt} \int f^p m^p &= \int \mathcal{K}[f] f^{p-1} m^p - K(x, v) f^p m^p \\ &\le \|\mathcal{K}[f]m\|_{L^p} \left( \int f^p m^p \right)^{1-1/p} - \int K(x, v) f^p m^p \\ &\le rk_{\infty} \int f^p m^p - K_0 \int \langle v \rangle^{\gamma} f^p m^p, \end{aligned} $$

where we have used twice the Holder inequality. This differential inequality together with the Gronwall lemma provides an a priori estimates about the growth of the $L_m^p$ norm. As a consequence, the same arguments as in section 8.3 imply that $S_\mathcal{L}$ is a positive semigroup in $L_m^p$ with growth bound $\omega(S_\mathcal{L}) = rk_\infty - K_0$. In particular, condition (H1) holds thanks to Lemma 2.2.
We are not aware of any result on the first eigentriplet problem for such linear Boltzmann like equation in the whole space. We may however compare our result to [200] where the corresponding mass conservative framework is considered. We present the proof of Theorem 10.4 in that situation by adapting the arguments presented in the previous section.

**Condition (H1).** Let us consider a weight function $m = \omega(\mathcal{H})$ as introduced before and let us fix $p \in [1, \infty)$. For a solution to the evolution equation (10.1), we classically compute

$$ 
\begin{aligned}
\frac{d}{dt} \int \frac{f^p}{p} m^p dv dx &= \int (\mathcal{L}f) f^{p-1} m^p dv dx \\
&= \int (f^p/p) \mathcal{T}^* m^p + \int (\mathcal{H}f) f^{p-1} m^p - \int f^p K m^p \\
&\le \frac{1}{p} \int (\mathcal{H}f)^p m^p + \int f^p \left(\frac{1}{p'} - K\right) m^p,
\end{aligned}
$$

by using an integration by parts and the Young inequality. For the first term, we have

$$ 
\begin{aligned}
\int (\mathcal{H}f)^p m^p dv dx &\le c_\omega \int \omega(\Phi) \left(\int f dv\right)^p dx \\
&= c_\omega \int \omega(\Phi) \int f^p \omega(|v|^2) dv dx \|\omega^{-1}(|v|^2)\|_{L^{p'}(\Omega)}^p \\
&\lesssim \int f^p m^p dv dx.
\end{aligned}
$$

All together and thanks to the Gronwall lemma, we have established an apriori estimate on the evolution of the norm $\|f\|_{L_m^p}$ and we deduce as in section 8 that $\mathcal{L}$ generates a positive semigroup on $L_m^p$. In particular, the condition (H1) is satisfied thanks to Lemma 2.2.

**Condition (H2).** We define $f_0 := e^{-\zeta \mathcal{H}}$ and we compute

$$ 
\begin{aligned}
\mathcal{L}f_0 &= \mathcal{C}f_0 = re^{-\zeta\Phi}\mathcal{H}[e^{-\mathcal{H}\zeta}] - Ke^{-\zeta\Phi}\mathcal{H}\zeta \\
&\ge (rc_\zeta - K_0)\langle v \rangle^\gamma e^{-\zeta\mathcal{H}} \ge 0,
\end{aligned}
$$

for $r > 0$ large enough. That implies that $\mathcal{I}$ is lower bounded by $\kappa_0 = 0$ by using Lemma 2.4-(ii), and we have thus established that $\mathcal{L}$ satisfies (H2).

**Condition (H3).** We introduce the collisionless operator

$$ \mathcal{B}f := \mathcal{T}f - Kf $$

and we define

$$ \mathcal{B}^{\mathcal{H}}\phi := \frac{1}{2}\mathcal{T}^*\phi - K\phi. $$

Our analysis is mainly a consequence of the following moment estimate.

**Lemma 10.5.** There exist some weight functions $w \lesssim \mathcal{H}$ and some real numbers $\alpha, c_\alpha, C_\alpha > 0$ such that

$$ (10.21) \qquad \mathcal{B}^{\mathcal{H}}w \le C_{\alpha}w - c_{\alpha}\mathcal{H}^{1+\alpha}. $$

*Proof of Lemma 10.5.* We split the proof into two steps.

**Step 1.** We first assume $\gamma \le \beta - 2$ and we define

$$ w := 1 + \frac{1}{2}[x]^{1+\gamma/2} \cdot v + \mathcal{H}, $$

with $[x]^\delta := x|x|^{\delta-1}$. We observe that

$$ |[x]^{1+\gamma/2} \cdot v| \le \frac{1}{2}|x|^{2+\gamma} + \frac{1}{2}|v|^2 \le \frac{1}{2}\mathcal{H} + \frac{1}{2}, $$

so that $w \sim \langle \mathcal{H} \rangle$. We now compute

$$ \mathcal{T}^*w \le \frac{\gamma+2}{4}|x|^{\gamma/2}|v|^2 - \frac{\beta}{2}|x|^{\beta+\gamma/2} $$

and thus

$$ \mathcal{B}^{\mathcal{H}}w \le C_1|x|^{\gamma/2}|v|^2 - \frac{1}{2}|x|^{\beta+\gamma/2} - \frac{1}{2}|v|^{2+\gamma}. $$
**Condition (H2).** For $f_0 := 1_{T^d \times B_1}$, where $B_1$ denotes the unit ball in $\mathbb{R}_v^d$, we compute

$$ \mathcal{L}f_0 = \mathcal{H}[f_0] - Kf_0 \geq \inf_{v \in B_1} \{r\mathcal{H}_1[f_0] - K\}f_0. $$

Using (10.5) and the strong positivity condition (10.7), we get

$$ (10.12) \qquad \inf_{v \in B_1} \{\mathcal{H}[f_0] - K\} \geq rk_1 - 2^{\gamma/2}K_0 =: \kappa_0, $$

which provides a constructive lower bound of the set $\mathcal{I}$ defined in (2.15) thanks to Lemma 2.4-(ii). We have thus established that $\mathcal{L}$ satisfies (H2).

**Condition (H3).** We define the operator

$$ \mathcal{B}f := -v \cdot \nabla_x f - K(x, v)f, $$

and we assume $\kappa_B := -\inf K \leq -K_0 < \kappa_0$, what holds whenever $r \geq r^*$, with $r^* > 0$ large enough thanks to (10.12). In the sequel, we assume $p = 2$ and we work in $X = L_m^2$. We immediately deduce that $\mathcal{B} - \kappa$ is dissipative for any $\kappa > \kappa_B$, and thus $\mathcal{R}_B(z)$ is bounded in $\mathcal{B}(L_m^2)$, uniformly in $z \in \Delta_\kappa$. For $\kappa > \kappa_B$ and $g \in L_m^2$, the function $f = \mathcal{R}_B(\kappa)g$ satisfies

$$ v \cdot \nabla_x f + (\kappa + K)f = g \quad \text{in } \mathcal{O}, $$

from what we deduce

$$ (\kappa - \kappa_B) \int_{\mathcal{O}} f^2 m^2 \leq \int_{\mathcal{O}} (\kappa + K) f^2 m^2 = \int_{\mathcal{O}} fgm^2, $$

and finally

$$ \|f\|_{L_m^2}^2 \leq \frac{1}{\kappa - \kappa_B} \|g\|_{L_m^2}^2. $$

Because of assumption (10.8), and defining $\mathcal{A} := \mathcal{H}$, we immediately deduce that

$$ (10.13) \qquad \mathcal{A}\mathcal{R}_B(\kappa) : L_m^2 \to L_{m_1}^2. $$

On the other hand, from the classical averaging lemma [181], we know that

$$ (10.14) \qquad \mathcal{A}_{\varphi}\mathcal{R}_B(\kappa) : L^2(\mathcal{O}) \to H^{1/2}(\mathcal{O}), $$

where for $\varphi = \varphi_1 \otimes \varphi_2 \in C_c^1(\mathcal{O}) \otimes C_c^1(\mathbb{R}^d)$, we have defined the mapping $\mathcal{A}_{\varphi} : L^2(\mathcal{O}) \to L^2(\mathcal{O})$ by

$$ \mathcal{A}_{\varphi}(f)(x) := \varphi_1(x, v) \int_{\mathbb{R}^d} f(x, v_*)\varphi_2(v_*) dv_*. $$

By classical approximation arguments, there exists a sequence $(\varphi_n)$ such that $\varphi_n \to k$ in the space $L^\infty(\mathbb{T}^d; L_{m_1 \times m^{-1}}^2(\mathbb{R}^d \times \mathbb{R}^d))$ and such that $\varphi_n$ is a linear combination of functions of $C_c^1(\mathcal{O}) \otimes C_c^1(\mathbb{R}^d)$. As a consequence of (10.13) and (10.14), we deduce that $\mathcal{A}\mathcal{R}_B(\kappa) \in \mathcal{H}(L_m^2)$ and next $(\mathcal{R}_B(\kappa)\mathcal{A})^2 \in \mathcal{H}(L_m^2)$ for any $\kappa > \kappa_B$. We may use Lemma 2.8 (and Remark 2.9-(2)) with $N = 2$ in $X = L_m^2$, and deduce that (H3) holds.

**Condition (H4).** We start with a result of independent interest about strict positivity. Such an argument is reminiscent from [87, 326] in the study of the Boltzmann equation and has been used for instance in [292, 78].

**Lemma 10.2.** For any $\varrho, \varrho_*, t > 0$, there exists $c > 0$ such that

$$ (10.15) \qquad (S_{\mathcal{L}}(t)f_0)(x, v) \geq c \mathbf{1}_{B_\varrho}(v) \int_{T^d \times B_{\varrho_*}} f_0 \, dv_* \, dx_*, $$

for all $f_0 \geq 0$ and $(x, v) \in \mathbb{T}^d \times \mathbb{R}^d$.

**Proof of Lemma 10.2.** We observe that the semigroup $S_B$ has explicit representation

$$ (S_B(t)f_0)(x, v) = f_0(x - vt, v)e^{-\int_0^t K(x - \tau v, v)d\tau}. $$

We next write the associated iterated Duhamel formula

$$ S_{\mathcal{L}} = S_{\mathcal{B}} + S_{\mathcal{B}}\mathcal{H} * S_{\mathcal{B}} + S_{\mathcal{B}}\mathcal{H} * S_{\mathcal{B}}\mathcal{H} * S_{\mathcal{B}} + S_{\mathcal{B}}\mathcal{H} * S_{\mathcal{B}}\mathcal{H} * S_{\mathcal{B}}\mathcal{H} * S_{\mathcal{B}}. $$

Since all the terms are nonnegative, we may through away the first terms and the last one, and we get

$$ S_{\mathcal{L}} \geq S_{\mathcal{B}} * \mathcal{H} S_{\mathcal{B}} * \mathcal{H} S_{\mathcal{B}}. $$
Using that

$$C_1|x|^{\gamma/2} \le C_1^2 + \frac{1}{4}|v|^{\gamma},$$

if $|x| \le |v|$ and

$$C_1|v|^2 \le (4C_1)^{\beta/(\beta-2)} + \frac{1}{4}|x|^{\beta},$$

if $|v| \le |x|$, we obtain

$$B^{\sharp}w \le (C_1^2 + (4C_1)^{\beta/(\beta-2)})\mathcal{H} - \frac{1}{4}|x|^{\beta+\gamma/2} - \frac{1}{4}|v|^{2+\gamma},$$

from what we conclude with $\alpha := \gamma/(2\beta)$.

Step 2. We now assume $\beta < \gamma + 2$ and we define

$$w := 1 + \frac{1}{2}[x]^{\beta/2} \cdot v + \mathcal{H},$$

so that again $w \sim \langle \mathcal{H} \rangle$. We easily compute

$$B^{\sharp}w \le C_0|x|^{\beta/2-1}|v|^2 - \frac{1}{2}|x|^{\frac{3}{2}\beta-1} - \frac{1}{2}|v|^{\gamma+2}.$$

Using Young's inequality similarly as in the step 1, we get that

$$B^{\sharp}w \le C\mathcal{H} - c|x|^{\frac{3}{2}\beta-1} - c|v|^{2+\gamma},$$

which in turn implies (10.21) with $\alpha := \min(\gamma/2, 1/2 - 1/\beta)$. $\square$

We classically deduce the following resolvent estimate.

**Lemma 10.6.** For any weight function $m_0 := \omega(\mathcal{H})$, there exists a weight function $m_1 := \omega_1(\mathcal{H})$ such that $m_1/m_0 \to \infty$ as $\mathcal{H} \to \infty$ and for any $\kappa > \kappa_B > -K_0$ there holds

$$(10.22) \qquad \mathcal{R}_B : L^2(m_0) \to L^2(m_1).$$

*Proof of Lemma 10.6.* We split the proof into two steps.

Step 1. We fix $\kappa_B \in (-K_0, \kappa_0)$ and $m_0 := \omega_0(\mathcal{H})$ with $\omega_0$ a function as defined above. We observe that

$$B^{\sharp}m_0 = -Km_0 \le -m_0,$$

so that

$$\int (\kappa - \mathcal{B})f.fm_0 = \int f^2(\kappa - \mathcal{B}^{\sharp})m_0 \le 0,$$

which means that $\kappa - \mathcal{B}$ is a dissipative operator in $L^2_{m_0}$. We deduce that $\mathcal{R}_B(\kappa) : L^2_{m_0} \to L^2_{m_0}$ for any $\kappa > -1$.

Step 2. We take

$$m := \omega(\mathcal{H})w, \quad \omega(\mathcal{H}) := \omega_0(\mathcal{H})/\mathcal{H},$$

where $w$ is defined as in Step 1 of Lemma 10.5 when $\gamma \le \beta - 2$ and as in Step 2 of Lemma 10.5 when $\gamma > \beta - 2$. In any cases $m \lesssim m_0$. On the other hand, Lemma 10.5 and $T^*\omega(\mathcal{H}) = 0$ imply together that

$$B^{\sharp}m \le C_{\alpha}m - c_{\alpha}\omega(\mathcal{H})\mathcal{H}^{1+\alpha}, \quad \alpha > 0.$$

This apriori estimate implies $\mathcal{R}_B(C_\alpha) : L^2_m \to L^2_{m_1}$, with $m_1 := m_0\mathcal{H}^\alpha$. For $g \in L^2_{m_0}$ and $\kappa > \kappa_B := -K_0$, the function $f := \mathcal{R}_B(\kappa)g \in L^2_{m_0} \subset L^2(m)$ also satisfies

$$(C_\alpha - \mathcal{B})f = g + (C_\alpha - \kappa)f.$$

We deduce $\|f\|_{L^2(m_1)} \lesssim \|g\|_{L^2(m)} + \|f\|_{L^2(m)} \lesssim \|f\|_{L^2(m_0)}$, which is nothing but (10.22). $\square$

We argue as during the proof of (H3) in Section 10.1. By a localization argument and the averaging lemma, we have $\mathcal{A}\mathcal{R}_B(\kappa) : L^2_{m_0} \to L^2(B_R \times B_R)$ with compact injection for any $R > 0$. Together with Lemma 10.6, we deduce that $\mathcal{R}_B(\kappa) \in \mathcal{K}(L^2_{m_0})$ for any $\kappa > \kappa_B$, and we conclude exactly as in Section 10.1.
On the one hand, using the explicit expression of $S_B$ and (10.7), we have

$$ (\mathcal{H}_1 S_B(s)f_0)(y, w) \geq k_{\varrho'} e^{-sK_{\varrho_*}} \mathbf{1}_{B_{\varrho'}}(w) \int_{B_{\varrho_*}} f_0(y - w_*s, w_*) dw_* =: g(s, y, w), $$

for any $s > 0$ and any $\varrho' \geq \varrho_* > 0$, with $K_\varrho := \sup_{z \in \mathbb{T}^d, |v| \leq \varrho} K(z, v)$. On the other hand, for the same reasons, we have

$$ \begin{aligned} (\mathcal{H}_1 S_B * g(t))(x, v) &= \int_0^t \int_{\mathbb{R}^d} k(x, v, v_*) g(s, x - v_*(t-s), v_*) e^{-\int_0^{t-s} K(x-\tau v_*, v_*) d\tau} dv_* ds \\ &\geq k_{\varrho'} \mathbf{1}_{B_\varrho}(v) \int_0^t \int_{B_{\varrho'}} g(s, x - v_*(t-s), v_*) e^{-(t-s)K_{\varrho'}} dv_* ds \\ &\geq k_{\varrho'}^2 e^{-tK_{\varrho'}} \mathbf{1}_{B_\varrho}(v) \int_0^t \int_{B_{\varrho'}} \int_{B_{\varrho_*}} f_0(x - v_*(t-s) - w_*s, w_*) dw_* dv_* ds \\ &\geq k_{\varrho'}^2 e^{-tK_{\varrho'}} \mathbf{1}_{B_\varrho}(v) \int_0^{t/2} \int_{B_{\varrho_*}} \int_{\mathbb{T}^d} f_0(y_*, w_*) \frac{dy_*}{(t-s)^d} dw_* ds \\ &\geq k_{\varrho'}^2 \frac{e^{-tK_{\varrho'}}}{(t/2)^d} \mathbf{1}_{B_\varrho}(v) \int_0^{t/2} \int_{B_{\varrho_*}} \int_{\mathbb{T}^d} f_0(y_*, w_*) dy_* dw_* ds \\ &\geq k_{\varrho'}^2 \frac{e^{-tK_{\varrho'}}}{(t/2)^d} \mathbf{1}_{B_\varrho}(v) \frac{t}{2} \int_{B_{\varrho_*}} \int_{\mathbb{T}^d} f_0(y_*, w_*) dy_* dw_* \end{aligned} $$

for any $t > 0$ and $\varrho' \geq \max(\varrho, \varrho_*)$ such that $t\varrho' \geq 2$, in such a way that $B(z, (t/2)\varrho') \supset \mathbb{T}^d$. We then have

$$ (\mathcal{H}_1 S_B * \mathcal{H}_1 S_B(t))(x, v) \geq k_{\varrho'}^2 \frac{e^{-tK_{\varrho'}}}{(t/2)^{d-1}} \mathbf{1}_{B_\varrho}(v) \int_{\mathbb{T}^d} \int_{B_{\varrho_*}} f_{0*} dw_* dx_* $$

for any $t \geq 2/\varrho'$. We finally conclude

$$ S_{\mathcal{L}}(t)f_0(x, v) \geq r^2 k_{\varrho'}^2 e^{-tK_{\varrho'}} \int_{2/\varrho'}^t \frac{ds}{(s/2)^{d-1}} \mathbf{1}_{B_\varrho}(v) \int_{\mathbb{T}^d} \int_{B_{\varrho_*}} f_{0*} dw_* dx_* $$

from what we deduce (10.15) by choosing $\varrho' = 8/t$. $\square$

<figure><img src="image_12.png" /></figure>

We now consider $\lambda \geq \lambda_1$ and $0 \leq f \in L_m^2$, $f \neq 0$, such that

$$ \lambda f + v \cdot \nabla_x f + K f - \mathcal{H}[f] \geq 0 \quad \text{in} \quad \mathbb{T}^d \times \mathbb{R}^d. $$

We fix $\varrho_* > 0$ such that $f \neq 0$ on $B_{\varrho_*}$. From (2.13), we have

$$ f \geq \int_{0}^{\infty} e^{-(1+\lambda)t} S_{\mathcal{L}}(t) f dt, $$

and we conclude that $f > 0$ a.e. on any set $\mathbb{T}^d \times B_\varrho$ thanks to Lemma 10.2. We have established that the strong maximum holds true, and thus (H4).

**Condition (H5).** Assume that $(\lambda, f) \in \mathbb{C} \times X \setminus \{0\}$ satisfies

$$ \mathcal{L}f = \lambda f, \quad \mathcal{L}|f| = (\Re e\lambda)|f| = \Re e(\operatorname{sign} f)\mathcal{L}f. $$

From (H4) and the first identity satisfied by $|f|$, we know that $|f| > 0$ a.e. on $\mathbb{T}^d \times \mathbb{R}^d$. Using the second identity, we get

$$ \mathcal{H}[|f|] = \Re e(\operatorname{sign} f)\mathcal{H}[f]. $$

Writing $f = e^{i\alpha}|f|$, we deduce

$$ \int_{\mathbb{R}^d} k(x, v, v_*) |f(x, v_*)| (1 - \cos(\alpha - \alpha_*)) dv_* = 0 \quad \text{a.e. on } \mathbb{T}^d \times \mathbb{R}^d, $$

and thus $\alpha = \alpha(x)$ thanks to (10.7). Next, coming back to the first equation, we have

$$ \begin{aligned} \lambda|f|e^{i\alpha} &= \mathcal{L}(|f|e^{i\alpha}) \\ &= e^{i\alpha}\mathcal{L}|f| - |f|e^{i\alpha}iv \cdot \nabla_x\alpha \\ &= e^{i\alpha}(\Re e\lambda)|f| - |f|e^{i\alpha}iv \cdot \nabla_x\alpha. \end{aligned} $$
**Condition (H4) and (H5')**. We recall that it has been proved in [79, Lem. 4.5] that the semigroup $S_t$ associated to the operator $\mathcal{L}$ satisfies the Harris condition: for any $T > 0$ and $\rho > 0$, there exists $\alpha > 0$ such that

$$ (10.23) \qquad S_T f \ge \alpha \mathbf{1}_{B_\rho} \int_{B_{\rho\theta}} f dx dv, \quad \forall f \ge 0, $$

for some constant $\vartheta \in (0, 1)$ and where $B_r := \{(x, v) \in \mathbb{R}^{2d}; |x| < r, |v| < r\}$. Although the statement of [79, Lem. 4.5] is not written in that way, one may easily track the constants appearing in Lemmas 3.5, 3.6, 3.7 and 4.1 in [79] and one obtain (10.23) with $\vartheta := 1/2$. Now, (10.23) immediately implies (H5') which in turn implies (H4) thanks to Lemma 4.8-(2)-(3).

Because $\mathcal{L}$ is the generator of a semigroup it also satisfies the weak maximum principle and Kato's inequalities (H1'). We are then in position to apply Theorem 2.21, Theorem 4.13, Theorem 5.18 and Theorem 5.23-(3) and thus complete the proof of Theorem 10.4.

## 11. THE KINETIC FOKKER-PLANCK EQUATION

In this part, we consider the kinetic Fokker-Planck evolution equation associated to the operator

$$ (11.1) \qquad \mathcal{L}f := -v \cdot \nabla_x f + \Delta_v f + b \cdot \nabla_v f + cf, $$

on functions $f: \mathcal{O} \to \mathbb{R}$, where $\mathcal{O} := \Omega \times \mathbb{R}^d$, $\Omega \subset \mathbb{R}^d$ is a domain, $b: \mathcal{O} \to \mathbb{R}^d$ is a given vector field and $c: \mathcal{O} \to \mathbb{R}$ is a given function. In contrast with the previous part, collisions are typically modelized by a Fokker-Planck operator $\Delta_v f + \text{div}_v(vf)$ (when $b = v$ and $c = d$) which takes into account a thermal bath of (Gaussian) whitenoise instead of the integral collisional operator $\mathcal{K}[f] - Kf$ in the linear Boltzmann equation (10.1).

We will consider the case when $\Omega$ is a bounded domain and the equation is complemented with a boundary condition. More precisely, we assume the classical balance between the values of the trace $\gamma f$ of $f$ on the outgoing and incoming velocities subsets of the boundary

$$ (11.2) \qquad (\gamma_f)(x, v) = \mathcal{R}_x(\gamma_+ f(x, .))(v) \quad \text{on } \Sigma_-, $$

where in this context we define $\Sigma_{\pm}^x := \{v \in \mathbb{R}^3; \pm v \cdot \nu_x > 0\}$ the sets of outgoing $(\Sigma_+^x)$ and incoming $(\Sigma_-^x)$ velocities at point $x \in \partial\Omega$, next the sets

$$ \Sigma_{\pm} = \{(x, v) \in \Sigma; \pm \nu_x \cdot v > 0\} = \{(x, v); x \in \partial\Omega, v \in \Sigma_{\pm}^x\}, $$

and finally the outgoing and incoming trace functions $\gamma_{\pm}f := \mathbf{1}_{\Sigma_{\pm}} \gamma f$. Here and in the sequel, $\nu_x$ denotes the unit normal outward vector field defined on the boundary set $\partial\Omega$. We similarly define the grazing velocity set

$$ \Sigma_0 = \{(x, v) \in \Sigma; \nu_x \cdot v = 0\}. $$

The reflection operator $\mathcal{R}_x$ is local in position, but can be local or nonlocal in the velocity variable, so that it writes

$$ (\mathcal{R}_x g)(v) := \int_{\Sigma_+^x} r(x, v, v_*) g(v_*) v_* \cdot \nu_x dv_*, $$

for a reflection kernel $r: \partial\Omega \times \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}$. Some classical general assumptions on $r$ are

$$ (11.3) \qquad r \ge 0, \quad \mathcal{R}_x^* 1 = 1, \quad \mathcal{R}_x \mathcal{M} = \mathcal{M}, $$

for some positive function $\mathcal{M} = \mathcal{M}(v)$, see for instance [95, 97, 98]. The second (normalisation) condition corresponds to the fact that all the particles reaching the outgoing boundary are put back on the incoming boundary (no mass is lost) while the third (reciprocity) condition means (when $\mathcal{M}$ is a Gaussian function) that the wall is in a local equilibrium state and is not influenced by the incoming particles. The normalization condition implies the local mass conservation

$$ (11.4) \qquad \int_{\Sigma_-^x} \mathcal{R}_x g |\nu \cdot v| dv = \int_{\Sigma_+^x} g \nu \cdot v dv, $$
The equation simplifies into

$$v \cdot \nabla_x \alpha = \Im m\lambda,$$

so that $\alpha(x) = \alpha$ is a constant and the reverse Kato's inequality holds.

Alternatively to (H5), we readily infer from Lemma 10.2 that the variant condition (H5') is verified.

At this stage, because of Theorem 2.21, Theorem 4.13 and Theorem 5.16 (or Theorem 5.18), we deduce the conclusions (S1), (S2) and (S32) about the existence and uniqueness of the eigentriplet $(\lambda_1, f_1, \phi_1)$ which satisfies $f_1 > 0$, $\phi_1 > 0$, $\lambda_1$ is algebraically simple and on the triviality of the boundary punctual spectrum. We now establish the exponential asymptotic stability with constructive constants.

We start with a gain of uniform boundedness estimate.

**Lemma 10.3.** There exists $N \ge 1$ such that $(AR_B(\kappa))^N : L_m^1 \to L_m^\infty$, for any $\kappa > \kappa_B$. As a consequence, $\phi_1 \in L_{m-1}^\infty$.

*Proof of Lemma 10.3. Step 1.* We argue similarly as in [278, Sec. 3.1]. On the one hand, denoting $A_1 = \mathcal{H}_1$, so that $A = rA_1$, we have for any $f_0 \in L_m^1$

$$(A_1 S_B(t) f_0)(x, v) = \int_{\mathbb{R}^d} k(x, v, v_*) f_0(x - v_*t, v_*) e^{-\int_0^t K(x-v_*\tau, v_*) d\tau} dv_*$$

and, using estimate (10.11), we deduce that

$$\begin{aligned} \|m_1 A_1 S_B(t) f_0\|_{L_x^1 L_v^\infty} &\le \|km_*^{-1} m_1\|_{L_{xv\nu_*}^\infty} \int_{\mathcal{O}} |f_0(x - v_*t, v_*)| m_* dv_* dx e^{t\kappa_B} \\ &\lesssim \|f_0\|_{L_m^1} e^{t\kappa_B}, \end{aligned}$$

for any $t \ge 0$. Now, we consider $f_0 \in L_x^1 L_v^\infty$, we write

$$(A_1 S_B(t) f_0)(x, v) = \sum_{u \in \mathbb{Z}^d} \int_{\mathbb{T}^d} k(x, v, v_*) f_0(x - ut - v_*t, u + v_*) e^{-\int_0^t K(x - (u+v_*)\tau, u+v_*) d\tau} dv_*$$

and using estimate (10.11) again, we compute

$$\begin{aligned} (A_1 S_B(t) f_0)(x, v) m(v) &\le \|km_{1*}^{-1/2} m\|_{L_{xv\nu_*}^\infty} \sum_{u \in \mathbb{Z}^d} \int_{\mathbb{T}^d} m_1^{1/2}(u+v_*) f_0(x-ut-v_*t, u+v_*) e^{t\kappa_B} dv_* \\ &\lesssim e^{t\kappa_B} \left( \sum_{u \in \mathbb{Z}^d} \|m_1^{-1/2}(u+\cdot)\|_{L^\infty(\mathbb{T}^d)} \right) \int_{\mathbb{T}^d} \|f_0(y, \cdot)\|_{L_m^\infty} \frac{dy}{t^d} \\ &\lesssim e^{t\kappa_B} \left(1+\frac{1}{t^d}\right) \|f_0\|_{L_m^1 L_{v\mu_1}^\infty}. \end{aligned}$$

Defining $\tilde{u}(t) := e^{-\kappa t} A_1 S_B(t)$, $\kappa > \kappa_B$, we have first established $\tilde{u} : L_m^1 \to L_m^1 L_{vm_1}^\infty$ uniformly in time, and thus $\tilde{u} : L_x^1 L_{vm_1}^\infty \to L_x^1 L_{vm_1}^\infty$ uniformly in time because $L_x^1 L_{vm_1}^\infty \subset L_m^1$ (we use here the fact that $m/m_1 \in L^1$). On the other hand, we have established that $t^d \tilde{u} : L_x^1 L_{vm_1}^\infty \to L_m^\infty$ uniformly in time. Using [275, Prop. 2.5] with $X := L_x^1 L_{vm_1}^\infty$ and $Y := L_m^\infty$, we deduce $\tilde{u}^{*(d+1)} : L_x^1 L_{vm_1}^\infty \to L_m^\infty$ uniformly in time, and we thus conclude that $\tilde{u}^{*N} : L_m^1 \to L_m^\infty$ uniformly in time, for $N := d+2$. Using formula (2.13), we deduce that $(AR_B)^N(z) : L_m^1 \to L_m^\infty$, uniformly for any $z \in \Delta_\kappa$.

**Step 2.** In particular, $(AR_B)^N(z) : L_m^1 \to L_m^2$ because $L_{m_1}^\infty \subset L_m^2$ (we use here the fact that $m/m_1 \in L^2$). By duality, we deduce that $(R_{B^*} A^*)^N(z) : L_{m-1}^2 \to L_{m-1}^\infty$. Coming back to the eigenvalue equation

$$A^* \phi_1 + B^* \phi_1 = \lambda_1 \phi_1,$$

we deduce

$$(10.16) \qquad \phi_1 = R_{B^*}(\lambda_1) A^* \phi = (R_{B^*}(\lambda_1) A^*)^N \phi_1.$$

By construction $\phi_1 \in L_{m-1}^2$ and we thus conclude that $\phi_1 \in L_{m-1}^\infty$. $\square$

<figure><img src="image_13.png" /></figure>
while the three assumptions (11.3) on $r$ together also imply

$$ 
\begin{aligned}
\int_{\Sigma_{-}^x} (\mathcal{R}_x g)^2 \mathcal{M}^{-1} |\nu \cdot v| dv &\le \int_{\Sigma_{-}^x} (\mathcal{R}_x (g^2/\mathcal{M})) (\mathcal{R}_x \mathcal{M}) \mathcal{M}^{-1} |\nu \cdot v| dv \\
&= \int_{\Sigma_{+}^x} g_*^2 \mathcal{M}_*^{-1} (\mathcal{R}^* 1) \nu \cdot v_* dv_*,
\end{aligned}
$$

where we have used the Cauchy-Schwarz inequality (and the fact that $r \ge 0$) in the first line and the reciprocity condition in the second line. As a consequence, we have

$$ (11.5) \qquad \int_{\Sigma_{-}^x} (\mathcal{R}_x g)^2 \mathcal{M}^{-1} |\nu \cdot v| dv \le \int_{\Sigma_{+}^x} g^2 \mathcal{M}^{-1} \nu \cdot v dv, $$

where we have used the normalization condition in that last step. In the sequel, we will rather consider the possibly position dependent Maxwell boundary condition operator

$$ (11.6) \qquad \mathcal{R}_x g = \alpha(x) \mathcal{D}_x g + \beta(x) \Gamma_x g, $$

where the accommodation coefficients $\alpha, \beta : \partial\Omega \to [0, 1]$ satisfy $\alpha(x) + \beta(x) =: \zeta(x) \le 1$, $\Gamma_x$ is the specular reflection operator

$$ (11.7) \qquad \Gamma_x(g(x, \cdot))(v) = g(x, \mathcal{V}_x v), \quad \mathcal{V}_x v = v - 2\nu(x)(\nu(x) \cdot v), $$

and $\mathcal{D}_x$ is the diffusive operator

$$ (11.8) \qquad \mathcal{D}_x(g(x, \cdot))(v) = c_{\mathcal{M}} \mathcal{M}(v) \tilde{g}(x), \quad \tilde{g}(x) = \int_{\Sigma_{+}^x} g(x, w) \nu(x) \cdot w \, dw. $$

Here the constant $c_{\mathcal{M}} := (2\pi)^{1/2}$ is such that $c_{\mathcal{M}} \tilde{\mathcal{M}} = 1$ and $\mathcal{M}$ stands for the standard Maxwellian

$$ (11.9) \qquad \mathcal{M}(v) := (2\pi)^{-d/2} \exp(-|v|^2/2), $$

or, more generally, $\mathcal{M} = \mathcal{M}(|v|) \ge 0$ is such that

$$ (11.10) \qquad \mathcal{D}_x^* 1 = 1, \quad \mathcal{D}_x \mathcal{M} = \mathcal{M}, \quad \langle v \rangle^\vartheta \mathcal{M} \in L^1(\mathbb{R}^d), $$

with $\vartheta \ge 1$ (that last condition is necessary in order that the second relation above makes sense). The boundary condition (11.6) corresponds to the *pure specular reflection* boundary condition when $\beta = 1$ and it corresponds to the *pure diffusive* boundary condition when $\alpha = 1$. When $\zeta = 1$, the Maxwell boundary condition operator (11.6) satisfies (11.3). On the contrary, when $\zeta \ne 1$, the $L^2$ estimate (11.5) holds but not anymore the mass conservation (11.4). However, the following $L^1$ estimate

$$ (11.11) \qquad \int_{\Sigma_{-}^x} |\mathcal{R}_x g| |\nu \cdot v| dv \le \zeta^* \int_{\Sigma_{+}^x} |g| |\nu \cdot v| dv $$

holds, with $0 \le \sup \zeta \le \zeta^* \le 1$. Finally, the case $\zeta = 0$ corresponds to the zero inflow problem.

Let us finally mention that similarly as in Part 8, the regularity needed on the domain $\Omega$ may be formulated in the following way: we assume that $\Omega$ is locally on one side of $\partial\Omega$ and there exists a function $\delta = \delta_\Omega \in W^{2,\infty}(\mathbb{R}^d)$ such that for all $x$ in an interior neighborhood of $\partial\Omega$ one has $\delta(x) = \text{dist}(x, \partial\Omega)$ and the vector field $\nu$ defined on $\mathbb{R}^d$ by $x \mapsto \nu(x) = \nu_x := -\nabla_x \delta(x)$ coincides with the previously defined unit outward normal vector field on $\partial\Omega$ and satisfies $\|\nu\|_{L^\infty} = 1$. We also assume that the Lebesgue measure on $\partial\Omega$ is well defined and it is denoted by $d\sigma_x$.

### 11.1. The trace problem.

We consider in this section the trace problem for a solution $g = g(x, v)$ to the stationary Vlasov-Fokker-Planck equation

$$ (11.12) \qquad \mathcal{M}g := v \cdot \nabla_x g - b \cdot \nabla_v g - \Delta_v g = G \quad \text{in } \mathcal{O}, $$

for a given a vector field $b = b(x, v)$, a source term $G = G(x, v)$ and for a solution $g = g(t, x, v)$ to the evolution Vlasov-Fokker-Planck equation

$$ (11.13) \qquad \partial_t g + v \cdot \nabla_x g - b \cdot \nabla_v g - \Delta_v g = G \quad \text{in } (0, T) \times \mathcal{O}, $$

for a given a vector field $b = b(t, x, v)$, a source term $G = G(t, x, v)$. The second problem has been considered first in [91] and next in [272, Sec. 4], where a strong (renormalized) trace function is
From now on, we choose the normalization convention $\|\phi_1\|_{L_{m-1}^\infty} = 1$ and $\langle f_1, \phi_1 \rangle = 1$.

Because of (10.9) and proceeding similarly as during the proof of condition (H3), we have

$$ \mathcal{A}\mathcal{R}_B(\kappa) : L_{m_0}^1 \to L_m^1, \quad \forall \kappa > \kappa_B, $$

so that

$$ \mathcal{R}_{B^*}(\kappa)\mathcal{A}^* : L_{m-1}^\infty \to L_{m_0-1}^\infty, \quad \forall \kappa > \kappa_B. $$

From the first identity in (10.16), we deduce

$$ \|\phi_1\|_{L_{m_0-1}^\infty} \le C_{01}\|\phi_1\|_{L_{m-1}^\infty}, $$

with constructive constant $C_{01} \in (0, \infty)$. We may here proceed along an already used argument. Consider $0 \le f \in L_m^1$ and assume $\|f\|_{L_m^1} \le A[f]_{\phi_1}$. We then compute

$$ \begin{aligned} \int f \phi_1 &= \int_{\mathcal{O}_\varrho} f \frac{\phi_1}{m} m + \int_{\mathcal{O}_\varrho^c} f m \frac{\phi_1}{m_0} \frac{m_0}{m} \\ &\le \langle f, \mathbf{1}_{\mathcal{O}_\varrho} \rangle \sup_{\mathcal{O}_\varrho} m + \|f\|_{L_m^1} C_{01} \sup_{\mathcal{O}_\varrho^c} \frac{m_0}{m} \\ &\le \langle f, \mathbf{1}_{\mathcal{O}_\varrho} \rangle \sup_{\mathcal{O}_\varrho} m + \frac{1}{2} [f]_{\phi_1} \end{aligned} $$

by choosing $\varrho = \varrho(A)$ large enough, where we denote $\mathcal{O}_\varrho := \mathbb{T}^d \times B_\varrho$. Together with Lemma 10.2, we deduce that there exists $T > 0$ and $g_A \ge 0$, $g_A \ne 0$, such that

$$ (10.17) \qquad S_{\mathcal{L}}(T) f \ge g_A[f]_{\phi_1}, $$

what is nothing but the Doblin-Harris condition (6.9). On the other hand, from the above regularization estimate, we have in the same time

$$ 1 = \|\phi_1\|_{L_{m-1}^\infty} \le C_0 \|\phi_1\|_{L_{m-1}^1}, \quad \|\phi_1\|_{L_{m_0-1}^1} \le C_1 \|\phi_1\|_{L_{m-1}^1}, $$

for some constructive constants $C_i \in (0, \infty)$. We may thus compute

$$ \begin{aligned} \int \phi_1 m^{-1} &\le \int_{B_\varrho} \phi_1 m^{-1} + \sup_{B_\varrho^c} \frac{m_0}{m} \int \phi_1 m_0^{-1} \\ &\le \int_{B_\varrho} \phi_1 m^{-1} + \sup_{B_\varrho^c} \frac{m_0}{m} C_1 \int \phi_1 m^{-1}, \end{aligned} $$

so that for $\varrho > 0$ large enough, we deduce

$$ (10.18) \qquad C_0^{-1} \le \|\phi_1\|_{L_{m-1}^1} \le 2 \int_{B_\varrho} \phi_1 m^{-1}. $$

Together with the definition of $g_A$, we deduce that the positivity condition (6.3) holds.

Finally, as during the proof of (H3) above, for $0 \le f_0 \in L_m^1$ and denoting $f := S_{\mathcal{L}}(t)f_0$, we compute

$$ \begin{aligned} \frac{d}{dt} \int fmdvdx &= \int \mathcal{H}[f]mdvdx - \int Kfmdxdv \\ &\le C_0 \int fm_0dvdx + \kappa_B \int fmdxdv, \end{aligned} $$

with $C_0 := \|kmm_{0*}^{-1}\|_{L_{xv_*}^\infty L_v^1} < \infty$ and $m_0/m \to 0$ as $v \to \infty$. Observing that

$$ \int fm_0 \le \int_{B_\varrho} f \phi_1 \sup_{B_\varrho} \frac{m_0}{\phi_1} + \int_{B_\varrho^c} f m \sup_{B_\varrho^c} \frac{m_0}{m}, $$

for any $\kappa > \kappa_B$, we may choose $\varrho > 0$ large enough in such a way that $\sup_{B_\varrho^c} \frac{m_0}{m} \le (\kappa - \kappa_B)/C_0$ and we deduce that

$$ \frac{d}{dt} \int fmdvdx \le C_1 \int f\phi_1 dvdx + \kappa \int fmdxdv $$
proved to exist. In the sequel, we recall these results and slightly extending them by considering a possible $L^2H^{-1}$ source term. We introduce some notations. We denote

$$d\xi := |\nu(x) \cdot v| \, dvd\sigma_x \quad \text{and} \quad d\xi^2 := (\nu(x) \cdot \hat{v})^2 \, dvd\sigma_x$$

the measures on the boundary set $\Sigma$. We denote by $B_1$ the class of renormalized functions $\beta \in W_{\text{loc}}^{2,\infty}(\mathbb{R})$ such that $\beta''$ has a compact support, by $B_2$ the class of functions $\beta \in W_{\text{loc}}^{2,\infty}(\mathbb{R})$ such that $\beta'' \in L^\infty(\mathbb{R})$ and by $D_0(\bar{O})$ the space of test functions $\varphi \in D(\bar{O})$ such that $\varphi = 0$ on $\Sigma_0$. We finally define the dual operator

$$M^*\varphi := -v \cdot \nabla_x\varphi + \operatorname{div}_v(b\varphi) - \Delta_v\varphi.$$

**Theorem 11.1.** We consider $g, b \in L_{\text{loc},x}^2 H_{\text{loc},v}^1$, $G \in L_{\text{loc},x}^2 H_{\text{loc},v}^{-1}$ and we assume that $g$ is a solution to the stationary Vlasov-Fokker-Planck equation (11.12). Then there exists $\gamma g \in L_{\text{loc}}^2(\Sigma, d\xi^2)$ such that the following Green renormalized formula

$$(11.14) \qquad \iint_{\bar{O}} (\beta(g) M^*\varphi - \beta''(g) |\nabla_v g|^2 \varphi) dvdx + \langle G, \beta'(g)\varphi \rangle = \\ = \iint_{\Sigma} \beta(\gamma g) \varphi \, \nu(x) \cdot v \, dvd\sigma_x$$

holds for any renormalized function $\beta \in B_1$ and any test functions $\varphi \in D(\bar{O})$, as well as for any renormalized function $\beta \in B_2$ and any test functions $\varphi \in D_0(\bar{O})$. It is worth emphasizing that $\beta'(g)\varphi \in L_x^2 H_v^1$ so that the duality product $\langle G, \beta'(g)\varphi \rangle$ is well defined.

If furthermore $\gamma g \in L_{\text{loc}}^2(\Sigma; d\xi)$ then $\gamma \pm g \in L_{\text{loc}}^2(\Sigma; d\xi)$ and (11.14) holds for any renormalized function $\beta \in B_2$ and any test functions $\varphi \in D(\bar{O})$.

**Proof of Theorem 11.1.** We only allude the proof which uses very similar arguments as those presented in Section 10 and that can also be partially found in [136, 272]. In the one hand, considering the mollifier $(\rho_\epsilon)_{\epsilon>0}$ defined in (8.19) with $z := (x, y)$, we get that $g_\epsilon$ is smooth and satisfies

$$g_\epsilon \to g \text{ in } L_{\text{loc},x}^2 H_{\text{loc},v}^1, \quad Mg_\epsilon = G_\epsilon \to G \text{ in } L_{\text{loc},x}^2 H_{\text{loc},v}^{-1},$$

which is nothing but a variant of [139, Lem. II.1]. The function $g_\epsilon$ being smooth, for any $\beta \in C^2$ such that $\beta' \in C_b^1$, we may differentiate $\beta(g_\epsilon)$ and we get

$$M\beta(g_\epsilon) + \beta''(g_\epsilon)|\nabla_v g_\epsilon|^2 = \beta'(g_\epsilon)G_\epsilon \quad \text{in } \mathcal{O},$$

with. We may thus pass to the limit as $\epsilon \to 0$ and we obtain (11.14). $\square$

We also write without proof (since this one is similar to the proof of Proposition 8.10) a stability result that we will use several times in the sequel.

**Proposition 11.2.** Let us consider three sequences $(g_k)$, $(b_k)$ of $L_{\text{loc},x}^2 H_{\text{loc},v}^1$ and $(G_k)$ of $L_{\text{loc},x}^2 H_{\text{loc},v}^{-1}$ such that

$$v \cdot \nabla_x g_k - b_k \cdot \nabla_v g_k - \Delta_v g_k = G_k \quad \text{in } \mathcal{D}'(\bar{O})$$

for any $k \ge 1$ and three functions $g, b \in L_{\text{loc},x}^2 H_{\text{loc},v}^1$ and $G \in L_{\text{loc},x}^2 H_{\text{loc},v}^{-1}$ such that $g_k \to g$ strongly in $L_{\text{loc},x}^2 H_{\text{loc},v}^1$, $b_k \to b$ weakly in $L_{\text{loc}}^2(\bar{O})$ and $G_k \to G$ strongly in $L_{\text{loc},x}^2 H_{\text{loc},v}^{-1}$. Then $g$ satisfies (11.12) and, up to the extaction of a subsequence, $\gamma g_k \to \gamma g$ a.e. on $\Sigma \setminus \Sigma_0$.

(2) If $g_k \to g$ weakly in $L_{\text{loc}}^1(\bar{O})$ then $g$ satisfies (11.14) and, up to the extaction of a subsequence, $\gamma g_k \rightharpoonup \gamma g$ on $\Sigma \setminus \Sigma_0$ (we recall that the renormalized convergence has been defined in (8.51)).

**11.2. Well-posedness problem with inflow term at the boundary.** We consider the kinetic Fokker-Planck operator $\mathcal{L}$ defined in (11.1) and we start revisiting the well posedness problem

$$(11.15) \qquad (\lambda - \mathcal{L})f = \mathfrak{F} \quad \text{in } \mathcal{O}, \quad \gamma_-f = \mathfrak{g} \quad \text{on } \Sigma_-,$$

for given data $\mathfrak{F}: \mathcal{O} \to \mathbb{R}$ and $\mathfrak{g}: \Sigma_- \to \mathbb{R}$.

For a given weight function $m: \mathbb{R}^d \to [1, \infty)$, we define the measure $d\xi_m := m^2|\nu(x) \cdot v| \, dvd\sigma_x$ on the boundary $\Sigma$. We next define $L^2 H_m^1 = L^2 H_m^1(\mathcal{O})$ the space associated to the Hilbert norm defined by

$$\|f\|_{L^2 H_m^1}^2 := \|f\|_{L_m^2}^2 + \|\nabla_v f\|_{L_m^2}^2,$$
with $C_1 = \sup_{B_\rho} \frac{m_0}{\phi_1}$. From the Gronwall lemma, we obtain

$$ \|f_t\|_{L_m^1} \le e^{\kappa t} \|f_0\|_{L_m^1} + \frac{e^{\lambda_1 t} - e^{\kappa t}}{\lambda_1 - \kappa} C_1 \int f_0 \phi_1, $$

from what we immediately deduce that $S_\mathcal{L}$ satisfies the Lyapunov condition (6.8) for any $t > 0$. It remains to quantify the constant $C_1$. The dual formulation of (10.15) applied to the dual eigenfunction $\phi_1$ with $t = 1$ and $\varrho_* = \varrho$ yields

$$ \phi_1 = e^{-\lambda_1} S_{\mathcal{L}}^*(1) \phi_1 \ge e^{-\lambda_1} c \mathbf{1}_{\mathbb{T}^d \times B_\rho} \int_{\mathbb{T}^d \times B_\rho} \phi_1 dv_* dx_* $$

Together with Equation (10.18), this provides the explicit bound $C_1 \le 2C_0e^{\lambda_1}c^{-1} \frac{\sup_{B_\rho} m_0}{\inf_{B_\rho} m}$.

We have established that the three conditions (6.9), (6.8) and (6.10) hold, so that we conclude the proof of Theorem 10.1 by just applying Theorem 6.4.

**10.2. The whole space case.** In this section, we assume that $\Omega := \mathbb{R}^d$ and we consider the kinetic equation (10.1) with an additional force field confinement $F = -\nabla_x \Phi$ associated to a potential $\Phi$. More precisely from now-on, we assume that

$$ (10.19) \qquad \Phi(x) = |x|^{\beta}, \quad \beta > 2, \quad K(v) = \langle v \rangle^{\gamma}, \quad \gamma > 0, $$

that (10.6) holds (for $p = 2$) and that there exist $\zeta, c_\zeta > 0$ such that

$$ (10.20) \qquad \mathcal{H}[M^\zeta] \ge c_\zeta \langle v \rangle^\gamma M^\zeta, \quad M := e^{-|v|^2/2}. $$

Observe that condition (10.20) is satisfied when $\mathcal{H}$ is the positive part of the mass conservative operator (10.3). For further references, we write $\mathcal{L} := \mathcal{T} + \mathcal{C}$ with

$$ \mathcal{T} := -v \cdot \nabla_x f + \nabla_x \Phi \cdot \nabla_v, \quad \mathcal{C} f := \mathcal{H}[f] - K f $$

and we define the Hamiltonian

$$ \mathcal{H} := \Phi(x) + \frac{1}{2}|v|^2. $$

In the sequel, we will only consider some weight functions $m = \omega(\mathcal{H})$ with $\omega(y) = y^\rho$, $\varrho \ge 0$, or $\omega(y) = e^{\kappa y}$, $\kappa \in (0,1)$, so that $\omega(\mathcal{H}) \sim \omega(|v|^2)\omega(\Phi)$. For $p \in [1, \infty)$, we further assume that $v \mapsto \omega^{-1}(|v|^2) \in L^{p'}$ (which imposes $\varrho > d/(2p')$ for a polynomial weight).

**Theorem 10.4.** *Consider the kinetic equation (10.1) in the whole space with confinement force and under conditions (10.5)-(10.6)-(10.7)-(10.8)-(10.19)-(10.20) for some weight function $m = \omega(\mathcal{H})$ as discussed above. There exists $r^* > 0$ such that for any $r \ge r^*$, the conclusions (S1), (S2), (S3$_2$) about existence, uniqueness and positivity of the eigentriplet solution $(\lambda_1, f_1, \phi_1)$ hold in $L_m^p$ as well as the ergodicity (E2) for the weak convergence in $L_{\phi_1}^1$.*

We are not aware of any result on the first eigentriplet problem for such linear Boltzmann like equation in the whole space. We may however compare our result to [199] where the corresponding mass conservative framework is considered. We present the proof of Theorem 10.4 in that situation by adapting the arguments presented in the previous section.

**Condition (H1).** Let us consider a weight function $m = \omega(\mathcal{H})$ as introduced before and let us fix $p \in [1, \infty)$. For a solution to the evolution equation (10.1), we classically compute

$$ \begin{align*} \frac{d}{dt} \int \frac{f^p}{p} m^p dv dx &= \int (\mathcal{L}f) f^{p-1} m^p dv dx \\ &= \int (f^p/p) \mathcal{T}^* m^p + \int (\mathcal{H}f) f^{p-1} m^p - \int f^p K m^p \\ &\le \frac{1}{p} \int (\mathcal{H}f)^p m^p + \int f^p \left(\frac{1}{p'} - K\right) m^p, \end{align*} $$
and we assume that $m$ satisfies the Poincaré type inequality

$$ (11.16) \qquad \|f \frac{\nabla m}{m}\|_{L_m^2} \lesssim \|f\|_{L^2 H_m^1}, \quad \forall f \in L^2 H_m^1. $$

Such a Poincaré inequality is classically known to be true when $m := \mathcal{M}^{-\vartheta}$, $\mathcal{M}$ is the Maxwellian (11.9) and $\vartheta > 0$. We also define

$$ L^2 H_m^{-1} := \{ \mathfrak{F} = g + \operatorname{div}_v G; g, G_i \in L_m^2(\mathcal{O}) \}, $$

so that when $m = 1$ the space $L^2 H_m^{-1}$ is nothing but the space of continuous and linear mappings on $L^2 H^1$. For $\mathfrak{F} \in L^2 H_m^{-1}$ and $f \in L^2 H_m^1$, we may thus write

$$ \langle \mathfrak{F}, f m^2 \rangle \le \|\mathfrak{F}\|_{L^2 H_m^{-1}} \|f\|_{L^2 H_m^1}. $$

We finally define in this context

$$ W_2 := \{f \in L^2 H_m^1; \hat{v} \cdot \nabla_x f \in L^2 H_m^{-1}\}, $$

and

$$ W_{2,\Sigma} := \{g \in W_2; \gamma g \in L^2(\Sigma; d\xi_m)\}, $$

with $W_{2,\Sigma} \neq W_2$ in general.

**Theorem 11.3.** Let us fix a vector field $b \in H_{\mathrm{loc}}^1(\mathcal{O})$, a function $c \in L^\infty(\mathcal{O})$, a weight function $m: \mathbb{R}^d \to [1, \infty)$ and let us assume that $b/\langle v \rangle \in L^\infty(\mathcal{O})$, that (11.16) holds and that

$$ (11.17) \qquad \lambda^* := \ess \sup \varpi < \infty, \quad \varpi := c + \frac{\Delta m^2}{2m^2} - \frac{1}{2} \mathrm{div} b - b \cdot \frac{\nabla m}{m}. $$

For any $\mathfrak{F} \in L^2 H_m^{-1}$, $g \in L^2(\Sigma_-; d\xi_m)$ and $\lambda > \lambda^*$, there exists a unique solution $f \in W_{2,\Sigma}$ to the Dirichlet problem (11.15). We have furthermore $f \ge 0$ if $\mathfrak{F} \ge 0$ and $g \ge 0$.

A similar result is proved in [129, Appendix A] in the case $\Omega = \mathbb{R}^d$. Also observe that (11.17) holds with $m := \mathcal{M}^{-1/2}$ when $\mathcal{M}$ is the standard Maxwellian (11.9) and $b(v) = \vartheta v$, with $\vartheta > 1/2$. **Proof of Theorem 11.3.** We split the proof into five steps.

**Step 1.** A priori estimates. We argue similarly as in [90, 89]. Multiplying the first equation in (11.15) by $f m^2$, performing several integrations by part in the velocity variable and using the Green formula, we have

$$ \int_{\mathcal{O}} (\lambda - \varpi) f^2 m^2 + \frac{1}{2} \int_{\Sigma} (\gamma f)^2 m^2 \nu \cdot v + \int_{\mathcal{O}} |\nabla_v f|^2 m^2 = \langle \mathfrak{F}, f m^2 \rangle. $$

Fixing $\lambda > \lambda^*$, using the Young inequality

$$ \|\mathfrak{F}\|_{L^2 H_m^{-1}} \|f\|_{L^2 H_m^1} \le \left( \frac{1}{2(\lambda - \lambda^*)} + \frac{1}{2} \right) \|\mathfrak{F}\|_{L^2 H_m^{-1}}^2 + \frac{\lambda - \lambda^*}{2} \|f\|_{L_m^2}^2 + \frac{1}{2} \|\nabla_v f\|_{L_m^2}^2 $$

and the boundary condition on the incoming set $\Sigma_-$ in (11.15), we deduce

$$ (11.18) \quad (\lambda - \lambda^*) \int_{\mathcal{O}} f^2 m^2 + \int_{\Sigma_+} (\gamma_+ f)^2 d\xi_m + \int_{\mathcal{O}} |\nabla_v f|^2 m^2 \le \frac{1 + \lambda - \lambda^*}{\lambda - \lambda^*} \|\mathfrak{F}\|_{L^2 H_m^{-1}}^2 + \int_{\Sigma_-} g^2 d\xi_m. $$

• Because of the first equation in (11.15) and the above estimate, we find

$$ (11.19) \qquad \hat{v} \cdot \nabla_x f = \frac{1}{\langle v \rangle} (\mathfrak{F} - \lambda f + \Delta_v f + b \cdot \nabla_v f + cf) \in L^2 H_m^{-1}, $$

so that $f \in W_2$.

• Multiplying the first equation in (11.15) by $f\psi$, $\psi := \nu(x) \cdot \tilde{v}m^2$ where here and below we use the notations $\hat{v} := v/\langle v \rangle$, $\tilde{v} := v/\langle v \rangle^2$, $\langle v \rangle^2 := 1+|v|^2$, and using the Green formula and one integration by part in the velocity variable, we get

$$ \frac{1}{2} \int_{\Sigma} (\gamma f)^2 (\nu \cdot \hat{v})^2 m^2 = \frac{1}{2} \int_{\mathcal{O}} f^2 \hat{v} \cdot D_x \nu_x \hat{v} m^2 - \int_{\mathcal{O}} |\nabla_v f|^2 \psi \\ + \int_{\mathcal{O}} f \nabla_v f (b\psi - \nabla_v \psi) + \int_{\mathcal{O}} f^2 \psi (c - \lambda) + \langle \mathfrak{F}, f \psi \rangle. $$

Observing that

$$ |\langle \mathfrak{F}, f \psi \rangle| \le \|\mathfrak{F}\|_{L^2 H_m^{-1}} \|f \nu(x) \cdot \tilde{v}\|_{L^2 H_m^1} \lesssim \|\mathfrak{F}\|_{L^2 H_m^{-1}} \|f\|_{L^2 H_m^1} $$
by using an integration by parts and the Young inequality. For the first term, we have

$$
\begin{align*}
\int (\mathcal{H} f)^p m^p dv dx &\le c_\omega \int \omega(\Phi) \left(\int f dv\right)^p dx \\
&= c_\omega \int \omega(\Phi) \int f^p \omega(|v|^2) dv dx \|\omega^{-1}(|v|^2)\|_{L^{p'}(\Omega)}^p \\
&\lesssim \int f^p m^p dv dx.
\end{align*}
$$

All together and thanks to the Gronwall lemma, we have established an a priori estimate on the evolution of the norm $\|f\|_{L_m^p}$ and we deduce as in section 8 that $\mathcal{L}$ generates a positive semigroup on $L_m^p$. In particular, the condition (H1) is satisfied thanks to Lemma 2.2.

**Condition (H2).** We define $f_0 := e^{-\zeta\mathcal{H}}$ and we compute

$$
\begin{align*}
\mathcal{L} f_0 &= \mathcal{C} f_0 = re^{-\zeta\Phi} \mathcal{H}[e^{-\mathcal{M}\zeta}] - Ke^{-\zeta\Phi} \mathcal{M}\zeta \\
&\geq (rc_\zeta - K_0) \langle v \rangle^\gamma e^{-\zeta\mathcal{H}} \geq 0,
\end{align*}
$$

for $r > 0$ large enough. That implies that $\mathcal{I}$ is lower bounded by $\kappa_0 = 0$ by using Lemma 2.4-(ii), and we have thus established that $\mathcal{L}$ satisfies (H2).

**Condition (H3).** We introduce the collisionless operator

$$
\mathcal{B}f := \mathcal{T}f - Kf
$$

and we define

$$
\mathcal{B}^{\sharp}\phi := \frac{1}{2}\mathcal{T}^{*}\phi - K\phi.
$$

Our analysis is mainly a consequence of the following moment estimate.

**Lemma 10.5.** There exist some weight functions $w \lesssim \mathcal{H}$ and some real numbers $\alpha, c_\alpha, C_\alpha > 0$ such that

$$
(10.21) \qquad \mathcal{B}^2 w \le C_\alpha w - c_\alpha \mathcal{H}^{1+\alpha}.
$$

*Proof of Lemma 10.5.* We split the proof into two steps.

Step 1. We first assume $\gamma \le \beta - 2$ and we define

$$
w := 1 + \frac{1}{2}[x]^{1+\gamma/2} \cdot v + \mathcal{H},
$$

with $[x]^\delta := x|x|^{\delta-1}$. We observe that

$$
|[x]^{1+\gamma/2} \cdot v| \le \frac{1}{2}|x|^{2+\gamma} + \frac{1}{2}|v|^2 \le \frac{1}{2}\mathcal{H} + \frac{1}{2},
$$

so that $w \sim \langle \mathcal{H} \rangle$. We now compute

$$
\mathcal{T}^* w \le \frac{\gamma + 2}{4} |x|^{\gamma/2} |v|^2 - \frac{\beta}{2} |x|^{\beta + \gamma/2}
$$

and thus

$$
\mathcal{B}^{\sharp} w \le C_1 |x|^{\gamma/2} |v|^2 - \frac{1}{2} |x|^{\beta+\gamma/2} - \frac{1}{2} |v|^{2+\gamma}.
$$

Using that

$$
C_1 |x|^{\gamma/2} \le C_1^2 + \frac{1}{4} |v|^{\gamma},
$$

if $|x| \le |v|$ and

$$
C_1 |v|^2 \le (4C_1)^{\beta/(\beta-2)} + \frac{1}{4} |x|^{\beta},
$$

if $|v| \le |x|$, we obtain

$$
\mathcal{B}^{\sharp} w \le (C_1^2 + (4C_1)^{\beta/(\beta-2)}) \mathcal{H} - \frac{1}{4} |x|^{\beta+\gamma/2} - \frac{1}{4} |v|^{2+\gamma},
$$

from what we conclude with $\alpha := \gamma/(2\beta)$.

Step 2. We now assume $\beta < \gamma + 2$ and we define

$$
w := 1 + \frac{1}{2}[x]^{\beta/2} \cdot v + \mathcal{H},
$$
and

$$ \|f \nabla_v \psi\|_{L^2} \lesssim \|f\|_{L^2 H_m^1}, $$

recalling that $b/\langle v \rangle \in L^\infty(\mathcal{O})$ and using the Cauchy-Schwarz inequality, we deduce

$$ (11.20) \qquad \|\gamma f\|_{L^2(\Sigma; d\xi_m^2)}^2 \le C(1 + |\lambda|) \|f\|_{L^2 H_m^1}^2 + C \|\mathfrak{F}\|_{L^2 H_m^{-1}} \|f\|_{L^2 H_m^1}, $$

for some constant $C = C(b, c, m, \nu)$, with $d\xi_m^2 := (\nu \cdot \hat{v})^2 m^2 d\nu d\sigma_x$.

• For latter reference, we establish an estimate about the behaviour of the solution near the boundary. We now introduce the following Lions-Perthame [256] type weight function

$$ (11.21) \qquad \psi := 2\delta(x)^{1/2}\nu(x) \cdot \tilde{v}, $$

and we observe that $\psi = 0$ on $\Sigma$, $\langle v \rangle \psi \in L^\infty(\mathcal{O})$, $\nabla_v \psi \in L^\infty(\mathcal{O})$ and

$$ v \cdot \nabla_x \psi = \frac{1}{\delta(x)^{1/2}} (\hat{v} \cdot \nu(x))^2 + 2\delta(x)^{1/2} \hat{v} \cdot D_x \nu(x) \hat{v}. $$

Multiplying the first equation in (11.15) by $f\psi$, we have

$$ \frac{1}{2} \int_{\mathcal{O}} v \cdot \nabla_x f^2 \psi - \int_{\mathcal{O}} f \frac{b}{\langle v \rangle} \cdot \nabla_v f \langle v \rangle \psi + \int_{\mathcal{O}} \nabla_v (f \psi) \cdot \nabla_v f + \int_{\mathcal{O}} (\lambda - c) f^2 \psi = \langle F, f \psi \rangle. $$

Using Cauchy-Schwarz and Young inequalities, we deduce

$$ (11.22) \qquad \int_{\mathcal{O}} f^2 \frac{(\hat{v} \cdot \nu(x))^2}{\delta(x)^{1/2}} dv dx \le C(1 + |\lambda|) (\|f\|_{L^2 H^1}^2 + \|F\|_{L^2 H^{-1}}^2), $$

for some constant $C = C(b, c, n)$.

• We finally state a somehow classical regularity estimate when $\mathcal{F} \in L_m^2(\mathcal{O})$. Taking advantage of the fact that $\mathcal{F} \in L_m^2$ and $f \in L^2 H_m^1$ and localizing the problem by introducing the function $g := f\chi_\epsilon \in L_x^2 H_v^1(\mathbb{R}^d \times \mathbb{R}^d)$, $\chi_\epsilon \in C_c^2(\mathcal{O})$, $\mathbf{1}_{\mathcal{O}} \le \chi_\epsilon \le 1$, $\mathcal{O}_\epsilon := \{(x, v) \in \mathcal{O}; \delta(x) > \epsilon, |v| \le 1/\epsilon\}$, we have

$$ v \cdot \nabla_x g - \Delta_v g + \langle v \rangle^2 g = \mathcal{G} \text{ in } \mathcal{D}'(\mathbb{R}^d \times \mathbb{R}^d), $$

with

$$ \mathcal{G} := (\mathcal{F} - \lambda f - cf - b \cdot \nabla_v f)\chi_\epsilon - 2\nabla_v f \cdot \nabla_v \chi_\epsilon + \langle v \rangle^2 f\chi_\epsilon \in L^2(\mathbb{R}^d \times \mathbb{R}^d). $$

From the quantitative Hormander's hypoellipticity estimate of Hérau & Pravda-Starov [210, Prop. 2.1], we then have

$$ \|D_x^{2/3} g\|_{L^2} + \|D_v^2 g\|_{L^2} \lesssim \|\mathcal{G}\|_{L^2} + \|g\|_{L^2}. $$

Coming back to the function $f$ and using the previous estimates, we deduce

$$ (11.23) \qquad \|D_x^{2/3} f\|_{L^2(\mathcal{O}_\epsilon)} + \|D_v^2 f\|_{L^2(\mathcal{O}_\epsilon)} \le C(\|\mathcal{L}f\|_{L^2(\mathcal{O})} + \|f\|_{L^2(\mathcal{O})}), $$

for a constant $C = C(\lambda, \epsilon) > 0$.

**Step 2. Existence.** We assume $\mathfrak{g} = 0$. A possible way for proving the existence is to use Lions' variant of the Lax-Milgram theorem [250, Chap III, §1] as in [36, 129] and as we proceed now. Defining the bilinear form $\mathcal{E}: L^2 H_m^1(\mathcal{O}) \times C_c^1(\mathcal{O} \cup \Sigma_-) \to \mathbb{R}$, by

$$ \begin{aligned} \mathcal{E}(f, \varphi) &= \int_{\mathcal{O}} (\lambda - \mathcal{L}) f \varphi m^2 \\ &:= \int_{\mathcal{O}} (\lambda f - b \cdot \nabla_v f - cf) \varphi m^2 + \nabla_v f \cdot \nabla_v (\varphi m^2) - f(v \cdot \nabla_x \varphi) m^2, \end{aligned} $$

for any $f \in L^2 H_m^1(\mathcal{O})$ and $\varphi \in C_c^1(\mathcal{O} \cup \Sigma_-)$, we observe that this one is coercive, namely

$$ \begin{aligned} \mathcal{E}(\varphi, \varphi) &= \int_{\mathcal{O}} (\lambda - \varpi) \varphi^2 m^2 + \int_{\mathcal{O}} |\nabla_v \varphi|^2 m^2 + \frac{1}{2} \int_{\Sigma_-} (\gamma_- \varphi)^2 d\xi_m \\ &\ge \kappa \|\varphi\|_{L^2 H_m^1}^2, \end{aligned} $$

for any $\varphi \in C_c^1(\mathcal{O} \cup \Sigma_-)$, with $\kappa := \min(\lambda - \lambda^*, 1) > 0$. From the above mentioned Lions's theorem, for any $\mathfrak{F} \in L^2 H_m^{-1}$, there exists $f \in L^2 H_m^1$ such that

$$ (11.24) \qquad \mathcal{E}(f, \varphi) = \langle \mathfrak{F}, \varphi m^2 \rangle, \quad \forall \varphi \in C_c^1(\mathcal{O} \cup \Sigma_-). $$

In particular, $f$ satisfies the first equation in (11.15) in the distributional sense $\mathcal{D}'(\mathcal{O})$, and thus from (11.19), we deduce that $f \in W_2$. Thanks to the trace Theorem 11.1 and the estimate (11.20),
so that again $w \sim \langle \mathcal{H} \rangle$. We easily compute

$$B^{\sharp}w \leq C_0|x|^{\beta/2-1}|v|^2 - \frac{1}{2}|x|^{\frac{3}{2}\beta-1} - \frac{1}{2}|v|^{\gamma+2}.$$

Using Young's inequality similarly as in the step 1, we get that

$$B^{\sharp}w \leq C\mathcal{H} - c|x|^{\frac{3}{2}\beta-1} - c|v|^{2+\gamma},$$

which in turn implies (10.21) with $\alpha := \min(\gamma/2, 1/2 - 1/\beta)$.

□

We classically deduce the following resolvent estimate.

**Lemma 10.6.** For any weight function $m_0 := \omega(\mathcal{H})$, there exists a weight function $m_1 := \omega_1(\mathcal{H})$ such that $m_1/m_0 \to \infty$ as $\mathcal{H} \to \infty$ and for any $\kappa > \kappa_B > -K_0$ there holds

$$(10.22) \qquad \mathcal{R}_B : L^2(m_0) \to L^2(m_1).$$

*Proof of Lemma 10.6.* We split the proof into two steps.

**Step 1.** We fix $\kappa_B \in (-K_0, \kappa_0)$ and $m_0 := \omega_0(\mathcal{H})$ with $\omega_0$ a function as defined above. We observe that

$$B^{\sharp}m_0 = -Km_0 \leq -m_0,$$

so that

$$\int (\kappa - \mathcal{B})f.fm_0 = \int f^2(\kappa - \mathcal{B}^\sharp)m_0 \leq 0,$$

which means that $\kappa - \mathcal{B}$ is a dissipative operator in $L^2_{m_0}$. We deduce that $\mathcal{R}_B(\kappa) : L^2_{m_0} \to L^2_{m_0}$ for any $\kappa > -1$.

**Step 2.** We take

$$m := \omega(\mathcal{H})w, \quad \omega(\mathcal{H}) := \omega_0(\mathcal{H})/\mathcal{H},$$

where $w$ is defined as in Step 1 of Lemma 10.5 when $\gamma \leq \beta - 2$ and as in Step 2 of Lemma 10.5 when $\gamma > \beta - 2$. In any cases $m \lesssim m_0$. On the other hand, Lemma 10.5 and $T^*\omega(\mathcal{H}) = 0$ imply together that

$$B^{\sharp}m \leq C_{\alpha}m - c_{\alpha}\omega(\mathcal{H})\mathcal{H}^{1+\alpha}, \quad \alpha > 0.$$

This a priori estimate implies $\mathcal{R}_B(C_\alpha) : L^2_m \to L^2_{m_1}$, with $m_1 := m_0\mathcal{H}^\alpha$. For $g \in L^2_{m_0}$ and $\kappa > \kappa_B := -K_0$, the function $f := \mathcal{R}_B(\kappa)g \in L^2_{m_0} \subset L^2(m)$ also satisfies

$$(C_\alpha - \mathcal{B})f = g + (C_\alpha - \kappa)f.$$

We deduce $\|f\|_{L^2(m_1)} \lesssim \|g\|_{L^2(m)} + \|f\|_{L^2(m)} \lesssim \|f\|_{L^2(m_0)}$, which is nothing but (10.22).

□

We argue as during the proof of (H3) in Section 10.1. By a localization argument and the averaging lemma, we have $\mathcal{A}\mathcal{R}_B(\kappa) : L^2_{m_0} \to L^2(B_R \times B_R)$ with compact injection for any $R > 0$. Together with Lemma 10.6, we deduce that $\mathcal{R}_B(\kappa) \in \mathcal{K}(L^2_{m_0})$ for any $\kappa > \kappa_B$, and we conclude exactly as in Section 10.1.

**Condition (H4) and (H5')**. We recall that it has been proved in [78, Lem. 4.5] that the semigroup $S_t$ associated to the operator $\mathcal{L}$ satisfies the Dobl-in-Harris condition: for any $T > 0$ and $\varrho > 0$, there exists $\alpha > 0$ such that

$$(10.23) \qquad S_T f \geq \alpha \mathbf{1}_{B_\varrho} \int_{B_{\varrho e}} f dxdv, \quad \forall f \geq 0,$$

for some constant $\vartheta \in (0, 1)$ and where $B_\varrho := \{(x, v) \in \mathbb{R}^{2d}; |x| < \rho, |v| < \rho\}$. Although the statement of [78, Lem. 4.5] is not written in that way, one may easily track the constants appearing in Lemmas 3.5, 3.6, 3.7 and 4.1 in [78] and one obtain (10.23) with $\vartheta := 1/2$. Now, (10.23) immediately implies (H5') which in turn implies (H4) thanks to Lemma 4.8-(2)-(3).

Because $\mathcal{L}$ is the generator of a semigroup it also satisfies the weak maximum principle and Kato's inequalities (H1'). We are then in position to apply Theorem 2.21, Theorem 4.13, Theorem 5.18 and Theorem 5.23-(3) and thus complete the proof of Theorem 10.4.
the function $f$ admits a trace $\gamma f \in L^2(\Sigma; d\xi_m^2)$. Using the Green formula (11.14) with $\beta = \text{id} \in \mathcal{B}_1$, we have

$$ (11.25) \qquad \iint_{\mathcal{O}} (f(\mathcal{L}^* - \lambda)\varphi + \mathfrak{F}\varphi) \, dv \, dx = \iint_{\Sigma} \gamma f \varphi \, \nu(x) \cdot v \, dv \, d\sigma_x, $$

for any $\varphi \in \mathcal{D}(\bar{\mathcal{O}})$. Particularizing to $\varphi \in \mathcal{D}(\mathcal{O} \cup \Sigma_-)$ and comparing with (11.24), we deduce that $\gamma_-f = 0$.

**Step 3. Existence.** The general case $\mathfrak{g} \in L^2(\Sigma_-; d\xi_m)$. When $\mathfrak{g} \in C_c^2(\Sigma_-)$, there exists a function $\mathfrak{h} \in C_c^2(\mathcal{O} \cup \Sigma_-)$ such that $\mathfrak{h}|_{\Sigma_-} = \mathfrak{g}$ and we consider the source term $G := \mathfrak{F} + (\mathcal{L} - \lambda)\mathfrak{h} \in L^2 H_m^{-1}$ as well as the problem

$$ (\lambda - \mathcal{L})g = G \text{ in } \mathcal{O}, \quad \gamma_-g = 0 \text{ on } \Sigma_- $$

From Step 2, there exists a solution $g \in W_{2,\Sigma}$ to this problem and we set $f := g + \mathfrak{h}$, in such a way that $f \in W_{2,\Sigma}$ and satisfies

$$ \begin{aligned} \int_{\mathcal{O}} f(\lambda - \mathcal{L}^*)\varphi &= \int_{\mathcal{O}} g(\lambda - \mathcal{L}^*)\varphi + \int_{\mathcal{O}} \mathfrak{h}(\lambda - \mathcal{L}^*)\varphi \\ &= \int_{\mathcal{O}} G\varphi + \int_{\mathcal{O}} (\lambda - \mathcal{L})\mathfrak{h}\varphi - \int_{\Sigma} \mathfrak{h}|_{\Sigma}\varphi \nu \cdot v, \end{aligned} $$

and thus

$$ (11.26) \qquad \int_{\mathcal{O}} f(\lambda - \mathcal{L}^*)\varphi = \int_{\mathcal{O}} \mathfrak{F}\varphi - \int_{\Sigma_-} \mathfrak{g}\varphi \nu \cdot v, $$

for any $\varphi \in C_c^2(\mathcal{O} \cup \Sigma_-)$. Together with (11.25), we get that $\gamma_-f = \mathfrak{g}$ on $\Sigma_-$. In order to deal with the general case $\mathfrak{g} \in L^2(\Sigma_-; d\xi_m)$, we introduce a sequence $(\mathfrak{g}^n)$ of $C_c^2(\Sigma_-)$ such that $\mathfrak{g}^n \to \mathfrak{g}$ in $L^2(\Sigma_-, d\xi_m)$ and we next consider the associated sequence of solutions $(f^n)$ of $W_{2,\Sigma}$ just built above. Using the estimates exhibited in Step 1, we get that $(f^n)$ is a Cauchy sequence in $W_2$, so that it converges to a limit $f \in W_{2,\Sigma}$. We may pass to the limit in (11.26) written for the sequence $(f^n)$ and deduce that the same equation holds at the limit for $f$.

**Step 4. Uniqueness.** Consider two weak solutions $f_i \in W_2$ to the equation (11.15) in the sense that

$$ \mathcal{E}(f_i, \varphi) = \langle \mathfrak{F}, \varphi m^2 \rangle, \quad \forall \varphi \in C_c^1(\mathcal{O} \cup \Sigma_-). $$

In particular, the difference $f := f_2 - f_1 \in W_2$ satisfies

$$ \mathcal{E}(f, \varphi) = 0, \quad \forall \varphi \in C_c^1(\mathcal{O} \cup \Sigma_-), $$

and from the above discussion $\gamma_-f = 0 \in L^2(\Sigma_-; d\xi_m)$. Thanks to the trace Theorem 11.1, we deduce that $\gamma f \in L_{\text{loc}}^2(\Sigma; d\xi_m)$ and we may choose $\beta(s) = s^2$ in the Green formula (11.14): we get

$$ \int_{\mathcal{O}} f^2 \{v \cdot \nabla_x \varphi - \text{div}_v(b\varphi) + \Delta_v \varphi + 2f(c - \lambda)\varphi\} - 2|\nabla_v f|^2 \varphi = \int_{\Sigma_+} (\gamma f)^2 \nu \cdot v \varphi, $$

for any test function $\varphi \in C_c^2(\bar{\mathcal{O}})$. Choosing $\varphi = m^2\chi_\varrho$, with $\chi_\varrho(v) := \chi(v/\rho)$, $\chi \in C_c^2(\mathbb{R}^d)$, $\mathbf{1}_{B_1} \le \chi \le \mathbf{1}_{B_2}$, we deduce

$$ \int_{\mathcal{O}} f^2 m^2 \{ (\lambda - \varpi)\chi_{\varrho} + \frac{1}{2}b \cdot \nabla \chi_{\varrho} - \frac{\nabla m}{m} \cdot \nabla \chi_{\varrho} - \Delta \chi_{\varrho} \} \le 0. $$

Because $f \in L^2 H_m^1$, we may pass to the limit $\rho \to \infty$ thanks to the dominated convergence theorem and we obtain

$$ \int_{\mathcal{O}} f^2 m^2 (\lambda - \varpi) \le 0, $$

and thus $f = 0$.

**Step 5. Positivity.** We assume now that $\mathfrak{F} \ge 0$ and $\mathfrak{g} \ge 0$. We proceed similarly as in the previous step by considering $\beta(s) = s^2$, $\varphi = m^2\chi_M$. Letting $M \to \infty$, we deduce

$$ \int_{\mathcal{O}} f_-^2 m^2 (\lambda - \varpi) \le 0, $$

and thus $f_- = 0$.

□
## 11. THE KINETIC FOKKER-PLANCK EQUATION

In this part, we consider the kinetic Fokker-Planck evolution equation associated to the operator

$$ (11.1) \qquad \mathcal{L}f := -v \cdot \nabla_x f + \Delta_v f + b \cdot \nabla_v f + cf, $$

on functions $f: O \to \mathbb{R}$, where $O := \Omega \times \mathbb{R}^d$, $\Omega \subset \mathbb{R}^d$ is a domain, $b: O \to \mathbb{R}^d$ is a given vector field and $c: O \to \mathbb{R}$ is a given function. In contrast with the previous part, collisions are typically modeled by a Fokker-Planck operator $\Delta_v f + \text{div}_v(vf)$ (when $b=v$ and $c=d$) which takes into account a thermal bath of (Gaussian) whitenoise instead of the integral collisional operator $\mathcal{K}[f] - Kf$ in the linear Boltzmann equation (10.1).

We will consider the case when $\Omega$ is a bounded domain and the equation is complemented with a boundary condition. More precisely, we assume the classical balance between the values of the trace $\gamma f$ of $f$ on the outgoing and incoming velocities subsets of the boundary

$$ (11.2) \qquad (\gamma_{-}f)(x, v) = \mathcal{R}_x(\gamma_{+}f(x, .))(v) \text{ on } \Sigma_{-}, $$

where in this context we define $\Sigma_{\pm}^x := \{v \in \mathbb{R}^3; \pm v \cdot \nu_x > 0\}$ the sets of outgoing $(\Sigma_+^x)$ and incoming $(\Sigma_-^x)$ velocities at point $x \in \partial\Omega$, next the sets

$$ \Sigma_{\pm} = \{(x, v) \in \Sigma; \pm \nu_x \cdot v > 0\} = \{(x, v); x \in \partial\Omega, v \in \Sigma_{\pm}^x\}, $$

and finally the outgoing and incoming trace functions $\gamma_{\pm}f := \mathbf{1}_{\Sigma_{\pm}}\gamma f$. Here and in the sequel, $\nu_x$ denotes the unit normal outward vector field defined on the boundary set $\partial\Omega$. We similarly define the grazing velocity set

$$ \Sigma_0 = \{(x, v) \in \Sigma; \nu_x \cdot v = 0\}. $$

The reflection operator $\mathcal{R}_x$ is local in position, but can be local or nonlocal in the velocity variable, so that it writes

$$ (\mathcal{R}_x g)(v) := \int_{\Sigma_+^x} r(x, v, v_*)g(v_*)v_* \cdot \nu_x dv_*, $$

for a reflection kernel $r: \partial\Omega \times \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}$. Some classical general assumptions on $r$ are

$$ (11.3) \qquad r \ge 0, \quad \mathcal{R}_x^*1 = 1, \quad \mathcal{R}_x \mathcal{M} = \mathcal{M}, $$

for some positive function $\mathcal{M} = \mathcal{M}(v)$, see for instance [94, 96, 97]. The second (normalisation) condition corresponds to the fact that all the particles reaching the outgoing boundary are put back on the incoming boundary (no mass is lost) while the third (reciprocity) condition means (when $\mathcal{M}$ is a Gaussian function) that the wall is in a local equilibrium state and is not influenced by the incoming particles. The normalization condition implies the local mass conservation

$$ (11.4) \qquad \int_{\Sigma_-^x} \mathcal{R}_x g |\nu \cdot v| dv = \int_{\Sigma_+^x} g \nu \cdot v dv, $$

while the three assumptions (11.3) on $r$ together also imply

$$ \int_{\Sigma_-^x} (\mathcal{R}_x g)^2 \mathcal{M}^{-1} |\nu \cdot v| dv \le \int_{\Sigma_-^x} (\mathcal{R}_x (g^2/\mathcal{M})) (\mathcal{R}_x \mathcal{M}) \mathcal{M}^{-1} |\nu \cdot v| dv \\ = \int_{\Sigma_+^x} g_*^2 \mathcal{M}_*^{-1} (\mathcal{R}^*1) \nu \cdot v_* dv_*, $$

where we have used the Cauchy-Schwarz inequality (and the fact that $r \ge 0$) in the first line and the reciprocity condition in the second line. As a consequence, we have

$$ (11.5) \qquad \int_{\Sigma_-^x} (\mathcal{R}_x g)^2 \mathcal{M}^{-1} |\nu \cdot v| dv \le \int_{\Sigma_+^x} g^2 \mathcal{M}^{-1} \nu \cdot v dv, $$

where we have used the normalization condition in that last step. In the sequel, we will rather consider the possibly position dependent Maxwell boundary condition operator

$$ (11.6) \qquad \mathcal{R}_x g = \alpha(x) \mathcal{D}_x g + \beta(x) \Gamma_x g, $$

where the accommodation coefficients $\alpha, \beta: \partial\Omega \to [0, 1]$ satisfy $\alpha(x) + \beta(x) =: \zeta(x) \le 1$, $\Gamma_x$ is the specular reflection operator

$$ (11.7) \qquad \Gamma_x(g(x, \cdot))(v) = g(x, \mathcal{V}_x v), \quad \mathcal{V}_x v = v - 2\nu(x)(\nu(x) \cdot v), $$
Summing up, gathering the above estimates (11.18), (11.19), (11.20), (11.22), (11.23), we see that there exists a constant $C > 0$ such that any function $f \in D(\mathcal{L})$ satisfies

$$ (11.27) \qquad \begin{aligned} & \|f\|_{L^2 H_m^1} + \|\hat{v} \cdot \nabla_x f\|_{L^2 H_m^{-1}} + \|f \frac{\hat{v} \cdot \nu}{\delta^{1/4}}\|_{L^2} \\ & \qquad + \|\gamma f\|_{L^2(\Sigma; d\xi_m^2)} + \|\gamma_+ f\|_{L^2(\Sigma; d\xi_m)} \le C(\|f\|_{L^2} + \|\mathcal{L}f\|_{L^2}) \end{aligned} $$

and for any $\varepsilon > 0$ there exists a constant $C_\varepsilon$ such that any function $f \in D(\mathcal{L})$ satisfies

$$ \|D_x^{2/3} f\|_{L^2(\mathcal{O}_\varepsilon)} + \|D_v^2 f\|_{L^2(\mathcal{O}_\varepsilon)} \le C_\varepsilon (\|f\|_{L^2} + \|\mathcal{L}f\|_{L^2}). $$

**11.3. Well-posedness problem with reflection condition at the boundary.** We consider now the well posedness problem associated to the stationary equation

$$ (11.28) \qquad (\lambda - \mathcal{L})f = \mathfrak{F} \quad \text{in } \mathcal{O}, \quad \gamma_-f = \mathcal{R}\gamma_+f \quad \text{on } \Sigma_-, $$

for a given datum $\mathfrak{F}: \mathcal{O} \to \mathbb{R}$, where the kinetic Fokker-Planck operator $\mathcal{L}$ is still defined by (11.1) and the reflexion operator $\mathcal{R}$ is described in (11.6), (11.7), (11.8).

**Theorem 11.4.** Let us fix a vector field $b \in H_{\text{loc}}^1(\bar{\mathcal{O}})$ and a function $c \in L^\infty(\mathcal{O})$ which satisfy the assumptions of Theorem 11.3 with a given weight function $m: \mathbb{R}^d \to [1, \infty)$ for the pure specular reflection case $\alpha = 0$ and with the weight function $m := \mathcal{M}^{-1/2}$ when $\alpha \neq 0$, where $\mathcal{M}$ is the Gaussian function (11.9) or a more general equilibrium satisfying (11.10). In that last case, we furthermore assume one of the two following conditions

(i) $1 - \zeta + \alpha^2/2 \ge \delta_* > 0$, and we observe that $L^2(\Sigma; d\xi_m) \subset L^1(\Sigma; d\xi)$,

(ii) $\langle v \rangle^2 \mathcal{M} \in L^1$, and we observe that $L^2(\Sigma; d\xi_m^2) \subset L^1(\Sigma; d\xi)$,

where we recall that we have defined $d\xi_m := m^2|\nu(x) \cdot v|\text{dvd}\sigma_x$ and $d\xi_m^2 := m^2(\nu(x) \cdot \hat{v})^2\text{dvd}\sigma_x$.

For any $\mathfrak{F} \in L^2 H_m^{-1}$ and $\lambda > \lambda^*$, there exists at least one solution $f \in W_2$ to the Dirichlet problem (11.28). Assuming furthermore that

$$ (11.29) \qquad \lambda^{**} := \ess\sup (c - \text{div}b) < \infty, $$

and $\lambda > \lambda^{**}$, the solution $f$ is unique and $f \ge 0$ if $\mathfrak{F} \ge 0$.

It is worth emphasizing that the assumptions of Theorem 11.4 hold when $b = v$ and $m := \mathcal{M}^{-1/2}$. We also emphasize on the fact that the additional assumptions (i) or (ii) are made in order to prove the uniqueness of the solution during the proof.

**Proof of Theorem 11.4.** We split the proof into four steps.

**Step 1. A priori estimates.** We multiply the first equation in (11.28) by $fm^2$. As in Step 1 of the proof of Theorem 11.3, we get

$$ \int_{\mathcal{O}} (\lambda - \varpi) f^2 m^2 + \frac{1}{2} \int_{\Sigma} (\gamma f)^2 m^2 \nu \cdot v + \int_{\mathcal{O}} |\nabla_v f|^2 m^2 = \langle \mathfrak{F}, fm^2 \rangle. $$

Using for instance [55, Lem. 3.1], we have

$$ (11.30) \qquad \int_{\Sigma} (\gamma f)^2 m^2 \nu \cdot v \ge \int_{\Sigma_+} [(1 - \zeta)(\gamma_+ f)^2 + \alpha (\mathcal{D}^\perp \gamma_+ f)^2] d\xi_m :=: \mathcal{E}_{\zeta, \alpha}(\gamma_+ f) \ge 0, $$

with $\mathcal{D}^\perp g := g - \mathcal{D}g$. Using that the contribution of the boundary is nonnegative in the first estimate, we first deduce

$$ (\lambda - \lambda^*) \|f\|_{L_m^2}^2 + \|\nabla f\|_{L_m^2}^2 \le \|\mathfrak{F}\|_{L^2 H_m^{-1}} \|f\|_{L^2 H_m^1}, $$

for $\lambda > \lambda^*$, so that

$$ \min(\lambda - \lambda^*, 1) \|f\|_{L^2 H_m^1} \le \|\mathfrak{F}\|_{L^2 H_m^{-1}}. $$

From the three above estimates together, for $\lambda > \lambda^*$, we obtain

$$ (11.31) \qquad \int_{\mathcal{O}} (\lambda - \varpi)_+ f^2 m^2 + \int_{\mathcal{O}} |\nabla_v f|^2 m^2 + \frac{1}{2} \mathcal{E}_{\zeta, \alpha}(\gamma_+ f) \le \frac{1}{\min(\lambda - \lambda^*, 1)} \|\mathfrak{F}\|_{L^2 H_m^{-1}}^2. $$

There is no difficulty for also getting the pieces of information (11.19), (11.20), (11.22) and (11.23), so that in particular $f \in W_2$. It is worth emphasizing here that when $\langle v \rangle^2 \mathcal{M} \in L^1$, we have $L^2(d\xi_m^2) \subset L^1(\Sigma; d\xi)$ by using the Cauchy-Schwarz and (11.20), so that in particular the boundary condition is well defined.
and $\mathcal{D}_x$ is the diffusive operator

$$ (11.8) \qquad \mathcal{D}_x(g(x, \cdot))(v) = c_{\mathcal{M}} \mathcal{M}(v) \tilde{g}(x), \quad \tilde{g}(x) = \int_{\Sigma_+^x} g(x, w) \nu(x) \cdot w \, dw. $$

Here the constant $c_{\mathcal{M}} := (2\pi)^{1/2}$ is such that $c_{\mathcal{M}} \tilde{\mathcal{M}} = 1$ and $\mathcal{M}$ stands for the standard Maxwellian

$$ (11.9) \qquad \mathcal{M}(v) := (2\pi)^{-d/2} \exp(-|v|^2/2), $$

or, more generally, $\mathcal{M} = \mathcal{M}(|v|) \ge 0$ is such that

$$ (11.10) \qquad \mathcal{D}_x^* 1 = 1, \quad \mathcal{D}_x \mathcal{M} = \mathcal{M}, \quad \langle v \rangle^\vartheta \mathcal{M} \in L^1(\mathbb{R}^d), $$

with $\vartheta \ge 1$ (that last condition is necessary in order that the second relation above makes sense). The boundary condition (11.6) corresponds to the pure specular reflection boundary condition when $\beta = 1$ and it corresponds to the pure diffusive boundary condition when $\alpha = 1$. When $\zeta = 1$, the Maxwell boundary condition operator (11.6) satisfies (11.3). On the contrary, when $\zeta \ne 1$, the $L^2$ estimate (11.5) holds but not anymore the mass conservation (11.4). However, the following $L^1$ estimate

$$ (11.11) \qquad \int_{\Sigma_{-}^x} |\mathcal{R}_x g| |\nu \cdot v| dv \le \zeta^* \int_{\Sigma_{+}^x} |g| |\nu \cdot v| dv $$

holds, with $0 \le \sup \zeta \le \zeta^* \le 1$. Finally, the case $\zeta = 0$ corresponds to the zero inflow problem.

Let us finally mention that similarly as in Part 8, the regularity needed on the domain $\Omega$ may be formulated in the following way: we assume that $\Omega$ is locally on one side of $\partial\Omega$ and there exists a function $\delta = \delta_\Omega \in W^{2,\infty}(\mathbb{R}^d)$ such that for all $x$ in an interior neighborhood of $\partial\Omega$ one has $\delta(x) = \text{dist}(x, \partial\Omega)$ and the vector field $\nu$ defined on $\mathbb{R}^d$ by $x \mapsto \nu(x) = \nu_x := -\nabla_x \delta(x)$ coincides with the previously defined unit outward normal vector field on $\partial\Omega$ and satisfies $\|\nu\|_{L^\infty} = 1$. We also assume that the Lebesgue measure on $\partial\Omega$ is well defined and it is denoted by $d\sigma_x$.

### 11.1. The trace problem.

We consider in this section the trace problem for a solution $g = g(x, v)$ to the stationary Vlasov-Fokker-Planck equation

$$ (11.12) \qquad \mathcal{M}g := v \cdot \nabla_x g - b \cdot \nabla_v g - \Delta_v g = G \quad \text{in } \mathcal{O}, $$

for a given a vector field $b = b(x, v)$, a source term $G = G(x, v)$ and for a solution $g = g(t, x, v)$ to the evolution Vlasov-Fokker-Planck equation

$$ (11.13) \qquad \partial_t g + v \cdot \nabla_x g - b \cdot \nabla_v g - \Delta_v g = G \quad \text{in } (0, T) \times \mathcal{O}, $$

for a given a vector field $b = b(t, x, v)$, a source term $G = G(t, x, v)$. The second problem has been considered first in [90] and next in [271, Sec. 4], where a strong (renormalized) trace function is proved to exist. In the sequel, we recall these results and slightly extending them by considering a possible $L^2 H^{-1}$ source term. We introduce some notations. We denote

$$ d\xi := |\nu(x) \cdot v| \, dv \, d\sigma_x \quad \text{and} \quad d\xi^2 := (\nu(x) \cdot \hat{v})^2 \, dv \, d\sigma_x $$

the measures on the boundary set $\Sigma$. We denote by $B_1$ the class of renormalized functions $\beta \in W_{\text{loc}}^{2,\infty}(\mathbb{R})$ such that $\beta''$ has a compact support, by $B_2$ the class of functions $\beta \in W_{\text{loc}}^{2,\infty}(\mathbb{R})$ such that $\beta'' \in L^\infty(\mathbb{R})$ and by $D_0(\bar{O})$ the space of test functions $\varphi \in D(\bar{O})$ such that $\varphi = 0$ on $\Sigma_0$. We finally define the dual operator

$$ \mathcal{M}^* \varphi := -v \cdot \nabla_x \varphi + \text{div}_v(b\varphi) - \Delta_v \varphi. $$

**Theorem 11.1.** We consider $g \in L_{\text{loc},x}^2 H_{\text{loc},v}^1$, $b \in L_{\text{loc},x}^2 H_{\text{loc},v}^1$, $b, \text{div}_v b \in L_{\text{loc}}^\infty$, $G \in L_{\text{loc},x}^2 H_{\text{loc},v}^{-1}$ and we assume that $g$ is a solution to the stationary Vlasov-Fokker-Planck equation (11.12). Then there exists $\gamma g \in L_{\text{loc}}^2(\Sigma, d\xi^2)$ such that the following Green renormalized formula

$$ (11.14) \qquad \iint_{\mathcal{O}} (\beta(g) \mathcal{M}^* \varphi - \beta''(g) |\nabla_v g|^2 \varphi) \, dv \, dx + \langle G, \beta'(g) \varphi \rangle = \\ = \iint_{\Sigma} \beta(\gamma g) \varphi \, \nu(x) \cdot v \, dv \, d\sigma_x $$
Let us show now how the last conclusion also holds under condition (i) in the statement of the Theorem. We then assume $\vartheta = 1$ in (11.10) and we show how to establish an additional a priori estimate. We indeed know from (11.20) that

$$ \int_{\Sigma_{-}} (\alpha \mathcal{D}(\gamma_{+}f))^{2}(\nu \cdot \hat{v})^{2} m^{2} dv d\sigma_{x} \leq \int_{\Sigma} (\gamma f)^{2}(\nu \cdot \hat{v})^{2} m^{2} dv d\sigma_{x} \leq C_{\lambda} \|\mathfrak{F}\|_{L^{2}H_{m}^{-1}}^{2}, $$

and similarly as in [18] or [272, proof of Lemma 2.2] that

$$ 1 = \int_{\Sigma_{-}^{x}} |\nu(x) \cdot v| \mathcal{M} dv = C \int_{\Sigma_{-}^{x}} (\nu(x) \cdot \hat{v})^{2} \mathcal{M} dv, \quad \forall x \in \partial \Omega, $$

for some constant $C \in (0, \infty)$, so that

$$ (11.32) \qquad \int_{\Sigma_{-}} (\alpha \mathcal{D}(\gamma_{+}f))^{2} d\xi_{m} = C \int_{\Sigma_{-}} (\alpha \mathcal{D}(\gamma_{+}f))^{2} (\nu \cdot \hat{v})^{2} m^{2} \leq C C_{\lambda} \|\mathfrak{F}\|_{L^{2}H_{m}^{-1}}^{2}. $$

Summing up (11.31) and (11.32), and using that

$$ (\gamma_{+}f)^{2} \leq 2(\mathcal{D}^{\perp}\gamma_{+}f)^{2} + 2(\mathcal{D}\gamma_{+}f)^{2}, $$

we deduce that

$$ (11.33) \qquad \int_{\Sigma_{+}} [1 - \zeta + \alpha^{2}/2](\gamma_{+}f)^{2} d\xi_{m} \leq C_{\lambda} \|\mathfrak{F}\|_{L^{2}H_{m}^{-1}}^{2}. $$

Defining

$$ f \in W_{2, \mathcal{R}} := \{g \in W_{2}; \gamma_{-}g = \mathcal{R}\gamma_{+}g\}, $$

we see that $W_{2,\mathcal{R}} = W_{2,\Sigma}$ if $1 - \zeta + \alpha^2/2 \geq \delta_* > 0$, but it is worth emphasizing that we may have $W_{2,\mathcal{R}} \neq W_{2,\Sigma}$ in the general case.

**Step 2. Existence when $\mathfrak{F} \geq 0$.** With the help of Theorem 11.3, we define $f_0 = 0$ and, recursively for any $n \geq 1$, we define $f_n \in W_{2,\Sigma}$ as the solution of

$$ (11.34) \qquad (\lambda - \mathcal{L})f_n = \mathfrak{F} \quad \text{in } \mathcal{O}, \quad \gamma_{-}f_n = \mathcal{R}\gamma_{+}f_{n-1} \quad \text{on } \Sigma_{-}. $$

It is worth emphasizing here that $\gamma_+ f_{n-1} \in L^2(\Sigma_+; d\xi_m)$ implies $\mathcal{R}(\gamma_+ f_{n-1}) \in L^2(\Sigma_-; d\xi_m)$ because of (11.5). We also notice that $f_n \geq 0$ because $\mathfrak{F} \geq 0$. By linearity

$$ (\lambda - \mathcal{L})(f_{n+1} - f_n) = 0 \quad \text{in } \mathcal{O}, \quad \gamma_{-}(f_{n+1} - f_n) = \mathcal{R}\gamma_{+}(f_n - f_{n-1}) \quad \text{on } \Sigma_{-}, $$

and we thus show recursevily that $f_{n+1} - f_n \geq 0$. In other words, $(f_n)$ is an increasing sequence and thus also is $(\gamma f_n)$. From (11.30), we have

$$ \begin{aligned} \int_{\Sigma} (\gamma f_n)^2 d\xi_m &= \int_{\Sigma_+} (\gamma_+ f_n)^2 d\xi_m - \int_{\Sigma_-} (\mathcal{R}\gamma_- f_{n-1})^2 d\xi_m \\ &\geq \int_{\Sigma_+} (\gamma_+ f_n)^2 d\xi_m - \int_{\Sigma_-} (\mathcal{R}\gamma_- f_n)^2 d\xi_m \geq \mathcal{E}_{\zeta, \alpha}(\gamma_+ f_n), \end{aligned} $$

so that the estimate (11.31) holds true for $f_n$ (instead of $f$) uniformly in $n \geq 1$. From the monotonous convergence theorem, there exists $f \in L^2 H_m^1$ satisfying (11.31), (11.33), (11.20) and such that $f_n \nearrow f$ a.e. Thanks to Proposition 11.2, we have $\gamma f_n \nearrow \gamma f$ a.e. on $\Sigma$, from what we deduce that $\mathcal{R}\gamma_+ f_n \to \mathcal{R}\gamma_+ f$ in $L^2(\Sigma_-; d\xi_m^2)$ thanks to the monotonous convergence theorem. As a consequence, we may pass to the limit in the weak formulation of (11.34), and we get that $f$ is a solution of (11.28). We may also pass to the limitinf in the estimate (11.31) written for $f_n$, and we thus deduce that the same estimate holds for $f$.

**Step 3. Existence when $\mathfrak{F} \in L^2 H_m^{-1}$.** When $\mathfrak{F} \in L_m^2$, we may introduce the splitting $\mathfrak{F} = \mathfrak{F}_+ - \mathfrak{F}_-$, just use the previous step for $\mathfrak{F}_\pm$ and conclude by linearity of the equation. When $\mathfrak{F} \notin L_m^2$, we proceed similarly as in [272] and in the following way. We first assume $\zeta \leq \zeta^* \in [0, 1)$ and we consider the mapping $\Psi: W_{2,\Sigma} \to W_{2,\Sigma}$, $g \mapsto f = \Psi(g)$, where $f$ is the solution to the stationary problem

$$ (11.35) \qquad \begin{cases} (\lambda - \mathcal{L})f = \mathfrak{F} & \text{in } \mathcal{O} \\ \gamma_{-}f = \mathcal{R}\gamma_{+}g & \text{on } \Sigma_{-}. \end{cases} $$
holds for any renormalized function $\beta \in \mathcal{B}_1$ and any test functions $\varphi \in \mathcal{D}(\bar{O})$, as well as for any renormalized function $\beta \in \mathcal{B}_2$ and any test functions $\varphi \in \mathcal{D}_0(\bar{O})$. It is worth emphasizing that $\beta'(g)\varphi \in L_x^2 H_v^1$ so that the duality product $\langle G, \beta'(g)\varphi \rangle$ is well defined.
If furthermore $\gamma \neq g \in L_{\text{loc}}^2(\Sigma; d\xi)$ then $\gamma \pm g \in L_{\text{loc}}^2(\Sigma; d\xi)$ and (11.14) holds for any renormalized function $\beta \in \mathcal{B}_2$ and any test functions $\varphi \in \mathcal{D}(\bar{O})$.

**Proof of Theorem 11.1.** We only allude the proof which uses very similar arguments as those presented in Section 10 and that can also be partially found in [135, 271]. In the one hand, considering the mollifier $(\rho_\epsilon)_{\epsilon>0}$ defined in (8.19) with $z := (x, v)$, we get that $g_\epsilon$ is smooth and satisfies

$$g_{\epsilon} \to g \text{ in } L_{\text{loc},x}^{2} H_{\text{loc},v}^{1}, \quad \mathcal{M}g_{\epsilon} = G_{\epsilon} \to G \text{ in } L_{\text{loc},x}^{2} H_{\text{loc},v}^{-1},$$

which is nothing but a variant of [138, Lem. II.1]. The function $g_\epsilon$ being smooth, for any $\beta \in C^2$ such that $\beta' \in C_b^1$, we may differentiate $\beta(g_\epsilon)$ and we get

$$\mathcal{M}\beta(g_{\epsilon}) + \beta''(g_{\epsilon})|\nabla_{v}g_{\epsilon}|^{2} = \beta'(g_{\epsilon})G_{\epsilon} \text{ in } \mathcal{O}.$$

We may thus pass to the limit as $\epsilon \to 0$ and we obtain (11.14). $\square$

<figure><img src="image_14.png" /></figure>

We formulate a stability result that we will use several times in the sequel.

**Proposition 11.2.** Let us consider three sequences $(g_k)$, $(b_k)$, $(G_k)$ and three functions $g$, $b$, $G$ which all satisfy the required bounds of Theorem 11.1. We assume that

$$v \cdot \nabla_x g_k - b_k \cdot \nabla_v g_k - \Delta_v g_k = G_k \quad \text{in } \mathcal{D}'(\bar{O})$$

for any $k \ge 1$ and that $g_k \rightharpoonup g$ weakly in $L_{\text{loc},x}^2 H_{\text{loc},v}^1$, $b_k \rightharpoonup b$ strongly in $L_{\text{loc}}^2(\bar{O})$, $(b_k)$, $(\text{div}_v b_k)$ bounded in $L_{\text{loc}}^\infty$ and $G_k \rightharpoonup G$ strongly in $L_{\text{loc},x}^2 H_{\text{loc},v}^{-1}$. Then $g$ satisfies (11.12), so that it admits a trace $\gamma g \in L_{\text{loc}}^2(\Gamma; d\xi^2)$ and $\gamma g_k \rightharpoonup \gamma g$ weakly in $L_{\text{loc}}^2(\Sigma; d\xi^2)$.

**Proof of Proposition 11.2.** Using (11.14) with $\beta(s) := s^2$ and $\varphi := v \cdot \nu(x)\chi$, $0 \le \chi \in \mathcal{D}(\bar{O})$, we obtain that $(\gamma g_k)$ is bounded in $L_{\text{loc}}^2(\Sigma; d\xi^2)$, and thus, up to the extraction of a subsequence, $\gamma g_k \rightharpoonup \bar{\gamma}$ weakly in $L_{\text{loc}}^2(\Sigma; d\xi^2)$, for some $\bar{\gamma} \in L_{\text{loc}}^2(\Sigma; d\xi^2)$. We may pass to the limit in the weak formulation (11.14) written for $g_k$, $\beta(s) = s$ and $\varphi \in \mathcal{D}_0(\bar{O})$. We immediately obtain that $g$ satisfies (11.14) for $\beta(s) = s$ and any $\varphi \in \mathcal{D}_0(\bar{O})$ and thus is a solution to the stationary Vlasov-Fokker-Planck equation (11.12). Thanks to Theorem 11.1, we deduce that $g$ admits a trace $\gamma g$, that $\bar{\gamma} = \gamma g$ and that the whole sequence $(\gamma g_k)$ weakly converges to $\gamma g$. $\square$

<figure><img src="image_15.png" /></figure>

**11.2. Well-posedness problem with inflow term at the boundary.** We consider the kinetic Fokker-Planck operator $\mathcal{L}$ defined in (11.1) and we start revisiting the well posedness problem

$$(\lambda - \mathcal{L})f = \mathfrak{F} \quad \text{in } \mathcal{O}, \quad \gamma_{-}f = \mathfrak{g} \quad \text{on } \Sigma_{-},$$

for given data $\mathfrak{F} : \mathcal{O} \to \mathbb{R}$ and $\mathfrak{g} : \Sigma_{-} \to \mathbb{R}$.

For a given weight function $m : \mathbb{R}^d \to [1, \infty)$, we define the measure $d\xi_m := m^2|\nu(x) \cdot v| \, dv \, d\sigma_x$ on the boundary $\Sigma$. We next define $L^2 H_m^1 = L^2 H_m^1(\mathcal{O})$ the space associated to the Hilbert norm defined by

$$\|f\|_{L^2 H_m^1}^2 := \|f\|_{L_m^2}^2 + \|\nabla_v f\|_{L_m^2}^2,$$

and we assume that $m$ satisfies the Poincaré type inequality

$$(11.16) \qquad \|f \frac{\nabla m}{m}\|_{L_m^2} \lesssim \|f\|_{L^2 H_m^1}, \quad \forall f \in L^2 H_m^1.$$

Such a Poincaré inequality is classically known to be true when $m := \mathcal{M}^{-\vartheta}$, $\mathcal{M}$ is the Maxwellian (11.9) and $\vartheta > 0$. We also define

$$L_m^2 H_m^{-1} := \{\mathfrak{F} = g + \text{div}_v G; g, G_i \in L_m^2(\mathcal{O})\},$$

so that when $m = 1$ the space $L^2 H_m^{-1}$ is nothing but the space of continuous and linear mappings on $L^2 H^1$. For $\mathfrak{F} \in L^2 H_m^{-1}$ and $f \in L^2 H_m^1$, we may thus write

$$\langle \mathfrak{F}, fm^2 \rangle \le \|\mathfrak{F}\|_{L^2 H_m^{-1}} \|f\|_{L^2 H_m^1}.$$

We finally define in this context

$$W_2 := \{f \in L^2 H_m^1; \hat{v} \cdot \nabla_x f \in L^2 H_m^{-1}\},$$
The space $W_{2,\Sigma}$ is endowed with the norm $\|\cdot\|_{W_{2,\Sigma}}$ defined by

$$ \|g\|_{W_{2,\Sigma}}^2 = \|g\|_{L_m^2}^2 + \|\nabla_v g\|_{L_m^2}^2 + \|\gamma_+ g\|_{L_m^2(d\xi_1)}^2. $$

From (11.18) and the estimate $\|\mathcal{R}g\|_{L^2(\Sigma_-; d\xi_m)} \le \zeta^* \|g\|_{L^2(\Sigma_+; d\xi_m)}$ what we obtain by repeating the proof of (11.5), we deduce

$$ \begin{aligned} \frac{1}{C_\lambda} \|f\|_{L^2 H_m^1}^2 + \|\gamma_+ f\|_{L^2(\Sigma_+; d\xi_m)}^2 &\le C_\lambda \|\mathfrak{F}\|_{L^2 H_m^{-1}}^2 + \|\mathcal{R}\gamma_+ g\|_{L^2(\Sigma_-; d\xi_m)} \\ &\le C_\lambda \|\mathfrak{F}\|_{L^2 H_m^{-1}}^2 + \zeta^* \|\gamma_+ g\|_{L^2(\Sigma_+; d\xi_m)}, \end{aligned} $$

for some constant $C_\lambda > 0$. By linearity of (11.35), we deduce that for two functions $g_1, g_2 \in W_{2,\Sigma}$, and denoting $f_i := \Psi(g_i)$, we have

$$ \frac{1}{C_\lambda} \|f_2 - f_1\|_{L^2 H_m^1}^2 + \|\gamma_+ f_2 - \gamma_+ f_1\|_{L^2(\Sigma_+; d\xi_m)}^2 \le \zeta^* \|\gamma_+ g_2 - \gamma_+ g_1\|_{L^2(\Sigma_+; d\xi_m)}^2, $$

so that $\Psi$ is a contraction in $W_{2,\Sigma}$. By the Banach fixed point theorem, we deduce that there exists a solution $f \in W_{2,\Sigma}$ to the equation (11.28) in that case. Finally, in order to deal with the case $\zeta^* = 1$, we consider a sequence $(\zeta_n^*)$ of $[0,1)$ such that $\zeta_n^* \nearrow 1$ and the associated sequence $(f_n)$ of solutions in $W_{2,\Sigma}$ associated to the equation (11.28) with the modified reflection kernel $\mathcal{R}_n g := \zeta_n^* \mathcal{R} g$. From (11.31) and (11.20), that sequence satisfies

$$ \|f_n\|_{L^2 H_m^1}^2 + \|\gamma f_n\|_{L^2(\Sigma; d\xi_m^2)}^2 + \mathcal{E}_{1,\alpha}(\gamma_+ f_n) \le C_\lambda \|\mathfrak{F}\|_{L^2 H_m^{-1}}^2. $$

When $\alpha \ne 0$, the above estimate or (11.33) also implies that $(\gamma_+ f_n)$ belongs to a weakly compact set of $L^1(\Sigma_+; d\xi)$. As a consequence, there exist $f \in W_2$ and $\bar{\gamma}_\pm$ two functions defined on $\Sigma_\pm$ such that, up to the extraction of a subsequence,

$$ \begin{aligned} & f_n \to f \quad L^2 H_m^1, \quad \gamma_\pm f_n \to \bar{\gamma}_\pm \quad L^2(\Sigma_\pm; d\xi_m^2), \\ & \gamma_+ f_n \to \bar{\gamma}_+ \quad L^1(\Sigma_+; d\xi), \quad \mathcal{R}\gamma_+ f_n \to \bar{\mathcal{R}}\bar{\gamma}_+ \quad L^1(\Sigma_-; d\xi), \end{aligned} $$

where we have used (11.11) for the last convergence. Using Proposition 11.2, we may thus pass to the limit in the equation (11.28) satisfied by $f_n$ with modified reflection kernel $\mathcal{R}_n$ and we get that $f$ is a solution of (11.28). In the pure specular reflection case $\alpha \equiv 0$, only the first line of convergences holds, but that it is enough in order to pass to the limit in the equations (we refer to [270, 272] for similar arguments).

**Step 4. Other properties.** We further assume $\lambda > \lambda^{**}$. We proceed similarly as in [261]. Consider two weak solutions $f_i \in W_2$ to the equation (11.28). In particular, the difference $f := f_2 - f_1 \in W_2$ satisfies

$$ (\lambda - \mathcal{L})f = 0 \text{ in } \mathcal{O}, \quad \gamma_-f = \mathcal{R}\gamma_+f \text{ on } \Sigma_- $$

Using the Green renormalized formula (11.14), we have

$$ 0 = \int_{\mathcal{O}} \beta'(f)(\lambda - c)f\varphi + \beta''(f)|\nabla f|^2\varphi + \beta(f)(\mathrm{div}_v(b\varphi) - v \cdot \nabla_x\varphi - \Delta_v\varphi) + \int_{\Sigma} \beta(\gamma f)\nu \cdot v\varphi. $$

for any $\beta \in C^2(\mathbb{R})$, $\beta' \in C_b^1(\mathbb{R})$ and any test function $\varphi \in C_c^2(\bar{\mathcal{O}})$. We choose $\varphi = \varphi(v) \ge 0$, $\beta \ge 0$ and $\beta'' \ge 0$, so that

$$ 0 \ge \int_{\mathcal{O}} \beta'(f)(\lambda - c)f\varphi + \beta(f)(\mathrm{div}_v(b\varphi) - \Delta_v\varphi) + \int_{\Sigma} \beta(\gamma f)\nu \cdot v\varphi. $$

By an approximation argument, we may now take $\beta(s) = |s|$, and we get

$$ 0 \ge \int_{\mathcal{O}} |f| \{ (\lambda - c)\varphi + (\mathrm{div}_v(b\varphi) - \Delta_v\varphi) \} + \int_{\Sigma} |\gamma f| \nu \cdot v\varphi. $$

We observe that in any cases we have $f \in L_m^2(\mathcal{O}) \subset L^1(\mathcal{O})$ and $\gamma f \in L^1(\Sigma; d\xi)$. By an approximation argument, we may now take $\varphi = 1$ and using the $L^1$ estimate (11.11) on $\mathcal{R}$ (with $\zeta^* = 1$), we get

$$ \begin{aligned} 0 &\ge \int_{\Sigma_-} |\mathcal{R}\gamma_+ f| |\nu \cdot v| - \int_{\Sigma_+} |\gamma_+ f| |\nu \cdot v| \\ &\ge \int_{\mathcal{O}} |f| \{\lambda - c + \mathrm{div}_v b\} \ge (\lambda - \lambda^{**}) \int_{\mathcal{O}} |f|. \end{aligned} $$
and

$$W_{2,\Sigma} := \{g \in W_2; \gamma g \in L^2(\Sigma; d\xi_m)\},$$

with $W_{2,\Sigma} \neq W_2$ in general.

**Theorem 11.3.** Let us fix a vector field $b \in H_{\text{loc}}^1(\bar{\mathcal{O}})$, a function $c \in L^\infty(\mathcal{O})$, a weight function $m: \mathbb{R}^d \to [1, \infty)$ and let us assume that $b/\langle v \rangle \in L^\infty(\mathcal{O})$, that (11.16) holds and that

$$(11.17) \qquad \lambda^* := \ess \sup \varpi < \infty, \quad \varpi := c + \frac{\Delta m^2}{2m^2} - \frac{1}{2} \text{div} b - b \cdot \frac{\nabla m}{m}.$$

For any $\mathfrak{F} \in L^2 H_m^{-1}$, $\mathfrak{g} \in L^2(\Sigma_-; d\xi_m)$ and $\lambda > \lambda^*$, there exists a unique solution $f \in W_{2,\Sigma}$ to the Dirichlet problem (11.15). We have furthermore $f \ge 0$ if $\mathfrak{F} \ge 0$ and $\mathfrak{g} \ge 0$.

A similar result is proved in [128, Appendix A] in the case $\Omega = \mathbb{R}^d$. Also observe that (11.17) holds with $m := \mathcal{M}^{-1/2}$ when $\mathcal{M}$ is the standard Maxwellian (11.9) and $b(v) = \vartheta v$, with $\vartheta > 1/2$.

*Proof of Theorem 11.3.* We split the proof into five steps.

**Step 1.** A priori estimates. We argue similarly as in [88, 89]. Multiplying the first equation in (11.15) by $f m^2$, performing several integrations by part in the velocity variable and using the Green formula, we have

$$\int_{\mathcal{O}} (\lambda - \varpi) f^2 m^2 + \frac{1}{2} \int_{\Sigma} (\gamma f)^2 m^2 \nu \cdot v + \int_{\mathcal{O}} |\nabla_v f|^2 m^2 = \langle \mathfrak{F}, f m^2 \rangle.$$

Fixing $\lambda > \lambda^*$, using the Young inequality

$$\|\mathfrak{F}\|_{L^2 H_m^{-1}} \|f\|_{L^2 H_m^1} \le \left( \frac{1}{2(\lambda - \lambda^*)} + \frac{1}{2} \right) \|\mathfrak{F}\|_{L^2 H_m^{-1}}^2 + \frac{\lambda - \lambda^*}{2} \|f\|_{L_m^2}^2 + \frac{1}{2} \|\nabla_v f\|_{L_m^2}^2$$

and the boundary condition on the incoming set $\Sigma_-$ in (11.15), we deduce

$$(11.18) \quad (\lambda - \lambda^*) \int_{\mathcal{O}} f^2 m^2 + \int_{\Sigma_+} (\gamma_+ f)^2 d\xi_m + \int_{\mathcal{O}} |\nabla_v f|^2 m^2 \le \frac{1 + \lambda - \lambda^*}{\lambda - \lambda^*} \|\mathfrak{F}\|_{L^2 H_m^{-1}}^2 + \int_{\Sigma_-} \mathfrak{g}^2 d\xi_m.$$

• Because of the first equation in (11.15) and the above estimate, we find

$$(11.19) \qquad \hat{v} \cdot \nabla_x f = \frac{1}{\langle v \rangle} (\mathfrak{F} - \lambda f + \Delta_v f + b \cdot \nabla_v f + cf) \in L^2 H_m^{-1},$$

so that $f \in W_2$.

• Multiplying the first equation in (11.15) by $f\psi$, $\psi := \nu(x) \cdot \tilde{v}m^2$ where here and below we use the notations $\hat{v} := v/\langle v \rangle$, $\tilde{v} := v/\langle v \rangle^2$, $\langle v \rangle^2 := 1 + |v|^2$, and using the Green formula and one integration by part in the velocity variable, we get

$$\frac{1}{2} \int_{\Sigma} (\gamma f)^2 (\nu \cdot \hat{v})^2 m^2 = \frac{1}{2} \int_{\mathcal{O}} f^2 \hat{v} \cdot D_x \nu_x \hat{v} m^2 - \int_{\mathcal{O}} |\nabla_v f|^2 \psi \\ + \int_{\mathcal{O}} f \nabla_v f (b\psi - \nabla_v \psi) + \int_{\mathcal{O}} f^2 \psi (c - \lambda) + \langle \mathfrak{F}, f \psi \rangle.$$

Observing that

$$|\langle \mathfrak{F}, f \psi \rangle| \le \|\mathfrak{F}\|_{L^2 H_m^{-1}} \|f \nu(x) \cdot \tilde{v}\|_{L^2 H_m^1} \lesssim \|\mathfrak{F}\|_{L^2 H_m^{-1}} \|f\|_{L^2 H_m^1}$$

and

$$\|f \nabla_v \psi\|_{L^2} \lesssim \|f\|_{L^2 H_m^1},$$

recalling that $b/\langle v \rangle \in L^\infty(\mathcal{O})$ and using the Cauchy-Schwarz inequality, we deduce

$$(11.20) \qquad \|\gamma f\|_{L^2(\Sigma; d\xi_m^2)}^2 \le C(1 + |\lambda|) \|f\|_{L^2 H_m^1}^2 + C \|\mathfrak{F}\|_{L^2 H_m^{-1}}^2 \|f\|_{L^2 H_m^1}^2,$$

for some constant $C = C(b, c, m, \nu)$, with $d\xi_m^2 := (\nu \cdot \hat{v})^2 m^2 dv d\sigma_x$.

• For latter reference, we establish an estimate about the behaviour of the solution near the boundary. We now introduce the following Lions-Perthame [255] type weight function

$$(11.21) \qquad \psi := 2\delta(x)^{1/2} \nu(x) \cdot \tilde{v},$$
We deduce that $f = 0$. The proof of the positivity property follows the same arguments but choosing $\beta(s) = s_-$.

For latter reference, we state the counterpart of the preceding result for the kinetic Fokker-Planck evolution equation.

**Theorem 11.5.** Let us make the same assumptions as in Theorem 11.4. For any $f_0 \in L_m^2$, there exists a unique solution $f \in C([0, T]; L_m^2) \cap L^2(0, T; H_m^1)$ for any $T > 0$ to the kinetic Fokker-Planck evolution equation

$$ (11.36) \qquad \left\{ \begin{array}{ll} \partial_t f = \mathcal{L}f & \text{in } (0, \infty) \times \mathcal{O} \\ \gamma_-f = \mathcal{R}\gamma_+f & \text{on } (0, \infty) \times \Sigma_-, \end{array} \right. $$

with $\mathcal{L}$ defined in (11.1) and $\mathcal{R}$ defined in (11.6).

The proof of Theorem 11.5 is skipped since it is a mere adaptation of the proof of Theorem 8.23 and Theorem 11.4. We refer to [365, Cor. 2 7, Lem. 2.8 and Cor. 2.8] where similar well-posedness results are established (see also [272] for the existence part).

### 11.4. The first eigenvalue problem in a domain with reflection at the boundary.

We consider now the first eigenvalue problem for the kinetic Fokker-Planck operator (11.1) in a domain with reflection at the boundary, namely

$$ (11.37) \qquad \left\{ \begin{array}{ll} \lambda f + v \cdot \nabla_x f - \Delta_v f - b \cdot \nabla_v f - cf = 0 & \text{in } \mathcal{O} \\ \gamma_- f = \mathcal{R}\gamma_+ f & \text{on } \Sigma_-, \end{array} \right. $$

and the associated dual problem. In this section, we assume that $b$ and $c$ satisfy the assumptions of Theorem 11.3 with the weight function $m := \mathcal{M}^{-1/2}$ when $\alpha \neq 0$ and for a given weight function $m: \mathbb{R}^d \to [1, \infty)$ when $\alpha = 0$ and $\mathcal{R}$ is given by (11.6). We additionally assume that

$$ (11.38) \qquad \liminf_{|(x,v)| \to \infty} \varpi(x, v) = -\infty, $$

where we recall that $\varpi$ is defined in (11.17). When $\mathcal{M}$ is the Gaussian function, we find

$$ \varpi = c + \frac{|v|^2 + d}{2} - \frac{1}{2} \mathrm{div} b - b \cdot v, $$

so that (11.38) holds when $b$ is typically a bounded perturbation of the vector field $b_0(v) = \vartheta_0 v$, $\vartheta_0 > 1/2$, and more precisely

$$ \mathrm{div}_v b \in L^\infty(\mathcal{O}) \quad \text{and} \quad \inf_{x \in \Omega} \liminf_{|v| \to \infty} (b \cdot v \langle v \rangle^{-2}) \ge \vartheta_0 > 1/2. $$

The above condition is quite technical but can be seen as a compatibility condition between the thermalization due to the boundary and to the Fokker-Planck collisional operator. We are then able to work in the functional space $X := L_m^2(\mathcal{O})$.

**Theorem 11.6.** Under the above conditions, the first eigentriplet problem associated to (11.1) has a unique solution $(\lambda_1, f_1, \phi_1) \in \mathbb{R} \times X \times X'$ with $f_1 > 0$ and $\phi_1 > 0$.

The proof of Theorem 11.6 follows from Theorem 2.21, Theorem 4.13 and Theorem 5.16 as a consequence of conditions (H1)-(H5). We prove now that each of these conditions is satisfied. Theorem 11.6 generalizes [248, Thm. 2.12] where the same problem is tackled for the zero inflow condition ($\alpha = \beta = 0$) with $b = v - F(x)$ and $c = 1$ by using the classical Krein-Rutman theorem [238] in the space $X = C_b(\bar{\mathcal{O}})$. We also refer to [193, Thm. 6.8] for a variant and somehow generalisation of [248].

**Condition (H1).** From Theorem 11.4, the operator $\mathcal{L}$ satisfies (H1) with

$$ \kappa_1 := \max(\lambda^*, \lambda^{**}), $$

with $\lambda^*$ defined by (11.17) and $\lambda^{**}$ defined by (11.29). For latter reference, let us state more precisely the available estimates for $f$. On the one hand, repeating the proof of Step 1 in the proof of Theorem 11.4, we establish that for any $\lambda > \kappa_1$ and $\mathfrak{F} \in L_m^2$, the solution $f \in W_2$ to the Dirichlet problem (11.28) satisfies

$$ (11.39) \qquad \int_{\mathcal{O}} (\lambda - \varpi)_+ f^2 m^2 + \int_{\mathcal{O}} |\nabla_v f|^2 m^2 + \frac{1}{2} \mathcal{E}_{\zeta, \alpha}(\gamma_+ f) \le \frac{1}{\lambda - \lambda^*} \|\mathfrak{F}\|_{L_m^2}^2. $$