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 $g := f\chi_\epsilon \in L_x^2 H_v^1(\mathbb{R}^d \times \mathbb{R}^d)$, $\chi_\epsilon \in C_c^2(\mathcal{O})$, $\mathbf{1}_{\mathcal{O}_\epsilon} \le \chi_\epsilon \le 1$, $\mathcal{O}_\epsilon := \{(x, v) \in \mathcal{O}; \delta(x) > \epsilon, |v| \le 1/\epsilon\}$, 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 [209, 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 $g = 0$. A possible way for proving the existence is to use Lions' variant of the Lax-Milgram theorem [249, Chap III, §1] as in [36, 128] 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), the function $f$ admits a trace $\gamma f \in L^2(\Sigma; d\xi_m^2)$. Using the Green formula (11.14) with $\beta = \mathrm{id} \in \mathcal{B}_1$, 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 $g \in L^2(\Sigma_-; d\xi_m)$. When $g \in C_c^2(\Sigma_-)$, there exists a function $h \in C_c^2(\mathcal{O} \cup \Sigma_-)$ such that $h|_{\Sigma_-} = g$ and we consider the source term $G := \mathfrak{F} + (\mathcal{L} - \lambda)h \in L^2 H_m^{-1}$ as well as the problem

$$(\lambda - \mathcal{L})g = G \text{ in } \mathcal{O}, \quad \gamma_- g = 0 \text{ on } \Sigma_-.$$
On the other hand, adapting the proof of (11.22), we straightforwardly obtain

$$ (11.40) \qquad \int_{\mathcal{O}} f^2 \frac{(\hat{v} \cdot \nu(x))^2}{\delta(x)^{1/2}} dv dx \le C \int \mathfrak{F}^2 m^2, $$

for some constant $C = C(b, c, \nu, \lambda)$. For $\varepsilon_x, \varepsilon_v, \varrho > 0$, let us now define

$$ (11.41) \qquad \mathcal{U} := \{(x, v) \in \mathcal{O}; d(x, \partial\Omega) > \varepsilon_x, |v| < \varrho\}, $$

and compute

$$ \int_{\mathcal{U}^c} f^2 m^2 \le \int f^2 m^2 \mathbf{1}_{|v| \ge \varrho} + \int f^2 m^2 \mathbf{1}_{A_x} + \int f^2 m^2 \mathbf{1}_B, $$

with

$$ A_x := \{v \in B_{\varrho}, (\hat{v} \cdot \nu(x))^2 \le \varepsilon_v^2\}, \quad B := \{(x, v); |v| \le \varrho, (\hat{v} \cdot n)^2 \ge \varepsilon_v^2, d(x, \partial\Omega) \le \varepsilon_x\}. $$

For the second term, we have

$$ \int f^2 m^2 \mathbf{1}_{A_x} \le \int |A_x|^{2/r'} \|f(x, \cdot)\|_{L^2_v}^2 dx \\ \lesssim (\varrho^{d-1} \varepsilon_v)^{2/r'} \|f\|_{L^2 H_m^1}^2, $$

where we have used the Holder inequality with $r \in (1, 2^*/2)$ in the first line and the Sobolev inequality in the second line. For the third term, we have

$$ \int f^2 m^2 \mathbf{1}_B \le m^2(\varrho) \frac{\varepsilon_x^{1/2}}{\varepsilon_v^2} \int_{\mathcal{O}} f_n^2 \frac{(\hat{v} \cdot \nu(x))^2}{\delta(x)^{1/2}}. $$

Gathering these last estimates with (11.39) and (11.40), we have established that the solution $f$ to equation (11.28) furthermore satisfies

$$ (11.42) \qquad \int_{\mathcal{U}^c} f^2 m^2 \le C \left( \frac{1}{(\varrho)^2} + \varrho^{d-1} \varepsilon_v + m^2 (\varrho) \frac{\varepsilon_x^{1/2}}{\varepsilon_v^2} \right) \int \mathfrak{F}^2 m^2, $$

for a constant $C = C(b, c, \Omega, \lambda)$ and for any $\varepsilon_x, \varepsilon_v, \varrho > 0$.

**The strong maximum principle.** Let us now consider a function $0 \le f \in W_2 \setminus \{0\}$ which satisfies the Dirichlet problem (11.28) associated to $\lambda > \kappa_1$ and a source term $0 \le \mathfrak{F} \in L_m^2 \cap L^\infty$. In order to simplify the discussion, we assume that the normalization $\|f\|_{L_m^2} = 1$ holds. For proving the strong maximum principle, we briefly explain how we may adapt the arguments we have presented for the diffusive equation in Part 7 by taking in particular advantage of the above established estimates, the regularity results established in [181, 192] and some spreading positivity results we learnt in [353, Cor. A.20]. We proceed in three steps.

**Step 1.** On the one hand, from (11.42), we may choose conveniently $\varrho^{-1}, \varepsilon_v, \varepsilon_x > 0$ small enough in such a way that

$$ \int_{\mathcal{U}^c} f^2 m^2 \le \frac{1}{2} \|f\|_{L_m^2}^2, $$

where $\mathcal{U}$ is defined by (11.41). Because of the normalization condition, we have

$$ (11.43) \qquad \int_{\mathcal{U}} f^2 m^2 \ge \frac{1}{2} \|f\|_{L_m^2}^2 $$

and consequently $f(x_0, v_0)^2 \ge \delta_0^2 := \|f\|_{L_m^2}^2 (2 \|\mathbf{1}_\mathcal{U}\|_{L_m^2}^2)^{-1}$ for at least one point $(x_0, v_0) \in \mathcal{U}$.

**Step 2.** On the other hand, let us recall some integrability and regularity results established in [181] for a solution $g$ to the kinetic Fokker-Planck evolution equation

$$ \partial_t g + v \cdot \nabla_x g = \Delta_v g + B \cdot \nabla_v g + s \quad \text{in } \mathcal{V}, $$

or a sub-solution

$$ \partial_t g + v \cdot \nabla_x g \le \Delta_v g + B \cdot \nabla_v g + s \quad \text{in } \mathcal{V}, $$

for some bounded set $\mathcal{V} \subset (0, T) \times \mathcal{O}$, $s \in L^2(\mathcal{V})$ and $B \in L^\infty(\mathcal{V})$. For that purpose, given some $(t^*, x^*, v^*)$, we define

$$ Q_r := \{(t, x, v); t \in (t^* - r^2, t^*], |x - x^* - (t - t^*)v^*| < r^3, |v - v^*| < r\}. $$
From Step 2, there exists a solution $g \in W_{2,\Sigma}$ to this problem and we set $f := g + h$, in such a way that $f \in W_{2,\Sigma}$ and satisfies

$$ 
\begin{aligned}
\int_{\mathcal{O}} f(\lambda - \mathcal{L}^*)\varphi &= \int_{\mathcal{O}} g(\lambda - \mathcal{L}^*)\varphi + \int_{\mathcal{O}} h(\lambda - \mathcal{L}^*)\varphi \\
&= \int_{\mathcal{O}} G\varphi + \int_{\mathcal{O}} (\lambda - \mathcal{L})h\varphi - \int_{\Sigma} h|_{\Sigma}\varphi \nu \cdot v,
\end{aligned}
$$

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 $\gamma_-f = g$ on $\Sigma_-$. In order to deal with the general case $g \in L^2(\Sigma_-; d\xi_m)$, we introduce a sequence $(g^n)$ of $C_c^2(\Sigma_-)$ such that $g^n \to g$ 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 $g \ge 0$. 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 $f_{-} = 0$.

□

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

$$ 
\begin{aligned}
& (11.27) \qquad \|f\|_{L^2 H_m^1} + \|\hat{v} \cdot \nabla_x f\|_{L^2 H_m^{-1}} + \|f \frac{\hat{v} \cdot \nu}{\delta^{1/4}}\|_{L^2} \\
& \qquad + \|\gamma f\|_{L^2(\Sigma; d\xi_m^2)} + \|\gamma_+ f\|_{L^2(\Sigma; d\xi_m^2)} \le C(\|f\|_{L^2} + \|\mathcal{L}f\|_{L^2})
\end{aligned}
$$

and for any $\varepsilon > 0$ there exists a constant $C_\varepsilon$ such that any function $f \in D(\mathcal{L})$ satisfies

$$ \|D_x^{2/3} f\|_{L^2(\mathcal{O}_\varepsilon)} + \|D_v^2 f\|_{L^2(\mathcal{O}_\varepsilon)} \le C_\varepsilon (\|f\|_{L^2} + \|\mathcal{L}f\|_{L^2}). $$
We claim then that there exist $2 < p < q < \infty$, $\alpha \in (0,1)$ and for any $0 < r_1 < r_0$ there exists $C$ such that

$$ (11.44) \qquad \|g\|_{L^p(Q_{r_1})} \le C (\|g\|_{L^2(Q_{r_0})} + \|s\|_{L^2(Q_{r_0})}) $$

for any nonnegative subsolution $g$ on $Q_{r_0}$ from [181, Thm. 6],

$$ (11.45) \qquad \|g\|_{L^\infty(Q_{r_1})} \le C (\|g\|_{L^2(Q_{r_0})} + \|s\|_{L^q(Q_{r_0})}) $$

for any nonnegative subsolution $g$ on $Q_{r_0}$ from [181, Thm. 12] and

$$ (11.46) \qquad \|g\|_{C^\alpha(Q_{r_1})} \le C (\|g\|_{L^2(Q_{r_0})} + \|s\|_{L^\infty(Q_{r_0})}) $$

for any solution $g$ on $Q_{r_0}$ from [181, Thm. 3]. As a consequence of (11.44) and a classical covering argument, for any bounded set $\mathcal{U} \subset \bar{\mathcal{U}} \subset \mathcal{O}$, there exist $C_0 = C_0(\mathcal{U})$ and $C_1 = C_1(\mathcal{U}, \lambda)$ such that

$$ \|f\|_{L^p(\mathcal{U})} \le C_0 (\|f\|_{L^2(\mathcal{O})} + \|\mathfrak{F} + cf - \lambda f\|_{L^2(\mathcal{O})}) \le C_1 (\|f\|_{L^2(\mathcal{O})} + \|\mathfrak{F}\|_{L^2(\mathcal{O})}). $$

Observing that for $\varrho = p/2 > 1$, we have

$$ v \cdot \nabla_x f^{\varrho} - \Delta_v f^{\varrho} - b \cdot \nabla_v f^{\varrho} + \varrho f^{\varrho-1} (\lambda f - cf - \mathfrak{F}) = -4 \frac{(\varrho-1)}{\varrho} |\nabla (f^{\varrho/2})|^2 \le 0, $$

so that $f^{\varrho}$ is a weak sub-solution to the kinetic Fokker-Planck equation, we may repeat the argument and obtain in that way that $f \in L^{p_k}(\mathcal{U})$ for any $k \ge 1$, with $p_k := \varrho^{k2}$. Now, choosing $k$ such that $p_k \ge q$ and using (11.45) (as well as again a classical covering argument), we get

$$ \|f\|_{L^\infty(\mathcal{U})} \lesssim \|f\|_{L^2(\mathcal{O})} + \|\mathfrak{F} + cf - \lambda f\|_{L^q(\mathcal{O})} \lesssim \|f\|_{L^2(\mathcal{O})} + \|\mathfrak{F}\|_{L^q(\mathcal{O})}. $$

Using finally (11.46), we deduce that there exists a constant $C = C(\mathcal{U}, \lambda)$ such that

$$ \|f\|_{C^\alpha(\mathcal{U})} \lesssim \|f\|_{L^2(\mathcal{O})} + \|\mathfrak{F}\|_{L^\infty(\mathcal{O})}. $$

Together with the conclusion of the first step, we deduce that there exists a constructive constant $r_0 > 0$ such that $f \ge \delta_0 \mathbf{1}_{B((x_0, v_0), r_0)}$.

**Step 3.** From [353, Cor. A.20], we deduce that for any bounded set $\mathcal{U} \subset \bar{\mathcal{U}} \subset \mathcal{O}$, there exists a constructive constant $\delta = \delta(\delta_0, r_0, \mathcal{U}) > 0$ such that

$$ f(x, v) \ge \delta \quad \text{for any} \quad (x, v) \in \mathcal{U}, $$

where it is worth emphasizing that the hypothesis $b, c \in C(\mathcal{O})$ made in [353, Cor. A.20] is not really necessary and can be replaced by $b, c \in L^\infty(\mathcal{U})$. Because $\mathcal{U}$ may be choosen arbitrary, we have established that $f > 0$ on $\mathcal{O}$ and the strong maximum principle.

**Condition (H2).** For a given function $0 \le h_0 \in C_c^2(\mathcal{O})$ normalized by $\|h_0\|_{L_m^2} = 1$, we define $f_0 \in D(\mathcal{L})$ as the solution to

$$ (\kappa_1 - \mathcal{L})f_0 = h_0 \quad \text{in} \quad \mathcal{O}, \quad \gamma_-f_0 = \mathcal{R}\gamma_+f_0 \quad \text{on} \quad \Sigma_- $$

Taking advantage of the fact that $h_0$ has compact support, we compute

$$ 1 = \int_{\mathcal{O}} h_0^2 m^2 = \int_{\mathcal{O}} (\kappa_1 - \mathcal{L}) f_0 h_0 m^2 = \int_{\mathcal{O}} f_0 (\kappa_1 - \mathcal{L}^*)(h_0 m^2) \le C_1 \|f_0\|_{L_m^2}, $$

with $C_1 := \|m^{-1}(\kappa_1 - \mathcal{L}^*)(h_0 m^2)\|_{L^2}$. On the other hand, from (11.27), we have

$$ (11.47) \qquad \|f_0\|_{L^2 H_m^1} + \|f_0 \frac{\hat{v} \cdot \nu}{\delta^{1/4}}\|_{L^2} \le C_2, $$

for a constant $C_2$ only depending on $\|h_0\|_{L_m^2}$, $\kappa_1$ and the constant $C$ which appears in (11.27). Arguing as in (11.43), we deduce that

$$ (11.48) \qquad \int_{\mathcal{U}} f_0^2 m^2 \ge (2C_1)^{-1}, \quad \text{supp} h_0 \subset \mathcal{U}, $$

with $\mathcal{U} = \mathcal{U}_\varrho$ defined in (11.41) and $\varrho > 0$ small enough (chosen constructively from $C_2$ and $C_1$). From the above constructive strong maximum principle, we deduce that $f_0 \ge \varepsilon \mathbf{1}_{\mathcal{U}} \ge 1/C_0 h_0$ for some $\varepsilon, C_0 > 0$. We conclude as in the second constructive argument for (H2) in Section 7.1. Coming back indeed to the equation, we have

$$ \mathcal{L}f_0 = \kappa_1 f_0 - h_0 \ge \kappa_1 f_0 - \|h_0\|_{L^\infty} \mathbf{1}_{\mathcal{U}} \ge (\kappa_1 - \|h_0\|_{L^\infty} C_0) f_0, $$

so that (H2) holds with $\kappa_0 := \kappa_1 - \|h_0\|_{L^\infty} C_0$ from Lemma 2.4-(ii).
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 $b \in H_{\text{loc}}^{1}(\bar{\mathcal{O}})$, $b, \text{div}_v b \in L_{\text{loc}}^{\infty}(\bar{\mathcal{O}})$, 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 $\alpha \equiv 0$ and with the weight function $m := \mathcal{M}^{-1/2}$ when $\alpha \not\equiv 0$, 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 $d\xi_m := m^2|\nu(x) \cdot v|dvd\sigma_x$ and $d\xi_m^2 := m^2(\nu(x) \cdot \hat{v})^2dvd\sigma_x$.

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 $f m^2$. As in Step 1 of the proof of Theorem 11.3, we get

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

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

$$ (11.31) \qquad \int_{\mathcal{O}} (\lambda - \bar{\varpi})_+ f^2 m^2 + \int_{\mathcal{O}} |\nabla_v f|^2 m^2 + \frac{1}{2} \mathcal{E}_{\zeta, \alpha}(\gamma_+ f) \le \frac{1}{\min(\lambda - \lambda^*, 1)} \|\mathfrak{F}\|_{L^2 H_m^{-1}}^2. $$

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.

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

$$ \int_{\Sigma_{-}} (\alpha \mathcal{D}(\gamma_{+}f))^2 (\nu \cdot \hat{v})^2 m^2 dv d\sigma_x \le \int_{\Sigma} (\gamma f)^2 (\nu \cdot \hat{v})^2 m^2 dv d\sigma_x \le C_{\lambda} \|\mathfrak{F}\|_{L^2 H_m^{-1}}^2, $$

and similarly as in [18] or [271, proof of Lem. 2.2] that

$$ 1 = \int_{\Sigma_{-}^x} |\nu(x) \cdot v| \mathcal{M} dv = C \int_{\Sigma_{-}^x} (\nu(x) \cdot \hat{v})^2 \mathcal{M} dv, \quad \forall x \in \partial \Omega, $$
**Condition (H3).** Let us fix $\kappa < \kappa_0$ arbitrary. We define $\mathcal{B}f := \mathcal{L}f - n\chi_R(v)f$ for any $f \in W_{2,\mathcal{R}}$, with $\chi_R \in \mathcal{D}(\mathbb{R}^d)$ such that $\mathbf{1}_{B_R} \le \chi_R \le \mathbf{1}_{B_{2R}}$ and for some given $n, R \ge 0$ to be specified below. We observe that, at least formally,

$$ \int f m^2(\mathcal{B} - \kappa)f = \int_{\mathcal{O}} (\varpi - \kappa - n\chi_R)f^2 m^2 - \frac{1}{2} \int_{\Sigma} (\gamma f)^2 m^2 \nu \cdot v - \int_{\mathcal{O}} |\nabla_v f|^2 m^2. $$

Thanks to (11.38), there exists a constant $R > 0$ such that

$$ \sup_{v \in \mathbb{R}^d \setminus B_R} \varpi \le \kappa. $$

Choosing $n := \sup \varpi_+ - \kappa$, we deduce that $\varpi - \kappa - n\chi_R \le 0$. On the other hand, because of (11.30), the contribution of the boundary term in the above identity is non positive. We thus deduce that $(\mathcal{B} - \kappa)$ is dissipative in $L_m^2$. We now establish that the associated operator $\mathcal{B}$ has compact resolvent. For $\mathfrak{F} \in L_m^2$, we consider $f \in L_m^2$ the solution to

$$ (11.49) \qquad -\mathcal{B}f = \mathfrak{F} \quad \text{in } \mathcal{O}, \quad \gamma_-f = \mathcal{R}\gamma_+f \quad \text{on } \Sigma_-, $$

which existence follows from Theorem 11.4. From the above discussion (with $\kappa = -1$) and the same arguments as in Step 1 of the proof of Theorem 11.3, we have

$$ (11.50) \qquad \int f^2 \langle \varpi \rangle_{-} m^2 + 2 \int |\nabla_v f|^2 m^2 \le \int \mathfrak{F}^2 m^2. $$

Together with the regularity estimate (11.23) and the compact imbedding $H^{2/3}(\mathcal{U}) \subset L^2(\mathcal{U})$, we conclude that $\mathcal{B}$ has compact resolvent. The operator $\mathcal{A}$ on $L_m^2$ defined by $\mathcal{A}f := n\chi_R(v)f$ being bounbded, we may apply Lemma 2.8-(2) and we deduce that (H3) holds for both the primal and the dual problems.

**Condition (H4)** is nothing but the yet established strong maximum principle.

**A variant of condition (H5).** Consider $(f, \lambda)$ a pair of eigenfunction and eigenvalue such that $\lambda \in \Sigma_{P+}(\mathcal{L})$. Arguing similarly as in the proof of condition (H5) in Section 7.1, we know that

$$ \tilde{\mathcal{L}}f = i\vartheta f, \quad \vartheta \in \mathbb{R}, \quad \tilde{\mathcal{L}}|f| = 0 $$

and introducing the real and complex part decomposition $f = g + ih$, we have

$$ \int_{\mathcal{O}} \frac{1}{|f|^2} |g \nabla_v h - h \nabla_v g|^2 = 0, $$

and finally $g\nabla_v h - h\nabla_v g = 0$ a.e. on $\mathcal{O}$. Because of the regularity estimate presented during the above proof of the strong maximum principle, the functions $f$ has Hölder regularity, and thus $g$ and $h$ are continuous on $\mathcal{O}$. Because $|f| \ne 0$, we may claim that there exists a point $(x_0, v_0) \in \mathcal{O}$ such that $h(x_0, v_0) > 0$ for instance. Denoting by $\omega$ the connected component of $\{(x, v) \in \mathcal{O}; h(x, v) > 0\}$ containing $(x_0, v_0)$, we have $\nabla(g/h) = 0$ on $\omega$, and thus $g = \alpha(x)h$ on $\omega$ for some continuous function $\alpha: \Omega \to \mathbb{R}$. Coming back to the eigenvalue equation that we may write in the following system form

$$ \tilde{\mathcal{L}}g = -\vartheta h, \quad \tilde{\mathcal{L}}h = \vartheta g, $$

we compute

$$ -\vartheta h = \tilde{\mathcal{L}}(\alpha h) = \alpha \tilde{\mathcal{L}}h - hv \cdot \nabla_x \alpha = \alpha \vartheta g - hv \cdot \nabla_x \alpha \quad \text{on } \omega, $$

so that

$$ -\vartheta = \alpha^2 \vartheta - v \cdot \nabla_x \alpha \quad \text{on } \omega. $$

We deduce that $\alpha$ is a constant on $\omega$ and finally $\vartheta = 0$. We have thus established that $\lambda = \lambda_1$.

At this stage, we may use Theorem 2.21, Theorem 4.13 and Theorem 5.16, in order to get 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 briefly explain how we may deduce the stability of $f_1$ by adapting some arguments developped in [269] and already mentioned. On the one hand, we know from [269, Lem. 1.1] that any solution $f$ to the rescaled evolution equation (11.36) with $\mathcal{L}$ replaced by $\tilde{\mathcal{L}} = \mathcal{L} - \lambda_1$ satisfies

$$ \partial_t(H(X)f_1\phi_1) + \text{div}_x(vH(X)f_1\phi_1) - \text{div}_v(\phi_1^2\nabla_v(H(X)f_1/\phi_1)) = -H''(X)f_1\phi_1|\nabla_v X|^2, $$
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 $1 - \zeta + \alpha^2/2 \ge \delta_*> 0$, but it is worth emphasizing that we may have $W_{2,\mathcal{R}} \ne W_{2,\Sigma}$ in the general case.

**Step 2. Existence when $\mathfrak{F} \ge 0$.** With the help of Theorem 11.3, we define $f_0 = 0$ and, recursively for any $n \ge 1$, 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 $f_n \ge 0$ because $\mathfrak{F} \ge 0$. 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 recursively that $f_{n+1} - f_n \ge 0$. 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 $n \ge 1$. 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 $\mathfrak{F} \in L^2_m$, 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 $\mathfrak{F} \notin L^2_m$, we proceed similarly as in [271] and in the following way. We first assume $\zeta \le \zeta^* \in [0, 1)$ 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

$$ (11.35) \qquad \left\{ \begin{array}{ll} (\lambda - \mathcal{L})f = \mathfrak{F} & \text{in } \mathcal{O} \\ \gamma_{-}f = \mathcal{R}\gamma_{+}g & \text{on } \Sigma_{-}. \end{array} \right. $$

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

$$ \begin{aligned} \frac{1}{C_{\lambda}} \|f\|_{L^2 H_m^1}^2 + \|\gamma_+ f\|_{L^2(\Sigma_+; d\xi_m)}^2 &\leq C_{\lambda} \|\mathfrak{F}\|_{L^2 H_m^{-1}}^2 + \|\mathcal{R}\gamma_+ g\|_{L^2(\Sigma_-; d\xi_m)} \\ &\leq C_{\lambda} \|\mathfrak{F}\|_{L^2 H_m^{-1}}^2 + \zeta^* \|\gamma_+ g\|_{L^2(\Sigma_+; d\xi_m)}, \end{aligned} $$
for any convex function $H: \mathbb{R} \to \mathbb{R}$ and with $X := f/f_1$. After integration, we get

$$ (11.51) \qquad \frac{d}{dt} \int_{\mathcal{O}} H(X) f_1 \phi_1 + \int_{\Sigma} \nu \cdot v H(X) f_1 \phi_1 = - \int_{\mathcal{O}} H''(X) f_1 \phi_1 |\nabla_v X|^2, $$

When $H(s) := |s|$, the boundary term is

$$ \begin{aligned} \int_{\Sigma} |\gamma f| \gamma \phi_1 \nu \cdot v &= \int_{\Sigma_+} |\gamma_+ f| \mathcal{R}^* \gamma_- \phi_1 \nu \cdot v - \int_{\Sigma_-} |\mathcal{R} \gamma_+ f| \gamma_- \phi_1 |\nu \cdot v| \\ &\geq \int_{\Sigma_+} |\gamma_+ f| \mathcal{R}^* \gamma_- \phi_1 |\nu \cdot v| - \int_{\Sigma_-} \mathcal{R} |\gamma_+ f| \gamma_- \phi_1 |\nu \cdot v| = 0, \end{aligned} $$

from what we deduce the non expansive property

$$ (11.52) \qquad \int_{\mathcal{O}} |f_{t_1}| \phi_1 \leq \int_{\mathcal{O}} |f_{t_0}| \phi_1, \quad \forall t_1 \geq t_0 \geq 0. $$

On the other hand, from the Cauchy-Schwarz inequality, we have

$$ (\mathcal{R}\gamma_+ f)^2 \leq (\mathcal{R}\gamma_+ f_1)\mathcal{R}(\gamma_+ f^2/\gamma_+ f_1), $$

so that

$$ \int_{\Sigma_{-}} \frac{(\mathcal{R}\gamma_{+}f)^2}{\mathcal{R}\gamma_{+}f_1} \gamma_{-}\phi_1 |\nu \cdot v| \leq \int_{\Sigma_{-}} \mathcal{R}(\gamma_{+}f^2/\gamma_{+}f_1) \gamma_{-}\phi_1 |\nu \cdot v| $$

and finally

$$ \int_{\Sigma} (\gamma f)^2 (\gamma f_1)^{-1} \gamma \phi_1 \nu \cdot v \leq 0. $$

When $H(s) = s^2$, the equation (11.51) and the last inequality imply

$$ (11.53) \qquad \frac{dt}{dt} \int_{\mathcal{O}} f_1 \phi_1 (f/f_1)^2 + 2 \int_{\mathcal{O}} f_1 \phi_1 |\nabla_v (f/f_1)|^2 \leq 0. $$

We next recall a classical compactness result.

**Lemma 11.7.** Let $(g_n)$ be a sequence of functions such that

$$ (g_n) \text{ is bounded in } L^\infty(0, T; L^2_{x,\text{loc}}) \cap L^2(0, T; L^2_{x,\text{loc}} H^1_{v,\text{loc}}) $$

and

$$ \partial_t g_n + v \cdot \nabla_x g - \Delta_v g_n = G_n \text{ bounded in } L^2_{\text{loc}}, $$

then $(g_n)$ belongs to a strong compact set of $L^2_{\text{loc}}$.

**Proof of Lemma 11.7.** We just sketch it. Because

$$ \partial_t g_n + v \cdot \nabla_x g = \Delta_v g_n + G_n \text{ bounded in } L^2_{tx} H_v^{-1}, $$

the usual averaging lemma in [182, 137] implies that

$$ (g_n * \rho) \text{ belongs to a strong compact set of } L^2_{\text{loc}}, $$

for any $\rho \in \mathcal{D}(\mathbb{R}^d)$. On the other hand, introducing a mollifiers sequence $(\rho_\varepsilon)$ and writing then

$$ g_n = (g_n - g_n * \rho_\varepsilon) + g_n * \rho_\varepsilon, $$

we see that the first term is small uniformly in $n$ as $\varepsilon \to 0$ and the second term is relatively compact thanks to the first step, from what we immediately conclude. $\square$

Now, for $0 \le f_0 \in L^1_{\phi_1}$, we introduce the sequence $f_{0,k} := (f_0 \wedge k)\mathbf{1}_{U_k} \in L^2(f_1^{-1}\phi_1) \cap L^2$, with $U_k := \{(x,v) \in \mathcal{O}; \delta(x) > 1/k, |v| \le k\}$, and the associated solution $f_k \in L^\infty(0,T; L^2) \cap L^2(0, \infty; L^2 H^1_v)$. Because of (11.53), for any increasing sequence $(t_n)$ which converges to $\infty$ and for any function $\varphi_m \in \mathcal{D}(\mathcal{O})$, $\mathbf{1}_{U_m} \le \varphi_m \le 1$, the rescaled and truncated function $g_n(t) := f_k(t+t_n)f_1^{-1}e^{-\lambda_1(t+t_n)}\varphi_m$ meet the hypothesis of Lemma 11.7, from what we classically deduce that the sequence of $\tilde{f}_n(t) := f_k(t+t_n)f_1^{-1}e^{-\lambda_1(t+t_n)}$ is relatively strongly compact in $L^2_{\text{loc}}$. Repeating the proof of Theorem 4.23 and Theorem 5.23 (see also [269, Thm. 3.2]), we deduce that $\tilde{f}_n(t) \to \langle f_{0,k}, \phi_1 \rangle f_1$ as $t \to \infty$. Together with the above non expansive property (11.53), we deduce that $f_t \to \langle f_0, \phi_1 \rangle f_1$ in $L^1_{\phi_1}$ as $t \to \infty$.

We summarize our convergence result in the following theorem.

**Theorem 11.8.** For any $f_0 \in L^2_m$, the holds $f_t \to \langle f_0, \phi_1 \rangle f_1$ in $L^1_{\phi_1}$ as $t \to \infty$.
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

$$ \|f_n\|_{L^2 H_m^1}^2 + \|\gamma f_n\|_{L^2(\Sigma; d\xi_m^2)}^2 + \mathcal{E}_{1,\alpha}(\gamma_+ f_n) \le C_\lambda \|\mathcal{F}\|_{L^2 H_m^{-1}}^2. $$

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,

$$ \begin{aligned} & f_n \to f \quad L^2 H_m^1, \quad \gamma_\pm f_n \to \bar{\gamma}_\pm \quad L^2(\Sigma_\pm; d\xi_m^2), \\ & \gamma_+ f_n \to \bar{\gamma}_+ \quad L^1(\Sigma_+; d\xi), \quad \mathcal{R}\gamma_+ f_n \to \mathcal{R}\bar{\gamma}_+ \quad L^1(\Sigma_-; d\xi), \end{aligned} $$

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 to the first equation in (11.28), that $\bar{\gamma}_\pm = \gamma_\pm f$ and then that $\gamma f$ satisfies the second equation in (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 [269, 271] for similar arguments).

**Step 4. Other properties.** We further assume $\lambda > \lambda^{**}$. We proceed similarly as in [260]. 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

$$ 0 = \int_{\mathcal{O}} \beta'(f)(\lambda - c)f\varphi + \beta''(f)\lvert\nabla f\rvert^2\varphi + \beta(f)(\mathrm{div}_v(b\varphi) - v \cdot \nabla_x\varphi - \Delta_v\varphi) + \int_{\Sigma} \beta(\gamma f)\nu \cdot v\varphi. $$

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

$$ \begin{aligned} 0 &\ge \int_{\Sigma_-} |\mathcal{R}\gamma_+ f| |\nu \cdot v| - \int_{\Sigma_+} |\gamma_+ f| |\nu \cdot v| \\ &\ge \int_{\mathcal{O}} |f| \{\lambda - c + \mathrm{div}_v b\} \ge (\lambda - \lambda^{**}) \int_{\mathcal{O}} |f|. \end{aligned} $$

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 11.8 generalizes [248, Thm. 2.18] for the zero inflow condition and [5, Thms. 1.6 & 1.7] for the torus case. It is worth emphasizing that in these papers the longtime convergence is established with exponential rate (with constructive estimate in [5]). In [248] the proof is based on a representation formula for the associated semigroup S which is proved to have a kernel $p_t \in (L^1 \cap L^\infty \cap C^\infty)(\mathcal{O})$ for any $t > 0$ (see [248, Thms. 2.4 & 2.6] as well as [330, 213, 247]). One then classically deduces that $S_t \in \mathcal{X}(X)$ for any $t > 0$ and $X = L^p, p \in [1, \infty]$, or $X = C_0$ (see [248, Thm. 2.18]), and next one may apply Theorem 5.28. We also refer to [193, Thm. 6.8], [299] and [218, 219, 220] for related results.

We follow now a similar approach as in [248, 193]. We start with a series of technical results. Here, we make the additional assumption

$$ (11.54) \qquad \varpi^\#(x, v) := \sup_{1 \le p \le \infty} w_p(x, v) \le \kappa_2 < \infty, $$

with

$$ \varpi_p := \frac{(2-p) \Delta m_p}{p m_p} + \frac{2 |\nabla m_p|^2}{p' m_p^2} + c - \frac{1}{p} \frac{\operatorname{div}(bm_p^p)}{m_p^p}, $$

and $m_p := \mathcal{M}^{-1+1/p}$.

**Lemma 11.9.** For any fixed $\kappa < \kappa_0$ there exists $\varrho_x > 0$, $\varrho_v > 0$ and $\kappa_2 \in \mathbb{R}$ such that defining $\mathcal{A}f := \xi_{\varrho_v}(v)\zeta_{\varrho_x}(x)f$ with $\xi_{\rho_v} \in \mathcal{D}(\mathbb{R}^d)$, $\mathbf{1}_{|v| \le 2\rho_v} \le \xi_{\rho_v} \le \mathbf{1}_{|v| \le 2\rho_v}$, $\zeta_{\rho_x} \in \mathcal{D}(\Omega)$, $\mathbf{1}_{\delta(x) \ge \rho_x/2} \le \zeta_{\varrho_x} \le \mathbf{1}_{\delta(x) \ge \varrho_x}$, and next $\mathcal{B} := \mathcal{L} - \mathcal{A}$, there hold

$$ (11.55) \qquad \|S_{\mathcal{B}}(t)\|_{\mathcal{B}(L_m^2)} \lesssim e^{\kappa t}, \quad \forall t \ge 0, $$

$$ (11.56) \qquad \|S_{\mathcal{B}}(t)\|_{\mathcal{B}(L_{m_p}^p)} \lesssim e^{\kappa_2 t}, \quad \forall t \ge 0, \forall p \in (2, \infty]. $$

**Proof of Lemma 11.9.** We first recall from Step 1 of the proof of Theorem 11.3 and (11.30) that

$$ \begin{aligned} (\mathcal{L}f, f)_{L_m^2} &= - \int |\nabla f|^2 m^2 - \frac{1}{2} \int_{\Sigma} (\gamma f)^2 m^2 \nu \cdot v + \int f^2 \varpi m^2 \\ &\le - \int |\nabla f|^2 m^2 + \int f^2 \varpi m^2 \end{aligned} $$

and, with $\psi$ defined in (11.21),

$$ \begin{aligned} (-\mathcal{L}f, f)_{L_{\psi}^2} &= -\frac{1}{2} \int f^2 (v \cdot \nabla_x \psi) - \int f \frac{b}{\langle v \rangle} \cdot \nabla_v f \langle v \rangle \psi + \int \nabla_v (f \psi) \cdot \nabla_v f - \int c f^2 \psi \\ &\le - \int f^2 \frac{(\hat{v} \cdot \nu(x))^2}{\delta(x)^{1/2}} dv dx + C \int (f^2 + |\nabla f|^2). \end{aligned} $$

Defining then $\tilde{m} := m - \beta\psi$, with $\beta > 0$ small enough, and summing up the two previous estimates, we get

$$ (\mathcal{L}f, f)_{L_m^2} \le -\beta \int f^2 \frac{(\hat{v} \cdot \nu(x))^2}{\delta(x)^{1/2}} - \frac{1}{2} \int |\nabla f|^2 m^2 + \int f^2 (\varpi m^2 + 1). $$

Similarly as in (11.41), we define

$$ \mathcal{U} := \{(x, v) \in \mathcal{O}; \delta(x) > \varrho_x, |v| < \varrho_v\}, $$

and we observe that

$$ \mathcal{U}^c \subset A \cup B \cup C, $$

with

$$ A := \{v \in B_{\varrho_v}, |\hat{v} \cdot \nu(x)| \le \varepsilon_v\}, \quad B := \{v \in B_{\varrho_v}, |\hat{v} \cdot n| \ge \varepsilon_v, \delta(x) \le \varrho_x\}, $$

for some $\varepsilon_x > 0$, and $C := B_{\rho_v}^c$. We next repeat the proof of (11.42), and we get

$$ \int_{\mathcal{U}^c} f^2 m^2 \lesssim (\varrho_v^{d-1} \varepsilon_v)^{2/r'} \int |\nabla_v f|^2 + m(\varrho_v)^2 \frac{\varrho_x^{1/2}}{\varepsilon_v^2} \int f^2 \frac{(\hat{v} \cdot \nu(x))^2}{\delta(x)^{1/2}} + \frac{1}{\varpi_-(\rho_v)} \int f^2 \varpi_- m^2. $$

Observing that

$$ \int f^2 (\varpi m^2 + 1) \le \kappa \int f^2 \tilde{m}^2 + C_\kappa \int_{\mathcal{U}} f^2 m^2 + C_\kappa \int_{\mathcal{U}^c} f^2 m^2 $$
**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

$$ (11.36) \qquad \begin{cases} \partial_t f = \mathcal{L}f \quad \text{in } (0, \infty) \times \mathcal{O} \\ \gamma_-f = \mathcal{R}\gamma_+f \quad \text{on } (0, \infty) \times \Sigma_-, \end{cases} $$

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 [364, Cor. 2.7, Lem. 2.8 and Cor. 2.8] where similar well-posedness results are established (see also [271] 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

$$ (11.37) \qquad \begin{cases} \lambda f + v \cdot \nabla_x f - \Delta_v f - b \cdot \nabla_v f - cf = 0 & \text{in } \mathcal{O} \\ \gamma_- f = \mathcal{R}\gamma_+ f & \text{on } \Sigma_-, \end{cases} $$

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 [247, 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 [237] in the space $X = C_b(\bar{\mathcal{O}})$. We also refer to [192, Thm. 6.8] for a variant and somehow generalisation of [247].

**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 later 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

$$ (11.39) \qquad \int_{\mathcal{O}} (\lambda - \varpi)_+ f^2 m^2 + \int_{\mathcal{O}} |\nabla_v f|^2 m^2 + \frac{1}{2} \mathcal{E}_{\zeta, \alpha}(\gamma_+ f) \le \frac{1}{\lambda - \lambda^*} \|\mathfrak{F}\|_{L_m^2}^2. $$

On the other hand, adapting the proof of (11.22), we straightforwardly obtain

$$ (11.40) \qquad \int_{\mathcal{O}} f^2 \frac{(\hat{v} \cdot \nu(x))^2}{\delta(x)^{1/2}} dv dx \le C \int \mathfrak{F}^2 m^2, $$
with $C_\kappa := \sup(\varpi + 2 - \kappa)_+ < \infty$, and $\mathcal{A} \ge C_\kappa \mathbf{1}_\mathcal{U}$ for $n := C_\kappa$, altogether, we conclude with

$$ (\mathcal{B}f, f)_{L_m^2} \le \kappa \|f\|_{L_m^2}. $$

We then classically deduce that (11.55) holds.

Similarly as for the first estimate and in the proof of [274, Lem. 3.8], for any smooth, rapidly decaying and positive function $f$, we have

$$ \int (\mathcal{L}f) f^{p-1} m_p^p = - \int_{\Sigma} \frac{(m_p \gamma f)^p}{p} \nu \cdot v - (p-1) \int |\nabla(m_p f)|^2 (m_p f)^{p-2} dx, + \int f^p \varpi_p m_p^p. $$

From Darozès-Guiraud (or Jensen) inequality, we know that the first (boundary) term is nonpositive (see [124] or [272, Rem. 6.4]) and we then classically conclude to (11.56). $\square$

**Lemma 11.10.** There exists a finite family $2 = p_0 < p_1 < \dots < p_k < \infty$ and $\alpha \in (0, 1)$ such that such that both $\mathcal{C} = \mathcal{B}, \mathcal{L}$, for any $T > \tau > 0$ and $\mathcal{V} \subset \subset \mathcal{O}$,

$$ (11.57) \qquad \int_{\tau}^{T} \|AS_{\mathcal{C}}(t)f_0\|_{L_{mp_j}^{p_j}} dt \le C_{p_j-1}^{p_j} \|f_0\|_{L_{mp_{j-1}}^{p_{j-1}}}, \quad j = 1, \dots, k, $$

$$ (11.58) \qquad \sup_{t \in [\tau, T]} \|AS_{\mathcal{B}}(t)f_0\|_{L^{\infty}} \le C_{p_k}^{\infty} \|f_0\|_{L^{p_k}}, $$

$$ (11.59) \qquad \sup_{t \in [\tau, T]} \|S_{\mathcal{B}}(t)f_0\|_{C^{\alpha}(\mathcal{V})} \le C_{\infty}^{\alpha} \|f_0\|_{L^{\infty}}. $$

*Proof of Lemma 11.10.* For $0 \le f_0 \in L_m^2$, let us denote $f := S_{\mathcal{B}}f_0$ which thus satisfies the PDE

$$ \partial_t f - \mathcal{B}f = s := cf \quad \text{in } \mathcal{D}'((0, T) \times \mathcal{O}). $$

Let us fix two open sets $U_i$ such that $[\tau, T] \times \text{supp}\xi \times \text{supp}\zeta \subset U_0 \subset \subset U_1 \subset \subset (0, T) \times \mathcal{O}$. From [181, Thm. 6] and a covering lemma, there exists a constant $\bar{C}_0 > 0$ and $p_1 > 2$ such that

$$ \|f\|_{L^{p_1}(U_0)} \le \bar{C}_0 (\|f\|_{L^2(U_1)} + \|s\|_{L^2(U_1)}). $$

The estimate (11.57) for $j = 1$ then follows from Theorem 11.5 (and the classical underlying energy estimate). On the other hand, [181, Thm. 12] similarly implies that there exists a constant $\bar{C}_k > 0$ and $p_k \in (p_1, \infty)$ such that

$$ \|f\|_{L^{\infty}(U_0)} \le \bar{C}_k (\|f\|_{L^2(U_1)} + \|s\|_{L^{p_k}(U_1)}), $$

and interpolating with the previous estimate, we get

$$ \|f\|_{L^{p_j}(U_0)} \le \bar{C}_{j-1} (\|f\|_{L^2(U_1)} + \|s\|_{L^{p_{j-1}}(U_1)}), \quad \forall j, 2 \le j \le k-1. $$

The growth bound (11.56) and the two last estimates imply (11.58) and (11.57) for any $2 \le j \le k-1$. Finally, [181, Thm. 3] similarly implies that there exists a constant $\bar{C}_{k+1} > 0$ and $\alpha \in (0, 1)$ such that

$$ \|f\|_{C^{\alpha}(U_0)} \le \bar{C}_{k+1} (\|f\|_{L^2(U_1)} + \|s\|_{L^{\infty}(U_1)}), $$

from what we deduce (11.59) in the same way. $\square$

**Theorem 11.11.** Under the conditions of Theorem 11.6 and the additional assumption (11.54), the conclusion (E31) holds in $L_m^2$ with non constructive rate.

*Proof of Theorem 11.11.* We introduce the splitting

$$ \mathcal{A}g := M\Upsilon_g, \quad \Upsilon_g := \chi_\varepsilon g, \quad \mathcal{B} := \mathcal{L} - \mathcal{A}, $$

with $\chi_\varepsilon \in C_c^2(\mathcal{O})$, $\mathbf{1}_{\mathcal{U}_2\varepsilon} \le \chi_\varepsilon \le \mathbf{1}_{\mathcal{U}_\varepsilon}$ and $\mathcal{U}_\varepsilon := \{|v| \le 1/\varepsilon, \delta(x) > \varepsilon\}$. We next write the iterated Duhamel formulas (with $N := k + 2$)

$$ S_{\mathcal{L}} = V + W * S_{\mathcal{L}}, $$

with the usual notations (3.41) for $V$ and $W$ associated to the integer $N := k + 2$ and $k \ge 1$ has been introduced in Lemma 11.10. Next for $T > 0$ large, $\tau \in (0, T)$ small and two functions (of operators) $a$ and $b$, we define the modified convolution operator

$$ \left\{ \begin{array}{ll} (a *_{\tau} b)(t) := \int_{\tau}^{t-\tau} a(t-s)b(s) ds & \text{if } t \in [\tau, T-\tau] \\ (a *_{\tau} b)(t) := 0 & \text{if } t \in [\tau, T-\tau]^c, \end{array} \right. $$
for some constant $C = C(b, c, \nu, \lambda)$. For $\varepsilon_x, \varepsilon_v, \varrho > 0$, let us now define

$$ (11.41) \qquad \mathcal{U} := \{(x, v) \in \mathcal{O}; d(x, \partial\Omega) > \varepsilon_x, |v| < \varrho\}, $$

and compute

$$ \int_{\mathcal{U}^c} f^2 m^2 \le \int f^2 m^2 \mathbf{1}_{|v| \ge \varrho} + \int f^2 m^2 \mathbf{1}_{A_x} + \int f^2 m^2 \mathbf{1}_B, $$

with

$$ A_x := \{v \in B_{\varrho}, (\hat{v} \cdot \nu(x))^2 \le \varepsilon_v^2\}, \quad B := \{(x, v); |v| \le \varrho, (\hat{v} \cdot n)^2 \ge \varepsilon_v^2, d(x, \partial\Omega) \le \varepsilon_x\}. $$

For the second term, we have

$$ \int f^2 m^2 \mathbf{1}_{A_x} \le \int |A_x|^{2/r'} \|f(x, \cdot)\|_{L^v}^2 dx \\ \lesssim (\varrho^{d-1} \varepsilon_v)^{2/r'} \|f\|_{L^2 H_m^1}^2, $$

where we have used the Holder inequality with $r \in (1, 2^*/2)$ in the first line and the Sobolev inequality in the second line. For the third term, we have

$$ \int f^2 m^2 \mathbf{1}_B \le m^2(\varrho) \frac{\varepsilon_x^{1/2}}{\varepsilon_v^2} \int_{\mathcal{O}} f_n^2 \frac{(\hat{v} \cdot \nu(x))^2}{\delta(x)^{1/2}}. $$

Gathering these last estimates with (11.39) and (11.40), we have established that the solution $f$ to equation (11.28) furthermore satisfies

$$ (11.42) \qquad \int_{\mathcal{U}^c} f^2 m^2 \le C \left( \frac{1}{\langle \varrho \rangle^2} + \varrho^{d-1} \varepsilon_v + m^2(\varrho) \frac{\varepsilon_x^{1/2}}{\varepsilon_v^2} \right) \int \mathfrak{F}^2 m^2, $$

for a constant $C = C(b, c, \Omega, \lambda)$ and for any $\varepsilon_x, \varepsilon_v, \varrho > 0$.

**The strong maximum principle.** Let us now consider a function $0 \le f \in W_2 \setminus \{0\}$ which satisfies the Dirichlet problem (11.28) associated to $\lambda > \kappa_1$ and a source term $0 \le \mathfrak{F} \in L_m^2 \cap L^\infty$. In order to simplify the discussion, we assume that the normalization $\|f\|_{L_m^2} = 1$ holds. For proving the strong maximum principle, we briefly explain how we may adapt the arguments we have presented for the diffusive equation in Part 7 by taking in particular advantage of the above established estimates, the regularity results established in [180, 191] and some spreading positivity results we learnt in [352, Cor. A.20]. We proceed in three steps.

**Step 1.** On the one hand, from (11.42), we may choose conveniently $\varrho^{-1}, \varepsilon_v, \varepsilon_x > 0$ small enough in such a way that

$$ \int_{\mathcal{U}^c} f^2 m^2 \le \frac{1}{2} \|f\|_{L_m^2}^2, $$

where $\mathcal{U}$ is defined by (11.41). Because of the normalization condition, we have

$$ (11.43) \qquad \int_{\mathcal{U}} f^2 m^2 \ge \frac{1}{2} \|f\|_{L_m^2}^2 $$

and consequently $f(x_0, v_0)^2 \ge \delta_0^2 := \|f\|_{L_m^2}^2 (2 \|\mathbf{1}_{\mathcal{U}}\|_{L_m^2}^2)^{-1}$ for at least one point $(x_0, v_0) \in \mathcal{U}$.

**Step 2.** On the other hand, let us recall some integrability and regularity results established in [180] for a solution $g$ to the kinetic Fokker-Planck evolution equation

$$ \partial_t g + v \cdot \nabla_x g = \Delta_v g + B \cdot \nabla_v g + s \quad \text{in } \mathcal{V}, $$

or a sub-solution

$$ \partial_t g + v \cdot \nabla_x g \le \Delta_v g + B \cdot \nabla_v g + s \quad \text{in } \mathcal{V}, $$

for some bounded set $\mathcal{V} \subset (0, T) \times \mathcal{O}$, $s \in L^2(\mathcal{V})$ and $B \in L^\infty(\mathcal{V})$. For that purpose, given some $(t^*, x^*, v^*)$, we define

$$ Q_r := \{(t, x, v); t \in (t^* - r^2, t^*], |x - x^* - (t - t^*)v^*| < r^3, |v - v^*| < r\}. $$

We claim then that there exist $2 < p < q < \infty$, $\alpha \in (0, 1)$ and for any $0 < r_1 < r_0$ there exists $C$ such that

$$ (11.44) \qquad \|g\|_{L^p(Q_{r_1})} \le C (\|g\|_{L^2(Q_{r_0})} + \|s\|_{L^2(Q_{r_0})}) $$
(with these notations $*_0 = *$) and by induction $a^{*\tau 1} := a$, $a^{*\tau k} := a^{*\tau(k-1)}*_\tau a$ for $k \ge 2$. With these notations, we define the new splitting

$$S_{\mathcal{L}} = V + K_1^c + K_2^c + K,$$

with

$$K := \Upsilon_{\nu} W_{\tau} *_{\tau} S_{\mathcal{L}}, \quad K_1^c := W * S_{\mathcal{L}} - W_{\tau} *_{\tau} S_{\mathcal{L}}, \quad K_2^c := (1 - \Upsilon_{\nu}) W_{\tau} *_{\tau} S_{\mathcal{L}},$$

where $W_\tau := (S_B A)^{*\tau N}$ and $\nu > 0$. For later references, we also define recursively $\Xi_0 := S_\mathcal{L}$, $\Xi_\ell := S_B A *_{\tau} \Xi_{\ell-1}$ for $\ell \ge 1$, so that $K = \Upsilon_\epsilon \Xi_N$. The sequel of the proof is split into two steps.

Step 1. On the one hand, we compute

$$\begin{aligned} \|\Xi_N(T)f_0\|_{L_{m_1}^{p_1}} &\le \|S_B\|_{L^{\infty}(\mathcal{B}(L_{m_1}^{p_1}))} \int_{\tau}^{T-\tau} \left\|\int_{\tau}^{t-\tau} \mathcal{A} S_B(t-s) \mathcal{A} \Xi_{k-1}(s) ds f_0\right\|_{L_{m_1}^{p_1}} dt \\ &\le C_T \int_{\tau}^{T} \int_{\tau}^{T} \|\mathcal{A} S_B(t) \mathcal{A} \Xi_{k-1}(s) f_0\|_{L_m^{p_1}} dt ds \\ &\le C_T C_2^{p_1} \int_{\tau}^{T} \|\mathcal{A} \Xi_{k-1}(s) f_0\|_{L_{m_1}^2} ds, \end{aligned}$$

and thus

$$(11.60) \qquad \|\Xi_N(T)f_0\|_{L_{m_1}^{p_1}} \le C_T \|f_0\|_{L_m^2},$$

where we have used (11.56) in the first line, the Fubini theorem in the second line, (11.57) with $j = 1$ in the third line and several times (11.55) in the last line.

For $\kappa < \kappa_0$, we may choose $\varepsilon > 0$ small enough such that (11.55) holds. From the very definition of $A$ and $S_B$, we may thus fix $\kappa_B \in (\kappa, \kappa_0)$ arbitrary and next $T > 0$ large enough such that $\|V(T)\|_{\mathcal{B}(L_m^2)} \le \frac{1}{3}e^{\kappa_B T}$. We may next use (11.55) and fix $\tau > 0$ small enough such that

$$\|K_1^c(T)\|_{\mathcal{B}(L_m^2)} \le \tau C_T \le \frac{1}{3} e^{\kappa_B T}.$$

Last, because of (11.60), we may fix $\nu > 0$ small enough, in such a way that

$$\|K_2^c(T)f_0\|_{L_m^{p_1}} \le \eta(\nu) \|\Xi_N(T)f_0\|_{L_{m_1}^{p_1}} \le \frac{1}{3} e^{\kappa_B T} \|f_0\|_{L_m^2}.$$

The three last estimates together, we have established

$$(11.61) \qquad \|(V + K_1^c + K_1^c)(T)\|_{\mathcal{B}(L_m^2)} \le e^{\kappa_B T}.$$

Step 2. Performing the same kind of computations as for proving (11.60) and in particular using (11.57), we get

$$\begin{aligned} \int_0^T \|\mathcal{A}\Xi_j(s)f_0\|_{L_{m_{p_j+1}}^{p_{j+1}}} ds &\le \int_0^T \int_{\tau}^{T-\tau} \|\mathcal{A}S_B(t)\mathcal{A}\Xi_{j-1}(s)f_0\|_{L_{m_{p_j+1}}^{p_{j+1}}} dt ds \\ &\le C_{p_j}^{p_{j+1}} \int_0^T \|\mathcal{A}\Xi_{j-1}(s)f_0\|_{L_{m_{p_j}}^{p_j}} ds, \end{aligned}$$

for $j = 1, \dots, k$, and with $p_{k+1} := \infty$. Iterating and using (11.57) with $j = 0$, we get

$$\int_0^T \|\mathcal{A}\Xi_k(s)f_0\|_{L_m^\infty} ds \lesssim \|f_0\|_{L_m^2}.$$

Similarly, we may write

$$\begin{aligned} \sup_{[\tau,T]} \|\mathcal{A}\Xi_{k+1}f_0\|_{L_m^\infty} &\le \sup_{t \in [\tau,T]} \int_{\tau}^{t} \|\mathcal{A}S_B(t-s)\mathcal{A}\Xi_k(s)f_0\|_{L_m^\infty} ds \\ &\le \sup_{t \in [\tau,T]} \|\mathcal{A}S_B(s)\|_{\mathcal{B}(L_m^\infty)} \int_{\tau}^{T} \|\mathcal{A}\Xi_k(s)f_0\|_{L_m^\infty} ds, \end{aligned}$$

thanks to (11.58), and

$$\begin{aligned} \|K f_0\|_{C^\alpha(\mathcal{O})} &\le \int_{\tau}^{T-\tau} \|S_B(T-s)\mathcal{A}\Xi_{k+1}(s)f_0\|_{C^\alpha(\mathcal{U}_\nu)} ds \\ &\le C_\infty^\alpha T \sup_{[\tau,T]} \|\mathcal{A}\Xi_{k+1}f_0\|_{L_m^\infty}, \end{aligned}$$
for any nonnegative subsolution $g$ on $Q_{r_0}$ from [180, Thm. 6],

$$ (11.45) \qquad \|g\|_{L^{\infty}(Q_{r_1})} \le C (\|g\|_{L^2(Q_{r_0})} + \|s\|_{L^q(Q_{r_0})}) $$

for any nonnegative subsolution $g$ on $Q_{r_0}$ from [180, Thm. 12] and

$$ (11.46) \qquad \|g\|_{C^\alpha(Q_{r_1})} \le C (\|g\|_{L^2(Q_{r_0})} + \|s\|_{L^\infty(Q_{r_0})}) $$

for any solution $g$ on $Q_{r_0}$ from [180, Thm. 3]. As a consequence of (11.44) and a classical covering argument, for any bounded set $\mathcal{U} \subset \bar{\mathcal{U}} \subset \mathcal{O}$, there exist $C_0 = C_0(\mathcal{U})$ and $C_1 = C_1(\mathcal{U}, \lambda)$ such that

$$ \|f\|_{L^p(\mathcal{U})} \le C_0 (\|f\|_{L^2(\mathcal{O})} + \|\mathfrak{F} + cf - \lambda f\|_{L^2(\mathcal{O})}) \le C_1 (\|f\|_{L^2(\mathcal{O})} + \|\mathfrak{F}\|_{L^2(\mathcal{O})}). $$

Observing that for $\varrho = p/2 > 1$, we have

$$ v \cdot \nabla_x f^\varrho - \Delta_v f^\varrho - b \cdot \nabla_v f^\varrho + \varrho f^{\varrho-1} (\lambda f - cf - \mathfrak{F}) = -4 \frac{(\varrho-1)}{\varrho} |\nabla (f^{\varrho/2})|^2 \le 0, $$

so that $f^\varrho$ is a weak sub-solution to the kinetic Fokker-Planck equation, we may repeat the argument and obtain in that way that $f \in L^{p_k}(\mathcal{U})$ for any $k \ge 1$, with $p_k := \varrho^k 2$. Now, choosing $k$ such that $p_k \ge q$ and using (11.45) (as well as again a classical covering argument), we get

$$ \|f\|_{L^\infty(\mathcal{U})} \lesssim \|f\|_{L^2(\mathcal{O})} + \|\mathfrak{F} + cf - \lambda f\|_{L^q(\mathcal{O})} \lesssim \|f\|_{L^2(\mathcal{O})} + \|\mathfrak{F}\|_{L^q(\mathcal{O})}. $$

Using finally (11.46), we deduce that there exists a constant $C = C(\mathcal{U}, \lambda)$ such that

$$ \|f\|_{C^\alpha(\mathcal{U})} \lesssim \|f\|_{L^2(\mathcal{O})} + \|\mathfrak{F}\|_{L^\infty(\mathcal{O})}. $$

Together with the conclusion of the first step, we deduce that there exists a constructive constant $r_0 > 0$ such that $f \ge \delta_0 \mathbf{1}_{B((x_0, v_0), r_0)}$.

**Step 3.** From [352, Cor. A.20], we deduce that for any bounded set $\mathcal{U} \subset \bar{\mathcal{U}} \subset \mathcal{O}$, there exists a constructive constant $\delta = \delta(\delta_0, r_0, \mathcal{U}) > 0$ such that

$$ f(x, v) \ge \delta \quad \text{for any} \quad (x, v) \in \mathcal{U}, $$

where it is worth emphasizing that the hypothesis $b, c \in C(\mathcal{O})$ made in [352, Cor. A.20] is not really necessary and can be replaced by $b, c \in L^\infty(\mathcal{U})$. Because $\mathcal{U}$ may be chosen arbitrary, we have established that $f > 0$ on $\mathcal{O}$ and the strong maximum principle.

**Condition (H2).** For a given function $0 \le h_0 \in C_c^2(\mathcal{O})$ normalized by $\|h_0\|_{L_m^2} = 1$, we define $f_0 \in D(\mathcal{L})$ as the solution to

$$ (\kappa_1 - \mathcal{L})f_0 = h_0 \quad \text{in} \quad \mathcal{O}, \quad \gamma_-f_0 = \mathcal{R}\gamma_+f_0 \quad \text{on} \quad \Sigma_-. $$

Taking advantage of the fact that $h_0$ has compact support, we compute

$$ 1 = \int_{\mathcal{O}} h_0^2 m^2 = \int_{\mathcal{O}} (\kappa_1 - \mathcal{L}) f_0 h_0 m^2 = \int_{\mathcal{O}} f_0 (\kappa_1 - \mathcal{L}^*)(h_0 m^2) \le C_1 \|f_0\|_{L_m^2}, $$

with $C_1 := \|m^{-1}(\kappa_1 - \mathcal{L}^*)(h_0 m^2)\|_{L^2}$. On the other hand, from (11.27), we have

$$ (11.47) \qquad \|f_0\|_{L^2 H_m^1} + \|f_0 \frac{\hat{v} \cdot \nu}{\delta^{1/4}}\|_{L^2} \le C_2, $$

for a constant $C_2$ only depending on $\|h_0\|_{L_m^2}$, $\kappa_1$ and the constant $C$ which appears in (11.27). Arguing as in (11.43), we deduce that

$$ (11.48) \qquad \int_{\mathcal{U}} f_0^2 m^2 \ge (2C_1)^{-1}, \quad \text{supp } h_0 \subset \mathcal{U}, $$

with $\mathcal{U} = \mathcal{U}_\varrho$ defined in (11.41) and $\varrho > 0$ small enough (chosen constructively from $C_2$ and $C_1$). From the above constructive strong maximum principle, we deduce that $f_0 \ge \varepsilon \mathbf{1}_\mathcal{U} \ge 1/C_0 h_0$ for some $\varepsilon, C_0 > 0$. We conclude as in the second constructive argument for (H2) in Section 7.1. Coming back indeed to the equation, we have

$$ \mathcal{L} f_0 = \kappa_1 f_0 - h_0 \ge \kappa_1 f_0 - \|h_0\|_{L^\infty} \mathbf{1}_\mathcal{U} \ge (\kappa_1 - \|h_0\|_{L^\infty} C_0) f_0, $$

so that (H2) holds with $\kappa_0 := \kappa_1 - \|h_0\|_{L^\infty} C_0$ from Lemma 2.4-(ii).
thanks to (11.59). The three last estimates together and the compact support property $\text{supp}\chi_\nu \subset \mathcal{O}$ imply

$$ \|K f_0\|_{C^\alpha \cap L_{m_{p_1}}^2} \lesssim \|f_0\|_{L_m^2}, \quad \forall f_0 \in L_m^2, $$

from what we deduce that $K \in \mathcal{K}(L_m^2)$. We may apply Theorem 5.28 in order to conclude. $\square$

## 12. A MUTATION-SELECTION MODEL

In this section, we consider the mutation-selection evolution equation associated to the mutation-selection operator

$$ (12.1) \qquad \mathcal{L}f := J * f - W(x)f $$

defined on functions $f: \mathbb{R}^d \to \mathbb{R}$, where $J$ is a the mutation kernel, $*$ stands for the convolution operator and $W$ is a confining potential.

**12.1. Almost regular mutation kernel.** We assume that the mutation kernel $J$ is a positive finite measure of $\mathbb{R}^d$ which is lower bounded on a neighborhood of the origin, or in other words

$$ (12.2) \qquad 0 \le J \in M^1(\mathbb{R}^d), \quad J \ge J_* \mathbf{1}_{B_r}, $$

for some constants $J_*, r > 0$. We also assume that the selection potential $W: \mathbb{R}^d \to \mathbb{R}$ is continuous and satisfies

$$ (12.3) \qquad W(x) > W(0) = 0, \forall x \neq 0, \quad W(x) \to +\infty \text{ as } |x| \to \infty. $$

We finally assume the following compatibility condition between mutation and selection: there exist $\beta > 0$ and a bounded Borel set $A \subset \mathbb{R}^d$ such that

$$ (12.4) \qquad a := \essinf_{x \in A_\beta} \int_{x-A_\beta} \frac{J(dz)}{W(x-z)} > 1, $$

$$ (12.5) \qquad J = J_1 + J_2, \quad J_1 \in C_c^1(\mathbb{R}^d), \quad \kappa_* := \|J_2\|_1 := \int_{\mathbb{R}^d} dJ_2 < \kappa_0 := (a-1)\beta, $$

where we use the notation $A_\beta = A \cap \{W \ge \beta\}$. In the sequel, we work in the Banach lattice $X := L^1(\mathbb{R}^d)$.

**Theorem 12.1.** Under the above assumptions, we have

(1) The first eigentriplet problem (1.1)-(1.2) 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 $\lambda_1 \ge \kappa_0, 0 < f_1 \in L^1_{\langle W \rangle}(\mathbb{R}^d) \cap L^\infty_{\langle W \rangle}(\mathbb{R}^d)$ and $0 < \phi_1 \in L^1_{\langle W \rangle}(\mathbb{R}^d) \cap L^\infty_{\langle W \rangle}(\mathbb{R}^d)$.

(2) Moreover, $\mathcal{L}$ generates a semigroup $S_\mathcal{L}$ on $X$ and for any $f_0 \in X$, there holds

$$ (12.6) \qquad \|e^{-\lambda_1 t} S_\mathcal{L}(t) f_0 - \langle \phi_1, f_0 \rangle f_1\|_{L^1} \le C e^{-\alpha t} \|f_0 - \langle \phi_1, f_0 \rangle f_1\|_{L^1}, \\ \text{for any } t \ge 0 \text{ and for some constructive constants } C \ge 1, \alpha > 0. $$

Let us comment on the above result.

## Remark 12.2.

(1) **Assumption (12.4) is satisfied** when $W$ is small enough in a neighborhood of the origin. It is for instance satisfied if $W^{-1} \notin L^1(B_1)$. That is in particular the case in dimension $d=1$ when $W$ is Lipschitz, because of the condition $W(0) = 0$.

(2) Assume $J(x) = \varepsilon^{-d}\rho(\varepsilon^{-1}x)$ with $\rho \in C_c^1(\mathbb{R}^d) \cap \mathcal{P}(\mathbb{R}^d)$ and $\rho \ge \rho_* \mathbf{1}_{B_1}$, $\rho_* > 0$, so that $J = J_1$ and $J_2 = 0$, and $W = W(|x|)$. We may observe that for $\beta > 0$ and $\varepsilon > 0$ small enough

$$ \inf_{\beta \le W(x) < 2\beta} \int_{\beta \le W(y) < 2\beta} \frac{J(x-y)}{W(y)} dy :=: a \ge \frac{\rho_*}{2\beta} \text{meas}\{\mathbb{R}_+^d \cap B_1\} > 1, $$

so that (12.4) holds with $A := \{W(x) < 2\beta\}$.

(3) **Assumption (12.4) is similar to [249, Condition (2.3)], see also [6, Assumption 2.6] and the comparison with [6, Assumption 2.4], as well as [75, Condition (3.7)-(3.8)] and [77, p. 250, Note added in proof.]. On the other hand, the conditions on $J$ are relaxed here since $J$ may have singular part in (12.5).**
**Condition (H3).** Let us fix $\kappa < \kappa_0$ arbitrary. We define $\mathcal{B}f := \mathcal{L}f - n\chi_R(v)f$ for any $f \in W_{2,\mathcal{R}}$, with $\chi_R \in \mathcal{D}(\mathbb{R}^d)$ such that $\mathbf{1}_{B_R} \le \chi_R \le \mathbf{1}_{B_{2R}}$ and for some given $n, R \ge 0$ to be specified below. We observe that, at least formally,

$$ \int fm^2(\mathcal{B} - \kappa)f = \int_{\mathcal{O}} (\varpi - \kappa - n\chi_R)f^2 m^2 - \frac{1}{2} \int_{\Sigma} (\gamma f)^2 m^2 \nu \cdot v - \int_{\mathcal{O}} |\nabla_v f|^2 m^2. $$

Thanks to (11.38), there exists a constant $R > 0$ such that

$$ \sup_{v \in \mathbb{R}^d \setminus B_R} \varpi \le \kappa. $$

Choosing $n := \sup \varpi_+ - \kappa$, we deduce that $\varpi - \kappa - n\chi_R \le 0$. On the other hand, because of (11.30), the contribution of the boundary term in the above identity is non positive. We thus deduce that $(\mathcal{B} - \kappa)$ is dissipative in $L_m^2$. We now establish that the associated operator $\mathcal{B}$ has compact resolvent. For $\mathfrak{F} \in L_m^2$, we consider $f \in L_m^2$ the solution to

$$ (11.49) \qquad -\mathcal{B}f = \mathfrak{F} \quad \text{in } \mathcal{O}, \quad \gamma_-f = \mathcal{R}\gamma_+f \quad \text{on } \Sigma_-, $$

which existence follows from Theorem 11.4. From the above discussion (with $\kappa = -1$) and the same arguments as in Step 1 of the proof of Theorem 11.3, we have

$$ (11.50) \qquad \int f^2 \langle \varpi \rangle_{-} m^2 + 2 \int |\nabla_v f|^2 m^2 \le \int \mathfrak{F}^2 m^2. $$

Together with the regularity estimate (11.23) and the compact imbedding $H^{2/3}(U) \subset L^2(U)$, we conclude that $\mathcal{B}$ has compact resolvent. The operator $\mathcal{A}$ on $L_m^2$ defined by $\mathcal{A}f := n\chi_R(v)f$ being bounded, we may apply Lemma 2.8-(2) and we deduce that (H3) holds for both the primal and the dual problems.

**Condition (H4)** is nothing but the yet established strong maximum principle.

**A variant of condition (H5).** Consider $(f, \lambda)$ a pair of eigenfunction and eigenvalue such that $\lambda \in \Sigma_{P+}(\mathcal{L})$. Arguing similarly as in the proof of condition (H5) in Section 7.1, we know that

$$ \tilde{\mathcal{L}}f = i\vartheta f, \quad \vartheta \in \mathbb{R}, \quad \tilde{\mathcal{L}}|f| = 0 $$

and introducing the real and complex part decomposition $f = g + ih$, we have

$$ \int_{\mathcal{O}} \frac{1}{|f|^2} |g \nabla_v h - h \nabla_v g|^2 = 0, $$

and finally $g\nabla_v h - h\nabla_v g = 0$ a.e. on $\mathcal{O}$. Because of the regularity estimate presented during the above proof of the strong maximum principle, the functions $f$ has Hölder regularity, and thus $g$ and $h$ are continuous on $\mathcal{O}$. Because $|f| \ne 0$, we may claim that there exists a point $(x_0, v_0) \in \mathcal{O}$ such that $h(x_0, v_0) > 0$ for instance. Denoting by $\omega$ the connected component of $\{(x, v) \in \mathcal{O}; h(x, v) > 0\}$ containing $(x_0, v_0)$, we have $\nabla(g/h) = 0$ on $\omega$, and thus $g = \alpha(x)h$ on $\omega$ for some continuous function $\alpha: \Omega \to \mathbb{R}$. Coming back to the eigenvalue equation that we may write in the following system form

$$ \tilde{\mathcal{L}}g = -\vartheta h, \quad \tilde{\mathcal{L}}h = \vartheta g, $$

we compute

$$ -\vartheta h = \tilde{\mathcal{L}}(\alpha h) = \alpha \tilde{\mathcal{L}}h - hv \cdot \nabla_x \alpha = \alpha \vartheta g - hv \cdot \nabla_x \alpha \quad \text{on } \omega, $$

so that

$$ -\vartheta = \alpha^2 \vartheta - v \cdot \nabla_x \alpha \quad \text{on } \omega. $$

We deduce that $\alpha$ is a constant on $\omega$ and finally $\vartheta = 0$. We have thus established that $\lambda = \lambda_1$.

At this stage, we may use Theorem 2.21, Theorem 4.13 and Theorem 5.16, in order to get 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 briefly explain how we may deduce the stability of $f_1$ by adapting some arguments developed in [268] and already mentioned. On the one hand, we know from [268, Lem. 1.1] that any solution $f$ to the rescaled evolution equation (11.36) with $\mathcal{L}$ replaced by $\tilde{\mathcal{L}} = \mathcal{L} - \lambda_1$ satisfies

$$ \partial_t(H(X)f_1\phi_1) + \operatorname{div}_x(vH(X)f_1\phi_1) - \operatorname{div}_v(\phi_1^2\nabla_v(H(X)f_1/\phi_1)) = -H''(X)f_1\phi_1|\nabla_v X|^2, $$
(4) Optimal conditions linking J and W for the existence of a spectral gap are still unknown. In the recent paper [6], using variational methods in a L² framework, the authors obtain a quantified spectral gap and the associated exponential stability when the mutation kernel J is additionally assumed to be symmetric. Up to our knowledge, Theorem 12.1 is the very first result providing a quantified spectral gap for a non-symmetric mutation kernel J.

(5) Condition (12.4) can be compared to the condition

$$ \bar{a} := \ess \sup_{x \in \mathbb{R}^d} \int_{\mathbb{R}^d} \frac{J(x-y)}{W(y)} dy < 1, $$

under which no first eigenfunction may exist in X. First, we claim that $\lambda_1 \ge 0$. Indeed, considering $\varepsilon > 0$ and $f_\varepsilon = \mathbf{1}_{B_\varepsilon}$, we have

$$ \mathcal{L}f_\varepsilon \ge -(\inf_{B_\varepsilon} W)f_\varepsilon, $$

so that the condition (H2) holds for $\kappa_0 = -\inf_{B_\varepsilon} W$ for any $\varepsilon > 0$. Since W is continuous and $W(0) = 0$, we deduce that $\lambda_1 \ge 0$ by passing to the limit $\varepsilon \to 0$. Assume now by contradiction that there exists $f_1 \in X_+ \setminus \{0\}$ such that

$$ (12.7) \qquad \lambda_1 f_1 = \mathcal{L}f_1 = J * f_1 - Wf_1 $$

and define, for any $\varepsilon > 0$, the function $\varphi_\varepsilon(x) = \frac{1}{\varepsilon+W(x)} \in L^\infty(\mathbb{R}^d)$. Testing (12.7) against $\varphi_\varepsilon$ we get for any $\varepsilon \in (0,1)$

$$ \begin{aligned} 0 \le \lambda_1 \langle f_1, \varphi_\varepsilon \rangle &\le \lambda_1 \langle f_1, \varphi_\varepsilon \rangle = \iint \frac{J(x-y)}{\varepsilon+W(x)} f_1(y) \, dx \, dy - \int \frac{W(x)}{\varepsilon+W(x)} f_1(x) \, dx \\ &\le \bar{a} \int f_1 - \int \frac{W(x)}{\varepsilon+W(x)} f_1(x) \, dx, \end{aligned} $$

and passing to the limit $\varepsilon \to 0$ we obtain the contradiction $0 \le \lambda_1 \langle f_1, \varphi_1 \rangle \le (\bar{a} - 1) \int f_1 < 0$. However, there always exists a principal eigenvector $f_1$ in $M^1(\mathbb{R}^d)$, which might have an atom at the origin when $\bar{a} < 1$, see for instance [77].

The proof of Theorem 12.1 follows from Theorem 2.21, Theorem 4.13 and Theorem 5.16 as a consequence of conditions (H1)-(H5) that we establish now. Setting $D(\mathcal{L}) := L^1_{\langle W \rangle}(\mathbb{R}^d)$, we observe that $\mathcal{L}$ is an unbounded closed operator with dense domain $D(\mathcal{L})$.

**Condition (H1) and (H1')**. We define the semigroup

$$ S_W(t)f(x) := e^{-W(x)t}f(x), \quad \forall f \in L^p, p \in [1, \infty], $$

which is clearly a positive semigroup of contractions. We next define $S_\mathcal{L}$ as a bounded perturbation of $S_W$. It is also positive and it satisfies the growth estimate $\|S_\mathcal{L}(t)\|_{\mathcal{B}(L^p)} \le e^{\|J\|_1 t}$, where we recall that $\|J\|_1$ stands for the $L^1$ norm or the total variation norm of $J$. We deduce that (H1) holds true with $\kappa_1 := \|J\|_1$ thanks to Lemma 2.2-(i). Multiplying $\mathcal{L}f$ by sign $f$, for $f \in D(\mathcal{L})$, we immediately get Kato's inequality

$$ (\operatorname{sign} f) \mathcal{L} f = (\operatorname{sign} f) J * f - W |f| \le J * |f| - W |f| = \mathcal{L} |f|. $$

**Condition (H2).** Let us define $f_0 := \frac{1}{W(x)}\mathbf{1}_{A_\beta}$, where $A_\beta$ is introduced in condition (12.4). We compute

$$ \begin{aligned} \mathcal{L}f_0 &= J * (\mathbf{1}_{A_\beta} \frac{1}{W}) - \mathbf{1}_{A_\beta} \ge (J * (\mathbf{1}_{A_\beta} \frac{1}{W}) - 1)\mathbf{1}_{A_\beta} \\ &\ge (\operatorname{ess\,inf}_{x \in A_\beta} [J * (\mathbf{1}_{A_\beta} \frac{1}{W})] - 1)\mathbf{1}_{A_\beta} \\ &= (a-1)\mathbf{1}_{A_\beta} \ge (a-1)\frac{\beta}{W}\mathbf{1}_{A_\beta} = \kappa_0 f_0, \end{aligned} $$

where in the second equality we have used the very definition of $a$ in assumption (12.4). We conclude that (H2) holds thanks to Lemma 2.4-(ii).

**Condition (H3).** We introduce the splitting

$$ (12.8) \qquad \mathcal{L} = \mathcal{A} + \mathcal{B}, \quad \mathcal{A}f := J_1 * f, \quad \mathcal{B}f := J_2 * f - W(x)f. $$
for any convex function $H: \mathbb{R} \to \mathbb{R}$ and with $X := f/f_1$. After integration, we get

$$ (11.51) \qquad \frac{d}{dt} \int_{\mathcal{O}} H(X) f_1 \phi_1 + \int_{\Sigma} \nu \cdot v H(X) f_1 \phi_1 = - \int_{\mathcal{O}} H''(X) f_1 \phi_1 |\nabla_v X|^2, $$

When $H(s) := |s|$, the boundary term is

$$ \begin{aligned} \int_{\Sigma} |\gamma f| \gamma \phi_1 \nu \cdot v &= \int_{\Sigma_+} |\gamma_+ f| \mathcal{R}^* \gamma_- \phi_1 \nu \cdot v - \int_{\Sigma_-} |\mathcal{R} \gamma_+ f| \gamma_- \phi_1 |\nu \cdot v| \\ &\geq \int_{\Sigma_+} |\gamma_+ f| \mathcal{R}^* \gamma_- \phi_1 |\nu \cdot v| - \int_{\Sigma_-} |\mathcal{R} \gamma_+ f| \gamma_- \phi_1 |\nu \cdot v| = 0, \end{aligned} $$

from what we deduce the non expansive property

$$ (11.52) \qquad \int_{\mathcal{O}} |f_{t_1}| \phi_1 \leq \int_{\mathcal{O}} |f_{t_0}| \phi_1, \quad \forall t_1 \geq t_0 \geq 0. $$

On the other hand, from the Cauchy-Schwarz inequality, we have

$$ (\mathcal{R}\gamma_+ f)^2 \leq (\mathcal{R}\gamma_+ f_1)\mathcal{R}(\gamma_+ f^2/\gamma_+ f_1), $$

so that

$$ \int_{\Sigma_{-}} \frac{(\mathcal{R}\gamma_{+}f)^2}{\mathcal{R}\gamma_{+}f_1} \gamma_{-}\phi_1 |\nu \cdot v| \leq \int_{\Sigma_{-}} \mathcal{R}(\gamma_{+}f^2/\gamma_{+}f_1) \gamma_{-}\phi_1 |\nu \cdot v| $$

and finally

$$ \int_{\Sigma} (\gamma f)^2 (\gamma f_1)^{-1} \gamma \phi_1 \nu \cdot v \leq 0. $$

When $H(s) = s^2$, the equation (11.51) and the last inequality imply

$$ (11.53) \qquad \frac{d}{dt} \int_{\mathcal{O}} f_1 \phi_1 (f/f_1)^2 + 2 \int_{\mathcal{O}} f_1 \phi_1 |\nabla_v (f/f_1)|^2 \leq 0. $$

We next recall a classical compactness result.

**Lemma 11.7.** Let $(g_n)$ be a sequence of functions such that

$$ (g_n) \text{ is bounded in } L^\infty(0, T; L^2_{xv,\text{loc}}) \cap L^2(0, T; L^2_{x,\text{loc}} H^1_{v,\text{loc}}) $$

and

$$ \partial_t g_n + v \cdot \nabla_x g - \Delta_v g_n = G_n \text{ bounded in } L^2_{\text{loc}}, $$

then $(g_n)$ belongs to a strong compact set of $L^2_{\text{loc}}$.

*Proof of Lemma 11.7.* We just sketch it. Because

$$ \partial_t g_n + v \cdot \nabla_x g = \Delta_v g_n + G_n \text{ bounded in } L^2_{tx} H_v^{-1}, $$

the usual averaging lemma in [181, 136] implies that

$$ (g_n * \rho) \text{ belongs to a strong compact set of } L^2_{\text{loc}}, $$

for any $\rho \in \mathcal{D}(\mathbb{R}^d)$. On the other hand, introducing a mollifiers sequence $(\rho_\epsilon)$ and writing then

$$ g_n = (g_n - g_n * \rho_\epsilon) + g_n * \rho_\epsilon, $$

we see that the first term is small uniformly in $n$ as $\epsilon \to 0$ and the second term is relatively compact thanks to the first step, from what we immediately conclude. $\square$

Now, for $0 \le f_0 \in L^1_{\phi_1}$, we introduce the sequence $f_{0,k} := (f_0 \wedge k)1_{U_k} \in L^2(f_1^{-1}\phi_1) \cap L^2$, with $U_k := \{(x, v) \in \mathcal{O}; \delta(x) > 1/k, |v| \le k\}$, and the associated solution $f_k \in L^\infty(0, T; L^2) \cap L^2(0, \infty; L^2 H^1_v)$. Because of (11.53), for any increasing sequence $(t_n)$ which converges to $\infty$ and for any function $\varphi_m \in \mathcal{D}(\mathcal{O})$, $1_{U_m} \le \varphi_m \le 1$, the rescaled and truncated function $g_n(t) := f_k(t+t_n)f_1^{-1}e^{-\lambda_1(t+t_n)}\varphi_m$ meet the hypothesis of Lemma 11.7, from what we classically deduce that the sequence of $\tilde{f}_n(t) := f_k(t+t_n)f_1^{-1}e^{-\lambda_1(t+t_n)}$ is relatively strongly compact in $L^2_{\text{loc}}$. Repeating the proof of Theorem 4.22 and Theorem 5.23 (see also [268, Thm. 3.2]), we deduce that $\tilde{f}_n(t) \to \langle f_{0,k}, \phi_1 \rangle f_1$ as $t \to \infty$. Together with the above non expansive property (11.52), we deduce that $f_t \to \langle f_0, \phi_1 \rangle f_1$ in $L^1_{\phi_1}$ as $t \to \infty$.

We summarize our convergence result in the following theorem.
Arguing as in the proof of condition (H1), we see that $\mathcal{B}$ is the generator a positive semigroup in $L^p(\mathbb{R}^d)$, $1 \le p \le \infty$, with growth bound $\omega(S_\mathcal{B}) \le \kappa_*$ and thus $(\alpha - \mathcal{B})$ is invertible for any $\alpha \ge \kappa_0 > \kappa_*$, with

$$ (12.9) \qquad \|(\alpha - \mathcal{B})^{-1}\|_{\mathcal{B}(L^p)} \le \frac{1}{\alpha - \kappa_*}. $$

Next, observing that

$$ (W + \alpha)h = (\alpha - \mathcal{B})h + J_2 * h, $$

for any $h \in \mathcal{D}(\mathcal{L})$ and $\alpha \ge \kappa_0$, we deduce that

$$ (12.10) \qquad (W + \alpha)(\alpha - \mathcal{B})^{-1}g = g + J_2 * ((\alpha - \mathcal{B})^{-1}g), $$

for any $g \in X$ and $\alpha \ge \kappa_0$. Together with (12.9), we deduce

$$ (12.11) \qquad \|(\alpha - \mathcal{B})^{-1}g\|_{L_W^p} \le \|g\|_{L^p} + \|J_2 * ((\alpha - \mathcal{B})^{-1}g)\|_{L^p} \le \frac{\alpha}{\alpha - \kappa_*} \|g\|_{L^p}, $$

for any $g \in L^p$ and $\alpha \ge \kappa_0$. Defining $\mathcal{W}(\alpha) := (\alpha - \mathcal{B})^{-1}\mathcal{A}$, we finally deduce from (12.10) the identity

$$ \mathcal{W}(\alpha)f = \frac{1}{W + \alpha}\mathcal{A}f + \frac{1}{W + \alpha}J_2 * ((\alpha - \mathcal{B})^{-1}\mathcal{A}f), $$

for any $f \in X$ and $\alpha \ge \kappa_0$. We may then compute

$$ \|\mathcal{W}(\alpha)f\|_{L^\infty} \le \frac{1}{\alpha}\|\mathcal{A}f\|_{L^\infty} + \frac{1}{\alpha}\|J_2\|_1\|(\alpha - \mathcal{B})^{-1}\mathcal{A}f\|_{L^\infty}, $$

and together with (12.9) for $p = \infty$ and (12.11), we deduce

$$ (12.12) \qquad \|\mathcal{W}(\alpha)f\|_{L^\infty} \le \|J_1\|_\infty \frac{1}{\alpha - \kappa_*} \|f\|_{L^1}, $$

for any $f \in X$ and $\alpha \ge \kappa_0$. Starting from the same identity, we prove in a similar way

$$ (12.13) \qquad \|\mathcal{W}(\alpha)f\|_{L_W^\infty} \le \|J_1\|_\infty \frac{\alpha}{\alpha - \kappa_*} \|f\|_{L^1}, $$

for any $f \in X$ and $\alpha \ge \kappa_0$. As a conclusion and gathering (12.9), (12.11), (12.12) and (12.13), we have established that

$$ (12.14) \qquad \mathcal{W}(\alpha) : L^1 \to L^1_{\langle W \rangle} \cap L^\infty_{\langle W \rangle}, $$

with uniform bound for any $\alpha \ge \kappa_0$. Observing that $L^1_{\langle W \rangle} \cap L^\infty_{\langle W \rangle} \subset L^1$ is weakly compact and using Lemma 2.13 with $p = 1$, we deduce that (H3) holds. We can actually strengthen the compactness by noticing that $\mathcal{A} : L^1 \to L^1_W \cap W^{1,1}$ is bounded because of assumption (12.5). This ensures that $\mathcal{A} : L^1 \to L^1$ is compact, from what we deduce that $\mathcal{W}(\alpha) : L^1 \to L^1$ is strongly compact for all $\alpha \ge \kappa_0$. We may thus apply Lemma 2.8-(2) to infer that condition (H3) holds for both the primal and the dual problems.

**Condition (H4).** Assume that $\lambda \ge \lambda_1$ and $f \in \mathcal{D}(\mathcal{L}) = L^1_{\langle W \rangle}$ satisfy

$$ (12.15) \qquad \|f\|_{L^1} = 1, \quad f \ge 0, \quad (\lambda - \mathcal{L})f \ge 0. $$

Denoting $W_R := \inf_{B_R^c} W$, we compute

$$ \int_{B_R} f \ge \int_{\mathbb{R}^d} f - \frac{1}{W_R} \int_{B_R^c} fW \ge 1 - \frac{1}{W_R} \|f\|_{L^1_{\langle W \rangle}} \ge 1/2, $$

for $R > 0$ large enough by taking advantage of the fact that $W(x)$ tend to infinity when $|x| \to \infty$. In particular, there exists $x_0^f \in B_R$ such that

$$ \int_{B_{r/2}(x_0^f)} f \ge \delta := \frac{1}{2} \left( \frac{r}{2R} \right)^d > 0, $$

where we recall that $r$ is defined in (12.2). We deduce that

$$ (J * f)(x) \ge J_* \int_{B_{r/2}(x_0^f)} f(y)dy \mathbf{1}_{B_{r/2}(x_0^f)}(x) \ge J_* \delta \mathbf{1}_{B_{r/2}(x_0^f)}(x). $$
**Theorem 11.8.** For any $f_0 \in L_m^2$, the holds $f_t \to \langle f_0, \phi_1 \rangle f_1$ in $L_{\phi_1}^1$ as $t \to \infty$.

Theorem 11.8 generalizes [247, Thm. 2.18] for the zero inflow condition and [5, Thms. 1.6 & 1.7] for the torus case. It is worth emphasizing that in these papers the longtime convergence is established with exponential rate (with constructive estimate in [5]). In [247] the proof is based on a representation formula for the associated semigroup $S$ which is proved to have a kernel $p_t \in (L^1 \cap L^\infty \cap C^\infty)(\mathcal{O})$ for any $t > 0$ (see [247, Thms. 2.4 & 2.6] as well as [329, 212, 246]). One then classically deduces that $S_t \in \mathcal{H}(X)$ for any $t > 0$ and $X = L^p, p \in [1, \infty]$, or $X = C_0$ (see [247, Thm. 2.18]), and next one may apply Theorem 5.28. We also refer to [192, Thm. 6.8], [298] and [217, 218, 219] for related results.

We follow now a similar approach as in [247, 192]. We start with a series of technical results. Here, we make the additional assumption

$$ (11.54) \qquad \bar{\omega}^\sharp(x, v) := \sup_{1 \le p \le \infty} w_p(x, v) \le \kappa_2 < \infty, $$

with

$$ \bar{\omega}_p := \frac{(2-p)}{p} \frac{\Delta m_p}{m_p} + \frac{2}{p'} \frac{|\nabla m_p|^2}{m_p^2} + c - \frac{1}{p} \frac{\operatorname{div}(bm_p^p)}{m_p^p}, $$

and $m_p := \mathcal{M}^{-1+1/p}$.

**Lemma 11.9.** For any fixed $\kappa < \kappa_0$ there exists $\varrho_x > 0$, $\varrho_v > 0$ and $\kappa_2 \in \mathbb{R}$ such that defining $\mathcal{A}f := \xi_{\varrho_v}(v)\zeta_{\varrho_x}(x)f$ with $\xi_{\varrho_v} \in \mathcal{D}(\mathbb{R}^d)$, $\mathbf{1}_{|v| \le \varrho_v} \le \xi_{\varrho_v} \le \mathbf{1}_{|v| \le 2\varrho_v}$, $\zeta_{\varrho_x} \in \mathcal{D}(\Omega)$, $\mathbf{1}_{\delta(x) \ge \varrho_x/2} \le \zeta_{\varrho_x} \le \mathbf{1}_{\delta(x) \ge \varrho_x}$, and next $\mathcal{B} := \mathcal{L} - \mathcal{A}$, there hold

$$ (11.55) \qquad \|S_{\mathcal{B}}(t)\|_{\mathcal{B}(L_m^2)} \lesssim e^{\kappa t}, \quad \forall t \ge 0, $$

$$ (11.56) \qquad \|S_{\mathcal{B}}(t)\|_{\mathcal{B}(L_{m_p}^p)} \lesssim e^{\kappa_2 t}, \quad \forall t \ge 0, \forall p \in (2, \infty]. $$

*Proof of Lemma 11.9.* We first recall from Step 1 of the proof of Theorem 11.3 and (11.30) that

$$ \begin{aligned} (\mathcal{L}f, f)_{L_m^2} &= - \int |\nabla f|^2 m^2 - \frac{1}{2} \int_{\Sigma} (\gamma f)^2 m^2 \nu \cdot v + \int f^2 \bar{\omega} m^2 \\ &\le - \int |\nabla f|^2 m^2 + \int f^2 \bar{\omega} m^2 \end{aligned} $$

and, with $\psi$ defined in (11.21),

$$ \begin{aligned} (-\mathcal{L}f, f)_{L_{\psi}^2} &= -\frac{1}{2} \int f^2 (v \cdot \nabla_x \psi) - \int f \frac{b}{\langle v \rangle} \cdot \nabla_v f \langle v \rangle \psi + \int \nabla_v (f \psi) \cdot \nabla_v f - \int c f^2 \psi \\ &\le - \int f^2 \frac{(\hat{v} \cdot \nu(x))^2}{\delta(x)^{1/2}} dv dx + C \int (f^2 + |\nabla f|^2). \end{aligned} $$

Defining then $\tilde{m} := m - \beta\psi$, with $\beta > 0$ small enough, and summing up the two previous estimates, we get

$$ (\mathcal{L}f, f)_{L_m^2} \le -\beta \int f^2 \frac{(\hat{v} \cdot \nu(x))^2}{\delta(x)^{1/2}} - \frac{1}{2} \int |\nabla f|^2 m^2 + \int f^2 (\bar{\omega} m^2 + 1). $$

Similarly as in (11.41), we define

$$ \mathcal{U} := \{(x, v) \in \mathcal{O}; \delta(x) > \varrho_x, |v| < \varrho_v\}, $$

and we observe that

$$ \mathcal{U}^c \subset A \cup B \cup C, $$

with

$$ A := \{v \in B_{\varrho_v}, |\hat{v} \cdot \nu(x)| \le \varepsilon_v\}, \quad B := \{v \in B_{\varrho_v}, |\hat{v} \cdot n| \ge \varepsilon_v, \delta(x) \le \varrho_x\}, $$

for some $\varepsilon_x > 0$, and $C := B_{\varrho_v}^c$. We next repeat the proof of (11.42), and we get

$$ \int_{\mathcal{U}^c} f^2 m^2 \lesssim (\varrho_v^{d-1} \varepsilon_v)^{2/r'} \int |\nabla_v f|^2 + m(\varrho_v)^2 \frac{\varrho_x^{1/2}}{\varepsilon_v^2} \int f^2 \frac{(\hat{v} \cdot \nu(x))^2}{\delta(x)^{1/2}} + \frac{1}{\bar{\omega}_-(\rho_v)} \int f^2 \bar{\omega}_- m^2. $$

Observing that

$$ \int f^2 (\bar{\omega} m^2 + 1) \le \kappa \int f^2 \tilde{m}^2 + C_\kappa \int_{\mathcal{U}} f^2 m^2 + C_\kappa \int_{\mathcal{U}^c} f^2 m^2 $$
Using the equation (12.15), we obtain

$$f(x) \geq \frac{(J * f)(x)}{W(x) + \lambda} \geq \frac{J_* \delta}{W[R] + \lambda} \mathbf{1}_{B_{r/2}(x_0^f)}(x),$$

for $W[R] = \sup_{B_R} W$. With that last information and (12.2) again, we have now

$$J * f \geq \frac{J_* \delta}{2^d W[R] + \lambda} \mathbf{1}_{B_r(x_0^f)},$$

and, iterating the argument, we deduce

$$f \geq \frac{J_*^m}{2^{(m-1)d}(W[R] + \lambda)^{m-1}} \delta \mathbf{1}_{B_{mr/2}(x_0^f)} \geq \bar{\gamma} \mathbf{1}_{B_R},$$

with $\bar{\gamma} = \bar{\gamma}(R) > 0$ for $m = m(R)$ large enough. Choosing $R$ an integer, we have proved that

$$(12.16) \qquad f \geq h_0 := \bar{\gamma}(R) \mathbf{1}_{B_R} + \sum_{n \geq R} \bar{\gamma}(n+1) \mathbf{1}_{B_{n+1} \setminus B_n} > 0.$$

That means that the (H4) holds, with constructive lower bound.

**Condition (H5).** Let us consider $f \in L_{\langle W \rangle}^1 \setminus \{0\}$ and $\lambda \in \mathbb{C}$ such that (5.16) holds, in particular

$$(12.17) \qquad \mathcal{L}|f| = (\Re e \lambda)|f| \quad \text{and} \quad \mathcal{L}|f| = \Re e(\text{sign} f)\mathcal{L}f.$$

The first equality means that $\Re e \lambda$ is an eigenvalue associated to a positive eigenfunction, and Lemma 4.17 then enforces $\Re e \lambda = \lambda_1$. Lemma 4.18 subsequently ensures that $|f| \in (\text{Span } f_1)_+ \setminus \{0\}$, and in particular $|f| > 0$. Throwing away the term $W|f|$ in each side of the second identity in (12.17), we have

$$\Re e \frac{\bar{f}}{|f|} (J * f) = J * |f|.$$

Integrating this equation, we get

$$\int_{\mathbb{R}^{2d}} J(x-y) \Re e \left[ |f(y)| - \frac{\bar{f}(x)}{|f(x)|} f(y) \right] dy = 0.$$

From the positivity condition (12.2) on $J$, we deduce

$$|f(y)| - \frac{\bar{f}(x)}{|f(x)|} f(y) = \Re e \left[ |f(y)| - \frac{\bar{f}(x)}{|f(x)|} f(y) \right] = 0, \quad \forall x, y \in \mathbb{R}^d, |x - y| < r,$$

and thus $\bar{f}(x)/|f(x)| = \bar{u}$ for any $x \in \mathbb{R}^d$ for a constant $u \in \mathbb{C}$. That ends the proof of the reverse Kato's inequality (H5).

*Proof of theorem 12.1 part (1).* We may use Theorem 2.21 in order to establish the existence of a solution $(\lambda_1, f_1, \phi_1) \in (0, +\infty) \times L^1 \times L^\infty$ to the first eigentriplet problem (1.1)-(1.2). From Theorem 4.13 and Theorem 5.16, this solution is unique, $f_1 > 0$, $\phi_1 > 0$, $\lambda_1$ is algebraically simple for both $\mathcal{L}$ and $\mathcal{L}^*$ and it is the unique eigenvalue in $\Sigma_+(\mathcal{L})$.

Due to (12.14), we actually have $f_1 \in L_{\langle W \rangle}^1 \cap L_{\langle W \rangle}^\infty$. Observing that $\mathcal{L}^*$ is of the same type as $\mathcal{L}$,

$$\mathcal{L}^* \phi = \check{J} * \phi - W \phi, \quad \check{J}(x) := J(-x),$$

and considering the dual problem as a primal problem in $L^1$, Theorem 2.21 also provides the existence of $\lambda_1^* > 0$ and $0 < \phi_1^* \in L_{\langle W \rangle}^1 \cap L_{\langle W \rangle}^\infty$ such that

$$\mathcal{L}^* \phi_1^* = \lambda_1^* \phi_1^*.$$

Because of Remark 4.16, we have in fact $\lambda_1^* = \lambda_1$ and the simplicity of $\lambda_1$ then yields that $\text{Span } \phi_1^* = \text{Span } \phi_1$. This ensures that $\phi_1 \in L_{\langle W \rangle}^1 \cap L_{\langle W \rangle}^\infty$ and also that $\phi_1$ enjoys the explicit lower bound (12.16). Besides, we can prove

$$\|\phi_1\|_{L_{\langle W \rangle}^{\infty}} \leq \|J_1\|_{L^1} \frac{\lambda_1}{\lambda_1 - \kappa_*} \|\phi_1\|_{L^{\infty}} \leq \|J_1\|_{L^1} \frac{\kappa_1}{\kappa_0 - \kappa_*} \|\phi_1\|_{L^{\infty}}$$

by arguing similarly as for (12.13).

□

In order to prove Theorem 12.1 part (2) with constructive constants we use a Doeblin-Harris type argument
with $C_\kappa := \sup(\varpi + 2 - \kappa)_+ < \infty$, and $\mathcal{A} \ge C_\kappa \mathbf{1}_\mathcal{U}$ for $n := C_\kappa$, altogether, we conclude with

$$ (\mathcal{B}f, f)_{L^2_m} \le \kappa \|f\|_{L^2_m}. $$

We then classically deduce that (11.55) holds.

Similarly as for the first estimate and in the proof of [273, Lem. 3.8], for any smooth, rapidly decaying and positive function $f$, we have

$$ \int (\mathcal{L}f) f^{p-1} m_p^p = - \int_{\Sigma} \frac{(m_p \gamma f)^p}{p} \nu \cdot v - (p-1) \int |\nabla(m_p f)|^2 (m_p f)^{p-2} dx, + \int f^p \varpi_p m_p^p. $$

From Darozès-Guiraud (or Jensen) inequality, we know that the first (boundary) term is nonpositive (see [123] or [271, Rem. 6.4]) and we then classically conclude to (11.56). □

**Lemma 11.10.** There exists a finite family $2 = p_0 < p_1 < \dots < p_k < \infty$ and $\alpha \in (0, 1)$ such that for both $\mathcal{C} = \mathcal{B}, \mathcal{L}$, for any $T > \tau > 0$ and $\mathcal{V} \subset \subset \mathcal{O}$,

$$ (11.57) \qquad \int_{\tau}^{T} \| \mathcal{A} S_{\mathcal{C}}(t) f_0 \|_{L_{m_p}^{p_j}} dt \le C_{p_j-1}^{p_j} \| f_0 \|_{L_{m_p}^{p_{j-1}}} , \quad j = 1, \dots, k, $$

$$ (11.58) \qquad \sup_{t \in [\tau, T]} \| \mathcal{A} S_{\mathcal{B}}(t) f_0 \|_{L^{\infty}} \le C_{p_k}^{\infty} \| f_0 \|_{L^{p_k}}, $$

$$ (11.59) \qquad \sup_{t \in [\tau, T]} \| S_{\mathcal{B}}(t) f_0 \|_{C^{\alpha}(\mathcal{V})} \le C_{\infty}^{\alpha} \| f_0 \|_{L^{\infty}}. $$

*Proof of Lemma 11.10.* For $0 \le f_0 \in L_m^2$, let us denote $f := S_{\mathcal{B}}f_0$ which thus satisfies the PDE

$$ \partial_t f - \mathcal{B}f = s := cf \quad \text{in } \mathcal{D}'((0, T) \times \mathcal{O}). $$

Let us fix two open sets $U_i$ such that $[\tau, T] \times \text{supp}\xi \times \text{supp}\zeta \subset U_0 \subset \subset U_1 \subset \subset (0, T) \times \mathcal{O}$. From [180, Thm. 6] and a covering lemma, there exists a constant $\bar{C}_0 > 0$ and $p_1 > 2$ such that

$$ \|f\|_{L^{p_1}(U_0)} \le \bar{C}_0 (\|f\|_{L^2(U_1)} + \|s\|_{L^2(U_1)}). $$

The estimate (11.57) for $j = 1$ then follows from Theorem 11.5 (and the classical underlying energy estimate). On the other hand, [180, Thm. 12] similarly implies that there exists a constant $\bar{C}_k > 0$ and $p_k \in (p_1, \infty)$ such that

$$ \|f\|_{L^{\infty}(U_0)} \le \bar{C}_k (\|f\|_{L^2(U_1)} + \|s\|_{L^{p_k}(U_1)}), $$

and interpolating with the previous estimate, we get

$$ \|f\|_{L^{p_j}(U_0)} \le \bar{C}_{j-1} (\|f\|_{L^2(U_1)} + \|s\|_{L^{p_{j-1}}(U_1)}), \quad \forall j, 2 \le j \le k-1. $$

The growth bound (11.56) and the two last estimates imply (11.58) and (11.57) for any $2 \le j \le k-1$. Finally, [180, Thm. 3] similarly implies that there exists a constant $\bar{C}_{k+1} > 0$ and $\alpha \in (0, 1)$ such that

$$ \|f\|_{C^{\alpha}(U_0)} \le \bar{C}_{k+1} (\|f\|_{L^2(U_1)} + \|s\|_{L^{\infty}(U_1)}), $$

from what we deduce (11.59) in the same way. □

**Theorem 11.11.** Under the conditions of Theorem 11.6 and the additional assumption (11.54), the conclusion (E31) holds in $L_m^2$ with non constructive rate.

*Proof of Theorem 11.11.* We introduce the splitting

$$ \mathcal{A}g := M\Upsilon_g, \quad \Upsilon_g := \chi_\epsilon g, \quad \mathcal{B} := \mathcal{L} - \mathcal{A}, $$

with $\chi_\epsilon \in C_c^2(\mathcal{O})$, $\mathbf{1}_{U_{2\epsilon}} \le \chi_\epsilon \le \mathbf{1}_{U_\epsilon}$ and $U_\epsilon := \{|v| \le 1/\epsilon, \delta(x) > \epsilon\}$. We next write the iterated Duhamel formulas (with $N := k+2$)

$$ S_{\mathcal{L}} = V + W * S_{\mathcal{L}}, $$

with the usual notations (3.40) for $V$ and $W$ associated to the integer $N := k+2$ and $k \ge 1$ has been introduced in Lemma 11.10. Next for $T > 0$ large, $\tau \in (0, T)$ small and two functions (of operators) $a$ and $b$, we define the modified convolution operator

$$ \left\{ \begin{array}{ll} (a *_{\tau} b)(t) := \int_{\tau}^{t-\tau} a(t-s)b(s) ds & \text{if } t \in [\tau, T-\tau] \\ (a *_{\tau} b)(t) := 0 & \text{if } t \in [\tau, T-\tau]^c, \end{array} \right. $$
**Lemma 12.3** (Lyapunov Condition). Under the above assumptions, for any $T > 0$, there are $\gamma_L \in (0, 1)$ and $K > 0$ such that

$$ \|\tilde{S}_T f\|_{L^1} \le \gamma_L \|f\|_{L^1} + K \|f\|_{\phi_1}. $$

*Proof of Lemma 12.3.* Writing $f_t = \tilde{S}_t f = e^{-\lambda_1 t} S_L(t) f$, we have, since $\lambda_1 \ge 0$,

$$ \begin{aligned} \frac{d}{dt} \int_{\mathbb{R}^d} |f_t| &\le \|J\|_1 \int_{\mathbb{R}^d} |f_t| - \int_{\mathbb{R}^d} W|f_t| \\ &\le \int_{B_R^c} (\|J\|_1 - W)|f_t| + \frac{\|J\|_1}{\alpha_R} \int_{B_R} |f_t|\phi_1, \end{aligned} $$

for any $R > 0$ and some $\alpha_R$ the bound by below of $\phi_1$ in $B_R$. Choosing $R$ large enough so that $W(x) \ge \|J\|_1 + 1$ for $|x| \ge R$, we get

$$ \frac{d}{dt} \int_{\mathbb{R}^d} |f_t| \le - \int_{\mathbb{R}^d} |f_t| + \frac{\|J\|_1 + 1}{\alpha_R} \int_{\mathbb{R}^d} |f_t|\phi_1. $$

Since

$$ \int_{\mathbb{R}^d} |f_t|\phi_1 \le \int_{\mathbb{R}^d} \tilde{S}_t |f_0|\phi_1 = \int_{\mathbb{R}^d} |f_0|\phi_1, $$

we infer

$$ \|\tilde{S}_t f\| \le e^{-t} \|f\| + \frac{\|J\|_1 + 1}{\alpha_R} (1 - e^{-t}) \|f\|_{\phi_1}, $$

by Grönwall's lemma.

□

**Lemma 12.4** (Harris's condition). Under the assumption above, there exist $\psi_0 \in X_{++}^\prime$, $g_0 \in X_+$ and $T > 0$ such that

$$ (12.18) \qquad S_T f \ge g_0 \langle f, \psi_0 \rangle, \quad \forall f \in X_+. $$

*Proof of Lemma 12.4.* Step 1. *proof of* (12.18). From Duhamel's formula (3.9) we have

$$ S_{\mathcal{L}} = S_{\mathcal{B}} + \dots + (S_{\mathcal{B}} \mathcal{A})^{*(N-1)} * S_{\mathcal{B}} + (S_{\mathcal{B}} \mathcal{A})^{*(N)} * S_{\mathcal{L}}. $$

We note that

$$ (S_{\mathcal{B}} \mathcal{A} * S_{\mathcal{B}}) f(x) = \int_{0}^{t} S_{\mathcal{B}}(t-s) \mathcal{A} S_{\mathcal{B}}(s) f ds = \int_{0}^{t} [\mathcal{A}(f e^{W(x)s})] e^{-W(x)(t-s)} ds. $$

For any $R > r$, $x \in B_R$, it is satisfied that

$$ \mathcal{A}(f e^{Ws})(x) = \int_{\mathbb{R}^d} J(x-y)f(y)e^{-W(y)s}dy \ge J_*e^{-W[2R]s} \int_{B_r(x)} f(y)dy $$

with $W[R]$ defined as in the proof of (12.16). Then we get

$$ (S_{\mathcal{B}} \mathcal{A} * S_{\mathcal{B}}) f(x) \ge \mathbf{1}_{B_R}(x) J_* t e^{-W[2R]t} \int_{B_r(x)} f(y)dy. $$

Subsequently, we obtain that

$$ S_{\mathcal{B}} \mathcal{A} * (S_{\mathcal{B}} \mathcal{A} * S_{\mathcal{B}}) f(x) \ge \mathbf{1}_{B_{R-r}}(x) \int_{0}^{t} J_* s e^{-W[2R]t} \mathcal{A} \left( \mathbf{1}_{B_R}(x) \int_{B_r(x)} f(y) dy \right) ds, $$

with

$$ \mathcal{A} \left( \mathbf{1}_{B_R}(x) \int_{B_r(x)} f(y) dy \right) = \int_{\mathbb{R}^d} J(x-y) \mathbf{1}_{B_R}(y) \int_{B_r(y)} f(z) dz dy \ge J_* \int_{B_r(x)} \int_{B_r(y)} f(z) dz dy. $$

We claim that for all $a \ge r$,

$$ \int_{B_r(x)} \int_{B_a(y)} f(z) dz dy \ge |B_{r/4}| \int_{B_{a+r/2}(x)} f(z) dz. $$

Indeed, we deduce

$$ \int_{B_r(x)} \int_{B_a(y)} f(z) dz dy = \int_{B_r(x)} \int_{\mathbb{R}^d} \mathbf{1}_{B_a(y)}(z) f(z) dz dy = \int_{\mathbb{R}^d} f(z) \int_{B_r(x)} \mathbf{1}_{B_a(z)}(y) dy dz $$
(with these notations $*_0 = *$) and by induction $a^{*\tau 1} := a$, $a^{*\tau k} := a^{*\tau(k-1)}*_\tau a$ for $k \ge 2$. With these notations, we define the new splitting

$$S_{\mathcal{L}} = V + K_1^c + K_2^c + K,$$

with

$$K := \Upsilon_{\nu} W_{\tau} *_{\tau} S_{\mathcal{L}}, \quad K_1^c := W * S_{\mathcal{L}} - W_{\tau} *_{\tau} S_{\mathcal{L}}, \quad K_2^c := (1 - \Upsilon_{\nu}) W_{\tau} *_{\tau} S_{\mathcal{L}},$$

where $W_\tau := (S_B A)^{*\tau N}$ and $\nu > 0$. For later references, we also define recursively $\Xi_0 := S_\mathcal{L}$, $\Xi_\ell := S_B A *_{\tau} \Xi_{\ell-1}$ for $\ell \ge 1$, so that $K = \Upsilon_\varepsilon \Xi_N$. The sequel of the proof is split into two steps.

Step 1. On the one hand, we compute

$$\begin{aligned} \|\Xi_N(T)f_0\|_{L_{m_1}^{p_1}} &\le \|S_B\|_{L^{\infty}(\mathcal{B}(L_{m_1}^{p_1}))} \int_{\tau}^{T-\tau} \left\|\int_{\tau}^{t-\tau} \mathcal{A} S_B(t-s) \mathcal{A} \Xi_{k-1}(s) ds f_0\right\|_{L_{m_1}^{p_1}} dt \\ &\le C_T \int_{\tau}^{T} \int_{\tau}^{T} \|\mathcal{A} S_B(t) \mathcal{A} \Xi_{k-1}(s) f_0\|_{L_{m_1}^{p_1}} dt ds \\ &\le C_T C_2^{p_1} \int_{\tau}^{T} \|\mathcal{A} \Xi_{k-1}(s) f_0\|_{L_{m_1}^{2}} ds, \end{aligned}$$

and thus

$$(11.60) \qquad \|\Xi_N(T)f_0\|_{L_{m_1}^{p_1}} \le C_T \|f_0\|_{L_m^2},$$

where we have used (11.56) in the first line, the Fubini theorem in the second line, (11.57) with $j=1$ in the third line and several times (11.55) in the last line.

For $\kappa < \kappa_0$, we may choose $\varepsilon > 0$ small enough such that (11.55) holds. From the very definition of $\mathcal{A}$ and $S_B$, we may thus fix $\kappa_B \in (\kappa, \kappa_0)$ arbitrary and next $T > 0$ large enough such that $\|V(T)\|_{\mathcal{B}(L_m^2)} \le \frac{1}{3}e^{\kappa_B T}$. We may next use (11.55) and fix $\tau > 0$ small enough such that

$$\|K_1^c(T)\|_{\mathcal{B}(L_m^2)} \le \tau C_T \le \frac{1}{3} e^{\kappa_B T}.$$

Last, because of (11.60), we may fix $\nu > 0$ small enough, in such a way that

$$\|K_2^c(T)f_0\|_{L_m^2} \le \eta(\nu) \|\Xi_N(T)f_0\|_{L_{m_1}^{p_1}} \le \frac{1}{3} e^{\kappa_B T} \|f_0\|_{L_m^2}.$$

The three last estimates together, we have established

$$(11.61) \qquad \|(V + K_1^c + K_1^c)(T)\|_{\mathcal{B}(L_m^2)} \le e^{\kappa_B T}.$$

Step 2. Performing the same kind of computations as for proving (11.60) and in particular using (11.57), we get

$$\begin{aligned} \int_0^T \|\mathcal{A}\Xi_j(s)f_0\|_{L_{m_{pj+1}}^{p_{j+1}}} ds &\le \int_0^T \int_{\tau}^{T-\tau} \|\mathcal{A}S_B(t)\mathcal{A}\Xi_{j-1}(s)f_0\|_{L_{m_{pj+1}}^{p_{j+1}}} dt ds \\ &\le C_{pj}^{p_{j+1}} \int_0^T \|\mathcal{A}\Xi_{j-1}(s)f_0\|_{L_{m_{pj}}^{p_j}} ds, \end{aligned}$$

for $j=1, \dots, k$, and with $p_{k+1} := \infty$. Iterating and using (11.57) with $j=0$, we get

$$\int_0^T \|\mathcal{A}\Xi_k(s)f_0\|_{L_m^\infty} ds \lesssim \|f_0\|_{L_m^2}.$$

Similarly, we may write

$$\begin{aligned} \sup_{[\tau,T]} \|\mathcal{A}\Xi_{k+1}f_0\|_{L_m^\infty} &\le \sup_{t \in [\tau,T]} \int_{\tau}^{t} \|\mathcal{A}S_B(t-s)\mathcal{A}\Xi_k(s)f_0\|_{L_m^\infty} ds \\ &\le \sup_{t \in [\tau,T]} \|\mathcal{A}S_B(s)\|_{\mathcal{B}(L_m^\infty)} \int_{\tau}^{T} \|\mathcal{A}\Xi_k(s)f_0\|_{L_m^\infty} ds, \end{aligned}$$

thanks to (11.58), and

$$\begin{aligned} \|K f_0\|_{C^\alpha(\mathcal{O})} &\le \int_{\tau}^{T-\tau} \|S_B(T-s)\mathcal{A}\Xi_{k+1}(s)f_0\|_{C^\alpha(\mathcal{U}_\nu)} ds \\ &\le C_\infty^\alpha T \sup_{[\tau,T]} \|\mathcal{A}\Xi_{k+1}f_0\|_{L_m^\infty}, \end{aligned}$$
and, since for all $z \in B_{a+r/2}(x)$,

$$B_{\frac{1}{4}}\left(\frac{z-x}{|z-x|}\frac{3r}{4}+x\right) \subset B_r(x) \cap B_a(z),$$

we have

$$\int_{B_r(x)} \mathbf{1}_{B_a(z)}(y) dy \geq |B_{r/4}| \mathbf{1}_{B_{a+r/2}(x)}(z),$$

and consequently,

$$\int_{B_r(x)} \int_{B_a(y)} f(z) dz dy \geq |B_{r/4}| \int_{B_{a+r/2}(x)} f(z) dz.$$

We have obtained that

$$S_B A * (S_B A * S_B) f(x) \geq \mathbf{1}_{B_{R-r}(x)} J_*^2 t^2 / 2e^{-W[2R]t} \int_{B_{r+r/2}(x)} f(y) dy.$$

Iterating the same argument we arrive to

$$(S_B A)^{(n)} * S_B f(x) \geq \mathbf{1}_{B_{R-nr}(x)} J_*^n \frac{t^n}{n!} e^{-W[2R]t} \int_{B_{r+(n-1)r/2}(x)} f(y) dy.$$

In consequence, for $R = (n+1)r$, we get

$$(S_B A)^{(n)} * S_B f(x) \geq \mathbf{1}_{B_r(x)} J_*^n \frac{t^n}{n!} e^{-W[2(n+1)r]t} \int_{B_{(n-1)r/2}(0)} f(y) dy.$$

Coming back to the Duhamel formula (3.9), we deduce

$$S_L f(x) \geq \mathbf{1}_{B_r}(x) \sum_{n=2}^{\infty} \frac{(J_* t)^n}{n!} e^{-W[2(n+1)r]t} \int_{B_{(n-1)r/2}} f(y) dy,$$

from where (12.18) follows with

$$\psi_0 := \sum_{n=2}^{\infty} \frac{(J_* T)^n}{n!} e^{-W[2(n+1)r]T} \mathbf{1}_{B_{(n-1)r/2}}$$

and $g_0 := \mathbf{1}_{B_r}$.

□

*Proof of Theorem 12.1 part (2).* Let us consider $A > 0$ and $f \in X_+$ such that $\|f\| \leq A[f]_{\phi_1}$. For any integer $n \geq 1$, we have

$$[f]_{\phi_1} = \int_{B_n} f \phi_1 + \int_{B_n^c} f \phi_1 \leq \alpha_n \langle f, \psi_0 \rangle + \beta_n \|f\| \\ \leq \alpha_n \langle f, \psi_0 \rangle + \beta_n A[f]_{\phi_1},$$

with $\alpha_n = \|\phi_1\|_{L^\infty} / \inf_{B_n} \psi_0$ and $\beta_n = \|\phi_1\|_{L^\infty(\mathbb{W})} / \inf_{B_n^c} W$. Choosing $n_A$ such that $\beta_{n_A} A \leq 1/2$, we deduce the constructive estimate

$$[f]_{\phi_1} \leq 2\alpha_{n_A} \langle f, \psi_0 \rangle,$$

and thus that (6.8) holds with $g_A := (2\alpha_{n_A})^{-1}g_0$. Because of the constructive lower bound (12.16) on $\phi_1$, we have

$$\langle \phi_1, g_R \rangle \geq (2\alpha_{n_A})^{-1} \langle h_0, g_0 \rangle =: r_A,$$

which provides (6.9) in a quantified way. The two above estimates and the Lyapunov condition established in Lemma 12.3 ensure that we may apply the Harris-Doblin Theorem 6.3 and thus conclude to (12.6) with constructive rate. □
thanks to (11.59). The three last estimates together and the compact support property $\text{supp}\chi_\nu \subset \mathcal{O}$ imply

$$ \|K f_0\|_{C^\alpha \cap L^2_{m_{p_1}}} \lesssim \|f_0\|_{L^2_m}, \quad \forall f_0 \in L^2_m, $$

from what we deduce that $K \in \mathcal{K}(L_m^2)$. We may apply Theorem 5.28 in order to conclude. $\square$

## 12. A MUTATION-SELECTION MODEL

In this section, we consider the mutation-selection evolution equation associated to the mutation-selection operator

$$ (12.1) \qquad \mathcal{L}f := J \ast f - W(x)f $$

defined on functions $f: \mathbb{R}^d \to \mathbb{R}$, where $J$ is a the mutation kernel, $\ast$ stands for the convolution operator and $W$ is a confining potential.

**12.1. Almost regular mutation kernel.** We assume that the mutation kernel $J$ is a positive finite measure of $\mathbb{R}^d$ which is lower bounded on a neighborhood of the origin, or in other words

$$ (12.2) \qquad 0 \le J \in M^1(\mathbb{R}^d), \quad J \ge J_* \mathbf{1}_{B_r}, $$

for some constants $J_*, r > 0$. We also assume that the selection potential $W: \mathbb{R}^d \to \mathbb{R}$ is continuous and satisfies

$$ (12.3) \qquad W(x) > W(0) = 0, \forall x \neq 0, \quad W(x) \to +\infty \text{ as } |x| \to \infty. $$

We finally assume the following compatibility condition between mutation and selection: there exist $\beta > 0$ and a bounded Borel set $A \subset \mathbb{R}^d$ such that

$$ (12.4) \qquad a := \essinf_{x \in A_\beta} \int_{x-A_\beta} \frac{J(dz)}{W(x-z)} > 1, $$

$$ (12.5) \qquad J = J_1 + J_2, \quad J_1 \in C_c^1(\mathbb{R}^d), \quad \kappa_* := \|J_2\|_1 := \int_{\mathbb{R}^d} dJ_2 < \kappa_0 := (a-1)\beta, $$

where we use the notation $A_\beta = A \cap \{W \ge \beta\}$. In the sequel, we work in the Banach lattice $X := L^1(\mathbb{R}^d)$.

**Theorem 12.1.** Under the above assumptions, we have

(1) The first eigentriplet problem (1.1)-(1.2) 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 $\lambda_1 \ge \kappa_0$, $0 < f_1 \in L^1_{\langle W \rangle}(\mathbb{R}^d) \cap L^\infty_{\langle W \rangle}(\mathbb{R}^d)$ and $0 < \phi_1 \in L^1_{\langle W \rangle}(\mathbb{R}^d) \cap L^\infty_{\langle W \rangle}(\mathbb{R}^d)$.

(2) Moreover, $\mathcal{L}$ generates a semigroup $S_\mathcal{L}$ on $X$ and for any $f_0 \in X$, there holds

$$ (12.6) \qquad \|e^{-\lambda_1 t} S_\mathcal{L}(t) f_0 - \langle \phi_1, f_0 \rangle f_1\|_{L^1} \le C e^{-\alpha t} \|f_0 - \langle \phi_1, f_0 \rangle f_1\|_{L^1}, $$
for any $t \ge 0$ and for some constructive constants $C \ge 1, \alpha > 0$.

In particular, the conclusions (S1), (S2), (S33) and (E31) hold with constructive constants in $L^1$.

Let us comment on the above result.

## Remark 12.2.

(1) Assumption (12.4) is satisfied when $W$ is small enough in a neighborhood of the origin. It is for instance satisfied if $W^{-1} \notin L^1(B_1)$. That is in particular the case in dimension $d = 1$ when $W$ is Lipschitz, because of the condition $W(0) = 0$.

(2) Assume $J(x) = \varepsilon^{-d}\rho(\varepsilon^{-1}x)$ with $\rho \in C_c^1(\mathbb{R}^d) \cap \mathcal{P}(\mathbb{R}^d)$ and $\rho \ge \rho_* \mathbf{1}_{B_1}$, $\rho_* > 0$, so that $J = J_1$ and $J_2 = 0$, and $W = W(|x|)$. We may observe that for $\beta > 0$ and $\varepsilon > 0$ small enough

$$ \inf_{\beta \le W(x) < 2\beta} \int_{\beta \le W(y) < 2\beta} \frac{J(x-y)}{W(y)} dy :=: a \ge \frac{\rho_*}{2\beta} \text{meas}\{\mathbb{R}_+^d \cap B_1\} > 1, $$

so that (12.4) holds with $A := \{W(x) < 2\beta\}$.

(3) Assumption (12.4) is similar to [248, Condition (2.3)], see also [6, Assumption 2.6] and the comparison with [6, Assumption 2.4], as well as [74, Condition (3.7)-(3.8)] and [76, p. 250, Note
12.2. **A singular mutation kernel.** Here we consider a mutation kernel supported by a set of zero Lebesgue measure, which thus does not satisfy (12.2). The kernel $J \in M_+^1(\mathbb{R}^d)$ is defined for any test function $\varphi \in C_0(\mathbb{R}^d)$ by

$$ \langle J, \varphi \rangle = \varepsilon^{-1} \sum_{i=1}^{d} \int_{\mathbb{R}} \varphi(0, \dots, 0, x_i, 0, \dots, 0) J_i(\varepsilon^{-1} x_i) dx_i, $$

where $(J_i)_{1 \le i \le d}$ is a family of $L^1$ positive kernels on $\mathbb{R}$ and $\varepsilon > 0$ is a variance parameter. The operator $\mathcal{L}$ then reads

$$ \mathcal{L} f(x) = \varepsilon^{-1} \sum_{i=1}^{d} \int_{\mathbb{R}} f(x - z \mathbf{e}_i) J_i(\varepsilon^{-1} z) dz - W(x) f(x), $$

where $\mathbf{e}_i$ is the $i$-th unit vector of the canonical basis of $\mathbb{R}^d$. This model was recently considered and studied by [350] through a probabilistic approach. It shares similarities with a model of telomere shortening which is under study in [147]. We show that the method developed in the first sections of the present paper allows us to handle this model, under similar yet slightly different assumptions on the $J_i$ and $W$ than in [350]. In particular we consider more general fitness functions $W$ than quadratic ones. More precisely, we assume that $W$ is a continuous function that satisfies (12.3) and

$$ (12.19) \qquad \log W(x) = O(|x|^2) \quad \text{as } |x|^2 := \sum_{i=1}^{d} x_i^2 \to \infty. $$

The kernels $J_i$ are supposed to be centered Gaussian distributions

$$ J_i(z) = M_i G_{\sigma_i}(z) := \frac{M_i}{\sigma_i \sqrt{2\pi}} e^{-\frac{z^2}{2\sigma_i^2}}, $$

for given masses $(M_i)_{1 \le i \le d} \in (0, +\infty)^d$ and variances $(\sigma_i)_{1 \le i \le d} \in (0, +\infty)^d$. Similarly as in Section 12.1, we work in the Banach lattice $X = L^1(\mathbb{R}^d)$ and we may prove the following result.

**Theorem 12.5.** Under the above assumptions, there exists a constructive $\varepsilon_0 > 0$ small enough, such that for any $\varepsilon \in (0, \varepsilon_0)$ the following conclusions hold

(1) The first eigentriplet problem (1.1)-(1.2) 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 $\phi_1 > 0$.

(2) Moreover, $\mathcal{L}$ generates a semigroup $S_{\mathcal{L}}$ on $X$ and for any $f_0 \in X$, there holds

$$ (12.20) \qquad \|e^{-\lambda_1 t} S_{\mathcal{L}}(t) f_0 - \langle \phi_1, f_0 \rangle f_1\|_{L^1} \le C e^{-\alpha t} \|f_0 - \langle \phi_1, f_0 \rangle f_1\|_{L^1}, $$
for any $t \ge 0$ and for some constructive constants $C, \alpha > 0$.

**Remark 12.6.** The assumption of small variance $\varepsilon$ in Theorem 12.5 replaces (12.4)-(12.5) as a condition which guarantees the strict positivity of $\kappa_0$ in the condition (H2), and so the strict positivity of $\lambda_1$. This property is fundamental for ensuring the existence of $f_1$ in $L^1$ and for the existence of a spectral gap. On the contrary, for large values of $\varepsilon$, there cannot exist $f_1 \in L^1$, as it is proved in Remark 12.2-(5). The reason is a concentration phenomenon which creates an atom at the origin for the principal eigenvector when the dispersion due to the mutations is too big. This is already noticed in [350, Rk. 5.3.1], and we refer to [64, 77, 110] for more details about the singularity of $f_1$ and the concentration phenomenon.

For proving Theorem 12.5, we first show that the conditions (H1), (H2) and (H3) are verified for the dual problem in $L^\infty = X' = (L^1)'$. Then we check that the Harris conditions are satisfied, thus ensuring the existence, uniqueness and exponential stability for the primal problem. It is worth noticing that since the $J_i$ are symmetric, we have $\mathcal{L}^* = \mathcal{L}$ and the only difference between the primal and dual problems is the Banach lattice in which it is posed.

**Condition (H1) and (H1')**. With the same proof as in Section 12.1, $\mathcal{L}$ generates a positive semigroup $S$ in $L^1$ with $\omega(S) \le \|J\|_1$ and satisfies Kato's inequality. We deduce that (H1) and
added in proof.]. On the other hand, the conditions on J are relaxed here since J may have singular part in (12.5).

(4) Optimal conditions linking J and W for the existence of a spectral gap are still unknown. In the recent paper [6], using variational methods in a L² framework, the authors obtain a quantified spectral gap and the associated exponential stability when the mutation kernel J is additionally assumed to be symmetric. Up to our knowledge, Theorem 12.1 is the very first result providing a quantified spectral gap for a non-symmetric mutation kernel J.

(5) Condition (12.4) can be compared to the condition

$$ \bar{a} := \underset{x \in \mathbb{R}^d}{\operatorname{ess sup}} \int_{\mathbb{R}^d} \frac{J(x-y)}{W(y)} dy < 1, $$

under which no first eigenfunction may exist in X. First, we claim that $\lambda_1 \ge 0$. Indeed, considering $\varepsilon > 0$ and $f_\varepsilon = \mathbf{1}_{B_\varepsilon}$, we have

$$ \mathcal{L}f_\varepsilon \ge -\left(\inf_{B_\varepsilon} W\right) f_\varepsilon, $$

so that the condition (H2) holds for $\kappa_0 = -\inf_{B_\varepsilon} W$ for any $\varepsilon > 0$. Since W is continuous and $W(0) = 0$, we deduce that $\lambda_1 \ge 0$ by passing to the limit $\varepsilon \to 0$. Assume now by contradiction that there exists $f_1 \in X_+ \setminus \{0\}$ such that

$$ (12.7) \qquad \lambda_1 f_1 = \mathcal{L}f_1 = J * f_1 - Wf_1 $$

and define, for any $\varepsilon > 0$, the function $\varphi_\varepsilon(x) = \frac{1}{\varepsilon+W(x)} \in L^\infty(\mathbb{R}^d)$. Testing (12.7) against $\varphi_\varepsilon$ we get for any $\varepsilon \in (0,1)$

$$ \begin{aligned} 0 \le \lambda_1 \langle f_1, \varphi_1 \rangle &\le \lambda_1 \langle f_1, \varphi_\varepsilon \rangle = \iint \frac{J(x-y)}{\varepsilon + W(x)} f_1(y) dx dy - \int \frac{W(x)}{\varepsilon + W(x)} f_1(x) dx \\ &\le \bar{a} \int f_1 - \int \frac{W(x)}{\varepsilon + W(x)} f_1(x) dx, \end{aligned} $$

and passing to the limit $\varepsilon \to 0$ we obtain the contradiction $0 \le \lambda_1 \langle f_1, \varphi_1 \rangle \le (\bar{a} - 1) \int f_1 < 0$. However, there always exists a principal eigenvector $f_1$ in $M^1(\mathbb{R}^d)$, which might have an atom at the origin when $\bar{a} < 1$, see for instance [76].

The proof of Theorem 12.1 follows from Theorem 2.21, Theorem 4.13 and Theorem 5.16 as a consequence of conditions (H1)-(H5) that we establish now. Setting $D(\mathcal{L}) := L^1_{\langle W \rangle}(\mathbb{R}^d)$, we observe that $\mathcal{L}$ is an unbounded closed operator with dense domain $D(\mathcal{L})$.

**Condition (H1) and (H1')**. We define the semigroup

$$ S_W(t)f(x) := e^{-W(x)t}f(x), \quad \forall f \in L^p, p \in [1, \infty], $$

which is clearly a positive semigroup of contractions. We next define $S_\mathcal{L}$ as a bounded perturbation of $S_W$. It is also positive and it satisfies the growth estimate $\|S_\mathcal{L}(t)\|_{\mathcal{B}(L^p)} \le e^{\|J\|_1 t}$, where we recall that $\|J\|_1$ stands for the $L^1$ norm or the total variation norm of $J$. We deduce that (H1) holds true with $\kappa_1 := \|J\|_1$ thanks to Lemma 2.2-(i). Multiplying $\mathcal{L}f$ by sign $f$, for $f \in D(\mathcal{L})$, we immediately get Kato's inequality

$$ (\operatorname{sign} f) \mathcal{L} f = (\operatorname{sign} f) J * f - W |f| \le J * |f| - W |f| = \mathcal{L} |f|. $$

**Condition (H2).** Let us define $f_0 := \frac{1}{W(x)} \mathbf{1}_{A_\beta}$, where $A_\beta$ is introduced in condition (12.4). We compute

$$ \begin{aligned} \mathcal{L}f_0 &= J * (\mathbf{1}_{A_\beta} \frac{1}{W}) - \mathbf{1}_{A_\beta} \ge (J * (\mathbf{1}_{A_\beta} \frac{1}{W}) - 1) \mathbf{1}_{A_\beta} \\ &\ge (\operatorname{ess\,inf}_{x \in A_\beta} [J * (\mathbf{1}_{A_\beta} \frac{1}{W})] - 1) \mathbf{1}_{A_\beta} \\ &= (a-1) \mathbf{1}_{A_\beta} \ge (a-1) \frac{\beta}{W} \mathbf{1}_{A_\beta} = \kappa_0 f_0, \end{aligned} $$

where in the second equality we have used the very definition of $a$ in assumption (12.4). We conclude that (H2) holds thanks to Lemma 2.4-(ii).

**Condition (H3).** We introduce the splitting

$$ (12.8) \qquad \mathcal{L} = \mathcal{A} + \mathcal{B}, \quad \mathcal{A}f := J_1 * f, \quad \mathcal{B}f := J_2 * f - W(x)f. $$
(H1') are verified for both $\mathcal{L}$ in $X$ and $\mathcal{L}^*$ in $X'$ with

$$ \kappa_1 = \|J\|_1 = \sum_{i=1}^{d} M_i. $$

**Condition (H2).** In view of condition (H3), we aim at verifying (H2) with $\kappa_0$ close enough to $\kappa_1$. More precisely, we define $\rho \in (0, 1]$ the ratio between the geometric and arithmetic means of the masses $M_i$, namely

$$ \rho := \frac{\left(\prod_{i=1}^{d} M_i\right)^{1/d}}{\frac{1}{d} \sum_{i=1}^{d} M_i}, $$

we set

$$ \zeta := \frac{d \prod_{i=1}^{d} M_i}{2 \kappa_1^d} = \frac{d^{1-d} \rho^d}{2} \in (0, 1/2], $$

and we prove that there exists $\varepsilon_0$ such that if $\varepsilon \in (0, \varepsilon_0)$, then (H2) is verified with

$$ \kappa_0 = \theta\kappa_1 \quad \text{with} \quad \theta := (1 - \zeta^2)^{1/d} \in (0, 1). $$

Let us fix $\eta > 0$ small enough so that

$$ 1 + (\eta\sigma_i)^2 \le \left(\frac{2}{1 + \theta}\right)^2 $$

for all $i \in \{1, \dots, d\}$. We then define

$$ f_0(x) = \prod_{j=1}^{d} G_{\varepsilon/\eta}(x_j), $$

and we compute

$$ \begin{aligned} \frac{\mathcal{L}f_0(x)}{f_0(x)} &= \sum_{i=1}^{d} M_i \frac{G_{\varepsilon/\eta} * G_{\varepsilon\sigma_i}(x_i)}{G_{\varepsilon/\eta}(x_i)} - W(x) = \sum_{i=1}^{d} M_i \frac{G_{\varepsilon\sqrt{\eta^{-2}+\sigma_i^2}}(x_i)}{G_{\varepsilon/\eta}(x_i)} - W(x) \\ &= \sum_{i=1}^{d} \frac{M_i}{\sqrt{1+(\eta\sigma_i)^2}} \exp\left(\frac{\eta^2(\eta\sigma_i)^2}{1+(\eta\sigma_i)^2} \frac{x_i^2}{2\varepsilon^2}\right) - W(x) \\ &\ge \frac{1+\theta}{2} \sum_{i=1}^{d} M_i \exp\left(\frac{\eta^2(\eta\sigma_i)^2}{1+(\eta\sigma_i)^2} \frac{x_i^2}{2\varepsilon^2}\right) - W(x). \end{aligned} $$

Due to Assumptions (12.3) and (12.19) on $W$ and using Jensen's inequality, we have

$$ W(x) \le \frac{1-\theta}{2} \left( \min_{1 \le i \le d} M_i \right) d e^{C|x|^2/d} \le \frac{1-\theta}{2} \sum_{i=1}^{d} M_i e^{Cx_i^2} $$

for some $C > 0$ large enough. Choosing $\varepsilon_0 > 0$ small enough so that

$$ 2\varepsilon_0^2 \le \frac{\eta^2(\eta\sigma_i)^2}{(1+(\eta\sigma_i)^2)C} $$

for all $i \in \{1, \dots, d\}$, we obtain that

$$ \frac{\mathcal{L}f_0(x)}{f_0(x)} \ge \theta \sum_{i=1}^{d} M_i e^{Cx_i^2} \ge \theta \kappa_1 = \kappa_0 $$

for any $\varepsilon \in (0, \varepsilon_0]$. By virtue of Lemma 2.4-(ii), this proves the announced result.

**Condition (H3) in $X' = L^\infty$.** We use the splitting $\mathcal{L} = \mathcal{A} + \mathcal{B}$ with $\mathcal{B}\phi = -W\phi$, and we aim at proving that (2.29) holds with $N = d$ in order to apply Lemma 2.19. More precisely, we want to find $\varphi \in L^1$ and $\gamma \in (0, 1)$ such that for any $\alpha \ge \kappa_0$, there holds

$$ (12.21) \qquad \|(\mathcal{R}_B(\alpha)\mathcal{A})^d\phi\|_{L^\infty} \le \gamma \|\phi\|_{L^\infty} + \int_{\mathbb{R}^d} \phi(x)\varphi(x) dx $$

for all $\phi \in L_+^\infty$. We have

$$ \mathcal{R}_B(\alpha)\phi = \frac{\phi}{\alpha + W} \le \frac{\phi}{\kappa_0} $$
Arguing as in the proof of condition (H1), we see that $\mathcal{B}$ is the generator a positive semigroup in $L^p(\mathbb{R}^d)$, $1 \le p \le \infty$, with growth bound $\omega(S_{\mathcal{B}}) \le \kappa_*$ and thus $(\alpha - \mathcal{B})$ is invertible for any $\alpha \ge \kappa_0 > \kappa_*$, with

$$ (12.9) \qquad \|(\alpha - \mathcal{B})^{-1}\|_{\mathcal{B}(L^p)} \le \frac{1}{\alpha - \kappa_*}. $$

Next, observing that

$$ (W + \alpha)h = (\alpha - \mathcal{B})h + J_2 * h, $$

for any $h \in \mathcal{D}(\mathcal{L})$ and $\alpha \ge \kappa_0$, we deduce that

$$ (12.10) \qquad (W + \alpha)(\alpha - \mathcal{B})^{-1}g = g + J_2 * ((\alpha - \mathcal{B})^{-1}g), $$

for any $g \in X$ and $\alpha \ge \kappa_0$. Together with (12.9), we deduce

$$ (12.11) \qquad \|(\alpha - \mathcal{B})^{-1}g\|_{L_W^p} \le \|g\|_{L^p} + \|J_2 * ((\alpha - \mathcal{B})^{-1}g)\|_{L^p} \le \frac{\alpha}{\alpha - \kappa_*} \|g\|_{L^p}, $$

for any $g \in L^p$ and $\alpha \ge \kappa_0$. Defining $\mathcal{W}(\alpha) := (\alpha - \mathcal{B})^{-1}\mathcal{A}$, we finally deduce from (12.10) the identity

$$ \mathcal{W}(\alpha)f = \frac{1}{W + \alpha}\mathcal{A}f + \frac{1}{W + \alpha}J_2 * ((\alpha - \mathcal{B})^{-1}\mathcal{A}f), $$

for any $f \in X$ and $\alpha \ge \kappa_0$. We may then compute

$$ \|\mathcal{W}(\alpha)f\|_{L^\infty} \le \frac{1}{\alpha}\|\mathcal{A}f\|_{L^\infty} + \frac{1}{\alpha}\|J_2\|_1\|(\alpha - \mathcal{B})^{-1}\mathcal{A}f\|_{L^\infty}, $$

and together with (12.9) for $p = \infty$ and (12.11), we deduce

$$ (12.12) \qquad \|\mathcal{W}(\alpha)f\|_{L^\infty} \le \|J_1\|_{\infty} \frac{1}{\alpha - \kappa_*} \|f\|_{L^1}, $$

for any $f \in X$ and $\alpha \ge \kappa_0$. Starting from the same identity, we prove in a similar way

$$ (12.13) \qquad \|\mathcal{W}(\alpha)f\|_{L_W^\infty} \le \|J_1\|_{\infty} \frac{\alpha}{\alpha - \kappa_*} \|f\|_{L^1}, $$

for any $f \in X$ and $\alpha \ge \kappa_0$. As a conclusion and gathering (12.9), (12.11), (12.12) and (12.13), we have established that

$$ (12.14) \qquad \mathcal{W}(\alpha) : L^1 \to L^1_{\langle W \rangle} \cap L^\infty_{\langle W \rangle}, $$

with uniform bound for any $\alpha \ge \kappa_0$. Observing that $L^1_{\langle W \rangle} \cap L^\infty_{\langle W \rangle} \subset L^1$ is weakly compact and using Lemma 2.13 with $p = 1$, we deduce that (H3) holds. We can actually strengthen the compactness by noticing that $\mathcal{A} : L^1 \to L^1_W \cap W^{1,1}$ is bounded because of assumption (12.5). This ensures that $\mathcal{A} : L^1 \to L^1$ is compact, from what we deduce that $\mathcal{W}(\alpha) : L^1 \to L^1$ is strongly compact for all $\alpha \ge \kappa_0$. We may thus apply Lemma 2.8-(2) to infer that condition (H3) holds for both the primal and the dual problems.

**Condition (H4).** Assume that $\lambda \ge \lambda_1$ and $f \in \mathcal{D}(\mathcal{L}) = L^1_{\langle W \rangle}$ satisfy

$$ (12.15) \qquad \|f\|_{L^1} = 1, \quad f \ge 0, \quad (\lambda - \mathcal{L})f \ge 0. $$

Denoting $W_R := \inf_{B_R^c} W$, we compute

$$ \int_{B_R} f \ge \int_{\mathbb{R}^d} f - \frac{1}{W_R} \int_{B_R^c} fW \ge 1 - \frac{1}{W_R} \|f\|_{L^1_{\langle W \rangle}} \ge 1/2, $$

for $R > 0$ large enough by taking advantage of the fact that $W(x)$ tend to infinity when $|x| \to \infty$. In particular, there exists $x_0^f \in B_R$ such that

$$ \int_{B_{r/2}(x_0^f)} f \ge \delta := \frac{1}{2} \left( \frac{r}{2R} \right)^d > 0, $$

where we recall that $r$ is defined in (12.2). We deduce that

$$ (J * f)(x) \ge J_* \int_{B_{r/2}(x_0^f)} f(y)dy \mathbf{1}_{B_{r/2}(x_0^f)}(x) \ge J_* \delta \mathbf{1}_{B_{r/2}(x_0^f)}(x). $$
and, defining

$$
\mathcal{A}_r \phi(x) := d \epsilon^{-d} \int_{\mathbb{R}^d} \phi(x-y) J^{\otimes}(y/\epsilon) dy \quad \text{with} \quad J^{\otimes}(y) := \prod_{i=1}^{d} J_i(y_i),
$$

we have

$$
\mathcal{A}^d = \mathcal{A}_r + \mathcal{A}_s
$$

with both $\mathcal{A}_r$ and $\mathcal{A}_s$ positive operators. Positivity ensures that

$$
\| \mathcal{A}_r \|_{\mathcal{B}(X')} = \mathcal{A}_r \mathbf{1} = d \prod_{i=1}^{d} M_i
$$

and

$$
\| \mathcal{A}_s \|_{\mathcal{B}(X')} = \mathcal{A}_s \mathbf{1} = \| \mathcal{A}^d \| - \| \mathcal{A}_r \| \le \| J \|_1^d - d \prod_{i=1}^d M_i.
$$

We deduce that for any $\alpha \ge \kappa_0$,

$$
\begin{aligned}
\| (\mathcal{R}_B(\alpha) \mathcal{A})^d \phi \|_{L^\infty} &\le \kappa_0^{-d} \| \mathcal{A}_s \phi \|_{L^\infty} + \kappa_0^{1-d} \| \mathcal{R}_B(\alpha) \mathcal{A}_r \phi \|_{L^\infty} \\
&\le \frac{\kappa_1^d - d \prod_{i=1}^d M_i}{\kappa_0^d} \| \phi \|_{L^\infty} + \kappa_0^{1-d} \left\| \frac{\mathcal{A}_r \phi}{\kappa_0 + W} \right\|_{L^\infty}.
\end{aligned}
$$

For any $R > 0$ we have

$$
\begin{align*}
\frac{\mathcal{A}_r \phi(x)}{\kappa_0 + W(x)} &= \frac{d \varepsilon^{-d} \mathbf{1}_{B_R}(x)}{\kappa_0 + W(x)} \int_{B_R} \phi(x-y) J^{\otimes}(y/\varepsilon) dy \\
&\quad + \frac{d \varepsilon^{-d} \mathbf{1}_{B_R}(x)}{\kappa_0 + W(x)} \int_{B_R^c} \phi(x-y) J^{\otimes}(y/\varepsilon) dy + \frac{d \varepsilon^{-d} \mathbf{1}_{B_R^c}(x)}{\kappa_0 + W(x)} \int_{\mathbb{R}^d} \phi(x-y) J^{\otimes}(y/\varepsilon) dy \\
&\le \frac{d \varepsilon^{-d}}{\kappa_0} \prod_{i=1}^d \frac{M_i}{\sigma_i \sqrt{2\pi}} \int_{B_{2R}} \phi(y) dy + \frac{d}{\kappa_0} \int_{B_{R/\epsilon_0}^c} J^{\otimes}(y) dy \|\phi\|_{L^\infty} + \frac{d \prod_{i=1}^d M_i}{\kappa_0 + W_R} \|\phi\|_{L^\infty} \\
&\le \kappa_0^{d-1} \int_{\mathbb{R}^d} \phi(x) \varphi_R(y) dy + \frac{\eta_R}{\kappa_0} \|\phi\|_{L^\infty},
\end{align*}
$$

where

$$
\varphi_R = \frac{d \prod_{i=1}^{d} M_i / \sigma_i}{\sqrt{2\pi} (\varepsilon \kappa_0)^d} \mathbf{1}_{B_{2R}} \quad \text{and} \quad \eta_R = d \int_{B_{R/\epsilon_0}^c} J^{\otimes}(y) dy + \frac{d \kappa_0}{W_R} \prod_{i=1}^{d} M_i,
$$

and with $W_R = \inf_{B_R^c} W$. We may therefore infer that

$$
\| (\mathcal{R}_B(\alpha) \mathcal{A})^d \phi \|_{L^\infty} \le \frac{\kappa_1^d - d \prod_{i=1}^d M_i + \eta_R}{\kappa_0^d} \|\phi\|_{L^\infty} + \langle \phi, \varphi_R \rangle.
$$

Since $W(x) \to +\infty$ and $J^{\otimes}(x) \to 0$ as $|x| \to \infty$, we can find $R$ large enough so that

$$
\eta_R \le \frac{d}{2} \prod_{i=1}^{d} M_i = \zeta \kappa_1^d.
$$

Recalling that $\kappa_0^d = (1 - \zeta)\kappa_1^d$, we then obtain (12.21) with

$$
\gamma = \frac{\kappa_1^d - \frac{d}{2} \prod_{i=1}^d M_i}{\kappa_0^d} = \frac{1 - \zeta}{1 - \zeta^2} = \frac{1}{1 + \zeta} < 1 \quad \text{and} \quad \varphi = \varphi_R \in L^1.
$$

Invoking Lemma 2.19, we deduce that (H3) holds true for $\mathcal{L}^* = \mathcal{L}$ in $X' = L^\infty$.

From conditions (H1)-(H2)-(H3), we infer the existence of a solution to the dual problem.

**Lemma 12.7.** If $\varepsilon < \varepsilon_0$, where $\varepsilon_0$ is defined in the paragraph about Condition (H2) above, then there exist $\lambda_1 \ge \kappa_0$ and $\phi_1 \in X'_+$, $\|\phi_1\|_{L^\infty} = 1$, such that $\mathcal{L}^*\phi_1 = \lambda_1\phi_1$. Moreover, $\phi_1 \in L^\infty_W$ and $\langle\phi_1, \varphi\rangle \ge 1 - \gamma$.
Using the equation (12.15), we obtain

$$f(x) \geq \frac{(J * f)(x)}{W(x) + \lambda} \geq \frac{J_* \delta}{W[R] + \lambda} \mathbf{1}_{B_{r/2}(x_0^f)}(x),$$

for $W[R] = \sup_{B_R} W$. With that last information and (12.2) again, we have now

$$J * f \geq \frac{J_*}{2^d} \frac{J_* \delta}{W[R] + \lambda} \mathbf{1}_{B_r(x_0^f)},$$

and, iterating the argument, we deduce

$$f \geq \frac{J_*^m}{2^{(m-1)d}(W[R] + \lambda)^{m-1}} \delta \mathbf{1}_{B_{m r/2}(x_0^f)} \geq \bar{\gamma} \mathbf{1}_{B_R},$$

with $\bar{\gamma} = \bar{\gamma}(R) > 0$ for $m = m(R)$ large enough. Choosing $R$ an integer, we have proved that

$$(12.16) \qquad f \geq h_0 := \bar{\gamma}(R) \mathbf{1}_{B_R} + \sum_{n \geq R} \bar{\gamma}(n+1) \mathbf{1}_{B_{n+1} \setminus B_n} > 0.$$

That means that the (H4) holds, with constructive lower bound.

**Condition (H5).** Let us consider $f \in L_{\langle W \rangle \setminus \{0\}}^1$ and $\lambda \in \mathbb{C}$ such that (5.15) holds, in particular

$$(12.17) \qquad \mathcal{L}|f| = (\Re e \lambda)|f| \quad \text{and} \quad \mathcal{L}|f| = \Re e (\text{sign} f) \mathcal{L}f.$$

The first equality means that $\Re e \lambda$ is an eigenvalue associated to a positive eigenfunction, and Lemma 4.16 then enforces $\Re e \lambda = \lambda_1$. Lemma 4.17 subsequently ensures that $|f| \in (\text{Span } f_1)_+ \setminus \{0\}$, and in particular $|f| > 0$. Throwing away the term $W|f|$ in each side of the second identity in (12.17), we have

$$\Re e \frac{\bar{f}}{|f|} (J * f) = J * |f|.$$

Integrating this equation, we get

$$\int_{\mathbb{R}^{2d}} J(x - y) \Re e \left[ |f(y)| - \frac{\bar{f}(x)}{|f(x)|} f(y) \right] dy = 0.$$

From the positivity condition (12.2) on $J$, we deduce

$$|f(y)| - \frac{\bar{f}(x)}{|f(x)|} f(y) = \Re e \left[ |f(y)| - \frac{\bar{f}(x)}{|f(x)|} f(y) \right] = 0, \quad \forall x, y \in \mathbb{R}^d, |x - y| < r,$$

and thus $\bar{f}(x)/|f(x)| = \bar{u}$ for any $x \in \mathbb{R}^d$ for a constant $u \in \mathbb{C}$. That ends the proof of the reverse Kato's inequality (H5).

*Proof of theorem 12.1 part (1).* We may use Theorem 2.21 in order to establish the existence of a solution $(\lambda_1, f_1, \phi_1) \in (0, +\infty) \times L^1 \times L^\infty$ to the first eigentriplet problem (1.1)-(1.2). From Theorem 4.13 and Theorem 5.16, this solution is unique, $f_1 > 0$, $\phi_1 > 0$, $\lambda_1$ is algebraically simple for both $\mathcal{L}$ and $\mathcal{L}^*$ and it is the unique eigenvalue in $\Sigma_+(\mathcal{L})$.

Due to (12.14), we actually have $f_1 \in L_{\langle W \rangle}^1 \cap L_{\langle W \rangle}^\infty$. Observing that $\mathcal{L}^*$ is of the same type as $\mathcal{L}$,

$$\mathcal{L}^* \phi = \tilde{J} * \phi - W \phi, \quad \tilde{J}(x) := J(-x),$$

and considering the dual problem as a primal problem in $L^1$, Theorem 2.21 also provides the existence of $\lambda_1^* > 0$ and $0 < \phi_1^* \in L_{\langle W \rangle}^1 \cap L_{\langle W \rangle}^\infty$ such that

$$\mathcal{L}^* \phi_1^* = \lambda_1^* \phi_1^*.$$

Because of Theorem 4.13, we have in fact $\lambda_1^* = \lambda_1$ and the simplicity of $\lambda_1$ then yields that $\text{Span } \phi_1^* = \text{Span } \phi_1$. This ensures that $\phi_1 \in L_{\langle W \rangle}^1 \cap L_{\langle W \rangle}^\infty$ and also that $\phi_1$ enjoys the explicit lower bound (12.16). Besides, we can prove

$$\|\phi_1\|_{L_{\langle W \rangle}^{\infty}} \leq \|J_1\|_{L^1} \frac{\lambda_1}{\lambda_1 - \kappa_*} \|\phi_1\|_{L^{\infty}} \leq \|J_1\|_{L^1} \frac{\kappa_1}{\kappa_0 - \kappa_*} \|\phi_1\|_{L^{\infty}}$$

by arguing similarly as for (12.13).

<figure><img src="image_16.png" /></figure>

In order to prove Theorem 12.1 part (2) with constructive constants we use a Doblin-Harris type argument
**Proof.** The existence of $(\lambda_1, \phi_1)$ follows from applying Theorem 2.21. The equation $\mathcal{L}\phi_1 = \lambda_1\phi$ readily gives that

$$ \|\phi_1\|_{L_W^\infty} \le \|J * \phi_1\|_{L^\infty} + \lambda_1 \|\phi_1\|_{L^\infty} \le \|J\|_1 + \lambda_1, $$
and the estimate $\langle \phi_1, \varphi \rangle \ge 1 - \gamma$ comes from Lemma 2.19 $\square$

We now aim at verifying (6.8), (6.7) and (6.9) in order to apply Theorem 6.3.

**Lemma 12.8 (Lyapunov Condition).** Under the above assumptions, for any $T > 0$, there are $\gamma_L \in (0, 1)$ and $K > 0$ such that

$$ \|\tilde{S}_T f\|_{L^1} \le \gamma_L \|f\|_{L^1} + K \|f\|_{\phi_1}. $$

*Proof.* The proof is exactly the same as for Lemma 12.3 in Section 12.1. $\square$

**Lemma 12.9 (Harris's condition).** Under the above assumptions, there exists $\psi_0 \in X_{++}'$, $g_0 \in X_+$ and $T > 0$ such that

$$ (12.22) \qquad S_T f \ge \langle f, \psi_0 \rangle g_0, \quad \forall f \in X_+. $$

*Proof.* We prove the dual version of (12.22), namely

$$ (12.23) \qquad S_T \phi \ge \langle \phi, g_0 \rangle \psi_0, \quad \forall \phi \in X_+'. $$

where we have used that $S_T^* = S_{\mathcal{L}}^*(T) = S_{\mathcal{L}}(T) = S_T$, since $\mathcal{L}^* = \mathcal{L}$ due to the symmetry of $J$. The iterated Duhamel formula (3.9) and the positivity of $\mathcal{A}$ and $S_B$ ensure that

$$ S_{\mathcal{L}} \ge (S_B \mathcal{A})^{(d)} * S_B. $$

We start by estimating $(S_B \mathcal{A} * S_B)(t)\phi$ for $\phi \ge 0$. Since

$$ \begin{aligned} \mathcal{A} S_B(s) \phi(x) &\ge \varepsilon^{-1} \int_{\mathbb{R}} \phi(x - z \mathbf{e}_1) e^{-s W(x - z \mathbf{e}_1)} J_1(z/\varepsilon) \, dz \\ &\ge \varepsilon^{-1} \int_{x_1-1}^{x_1+1} \phi(x - z \mathbf{e}_1) e^{-s W(x - z \mathbf{e}_1)} J_1(z/\varepsilon) \, dz \\ &\ge \varepsilon^{-1} e^{-s W(|x|+1)} J_1 \left( \frac{|x_1|+1}{\varepsilon} \right) \int_{x_1-1}^{x_1+1} \phi(x - z \mathbf{e}_1) \, dz, \end{aligned} $$

where we recall the notation $W[R] = \sup_{B_R} W$, we get

$$ (S_B \mathcal{A} * S_B)(t) \phi(x) \ge \frac{t}{\varepsilon} e^{-tW(|x|+1)} J_1 \left( \frac{|x_1|+1}{\varepsilon} \right) \int_{x_1-1}^{x_1+1} \phi(x - z \mathbf{e}_1) \, dz. $$

Using now the part $J_2$ of $J$ we obtain

$$ \begin{aligned} \mathcal{A}(S_B \mathcal{A} * S_B)(s)\phi(x) &\ge \frac{s}{\varepsilon^2} J_1\left(\frac{|x_1|+1}{\varepsilon}\right) \int_{\mathbb{R}} e^{-sW(|x-z_2\mathbf{e}_2|+1)} \int_{x_1-1}^{x_1+1} \phi(x-z_1\mathbf{e}_1-z_2\mathbf{e}_1) \, dz_1 J_2(z_2/\varepsilon) \, dz_2 \\ &\ge \frac{s}{\varepsilon^2} J_1\left(\frac{|x_1|+1}{\varepsilon}\right) J_2\left(\frac{|x_2|+1}{\varepsilon}\right) e^{-sW(|x|+2)} \int_{x_2-1}^{x_2+1} \int_{x_1-1}^{x_1+1} \phi(x-z_1\mathbf{e}_1-z_2\mathbf{e}_1) \, dz_1 dz_2 \end{aligned} $$

and then

$$ \begin{aligned} &((S_B \mathcal{A})^{(2)} * S_B)(t)\phi(x) \\ &\ge \frac{t^2}{2\varepsilon^2} e^{-tW(|x|+2)} J_1\left(\frac{|x_1|+1}{\varepsilon}\right) J_2\left(\frac{|x_2|+1}{\varepsilon}\right) \int_{x_2-1}^{x_2+1} \int_{x_1-1}^{x_1+1} \phi(x - z_1\mathbf{e}_1 - z_2\mathbf{e}_1) \, dz_1 dz_2. \end{aligned} $$

Iterating and using the successive $J_i$'s parts of $J$ we finally get

$$ \begin{aligned} S_{\mathcal{L}}(t)\phi(x) &\ge ((S_B \mathcal{A})^{(d)} * S_B)(t)\phi(x) \\ &\ge \frac{t^d}{d!} \varepsilon^{-d} e^{-tW(|x|+d)} J^{\otimes} \left(\frac{|x|+1}{\varepsilon}\right) \int_{[-1,1]^d} \phi(y) \, dy, \end{aligned} $$

which yields (12.23), and so (12.22), with

$$ \psi_0(x) = \frac{T^d}{d!} \varepsilon^{-d} e^{-TW(|x|+d)} J^{\otimes} \left( \frac{|x|+1}{\varepsilon} \right) $$
**Lemma 12.3** (Lyapunov Condition). Under the above assumptions, for any $T > 0$, there are $\gamma_L \in (0, 1)$ and $K > 0$ such that

$$ \|\tilde{S}_T f\|_{L^1} \le \gamma_L \|f\|_{L^1} + K \|f\|_{\phi_1}. $$

*Proof of Lemma 12.3.* Writing $f_t = \tilde{S}_t f = e^{-\lambda_1 t} S_L(t) f$, we have, since $\lambda_1 \ge 0$,

$$ \begin{aligned} \frac{d}{dt} \int_{\mathbb{R}^d} |f_t| &\le \|J\|_1 \int_{\mathbb{R}^d} |f_t| - \int_{\mathbb{R}^d} W|f_t| \\ &\le \int_{B_R^*} (\|J\|_1 - W)|f_t| + \frac{\|J\|_1}{\alpha_R} \int_{B_R} |f_t|\phi_1, \end{aligned} $$

for any $R > 0$ and some $\alpha_R$ the bound by below of $\phi_1$ in $B_R$. Choosing $R$ large enough so that $W(x) \ge \|J\|_1 + 1$ for $|x| \ge R$, we get

$$ \frac{d}{dt} \int_{\mathbb{R}^d} |f_t| \le - \int_{\mathbb{R}^d} |f_t| + \frac{\|J\|_1 + 1}{\alpha_R} \int_{\mathbb{R}^d} |f_t|\phi_1. $$

Since

$$ \int_{\mathbb{R}^d} |f_t|\phi_1 \le \int_{\mathbb{R}^d} \tilde{S}_t |f_0|\phi_1 = \int_{\mathbb{R}^d} |f_0|\phi_1, $$

we infer

$$ \|\tilde{S}_t f\| \le e^{-t} \|f\| + \frac{\|J\|_1 + 1}{\alpha_R} (1 - e^{-t}) \|f\|_{\phi_1}, $$

by Grönwall's lemma.

□

**Lemma 12.4** (Harris's condition). Under the assumption above, there exist $\psi_0 \in X_{++}'$, $g_0 \in X_+$ and $T > 0$ such that

$$ (12.18) \qquad S_T f \ge g_0 \langle f, \psi_0 \rangle, \quad \forall f \in X_+. $$

*Proof of Lemma 12.4.* Step 1. proof of (12.18). From Duhamel's formula (3.8) we have

$$ S_L = S_B + \dots + (S_B A)^{(N-1)} * S_B + (S_B A)^{(N)} * S_L. $$

We note that

$$ (S_B A * S_B) f(x) = \int_0^t S_B(t-s) A S_B(s) f ds = \int_0^t [A(f e^{W(x)s})] e^{-W(x)(t-s)} ds. $$

For any $R > r$, $x \in B_R$, it is satisfied that

$$ A(f e^{Ws})(x) = \int_{\mathbb{R}^d} J(x-y)f(y)e^{-W(y)s}dy \ge J_*e^{-W[2R]s} \int_{B_r(x)} f(y)dy $$

with $W[R]$ defined as in the proof of (12.16). Then we get

$$ (S_B A * S_B) f(x) \ge \mathbf{1}_{B_R}(x) J_* t e^{-W[2R]t} \int_{B_r(x)} f(y) dy. $$

Subsequently, we obtain that

$$ S_B A * (S_B A * S_B) f(x) \ge \mathbf{1}_{B_{R-r}}(x) \int_0^t J_* s e^{-W[2R]t} A \left( \mathbf{1}_{B_R}(x) \int_{B_r(x)} f(y) dy \right) ds, $$

with

$$ A \left( \mathbf{1}_{B_R}(x) \int_{B_r(x)} f(y) dy \right) = \int_{\mathbb{R}^d} J(x-y) \mathbf{1}_{B_R}(y) \int_{B_r(y)} f(z) dz dy \ge J_* \int_{B_r(x)} \int_{B_r(y)} f(z) dz dy. $$

We claim that for all $a \ge r$,

$$ \int_{B_r(x)} \int_{B_a(y)} f(z) dz dy \ge |B_r/4| \int_{B_{a+r/2}(x)} f(z) dz. $$

Indeed, we deduce

$$ \int_{B_r(x)} \int_{B_a(y)} f(z) dz dy = \int_{B_r(x)} \int_{\mathbb{R}^d} \mathbf{1}_{B_a(y)}(z) f(z) dz dy = \int_{\mathbb{R}^d} f(z) \int_{B_r(x)} \mathbf{1}_{B_a(z)}(y) dy dz $$
and $g_0 = \mathbf{1}_{[-1,1]^d}$.

□

**Corollary 12.10.** For $\varepsilon \in (0, \varepsilon_0)$, there exists $f_1 \in X_+$ such that $\mathcal{L}f_1 = \lambda_1 f_1$ with $\langle f_1, \phi_1 \rangle = 1$. Moreover, the exponential convergence (12.20) holds for some constructive constants $C \ge 1$ and $\alpha > 0$.

*Proof.* Similarly as in the proof of Theorem 12.1 part (2), we can infer from Lemma 12.9 that (6.8) holds with $g_R = C_R g_0$ where $C_R > 0$ is an explicit constant. The Lyapunov condition (6.7) is established in Lemma 12.8, and the positivity condition (6.9) readily follows from the estimate $\langle \phi_1, \varphi \rangle \ge 1 - \gamma$ established in Lemma 12.7. We can thus apply Theorem 6.3 which, together with its attached Remark 6.4, gives the conclusion.
□

*Proof of Theorem 12.5.* It only remains to prove the uniqueness and strict positivity properties. Combining (12.22) and (12.22) with $\phi = g_0$, we get that

$$S_{2T}f = S_T(S_Tf) \ge \langle f, \psi_0 \rangle S_T g_0 \ge \left( \int g_0^2 \right) \langle f, \psi_0 \rangle \psi_0 = 2^d \langle f, \psi_0 \rangle \psi_0.$$

for all $f \in X_+$. Since $\psi_0 > 0$, this ensures that (4.12) is verified, and then (H4) because of point (4) in Lemma 4.8. This gives the result of uniqueness and strict positivity by using Theorem 4.13.
□

**Acknowledgments.** The authors are very grateful to Philippe Cieutat for the crucial discussions and references about the almost periodic functions, to Boyan Sirakov for having pointed out relevant references about the Krein-Rutman theorem for parabolic equations and to Otared Kavian for the enlightening discussions in many occasions. The authors would also like to warmly thank Jochen Glück for his relevant and useful comments, suggestions, and corrections on the first version of the paper. CFS acknowledge funding by the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 754362. PG has been supported by the ANR project NOLO (ANR-20-CE40-0015), funded by the French Ministry of Research. This work was partially funded by the European Union (ERC, SINGER, 101054787). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

## REFERENCES

[1] N. B. Abdallah. Weak solutions of the initial-boundary value problem for the Vlasov-Poisson system. *Math. Methods Appl. Sci.*, 17(6):451–476, 1994.

[2] V. I. Agoshkov. Problema sledov funktsiž iz prostranstv $H_p^1(\Omega \times D)$ i issledovanie nekotorykh variatsionnykh kraevykh zadach perenosa, volume 85 of *Preprint*. Akad. Nauk SSSR, Otdel Vychisl. Mat., Moscow, 1985.

[3] V. I. Agoshkov. The existence of traces of functions in spaces used in problems of transport theory. *Dokl. Akad. Nauk SSSR*, 288(2):265–269, 1986.

[4] V. I. Agoshkov. Some trace and extension theorems for functions from $W_{x_i,p}^1(D)$ in the case of a bounded domain, volume 243 of *Preprint*. Akad. Nauk SSSR, Otdel Vychisl. Mat., Moscow, 1989.

[5] D. Albritton, S. Armstrong, J.-C. Mourrat, and M. Novack. Variational methods for the kinetic fokker-planck equation. *arXiv:1902.04037* (2021), 2021.

[6] M. Alfaro, P. Gabriel, and O. Kavian. Confining integro-differential equations originating from evolutionary biology: ground states and long time dynamics. *Discrete Contin. Dyn. Syst. Ser. B*, 28(12):5905–5933, 2023.

[7] W. Alt. Orientation of cells migrating in a chemotactic gradient. In *Biological growth and spread (Proc. Conf., Heidelberg, 1979)*, volume 38 of *Lecture Notes in Biomath.*, pages 353–366. Springer, Berlin-New York, 1980.

[8] L. Ambrosio. Transport equation and Cauchy problem for BV vector fields. *Invent. Math.*, 158(2):227–260, 2004.

[9] L. Ambrosio and G. Crippa. Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields. In *Transport equations and multi-D hyperbolic conservation laws*, volume 5 of *Lect. Notes Unione Mat. Ital.*, pages 3–57. Springer, Berlin, 2008.

[10] L. Ambrosio, G. Crippa, and S. Maniglia. Traces and fine properties of a BD class of vector fields and applications. *Ann. Fac. Sci. Toulouse Math. (6)*, 14(4):527–561, 2005.

[11] W. Arendt. Kato's equality and spectral decomposition for positive $C_0$-groups. *Manuscripta Math.*, 40(2-3):277–298, 1982.

[12] W. Arendt. Generators of positive semigroups. In *Infinite-dimensional systems (Retzhof, 1983)*, volume 1076 of *Lecture Notes in Math.*, pages 1–15. Springer, Berlin, 1984.
and, since for all $z \in B_{a+r/2}(x)$,

$$B_{\frac{z}{4}}\left(\frac{z-x}{|z-x|}\frac{3r}{4} + x\right) \subset B_r(x) \cap B_a(z),$$

we have

$$\int_{B_r(x)} \mathbf{1}_{B_a(z)}(y) dy \geq |B_{r/4}| \mathbf{1}_{B_{a+r/2}(x)}(z),$$

and consequently,

$$\int_{B_r(x)} \int_{B_a(y)} f(z) dz dy \geq |B_{r/4}| \int_{B_{a+r/2}(x)} f(z) dz.$$

We have obtained

$$S_B A * (S_B A * S_B) f(x) \geq \mathbf{1}_{B_{R-r}(x)} J_*^2 t^2 / 2e^{-W[2R]t} \int_{B_{r+r/2}(x)} f(y) dy.$$

Iterating the same argument, we arrive to

$$(S_B A)^{(n)} * S_B f(x) \geq \mathbf{1}_{B_{R-nr}(x)} J_*^n \frac{t^n}{n!} e^{-W[2R]t} \int_{B_{r+(n-1)r/2}(x)} f(y) dy.$$

In consequence, for $R = (n+1)r$, we get

$$(S_B A)^{(n)} * S_B f(x) \geq \mathbf{1}_{B_r(x)} J_*^n \frac{t^n}{n!} e^{-W[2(n+1)r]t} \int_{B_{(n-1)r/2}(0)} f(y) dy.$$

Coming back to the Duhamel formula (3.8), we deduce

$$S_L f(x) \geq \mathbf{1}_{B_r(x)} \sum_{n=2}^{\infty} \frac{(J_* t)^n}{n!} e^{-W[2(n+1)r]t} \int_{B_{(n-1)r/2}} f(y) dy,$$

from where (12.18) follows with

$$\psi_0 := \sum_{n=2}^{\infty} \frac{(J_* T)^n}{n!} e^{-W[2(n+1)r]T} \mathbf{1}_{B_{(n-1)r/2}}$$

and $g_0 := \mathbf{1}_{B_r}$.

□

*Proof of Theorem 12.1 part (2).* Let us consider $A > 0$ and $f \in X_+$ such that $\|f\| \leq A[f]_{\phi_1}$. For any integer $n \geq 1$, we have

$$[f]_{\phi_1} = \int_{B_n} f \phi_1 + \int_{B_n^c} f \phi_1 \leq \alpha_n \langle f, \psi_0 \rangle + \beta_n \|f\| \\ \leq \alpha_n \langle f, \psi_0 \rangle + \beta_n A[f]_{\phi_1},$$

with $\alpha_n = \|\phi_1\|_{L^\infty} / \inf_{B_n} \psi_0$ and $\beta_n = \|\phi_1\|_{L^\infty(W)} / \inf_{B_n^c} W$. Choosing $n_A$ such that $\beta_{n_A} A \leq 1/2$, we deduce the constructive estimate

$$[f]_{\phi_1} \leq 2\alpha_{n_A} \langle f, \psi_0 \rangle,$$

and thus that (6.9) holds with $g_A := (2\alpha_{n_A})^{-1}g_0$. Because of the constructive lower bound (12.16) on $\phi_1$, we have

$$\langle \phi_1, g_R \rangle \geq (2\alpha_{n_A})^{-1} \langle h_0, g_0 \rangle := r_A,$$

which provides (6.10) in a quantified way. The two above estimates and the Lyapunov condition established in Lemma 12.3 ensure that we may apply the Doblin-Harris Theorem 6.4 and thus conclude to (12.6) with constructive rate. □
[13] W. Arendt. Kato's inequality: a characterisation of generators of positive semigroups. *Proc. Roy. Irish Acad. Sect.* A, 84(2):155-174, 1984.

[14] W. Arendt. Resolvent positive operators. *Proc. London Math. Soc.* (3), 54(2):321-349, 1987.

[15] W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander, and U. Schlotterbeck. *One-parameter semigroups of positive operators*, volume 1184 of *Lecture Notes in Mathematics*. Springer-Verlag, Berlin, 1986.

[16] L. Arkeryd and C. Cercignani. A global existence theorem for the initial-boundary value problem for the Boltzmann equation when the boundaries are not isothermal. *Arch. Rational Mech. Anal.*, 125(3):271-287, 1993.

[17] L. Arkeryd and A. Heintz. On the solvability and asymptotics of the Boltzmann equation in irregular domains. *Comm. Partial Differential Equations*, 22(11-12):2129-2152, 1997.

[18] L. Arkeryd and N. Maslova. On diffuse reflection at the boundary for the Boltzmann equation and related equations. *J. Statist. Phys.*, 77(5-6):1051-1077, 1994.

[19] R. Arking. *Biology of Aging*. Sinauer, 1998.

[20] L. Arlotti, J. Banasiak, and B. Lods. On transport equations driven by a non-divergence-free force field. *Math. Methods Appl. Sci.*, 30(17):2155-2177, 2007.

[21] L. Arlotti, J. Banasiak, and B. Lods. A new approach to transport equations associated to a regular field: trace results and well-posedness. *Mediterr. J. Math.*, 6(4):367-402, 2009.

[22] L. Arlotti, J. Banasiak, and B. Lods. On general transport equations with abstract boundary conditions. The case of divergence free force field. *Mediterr. J. Math.*, 8(1):1-35, 2011.

[23] L. Arlotti and B. Lods. An $L^p$-approach to the well-posedness of transport equations associated to a regular field: Part II. *Mediterr. J. Math.*, 16(6):Paper No. 145, 30, 2019.

[24] L. Arlotti and B. Lods. An $L^p$-approach to the well-posedness of transport equations associated with a regular field: Part I. *Mediterr. J. Math.*, 16(6):Paper No. 152, 25, 2019.

[25] D. G. Aronson and J. Serrin. Local behavior of solutions of quasilinear parabolic equations. *Arch. Rational Mech. Anal.*, 25:81-122, 1967.

[26] D. Bakry. L'hypercontractivité et son utilisation en théorie des semigroupes. In *Lectures on probability theory (Saint-Flour, 1992)*, volume 1581 of *Lecture Notes in Math.*, pages 1-114. Springer, Berlin, 1994.

[27] D. Bakry, F. Barthe, P. Cattiaux, and A. Guillin. A simple proof of the Poincaré inequality for a large class of probability measures including the log-concave case. *Electron. Commun. Probab.*, 13:60-66, 2008.

[28] D. Bakry, P. Cattiaux, and A. Guillin. Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré. *J. Funct. Anal.*, 254(3):727-759, 2008.

[29] D. Bakry and M. Émery. Diffusions hypercontractives. In *Séminaire de probabilités, XIX, 1983/84*, volume 1123 of *Lecture Notes in Math.*, pages 177-206. Springer, Berlin, 1985.

[30] D. Bakry, I. Gentil, and M. Ledoux. *Analysis and geometry of Markov diffusion operators*, volume 348 of *Grundlehren der mathematischen Wissenschaften* [Fundamental Principles of Mathematical Sciences]. Springer, Cham, 2014.

[31] D. Balagué, J. A. Cañizo, and P. Gabriel. Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates. *Kinet. Relat. Models*, 6(2):219-243, 2013.

[32] J. Banasiak and L. Arlotti. *Perturbations of positive semigroups with applications*. Springer Monographs in Mathematics. Springer-Verlag London Ltd., London, 2006.

[33] J. Banasiak, K. Pichór, and R. Rudnicki. Asynchronous exponential growth of a general structured population model. *Acta Appl. Math.*, 119:149-166, 2012.

[34] V. Bansaye, B. Cloez, and P. Gabriel. Ergodic behavior of non-conservative semigroups via generalized Doeblin's conditions. *Acta Applicandae Mathematicae*, 166:1-44, 2019.

[35] V. Bansaye, B. Cloez, P. Gabriel, and A. Marguet. A non-conservative Harris ergodic theorem. *J. Lond. Math. Soc.* (2), 106(3):2459-2510, 2022.

[36] M. S. Baouendi and P. Grisvard. Sur une équation d'évolution changeant de type. *J. Functional Analysis*, 2:352-367, 1968.

[37] C. Bardos. Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d'approximation; application à l'équation de transport. *Ann. Sci. École Norm. Sup.* (4), 3:185-233, 1970.

[38] C. Bardos, F. Golse, and B. Perthame. The Rosseland approximation for the radiative transfer equations. *Comm. Pure Appl. Math.*, 40(6):691-721, 1987.

[39] C. Bardos, F. Golse, B. Perthame, and R. Sentis. The nonaccretive radiative transfer equations: existence of solutions and Rosseland approximation. *J. Funct. Anal.*, 77(2):434-460, 1988.

[40] C. Bardos, R. Santos, and R. Sentis. Diffusion approximation and computation of the critical size. *Trans. Amer. Math. Soc.*, 284(2):617-649, 1984.

[41] A. Bátkai, M. Kramar Fijavž, and A. Rhandi. *Positive operator semigroups*, volume 257 of *Operator Theory: Advances and Applications*. Birkhäuser/Springer, Cham, 2017. From finite to infinite dimensions, With a foreword by Rainer Nagel and Ulf Schlotterbeck.

[42] F. L. Bauer. An elementary proof of the Hopf inequality for positive operators. *Numer. Math.*, 7:331-337, 1965.

[43] R. Beals and V. Protopopescu. Abstract time-dependent transport equations. *J. Math. Anal. Appl.*, 121(2):370-405, 1987.

[44] W. Beckner. A generalized Poincaré inequality for Gaussian measures. *Proc. Amer. Math. Soc.*, 105(2):397-400, 1989.
12.2. **A singular mutation kernel.** Here we consider a mutation kernel supported by a set of zero Lebesgue measure, which thus does not satisfy (12.2). The kernel $J \in M_+^1(\mathbb{R}^d)$ is defined for any test function $\varphi \in C_0(\mathbb{R}^d)$ by

$$ \langle J, \varphi \rangle = \varepsilon^{-1} \sum_{i=1}^{d} \int_{\mathbb{R}} \varphi(0, \dots, 0, x_i, 0, \dots, 0) J_i(\varepsilon^{-1} x_i) dx_i, $$

where $(J_i)_{1 \le i \le d}$ is a family of $L^1$ positive kernels on $\mathbb{R}$ and $\varepsilon > 0$ is a variance parameter. The operator $\mathcal{L}$ then reads

$$ \mathcal{L} f(x) = \varepsilon^{-1} \sum_{i=1}^{d} \int_{\mathbb{R}} f(x - z e_i) J_i(\varepsilon^{-1} z) dz - W(x) f(x), $$

where $e_i$ is the $i$-th unit vector of the canonical basis of $\mathbb{R}^d$. This model was recently considered and studied by [349] through a probabilistic approach. It shares similarities with a model of telomere shortening which is under study in [146]. We show that the method developed in the first sections of the present paper allows us to handle this model, under similar yet slightly different assumptions on the $J_i$ and $W$ than in [349]. In particular we consider more general fitness functions $W$ than quadratic ones. More precisely, we assume that $W$ is a continuous function that satisfies (12.3) and

$$ (12.19) \qquad \log W(x) = O(|x|^2) \quad \text{as } |x|^2 := \sum_{i=1}^{d} x_i^2 \to \infty. $$

The kernels $J_i$ are supposed to be centered Gaussian distributions

$$ J_i(z) = M_i G_{\sigma_i}(z) := \frac{M_i}{\sigma_i \sqrt{2\pi}} e^{-\frac{z^2}{2\sigma_i^2}}, $$

for given masses $(M_i)_{1 \le i \le d} \in (0, +\infty)^d$ and variances $(\sigma_i)_{1 \le i \le d} \in (0, +\infty)^d$. Similarly as in Section 12.1, we work in the Banach lattice $X = L^1(\mathbb{R}^d)$ and we may prove the following result.

**Theorem 12.5.** Under the above assumptions, there exists a constructive $\varepsilon_0 > 0$ small enough, such that for any $\varepsilon \in (0, \varepsilon_0)$ the following conclusions hold

(1) The first eigentriplet problem (1.1)-(1.2) 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 $\phi_1 > 0$.

(2) Moreover, $\mathcal{L}$ generates a semigroup $S_{\mathcal{L}}$ on $X$ and for any $f_0 \in X$, there holds

$$ (12.20) \qquad \|e^{-\lambda_1 t} S_{\mathcal{L}}(t) f_0 - \langle \phi_1, f_0 \rangle f_1\|_{L^1} \le C e^{-\alpha t} \|f_0 - \langle \phi_1, f_0 \rangle f_1\|_{L^1}, $$
for any $t \ge 0$ and for some constructive constants $C, \alpha > 0$.

In other words, the conclusions (S1), (S2), (S3) and (E3) hold with constructive constants in $L^1$.

**Remark 12.6.** The assumption of small variance $\varepsilon$ in Theorem 12.5 replaces (12.4)-(12.5) as a condition which guarantees the strict positivity of $\kappa_0$ in the condition (H2), and so the strict positivity of $\lambda_1$. This property is fundamental for ensuring the existence of $f_1$ in $L^1$ and for the existence of a spectral gap. On the contrary, for large values of $\varepsilon$, there cannot exist $f_1 \in L^1$, as it is proved in Remark 12.2-(5). The reason is a concentration phenomenon which creates an atom at the origin for the principal eigenvector when the dispersion due to the mutations is too big. This is already noticed in [349, Rk. 5.3.1], and we refer to [64, 76, 109] for more details about the singularity of $f_1$ and the concentration phenomenon.

For proving Theorem 12.5, we first show that the conditions (H1), (H2) and (H3) are verified for the dual problem in $L^\infty = X' = (L^1)'$. Then we check that the Doblin-Harris conditions are satisfied, thus ensuring the existence, uniqueness and exponential stability for the primal problem. It is worth noticing that since the $J_i$ are symmetric, we have $\mathcal{L}^* = \mathcal{L}$ and the only difference between the primal and dual problems is the Banach lattice in which it is posed.
[45] G. I. Bell and S. Glasstone. Nuclear reactor theory.

[46] H. Berestycki, J. Coville, and H.-H. Vo. On the definition and the properties of the principal eigenvalue of some nonlocal operators. *J. Funct. Anal.*, 271(10):2701-2751, 2016.

[47] H. Berestycki, L. Nirenberg, and S. R. S. Varadhan. The principal eigenvalue and maximum principle for second-order elliptic operators in general domains. *Comm. Pure Appl. Math.*, 47(1):47-92, 1994.

[48] H. Berestycki and L. Rossi. Generalizations and properties of the principal eigenvalue of elliptic operators in unbounded domains. *Comm. Pure Appl. Math.*, 68(6):1014-1065, 2015.

[49] E. Bernard. Loskot-Rudnicki's Inequality and General Relative Entropy Inequality for Cauchy Problems Pre-serving Positivity. *Acta Appl. Math.*, 180:Paper No. 13, 2022.

[50] E. Bernard, M. Doumic, and P. Gabriel. Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts. *Kinet. Relat. Models*, 12(3):551-571, 2019.

[51] E. Bernard and P. Gabriel. Asymptotic behavior of the growth-fragmentation equation with bounded frag-mentation rate. *J. Funct. Anal.*, 272(8):3455-3485, 2017.

[52] E. Bernard and P. Gabriel. Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate. *J. Evol. Equ.*, 20(2):375-401, 2020.

[53] E. Bernard and F. Salvarani. On the exponential decay to equilibrium of the degenerate linear Boltzmann equation. *J. Funct. Anal.*, 265(9):1934-1954, 2013.

[54] E. Bernard and F. Salvarani. Optimal estimate of the spectral gap for the degenerate Goldstein-Taylor model. *J. Stat. Phys.*, 153(2):363-375, 2013.

[55] A. Bernoulli, and I. Tristani. Hypocoercivity for kinetic linear equations in bounded domains with general Maxwell boundary condition. *Ann. Inst. H. Poincaré C Anal. Non Linéaire*, 40(2):287-338, 2023.

[56] J. Bertoin and A. R. Watson. A probabilistic approach to spectral analysis of growth-fragmentation equations. *J. Funct. Anal.*, 274(8):2163-2204, 2018.

[57] J. Bertoin and A. R. Watson. The strong Malthusian behavior of growth-fragmentation processes. *Annales Henri Lebesgue*, 3:795-823, 2020.

[58] G. Birkhoff. Extensions of Jentzsch's theorem. *Trans. Amer. Math. Soc.*, 85:219-227, 1957.

[59] G. Birkhoff. Positivity and criticality. In *Proc. Sympos. Appl. Math., Vol. XI*, pages 116-126. American Mathematical Society, Providence, R.I., 1961.

[60] G. D. Birkhoff. Proof of the ergodic theorem. *In : Proceedings of the National Academy of Sciences of the United States of America*, 17:656-660, 1931.

[61] M. D. Blake. A spectral bound for asymptotically norm-continuous semigroups. *J. Operator Theory*, 45(1):111-130, 2001.

[62] L. Boccardo and T. Gallouët. Nonlinear elliptic and parabolic equations involving measure data. *J. Funct. Anal.*, 87(1):149-169, 1989.

[63] M. Bôcher. The theorems of oscillation of Sturm and Klein. *Bull. Amer. Math. Soc.*, 4,5(8,1):295-313,365-376,22-43., 1898.

[64] O. Bonnefon, J. Coville, and G. Legendre. Concentration phenomenon in some non-local equation. *Discrete and Continuous Dynamical Systems. Series B*, 22(3):763-781, 2017.

[65] F. F. Bonsall. Linear operators in complete positive cones. *Proc. London Math. Soc. (3)*, 8:53-75, 1958.

[66] F. Bouchut. Existence and uniqueness of a global smooth solution for the Vlasov-Poisson-Fokker-Planck system in three dimensions. *J. Funct. Anal.*, 111(1):239-258, 1993.

[67] F. Bouchut. Hypoelliptic regularity in kinetic equations. *J. Math. Pures Appl. (9)*, 81(11):1135-1159, 2002.

[68] E. Bouin, J. Dolbeault, S. Mischler, C. Mouhot, and C. Schmeiser. Hypocoercivity without confinement. *Pure Appl. Anal.*, 2(2):203-232, 2020.

[69] N. Bourbaki. *Éléments de mathématique, Fasc. XIII. Livre VI: Intégration. Chapitres 1, 2, 3 et 4: Inégalités de convexité, Espaces de Riesz, Mesures sur les espaces localement compacts, Prolongement d'une mesure, Espaces Lp*. Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1175. Hermann, Paris, 1965. Deuxième édition revue et augmentée.

[70] F. Boyer. Trace theorems and spatial continuity properties for the solutions of the transport equation. *Differential Integral Equations*, 18(8):891-934, 2005.

[71] H. Brezis. *Functional analysis, Sobolev spaces and partial differential equations*. Universitext. Springer, New York, 2011.

[72] H. Brézis and E. Lieb. A relation between pointwise convergence of functions and convergence of functionals. *Proc. Amer. Math. Soc.*, 88(3):486-490, 1983.

[73] R. M. Brooks and K. Schmitt. *The contraction mapping principle and some applications*, volume 9 of *Electronic Journal of Differential Equations. Monograph*. Texas State University–San Marcos, Department of Mathematics, San Marcos, TX, 2009. Available electronically at http://ejde.math.txstate.edu/.

[74] J. Burbea and C. R. Rao. On the convexity of some divergence measures based on entropy functions. *IEEE Trans. Inform. Theory*, 28(3):489-495, 1982.

[75] R. Bürger. Perturbations of positive semigroups and applications to population genetics. *Math. Z.*, 197(2):259-272, 1988.

[76] R. Bürger. *The mathematical theory of selection, recombination, and mutation*. Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2000.

[77] R. Bürger and I. M. Bomze. Stationary distributions under mutation-selection balance: Structure and properties. *Adv. Appl. Probab.*, 28(1):227-251, 1996.

[78] P. J. Bushell. Hilbert's metric and positive contraction mappings in a Banach space. *Arch. Rational Mech. Anal.*, 52:330-338, 1973.
**Condition (H1) and (H1')**. With the same proof as in Section 12.1, $\mathcal{L}$ generates a positive semigroup $S$ in $L^1$ with $\omega(S) \le \|J\|_1$ and satisfies Kato's inequality. We deduce that (H1) and (H1') are verified for both $\mathcal{L}$ in $X$ and $\mathcal{L}^*$ in $X'$ with

$$ \kappa_1 = \|J\|_1 = \sum_{i=1}^{d} M_i. $$

**Condition (H2).** In view of condition (H3), we aim at verifying (H2) with $\kappa_0$ close enough to $\kappa_1$. More precisely, we define $\rho \in (0, 1]$ the ratio between the geometric and arithmetic means of the masses $M_i$, namely

$$ \rho := \frac{\left(\prod_{i=1}^{d} M_i\right)^{1/d}}{\frac{1}{d} \sum_{i=1}^{d} M_i}, $$

we set

$$ \zeta := \frac{d \prod_{i=1}^{d} M_i}{2 \kappa_1^d} = \frac{d^{1-d} \rho^d}{2} \in (0, 1/2], $$

and we prove that there exists $\varepsilon_0$ such that if $\varepsilon \in (0, \varepsilon_0)$, then (H2) is verified with

$$ \kappa_0 = \theta \kappa_1 \quad \text{with} \quad \theta := (1 - \zeta^2)^{1/d} \in (0, 1). $$

Let us fix $\eta > 0$ small enough so that

$$ 1 + (\eta \sigma_i)^2 \le \left( \frac{2}{1 + \theta} \right)^2 $$

for all $i \in \{1, \dots, d\}$. We then define

$$ f_0(x) = \prod_{j=1}^{d} G_{\varepsilon/\eta}(x_j), $$

and we compute

$$ \begin{aligned} \frac{\mathcal{L} f_0(x)}{f_0(x)} &= \sum_{i=1}^{d} M_i \frac{G_{\varepsilon/\eta} * G_{\varepsilon\sigma_i}(x_i)}{G_{\varepsilon/\eta}(x_i)} - W(x) = \sum_{i=1}^{d} M_i \frac{G_{\varepsilon\sqrt{\eta^{-2}+\sigma_i^2}}(x_i)}{G_{\varepsilon/\eta}(x_i)} - W(x) \\ &= \sum_{i=1}^{d} \frac{M_i}{\sqrt{1+(\eta\sigma_i)^2}} \exp\left(\frac{\eta^2(\eta\sigma_i)^2}{1+(\eta\sigma_i)^2} \frac{x_i^2}{2\varepsilon^2}\right) - W(x) \\ &\ge \frac{1+\theta}{2} \sum_{i=1}^{d} M_i \exp\left(\frac{\eta^2(\eta\sigma_i)^2}{1+(\eta\sigma_i)^2} \frac{x_i^2}{2\varepsilon^2}\right) - W(x). \end{aligned} $$

Due to Assumptions (12.3) and (12.19) on $W$ and using Jensen's inequality, we have

$$ W(x) \le \frac{1-\theta}{2} \left( \min_{1 \le i \le d} M_i \right) d e^{C|x|^2/d} \le \frac{1-\theta}{2} \sum_{i=1}^{d} M_i e^{Cx_i^2} $$

for some $C > 0$ large enough. Choosing $\varepsilon_0 > 0$ small enough so that

$$ 2\varepsilon_0^2 \le \frac{\eta^2(\eta\sigma_i)^2}{(1+(\eta\sigma_i)^2)C} $$

for all $i \in \{1, \dots, d\}$, we obtain that

$$ \frac{\mathcal{L} f_0(x)}{f_0(x)} \ge \theta \sum_{i=1}^{d} M_i e^{Cx_i^2} \ge \theta \kappa_1 = \kappa_0 $$

for any $\varepsilon \in (0, \varepsilon_0]$. By virtue of Lemma 2.4-(ii), this proves the announced result.

**Condition (H3) in $X' = L^\infty$.** We use the splitting $\mathcal{L} = \mathcal{A} + \mathcal{B}$ with $\mathcal{B}\phi = -W\phi$, and we aim at proving that (2.28) holds with $N = d$ in order to apply Lemma 2.19. More precisely, we want to find $\varphi \in L^1$ and $\gamma \in (0, 1)$ such that for any $\alpha \ge \kappa_0$, there holds

$$ (12.21) \qquad \|(\mathcal{R}_B(\alpha)\mathcal{A})^d\phi\|_{L^\infty} \le \gamma \|\phi\|_{L^\infty} + \int_{\mathbb{R}^d} \phi(x)\varphi(x) dx $$
[79] J. A. Cañizo, C. Cao, J. Evans, and H. Yoldaş. Hypocoercivity of linear kinetic equations via Harris's theorem. *Kinet. Relat. Models*, 13(1):97-128, 2020.

[80] J. A. Cañizo, P. Gabriel, and H. Yoldaş. Spectral gap for the growth-fragmentation equation via Harris's theorem. *SIAM J. Math. Anal.*, 53(5):5185-5214, 2021.

[81] J. A. Cañizo and S. Mischler. Harris-type results on geometric and subgeometric convergence to equilibrium for stochastic semigroups. *J. Funct. Anal.*, 284(7):Paper No. 109830, 2023.

[82] J. A. Cañizo and H. Yoldaş. Asymptotic behaviour of neuron population models structured by elapsed-time. *Nonlinearity*, 32(2):464-495, 2019.

[83] M. J. Cáceres, J. A. Cañizo, and S. Mischler. Rate of convergence to self-similarity for the fragmentation equation in $L^1$ spaces. *Commun. Appl. Ind. Math.*, 1(2):299-308, 2010.

[84] M. J. Cáceres, J. A. Cañizo, and S. Mischler. Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations. *J. Math. Pures Appl. (9)*, 96(4):334-362, 2011.

[85] A. P. Calderon and A. Zygmund. On the existence of certain singular integrals. *Acta Math.*, 88:85-139, 1952.

[86] M. Cannone and C. Cercignani. A trace theorem in kinetic theory. *Appl. Math. Lett.*, 4(6):63-67, 1991.

[87] C. Cao. The kinetic Fokker-Planck equation with weak confinement force. *Commun. Math. Sci.*, 17(8):2281-2308, 2019.

[88] T. Carleman. Sur la théorie de l'équation intégrodifférentielle de Boltzmann. *Acta Math.*, 60(1):91-146, 1933.

[89] K. Carrapatoso and S. Mischler. The Boltzmann equation in a domain, in preparation.

[90] K. Carrapatoso and S. Mischler. The Kinetic Fokker-Planck equation in a domain: ultracontractivity and long time asymptotic, in preparation.

[91] J. A. Carrillo. Global weak solutions for the initial-boundary-value problems to the Vlasov-Poisson-Fokker-Planck system. *Math. Methods Appl. Sci.*, 21(10):907-938, 1998.

[92] G. Carron. Inégalités isopérimétriques de Faber-Krahn et conséquences. In *Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992)*, volume 1 of *Sémin. Congr.*, pages 205-232. Soc. Math. France, Paris, 1996.

[93] K. M. Case and P. F. Zweifel. *Linear transport theory*. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1967.

[94] V. Caselles. On the peripheral spectrum of positive operators. *Israel J. Math.*, 58(2):144-160, 1987.

[95] C. Cercignani. *The Boltzmann equation and its applications*, volume 67 of *Applied Mathematical Sciences*. Springer-Verlag, New York, 1988.

[96] C. Cercignani. On the initial-boundary value problem for the Boltzmann equation. *Arch. Rational Mech. Anal.*, 116(4):307-315, 1992.

[97] C. Cercignani, R. Illner, and M. Pulvirenti. *The mathematical theory of dilute gases*, volume 106 of *Applied Mathematical Sciences*. Springer-Verlag, New York, 1994.

[98] C. Cercignani, M. Lampis, and A. Lentati. A new scattering kernel in kinetic theory of gases. *Transp. Theory Stat. Phys.*, 24(9):1319-1336, 1995.

[99] M. Cessenat. Théorèmes de trace $L^p$ pour des espaces de fonctions de la neutronique. *C. R. Acad. Sci. Paris Sér. I Math.*, 299(16):831-834, 1984.

[100] M. Cessenat. Théorèmes de trace pour des espaces de fonctions de la neutronique. *C. R. Acad. Sci. Paris Sér. I Math.*, 300(3):89-92, 1985.

[101] D. Chafaï. Entropies, convexity, and functional inequalities: on $\Phi$-entropies and $\Phi$-Sobolev inequalities. *J. Math. Kyoto Univ.*, 44(2):325-363, 2004.

[102] F. A. C. C. Chalub, P. A. Markowich, B. Perthame, and C. Schmeiser. Kinetic models for chemotaxis and their drift-diffusion limits. *Monatsh. Math.*, 142(1-2):123-141, 2004.

[103] N. Champagnat and D. Villemonais. Exponential convergence to quasi-stationary distribution and $Q$-process. *Probab. Theory Related Fields*, 164(1-2):243-283, 2016.

[104] N. Champagnat and D. Villemonais. Practical criteria for $R$-positive recurrence of unbounded semigroups. *Electron. Commun. Probab.*, 25:Paper No. 6, 11, 2020.

[105] N. Champagnat and D. Villemonais. General criteria for the study of quasi-stationarity. *Electron. J. Probab.*, 2023.

[106] M. Chicco. Principio di massimo generalizzato e valutazione del primo autovalore per problemi ellittici del secondo ordine di tipo variazionale. *Ann. Mat. Pura Appl. (4)*, 87:1-9, 1970.

[107] M. Chicco. Some properties of the first eigenvalue and the first eigenfunction of linear second order elliptic partial differential equations in divergence form. *Boll. Un. Mat. Ital. (4)*, 5:245-254, 1972.

[108] B. Cloez, B. de Saporta, and T. Roget. Long-time behavior and Darwinian optimality for an asymmetric size-structured branching process. *Journal of Mathematical Biology*, 83(6-7):30-2021. Id/No 69.

[109] B. Cloez and P. Gabriel. On an irreducibility type condition for the ergodicity of nonconservative semigroups. *C. R. Math. Acad. Sci. Paris*, 358(6):733-742, 2020.

[110] B. Cloez and P. Gabriel. Fast, slow convergence, and concentration in the house of cards replicator-mutator model. *Differential Integral Equations*, 37(7-8):547-584, 2024.

[111] P. Collet, S. Martínez, S. Méléard, and J. San Martín. Quasi-stationary distributions for structured birth and death processes with mutations. *Probab. Theory Related Fields*, 151(1-2):191-231, 2011.

[112] P. Collet, S. Martínez, S. Méléard, and J. San Martín. Stochastic models for a chemostat and long-time behavior. *Adv. in Appl. Probab.*, 45(3):822-836, 2013.

[113] C. Corduneanu. *Almost periodic oscillations and waves*. Springer, New York, 2009.

[114] R. Courant. Über die Eigenwerte bei den Differentialgleichungen der mathematischen Physik. *Math. Z.*, 7(1-4):1-57, 1920.
for all $\phi \in L_+^\infty$. We have

$$ \mathcal{R}_B(\alpha)\phi = \frac{\phi}{\alpha + W} \le \frac{\phi}{\kappa_0} $$

and, defining

$$ \mathcal{A}_r \phi(x) := d \epsilon^{-d} \int_{\mathbb{R}^d} \phi(x-y) J^{\otimes}(y/\epsilon) dy \quad \text{with} \quad J^{\otimes}(y) := \prod_{i=1}^{d} J_i(y_i), $$

we have

$$ \mathcal{A}^d = \mathcal{A}_r + \mathcal{A}_s $$

with both $\mathcal{A}_r$ and $\mathcal{A}_s$ positive operators. Positivity ensures that

$$ \|\mathcal{A}_r\|_{\mathcal{B}(L^\infty)} = \mathcal{A}_r \mathbf{1} = d \prod_{i=1}^{d} M_i $$

and

$$ \|\mathcal{A}_s\|_{\mathcal{B}(L^\infty)} = \mathcal{A}_s \mathbf{1} = \|\mathcal{A}^d\| - \|\mathcal{A}_r\| \le \|J\|_1^d - d \prod_{i=1}^{d} M_i. $$

We deduce that for any $\alpha \ge \kappa_0$,

$$ \begin{aligned} \| (\mathcal{R}_B(\alpha)\mathcal{A})^d \phi \|_{L^\infty} &\le \kappa_0^{-d} \| \mathcal{A}_s \phi \|_{L^\infty} + \kappa_0^{1-d} \| \mathcal{R}_B(\alpha) \mathcal{A}_r \phi \|_{L^\infty} \\ &\le \frac{\kappa_1^d - d \prod_{i=1}^d M_i}{\kappa_0^d} \| \phi \|_{L^\infty} + \kappa_0^{1-d} \left\| \frac{\mathcal{A}_r \phi}{\kappa_0 + W} \right\|_{L^\infty}. \end{aligned} $$

For any $R > 0$ we have

$$ \begin{aligned} \frac{\mathcal{A}_r \phi(x)}{\kappa_0 + W(x)} &= \frac{d \epsilon^{-d} \mathbf{1}_{B_R}(x)}{\kappa_0 + W(x)} \int_{B_R} \phi(x-y) J^{\otimes}(y/\epsilon) dy \\ &\quad + \frac{d \epsilon^{-d} \mathbf{1}_{B_R}(x)}{\kappa_0 + W(x)} \int_{B_R^c} \phi(x-y) J^{\otimes}(y/\epsilon) dy + \frac{d \epsilon^{-d} \mathbf{1}_{B_R^c}(x)}{\kappa_0 + W(x)} \int_{\mathbb{R}^d} \phi(x-y) J^{\otimes}(y/\epsilon) dy \\ &\le \frac{d \epsilon^{-d}}{\kappa_0} \prod_{i=1}^d \frac{M_i}{\sigma_i \sqrt{2\pi}} \int_{B_{2R}} \phi(y) dy + \frac{d}{\kappa_0} \int_{B_{R/\epsilon_0}^c} J^{\otimes}(y) dy \|\phi\|_{L^\infty} + \frac{d \prod_{i=1}^d M_i}{\kappa_0 + W_R} \|\phi\|_{L^\infty} \\ &\le \kappa_0^{d-1} \int_{\mathbb{R}^d} \phi(x) \varphi_R(y) dy + \frac{\eta_R}{\kappa_0} \|\phi\|_{L^\infty}, \end{aligned} $$

where

$$ \varphi_R = \frac{d \prod_{i=1}^d M_i / \sigma_i}{\sqrt{2\pi} (\varepsilon \kappa_0)^d} \mathbf{1}_{B_{2R}} \quad \text{and} \quad \eta_R = d \int_{B_{R/\epsilon_0}^c} J^{\otimes}(y) dy + \frac{d \kappa_0}{W_R} \prod_{i=1}^d M_i, $$

and with $W_R = \inf_{B_R^c} W$. We may therefore infer that

$$ \| (\mathcal{R}_B(\alpha)\mathcal{A})^d \phi \|_{L^\infty} \le \frac{\kappa_1^d - d \prod_{i=1}^d M_i + \eta_R}{\kappa_0^d} \|\phi\|_{L^\infty} + \langle \phi, \varphi_R \rangle. $$

Since $W(x) \to +\infty$ and $J^{\otimes}(x) \to 0$ as $|x| \to \infty$, we can find $R$ large enough so that

$$ \eta_R \le \frac{d}{2} \prod_{i=1}^{d} M_i = \zeta \kappa_1^d. $$

Recalling that $\kappa_0^d = (1 - \zeta)\kappa_1^d$, we then obtain (12.21) with

$$ \gamma = \frac{\kappa_1^d - \frac{d}{2} \prod_{i=1}^d M_i}{\kappa_0^d} = \frac{1 - \zeta}{1 - \zeta^2} = \frac{1}{1 + \zeta} < 1 \quad \text{and} \quad \varphi = \varphi_R \in L^1. $$

Invoking Lemma 2.19, we deduce that (H3) holds true for $\mathcal{L}^* = \mathcal{L}$ in $X' = L^\infty$.

From conditions (H1)-(H2)-(H3), we infer the existence of a solution to the dual problem.
[115] R. Courant and D. Hilbert. *Methods of mathematical physics. Vol. I.* Interscience Publishers, Inc., New York, N.Y., 1953.

[116] J. Coville. On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators. *J. Differential Equations*, 249(11):2921-2953, 2010.

[117] J. Coville. Singular measure as principal eigenfunctions of some nonlocal operators. *Appl. Math. Lett.*, 26(8):831-835, 2013.

[118] J. Coville and F. Hamel. On generalized principal eigenvalues of nonlocal operators with a drift. *Nonlinear Anal.*, 193:111569, 20, 2020.

[119] G. Crippa, C. Donadello, and L. V. Spinolo. Initial-boundary value problems for continuity equations with BV coefficients. *J. Math. Pures Appl. (9)*, 102(1):79-98, 2014.

[120] G. Crippa, C. Donadello, and L. V. Spinolo. A note on the initial-boundary value problem for continuity equations with rough coefficients. In *Hyperbolic problems: theory, numerics, applications*, volume 8 of *AIMS Ser. Appl. Math.*, pages 957-966. Am. Inst. Math. Sci. (AIMS), Springfield, MO, 2014.

[121] I. Csiszár. Eine informationstheoretische Ungleichung und ihre Anwendung auf den Beweis der Ergodizität von Markoffsschen Ketten. *Magyar Tud. Akad. Mat. Kutató Int. Közl.*, 8:85-108, 1963.

[122] I. Csiszár. A class of measures of informativity of observation channels. *Period. Math. Hungar.*, 2:191-213, 1972.

[123] J. M. Cushing. *An introduction to structured population dynamics*, volume 71 of *CBMS-NSF Regional Conference Series in Applied Mathematics*. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998.

[124] J. Darrozès and J. Guiraud. Généralisation formelle du théorème h en présence de parois. *Note C. R. Acad. Sci. Paris Série I*, 262(A):368-371, 1966.

[125] R. Dautray and J.-L. Lions. *Mathematical analysis and numerical methods for science and technology. Vol. 3. Springer-Verlag, Berlin, 1990. Spectral theory and applications, With the collaboration of Michel Artola and Michel Cessenat, Translated from the French by John C. Amson.*

[126] R. Dautray and J.-L. Lions. *Mathematical analysis and numerical methods for science and technology. Vol. 6. Springer-Verlag, Berlin, 1993. Evolution problems. II, With the collaboration of Claude Bardos, Michel Cessenat, Alain Kavenoky, Patrick Lascaux, Bertrand Mercier, Olivier Pironneau, Bruno Scheurer and Rémi Sentis, Translated from the French by Alan Craig.*

[127] E. De Giorgi. Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari. *Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3)*, 3:25-43, 1957.

[128] B. de Pagter. Irreducible compact operators. *Math. Z.*, 192:149-153, 1986.

[129] P. Degond. Global existence of smooth solutions for the Vlasov-Fokker-Planck equation in 1 and 2 space dimensions. *Ann. Sci. École Norm. Sup. (4)*, 19(4):519-542, 1986.

[130] P. Del Moral and A. Guionnet. On the stability of interacting processes with applications to filtering and genetic algorithms. *Ann. Inst. H. Poincaré Probab. Statist.*, 37(2):155-194, 2001.

[131] P. Del Moral and L. Miclo. On the stability of nonlinear Feynman-Kac semigroups. *Ann. Fac. Sci. Toulouse Math. (6)*, 11(2):135-175, 2002.

[132] L. Desvillettes. Hypocoercivity: the example of linear transport. In *Recent trends in partial differential equations*, volume 409 of *Contemp. Math.*, pages 33-53. Amer. Math. Soc., Providence, RI, 2006.

[133] L. Desvillettes and C. Villani. On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation. *Comm. Pure Appl. Math.*, 54(1):1-42, 2001.

[134] O. Diekmann, H. J. A. M. Heijmans, and H. R. Thieme. On the stability of the cell size distribution. *J. Math. Biol.*, 19(2):227-248, 1984.

[135] H. Dietert, F. Hérau, H. Hutridurga, and C. Mouhot. Quantitative geometric control in linear kinetic theory, 2023.

[136] R. J. DiPerna and P.-L. Lions. On the Fokker-Planck-Boltzmann equation. *Comm. Math. Phys.*, 120(1):1-23, 1988.

[137] R. J. DiPerna and P.-L. Lions. Global weak solutions of Vlasov-Maxwell systems. *Comm. Pure Appl. Math.*, 42(6):729-757, 1989.

[138] R. J. DiPerna and P.-L. Lions. On the Cauchy problem for Boltzmann equations: global existence and weak stability. *Ann. of Math. (2)*, 130(2):321-366, 1989.

[139] R. J. DiPerna and P.-L. Lions. Ordinary differential equations, transport theory and Sobolev spaces. *Invent. Math.*, 98(3):511-547, 1989.

[140] W. Doblin. Éléments d'une théorie générale des chaînes simples constantes de Markoff. *Ann. École Norm. (3)*, 57:61-111, 1940.

[141] J. Dolbeault, C. Mouhot, and C. Schmeiser. Hypocoercivity for linear kinetic equations conserving mass. *Trans. Amer. Math. Soc.*, 367(6):3807-3828, 2015.

[142] M. D. Donsker and S. R. S. Varadhan. On a variational formula for the principal eigenvalue for operators with maximum principle. *Proc. Nat. Acad. Sci. U.S.A.*, 72:780-783, 1975.

[143] M. D. Donsker and S. R. S. Varadhan. On the principal eigenvalue of second-order elliptic differential operators. *Comm. Pure Appl. Math.*, 29(6):595-621, 1976.

[144] R. Douc, G. Fort, and A. Guillin. Subgeometric rates of convergence of $f$-ergodic strong Markov processes. *Stochastic Process. Appl.*, 119(3):897-923, 2009.

[145] R. Douc, G. Fort, E. Moulines, and P. Soulier. Practical drift conditions for subgeometric rates of convergence. *Ann. Appl. Probab.*, 14(3):1353-1377, 08 2004.
**Lemma 12.7.** If $\varepsilon < \varepsilon_0$, where $\varepsilon_0$ is defined in the paragraph about Condition (H2) above, then there exist $\lambda_1 \ge \kappa_0$ and $\phi_1 \in X'_+$, $\|\phi_1\|_{L^\infty} = 1$, such that $\mathcal{L}^*\phi_1 = \lambda_1\phi_1$. Moreover, $\phi_1 \in L_W^\infty$ and $\langle\phi_1, \varphi\rangle \ge 1 - \gamma$.

*Proof of Lemma 12.7.* The existence of $(\lambda_1, \phi_1)$ follows from applying Theorem 2.21. The equation $\mathcal{L}\phi_1 = \lambda_1\phi_1$ readily gives that

$$ \|\phi_1\|_{L_W^\infty} \le \|J * \phi_1\|_{L^\infty} + \lambda_1 \|\phi_1\|_{L^\infty} \le \|J\|_1 + \lambda_1, $$

and the estimate $\langle\phi_1, \varphi\rangle \ge 1 - \gamma$ comes from Lemma 2.19

<figure><img src="image_17.png" /></figure>

We now aim at verifying (6.9), (6.8) and (6.10) in order to apply Theorem 6.4.

**Lemma 12.8 (Lyapunov Condition).** Under the above assumptions, for any $T > 0$, there are $\gamma_L \in (0, 1)$ and $K > 0$ such that

$$ \|\tilde{S}_T f\|_{L^1} \le \gamma_L \|f\|_{L^1} + K \|f\|_{\phi_1}. $$

*Proof of Lemma 12.8.* The proof is exactly the same as for Lemma 12.3 in Section 12.1.

<figure><img src="image_18.png" /></figure>

**Lemma 12.9 (Harris's condition).** Under the above assumptions, there exists $\psi_0 \in X'_{++}$, $g_0 \in X_+$ and $T > 0$ such that

$$ (12.22) \qquad S_T f \ge \langle f, \psi_0 \rangle g_0, \quad \forall f \in X_+. $$

*Proof of Lemma 12.9.* We prove the dual version of (12.22), namely

$$ (12.23) \qquad S_T \phi \ge \langle \phi, g_0 \rangle \psi_0, \quad \forall \phi \in X'_+, $$

where we have used that $S_T^* = S_{\mathcal{L}^*}(T) = S_{\mathcal{L}}(T) = S_T$, since $\mathcal{L}^* = \mathcal{L}$ due to the symmetry of $J$. The iterated Duhamel formula (3.8) and the positivity of $\mathcal{A}$ and $S_B$ ensure that

$$ S_{\mathcal{L}} \ge (S_B \mathcal{A})^{(d)} * S_B. $$

We start by estimating $(S_B \mathcal{A} * S_B)(t)\phi$ for $\phi \ge 0$. Since

$$ \begin{aligned} \mathcal{A} S_B(s) \phi(x) &\ge \varepsilon^{-1} \int_{\mathbb{R}} \phi(x - z \mathbf{e}_1) e^{-s W(x - z \mathbf{e}_1)} J_1(z/\varepsilon) \, dz \\ &\ge \varepsilon^{-1} \int_{x_1-1}^{x_1+1} \phi(x - z \mathbf{e}_1) e^{-s W(x - z \mathbf{e}_1)} J_1(z/\varepsilon) \, dz \\ &\ge \varepsilon^{-1} e^{-s W(|x|+1)} J_1\left(\frac{|x_1|+1}{\varepsilon}\right) \int_{x_1-1}^{x_1+1} \phi(x - z \mathbf{e}_1) \, dz, \end{aligned} $$

where we recall the notation $W[R] = \sup_{B_R} W$, we get

$$ (S_B \mathcal{A} * S_B)(t) \phi(x) \ge \frac{t}{\varepsilon} e^{-tW(|x|+1)} J_1\left(\frac{|x_1|+1}{\varepsilon}\right) \int_{x_1-1}^{x_1+1} \phi(x - z\mathbf{e}_1) \, dz. $$

Using now the part $J_2$ of $J$ we obtain

$$ \begin{aligned} \mathcal{A}(S_B \mathcal{A} * S_B)(s) \phi(x) &\ge \frac{s}{\varepsilon^2} J_1\left(\frac{|x_1|+1}{\varepsilon}\right) \int_{\mathbb{R}} e^{-sW(|x-z_2\mathbf{e}_2|+1)} \int_{x_1-1}^{x_1+1} \phi(x - z_1\mathbf{e}_1 - z_2\mathbf{e}_1) \, dz_1 J_2(z_2/\varepsilon) \, dz_2 \\ &\ge \frac{s}{\varepsilon^2} J_1\left(\frac{|x_1|+1}{\varepsilon}\right) J_2\left(\frac{|x_2|+1}{\varepsilon}\right) e^{-sW(|x|+2)} \int_{x_2-1}^{x_2+1} \int_{x_1-1}^{x_1+1} \phi(x - z_1\mathbf{e}_1 - z_2\mathbf{e}_1) \, dz_1 \, dz_2 \end{aligned} $$

and then

$$ \begin{aligned} &((S_B \mathcal{A})^{(2)} * S_B)(t) \phi(x) \\ &\ge \frac{t^2}{2\varepsilon^2} e^{-tW(|x|+2)} J_1\left(\frac{|x_1|+1}{\varepsilon}\right) J_2\left(\frac{|x_2|+1}{\varepsilon}\right) \int_{x_2-1}^{x_2+1} \int_{x_1-1}^{x_1+1} \phi(x - z_1\mathbf{e}_1 - z_2\mathbf{e}_1) \, dz_1 \, dz_2. \end{aligned} $$

Iterating and using the successive $J_i$'s parts of $J$ we finally get

$$ \begin{aligned} S_{\mathcal{L}}(t)\phi(x) &\ge ((S_B \mathcal{A})^{(d)} * S_B)(t)\phi(x) \\ &\ge \frac{t^d}{d!} \varepsilon^{-d} e^{-tW(|x|+d)} J^\otimes\left(\frac{|x|+1}{\varepsilon}\right) \int_{[-1,1]^d} \phi(y) \, dy, \end{aligned} $$
[146] M. Doumic, M. Hoffmann, N. Krell, and L. Robert. Statistical estimation of a growth-fragmentation model observed on a genealogical tree. *Bernoulli*, 21(3):1760–1799, 2015.

[147] M. Doumic, J. Olayé, and M. Tomasevic. Ergodic behaviour of a multidimensional Markov jump branching process modelling telomere shortening in a non-compact setting. In preparation.

[148] M. Doumic, Jauffret and P. Gabriel. Eigenelements of a general aggregation-fragmentation model. *M³AS. Mathematical Models & Methods in Applied Sciences*, 20(5):757–783, 2010.

[149] R. J. Duffin. Lower bounds for eigenvalues. *Phys. Rev. (2)*, 71:827–828, 1947.

[150] G. Dumont and P. Gabriel. The mean-field equation of a leaky integrate-and-fire neural network: measure solutions and steady states. *Nonlinearity*, 33(12):6381–6420, 2020.

[151] J.-P. Eckmann and M. Hairer. Spectral properties of hypoelliptic operators. *Comm. Math. Phys.*, 235(2):233–253, 2003.

[152] K.-J. Engel and R. Nagel. *One-parameter semigroups for linear evolution equations*, volume 194 of *Graduate Texts in Mathematics*. Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt.

[153] M. Escobedo, S. Mischler, and M. Rodriguez Ricard. On self-similarity and stationary problem for fragmentation and coagulation models. *Ann. Inst. H. Poincaré Anal. Non Linéaire*, 22(1):99–125, 2005.

[154] J. Evans. Hypocoercivity in phi-entropy for the linear relaxation Boltzmann equation on the torus. *SIAM J. Math. Anal.*, 53(2):1357–1378, 2021.

[155] S. P. Eveson and R. D. Nussbaum. Applications of the Birkhoff-Hopf theorem to the spectral theory of positive linear operators. *Math. Proc. Cambridge Philos. Soc.*, 117(3):491–512, 1995.

[156] S. P. Eveson and R. D. Nussbaum. An elementary proof of the Birkhoff-Hopf theorem. *Math. Proc. Cambridge Philos. Soc.*, 117(1):31–55, 1995.

[157] G. Faber. Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt. Münch. Ber. 1923, 169-172 (1923)., 1923.

[158] W. Feller. On the integral equation of renewal theory. *Ann. Math. Statistics*, 12:243-267, 1941.

[159] W. Feller. *An introduction to probability theory and its applications. Vol. II*. John Wiley & Sons Inc., New York, 1966.

[160] E. Fischer. Über quadratische Formen mit reellen Koeffizienten. *Monatsh. Math. Phys.*, 16(1):234–249, 1905.

[161] W. H. Fleming. Equilibrium distributions of continuous polygenic traits. *SIAM J. Appl. Math.*, 36(1):148–168, 1979.

[162] R. Forien, J. Garnier, and F. Patout. Ancestral lineages in mutation selection equilibria with moving optimum. *Bulletin of Mathematical Biology*, 84(9):43, 2022. Id/No 93.

[163] E. Franco, O. Diekmann, and M. Gyllenberg. Modelling physiologically structured populations: renewal equations and partial differential equations. *J. Evol. Equ.*, 23(3):62, 2023. Id/No 46.

[164] I. Fredholm. Sur une classe d'équations fonctionnelles. *Acta Math.*, 27(1):365-390, 1903.

[165] K. O. Friedrichs. The identity of weak and strong extensions of differential operators. *Trans. Amer. Math. Soc.*, 55:132–151, 1944.

[166] K. O. Friedrichs. On the differentiability of the solutions of linear elliptic differential equations. *Comm. Pure Appl. Math.*, 6:299–326, 1953.

[167] G. Frobenius. Ueber matrizen aus nicht negativen elementen. *Sitzungsber. Königl. Preuss. Akad. Wiss.*, page 456–477, 1912.

[168] P. Gabriel. Measure solutions to the conservative renewal equation. In *CIMPA School on Mathematical Models in Biology and Medicine*, volume 62 of *ESAIM Proc. Surveys*, pages 68–78. EDP Sci., Les Ulis, 2018.

[169] P. Gabriel and H. Martin. Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation. *Ann. H. Lebesgue*, 5:275–301, 2022.

[170] E. Gagliardo. Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in $n$ variabili. *Rend. Sem. Mat. Univ. Padova*, 27:284–305, 1957.

[171] T. Gallay and C. E. Wayne. Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on $\mathbb{R}^2$. *Arch. Ration. Mech. Anal.*, 163(3):209–258, 2002.

[172] I. M. Gamba, V. Panferov, and C. Villani. On the Boltzmann equation for diffusively excited granular media. *Comm. Math. Phys.*, 246(3):503–541, 2004.

[173] L. Gårding. Dirichlet's problem for linear elliptic partial differential equations. *Math. Scand.*, 1:55–72, 1953.

[174] S. Gaubert and Z. Qu. Dobrushin's ergodicity coefficient for Markov operators on cones. *Integral Equations Operator Theory*, 81(1):127–150, 2015.

[175] L. Gearhart. Spectral theory for contraction semigroups on Hilbert space. *Trans. Amer. Math. Soc.*, 236:385–394, 1978.

[176] I. Gentil and C. Imbert. The Lévy-Fokker-Planck equation: $\Phi$-entropies and convergence to equilibrium. *Asymptot. Anal.*, 59(3-4):125–138, 2008.

[177] M. Gerlach. On the peripheral point spectrum and the asymptotic behavior of irreducible semigroups of Harris operators. *Positivity*, 17(3):875–898, 2013.

[178] M. Gerlach and J. Glück. On characteristics of the range of integral operators. May 2022. arxiv:2205.14397.

[179] D. Gilbarg and N. S. Trudinger. *Elliptic partial differential equations of second order*. Grundlehren der Mathematischen Wissenschaften, Vol. 224. Springer-Verlag, Berlin-New York, 1977.

[180] J. Glück and F. G. Martin. Uniform convergence of stochastic semigroups. *Israel Journal of Mathematics*, 247(1):1–19, 2022.
which yields (12.23), and so (12.22), with

$$ \psi_0(x) = \frac{T^d}{d!} \varepsilon^{-d} e^{-TW[|x|+d]} J^\otimes \left( \frac{|x|+1}{\varepsilon} \right) $$

and $g_0 = \mathbf{1}_{[-1,1]^d}$.

□

**Corollary 12.10.** For $\varepsilon \in (0, \varepsilon_0)$, there exists $f_1 \in X_+$ such that $\mathcal{L}f_1 = \lambda_1 f_1$ with $\langle f_1, \phi_1 \rangle = 1$. Moreover, the exponential convergence (12.20) holds for some constructive constants $C \ge 1$ and $\alpha > 0$.

*Proof of Corollary 12.10.* Similarly as in the proof of Theorem 12.1 part (2), we can infer from Lemma 12.9 that (6.9) holds with $g_R = C_R g_0$ where $C_R > 0$ is an explicit constant. The Lyapunov condition (6.8) is established in Lemma 12.8, and the positivity condition (6.10) readily follows from the estimate $\langle \phi_1, \varphi \rangle \ge 1 - \gamma$ established in Lemma 12.7. We can thus apply Theorem 6.4 which gives the conclusion. □

*Proof of Theorem 12.5.* It only remains to prove the uniqueness and strict positivity properties. Combining (12.22) and (12.22) with $\phi = g_0$, we get that

$$ S_{2T}f = S_T(S_Tf) \ge \langle f, \psi_0 \rangle S_T g_0 \ge \left( \int g_0^2 \right) \langle f, \psi_0 \rangle \psi_0 = 2^d \langle f, \psi_0 \rangle \psi_0. $$

for all $f \in X_+$. Since $\psi_0 > 0$, this ensures that (4.11) is verified, and then (H4) because of point (4) in Lemma 4.8. This gives the result of uniqueness and strict positivity by using Theorem 4.13. □

**Acknowledgments.** The authors are very grateful to Philippe Cieutat for the crucial discussions and references about the almost periodic functions, to Boyan Sirakov for having pointed out relevant references about the Krein-Rutman theorem for parabolic equations and to Otared Kavian for the enlightening discussions in many occasions. The authors would also like to warmly thank Jochen Glück for his relevant and useful comments, suggestions, and corrections on the first version of the paper. CFS acknowledge funding by the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 754362. PG has been supported by the ANR project NOLO (ANR-20-CE40-0015), funded by the French Ministry of Research. This work was partially funded by the European Union (ERC, SINGER, 101054787). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

## REFERENCES

[1] N. B. Abdallah. Weak solutions of the initial-boundary value problem for the Vlasov-Poisson system. *Math. Methods Appl. Sci.*, 17(6):451-476, 1994.

[2] V. I. Agoshkov. Problema sledov funktsií iz prostranstv $H_p^1(\Omega \times D)$ i issledovanie nekotorykh variatsionnykh kraevykh zadach perenosa, volume 85 of Preprint. Akad. Nauk SSSR, Otdel Vychisl. Mat., Moscow, 1985.

[3] V. I. Agoshkov. The existence of traces of functions in spaces used in problems of transport theory. Dokl. Akad. Nauk SSSR, 288(2):265-269, 1986.

[4] V. I. Agoshkov. Some trace and extension theorems for functions from $W_{x_i,p}^1(D)$ in the case of a bounded domain, volume 243 of Preprint. Akad. Nauk SSSR, Otdel Vychisl. Mat., Moscow, 1989.

[5] D. Albritton, S. Armstrong, J.-C. Mourrat, and M. Novack. Variational methods for the kinetic Fokker-Planck equation. *Anal. PDE*, 17(6):1953-2010, 2024.

[6] M. Alfaro, P. Gabriel, and O. Kavian. Confining integro-differential equations originating from evolutionary biology: ground states and long time dynamics. *Discrete Contin. Dyn. Syst. Ser. B*, 28(12):5905-5933, 2023.

[7] W. Alt. Orientation of cells migrating in a chemotactic gradient. In *Biological growth and spread (Proc. Conf., Heidelberg, 1979)*, volume 38 of *Lecture Notes in Biomath.*, pages 353-366. Springer, Berlin-New York, 1980.

[8] L. Ambrosio. Transport equation and Cauchy problem for BV vector fields. *Invent. Math.*, 158(2):227-260, 2004.

[9] L. Ambrosio and G. Crippa. Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields. In *Transport equations and multi-D hyperbolic conservation laws*, volume 5 of *Lect. Notes Unione Mat. Ital.*, pages 3-57. Springer, Berlin, 2008.

[10] L. Ambrosio, G. Crippa, and S. Maniglia. Traces and fine properties of a BD class of vector fields and applications. *Ann. Fac. Sci. Toulouse Math. (6)*, 14(4):527-561, 2005.
[181] F. Golse, C. Imbert, C. Mouhot, and A. F. Vasseur. Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation. *Ann. Sc. Norm. Super. Pisa Cl. Sci.* (5), 19(1):253–295, 2019.

[182] F. Golse, P.-L. Lions, B. Perthame, and R. Sentis. Regularity of the moments of the solution of a transport equation. *J. Funct. Anal.*, 76(1):110–125, 1988.

[183] F. Golse and B. Perthame. Generalized solutions of the radiative transfer equations in a singular case. *Comm. Math. Phys.*, 106(2):211–239, 1986.

[184] W. Greenberg, C. van der Mee, and V. Protopopescu. *Boundary value problems in abstract kinetic theory*, volume 23 of *Operator Theory: Advances and Applications*. Birkhäuser Verlag, Basel, 1987.

[185] G. Greiner. Zur Perron-Frobenius-Theorie stark stetiger Halbgruppen. *Math. Z.*, 177(3):401–423, 1981.

[186] G. Greiner. Spectral properties and asymptotic behavior of the linear transport equation. *Math. Z.*, 185(2):167–177, 1984.

[187] G. Greiner. A typical Perron-Frobenius theorem with applications to an age-dependent population equation. In *Infinite-dimensional systems (Retzhof, 1983)*, volume 1076 of *Lecture Notes in Math.*, pages 86–100. Springer, Berlin, 1984.

[188] G. Greiner and R. Nagel. Growth of cell populations via one-parameter semigroups of positive operators. In *Mathematics applied to science (New Orleans, La., 1986)*, pages 79–105. Academic Press, Boston, MA, 1988.

[189] G. Greiner, J. Voigt, and M. Wolff. On the spectral bound of the generator of semigroups of positive operators. *J. Operator Theory*, 5(2):245–256, 1981.

[190] M. P. Gualdani, S. Mischler, and C. Mouhot. Factorization of non-symmetric operators and exponential H-theorem. *Mém. Soc. Math. Fr. (N.S.)*, (153):137, 2017.

[191] J. Guerand. Quantitative regularity for parabolic De Giorgi classes. (2020) hal-02069086.

[192] J. Guerand and C. Mouhot. Quantitative De Giorgi methods in kinetic theory. *J. Éc. polytech. Math.*, 9:1159–1181, 2022.

[193] A. Guillin, B. Nectoux, and L. Wu. Quasi-stationary distribution for strongly feller markov processes by lyapunov functions and applications to hypoelliptic hamiltonian systems. hal-03068461 (2020), 2020.

[194] Y. Guo. Global weak solutions of the Vlasov-Maxwell system with boundary conditions. *Comm. Math. Phys.*, 154(2):245–263, 1993.

[195] Y. Guo. The Landau equation in a periodic box. *Comm. Math. Phys.*, 231(3):391–434, 2002.

[196] P. Gwiazda and B. Perthame. Invariants and exponential rate of convergence to steady state in the renewal equation. *Markov Process. Related Fields*, 12(2):413–424, 2006.

[197] M. Hairer. Convergence of Markov processes, January 2016. Unpublished lecture notes.

[198] M. Hairer and J. C. Mattingly. Yet another look at Harris’ ergodic theorem for Markov chains. In *Seminar on Stochastic Analysis, Random Fields and Applications VI*, volume 63 of *Progr. Probab.*, pages 109–117. Birkhäuser/Springer Basel AG, Basel, 2011.

[199] K. Hamdache. Initial-boundary value problems for the Boltzmann equation: global existence of weak solutions. *Arch. Rational Mech. Anal.*, 119(4):309–353, 1992.

[200] D. Han-Kwan and M. Léautaud. Geometric analysis of the linear Boltzmann equation I. Trend to equilibrium. *Ann. PDE*, 1(1):Art. 3, 84, 2015.

[201] A. Haraux. A simple almost-periodicity criterion and applications. *J. Differential Equations*, 66(1):51–61, 1987.

[202] T. E. Harris. The existence of stationary measures for certain Markov processes. In *Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. II*, pages 113–124. University of California Press, Berkeley and Los Angeles, Calif., 1956.

[203] M. Hauray and C. Le Bris. A new proof of the uniqueness of the flow for ordinary differential equations with $BV$ vector fields. *Ann. Mat. Pura Appl.* (4), 190(1):91–103, 2011.

[204] M. Hauray, C. Le Bris, and P.-L. Lions. Deux remarques sur les flots généralisés d’équations différentielles ordinaires. *C. R. Math. Acad. Sci. Paris*, 344(12):759–764, 2007.

[205] B. Helffer and F. Nier. *Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians*, volume 1862 of *Lecture Notes in Mathematics*. Springer-Verlag, Berlin, 2005.

[206] B. Helffer and J. Sjöstrand. Improving semigroup bounds with resolvent estimates. *Integral Equations Operator Theory*, 93(3):Paper No. 36, 41, 2021.

[207] B. Henry, S. Méléard, and V. C. Tran. Time reversal of spinal processes for linear and non-linear branching processes near stationarity. *Electron. J. Probab.*, 28:Paper No. 32, 27, 2023.

[208] F. Hérau. Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation. *Asymptot. Anal.*, 46(3-4):349–359, 2006.

[209] F. Hérau and F. Nier. Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential. *Arch. Ration. Mech. Anal.*, 171(2):151–218, 2004.

[210] F. Hérau and K. Pravda-Starov. Anisotropic hypoelliptic estimates for Landau-type operators. *J. Math. Pures Appl.* (9), 95(5):513–552, 2011.

[211] I. Herbst. The spectrum of Hilbert space semigroups. *J. Operator Theory*, 10(1):87–94, 1983.

[212] J. Hersch. Sur la fréquence fondamentale d'une membrane vibrante: évaluations par défaut et principe de maximum. *Z. Angew. Math. Phys.*, 11:387–413, 1960.

[213] D. P. Herzog and J. C. Mattingly. A practical criterion for positivity of transition densities. *Nonlinearity*, 28(8):2823–2845, 2015.

[214] T. Hillen, C. Rohde, and F. Lutscher. Existence of weak solutions for a hyperbolic model of chemosensitive movement. *J. Math. Anal. Appl.*, 260(1):173–199, 2001.