$\lambda \in \Sigma_L$ and so $\xi \in e^{t\Sigma_L}$. It thus only remains to prove that there exists $k \in \mathbb{Z}$ such that $g_k \neq 0$. Assume by contradiction that $g_k = 0$ for all $k \in \mathbb{Z}$. This means that the continuous and periodic complex-valued function $s \mapsto e^{-\lambda_0 s} \langle \phi, S_s f \rangle$ has all its Fourier coefficients equal to zero, which is not possible since this function is not equally zero (its value at $s = 0$ is $\langle \phi, f \rangle \neq 0$). $\square$

We next present a very classical result about the exponential stability of $f_1$ which is based on the quasi-compact semigroup framework of Voigt [354] (see also [15, B-IV-2] and [151, Sec. V.3]) and which is a more accurate version of Lemma 2.7 and Theorem 5.7.

**Theorem 5.28.** Let $(S_t)_{t \ge 0}$ be a positive irreducible semigroup on a Banach lattice $X$ satisfying the hypotheses of Lemma 2.7 and Theorem 5.7, in particular (H2) holds for a constant $\kappa_0 \in \mathbb{R}$ and there exists $T > 0$ such that the splitting

$$ (5.29) \qquad S_T = V_T + K_T, $$

holds with $\|V_T\|_{\mathcal{B}(X)} \le e^{\kappa T}$, $\kappa < \kappa_0$, and $K_T \in \mathcal{K}(X)$. Then there exists a unique solution $(\lambda_1, f_1, \phi_1)$ to the eigentriplet and the exponential stability (E31) holds (without constructive estimate).

**Remark 5.29.** In the splitting framework (5.26) the critical hypothesis $K_T \in \mathcal{K}(X)$ may be obtained by assuming that

$$ \|W(t)\|_{\mathcal{B}(X, \chi_1)} \lesssim e^{\kappa t}, \quad \forall t \ge 0, \quad \chi_1 \subset X \text{ compact.} $$

In fact, in many applications, we are also able to establish $\chi_1 \subset D(\mathcal{L}^\beta)$, for some $\beta > 0$, without too much more work.

Theorem 5.28 is in fact nothing but [41, Thm. 14.18] (see also [359, Sec. 2], [151, Thm. V.3.7] or [15, C-IV, Thm. 2.1 & Rk. 2.2]). We give however a short proof of Theorem 5.28 since it is simpler and more direct than the ones we usual find in the literature and in particular does not refer to subtle results about the spectrum and its essential part.

**Proof of Theorem 5.28.** First step. From Lemma 2.7, we already know that (H1), (H2) and (H3) hold. Together with the irreducibility which is nothing but (H4) from Lemma 4.8, we may apply Theorem 4.13 and conclude to the existence, uniqueness, strict positivity and algebraic simplicity result about the eigentriplet solution $(\lambda_1, f_1, \phi_1)$, with $\lambda_1 > \kappa$.

**Second step.** For any $\kappa' > \kappa$, the set $\Sigma(S_T) \cap \{z \in \mathbb{C}, |z| \ge e^{\kappa'T}\}$ is made of a finite number of isolated eigenvalues with finite algebraic multiplicity. Let us briefly present the argument and we refer to [330, 354] for more details. From (5.29), we readily have the identity

$$ (z - S_T)^{-1} = (z - V_T)^{-1} (I - K_T(z - V_T)^{-1})^{-1}, $$

as soon as the RHS term is well defined. On the one hand, the function $z \mapsto (z - V_T)^{-1}$ is holomorphic on $\{z \in \mathbb{C}; |z| > e^{\kappa t}\}$ with values in $\mathcal{B}(X)$ because of the boundedness assumption made on $V_T$, and thus $z \mapsto K_T(z - V_T)^{-1}$ is holomorphic with values in $\mathcal{K}(X)$. Using [330, Corollary II] (see also the proof of [354, Theorem 1.1]) we obtain that $z \mapsto (I - K_T(z - V_T)^{-1})^{-1}$ and thus $(z - S_T)^{-1} = (z - V_T)^{-1} (I - K_T(z - V_T)^{-1})^{-1}$ are degenerate-meromorphic functions. That means that $(z - S_T)^{-1}$ is a meromorphic function such that at any of its (isolated) poles, the coefficients of the (finite number of) singular terms in the corresponding Laurent series are finite rank operators. As a consequence, the Dunford integral on a small radius circle around a pole $\xi$ (which is equivalently a spectral value of $S_T$) defines a projection $\Pi_\xi$ with finite rank: in other words, $\Pi_\xi$ is the projection on the algebraic eigenspace associated to $\xi$. We have established that $\Sigma(S_T) \cap \{z \in \mathbb{C}, |z| > e^{\kappa T}\} \subset \Sigma_d(S_T)$, the set of isolated eigenvalues with finite algebraic multiplicity. Observing that $\Sigma(S_T) \cap \{z \in \mathbb{C}, |z| \ge e^{\kappa'T}\}$ is a compact subset of $\Sigma(S_T) \cap \{z \in \mathbb{C}, |z| > e^{\kappa T}\}$, since $S_T$ is a bounded operator, we readily conclude.

**Third step.** Since $S_T f_1 = e^{\lambda_1 T} f_1$, we have that $e^{\lambda_1 T} \in \Sigma(S_T)$. We now prove the existence of a spectral gap, namely the existence of $\epsilon > 0$ such that $\Sigma(S_T) \cap \{z \in \mathbb{C}, |z| \ge e^{(\lambda_1 - \epsilon)T}\} = \{e^{\lambda_1 T}\}$. Let us take $\kappa' \in (\kappa, \lambda_1)$. From the previous step, there must be some $\epsilon > 0$ such that

$$ \Sigma(S_T) \cap \{z \in \mathbb{C}, |z| \ge e^{(\lambda_1 - \epsilon)T}\} = \Sigma(S_T) \cap \{z \in \mathbb{C}, |z| \ge e^{\lambda_1 T}\}, $$
**Theorem 6.2.** Consider a semigroup S on a Banach lattice X such that its generator L enjoys the conclusion (C1) on the existence of the first eigentriplet $(\lambda_1, f_1, \phi_1)$. We assume furthermore the Doblin condition (6.2)-(6.4)-(6.3). Then the exponential stability (E3) in the norm $[\cdot]_{\psi_0}$ holds true, with constructive constants.

The proof closely follows the usual contraction argument in the Doblin result, see for instance [268], [168, Thm. 11] or [81, Thm. 2.1]. We do not explicitly assume the irreducibility of the semigroup, but the Doblin condition (6.2)-(6.4)-(6.3) is in many aspects a strong positivity condition. In particular, our result implies the uniqueness of the first eigentriplet $(\lambda_1, f_1, \phi_1)$ and the triviality of the boundary spectrum.

**Proof of Theorem 6.2.** The two conditions (6.2) and (6.4) together imply the modified Doblin condition

$$ \exists T > 0, \exists g_1 > 0, \forall f \ge 0, S_T f \ge g_1 \langle \phi_1, f \rangle, $$

with $g_1 := g_0/R_0$. Take $f$ such that $\langle \phi_1, f \rangle = 0$, so that $\langle \phi_1, f_{\pm} \rangle = r := \langle \phi_1, |f| \rangle/2 \ge 0$ and thus

$$ S_T f_{\pm} \ge g_1 \langle \phi_1, f_{\pm} \rangle = rg_1. $$

We write

$$ |S_T f| \le |S_T f_+ - rg_1| + |S_T f_- - rg_1| = S_T |f| - 2rg_1. $$

We deduce

$$ \langle \phi_1, |S_T f| \rangle \le \langle S_T^* \phi_1, |f| \rangle - 2r \langle \phi_1, g_1 \rangle = (e^{\lambda_1 T} - \langle \phi_1, g_1 \rangle) \langle \phi_1, |f| \rangle. $$

In other words, setting $\tilde{S}_t := e^{-\lambda_1 t} S_t$, we have

$$ [\tilde{S}_T f]_{\phi_1} \le \gamma[f]_{\phi_1}, $$

with $\gamma < 1$ which depends explicitly of $r_0$, $R_0$, $T$ and the estimates on $\lambda_1$. We then classically deduce the exponential convergence in the $[\cdot]_{\phi_1}$ norm. Now, the dual condition associated to the Doblin hypothesis (6.2) is

$$ \forall \psi \in X'_+, S_T^* \psi \ge \psi_0 \langle \psi, g_0 \rangle. $$

In particular, the first dual eigenvector $\phi_1$ satisfies

$$ (6.6) \qquad \phi_1 = e^{-\lambda_1 T} S_T^* \phi_1 \ge e^{-\lambda_1 T} \psi_0 \langle \phi_1, g_0 \rangle = e^{-\lambda_1 T} r_0 \psi_0. $$

Together with condition in (6.4), we see that $[\cdot]_{\phi_1}$ and $[\cdot]_{\psi_0}$ are equivalent norm, and we immediately obtain the exponential convergence in the $[\cdot]_{\psi_0}$ norm (with constructive constants). $\square$

**6.3. Asymptotic stability under Harris condition.** The Doblin condition (6.2)-(6.4)-(6.3) is too much demanding for many applications. In this section, we make the following somehow more general Harris type condition complemented with a Lyapunov condition. More precisely, we assume that there exists $T > 0$ such that $\tilde{S}_T := S_T e^{-\lambda_1 T}$ first satisfies the Lyapunov condition

$$ (6.7) \qquad \|\tilde{S}_T f\| \le \gamma_L \|f\| + K[f]_{\phi_1}, $$

with $\gamma_L \in (0, 1)$, $K \ge 0$. We next assume that $\tilde{S}_T$ satisfies the Harris condition

$$ (6.8) \qquad \begin{cases} \exists A > K/(1 - \gamma_L), \exists g_A > 0 \text{ such that} \\ \forall f \ge 0, \|f\| \le A[f]_{\phi_1} \text{ there holds } S_T f \ge g_A[f]_{\phi_1}. \end{cases} $$

We finally replace the positivity condition (6.3) by

$$ (6.9) \qquad \exists r_A > 0, \langle \phi_1, g_A \rangle \ge r_A. $$

As we have seen several times, condition (6.7) is some kind of regularity hypothesis which is natural under a splitting structure on the semigroup $S_L$. We emphasize that conditions (6.7)-(6.8)-(6.9) slightly generalize the usual set of hypothesizes for the Harris theorem, see for instance [81, Sect. 3]. We also point out that there is a connection between the condition (6.8) and the notion of *partial integral* or *partial kernel* operators, see for instance [178, Cor. 5.3]. The long term convergence of semigroups that contain a partially integral operator was studied in particular in [317, 177, 180].
and thus

$$ \begin{aligned} \Sigma(S_T) \cap \{z \in \mathbb{C}, |z| \ge e^{(\lambda_1 - \epsilon)T}\} &= \Sigma_P(S_T) \cap \{z \in \mathbb{C}, |z| \ge e^{\lambda_1 T}\} \\ &= e^{T(\Sigma_P(\mathcal{L}) \cap \{\lambda \in \mathbb{C}, \Re\lambda \ge \lambda_1\})} = e^{T\Sigma_P^+(\mathcal{L})}, \end{aligned} $$

where we have used the previous step again for the first equality, Lemma 5.27 for the second equality, and the definition of the boundary point spectrum $\Sigma_P^+(\mathcal{L})$ in the last one. We next claim that $\Sigma_P^+(\mathcal{L})$ is finite. Otherwise, it would exist a infinite family $\{(\lambda_k, f_k)\}$ of couples of eigenvalue-eingenvector of $\mathcal{L}$ with $\lambda_k \in \Sigma_P^+(\mathcal{L})$ and thanks to Lemma 5.27 a corresponding infinite family $\{(e^{T\lambda_k}, f_k)\}$ of couples of eigenvalue-eingenvector of $S_T$ with $|e^{T\lambda_k}| \ge e^{T\lambda_1}$, what is in contradiction with the conclusion of the second step. Since the translated boundary point spectrum $\Sigma_P^+(\mathcal{L}) - \lambda_1$ is a group from Theorem 5.5, we infer that the only possibility is $\Sigma_P^+(\mathcal{L}) = \{\lambda_1\}$, which finally gives $\Sigma(S_T) \cap \{z \in \mathbb{C}, |z| \ge e^{(\lambda_1 - \epsilon)T}\} = \{e^{\lambda_1 T}\}$.

**Fourth step.** We prove that $e^{\lambda_1 T}$ is (geometrically) simple with eigenspace spanned by $f_1$. Let $f_2$ be an eigenvector of $S_T$ with corresponding eigenvalue $e^{\lambda_1 T}$. Since $\langle \phi_1, f_1 \rangle \ne 0$, we can find $\alpha \in \mathbb{C}$ such that $f_3 := f_2 - \alpha f_1$ verifies $\langle \phi_1, \int_0^T e^{-\lambda_1 s} S_s f_3 ds \rangle = T(\langle \phi_1, f_2 \rangle - \alpha \langle \phi_1, f_1 \rangle) = 0$. We now prove that $f_3 = 0$, which will give the conclusion. Assume by contradiction that $f_3 \ne 0$. Then it is an eigenvector of $S_T$ with corresponding eigenvalue $e^{\lambda_1 T}$, as a linear combination of $f_1$ and $f_2$. By virtue of Lemma 5.27, there exists $\lambda \in \Lambda_T(e^{\lambda_1 T}) = \{\lambda_1\} + \frac{2i\pi}{T}\mathbb{Z}$ such that $\int_0^t e^{-\lambda s} S_s f_3 ds$ is an eigenvector of $\mathcal{L}$ with corresponding eigenvalue $\lambda$. We thus have

$$ \lambda \in \Sigma_P(\mathcal{L}) \cap \Lambda_T(e^{\lambda_1 T}) \subset \Sigma_P(\mathcal{L}) \cap (\{\lambda_1\} + i\mathbb{R}) = \Sigma_P^+(\mathcal{L}) = \{\lambda_1\}, $$

where the last equality comes from the third step of the proof, and $g := \int_0^t e^{-\lambda_1 s} S_s f_3 ds$ is an eigenvector of $\mathcal{L}$ with corresponding eigenvalue $\lambda_1$. Since we know from the first step of the proof that $\lambda_1$ is simple, we must have $g \in \text{Span}(f_1) \setminus \{0\}$, which is not possible since $\langle \phi_1, g \rangle = 0$ by construction of $f_3$. We have obtained a contradiction, from which we get that $f_3 = 0$, $f_2 = \alpha f_1$.

**Fifth step.** We conclude to the exponential stability (E31). We have proved in the two previous steps that $\Sigma(S_T) \cap \{z \in \mathbb{C}, |z| \ge e^{(\lambda_1 - \epsilon)T}\} = \{e^{\lambda_1 T}\}$ and $e^{\lambda_1 T}$ is simple with eigenspace spanned by $f_1$. The restriction $S_T^\perp$ of $S_T$ to the invariant subspace $X_\perp := \{f \in X, \langle \phi_1, f \rangle = 0\}$, which does not contain $f_1$, thus has a spectral radius smaller than $e^{(\lambda_1 - \epsilon)T}$. The spectral radius formula (see [333, Thm. 10.13] for instance) then ensures that

$$ \lim_{n \to \infty} \|S_{nT}^\perp\|^{1/n} = r(S_T^\perp) \le e^{(\lambda_1 - \epsilon)T}. $$

This guarantees, for any $\eta \in (0, \epsilon)$, the existence of a constant $C_\eta > 0$ such that for all $f \in X_\perp$ and all $t \ge 0$

$$ \|e^{-\lambda_1 t} S_t f\| \le C_\eta e^{-\eta t} \|f\|, $$

and the proof is complete.

□

Let us now present a variant of another classical result known as the Gearhart-Prüss Theorem in [174, 325], see also the contributions of Herbst [210] and Greiner [15, A-III.7] as well as the more constructive proof [151, Thm. V.1.11] and recently [205] based on techniques developed in or related to [363, 61].

**Theorem 5.30.** Consider a positive semigroup $S$ on a Banach lattice $X$ such that its generator $\mathcal{L}$ satisfies the conclusions (C2) about the existence, positivity and uniqueness of the first eigentriplet $(\lambda_1, f_1, \phi_1)$. We assume furthermore that $X$ is an Hilbert space and that there exist $\kappa < \lambda_1$ and $R > 0$ such that

$$ (i) \sup_{z \in \Delta_\kappa \setminus B_R} \|\mathcal{R}_\mathcal{L}(z)\|_{\mathcal{B}(X)} < \infty; $$

$$ (ii) \Sigma(\mathcal{L}) \cap \Delta_\kappa \subset \Sigma_d(\mathcal{L}) \cap B_R. $$

Then the exponential stability (E31) holds (without constructive estimate).

**Proof of Theorem 5.30.** The spectral information (C2) and (ii) together imply (C3) (because of Theorem 5.5) and that there exists $\kappa^* \in (\kappa, \lambda_1)$, such that $\Sigma(\mathcal{L}) \cap \Delta_{\kappa^*} = \{\lambda_1\}$. The operator $\mathcal{L}$ on $X_0 := (\text{vect}\{f_1\})^\perp$ thus satisfies $\sup_{z \in \Delta_{\kappa^*}} \|\mathcal{R}_\mathcal{L}(z)\|_{\mathcal{B}(X_0)} < \infty$, and we conclude thanks to [151, Thm. V.1.11]. The lack of constructively here only comes from the fact that our assumptions do not provide any information on the spectral gap $\lambda_1 - \kappa > 0$. □
**Theorem 6.3.** Consider a semigroup $S$ on a Banach lattice $X$ such that its generator $\mathcal{L}$ enjoys the conclusions (C1) on the existence of the first eigentriplet $(\lambda_1, f_1, \phi_1)$. We assume furthermore the Harris condition (6.8) together with the Lyapunov condition (6.7) and the positivity condition (6.9). Then the exponential stability (E3$_1$) in the norm of $X$ holds true, with constructive constants.

Of course, in order that Theorem 6.3 really gives a constructive convergence result, we have to establish (6.8), (6.7) and (6.9) in a constructive way.

*Proof of Theorem 6.3.* On the one hand, we have

$$ (6.10) \qquad [\tilde{S}_T f]_{\phi_1} \le \langle \tilde{S}_T |f|, \phi_1 \rangle = \langle |f|, \tilde{S}_T^* \phi_1 \rangle = [f]_{\phi_1}. $$

On the other hand, we wish to establish the coupling property

$$ (6.11) \qquad [\tilde{S}_T f]_{\phi_1} \le \gamma_H[f]_{\phi_1} \quad \text{if} \quad \|f\| \le A'[f]_{\phi_1} \text{ and } \langle f, \phi_1 \rangle = 0, $$

for some $\gamma_H \in (0,1)$ and with $A' := A/2$. We thus consider $f \in X$, such that $\langle f, \phi_1 \rangle = 0$ and $\|f\| \le A'[f]_{\phi_1}$, so that

$$ \|f_{\pm}\| \le \|f\| \le A'[f]_{\phi_1} = A[f_{\pm}]_{\phi_1}. $$

Using the Harris condition (6.8), we deduce

$$ \tilde{S}_T f_{\pm} \ge \vartheta g_A, \quad \vartheta := \frac{1}{2} e^{-\lambda_1 T} [f]_{\phi_1}. $$

Similarly as in the proof of Theorem 6.2, we next compute

$$ |\tilde{S}_T f| \le |\tilde{S}_T f_+ - \vartheta g_A| + |\tilde{S}_T f_- - \vartheta g_A| \le \tilde{S}_T |f| - 2\vartheta g_A $$

and then

$$ \begin{aligned} [\tilde{S}_T f]_{\phi_1} &\le \langle \tilde{S}_T |f| - 2\vartheta g_A, \phi_1 \rangle \\ &= \langle |f|, \tilde{S}_T^* \phi_1 \rangle - 2\vartheta \langle g_A, \phi_1 \rangle \\ &= (1 - e^{-\lambda_1 T} \langle g_A, \phi_1 \rangle) [f]_{\phi_1}, \end{aligned} $$

which in turn implies (6.11) with $\gamma_H := 1 - e^{-\kappa_0 T} r_A$.

Now, the two estimates (6.10) and (6.11) together give

$$ (6.12) \qquad [\tilde{S}_T f]_{\phi_1} \le \gamma_H[f]_{\phi_1} + \frac{1 - \gamma_H}{A'} \|f\|. $$

From (6.12) and the Lyapunov condition (6.7), we deduce that

$$ U^{n+1} = M U^n $$

with

$$ U^n := \begin{pmatrix} \|\tilde{S}_T^n f\| \\ [\tilde{S}_T^n f]_{\phi_1} \end{pmatrix} = \begin{pmatrix} \|\tilde{S}_{nT} f\| \\ [\tilde{S}_{nT} f]_{\phi_1} \end{pmatrix} \quad \text{and} \quad M := \begin{pmatrix} \gamma_L & K \\ \frac{1-\gamma_H}{A} & \gamma_H \end{pmatrix}. $$

The eigenvalues of $M$ are

$$ \mu_{\pm} := \frac{1}{2} (T \pm \sqrt{T^2 - 4D}), $$

with

$$ T := \operatorname{tr} M = \gamma_L + \gamma_H, \quad D := \operatorname{det} M = \gamma_L \gamma_H - (1 - \gamma_H) \frac{K}{A}. $$

We observe that

$$ \gamma_L \gamma_H > D > \gamma_L \gamma_H - (1 - \gamma_H)(1 - \gamma_L) = T - 1, $$

so that

$$ (\gamma_H - \gamma_L)^2 = T^2 - 4\gamma_L\gamma_H < T^2 - 4D < T^2 - 4(T-1) = (T-2)^2 $$

and finally

$$ \alpha := \max(|\mu_+|, |\mu_-|) < \max(\gamma_H, \gamma_L, |T-1|, 1) = 1. $$

We conclude that $\|M^n\| \lesssim \alpha^n$, from what we immediately conclude. $\square$
**Remark 5.31.** Except of the Hilbert space framework, the assumptions made in Theorem 5.30 are slightly weaker than those of Theorem 5.28, and are indeed established during the proof of Theorem 5.28: such an information at the level of the resolvent is a bit easier to establish than a similar estimate at the level of the semigroup. In the splitting framework (5.26) and its resolvent counterpart (2.21), we typically only have to show

$$ (5.30) \quad \sup_{\kappa \le \Re \mathrm{ez} \le \kappa_1} \|\mathcal{V}(z)\|_{\mathcal{B}(X)} < \infty, \quad \lim_{r \to \infty} \sup_{\kappa \le \Re \mathrm{ez} \le \kappa_1, |\Im mz| \ge r} \|\mathcal{W}(z)\|_{\mathcal{B}(X)} = 0, $$

for some $\kappa < \lambda_1$, and $\mathcal{W}(z) \in \mathcal{K}(X)$ for any $z \in \Delta_\kappa$. That last claim is classical (see for instance [189]) and we only briefly sketch the proof. On the one hand, from the first and the last estimates, we deduce that $\Sigma(\mathcal{L}) \cap \Delta_\kappa \subset \Sigma_d(\mathcal{L})$ thanks to Theorem 5.6. As in the proof of Theorem 5.6 and with the usual notations, we also have

$$ (I - \mathcal{W}(z))\mathcal{R}_{\mathcal{L}}(z) = \mathcal{V}(z), \quad \forall z \in \Delta_{\kappa}, $$

where $I - \mathcal{W}(z)$ is invertible and $\|(I - \mathcal{W}(z))^{-1}\|_{\mathcal{B}(X)} \le 2$ for any $z \in \mathbb{C}$ such that $\kappa \le \Re \mathrm{ez} \le \kappa_1$, $|\Im mz| \ge R$ and $R$ is large enough. We immediately deduce that the condition (i) in Theorem 5.30 holds.

We end this section by a more recent result which is similar to the Gearhart-Prüss Theorem but is not restricted to an Hilbert space.

**Theorem 5.32.** Consider a positive semigroup $S$ on a Banach lattice $X$ such that its generator $\mathcal{L}$ satisfies the conclusions (C2) about the existence, positivity and uniqueness of the first eigentriplet $(\lambda_1, f_1, \phi_1)$. We further assume that $\mathcal{L} = \mathcal{A} + \mathcal{B}$ with the associated operators $V$ and $W$ defined by (3.13) satisfy (5.26) for some $\kappa < \lambda_1$ and that the resolvent counterpart $\mathcal{W}$ defined by (2.21) satisfies (5.30) and more precisely

$$ \sup_{\kappa \le \Re \mathrm{ez} \le \kappa_1} \|\langle z \rangle^\alpha \mathcal{W}(z)\|_{\mathcal{B}(X)} < \infty, $$

with $\alpha > 1$. Then the exponential stability (E3$_1$) holds (without constructive estimate).

The proof of Theorem 5.32 is a mere adaptation of [277, Thm. 3.1] (see also [272]) and it is thus skipped. The needed estimates are a bit stronger than those of Remark 5.31, but in the applications, they are not really more demanding. They also hold at the level of the resolvent instead of what is assumed in the statement of Theorem 5.28.

We conclude by emphasizing again on the fact that all the above results are not constructive. We propose in the next part an alternative approach which is constructive.

Let us emphasize first that (C2) (which can be established using the material of Sections 2, 3 and 4) together with Theorem 5.16, Theorem 5.18 or Theorem 5.20-(1) imply the triviality of the principal point spectrum $\Sigma_P^1(\mathcal{L}) = \{\lambda_1\}$, what is a weak version of the answer (S3$_2$) about the geometry of the spectrum problem. With the additional property $\Sigma_P^1(\mathcal{L}) = \Sigma_P^+(\mathcal{L})$ (which can be established directly using the material of Section 2 or, for a positive semigroup, using the material of Sections 3 and 4, see in particular the discussion at the end of Section 4), we conclude to (S3$_2$). When furthermore $\mathcal{L}$ is also the generator of a positive semigroup, Theorem 5.21, Corollary 5.22 or Theorem 5.23 implies the ergodicity property (E2) in several topologies. Altogether, in that way, we thus complete the proof of Theorem 1.1-(3). Alternatively, we may use Theorem 5.28 or (C2) together with Theorem 5.30 or Theorem 5.32 in order to obtain the exponential stability (E3$_1$), and thus also the spectral gap property (S3$_3$). In particular, these arguments provide a proof of Theorem 1.1-(4).

## 6. QUANTITATIVE STABILITY

In this section we establish some quantitative stability results in the spirit of the Doblin, Harris, Meyn-Tweedie theory for Markov semigroup, and in particular the geometric part (S3$_2$) and the evolution part (E3$_1$) of the Krein-Rutman theorem in a quantitative way. We do not use the previous positivity and other structure conditions (Hi), (Xi), (HSi), but we rather start with the condition (C1) and introduce the additional necessary conditions, which are somehow stronger: they are mainly formulated in term of the semigroup and are always quantitative.
**Remark 6.4.** It is useful to emphasize that the existence of $f_1$ is not required in the proof of Theorem 6.3 for proving that $\|M^n\| \lesssim \alpha^n$, and this estimate can actually be used to derive the existence of $f_1$. In order to prove that last claim, we first observe that Theorem 6.3 ensures that $(\|\tilde{S}_{nT}f_0\|)_n$ is a Cauchy sequence for any $f_0 \in X$. Indeed, for any $p \in \mathbb{N}$, $f = f_0 - \tilde{S}_{pT}f_0$ verifies

$$ \langle f, \phi_1 \rangle = \langle f_0, \phi_1 \rangle - \langle f_0, \tilde{S}_{pT}^* \phi_1 \rangle = \langle f_0, \phi_1 \rangle - \langle f_0, \phi_1 \rangle = 0, $$

and we then have

$$ \|\tilde{S}_{nT}f_0 - \tilde{S}_{(n+p)T}f_0\| + [\tilde{S}_{nT}f_0 - \tilde{S}_{(n+p)T}f_0]_{\phi_1} \lesssim \alpha^n (\|f_0 - \tilde{S}_{pT}f_0\| + [f_0 - \tilde{S}_{pT}f_0]_{\phi_1}). $$

Choosing $f_0 \in X_+$ such that $[f_0]_{\phi_1} = 1$, we deduce that $(\tilde{S}_{nT}f)$ converges to a fixed point $f_1$ of $\tilde{S}_T$, which is not zero because

$$ [f_1]_{\phi_1} = \lim[\tilde{S}_{nT}f_0]_{\phi_1} = [f_0]_{\phi_1} = 1, $$

and $f_1$ is the unique fixed point with normalization $[f_1]_{\phi_1} = 1$. Besides, $f_1 \in X_+$ because of the positivity of $S$ and $f_0$. This ensures that

$$ [\tilde{S}_t f_1]_{\phi_1} = \langle \tilde{S}_t f_1, \phi_1 \rangle = \langle f_1, \tilde{S}_t^* \phi_1 \rangle = \langle f_1, \phi_1 \rangle = 1, $$

for any $t > 0$. Since on the other hand

$$ \tilde{S}_T \tilde{S}_t f_1 = \tilde{S}_{t+T} f_1 = \tilde{S}_t \tilde{S}_T f_1 = \tilde{S}_t f_1, $$

we deduce from the uniqueness of the fixed point that $\tilde{S}_t f_1 = f_1$, which yields that $f_1 \in D(\mathcal{L})$ and $\mathcal{L}f_1 = \lambda_1 f_1$.

**6.4. Quantified isolation of the first eigenvalue.** In terms of the geometry of the spectrum, an immediate consequence of Theorem 6.3 is that the conditions (6.8), (6.7) and (6.9) ensure the existence of a spectral gap, namely the existence of $\varepsilon > 0$ such that

$$ \Sigma(\mathcal{L}) \cap \Delta_{\lambda_1 - \varepsilon} = \{\lambda_1\}. $$

We still assume that the Lyapunov condition (6.7) holds for some $T > 0$, $\gamma_L \in (0, 1)$ and $K \ge 0$, but we relax (6.8) into the time-averaged condition

$$ (6.13) \qquad \left\{ \begin{array}{l} \exists A > K/(1 - \gamma_L), \exists g_A > 0 \text{ such that} \\ \forall f \ge 0, \|f\| \le A[f]_{\phi_1} \text{ there holds } \int_0^T S_t f dt \ge g_A[f]_{\phi_1}. \end{array} \right. $$

It is worth emphasizing that (6.13) does not imply anymore the existence of a spectral gap, and there can be a non-trivial boundary spectrum, see Section 9.2 for an example. However, it is strong enough for guaranteeing that $\lambda_1$ is isolated from the rest of the spectrum, in the sense that

$$ (6.14) \qquad \Sigma(\mathcal{L}) \cap B(\lambda_1, \varepsilon) = \{\lambda_1\}, $$

for some $\varepsilon > 0$. In particular, if not trivial, the boundary spectrum must be discrete from Theorem 5.5 (under the additional assumptions listed in the statement of this last result).

**Theorem 6.5.** Consider a semigroup $S$ on a Banach lattice $X$ such that its generator $\mathcal{L}$ enjoys the conclusions (C1) on the existence of the first eigentriplet $(\lambda_1, f_1, \phi_1)$. We assume furthermore the time-averaged Harris condition (6.13) together with the Lyapunov condition (6.7) and the positivity condition (6.9). Then (6.14) holds true for some constructive constant $\varepsilon > 0$.

*Proof.* First, we readily deduce from (6.13) and the inversion formula (2.13) that

$$ (6.15) \qquad \left\{ \begin{array}{l} \exists A > K/(1 - \gamma_L), \exists g_A > 0 \text{ such that} \\ \forall f \ge 0, \|f\| \le A[f]_{\phi_1} \text{ there holds } \tilde{\mathcal{R}}(\lambda)f \ge \tilde{g}_A[f]_{\phi_1}, \forall \lambda > \lambda_1, \end{array} \right. $$

where $\tilde{\mathcal{R}}(\lambda) := (\lambda - \lambda_1)\mathcal{R}_\mathcal{L}(\lambda)$ and $\tilde{g}_A := (\lambda - \lambda_1)e^{-\lambda T}g_A$. It is worth emphasizing that $\tilde{\mathcal{R}}(\lambda)f_1 = f_1$ and $1 \in \Sigma(\tilde{\mathcal{R}}(\lambda) \subset \bar{B}(0, 1)$. Next, we claim that the Lyapunov condition (6.7) ensures the existence of $\lambda > \lambda_1$ such that

$$ (6.16) \qquad \|\tilde{\mathcal{R}}(\lambda)f\| \le \gamma'_L \|f\| + K'[f]_{\phi_1} $$

for all $f \in X$ and some $\gamma'_L < 1$ and $K' > 0$. Indeed, by iteration of (6.7), we have

$$ \|\tilde{S}_{nT}f\| \le \gamma_L^n \|f\| + \frac{K}{1 - \gamma_L} [f]_{\phi_1}, $$
**6.1. About quantified positivity conditions.** We briefly discuss some positivity conditions related to the strong maximum principle and barriers techniques. The issue is about how to quantify the strong maximum principle

$$f \in X_+ \setminus \{0\}, (\kappa_1 - \mathcal{L})f \geq 0 \text{ imply } f > 0 \text{ or } f \gg 0$$

or the related strong positivity of the associated semigroup. A possible way can be achieved with the help of a barrier functions family $\mathcal{G} \subset X_+$ and a second weaker (semi)norm [·] used for normalization. Let us then introduce the two conditions

$$(6.1) \qquad \forall R > 0, \exists g_i \in \mathcal{G}, \forall f \in X_+, [f] = 1, \|f\| \leq R,$$

we have

(i) $S_T f \geq g_1$ (for some $T > 0$)

or

(ii) $f \geq g_2$ if $(\kappa_1 - \mathcal{L})f \geq 0$.

Point (ii) is a quantified version of the strong maximum principle when $\mathcal{G} \subset X_{++}$ and it is always a consequence of the positivity condition (i). Assume indeed that (i) holds (for some $T > 0$) and that $f$ satisfies the requirements (6.1) and $(\kappa_1 - \mathcal{L})f \geq 0$. We then write

$$\frac{d}{dt}(e^{(\mathcal{L}-\kappa_1)t}f) = e^{(\mathcal{L}-\kappa_1)t}(\mathcal{L}-\kappa_1)f \leq 0,$$

so that

$$f \geq e^{(\mathcal{L}-\kappa_1)T}f = e^{-\kappa_1 T}S_T f \geq e^{-\kappa_1 T}g_2 =: g_1,$$

with $g_2$ given by condition (i). The reciprocal implication is not clear, see however Lemma 4.8-(3).

Let us now make a list of possible quantified positivity conditions of Doblin-Harris type for a linear (and continuous) operator $A: X \to X$:

(P1') $\exists g_0 \in X_+ \setminus \{0\}, \exists \psi_0 \in X_+ \setminus \{0\}, \forall f \in X_+, Af \geq g_0 \langle f, \psi_0 \rangle$;

(P2') $\exists g_0 \in X_+ \setminus \{0\}, \exists \psi_0 \in X'_+, \forall f \in X_+, Af \geq g_0 \langle f, \psi_0 \rangle$;

(P3') $\exists g_0 \in X_{++}, \exists \psi_0 \in X'_+ \setminus \{0\}, \forall f \in X_+, Af \geq g_0 \langle f, \psi_0 \rangle$;

(P4') $\exists g_0 \in X_{++}, \exists \psi_0 \in X'_+, \forall f \in X_+, Af \geq g_0 \langle f, \psi_0 \rangle$.

We summarize some elementary relations between these conditions and those listed in Section 4.2.

**Lemma 6.1.** We have $(P2') \Rightarrow (P2) \Rightarrow (P1), (P3') \Rightarrow (P3) \Rightarrow (P1), (P4') \Rightarrow ((P4), (P3'))$, as well as $(P4) \Rightarrow ((P3'), (P2))$.

We also have: A satisfies $(P2')$ iff $A^*$ satisfies $(P3')$; A satisfies $(P3')$ iff $A^*$ satisfies $(P2')$; A satisfies $(P4')$ iff $A^*$ satisfies $(P4')$.

We finally have: A satisfies $(P2')$ implies $\exists g_0 \in X_+ \setminus \{0\}, \exists \kappa > 0, A g_0 \geq \kappa g_0$

**Proof of Lemma 6.1.** We assume $Af \geq g_0 \langle f, \psi_0 \rangle$ for any $f \in X_+$ and some $g_0 \in X_+, \psi_0 \in X'_+$. For any $\phi \in X' \setminus \{0\}$ and $f \in X_+$, we have

$$\langle A^* \phi, f \rangle = \langle \phi, Af \rangle \geq \langle \phi, g_0 \langle f, \psi_0 \rangle \rangle,$$

which implies $A^* \phi \geq \psi_0 \langle \phi, g_0 \rangle$. We thus deduce that A satisfies $(P2')$ (resp. $(P3')$, (P4)) implies that $A^*$ satisfies $(P3')$ (resp. $(P2')$, (P4)). The other implications can be established in a similar or even simpler way. $\square$

We conclude this introductory section by emphasizing on the fact (as already mentioned above) that $S_\mathcal{L}$ satisfies $(Pi')$ implies $\mathcal{R}_\mathcal{L}(\lambda)$ satisfies $(Pi')$ for any $\lambda \geq \lambda_1$ and $i = 1, \dots, 4$.

**6.2. Asymptotic stability under Doblin condition.** We start with a simple situation. We assume the Doblin condition, namely

$$(6.2) \qquad \exists T > 0, \exists \psi_0 \gg 0, \exists g_0 > 0, \forall f \geq 0, S_T f \geq g_0 \langle \psi_0, f \rangle,$$

together with the companion positivity condition

$$(6.3) \qquad \exists r_0 > 0, \langle \phi_1, g_0 \rangle \geq r_0,$$

as well as the strong additional boundedness assumption

$$(6.4) \qquad \exists R_0 > 0, \phi_1 \leq R_0 \psi_0.$$
for all integer $n$, from which we deduce

$$ \|\tilde{S}_t f\| \le C \gamma_L^{[t/T]} \|f\| + \frac{CK}{1 - \gamma_L} [f]_{\phi_1}, $$

for all $t \ge 0$ and where $C = \sup_{0 \le t \le T} \|\tilde{S}_t\|$. We finally infer from the inversion formula (2.13) that

$$ \|\tilde{\mathcal{R}}(\lambda)f\| \le \frac{C_1(\lambda - \lambda_1)}{\lambda - \lambda_1 + \log \frac{1}{\gamma_L}} \|f\| + \frac{C_2}{1 - \gamma_L} [f]_{\phi_1}, $$

for all $\lambda > \lambda_1$ and some $C_1, C_2 > 0$. Then we only need to choose $\lambda$ close enough to $\lambda_1$ so that $\frac{C_1(\lambda - \lambda_1)}{\lambda - \lambda_1 + \log \frac{1}{\gamma_L}} < 1$ and we obtain (6.16).

We have proved that $\tilde{\mathcal{R}}(\lambda)$ satisfies (6.16) and (6.15). Together with the positivity condition (6.9), we can thus repeat the proof of Theorem 6.3 for the operator $\tilde{\mathcal{R}}$ instead of $\tilde{S}$ and we obtain the existence of constructive constants $\alpha \in (0, 1)$ and $C \ge 1$ such that

$$ \|\tilde{\mathcal{R}}(\lambda)^n f\| \le C \alpha^n \|f\|, \quad \forall n \ge 1, $$

for any $f \in X$, $\langle f, \phi_1 \rangle = 0$. By the spectral radius formula, we deduce

$$ \Sigma(\tilde{\mathcal{R}}(\lambda)) \cap \{z \in \mathbb{C}, |z| > \alpha\} = \{1\}. $$

The spectral mapping theorem for the resolvent, which ensures that

$$ \Sigma(\tilde{\mathcal{R}}(\lambda)) \setminus \{0\} = \frac{\lambda - \lambda_1}{\lambda - \Sigma(\mathcal{L})}, $$

then yields (6.14) with $\varepsilon = (\alpha^{-1} - 1)(\lambda - \lambda_1)$. $\square$

<figure><img src="image_4.png" /></figure>

**6.5. The weak dissipativity case.** In this section, we consider a weak dissipative semigroup $(S_t)$ as considered in Section 3.3 and in a sense we make precise now. We consider four Banach lattices $X_3 \subset X_2 \subset X_1 \subset X_0 = X$. We first make the same kind of Harris type condition as in the previous section, namely

**Hypothesis (H)** (Doblin-Harris) condition (6.8) holds for the same time $T > 0$ and for both norms $\| \cdot \| = \| \cdot \|_{X_0}$ and $\| \cdot \| = \| \cdot \|_{X_2}$ as well as the companion positivity condition (6.9) holds. Instead of the strong Lyapunov condition (6.7), we assume

**Hypothesis (L)** (weak Lyapunov) there exist a constant $K \ge 0$ such that

$$ \|\tilde{S}f\|_1 + \|\tilde{S}f\|_0 \le \|f\|_1 + K[f]_{\phi_1}, \quad \forall f \in X_1, $$

$$ \|\tilde{S}f\|_3 + \|\tilde{S}f\|_2 \le \|f\|_3 + K[f]_{\phi_1}, \quad \forall f \in X_3, $$

with $\tilde{S} = S_T e^{-\lambda_1 T}$.

**Hypothesis (I)** (interpolation) there exists an increasing function $\xi : \mathbb{R}_+ \to \mathbb{R}_+$, $\lambda \mapsto \xi_\lambda$, such that

$$ \lambda \|f\|_1 \le \|f\|_0 + \xi_\lambda \|f\|_3, \quad \forall \lambda > 0, \quad \xi_\lambda/\lambda \to 0 \text{ as } \lambda \to 0. $$

**Theorem 6.6.** Consider a semigroup $S$ on a Banach lattice $X$ such that its generator $\mathcal{L}$ enjoys the conclusions (C1) on the existence of the first eigentriplet $(\lambda_1, f_1, \phi_1)$. We assume furthermore the three above conditions of weak confinement (L), Doblin-Harris strong irreducibility (H) and interpolation (I). Then, there exist some constructive decay rate functions $\Theta$ and $\tilde{\Theta}$ such that

$$ (6.17) \qquad \|S^n f\|_{X_1} \lesssim \Theta(n) \|f\|_{X_3}, \quad \forall n \ge 1, $$

and

$$ (6.18) \qquad \|S^n f\| \lesssim \tilde{\Theta}(n) \|f\|_{X_3}, \quad \forall n \ge 1, $$

for any $f \in X_3$, $\langle f, \phi_1 \rangle = 0$. More precisely, the decay rate functions $\Theta$ and $\tilde{\Theta}$ are defined by

$$ (6.19) \qquad \Theta(t) := \inf_{\lambda} \Theta_{\zeta\lambda}(t), \quad \tilde{\Theta}(t) := t^{-1}\Theta([t/2]), $$
When $\psi_0 := 1 \in X' \subset L^\infty$, the condition in (6.4) is automatically satisfied with $R_0 := \|\phi_1\| = 1$. Let us first emphasize that (6.3) is a natural condition when $S_\mathcal{L}^*$ enjoys a splitting structure similar to (5.18). More precisely, when

$$ \|\tilde{S}^*(t)\phi\| \le \Theta(t)\|\phi\| + \int_0^t \Theta(t-s)[\tilde{S}^*(s)\phi]_{g_0} ds, $$

with $\Theta \in L^1(\mathbb{R}_+) \cap C_0(\mathbb{R}_+)$, we deduce that

$$ 1 = \|\phi_1\| = \|\tilde{S}^*(t)\phi_1\| \le \Theta(t) + \int_0^t \Theta(t-s)[\phi_1]_{g_0} ds, \quad \forall t > 0. $$

Passing to the limit $t \to \infty$, we get (6.3) with $r_0 := \|\Theta\|_{L^1}^{-1}$. Also (6.4) can be deduced from a splitting structure condition on the dual problem. More precisely, we assume that $D(\mathcal{L}^\infty) \subset L_{\text{loc}}^1$ and the splitting property $\mathcal{L} = \mathcal{A} + \mathcal{B}$ with $\mathcal{A} \in \mathcal{B}(X)$, $\mathcal{R}_\mathcal{B}(\lambda) \in \mathcal{B}(X) \cap \mathcal{B}(X_+)$ for any $\lambda \ge \kappa$, with $\kappa < \kappa_0 \le \lambda_1$, and the additional regularity condition

$$ (6.5) \qquad (\mathcal{R}_{\mathcal{B}^*}(\lambda)\mathcal{A}^*)^N : L_{g_0}^1 \to L_{\psi_0^{-1}}^\infty, \quad \forall \lambda > \kappa. $$

Since the dual eigenvector $\phi_1$ satisfies

$$ (\lambda_1 - \mathcal{B}^*)\phi_1 = \mathcal{A}^*\phi_1, \quad \lambda_1 > \kappa, $$

and then $\phi_1 = (\mathcal{R}_{\mathcal{B}^*}(\lambda_1)\mathcal{A}^*)^N \phi_1$, we may use estimate (6.5) and we get that (6.3)-(6.4) holds with the normalization condition $r_0 := 1$ and $R_0 := \|(\mathcal{R}_{\mathcal{B}^*}(\lambda)\mathcal{A}^*)^N\|_{\mathcal{B}(L_{g_0}^1, L_{\psi_0^{-1}}^\infty)}$.

We are then able to formulate a first quantified stability result.

**Theorem 6.2.** Consider a positive semigroup $S$ on a Banach lattice $X$ such that its generator $\mathcal{L}$ enjoys the conclusion (C1) on the existence of the two eigenpairs $(\lambda_1, f_1)$ and $(\lambda_1^*, \phi_1)$. We assume furthermore the Doblin condition (6.2)-(6.4)-(6.3). Then $\lambda_1 = \lambda_1^*$ and the exponential stability (E31) in the norm $[\cdot]_{\psi_0}$ holds true, with constructive constants.

The proof closely follows the usual contraction argument in the Doblin result, see for instance [267], [167, Thm. 11] or [80, Thm. 2.1]. We do not explicitly assume the irreducibility of the semigroup, but the Doblin condition (6.2)-(6.4)-(6.3) is in many aspects a strong positivity condition. In particular, our result implies the uniqueness of the first eigentriplet $(\lambda_1, f_1, \phi_1)$ and the triviality of the boundary spectrum.

*Proof of Theorem 6.2.* The dual (equivalent) formulation of the Doblin condition (6.2) is

$$ \forall \psi \in X'_+, \quad S_T^*\psi \ge \psi_0\langle\psi, g_0\rangle. $$

Applied to the first dual eigenvector $\phi_1$ it yields, by using (6.3),

$$ (6.6) \qquad \phi_1 = e^{-\lambda_1^* T} S_T^* \phi_1 \ge e^{-\lambda_1^* T} \psi_0 \langle \phi_1, g_0 \rangle = e^{-\lambda_1^* T} r_0 \psi_0. $$

Together with (6.4), it implies that $[\cdot]_{\phi_1}$ and $[\cdot]_{\psi_0}$ are equivalent norms. Another important consequence is that $\lambda_1 = \lambda_1^*$ since we have

$$ \lambda_1 \langle \phi_1, f_1 \rangle = \langle \phi_1, \mathcal{L} f_1 \rangle = \langle \mathcal{L}^* \phi_1, f_1 \rangle = \lambda_1^* \langle \phi_1, f_1 \rangle $$

and $\langle \phi_1, f_1 \rangle \ne 0$ because $\phi_1 \in X'_{++}$, due to (6.6), and $f_1 \in X_+ \setminus \{0\}$.

We now prove constructively the exponential stability (E31) in the norm $[\cdot]_{\phi_1}$, which will finish the proof due to the (quantified) equivalence of the norms $[\cdot]_{\phi_1}$ and $[\cdot]_{\psi_0}$. The two conditions (6.2) and (6.4) together imply the modified Doblin condition

$$ \exists T > 0, \exists g_1 > 0, \forall f \ge 0, S_T f \ge g_1 \langle \phi_1, f \rangle, $$

with $g_1 := g_0/R_0$. Take $f$ such that $\langle \phi_1, f \rangle = 0$, so that $\langle \phi_1, f_\pm \rangle = r := \langle \phi_1, |f| \rangle / 2 \ge 0$ and thus

$$ S_T f_\pm \ge g_1 \langle \phi_1, f_\pm \rangle = rg_1. $$

We write

$$ |S_T f| \le |S_T f_+ - rg_1| + |S_T f_- - rg_1| = S_T |f| - 2rg_1. $$

We deduce

$$ \langle \phi_1, |S_T f| \rangle \le \langle S_T^* \phi_1, |f| \rangle - 2r \langle \phi_1, g_1 \rangle = (e^{\lambda_1 T} - \langle \phi_1, g_1 \rangle) \langle \phi_1, |f| \rangle. $$
for a constructive constant $\zeta \in (0,1)$, the infimum being taken over all the decreasing function $\lambda: \mathbb{R}_+ \to \mathbb{R}_+$, $t \mapsto \lambda_t$, and $\Theta_\lambda$ is defined by

$$ (6.20) \qquad \Theta(t) \coloneqq \inf_{\lambda>0} \left(e^{-\lambda t} + \frac{\xi_\lambda}{\lambda}\right). $$

The proof follows closely the proof of [81, Thm. 4.8]. We start with the following key argument of non expansive mapping result on a well chosen norm.

**Proposition 6.7.** Consider a positive semigroup $(S_t)$ which satisfies both above conditions of weak confinement (L) and Doblin-Harris strong irreducibility (H). There exist some equivalent norms $\|\cdot\|_1$ to $\|\cdot\|_1$ and $\|\cdot\|_3$ to $\|\cdot\|_3$ such that $\tilde{S}_t$ is a non expansive mapping for the two new norms $\|\cdot\|_1$ and $\|\cdot\|_3$. More precisely, there exists $\alpha > 0$ such that

$$ (6.21) \qquad \|\tilde{S}f\|_1 + \alpha \|\tilde{S}f\|_0 \le \|f\|_1, \quad \forall f \in X_1, \langle f, \phi_1 \rangle = 0, $$

$$ (6.22) \qquad \|\tilde{S}f\|_3 + \alpha \|\tilde{S}f\|_2 \le \|f\|_3, \quad \forall f \in X_3, \langle f, \phi_1 \rangle = 0. $$

**Proof of Proposition 6.7.** We define

$$ (6.23) \qquad \|f\|_1 := [f]_{\phi_1} + \delta \|f\|_0 + \beta \|f\|_1, $$

with $\beta > \delta > 0$ conveniently chosen. We take $\beta := (1 - \gamma_H)/K$, $\delta := (1 - \gamma_H)/A$. We define $\|\cdot\|_3$ in the same way. In what follows, we then only establish (6.21), the proof of (6.22) being exactly the same.

We fix $f \in X_1$, $\langle f, \phi_1 \rangle = 0$, and we recall

$$ (6.24) \qquad [\tilde{S}f]_{\phi_1} \le [f]_{\phi_1}. $$

We also recall that from (6.11), for any $A > 0$, there exists $\gamma_H = \gamma_H(A) \in (0,1)$ such that the following coupling property holds

$$ (6.25) \qquad [\tilde{S}f]_{\phi_1} \le \gamma_H[f]_{\phi_1} \quad \text{if} \quad \|f\|_0 \le A[f]_{\phi_1}. $$

We fix $A > K$ and we observe that the following alternative holds

$$ (6.26) \qquad \|f\|_0 \le A[f]_{\phi_1} $$

or

$$ (6.27) \qquad \|f\|_0 > A[f]_{\phi_1}. $$

**Case 1.** Under condition (6.26), we use (6.25) and the first estimate in (L), and we deduce

$$ \begin{aligned} \|\tilde{S}f\|_1 &= [\tilde{S}f]_{\phi_1} + \delta \|\tilde{S}f\|_0 + \beta \|\tilde{S}f\|_1 \\ &\le \gamma_H[f]_{\phi_1} + \beta \|f\|_1 + \beta K [f]_{\phi_1} - (\beta - \delta) \|\tilde{S}f\|_0. \end{aligned} $$

From our choice of $\beta > 0$ we have $\gamma_H + \beta K = 1$, and we conclude that (6.21) holds with $\alpha := \beta - \delta > 0$.

**Case 2.** Under condition (6.27), the first Lyapunov condition in (L) implies

$$ \|\tilde{S}f\|_1 + \|\tilde{S}f\|_0 \le \|f\|_1 + \frac{K}{A} \|f\|_0. $$

Together with the non expansivity estimate (6.24), we get

$$ [\tilde{S}f]_{\phi_1} + \beta \|\tilde{S}f\|_1 + \beta \|\tilde{S}f\|_0 \le [f]_{\phi_1} + \beta \|f\|_1 + \delta \|f\|_0, $$

and we conclude to (6.21) again. $\square$

The subgeometric convergence result is a straightforward consequence of Proposition 6.7 and an interpolation argument.

**Proposition 6.8.** Assume that S satisfies the hypotheses of Theorem 6.6. Then (6.17) and (6.18) hold true with the same decay rate functions $\Theta$ and $\tilde{\Theta}$ given by (6.19) (up to a modification of the constant $\zeta$).
In other words, setting $\tilde{S}_t := e^{-\lambda_1 t} S_t$, we have

$$ (6.7) \qquad [\tilde{S}_T f]_{\phi_1} \le \gamma[f]_{\phi_1}, $$

with $\gamma < 1$ which depends explicitly of $r_0$, $R_0$, $T$ and the estimates on $\lambda_1$. We then classically deduce the exponential convergence in the $[\cdot]_{\phi_1}$ norm. $\square$

**Remark 6.3.** The existence of $(\lambda_1, f_1)$ is not required in the argument for proving the contraction property (6.7) of $e^{-\lambda_1 T} S_T$. This contraction can even be used for proving the existence of a solution $(\lambda_1, f_1)$ with $\lambda_1 = \lambda_1^*$ to the direct problem (1.1) by only assuming the existence of $(\lambda_1^*, \phi_1)$ to the dual problem (1.2) together with Doblin's condition (6.2)-(6.4)-(6.3). We refer to the proof of Theorem 6.4 for more details about this approach.

**6.3. Asymptotic stability under Doblin-Harris condition.** The Doblin condition (6.2)-(6.4)-(6.3) is too much demanding for many applications. In this section, we make the following somehow more general Doblin-Harris type condition complemented with a Lyapunov condition. More precisely, we assume the existence of a solution $(\lambda_1^*, \phi_1)$ to the dual eigenproblem (1.2) and that there exists $T > 0$ such that $\tilde{S}_T := e^{-\lambda_1^* T} S_T$ first satisfies the Lyapunov condition

$$ (6.8) \qquad \|\tilde{S}_T f\| \le \gamma_L \|f\| + K[f]_{\phi_1}, $$

with $\gamma_L \in (0, 1)$, $K \ge 0$. We next assume that $\tilde{S}_T$ satisfies the Doblin-Harris condition

$$ (6.9) \qquad \begin{cases} \exists A > K/(1 - \gamma_L), \exists g_A > 0 \text{ such that} \\ \forall f \ge 0, \|f\| \le A[f]_{\phi_1} \text{ there holds } S_T f \ge g_A[f]_{\phi_1}. \end{cases} $$

We finally replace the positivity condition (6.3) by

$$ (6.10) \qquad \exists r_A > 0, \quad \langle \phi_1, g_A \rangle \ge r_A. $$

As we have seen several times, condition (6.8) is some kind of regularity hypothesis which is natural under a splitting structure on the semigroup $S_L$. We emphasize that conditions (6.8)-(6.9)-(6.10) slightly generalize the usual set of hypotheses for the Doblin-Harris theorem, see for instance [80, Sect. 3]. We also point out that there is a connection between the condition (6.9) and the notion of *partial integral* or *partial kernel* operators, see for instance [177, Cor. 5.3]. The long term convergence of semigroups that contain a partially integral operator was studied in particular in [316, 176, 179].

**Theorem 6.4.** Consider a positive semigroup $S$ on a Banach lattice $X$ such that its generator $L$ admits a solution $(\lambda_1^*, \phi_1)$ to the dual eigenproblem (1.2). We assume furthermore the Doblin-Harris condition (6.9) together with the Lyapunov condition (6.8) and the positivity condition (6.10). Then there exists a solution $(\lambda_1, f_1)$ with $\lambda_1 = \lambda_1^*$ to the direct eigenproblem (1.1), and the exponential stability (E31) holds true in the norm of $X$ with constructive constants.

Of course, in order that Theorem 6.4 really gives a constructive convergence result, we have to establish (6.9), (6.8) and (6.10) in a constructive way.

*Proof of Theorem 6.4.* On the one hand, we have

$$ (6.11) \qquad [\tilde{S}_T f]_{\phi_1} \le \langle \tilde{S}_T |f|, \phi_1 \rangle = \langle |f|, \tilde{S}_T^* \phi_1 \rangle = [f]_{\phi_1}. $$

On the other hand, we wish to establish the coupling property

$$ (6.12) \qquad [\tilde{S}_T f]_{\phi_1} \le \gamma_H[f]_{\phi_1} \quad \text{if} \quad \|f\| \le A'[f]_{\phi_1} \text{ and } \langle f, \phi_1 \rangle = 0, $$

for some $\gamma_H \in (0, 1)$ and with $A' := A/2$. We thus consider $f \in X$, such that $\langle f, \phi_1 \rangle = 0$ and $\|f\| \le A'[f]_{\phi_1}$, so that

$$ \|f_{\pm}\| \le \|f\| \le A'[f]_{\phi_1} = A[f_{\pm}]_{\phi_1}. $$

Using the Doblin-Harris condition (6.9), we deduce

$$ \tilde{S}_T f_{\pm} \ge \vartheta g_A, \quad \vartheta := \frac{1}{2} e^{-\lambda_1 T} [f]_{\phi_1}. $$

Similarly as in the proof of Theorem 6.2, we next compute

$$ |\tilde{S}_T f| \le |\tilde{S}_T f_+ - \vartheta g_A| + |\tilde{S}_T f_- - \vartheta g_A| \le \tilde{S}_T |f| - 2\vartheta g_A $$
Proof of Proposition 6.8. We recall that we have already proven (6.21) and (6.22). From (6.21) and the interpolation condition (I), we deduce

$$ \|\tilde{S}f\|_1 + \lambda\alpha\|\tilde{S}f\|_1 \le \|f\|_1 + \xi_\lambda\alpha\|\tilde{S}f\|_3. $$

We observe next that from the very definition of the $||\cdot||_1$ norm

$$ \|\tilde{S}f\|_1 + \frac{\alpha}{\lambda}\|\tilde{S}f\|_1 \ge Z_\lambda \|\tilde{S}f\|_1, \quad Z_\lambda = 1 + \kappa\lambda \in (1, 2], $$

for some $\kappa > 0$ and any $\lambda \in (0, \lambda_0)$, $\lambda_0 > 0$, and that from the very definition of the $||\cdot||_3$ norm

$$ \alpha\xi_\lambda \|\tilde{S}f\|_3 \le B\xi_\lambda \|\tilde{S}f\|_3, $$

for some $B > 0$. The three above estimates together imply

$$ Z_\lambda \|\tilde{S}f\|_1 \le \|f\|_1 + B\xi_\lambda \|\tilde{S}f\|_3. $$

Using the second estimate (6.22) and repeating the same proof, we have

$$ Z_{\lambda_{n+1}} \|\tilde{S}^{n+1} f\|_1 \le \|\tilde{S}^n f\|_1 + B\xi_{\lambda_{n+1}} \|f\|_3, $$

for any $n \ge 0$ and for any $\lambda_{n+1} > 0$. The discrete Grönwall lemma implies

$$ (6.28) \qquad \|\tilde{S}^n f\|_1 \le A_n \|f\|_1 + \sum_{k=1}^n A_{k,n} \xi_{\lambda_k} B \|f\|_3, \quad \forall n \ge 0, $$

where we have defined

$$ A_n := \prod_{k=1}^{n} a_k, \quad A_{k,n} = A_n/A_k = \prod_{i=k+1}^{n} a_i, \quad a_i := Z_{\lambda_i}^{-1}. $$

Observing that

$$ A_{k,n} \lesssim e^{-\kappa \sum_{i=k}^n \lambda_i} \lesssim e^{\kappa(\Lambda(k)-\Lambda(n))}, \quad \text{with} \quad \Lambda(t) := \int_0^t \lambda_s ds, $$

and $\lambda_s := \lambda_i$ if $s \in (i-1, i]$, we immediately conclude that the first estimate (6.17) holds true. We come back to the first inequality in (6.21) that we iterate and sum up in order to obtain

$$ \|\tilde{S}^n f\|_1 + \alpha \sum_{k=[n/2]+1}^{n} \|\tilde{S}^k f\|_0 \le \|\tilde{S}^{[n/2]} f\|_1, $$

for any $n \ge 1$. Together with the non expansion inequality

$$ [\tilde{S}^n f]_{\phi_1} \le [\tilde{S}^k f]_{\phi_1} \lesssim \|\tilde{S}^k f\|_0, \quad \forall n \ge k, $$

and the first estimate (6.17), we deduce

$$ (n - [n/2] - 1) \alpha [\tilde{S}^n f]_{\phi_1} \lesssim \Theta([n/2]) \|f\|_3, $$

which is nothing but (6.18). □

## 7. PARABOLIC EQUATIONS

In this part, we consider a general elliptic operator in divergence form

$$ (7.1) \qquad \mathcal{L}f := \partial_i(a_{ij}\partial_j f) + b_i\partial_i f + \partial_i(\beta_i f) + cf, \quad f \in H_0^1(\Omega), $$

where $\Omega \subset \mathbb{R}^d$ is a bounded domain (i.e. an open and connected set) or $\Omega = \mathbb{R}^d$, and we always assume $d \ge 3$ (in order to simplify the discussions when using the Sobolev inequality). We also always assume at least a boundedness and ellipticity condition on the $(a_{ij})$ matrix, namely

$$ (7.2) \qquad a_{ij} \in L^\infty(\Omega), \quad \exists \nu > 0, \forall \xi \in \mathbb{R}^d, a_{ij}\xi_i\xi_j \ge \nu|\xi|^2, $$

and some conditions on the coefficients $b_i$, $\beta_j$ and $c$ which will be described below.

We aim to establish the existence of $(\lambda_1, f_1, \phi_1)$ solution to the first eigentriplet problem

$$ (7.3) \qquad \lambda_1 \in \mathbb{R}, \quad 0 < f_1 \in H_0^1, \mathcal{L}f_1 = \lambda_1 f_1, \quad 0 < \phi_1 \in H_0^1, \mathcal{L}^*\phi_1 = \lambda_1 \phi_1, $$
and then

$$
\begin{align*}
[\tilde{S}_T f]_{\phi_1} &\le \langle \tilde{S}_T |f| - 2\vartheta g_A, \phi_1 \rangle \\
&= \langle |f|, \tilde{S}_T^* \phi_1 \rangle - 2\vartheta \langle g_A, \phi_1 \rangle \\
&= (1 - e^{-\lambda_1 T} \langle g_A, \phi_1 \rangle) [f]_{\phi_1},
\end{align*}
$$

which in turn implies (6.12) with $\gamma_H := 1 - e^{-\kappa_0 T} r_A$.

Now, the two estimates (6.11) and (6.12) together give

$$
(6.13) \qquad [\tilde{S}_T f]_{\phi_1} \le \gamma_H[f]_{\phi_1} + \frac{1 - \gamma_H}{A'} \|f\|.
$$

From (6.13) and the Lyapunov condition (6.8), we deduce that

$U^{n+1} = M U^n$

with

$$
U^n := \left( \frac{\| \tilde{S}_T^n f \|}{[\tilde{S}_T^n f]_{\phi_1}} \right) = \left( \frac{\| \tilde{S}_{nT} f \|}{[\tilde{S}_{nT} f]_{\phi_1}} \right) \quad \text{and} \quad M := \begin{pmatrix} \frac{\gamma_L}{1-\gamma_H} & K \\ \frac{1-\gamma_H}{A} & \gamma_H \end{pmatrix}.
$$

The eigenvalues $\mu_\pm$ of $M$ are

$$
\mu_{\pm} := \frac{1}{2} (T \pm \sqrt{T^2 - 4D}),
$$

with

$$
T := \operatorname{tr} M = \gamma_L + \gamma_H, \quad D := \det M = \gamma_L \gamma_H - (1 - \gamma_H) \frac{K}{A'}.
$$

We observe that

$$
\gamma_L \gamma_H > D > \gamma_L \gamma_H - (1 - \gamma_H)(1 - \gamma_L) = T - 1,
$$

so that

$$
(\gamma_H - \gamma_L)^2 = T^2 - 4\gamma_L\gamma_H < T^2 - 4D < T^2 - 4(T-1) = (T-2)^2
$$

and finally

$$
\alpha := \max(|\mu_+|, |\mu_-|) < \max(\gamma_H, \gamma_L, |T-1|, 1) = 1.
$$

We conclude that $\|M^n\| \lesssim \alpha^n$, from what we immediately deduce the existence of $C > 0$ such that

$$
(6.14) \qquad \| \tilde{S}_{nT} f \| \le C \alpha^n \| f \|
$$

for all $f \in X$ such that $\langle \phi_1, f \rangle = 0$. This classically yields (E31) and it only remains to prove the existence of $f_1$.

We first observe that (6.14) ensures that $(\|\tilde{S}_{nT}f_0\|)_n$ is a Cauchy sequence for any $f_0 \in X$. Indeed, for any $p \in \mathbb{N}$, $f = f_0 - \tilde{S}_{pT}f_0$ verifies

$$
\langle f, \phi_1 \rangle = \langle f_0, \phi_1 \rangle - \langle f_0, \tilde{S}_{pT}^* \phi_1 \rangle = \langle f_0, \phi_1 \rangle - \langle f_0, \phi_1 \rangle = 0,
$$

and we then have

$$
\| \tilde{S}_{nT} f_0 - \tilde{S}_{(n+p)T} f_0 \| + [\tilde{S}_{nT} f_0 - \tilde{S}_{(n+p)T} f_0]_{\phi_1} \lesssim \alpha^n (\| f_0 - \tilde{S}_{pT} f_0 \| + [f_0 - \tilde{S}_{pT} f_0]_{\phi_1}).
$$

Choosing $f_0 \in X_+$ such that $[f_0]_{\phi_1} = 1$, we deduce that $(\tilde{S}_{nT}f)$ converges to a fixed point $f_1$ of $\tilde{S}_T$, which is not zero because

$$
[f_1]_{\phi_1} = \lim[\tilde{S}_{nT}f_0]_{\phi_1} = [f_0]_{\phi_1} = 1,
$$

and, thanks to (6.14), $f_1$ is the unique fixed point with normalization $[f_1]_{\phi_1} = 1$. Besides, $f_1 \in X_+$ because of the positivity of $S$ and $f_0$. This ensures that

$$
[\tilde{S}_t f_1]_{\phi_1} = \langle \tilde{S}_t f_1, \phi_1 \rangle = \langle f_1, \tilde{S}_t^* \phi_1 \rangle = \langle f_1, \phi_1 \rangle = 1,
$$

for any $t > 0$. Since on the other hand

$$
\tilde{S}_T \tilde{S}_t f_1 = \tilde{S}_{t+T} f_1 = \tilde{S}_t \tilde{S}_T f_1 = \tilde{S}_t f_1,
$$

we deduce from the uniqueness of the fixed point that $\tilde{S}_t f_1 = f_1$, which yields that $f_1 \in D(\mathcal{L})$ and $\mathcal{L}f_1 = \lambda_1^* f_1$. $\square$
and the existence of some (constructive) rate function $\Theta$ such that the rescaled semigroup $\tilde{S}$ associated to the generator $\tilde{\mathcal{L}} = \mathcal{L} - \lambda_1$ satisfies

$$ (7.4) \qquad \|\tilde{S}(t)f - \langle f, \phi_1 \rangle f_1\|_{H_0} \le \Theta(t) \|f - \langle f, \phi_1 \rangle f_1\|_{H}, $$

for any $t \ge 0$ and any $f \in H$, with $H \subset H_0 \subset L^2$.

**7.1. Diffusion with rough coefficients in a bounded domain.** In this section, we consider the general elliptic operator in divergence form (7.1) in the case of a bounded and smooth enough domain $\Omega \subset \mathbb{R}^d$ with general elliptic condition on $a_{ij}$ as formulated above. We further assume that

$$ (7.5) \qquad b_i, \beta_j \in L^r(\Omega), \quad c \in L^{r/2}(\Omega), \quad r > d. $$

In that situation, the first eigentriplet problem (7.3) has been considered by Chicco in [106, 107] and revisited in a slightly less general framework (all the coefficients belong to $L^\infty$) in [252], where the conclusions (C2) are established. We explain with all details the existence proof by following more or less the arguments presented in [252] stressing on the constructive way for obtaining the estimates, and next we present a proof of the geometric part and the stability part by taking advantage of the abstract material developed in the previous sections. It is worth emphasizing that our proof of the uniqueness of the first eigenfunction significantly differs from the one presented in [252] which is based on a dissipativity argument, probably related to the reverse Kato's inequality condition. The framework considered here is the usual generalized solutions or weak solutions framework which goes back at least to Stampacchia [339, 340], but it is reminiscent of previous contributions by Friedrichs [165, 166], Gårding [173], De Giorgi [127], Nash [297], Morrey [289], Moser [290, 291, 292], Ladyzhenskaya, Solonnikov, Ural'ceva [244, 242], Oleinik, Kruzhkov [303] and many others. Lot of the functional arguments are picked up from the book of Gilbarg and Trudinger, and more specifically from [179, Chapter 8], and also in recent notes by Kavian [230] and Vasseur [349]. It is worth emphasizing that the present analysis does not apply directly to elliptic operators in non divergence form, although this framework is considered in [252]. We expect that all the results developed below can be generalized to a non divergence form framework, for example the one developed in [47], but we do not follow this line of research in the present work.

The proof of (7.3) and (7.4) are straightforward consequences of the abstract results developed in the previous sections once we have been able to check that the corresponding hypotheses are fulfilled. In the sequel, we will then show how these hypotheses are met in the present context.

**Condition (H1).** We recall that a weak (or variational) solution to the elliptic equation

$$ \mathcal{L}f = g \in H^{-1}(\Omega), \quad f \in H_0^1(\Omega), $$

is a function $f \in H_0^1(\Omega)$ such that

$$ (7.6) \qquad D_{\mathcal{L}}(f, w) = \langle g, w \rangle, \quad \forall w \in H_0^1(\Omega), $$

where the (negative) Dirichlet form $D_{\mathcal{L}}$ is defined by

$$ D_{\mathcal{L}}(f, w) := - \int_{\Omega} (a_{ij} \partial_j f + \beta_i f) \partial_i w + \int_{\Omega} (b_i \partial_i f w + cfw), $$

for any $f, w \in H_0^1(\Omega)$. Most of the time, we will simply write

$$ (7.7) \qquad \langle \mathcal{L}f, w \rangle = \langle g, w \rangle, \quad \forall w \in H_0^1(\Omega), $$

instead of (7.6). For the reader convenience, we repeat here some estimates picked up in [340]. For $\lambda \in \mathbb{R}$ and $f \in H_0^1(\Omega)$, we start with

$$ \begin{aligned} \langle (\lambda - \mathcal{L})f, f \rangle &= \int_{\Omega} a_{ij} \partial_i f \partial_j f + \int_{\Omega} (\beta_i - b_i) \partial_i f f + \int_{\Omega} (\lambda - c) f^2 \\ &\ge \|f \sqrt{c_-}\|_{L^2}^2 + \nu \|\nabla f\|_{L^2}^2 - \|\beta - b\|_f \|\nabla f\|_{L^2} - \|\sqrt{c_+} f\|_{L^2}^2 + \lambda \|f\|_{L^2}^2 \\ &\ge \|f \sqrt{c_-}\|_{L^2}^2 + \frac{\nu}{2} \|\nabla f\|_{L^2}^2 - \frac{1}{2\nu} \|\beta - b\|_f \|\nabla f\|_{L^2} - \|\sqrt{c_+} f\|_{L^2}^2 + \lambda \|f\|_{L^2}^2, \end{aligned} $$
**Remark 6.5.** We have seen in Theorem 6.2 that under Doblin's condition (6.2)-(6.4)-(6.3), if we have two eigenpairs $(\lambda_1, f_1)$ and $(\lambda_1^*, \phi_1)$ solutions to the direct and dual eigenvalue problems (1.1) and (1.2) respectively, then necessarily $\lambda_1 = \lambda_1^*$. This is not the case for the conditions (6.8)-(6.9)-(6.10), which only ensure that $\lambda_1 \le \lambda_1^*$.

**6.4. Quantified isolation of the first eigenvalue.** In terms of the geometry of the spectrum, an immediate consequence of Theorem 6.4 is that the conditions (6.9)-(6.8)-(6.10) ensure the existence of a spectral gap, namely the existence of $\varepsilon > 0$ such that

$$ \Sigma(\mathcal{L}) \cap \Delta_{\lambda_1 - \varepsilon} = \{\lambda_1\}. $$

We still assume that the Lyapunov condition (6.8) holds for some $T > 0$, $\gamma_L \in (0, 1)$ and $K \ge 0$, but we relax (6.9) into the time-averaged condition

$$ (6.15) \qquad \left\{ \begin{array}{l} \exists A > K/(1 - \gamma_L), \exists g_A > 0 \text{ such that} \\ \forall f \ge 0, \|f\| \le A[f]_{\phi_1} \text{ there holds } \int_0^T S_t f dt \ge g_A[f]_{\phi_1}. \end{array} \right. $$

It is worth emphasizing that (6.15) does not imply anymore the existence of a spectral gap, and there can be a non-trivial boundary spectrum, see Section 9.2 for an example. However, it is strong enough for guaranteeing that $\lambda_1$ is isolated from the rest of the spectrum, in the sense that

$$ (6.16) \qquad \Sigma(\mathcal{L}) \cap B(\lambda_1, \varepsilon) = \{\lambda_1\}, $$

for some $\varepsilon > 0$. In particular, if not trivial, the boundary spectrum must be discrete from Theorem 5.5 (under the additional assumptions listed in the statement of this last result).

**Theorem 6.6.** Consider a positive semigroup $S$ on a Banach lattice $X$ such that its generator $\mathcal{L}$ admits a solution $(\lambda_1^*, \phi_1)$ to the dual eigenproblem (1.2). We assume furthermore the time-averaged Doblin-Harris condition (6.15) together with the Lyapunov condition (6.8) and the positivity condition (6.10). Then there exists a solution $(\lambda_1, f_1)$ with $\lambda_1 = \lambda_1^*$ to the direct eigenproblem (1.1), and (6.16) holds true for some constructive constant $\varepsilon > 0$, providing a quantified information on (S31).

*Proof of Theorem 6.6.* We start by noticing that by iteration of (6.8) we have

$$ \|\tilde{S}_{nT}f\| \le \gamma_L^n \|f\| + \frac{K}{1 - \gamma_L}[f]_{\phi_1}, $$

for all integer $n$, from which we deduce

$$ (6.17) \qquad \|\tilde{S}_t f\| \le C \gamma_L^{\lfloor t/T \rfloor} \|f\| + \frac{CK}{1 - \gamma_L} [f]_{\phi_1}, $$

for all $t \ge 0$, where $C = \sup_{0 \le t \le T} \|\tilde{S}_t\|$. In particular

$$ \|\tilde{S}_t\| \le C + \frac{CK}{1 - \gamma_L} \|\phi_1\|, $$

which implies that $\omega(\tilde{S}) \le 0$ and consequently $\lambda - \mathcal{L}$ is invertible for all $\lambda > \lambda_1^*$ with inverse $\mathcal{R}_\mathcal{L}(\lambda)$ given by the inversion formula (2.13). Using this inversion formula and (6.15), we readily deduce that

$$ (6.18) \qquad \left\{ \begin{array}{l} \exists A > K/(1 - \gamma_L), \exists g_A > 0 \text{ such that} \\ \forall f \ge 0, \|f\| \le A[f]_{\phi_1} \text{ there holds } \tilde{\mathcal{R}}(\lambda)f \ge \tilde{g}_A[f]_{\phi_1}, \forall \lambda > \lambda_1^*, \end{array} \right. $$

where $\tilde{\mathcal{R}}(\lambda) := (\lambda - \lambda_1^*)\mathcal{R}_\mathcal{L}(\lambda)$ and $\tilde{g}_A := (\lambda - \lambda_1^*)e^{-\lambda T}g_A$. Next, we claim that the Lyapunov condition (6.8) ensures the existence of $\lambda > \lambda_1^*$ such that

$$ (6.19) \qquad \|\tilde{\mathcal{R}}(\lambda)f\| \le \gamma'_L \|f\| + K'[f]_{\phi_1} $$

for all $f \in X$ and some $\gamma'_L < 1$ and $K' > 0$. Indeed, we infer from (6.17) and the inversion formula (2.13) that

$$ \|\tilde{\mathcal{R}}(\lambda)f\| \le \frac{C_1(\lambda - \lambda_1^*)}{\lambda - \lambda_1^* + \log \frac{1}{\gamma_L}} \|f\| + \frac{C_2}{1 - \gamma_L} [f]_{\phi_1}, $$
using the Cauchy-Schwarz inequality and the Young inequality, and next

$$
\begin{aligned}
\langle (\lambda - \mathcal{L})f, f \rangle & \geq \|f\sqrt{c_{-}}\|_{L^2}^2 + \frac{\nu}{4}\|\nabla f\|_{L^2}^2 + (\lambda - \frac{M}{2\nu} - M^{1/2})\|f\|_{L^2}^2 \\
& \phantom{\geq} + \frac{\nu}{4}C_{\Omega}\|f\|_{L^{2*}}^2 - \frac{1}{2\nu}\|\beta - b|\mathbf{1}_{|\beta-b|\geq M}f\|_{L^2}^2 - \|\sqrt{c_{+}}\mathbf{1}_{c_{+}\geq M}f\|_{L^2}^2 \\
& \geq \|f\sqrt{c_{-}}\|_{L^2}^2 + \frac{\nu}{4}\|\nabla f\|_{L^2}^2 + (\lambda - \frac{M}{2\nu} - M^{1/2})\|f\|_{L^2}^2 \\
& \phantom{\geq} + (\frac{\nu}{4}C_{\Omega} - \frac{1}{2\nu}\|\beta - b|\mathbf{1}_{|\beta-b|\geq M}\|_{L^d}^2 - \|c_{+}\mathbf{1}_{c_{+}\geq M}\|_{L^{d/2}}\|f\|_{L^{2*}}^2,
\end{aligned}
$$

using the Sobolev inequality (with associated constant $C_\Omega$) and the Holder inequality. Choosing $M > 0$ large enough in such a way that the last term is positive, and next $\kappa_1 > 0$ large enough, we deduce for instance that

$$ (7.8) \qquad \langle (\lambda - \mathcal{L})f, f \rangle \geq \|f\sqrt{c_{-}}\|_{L^2}^2 + \frac{\nu}{4}\|\nabla f\|_{L^2}^2 + \|f\|_{L^2}^2, \quad \forall \lambda \geq \kappa_1. $$

Thanks to the Lax-Milgram theorem and the above coercivity estimate, we deduce that $\lambda - \mathcal{L}$ is invertible, and more precisely the mapping $(\lambda - \mathcal{L})^{-1} : H^{-1} \to H_0^1(\Omega)$ is well defined. We also claim that $\lambda - \mathcal{L}$ enjoys a weak principle maximum, and more precisely

$$ (7.9) \qquad f \in H_0^1(\Omega), \quad (\lambda - \mathcal{L})f \geq 0 \quad \text{imply} \quad f \geq 0. $$

Indeed, for such a function $f \in H_0^1(\Omega)$, we take $w = f_- \in H_0^1(\Omega)$, as a test function, and elementary Sobolev space calculus together with the previous estimate yields

$$ \begin{aligned} 0 & \leq \langle (\lambda - \mathcal{L})f, f_- \rangle = -\langle (\lambda - \mathcal{L})f_-, f_- \rangle \\ & \leq -\|f_-\sqrt{c_-}\|_{L^2}^2 - \frac{\nu}{4}\|\nabla f_-\|_{L^2}^2 - \|f_-\|_{L^2}^2 \leq 0, \end{aligned} $$

so that $f_- = 0$ and $f \geq 0$. We thus deduce $(\lambda - \mathcal{L})^{-1} : L_+^2 \to L_-^2$, and from J.-L. Lions theory on parabolic equation (see for instance [251, Chapter 3]), we next deduce that $\mathcal{L}$ is the generator in $L^2$ of a positive semigroup $S_\mathcal{L}$, so that (H1) holds. It is worth emphasizing at this point that the semigroup $S$ built thanks to Lions's theory is defined by $S(t)f_0 = f$ for any $f_0 \in L^2$, where $f \in \mathcal{E} := C([0, \infty); L^2) \cap L_{\text{loc}}^2([0, \infty); H_0^1) \cap H_{\text{loc}}^1([0, \infty); H^{-1})$ is the unique (variational) solution to the equation

$$ (7.10) \qquad (f(T), g(T))_{L^2} - (f_0, g(0))_{L^2} = \int_0^T \{\langle \partial_t g, f \rangle_{H^{-1}, H_0^1} + D_\mathcal{L}(f, g)\} ds, $$

for any $T > 0$ and $g \in \mathcal{E}$. Choosing $g = f$ in the above equation, we classically compute

$$ \frac{1}{2} \|f(t)\|_{L^2}^2 - \frac{1}{2} \|f_0\|_{L^2}^2 - \int_0^t D_{\mathcal{L}}(f, f) ds = 0, \quad \forall t > 0, $$

which together with (7.8) implies

$$ \frac{1}{t} \int_0^t \frac{\nu}{4} \|\nabla f\|_{L^2}^2 ds \leq - \left( \frac{f(t) - f_0}{t}, \frac{f(t) + f_0}{2} \right)_{L^2} + \frac{\kappa_1}{t} \int_0^t \|f\|_{L^2}^2 ds, \quad \forall t > 0. $$

When $f_0 \in D(\mathcal{L})$, the RHS is bounded and there thus exists a sequence $t_n \to 0$ such that $\|\nabla f(t_n)\|_{L^2}$ is bounded. That implies $f_0 \in H_0^1(\Omega)$ and thus $D(\mathcal{L}) \subset H_0^1(\Omega)$. Similarly, we may consider the dual Dirichlet form $D^*(f, g) := D_\mathcal{L}(g, f)$ and build an associated positive semigroup $S^*$ through Lions's theory described above. More precisely $S^*(t)g_0 = g$ for any $t \geq 0$ and $g_0 \in L^2$, where $g \in \mathcal{E}$ is the unique (variational) solution to the equation

$$ (g(t), f(t))_{L^2} - (g_0, f(0))_{L^2} = \int_0^t \{\langle \partial_t f, g \rangle_{H^{-1}, H_0^1} + D^*(g, f)\} ds, $$

for any $t > 0$ and $f \in \mathcal{E}$. Now, we fix $T > 0$, $g_T \in L^2$ and we set $g(t) := S^*(T - t)g_T$, so that $g$ is a solution to the backward evolution equation

$$ -\partial_t g = \mathcal{L}^* g, \quad g(T) = g_T, $$

with

$$ \mathcal{L}^* g := \partial_j (a_{ij} \partial_i g) - \partial_i (b_i g) - \beta_i \partial_i g + cg. $$
for all $\lambda > \lambda_1^*$ and some $C_1, C_2 > 0$. Then we only need to choose $\lambda$ close enough to $\lambda_1^*$ so that
$$ \frac{C_1(\lambda - \lambda_1^*)}{\lambda - \lambda_1^* + \log \frac{1}{\gamma_L}} < 1 $$
and we obtain (6.19).

We have proved that $\tilde{\mathcal{R}}(\lambda)$ satisfies (6.19) and (6.18). Together with the positivity condition (6.10), we can thus repeat the proof of Theorem 6.4 for the operator $\tilde{\mathcal{R}}$ instead of $\tilde{S}$ and we obtain the existence of $f_1$ such that $\tilde{\mathcal{R}}(\lambda)f_1 = f_1$, which is equivalent to $\mathcal{L}f_1 = \lambda_1^*f_1$, and the existence of constructive constants $\alpha \in (0, 1)$ and $C \ge 1$ such that

$$ \|\tilde{\mathcal{R}}(\lambda)^n f\| \le C \alpha^n \|f\|, \quad \forall n \ge 1, $$

for any $f \in X$, $\langle f, \phi_1 \rangle = 0$. By the spectral radius formula (see [333, Thm. 10.13] for instance), we deduce that

$$ r(\tilde{R}(\lambda) = \lim_{n \to \infty} \|\tilde{\mathcal{R}}(\lambda)^n\|^{1/n} \le \alpha $$

and so

$$ \Sigma(\tilde{\mathcal{R}}(\lambda)) \cap \{z \in \mathbb{C}, |z| > \alpha\} = \{1\}. $$

The spectral mapping theorem for the resolvent (see for instance [151, Thm. IV.1.13]) ensures that

$$ \Sigma(\tilde{\mathcal{R}}(\lambda)) \setminus \{0\} = \frac{\lambda - \lambda_1^*}{\lambda - \Sigma(\mathcal{L})}, $$

then yields (6.16) with $\varepsilon = (\alpha^{-1} - 1)(\lambda - \lambda_1^*)$. $\square$

<figure><img src="image_6.png" /></figure>

**6.5. The weak dissipativity case.** In this section, we consider a weak dissipative semigroup ($S_t$) as considered in Section 3.3 and in a sense we make precise now. We assume that its generator $\mathcal{L}$ admits a solution $(\lambda_1^*, \phi_1)$ to the dual eigenproblem (1.2). We consider four Banach lattices $X_3 \subset X_2 \subset X_1 \subset X_0 = X$. We first make the same kind of Doblin-Harris type condition as in the previous section, namely

**Hypothesis (H)** (Doblin-Harris) condition (6.9) holds for the same time $T > 0$ and for both norms $\| \cdot \| = \| \cdot \|_{X_0}$ and $\| \cdot \| = \| \cdot \|_{X_2}$ as well as the companion positivity condition (6.10) holds. Instead of the strong Lyapunov condition (6.8), we assume

**Hypothesis (L)** (weak Lyapunov) there exist a constant $K \ge 0$ such that

$$ \|\tilde{S}f\|_1 + \|\tilde{S}f\|_0 \le \|f\|_1 + K[f]_{\phi_1}, \quad \forall f \in X_1, $$

$$ \|\tilde{S}f\|_3 + \|\tilde{S}f\|_2 \le \|f\|_3 + K[f]_{\phi_1}, \quad \forall f \in X_3, $$

with $\tilde{S} = S_T e^{-\lambda_1^* T}$.

**Hypothesis (I)** (interpolation) there exists an increasing function $\xi : \mathbb{R}_+ \to \mathbb{R}_+$, $\lambda \mapsto \xi_\lambda$, such that

$$ \lambda \|f\|_1 \le \|f\|_0 + \xi_\lambda \|f\|_3, \quad \forall \lambda > 0, \quad \xi_\lambda / \lambda \to 0 \text{ as } \lambda \to 0. $$

**Theorem 6.7.** Consider a semigroup $S$ on a Banach lattice $X$ such that its generator $\mathcal{L}$ admits a solution $(\lambda_1^*, \phi_1)$ to the dual eigenproblem (1.2). We assume furthermore the three above conditions of weak confinement (L), Doblin-Harris strong irreducibility (H) and interpolation (I). Then, there exist some constructive decay rate functions $\Theta$ and $\tilde{\Theta}$ such that

$$ (6.20) \qquad \|\tilde{S}^n f\|_{X_1} \lesssim \Theta(n) \|f\|_{X_3}, \quad \forall n \ge 1, $$

and

$$ (6.21) \qquad \|\tilde{S}^n f\| \lesssim \tilde{\Theta}(n) \|f\|_{X_3}, \quad \forall n \ge 1, $$

for any $f \in X_3$, $\langle f, \phi_1 \rangle = 0$. More precisely, the decay rate functions $\Theta$ and $\tilde{\Theta}$ are defined by

$$ (6.22) \qquad \Theta(t) := \inf_{\lambda} \Theta_{\zeta\lambda}(t), \quad \tilde{\Theta}(t) := t^{-1}\Theta([t/2]), $$

for a constructive constant $\zeta \in (0, 1)$, the infimum being taken over all the decreasing function $\lambda : \mathbb{R}_+ \to \mathbb{R}_+$, $t \mapsto \lambda_t$, and $\Theta_\lambda$ is defined by

$$ (6.23) \qquad \Theta(t) \asymp \inf_{\lambda>0} \left(e^{-\lambda t} + \frac{\xi_\lambda}{\lambda}\right). $$
The variational formulation of this last problem is

$$ (7.11) \qquad (g_T, f(T))_{L^2} - (g(0), f(0))_{L^2} = \int_0^T \{\langle \partial_t f, g \rangle_{H^{-1}, H_0^1} - D^*(g, f)\} ds, $$

for any $f \in \mathcal{E}$. Summing up (7.10) and (7.11) with $f(t) := S(t)f_0$ for $f_0 \in L^2$ and $g(t) := S^*(T-t)g_T$ for $g_T \in L^2$, we deduce

$$ (S(T)f_0, g_T)_{L^2} = (S^*(T)g_T, f_0)_{L^2}. $$

In other words, we have established that $S^* = (S_\mathcal{L})^*$ and thus that $\mathcal{L}^*$ is the generator of the semigroup $S^*$.

**Condition (H2).** Let us consider a ball $B_R$, $R > 0$, such that $B_{4R} \subset \Omega$ and next the solution

$$ (7.12) \qquad f_0 \in H_0^1(\Omega), \quad (\kappa_1 - \mathcal{L})f_0 = \mathbf{1}_{B_R}, $$

which exists from the above discussion. We next recall some classical results. On the one hand, from [339, Sec. 3 & Sec. 4] or [179, Thm. 8.15] (see also the original papers [127, 297, 290]), the following global $L^\infty$ De Gorgi-Nash-Moser type estimate

$$ (7.13) \qquad \|f_+\|_{L^\infty(\Omega)} \lesssim \|f_+\|_{L^2(\Omega)} + \|g\|_{L^{r/2}(\Omega)} $$

holds for any subsolution

$$ f \in H_0^1(\Omega), \quad (\lambda - \mathcal{L})f \leq g \in L^{r/2}(\Omega). $$

The local estimate variant [179, Thm. 8.18] (or weak Harnack inequality)

$$ (7.14) \qquad \|f\|_{L^p(B_{2R})} \lesssim \inf_{B_R} f + \|g\|_{L^{r/2}(\Omega)}, \quad \forall p \in [1, 2^*/2), $$

also holds for a nonnegative supersolution

$$ f \in H^1(\Omega), \quad f \geq 0 \text{ on } B_{4R} \subset \Omega, \quad (\lambda - \mathcal{L})f \geq g \in L^{r/2}(\Omega), $$

from what one deduces that a strong maximum principle [179, Thm. 8.19] holds. More precisely, under the additional one side pointwise bound

$$ (7.15) \qquad c + \operatorname{div}\beta \leq c_0 \quad \text{or} \quad c - \operatorname{div}b \leq c_0, $$

for some $c_0 \in \mathbb{R}$, we have that, for any $f \in H_0^1(\Omega)$,

$$ (7.16) \qquad \mathcal{L}f \leq 0 \text{ in } \Omega, \quad f \geq 0 \text{ in } \Omega \quad \text{imply} \quad f \equiv 0 \text{ or } f > 0 \text{ a.e. in } \Omega. $$

When indeed $f \neq 0$, we may choose $B_{4R} \subset \Omega$ such that $\|f\|_{L^1(B_{2R})} > 0$ and thus $\inf_{B_R} f > 0$ from (7.14) (with $g = 0$) and because constants are supersolutions thanks to the first condition in (7.15). In the case only the second condition holds in (7.15), the same argument implies that $\mathcal{L}^*$ satisfies the strong maximum principle and thus also $\mathcal{L}$ thanks to Lemma 4.9. We conclude that $f$ is positive by a connectivity argument. An alternative and less demanding proof is presented in [106, Cor. 1] where (7.16) is established without the additional assumption (7.15).

On the other hand, the following Hölder regularity estimate [339, Théorème 7.1] and [179, Thm. 8.29] (see also the original papers [127, 297, 290]) of De Gorgi-Nash-Moser type

$$ (7.17) \qquad \|f\|_{C^\alpha(\Omega)} \leq C \|(\lambda - \mathcal{L})f\|_{L^\infty(\Omega)} $$

holds true for some $\alpha = \alpha(a_{ij}) \in (0, 1)$ and $C > 0$. These last two pieces of information together and the fact that $f_0 \neq 0$ imply that there exists a constant $\theta > 0$ such that $f_0 \geq \theta \mathbf{1}_{B_R}$, and thus

$$ \mathcal{L}f_0 \geq (\kappa_1 - \theta^{-1})f_0. $$

That is condition (i) in Lemma 2.4, so that condition (H2) holds thanks to Lemma 2.4. Presented in that way, the above estimate is not really constructive, but the constant $\theta := \inf_{B_R}(\kappa_1 - \mathcal{L})^{-1}\mathbf{1}_{B_R}$ can also be considered as a geometric quantity associated to geometric properties of the operator and the domain.

**First constructive argument for (H2).** In the case when $\mathcal{L}$ is self-adjoint, that corresponds to the case $a_{ij} = a_{ji}$ and $b_i + \beta_i = 0$, we classically know (that has been recalled in Section 2.3, see (2.35)) that

$$ \lambda_1 = \inf_{f \in X_+ \setminus \{0\}} \frac{\langle \mathcal{L}f, f \rangle}{\|f\|^2} = \inf_{f \in H_0^1, \|f\|_{L^2} = 1} \int_{\Omega} \{a \nabla f \cdot \nabla f + cf^2\}, $$
The proof follows closely the proof of [80, Thm. 4.8]. We start with the following key argument of non expansive mapping result on a well chosen norm.

**Proposition 6.8.** Consider a positive semigroup ($S_t$) which satisfies both above conditions of weak confinement (L) and Doblin-Harris strong irreducibility (H). There exist some equivalent norms $||| \cdot |||_1$ to $|| \cdot ||_1$ and $||| \cdot |||_3$ to $|| \cdot ||_3$ such that $\tilde{S}_t$ is a non expansive mapping for the two new norms $||| \cdot |||_1$ and $||| \cdot |||_3$. More precisely, there exists $\alpha > 0$ such that

$$ (6.24) \qquad \|\tilde{S}f\|_1 + \alpha \|\tilde{S}f\|_0 \le \|f\|_1, \quad \forall f \in X_1, \langle f, \phi_1 \rangle = 0, $$

$$ (6.25) \qquad \|\tilde{S}f\|_3 + \alpha \|\tilde{S}f\|_2 \le \|f\|_3, \quad \forall f \in X_3, \langle f, \phi_1 \rangle = 0. $$

*Proof of Proposition 6.8.* We define

$$ (6.26) \qquad \|f\|_1 := [f]_{\phi_1} + \delta\|f\|_0 + \beta\|f\|_1, $$

with $\beta > \delta > 0$ conveniently chosen. We take $\beta := (1 - \gamma_H)/K$, $\delta := (1 - \gamma_H)/A$. We define $||| \cdot |||_3$ in the same way. In what follows, we then only establish (6.24), the proof of (6.25) being exactly the same.

We fix $f \in X_1$, $\langle f, \phi_1 \rangle = 0$, and we recall

$$ (6.27) \qquad [\tilde{S}f]_{\phi_1} \le [f]_{\phi_1}. $$

We also recall that from (6.12), for any $A > 0$, there exists $\gamma_H = \gamma_H(A) \in (0, 1)$ such that the following coupling property holds

$$ (6.28) \qquad [\tilde{S}f]_{\phi_1} \le \gamma_H[f]_{\phi_1} \quad \text{if} \quad \|f\|_0 \le A[f]_{\phi_1}. $$

We fix $A > K$ and we observe that the following alternative holds

$$ (6.29) \qquad \|f\|_0 \le A[f]_{\phi_1} $$

or

$$ (6.30) \qquad \|f\|_0 > A[f]_{\phi_1}. $$

**Case 1.** Under condition (6.29), we use (6.28) and the first estimate in (L), and we deduce

$$ \begin{aligned} \|\tilde{S}f\|_1 &= [\tilde{S}f]_{\phi_1} + \delta\|\tilde{S}f\|_0 + \beta\|\tilde{S}f\|_1 \\ &\le \gamma_H[f]_{\phi_1} + \beta\|f\|_1 + \beta K[f]_{\phi_1} - (\beta - \delta)\|\tilde{S}f\|_0. \end{aligned} $$

From our choice of $\beta > 0$ we have $\gamma_H + \beta K = 1$, and we conclude that (6.24) holds with $\alpha := \beta - \delta > 0$.

**Case 2.** Under condition (6.30), the first Lyapunov condition in (L) implies

$$ \|\tilde{S}f\|_1 + \|\tilde{S}f\|_0 \le \|f\|_1 + \frac{K}{A} \|f\|_0. $$

Together with the non expansivity estimate (6.27), we get

$$ [\tilde{S}f]_{\phi_1} + \beta \|\tilde{S}f\|_1 + \beta \|\tilde{S}f\|_0 \le [f]_{\phi_1} + \beta \|f\|_1 + \delta \|f\|_0, $$

and we conclude to (6.24) again. $\square$

<figure><img src="image_7.png" /></figure>

The subgeometric convergence result is a straightforward consequence of Proposition 6.8 and an interpolation argument.

**Proposition 6.9.** Assume that S satisfies the hypotheses of Theorem 6.7. Then (6.20) and (6.21) hold true with the same decay rate functions $\Theta$ and $\tilde{\Theta}$ given by (6.22) (up to a modification of the constant $\zeta$).

*Proof of Proposition 6.9.* We recall that we have already proven (6.24) and (6.25). From (6.24) and the interpolation condition (I), we deduce

$$ \|\tilde{S}f\|_1 + \lambda\alpha \|\tilde{S}f\|_1 \le \|f\|_1 + \xi_\lambda\alpha \|\tilde{S}f\|_3. $$

We observe next that from the very definition of the $||| \cdot |||_1$ norm

$$ \|\tilde{S}f\|_1 + \frac{\alpha}{\lambda} \|\tilde{S}f\|_1 \ge Z_\lambda \|\tilde{S}f\|_1, \quad Z_\lambda = 1 + \kappa\lambda \in (1, 2], $$
from what and the Sobolev imbedding, we get

$$ \lambda_1 \geq \inf_{f \in H_0^1, \|f\|_{L^2}=1} \{(\nu C_{\Omega} - \|c-1_{c-\geq M}\|_{L^{d/2}})\|f\|_{L^{2^*}}^2 - M\} \geq -M, $$

by choosing $M$ large enough. That gives an explicit lower bound on $\lambda_1$.

**Second constructive argument for (H2).** We give another constructive argument without assuming any self-adjointness property. We rather assume

$$ (7.18) \qquad (\partial_i b_i - c)_+ \in M^1(\Omega), \quad b_i + \beta_i - \partial_j a_{ij} \in M^1(\Omega). $$

We fix $h_0 \in C_c^2(\Omega)$ such that $c_01_{B_\rho} \le h_0 \le c_01_{B_{3\rho/2}}$ with $B_{8\rho} \subset \Omega$ and $\|h_0\|_{L^2} = 1$. We next define $f_0$ as the (positive) solution to

$$ (7.19) \qquad f_0 \in H_0^1(\Omega), \quad (\kappa_1 - \mathcal{L})f_0 = h_0, $$

so that $f_0 \in C^\alpha(\Omega)$ from (7.13) and (7.17), and similarly

$$ (7.20) \qquad \tilde{f}_0 \in H_0^1(B_{2\rho}), \quad (\kappa_1 - \mathcal{L})\tilde{f}_0 = h_0, $$

so that $\tilde{f}_0 \in C^\alpha(B_{2\rho})$ from (7.13) and (7.17). We observe that $0 \le \tilde{f}_0 \le f_0$ thanks to the weak maximum principle. We then compute

$$ 1 = \|h_0\|_{L^2}^2 = \int_{B_{2\rho}} h_0(\kappa_1 - \mathcal{L})\tilde{f}_0 = \int_{B_{2\rho}} \tilde{f}_0(\kappa_1 - \mathcal{L}^*)h_0 \le \|\tilde{f}_0\|_{L^\infty} \|(\kappa_1 - \mathcal{L}^*)h_0\|_{M^1}, $$

where the last term is finite because of the additional hypothesis (7.18). We conclude to a first constructive lower bound $\|\tilde{f}_0\|_{L^\infty(B_{2\rho})} \ge c_1 > 0$. Because of the Holder continuity, we also have $\|\tilde{f}_0\|_{L^1(B_{2\rho})} \ge c_2$ with constructive constant $c_2 = c_2(c_1, \alpha, d) > 0$. Thanks to (7.14) (with $g=0$), we obtain

$$ \begin{aligned} f_0 &\ge \mathbf{1}_{B_{3\rho/2}} \inf_{B_{3\rho/2}} f_0 \ge \mathbf{1}_{B_{3\rho/2}} C_{wH} \|f_0\|_{L^1(B_{3\rho/2})} \\ &\ge \mathbf{1}_{B_{3\rho/2}} C_{wH} \|\tilde{f}_0\|_{L^1(B_{3\rho/2})} \ge C_{wH} c_2 c_0^{-1} h_0. \end{aligned} $$

Because all the inequalities are constructive and proceeding as above, we deduce that condition (ii) in Lemma 2.4 holds and thus also (H2) with constructive constant $\kappa_0 := \kappa_1 - C_{wH}^{-1}c_2^{-1}c_0$. Finally, because of $(\kappa_1 - \mathcal{L})f_0 = 0$ on $\Omega \setminus B_{3\rho/2}$, we may apply the Harnack inequality [179, Cor. 8.21], and we classically deduce there exist constructive constants $C > 0$ and $C_\varrho > 0$ for any $\varrho > 0$ such that

$$ (7.21) \qquad C_\varrho \mathbf{1}_{\omega_\varrho} \le f_0 \le C, $$

with $\omega_\varrho := \{x \in \Omega; \delta(x) > \varrho\}$ and $\delta(x) := d(x, \partial\Omega)$ is the distance to the boundary function.

We can also get a constructive argument for (H2) by asking that condition (i) in Lemma 2.4 holds. We may for instance verify that the dual counterpart of the above constructive argument holds when $(c + \partial_i \beta_i)_- \in M^1$ and $b_i + \beta_i + \partial_j a_{ji} \in M^1$. More precisely, we establish in a similar way as above that the solution to the problem

$$ (7.22) \qquad \phi_0 \in H_0^1(\Omega), \quad (\kappa_1 - \mathcal{L}^*)\phi_0 = h_0, $$

satisfies

$$ (7.23) \qquad \kappa_0 \phi_0 \le \mathcal{L}^* \phi_0 \le \kappa_1 \phi_0, $$

for some constructive constants $\kappa_0 \le \kappa_1$. Similarly as above again, there exist constructive constants $C > 0$ and $C_\varrho > 0$ for any $\varrho > 0$ such that

$$ (7.24) \qquad C_\varrho \mathbf{1}_{\omega_\varrho} \le \phi_0 \le C. $$

**Third constructive argument for (H2).** We write

$$ (7.25) \qquad \mathcal{L}f = a_{ij}\partial_{ij}^2 f + \tilde{b}_i\partial_i f + \tilde{c}f, $$

with $\tilde{b}_i := b_i + \partial_j a_{ji} + \beta_i$ and $\tilde{c} := c + \partial_i \beta_i$. We further assume $\tilde{b}_i, \tilde{c} \in L^\infty$. In that case, we may also obtain an explicit lower bound on $\lambda_1$ by proceeding in the following way. We define $f_0(x) := \chi(|x|)$ with $\chi \in C_c^1(\mathbb{R}_+) \cap W^{2,\infty}(\mathbb{R}_+)$, $\mathbf{1}_{[0,1/3]} \le \chi \le \mathbf{1}_{[0,1]}$, $\chi' \le 0$ on $[0, 1]$, $\chi(s) := n^2(1-s)^2/2$ on $[\iota_n, 1]$,
for some $\kappa > 0$ and any $\lambda \in (0, \lambda_0)$, $\lambda_0 > 0$, and that from the very definition of the $\|\cdot\|_3$ norm
$$ \alpha \xi_{\lambda} \|\tilde{S}f\|_3 \le B \xi_{\lambda} \|\tilde{S}f\|_3, $$

for some $B > 0$. The three above estimates together imply

$$ Z_{\lambda} \|\tilde{S}f\|_{1} \le \|f\|_{1} + B\xi_{\lambda} \|\tilde{S}f\|_{3}. $$

Using the second estimate (6.25) and repeating the same proof, we have

$$ Z_{\lambda_{n+1}} \|\tilde{S}^{n+1} f\|_{1} \le \|\tilde{S}^{n} f\|_{1} + B \xi_{\lambda_{n+1}} \|f\|_{3}, $$

for any $n \ge 0$ and for any $\lambda_{n+1} > 0$. The discrete Grönwall lemma implies

$$ (6.31) \qquad \|\tilde{S}^n f\|_1 \le A_n \|f\|_1 + \sum_{k=1}^n A_{k,n} \xi_{\lambda_k} B \|f\|_3, \quad \forall n \ge 0, $$

where we have defined

$$ A_n := \prod_{k=1}^{n} a_k, \quad A_{k,n} = A_n/A_k = \prod_{i=k+1}^{n} a_i, \quad a_i := Z_{\lambda_i}^{-1}. $$

Observing that

$$ A_{k,n} \lesssim e^{-\kappa \sum_{i=k}^n \lambda_i} \lesssim e^{\kappa(\Lambda(k)-\Lambda(n))}, \quad \text{with} \quad \Lambda(t) := \int_0^t \lambda_s ds, $$

and $\lambda_s := \lambda_i$ if $s \in (i-1, i]$, we immediately conclude that the first estimate (6.20) holds true. We come back to the first inequality in (6.24) that we iterate and sum up in order to obtain

$$ \|\tilde{S}^n f\|_1 + \alpha \sum_{k=[n/2]+1}^{n} \|\tilde{S}^k f\|_0 \le \|\tilde{S}^{[n/2]} f\|_1, $$

for any $n \ge 1$. Together with the non expansion inequality

$$ [\tilde{S}^n f]_{\phi_1} \le [\tilde{S}^k f]_{\phi_1} \lesssim \|\tilde{S}^k f\|_0, \quad \forall n \ge k, $$

and the first estimate (6.20), we deduce

$$ (n - [n/2] - 1) \alpha [\tilde{S}^n f]_{\phi_1} \lesssim \Theta([n/2]) \|f\|_3, $$

which is nothing but (6.21).

<figure><img src="image_8.png" /></figure>

Let us finally emphasize that (C2) together with Theorem 6.2 or Theorem 6.4 provide the exponential stability (E3$_1$) with constructive constants, and thus also the spectral gap property (S3$_3$). On the other hand, (C2) together with Theorem 6.7 provide a quantitative stability estimate (E3$_2$). Both together, we deduce Theorem 1.1-(5).

## 7. PARABOLIC EQUATIONS

In this part, we consider a general elliptic operator in divergence form

$$ (7.1) \qquad \mathcal{L}f := \partial_i(a_{ij}\partial_j f) + b_i\partial_i f + \partial_i(\beta_i f) + cf, \quad f \in H_0^1(\Omega), $$

where $\Omega \subset \mathbb{R}^d$ is a bounded domain (i.e. an open and connected set) or $\Omega = \mathbb{R}^d$, and we always assume $d \ge 3$ (in order to simplify the discussions when using the Sobolev inequality). We also always assume at least a boundedness and ellipticity condition on the $(a_{ij})$ matrix, namely

$$ (7.2) \qquad a_{ij} \in L^\infty(\Omega), \quad \exists \nu > 0, \forall \xi \in \mathbb{R}^d, a_{ij} \xi_i \xi_j \ge \nu |\xi|^2, $$

and some conditions on the coefficients $b_i$, $\beta_j$ and $c$ which will be described below.

We aim to establish the existence of $(\lambda_1, f_1, \phi_1)$ solution to the first eig triplet problem

$$ (7.3) \qquad \lambda_1 \in \mathbb{R}, \quad 0 < f_1 \in H_0^1, \mathcal{L}f_1 = \lambda_1 f_1, \quad 0 < \phi_1 \in H_0^1, \mathcal{L}^*\phi_1 = \lambda_1 \phi_1, $$

and the existence of some (constructive) rate function $\Theta$ such that the rescaled semigroup $\tilde{S}$ associated to the generator $\tilde{\mathcal{L}} = \mathcal{L} - \lambda_1$ satisfies

$$ (7.4) \qquad \|\tilde{S}(t)f - \langle f, \phi_1 \rangle f_1\|_{H_0} \le \Theta(t) \|f - \langle f, \phi_1 \rangle f_1\|_{H}, $$
$\iota_n := 1 - 1/(2n)$, for some $n \ge 1$ to be chosen. As a consequence, $\chi'' = n^2$ on $[\iota_n, 1]$, $|\chi'| \le n$ on $[\iota_n, 1]$ and $\chi \ge 1/2$ on $[0, \iota_n]$. Denoting $s := |x|$, we compute

$$ \mathcal{L}f_0 = a_{ij}\{\chi''(s)\hat{x}_i\hat{x}_j + \chi'(s)\frac{\delta_{ij} - \hat{x}_i\hat{x}_j}{s}\} + \tilde{b}(x) \cdot \hat{x}\chi'(s) + \tilde{c}(x)\chi(s). $$

For $n$ large enough, we get

$$ \mathcal{L}f_0 \ge n^2\nu - n2A - nB - C \ge 0 \quad \text{on } B_1 \setminus B_{\iota_n}, $$

$$ \mathcal{L}f_0 \ge -A\{\|\chi''\|_{L^\infty} + \|\chi'(s)/s\|_{L^\infty}\} - B\|\chi'\|_{L^\infty} - C \ge \kappa_0\chi \quad \text{on } B_{\iota_n}, $$

with $A := \|a\|_{L^\infty(B_1)}$, $B := \|\tilde{b}\|_{L^\infty(B_1)}$, $C := \|\tilde{c}\|_{L^\infty(B_1)}$ and $\kappa_0 \in \mathbb{R}_-$. As a conclusion, we have again established condition (ii) in Lemma 2.4, so that condition (H2) holds.

**Fourth constructive argument for (H2).** We present a last situation when we are able to prove a quantitative version of condition (H2). We assume that $a \in C^0(\bar{\Omega})$, $\text{div}\beta \in L^{r/2}$, as well as $\tilde{b}_i \in L^r$ and $\tilde{c} \in L^{r/2}$ in the definition of (7.25). We define $h_0$ and $f_0$ as in the second constructive argument for (H2), so that (7.18) holds. Choosing $p \in (1, 2)$ defined by $1/p := 1/r + 1/2 > 2/r + 1/2^*$, we observe that

$$ \begin{aligned} \|\kappa_1 f_0 - \tilde{b}_i \partial_i f_0 - \tilde{c} f_0 - h_0\|_{L^p} &\lesssim \kappa_1 \|f_0\|_{L^2} + \|\tilde{b}_i\|_{L^r} \|\partial_i f_0\|_{L^2} + \|\tilde{c}\|_{L^{r/2}} \|f_0\|_{L^{2^*}} + \|h_0\|_{L^2} \\ &\lesssim \|h_0\|_{L^2}, \end{aligned} $$

from equation (7.19) and the coercivity estimate (7.8). From the Calderon-Zygmond regularity theory [85] or [179, Thm. 9.14], we also know that

$$ (7.26) \qquad \|f_0\|_{W^{2,p}(\Omega)} \lesssim \|a_{ij}\partial_{ij}^2 f_0\|_{L^p(\Omega)}. $$

Writing $a_{ij}\partial_{ij}^2 f_0 = \kappa_1 f_0 - \tilde{b}_i \partial_i f_0 - \tilde{c} f_0 - h_0$ and using the two above estimates, we deduce

$$ (7.27) \qquad \|f_0\|_{W^{2,1}(\Omega)} \lesssim \|h_0\|_{L^2(\Omega)}. $$

On the other hand, from (7.7) and the Poincaré inequality, we have

$$ 1 = \|h_0\|_{L^2}^2 = \langle (\kappa_1 - \mathcal{L})f_0, h_0 \rangle \lesssim \|\nabla f_0\|_{L^2} \|\nabla h_0\|_{L^2}. $$

Together with the estimate (7.27) and the Gagliardo-Niremberg inequality

$$ \|\nabla f\|_{L^2} \lesssim \|D^2 f\|_{L^1}^{1/2} \|f\|_{L^\infty}^{1/2}, $$

we obtain a lower bound $\|f_0\|_{L^\infty} \ge C_0 > 0$. We then conclude as in the second constructive argument for (H2).

**Condition (H3).** Because of Rellich-Kondrachov theorem on the compact embedding $H_0^1 \subset L^2$, the mapping $(\lambda - \mathcal{L})^{-1} : L^2 \to L^2$ is compact for any $\lambda \ge \kappa_1$. As a consequence, introducing the splitting $\mathcal{L} = \mathcal{A} + \mathcal{B}$ with $\mathcal{A} := \kappa_1 - \kappa_B$, $\kappa_B \in \mathbb{R}$ arbitrary, the operator $\mathcal{R}_B(\lambda) = (\lambda + \kappa_1 - \kappa_B - \mathcal{L})^{-1}$ is bounded uniformly on $\lambda \ge \kappa_B$ and it is compact for any $\lambda \ge \kappa_B$. We deduce from Lemma 2.8-(2) that (H3) holds for both the primal and the dual problems.

We may thus apply Theorem 2.21 and deduce the existence of a solution $(\lambda_1, f_1, \phi_1)$ to the first eigentriplet problem

$$ (7.28) \qquad \lambda_1 \in \mathbb{R}, \quad 0 \le f_1 \in H_0^1, \mathcal{L}f_1 = \lambda_1 f_1, \quad 0 \le \phi_1 \in H_0^1, \mathcal{L}^*\phi_1 = \lambda_1 \phi_1, $$

where both equations must be understood in the variational sense as a consequence of the discussion at the end of the proof of condition (H1).

**Condition (H4).** The strong maximum principle holds as already mentioned in the paragraph dedicated to condition (H2). As a consequence and thanks to Theorem 4.13, we know that the first eigentriplet problem (7.3) has a unique solution $(\lambda_1, f_1, \phi_1)$ which satisfies $f_1 > 0$, $\phi_1 > 0$, $N(\mathcal{L} - \lambda_1)^k = \text{Span}(f_1)$ and $N(\mathcal{L}^* - \lambda_1)^k = \text{Span}(\phi_1)$ for any $k \ge 1$.

**Condition (H5).** Consider $f \in D(\mathcal{L}^\infty)$ such that $0 < |f| \in D(\mathcal{L}^\infty)$ and

$$ \mathcal{L}|f| = \Re(\text{sign}f)\mathcal{L}f, $$

so that multiplying both term of the equation by $|f|$ and integrating, we have

$$ \Re e \langle \mathcal{L}f, \bar{f} \rangle = \langle \mathcal{L}|f|, |f| \rangle. $$
for any $t \ge 0$ and any $f \in H$, with $H \subset H_0 \subset L^2$.

**7.1. Diffusion with rough coefficients in a bounded domain.** In this section, we consider the general elliptic operator in divergence form (7.1) in the case of a bounded and smooth enough domain $\Omega \subset \mathbb{R}^d$ with general elliptic condition on $a_{ij}$ as formulated above. We further assume that

$$ (7.5) \qquad b_i, \beta_j \in L^r(\Omega), \quad c \in L^{r/2}(\Omega), \quad r > d. $$

In that situation, the first eigentriplet problem (7.3) has been considered by Chicco in [105, 106] and revisited in a slightly less general framework (all the coefficients belong to $L^\infty$) in [251], where the conclusions (C2) are established. We explain with all details the existence proof by following more or less the arguments presented in [251] stressing on the constructive way for obtaining the estimates, and next we present a proof of the geometric part and the stability part by taking advantage of the abstract material developed in the previous sections. It is worth emphasizing that our proof of the uniqueness of the first eigenfunction significantly differs from the one presented in [251] which is based on a dissipativity argument, probably related to the reverse Kato's inequality condition. The framework considered here is the usual generalized solutions or weak solutions framework which goes back at least to Stampacchia [338, 339], but it is reminiscent of previous contributions by Friedrichs [164, 165], Gårding [172], De Giorgi [126], Nash [296], Morrey [288], Moser [289, 290, 291], Ladyzhenskaya, Solonnikov, Ural'ceva [243, 241], Oleinik, Kruzhkov [302] and many others. Lot of the functional arguments are picked up from the book of Gilbarg and Trudinger, and more specifically from [178, Chapter 8], and also in recent notes by Kavian [229] and Vasseur [348]. It is worth emphasizing that the present analysis does not apply directly to elliptic operators in non divergence form, although this framework is considered in [251]. We expect that all the results developed below can be generalized to a non divergence form framework, for example the one developed in [47], but we do not follow this line of research in the present work.

The proof of (7.3) and (7.4) are straightforward consequences of the abstract results developed in the previous sections once we have been able to check that the corresponding hypotheses are fulfilled. In the sequel, we will then show how these hypotheses are met in the present context.

**Condition (H1).** We recall that a weak (or variational) solution to the elliptic equation

$$ \mathcal{L}f = g \in H^{-1}(\Omega), \quad f \in H_0^1(\Omega), $$

is a function $f \in H_0^1(\Omega)$ such that

$$ (7.6) \qquad D_{\mathcal{L}}(f, w) = \langle g, w \rangle, \quad \forall w \in H_0^1(\Omega), $$

where the (negative) Dirichlet form $D_{\mathcal{L}}$ is defined by

$$ D_{\mathcal{L}}(f, w) := - \int_{\Omega} (a_{ij} \partial_j f + \beta_i f) \partial_i w + \int_{\Omega} (b_i \partial_i f w + cfw), $$

for any $f, w \in H_0^1(\Omega)$. Most of the time, we will simply write

$$ (7.7) \qquad \langle \mathcal{L}f, w \rangle = \langle g, w \rangle, \quad \forall w \in H_0^1(\Omega), $$

instead of (7.6). For the reader convenience, we repeat here some estimates picked up in [339]. For $\lambda \in \mathbb{R}$ and $f \in H_0^1(\Omega)$, we start with

$$ \begin{aligned} \langle (\lambda - \mathcal{L})f, f \rangle &= \int_{\Omega} a_{ij} \partial_i f \partial_j f + \int_{\Omega} (\beta_i - b_i) \partial_i f f + \int_{\Omega} (\lambda - c) f^2 \\ &\geq \|f \sqrt{c_-} \|_{L^2}^2 + \nu \|\nabla f\|_{L^2}^2 - \|\beta - b\|_f \|\nabla f\|_{L^2} - \|\sqrt{c_+} f\|_{L^2}^2 + \lambda \|f\|_{L^2}^2 \\ &\geq \|f \sqrt{c_-} \|_{L^2}^2 + \frac{\nu}{2} \|\nabla f\|_{L^2}^2 - \frac{1}{2\nu} \|\beta - b\|_f \|\nabla f\|_{L^2} - \|\sqrt{c_+} f\|_{L^2}^2 + \lambda \|f\|_{L^2}^2, \end{aligned} $$
We next compute

$$ \Re e\langle \mathcal{L}f, \bar{f} \rangle = - \int_{\Omega} a_{kj} \Re e(\partial_j f \partial_k \bar{f}) + \int_{\Omega} (b_k - \beta_k) \Re e(\bar{f} \partial_k f) + \int_{\Omega} c |f|^2, $$

and

$$ \langle \mathcal{L} |f|, |f| \rangle = - \int_{\Omega} a_{kj} \partial_j |f| \partial_k |f| + \int_{\Omega} (b_k - \beta_k) \Re e(\bar{f} \partial_k f) + \int_{\Omega} c |f|^2, $$

where in the last equality, we have used that $\partial_k |f| = \frac{1}{|f|} \Re e(\bar{f} \partial_k f)$. From the three above equations, we deduce

$$ \int_{\Omega} a_{kj} [\partial_j |f| \partial_k |f| - \Re e(\partial_j f \partial_k \bar{f})] = 0. $$

Introducing the real and complex part decomposition $f = u + iv$, and similarly as in [231, Proof of Theorem 5.1], we next compute

$$ \begin{aligned} & \partial_j |f| \partial_k |f| - \Re e(\partial_j f \partial_k \bar{f}) \\ &= \frac{1}{|f|^2} [uv(\partial_k u \partial_j v + \partial_k v \partial_j u) - u^2 \partial_j v \partial_k v - v^2 \partial_j u \partial_k u] \\ &= \frac{1}{|f|^2} (u \partial_j v - v \partial_j u)(u \partial_k v - v \partial_k u), \end{aligned} $$

so that from the ellipticity condition on $a$, we have $u \partial_k v - v \partial_k u = 0$ a.e. on $\Omega$. On the other hand, from De Girogi-Nash-Moser regularity estimates (7.13) and (7.17), $f$ has Hölder regularity. In particular both functions $u$ and $v$ are continuous. Because $|f| \neq 0$, one of the two function is not identically vanishing, say for instance $v \neq 0$. There exists some points $x_0 \in \Omega$ such that $v(x_0) \neq 0$, say for instance $v(x_0) > 0$. Denoting by $\omega$ the connected component of the set $\{x \in \Omega; v(x) > 0\}$ containing $x_0$, we have $\nabla(u/v) = 0$ on $\omega$. Hence $u = \alpha v$ on $\omega$ for some $\alpha \in \mathbb{R}$, which implies that there exists $\sigma \in \mathbb{S}^1$ such that $f = \sigma|f|$ on $\omega$. If $\omega \neq \Omega$, we would have $|f| = 0$ on $\partial\omega \cap \Omega \neq \emptyset$, which would be a contradiction with the fact that $|f| > 0$. We conclude that $\omega = \Omega$ and thus that $f = \sigma|f|$, which is nothing but the reverse Kato's inequality condition (H5).

At this stage, we may use Theorem 5.16, in order to get the conclusion (C3) on the triviality of the boundary punctual spectrum.

In order to go one step further and establish the asymptotic stability of $f_1$, we may use the two following approaches which are consequences respectively of Lemma 7.1 and Lemma 7.2.

**Lemma 7.1.** For any $R > 0$, the set

$$ \mathcal{K} := \{f \in D(\mathcal{L}); [f] \le R, [\mathcal{L}f] \le R\} $$

is strongly compact in $L_{\text{loc}}^1(\Omega)$, where $[g] := \|g\|_{L_{\phi_1}^1}$.

**Proof of Lemma 7.1.** Consider $f \in \mathcal{K}$ so that $f \in H_0^1(\Omega)$ and

$$ \partial_i (a_{ij} \partial_j f) + b_i \partial_i f + \partial_i (\beta_i f) + cf = g \in L^2(\Omega). $$

From the renormalization theory of elliptic equations and the GRE trick (see for instance [269] and the references therein) for any renormalizing function $H \in C^2(\mathbb{R})$, there holds

$$ H''(u) f_1 \phi_1 a \nabla u \cdot \nabla u = \operatorname{div}(a \phi_1 \nabla (H(u) f_1)) - \operatorname{div}(f_1 H(u) a \nabla \phi_1) \\ + \operatorname{div}((b + \beta) H(u) f_1 \phi_1) + g H'(u) f_1 \phi_1, $$

with $u := f/f_1$. Considering $H \in W^{2,\infty}$ the even (and convex) function such that $H(0) = 0$ and $H'' := 1_{[n,n+1]}$, so that in particular $|H'(s)| \le 1$, and integrating the previous equation, we deduce

$$ \nu \int_{|u| \in [n, n+1]} |\nabla u|^2 f_1 \phi_1 \le \int |g| f_1 \phi_1 \le \|f_1\|_{L^\infty} R. $$

We proceed along the line of the proof of [62, Thm. 1]. For a fixed $\omega \subset \subset \Omega$, we define $B_n := \{x \in \omega; |u(x)| \in [n, n+1]\}$. Using that $f_1 > 0$ and $\phi_1 > 0$, there exists a constructive constant $C_{\omega,R} > 0$ such that

$$ \int_{B_n} |\nabla u|^2 \le C_{\omega}^2, \quad \forall n \ge 0. $$
using the Cauchy-Schwarz inequality and the Young inequality, and next

$$
\begin{align*}
\langle (\lambda - \mathcal{L})f, f \rangle &\ge \|f\sqrt{c_-}\|_{L^2}^2 + \frac{\nu}{4}\|\nabla f\|_{L^2}^2 + (\lambda - \frac{M}{2\nu} - M^{1/2})\|f\|_{L^2}^2 \\
&\phantom{\ge} + \frac{\nu}{4}C_\Omega \|f\|_{L^{2^*}}^2 - \frac{1}{2\nu}\|\beta - b|\mathbf{1}_{|\beta-b|\ge M}f\|_{L^2}^2 - \|\sqrt{c_+}\mathbf{1}_{c_+\ge M}f\|_{L^2}^2 \\
&\ge \|f\sqrt{c_-}\|_{L^2}^2 + \frac{\nu}{4}\|\nabla f\|_{L^2}^2 + (\lambda - \frac{M}{2\nu} - M^{1/2})\|f\|_{L^2}^2 \\
&\phantom{\ge} + (\frac{\nu}{4}C_\Omega - \frac{1}{2\nu}\|\beta - b|\mathbf{1}_{|\beta-b|\ge M}\|_{L^d}^2 - \|c_+\mathbf{1}_{c_+\ge M}\|_{L^{d/2}}\|f\|_{L^{2^*}}^2,
\end{align*}
$$

using the Sobolev inequality (with associated constant $C_\Omega$) and the Holder inequality. Choosing $M > 0$ large enough in such a way that the last term is positive, and next $\kappa_1 > 0$ large enough, we deduce for instance that

$$
(7.8) \qquad \langle (\lambda - \mathcal{L})f, f \rangle \ge \|f\sqrt{c_-}\|_{L^2}^2 + \frac{\nu}{4}\|\nabla f\|_{L^2}^2 + \|f\|_{L^2}^2, \quad \forall \lambda \ge \kappa_1.
$$

Thanks to the Lax-Milgram theorem and the above coercivity estimate, we deduce that $\lambda - \mathcal{L}$ is
invertible, and more precisely the mapping $(\lambda - \mathcal{L})^{-1} : H^{-1} \to H_0^1(\Omega)$ is well defined. We also
claim that $\lambda - \mathcal{L}$ enjoys a weak principle maximum, and more precisely

$$
(7.9) \qquad f \in H_0^1(\Omega), \quad (\lambda - \mathcal{L})f \ge 0 \quad \text{imply} \quad f \ge 0.
$$

Indeed, for such a function $f \in H_0^1(\Omega)$, we take $w = f_- \in H_0^1(\Omega)$, as a test function, and elementary Sobolev space calculus together with the previous estimate yields

$$
\begin{align*}
0 &\le \langle (\lambda - \mathcal{L})f, f_- \rangle = -\langle (\lambda - \mathcal{L})f_-, f_- \rangle \\
  &\le -\|f_-\sqrt{c_-}\|_{L^2}^2 - \frac{\nu}{4}\|\nabla f_\-\|_{L^2}^2 - \|f_\-\|_{L^2}^2 \le 0,
\end{align*}
$$

so that $f_- = 0$ and $f \ge 0$. We thus deduce $(\lambda - \mathcal{L})^{-1} : L_+^2 \to L_-^2$, and from J.-L. Lions theory on parabolic equation (see for instance [250, Chapter 3]), we next deduce that $\mathcal{L}$ is the generator in $L^2$ of a positive semigroup $S_{\mathcal{L}}$, so that (H1) holds. It is worth emphasizing at this point that the semigroup $S$ built thanks to Lions's theory is defined by $S(t)f_0 = f$ for any $f_0 \in L^2$, where $f \in \mathcal{E} := C([0, \infty); L^2) \cap L_{\text{loc}}^2([0, \infty); H_0^1) \cap H_{\text{loc}}^1([0, \infty); H^{-1})$ is the unique (variational) solution to the equation

$$
(7.10) \qquad (f(T), g(T))_{L^2} - (f_0, g(0))_{L^2} = \int_0^T \{ \langle \partial_t g, f \rangle_{H^{-1}, H_0^1} + D_{\mathcal{L}}(f, g) \} ds,
$$

for any $T > 0$ and $g \in \mathcal{E}$. Choosing $g = f$ in the above equation, we classically compute

$$
\frac{1}{2} \|f(t)\|_{L^2}^2 - \frac{1}{2} \|f_0\|_{L^2}^2 - \int_0^t D_{\mathcal{L}}(f, f) ds = 0, \quad \forall t > 0,
$$

which together with (7.8) implies

$$
\frac{1}{t} \int_{0}^{t} \frac{\nu}{4} \|\nabla f\|_{L^2}^2 ds \le - \left( \frac{f(t) - f_0}{t}, \frac{f(t) + f_0}{2} \right)_{L^2} + \frac{\kappa_1}{t} \int_{0}^{t} \|f\|_{L^2}^2 ds, \quad \forall t > 0.
$$

When $f_0 \in D(\mathcal{L})$, the RHS is bounded and there thus exists a sequence $t_n \to 0$ such that
$\|\nabla f(t_n)\|_{L^2}$ is bounded. That implies $f_0 \in H_0^1(\Omega)$ and thus $D(\mathcal{L}) \subset H_0^1(\Omega)$. Similarly, we may
consider the dual Dirichlet form $D^*(f,g) := D_{\mathcal{L}}(g,f)$ and build an associated positive semigroup
$S^*$ through Lions's theory described above. More precisely $S^*(t)g_0 = g$ for any $t \ge 0$ and $g_0 \in L^2$,
where $g \in \mathcal{E}$ is the unique (variational) solution to the equation

$$
(g(t), f(t))_{L^2} - (g_0, f(0))_{L^2} = \int_0^t \{ \langle \partial_t f, g \rangle_{H^{-1}, H_0^1} + D^*(g, f) \} ds,
$$

for any $t > 0$ and $f \in \mathcal{E}$. Now, we fix $T > 0$, $g_T \in L^2$ and we set $g(t) := S^*(T - t)g_T$, so that $g$ is
a solution to the backward evolution equation

$$
-\partial_t g = \mathcal{L}^* g, \quad g(T) = g_T,
$$

with

$$
\mathcal{L}^* g := \partial_j (a_{ij} \partial_i g) - \partial_i (b_i g) - \beta_i \partial_i g + cg.
$$
From the Cauchy-Schwarz inequality, we have

$$ (7.29) \qquad \int_{B_n} |\nabla u| \le C_\omega \mathrm{meas}(B_n)^{1/2}, \quad \forall n \ge 0. $$

On the other hand, denoting by $1^* := d/(d-1)$ the Sobolev exponent, we have

$$ \int_{B_n} |\nabla u| \le C_{\omega,R} (n^{-1^*} \int_{B_n} |u|^{1^*})^{1/2}. $$

Summing up and using the Cauchy-Schwarz inequality again, we have

$$ \begin{aligned} \sum_{n \ge 1} \int_{B_n} |\nabla u| & \le C_{\omega,R} \left( \sum_{n \ge 1} n^{-1^*} \right)^{1/2} \left( \sum_{n \ge 1} \int_{B_n} |u|^{1^*} \right)^{1/2} \\ & \le C_{\omega,R} \left( \sum_{n \ge 1} n^{-1^*} \right)^{1/2} \|u\|_{L^{1^*} L^1}^{1^*/2}. \end{aligned} $$

Together with (7.29) for $n=0$, we deduce

$$ \|\nabla u\|_{L^1(\omega)} \le C'_{\omega,R} (1 + \|\nabla u\|_{L^1(\omega)}^{1^*/2}). $$

Because $1^*/2 \le 3/4 < 1$ (recall that $d \ge 3$), we can kill the last term, and we obtain the estimate

$$ \|\nabla(f/f_1)\|_{L^1(\omega)} \le C'', \quad \forall f \in \mathcal{K}, $$

for some constant $C'' := C''_{\omega,R} > 0$. We classically conclude thanks to the Rellich-Kondrachov theorem. □

From the above lemma and Theorem 5.23, we deduce that $\tilde{S}(t)f \to \langle f, \phi_1 \rangle f_1$ in the $L^1_{\phi_1}$ norm sense as $t \to \infty$ for any $f \in L^2(\Omega)$. The alternative approach is based on the following result.

**Lemma 7.2.** Setting $\kappa := \kappa_0 - 1$, there exist $A, \alpha, R > 0$ such that

$$ \text{(i)} \quad \sup_{z \in \Delta_{\kappa}} \langle y \rangle^{\alpha} \| \mathcal{R}_{\mathcal{B}}(z) \|_{\mathcal{B}(L^2; H_0^1)} + \sup_{z \in \Delta_{\kappa} \setminus B_R} \| \mathcal{R}_{\mathcal{L}}(z) \|_{\mathcal{B}(L^2; H_0^1)} < \infty, $$

$$ \text{(ii)} \quad \Sigma(\mathcal{L}) \cap \Delta_{\kappa} \subset \Sigma_d(\mathcal{L}) \cap B_R, $$

$$ \text{where } \mathcal{B} := \mathcal{L} - A \text{ and } z = x + iy, x, y \in \mathbb{R}. $$

**Proof of Lemma 7.2.** Let us consider an a priori solution to the stationary problem

$$ f \in H_0^1, \quad z = x + iy \in \Delta_{\kappa}, \quad (\mathcal{L} + z)f = g \in L^2. $$

This one satisfies

$$ \left| - \int (a \nabla f + \beta f) \cdot \nabla \bar{f} + \int b \cdot \nabla f \bar{f} + (c+z) |f|^2 \right| = \left| \int g \bar{f} \right|. $$

Using the elliptic condition, the Cauchy-Schwarz inequality and triangular inequalities, we get

$$ \begin{aligned} \left| \int g \bar{f} \right| & \ge \left| \int a \nabla f \nabla \bar{f} + ((c+x)_+ + iy) |f|^2 \right| - \left| \int b \cdot \nabla f \bar{f} - \beta f \cdot \nabla \bar{f} + (c+x)_- |f|^2 \right| \\ & \ge \frac{\nu}{2} \|\nabla f\|_{L^2}^2 + \left(\frac{|y|}{2} - x_-\right) \|f\|_{L^2}^2 - \|(|b| + |\beta|)f\|_{L^2} \|\nabla f\|_{L^2} - \|\sqrt{c_-} f\|_{L^2}^2. \end{aligned} $$

Using next similar arguments and those introduced in the paragraph dedicated to condition (H1) and with similar definition for the constant $M := M(b, \beta, c) > 0$, we deduce

$$ \left| \int g \bar{f} \right| \ge \left( \frac{|y|}{2} - x_- - M \right) \|f\|_{L^2}^2 + \frac{\nu}{4} \|\nabla f\|_{L^2}^2. $$

Defining the sectorial set

$$ \mathcal{S} := \{z = x + iy \in \mathbb{C}; |y| > 2x_- + M\}, $$

we have established the a priori estimates

$$ \|f\|_{L^2} \le \left( \frac{|y|}{2} - x_- - M \right)^{-1/2} \|g\|_{L^2}, $$

$$ \|\nabla f\|_{L^2} \le 2\nu^{-1/2} \left( \frac{|y|}{2} - x_- - M \right)^{-1/4} \|g\|_{L^2}, $$
The variational formulation of this last problem is

$$ (7.11) \qquad (g_T, f(T))_{L^2} - (g(0), f(0))_{L^2} = \int_0^T \{\langle \partial_t f, g \rangle_{H^{-1}, H_0^1} - D^*(g, f)\} ds, $$

for any $f \in \mathcal{E}$. Summing up (7.10) and (7.11) with $f(t) := S(t)f_0$ for $f_0 \in L^2$ and $g(t) := S^*(T-t)g_T$ for $g_T \in L^2$, we deduce

$$ (S(T)f_0, g_T)_{L^2} = (S^*(T)g_T, f_0)_{L^2}. $$

In other words, we have established that $S^* = (S_\mathcal{L})^*$ and thus that $\mathcal{L}^*$ is the generator of the semigroup $S^*$.

**Condition (H2).** Let us consider a ball $B_R$, $R > 0$, such that $B_{4R} \subset \Omega$ and next the solution

$$ (7.12) \qquad f_0 \in H_0^1(\Omega), \quad (\kappa_1 - \mathcal{L})f_0 = \mathbf{1}_{B_R}, $$

which exists from the above discussion. We next recall some classical results. On the one hand, from [338, Sec. 3 & Sec. 4] or [178, Thm. 8.15] (see also the original papers [126, 296, 289]), the following global $L^\infty$ De Gorgi-Nash-Moser type estimate

$$ (7.13) \qquad \|f_+\|_{L^\infty(\Omega)} \lesssim \|f_+\|_{L^2(\Omega)} + \|g\|_{L^{r/2}(\Omega)} $$

holds for any subsolution

$$ f \in H_0^1(\Omega), \quad (\lambda - \mathcal{L})f \le g \in L^{r/2}(\Omega). $$

The local estimate variant [178, Thm. 8.18] (or weak Harnack inequality)

$$ (7.14) \qquad \|f\|_{L^p(B_{2R})} \lesssim \inf_{B_R} f + \|g\|_{L^{r/2}(\Omega)}, \quad \forall p \in [1, 2^*/2), $$

also holds for a nonnegative supersolution

$$ f \in H^1(\Omega), \quad f \ge 0 \text{ on } B_{4R} \subset \Omega, \quad (\lambda - \mathcal{L})f \ge g \in L^{r/2}(\Omega), $$

from what one deduces that a strong maximum principle [178, Thm. 8.19] holds. More precisely, under the additional one side pointwise bound

$$ (7.15) \qquad c + \operatorname{div}\beta \le c_0 \quad \text{or} \quad c - \operatorname{div}b \le c_0, $$

for some $c_0 \in \mathbb{R}$, we have that, for any $f \in H_0^1(\Omega)$,

$$ (7.16) \qquad \mathcal{L}f \le 0 \text{ in } \Omega, \quad f \ge 0 \text{ in } \Omega \quad \text{imply} \quad f \equiv 0 \text{ or } f > 0 \text{ a.e. in } \Omega. $$

When indeed $f \ne 0$, we may choose $B_{4R} \subset \Omega$ such that $\|f\|_{L^1(B_{2R})} > 0$ and thus $\inf_{B_R} f > 0$ from (7.14) (with $g = 0$) and because constants are supersolutions thanks to the first condition in (7.15). In the case only the second condition holds in (7.15), the same argument implies that $\mathcal{L}^*$ satisfies the strong maximum principle and thus also $\mathcal{L}$ thanks to Lemma 4.9. We conclude that $f$ is positive by a connectedness argument. An alternative and less demanding proof is presented in [105, Cor. 1] where (7.16) is established without the additional assumption (7.15).

On the other hand, the following Hölder regularity estimate [338, Théorème 7.1] and [178, Thm. 8.29] (see also the original papers [126, 296, 289]) of De Gorgi-Nash-Moser type

$$ (7.17) \qquad \|f\|_{C^\alpha(\Omega)} \le C \|(\lambda - \mathcal{L})f\|_{L^\infty(\Omega)} $$

holds true for some $\alpha = \alpha(a_{ij}) \in (0, 1)$ and $C > 0$. These last two pieces of information together and the fact that $f_0 \ne 0$ imply that there exists a constant $\theta > 0$ such that $f_0 \ge \theta \mathbf{1}_{B_R}$, and thus

$$ \mathcal{L}f_0 \ge (\kappa_1 - \theta^{-1})f_0. $$

That is condition (i) in Lemma 2.4, so that condition (H2) holds thanks to Lemma 2.4. Presented in that way, the above estimate is not really constructive, but the constant $\theta := \inf_{B_R}(\kappa_1 - \mathcal{L})^{-1}\mathbf{1}_{B_R}$ can also be considered as a geometric quantity associated to geometric properties of the operator and the domain. In order to make the lower bound (H2) constructive, we propose several strategies below, each one requiring additional mild assumptions about the coefficients and none being really better than the others.
for any $z \in S$. We classically and immediately deduce that $\rho(\mathcal{L}) \supset S$ and the resolvent estimate
$$ \|\mathcal{R}_{\mathcal{L}}(z)\|_{\mathcal{B}(L^2, H_0^1)} \lesssim (\frac{|y|}{2} - x_- - M)^{-1/2} + (\frac{|y|}{2} - x_- - M)^{-1/4} $$
for any $z \in S$, and in particular the estimate (i) holds true.

On the other hand, because $\mathcal{L}$ has compact resolvent as established just above or during the proof of (H3) and using the Fredholm alternative, we have $\Sigma(\mathcal{L}) = \Sigma_d(\mathcal{L})$ and $\Sigma(\mathcal{L}) \cap \Delta_\kappa$ is finite for any $\kappa \in \mathbb{R}$, what is nothing but the property (ii). $\square$

From the above lemma and Theorem 5.30 or Theorem 5.32, we deduce that $\tilde{S}(t)f \to \langle f, \phi_1 \rangle f_1$ in the $L^2$ norm sense as $t \to \infty$ for any $f \in L^2(\Omega)$ with exponential rate.

We may summarize our analysis in the following result.

**Theorem 7.3.** Consider the elliptic operator (7.1) in a bounded domain and assume that the coefficients satisfy (7.2), (7.5) and (7.18). Then the conclusions (C3) holds as well as (E2) in $L_{\phi_1}^1$ norm and (E3$_1$) in $L^2$ with non constructive rate.

It is however worth emphasizing again that the above approach is definitively not constructive. We propose now an alternative approach which is constructive.

**Quantitative estimate of stability.**

Using the Doblin-Harris type approach presented in Section 6, we are able to establish a rate of convergence to the principal dynamic, at least in a regular framework. We thus make some regularity assumptions on $\Omega$ and additional regularity assumptions on the coefficients.

- For the domain, we assume that there exists a constant $r_\Omega > 0$ such that for any $x \in \Omega$ there is $y \in \Omega$ such that $x \in B(y, r_\Omega) \subset \Omega$, in particular, for any $x \in \partial\Omega$ there is $y \in \Omega$ such that $x \in \partial B(y, r_\Omega)$, $B(y, r_\Omega) \subset \Omega$. We also assume that $\Omega$ is $C^{1,1}$.

- For the coefficients, we assume $a_{ij} \in C(\bar{\Omega})$, $\tilde{b}_i, \tilde{c} \in L^\infty(\Omega)$, where $\tilde{b}_i$ and $\tilde{c}$ are defined in (7.25).

**Theorem 7.4.** Consider the elliptic operator (7.1) in a bounded domain and assume that the assumptions of Theorem 7.3 hold together with the above additional regularity assumptions on the coefficients and the boundary. Then the conclusion (E3$_1$) holds with constructive exponential rate.

The proof of Theorem 7.4 follows from Theorem 6.3. We split the proof into several steps.

**- Step 1. Regularity estimates.** Thanks to De Giorgi-Nash-Moser regularity technique for parabolic equations developed for instance in [243] (in Russian), [346, Thm. 1.3, Thm. 2.2] as well as more recently in [230, Lem. 2.7] and [191, Thm. 1.1], there exists $\alpha = \alpha(a_{ij}) \in (0, 1)$ and for any $T_1 > T_0 > 0$ and any $\varrho \in (0, 1)$, there exist constructive constants $C_i = C_i(\|f\|_{L^\infty_t L^2_x}, T, \tau, r)$ such that any solution $f \in L^\infty(0, \infty; L^2(\Omega))$ to the parabolic equation $\partial_t f = \mathcal{L}f$ satisfies

$$ (7.30) \qquad \|f\|_{L^\infty([T_0, T_1] \times \Omega)} \le C_1, \quad \|f\|_{C^\alpha([T_0, T_1] \times \omega_\varrho)} \le C_2, $$

with $\omega_r := \{x \in \Omega; d(x, \partial\Omega) > r\}$. More precisely, in order to establish the second estimate in (7.30) with constructive constant, one may observe that the proof of [191, Prop. 2.4] may be repeated in order to get that solutions to the parabolic equation considered in the present framework fall into De Giorgi classes as defined in [191, Definition 2.3], and thus [191, Thm. 1.1] applies.

On the other hand, in this context and because of the regularity assumptions, we may establish a more accurate regularity estimate. More precisely, by gathering the Sobolev inequality and the Calderon-Zygmond estimate (7.26), we obtain the classical constructive regularity estimate

$$ (7.31) \qquad \|u\|_{C^{0,1}(\Omega)} \lesssim \|u\|_{W^{2,d+1}(\Omega)} \lesssim \|(\kappa_1 - \mathcal{L})u\|_{L^{d+1}(\Omega)}, $$

see for instance Theorem 7.10, Theorem 7.25 and Lemma 9.17 in [179]. Iterating the same kind of arguments, we get

$$ (7.32) \qquad \|u\|_{C^{0,1}(\Omega)} \le C \|(\kappa_1 - \mathcal{L})^k u\|_{L^2(\Omega)}, $$

with constructive constants $C$ and $k$.

**- Step 2. Harnack estimate.** We claim that for any $T > t_0 > 0$ and $\varrho > 0$, there exist a constant $C_H > 0$ such that, for any $f_0 \in L^2$, the associated solution $f := S_{\mathcal{L}}f_0$ satisfies

$$ (7.33) \qquad \sup_{\omega_\varrho} f_{t_0} \le C_H \inf_{\omega_\varrho} f_T. $$
**First constructive argument for (H2).** In the case when $\mathcal{L}$ is self-adjoint, that corresponds to the case $a_{ij} = a_{ji}$ and $b_i + \beta_i = 0$, we classically know (that has been recalled in Section 2.3, see (2.34)) that

$$ \lambda_1 = \inf_{f \in X_+ \setminus \{0\}} \frac{\langle \mathcal{L}f, f \rangle}{\|f\|^2} = \inf_{f \in H_0^1, \|f\|_{L^2} = 1} \int_{\mathcal{O}} \{a \nabla f \cdot \nabla f + cf^2\}, $$

from what and the Sobolev imbedding, we get

$$ \lambda_1 \ge \inf_{f \in H_0^1, \|f\|_{L^2} = 1} \{(\nu C_{\Omega} - \|c - 1_{c \ge M}\|_{L^{d/2}})\|f\|_{L^{2*}}^2 - M\} \ge -M, $$

by choosing $M$ large enough. That gives an explicit lower bound on $\lambda_1$.

**Second constructive argument for (H2).** We give another constructive argument without assuming any self-adjointness property. We rather assume

$$ (7.18) \qquad (\partial_i b_i - c)_+ \in M^1(\Omega), \quad b_i + \beta_i - \partial_j a_{ij} \in M^1(\Omega). $$

We fix $h_0 \in C_c^2(\Omega)$ such that $c_01_{B_\rho} \le h_0 \le c_01_{B_{3\rho/2}}$ with $B_{8\rho} \subset \Omega$ and $\|h_0\|_{L^2} = 1$. We next define $f_0$ as the (positive) solution to

$$ (7.19) \qquad f_0 \in H_0^1(\Omega), \quad (\kappa_1 - \mathcal{L})f_0 = h_0, $$

so that $f_0 \in C^\alpha(\Omega)$ from (7.13) and (7.17), and similarly

$$ (7.20) \qquad \tilde{f}_0 \in H_0^1(B_{2\rho}), \quad (\kappa_1 - \mathcal{L})\tilde{f}_0 = h_0, $$

so that $\tilde{f}_0 \in C^\alpha(B_{2\rho})$ from (7.13) and (7.17). We observe that $0 \le \tilde{f}_0 \le f_0$ thanks to the weak maximum principle. We then compute

$$ 1 = \|h_0\|_{L^2}^2 = \int_{B_{2\rho}} h_0(\kappa_1 - \mathcal{L})\tilde{f}_0 = \int_{B_{2\rho}} \tilde{f}_0(\kappa_1 - \mathcal{L}^*)h_0 \le \|\tilde{f}_0\|_{L^\infty} \|(\kappa_1 - \mathcal{L}^*)h_0\|_{M^1}, $$

where the last term is finite because of the additional hypothesis (7.18). We conclude to a first constructive lower bound $\|\tilde{f}_0\|_{L^\infty(B_{2\rho})} \ge c_1 > 0$. Because of the Holder continuity, we also have $\|\tilde{f}_0\|_{L^1(B_{2\rho})} \ge c_2$ with constructive constant $c_2 = c_2(c_1, \alpha, d) > 0$. Thanks to (7.14) (with $g = 0$), we obtain

$$ \begin{aligned} f_0 &\ge \mathbf{1}_{B_{3\rho/2}} \inf_{B_{3\rho/2}} f_0 \ge \mathbf{1}_{B_{3\rho/2}} C_{wH} \|f_0\|_{L^1(B_{3\rho/2})} \\ &\ge \mathbf{1}_{B_{3\rho/2}} C_{wH} \|\tilde{f}_0\|_{L^1(B_{3\rho/2})} \ge C_{wH} c_2 c_0^{-1} h_0. \end{aligned} $$

Because all the inequalities are constructive and proceeding as above, we deduce that condition (ii) in Lemma 2.4 holds and thus also (H2) with constructive constant $\kappa_0 := \kappa_1 - C_{wH}^{-1}c_2^{-1}c_0$. Finally, because of $(\kappa_1 - \mathcal{L})f_0 = 0$ on $\Omega \setminus B_{3\rho/2}$, we may apply the Harnack inequality [178, Cor. 8.21], and we classically deduce there exist constructive constants $C > 0$ and $C_\varrho > 0$ for any $\varrho > 0$ such that

$$ (7.21) \qquad C_\varrho \mathbf{1}_{\omega_\varrho} \le f_0 \le C, $$

with $\omega_\varrho := \{x \in \Omega; \delta(x) > \varrho\}$ and $\delta(x) := d(x, \partial\Omega)$ is the distance to the boundary function.

We can also get a constructive argument for (H2) by asking that condition (i) in Lemma 2.4 holds. We may for instance verify that the dual counterpart of the above constructive argument holds when $(c + \partial_i \beta_i)_- \in M^1$ and $b_i + \beta_i + \partial_j a_{ji} \in M^1$. More precisely, we establish in a similar way as above that the solution to the problem

$$ (7.22) \qquad \phi_0 \in H_0^1(\Omega), \quad (\kappa_1 - \mathcal{L}^*)\phi_0 = h_0, $$

satisfies

$$ (7.23) \qquad \kappa_0 \phi_0 \le \mathcal{L}^* \phi_0 \le \kappa_1 \phi_0, $$

for some constructive constants $\kappa_0 \le \kappa_1$. Similarly as above again, there exist constructive constants $C > 0$ and $C_\varrho > 0$ for any $\varrho > 0$ such that

$$ (7.24) \qquad C_\varrho \mathbf{1}_{\omega_\varrho} \le \phi_0 \le C. $$

**Third constructive argument for (H2).** We write

$$ (7.25) \qquad \mathcal{L}f = a_{ij}\partial_{ij}^2 f + \tilde{b}_i\partial_i f + \tilde{c}f, $$
The proof mainly follows from Aronson-Serrin [25] (see also [292, 222, 224, 240, 223, 345, 346, 241] for similar results). First, we know from [25, Thm. 3] that

$$ (7.34) \qquad \max_{Q^*(\rho)} f \le C \min_{Q(\rho)} f, $$

for any $\rho > 0$, $t > 0$ such that $Q^*(3\rho) \subset (0, \infty) \times \Omega$, where $Q(\rho) := [t - \rho^2, t] \times C(\rho)$, $Q^*(\rho) := [t - 8\rho^2, t - 7\rho^2] \times C(\rho)$ and $C(\rho)$ is a cube with length $\rho$. To avoid technical issues we assume that $w_\rho$ is convex. In other case, the geometrical condition given above implies that there is $N \in \mathbb{Z}_+$ such that any two points $x, y \in \Omega$ can be connected by a polygonal path of at most $N$ segments, and we can argue as follows for any segment. We define $D := \sup_{a,b \in \Omega} d(a, b)$ the diameter of $\Omega$ and we choose $r' < \varrho/7$ such that

$$ 7(\lfloor \frac{D}{2r'} \rfloor + 1)(r')^2 < T - t_0. $$

For any $x, y \in \omega_\rho$, we also define $N_c = \lfloor \frac{|x-y|}{r'} \rfloor$. Since $\omega_\rho$ is convex, $r' < \varrho/7$, we have that the family of cubes $\{C(x_i, 2r')\}_{i=0, N_c}$ of center $x_i$ and length $2r'$ for $x_i = x + \frac{(x-y)i}{N_c}$ satisfy that $C(x_i, 6r') \subset \Omega$ and $C(x_i, 2r') \cap C(x_{i+1}, 2r') \neq \emptyset$ for any $i = 0, \dots, N_c$. As a consequence, we can apply Aronson-Serrin estimate (7.34) for each cube to obtain

$$ \max_{C(x_i, 2r')} f_{t_i} \le C_{2r'} \min_{C(x_i, 2r')} f_{t_{i+1}}, $$

with $t_i = t_0 + 7i(2r')^2$. Taking $y_i \in C(x_i, 2r') \cap C(x_{i+1}, 2r')$, we deduce

$$ \max_{C(x_i, 2r')} f_{t_i} \le C_{2r'} \min_{C(x_i, 2r')} f_{t_{i+1}} \le C_{2r'} f_{t_{i+1}}(y_i) \le C_{2r'} \max_{C(x_{i+1}, 2r')} f_{t_{i+1}} \le C_{2r'}^2 \min_{C(x_{i+1}, 2r')} f_{t_{i+2}}. $$

By induction, we obtain

$$ f_{t_0}(x) \le \max_{C(x_1, 2r')} f_{t_1} \le C_{2r'}^{N_c} \min_{C(x_{N_c}, 2r')} f_{t_{N_c}} \le C_{2r'}^{N_c} f_{t_{N_c}}(y), $$

with $t_{N_c} = t_0 + 7N_c r'^2 \le T$. Note that in any case the constant $C_{2r'}$ is the same since it only depends on the length $2r'$ and the coefficient of the equation. We have thus established (7.33) with $C_H := C_{2r'}^{\lfloor \frac{D}{2r'} \rfloor + 1}$.

On the other hand, we state an improved version of the already mentioned stationary Harnack inequality. Because of the interior ball condition the Hopf Lemma (see for instance the proof of [179, Lem. 3.4]) claims that for any $\varrho \in (0, r_\Omega/2]$ there exists a constructive constant $\alpha > 0$ such that if $u \in W^{2,p}(\Omega)$, $p > d$, is such that

$$ u \ge \mathbf{1}_{\omega_\rho}, \quad (\kappa_1 - \mathcal{L}^*)u \ge 0, $$

then $u$ satisfies

$$ (7.35) \qquad u \ge \chi(x) := e^{-\alpha(2\varrho - \delta(x))^2} - e^{-\alpha(2\varrho)^2} \quad \text{on } \omega_\rho^c. $$

Let us give two applications of the above sharp regularity and positivity estimates. First, recalling (7.24) and using (7.31) and (7.35), we deduce that there exist two constructive constants $c_i \in (0, \infty)$ such that

$$ (7.36) \qquad c_0\delta \le \phi_0 \le c_1\delta \quad \text{on } \Omega. $$

Consider now $f_1 \in H_0^1(\Omega)$ the positive first eigenfunction with normalization $\|f_1\|_{L^2} = 1$. Using the estimate of regularity (7.32) on the iterated equation $(\kappa_1 - \mathcal{L})^k f_1 = (\kappa_1 - \lambda_1)^k f_1$, we have

$$ \|f_1\|_{L^\infty(\Omega)} \le \|f_1\|_{C^{0,1}(\Omega)} \le C_1, $$

for some constructive constant $C_1 \in (0, \infty)$. Next using the elementary inequality

$$ 1 = \int_{\Omega} f_1^2 \le \|f_1\|_{L^{\infty}} \|f_1\|_{L^1} \le C_1 \|f_1\|_{L^1}, $$

we deduce

$$ \begin{aligned} |\Omega| \sup_{\omega_\rho} f_1 &\ge \int_{\omega_\rho} f_1 = \|f_1\|_{L^1} - \int_{\omega_\rho^c} f_1 \\ &\ge 1/C_1 - C_1|\omega_\rho^c| \ge 1/(2C_1), \end{aligned} $$
with $\tilde{b}_i := b_i + \partial_j a_{ji} + \beta_i$ and $\tilde{c} := c + \partial_i \beta_i$. We further assume $\tilde{b}_i, \tilde{c} \in L^\infty$. In that case, we may also obtain an explicit lower bound on $\lambda_1$ by proceeding in the following way. We define $f_0(x) := \chi(|x|)$ with $\chi \in C_c^1(\mathbb{R}_+) \cap W^{2,\infty}(\mathbb{R}_+)$, $\mathbf{1}_{[0,1/3]} \le \chi \le \mathbf{1}_{[0,1]}$, $\chi'(s) := n^2(1-s)^2/2$ on $[\iota_n, 1]$, $\iota_n := 1 - 1/(2n)$, for some $n \ge 1$ to be chosen. As a consequence, $\chi'' = n^2$ on $[\iota_n, 1]$, $|\chi'| \le n$ on $[\iota_n, 1]$ and $\chi \ge 1/2$ on $[0, \iota_n]$. Denoting $s := |x|$, we compute

$$ \mathcal{L}f_0 = a_{ij}\{\chi''(s)\hat{x}_i\hat{x}_j + \chi'(s)\frac{\delta_{ij} - \hat{x}_i\hat{x}_j}{s}\} + \tilde{b}(x) \cdot \hat{x}\chi'(s) + \tilde{c}(x)\chi(s). $$

For $n$ large enough, we get

$$ \mathcal{L}f_0 \ge n^2\nu - n2A - nB - C \ge 0 \quad \text{on } B_1 \setminus B_{\iota_n}, $$

$$ \mathcal{L}f_0 \ge -A\{\|\chi''\|_{L^\infty} + \|\chi'(s)/s\|_{L^\infty}\} - B\|\chi'\|_{L^\infty} - C \ge \kappa_0\chi \quad \text{on } B_{\iota_n}, $$

with $A := \|a\|_{L^\infty(B_1)}$, $B := \|\tilde{b}\|_{L^\infty(B_1)}$, $C := \|\tilde{c}\|_{L^\infty(B_1)}$ and $\kappa_0 \in \mathbb{R}_-$. As a conclusion, we have again established condition (ii) in Lemma 2.4, so that condition (H2) holds.

**Fourth constructive argument for (H2).** We present a last situation when we are able to prove a quantitative version of condition (H2). We assume that $a \in C^0(\bar{\Omega})$, $\text{div}\beta \in L^{r/2}$, as well as $\tilde{b}_i \in L^r$ and $\tilde{c} \in L^{r/2}$ in the definition of (7.25). We define $h_0$ and $f_0$ as in the second constructive argument for (H2), so that (7.18) holds. Choosing $p \in (1,2)$ defined by $1/p := 1/r + 1/2 > 2/r + 1/2^*$, we observe that

$$ \begin{aligned} \|\kappa_1 f_0 - \tilde{b}_i \partial_i f_0 - \tilde{c} f_0 - h_0\|_{L^p} &\lesssim \kappa_1 \|f_0\|_{L^2} + \|\tilde{b}_i\|_{L^r} \|\partial_i f_0\|_{L^2} + \|\tilde{c}\|_{L^{r/2}} \|f_0\|_{L^{2^*}} + \|h_0\|_{L^2} \\ &\lesssim \|h_0\|_{L^2}, \end{aligned} $$

from equation (7.19) and the coercivity estimate (7.8). From the Calderon-Zygmond regularity theory [84] or [178, Thm. 9.14], we also know that

$$ (7.26) \qquad \|f_0\|_{W^{2,p}(\Omega)} \lesssim \|a_{ij}\partial_{ij}^2 f_0\|_{L^p(\Omega)}. $$

Writing $a_{ij}\partial_{ij}^2 f_0 = \kappa_1 f_0 - \tilde{b}_i \partial_i f_0 - \tilde{c} f_0 - h_0$ and using the two above estimates, we deduce

$$ (7.27) \qquad \|f_0\|_{W^{2,1}(\Omega)} \lesssim \|h_0\|_{L^2(\Omega)}. $$

On the other hand, from (7.7) and the Poincaré inequality, we have

$$ 1 = \|h_0\|_{L^2}^2 = \langle (\kappa_1 - \mathcal{L})f_0, h_0 \rangle \lesssim \|\nabla f_0\|_{L^2} \|\nabla h_0\|_{L^2}. $$

Together with the estimate (7.27) and the Gagliardo-Niremberg inequality

$$ \|\nabla f\|_{L^2} \lesssim \|D^2 f\|_{L^1}^{1/2} \|f\|_{L^\infty}^{1/2}, $$

we obtain a lower bound $\|f_0\|_{L^\infty} \ge C_0 > 0$. We then conclude as in the second constructive argument for (H2).

**Condition (H3).** Because of Rellich-Kondrachov theorem on the compact embedding $H_0^1 \subset L^2$, the mapping $(\lambda - \mathcal{L})^{-1} : L^2 \to L^2$ is compact for any $\lambda \ge \kappa_1$. As a consequence, introducing the splitting $\mathcal{L} = \mathcal{A} + \mathcal{B}$ with $\mathcal{A} := \kappa_1 - \kappa_\mathcal{B}$, $\kappa_\mathcal{B} \in \mathbb{R}$ arbitrary, the operator $\mathcal{R}_\mathcal{B}(\lambda) = (\lambda + \kappa_1 - \kappa_\mathcal{B} - \mathcal{L})^{-1}$ is bounded uniformly on $\lambda \ge \kappa_\mathcal{B}$ and it is compact for any $\lambda \ge \kappa_\mathcal{B}$. We deduce from Lemma 2.8-(2) that (H3) holds for both the primal and the dual problems.

We may thus apply Theorem 2.21 and deduce the existence of a solution $(\lambda_1, f_1, \phi_1)$ to the first eigentriplet problem

$$ (7.28) \qquad \lambda_1 \in \mathbb{R}, \quad 0 \le f_1 \in H_0^1, \mathcal{L}f_1 = \lambda_1 f_1, \quad 0 \le \phi_1 \in H_0^1, \mathcal{L}^*\phi_1 = \lambda_1 \phi_1, $$

where both equations must be understood in the variational sense as a consequence of the discussion at the end of the proof of condition (H1).

**Condition (H4).** The strong maximum principle holds as already mentioned in the paragraph dedicated to condition (H2). As a consequence and thanks to Theorem 4.13, we know that the first eigentriplet problem (7.3) has a unique solution $(\lambda_1, f_1, \phi_1)$ which satisfies $f_1 > 0$, $\phi_1 > 0$, $N(\mathcal{L} - \lambda_1)^k = \text{Span}(f_1)$ and $N(\mathcal{L}^* - \lambda_1)^k = \text{Span}(\phi_1)$ for any $k \ge 1$.

**Condition (H5).** Consider $f \in D(\mathcal{L}^\infty)$ such that $0 < |f| \in D(\mathcal{L}^\infty)$ and

$$ \mathcal{L}|f| = \Re\langle \text{sign}f, \mathcal{L}f \rangle, $$
by choosing $\rho \in (0, r_{\Omega}/2)$ small enough. Then, from the Harnack inequality [179, Cor. 8.21], we deduce

$$ \inf_{\omega_{\rho}} f_1 \geq C_H \sup_{\omega_{\rho}} f_1 \geq C_H (2C_1|\Omega|)^{-1}. $$

Finally, from the above Hopf lemma and the above Lipschitz continuity, we have established

$$ (7.37) \qquad c_0\delta \le f_1 \le c_1\delta \quad \text{on } \Omega, $$

for two constructive constants $c_i \in (0, \infty)$. The same arguments on the normalized and positive first dual eigenfunction $\phi_1$ lead to the same estimate

$$ (7.38) \qquad c_0\delta \le \phi_1 \le c_1\delta \quad \text{on } \Omega. $$

In particular, for any such $\rho \in (0, r_{\Omega}/2)$, we have

$$ (7.39) \qquad \langle \phi_1, \mathbf{1}_{\omega_{\rho}} \rangle \ge r_{\rho}, $$

with constructive constant $r_{\rho}$, what is nothing but condition (6.9) in the Harris theorem that we will use below.

**- Step 3. Splitting of $\mathcal{L}$**. We introduce the splitting $\mathcal{L} = \mathcal{A} + \mathcal{B}$, with $\mathcal{A}f = \mathcal{M}\mathbf{1}_{\omega_{\rho}}f$, $\mathcal{M} \ge 0$ large enough and $\rho > 0$ small enough that we fix just below. Using (7.8), we observe that

$$ \begin{aligned} (\mathcal{B}f, f)_{L^2} &= (\mathcal{L}f, f)_{L^2} - \mathcal{M}\|f\|_{L^2}^2 + \mathcal{M} \int_{\omega_{\rho}^c} f^2 \\ &\le -\frac{\nu}{4} \|\nabla f\|_{L^2}^2 + (\kappa_1 - \mathcal{M}) \|f\|_{L^2}^2 + \mathcal{M} |\omega_{\rho}^c|^{4/d} \|f\|_{L^{2^*}}^2 \le \kappa_0 \|f\|_{L^2}^2, \end{aligned} $$

by choosing first $\mathcal{M} \ge \kappa_1 - \kappa_0$ and next $\rho > 0$ small enough in order to be able to throw away the last term using the negative first term and the Sobolev inequality. We deduce

$$ (7.40) \qquad S_{\mathcal{B}}(t) : L^2 \to L^2 \quad \text{with bound} \quad \mathcal{O}(e^{\kappa_0 t}). $$

On the other hand, denoting $f_t := S_{\mathcal{L}}(t)f$ for $f \in L^2(\Omega)$ and recalling that $\phi_0$ defined by (7.22) satisfies (7.23), we have

$$ \frac{d}{dt} \int |f_t| \phi_0 \le \int \mathcal{L} |f_t| \phi_0 \le \int |f_t| \mathcal{L}^* \phi_0 \le \kappa_1 \int |f_t| \phi_0, $$

so that

$$ (7.41) \qquad \int |f_t| \phi_0 \le e^{\kappa_1 t} \int |f_0| \phi_0. $$

Arguing in the same way for $S_{\mathcal{B}}$ and using (7.36), we have established

$$ (7.42) \qquad S_{\mathcal{L}}(t), S_{\mathcal{B}}(t) : L_{\delta}^{1} \to L_{\delta}^{1} \quad \text{with bound} \quad \mathcal{O}(e^{\kappa_1 t}). $$

For a solution to the evolution equation $\partial_t f = \mathcal{C}f$, $\mathcal{C} = \mathcal{L}$ or $\mathcal{C} = \mathcal{B}$, we also classically compute

$$ \begin{aligned} \frac{d}{dt} \int f^2 \phi_0 &= 2 \int (\mathcal{C}f) f \phi_0 \\ &= -2 \int (\nabla f \cdot a \nabla f) \phi_0 + \int f^2 \mathcal{C}^* \phi_0. \end{aligned} $$

Thanks to (7.23) again, we have

$$ (7.43) \qquad \frac{d}{dt} \int f^2 \phi_0 \le -2\nu \int |\nabla f|^2 \phi_0 + \kappa_1 \int f^2 \phi_0, $$

from what we deduce

$$ S_{\mathcal{L}}(t), S_{\mathcal{B}}(t) : L^2(\delta) \to L^2(\delta) \quad \text{with bound} \quad \mathcal{O}(e^{\kappa_1 t/2}). $$

In the sequel, we will need the following version of Nash inequality.

**Lemma 7.5** (weighted Nash inequality). There exists a constructive constant $C_N$ such that

$$ (7.44) \qquad \|f\|_{L^2(\delta)} \le C_N \|\nabla f\|_{L^2(\delta)}^{\frac{d+1}{2}} \|f\|_{L_{\delta}^{1}}^{\frac{d+2}{d-1}}, \quad \forall f \in H^1(\delta). $$
so that multiplying both term of the equation by $|f|$ and integrating, we have

$$ \Re e\langle \mathcal{L}f, \bar{f} \rangle = \langle \mathcal{L}|f|, |f| \rangle. $$

We next compute

$$ \Re e\langle \mathcal{L}f, \bar{f} \rangle = - \int_{\Omega} a_{kj} \Re e(\partial_j f \partial_k \bar{f}) + \int_{\Omega} (b_k - \beta_k) \Re e(\bar{f} \partial_k f) + \int_{\Omega} c |f|^2, $$

and

$$ \langle \mathcal{L} |f|, |f| \rangle = - \int_{\Omega} a_{kj} \partial_j |f| \partial_k |f| + \int_{\Omega} (b_k - \beta_k) \Re e(\bar{f} \partial_k f) + \int_{\Omega} c |f|^2, $$

where in the last equality, we have used that $\partial_k |f| = \frac{1}{|f|} \Re e(\bar{f} \partial_k f)$. From the three above equations, we deduce

$$ \int_{\Omega} a_{kj} [\partial_j |f| \partial_k |f| - \Re e(\partial_j f \partial_k \bar{f})] = 0. $$

Introducing the real and complex part decomposition $f = u + iv$, and similarly as in [230, Proof of Theorem 5.1], we next compute

$$ \begin{aligned} \partial_j |f| \partial_k |f| - \Re e(\partial_j f \partial_k \bar{f}) &= \frac{1}{|f|^2} [uv(\partial_k u \partial_j v + \partial_k v \partial_j u) - u^2 \partial_j v \partial_k v - v^2 \partial_j u \partial_k u] \\ &= \frac{1}{|f|^2} (u \partial_j v - v \partial_j u)(u \partial_k v - v \partial_k u), \end{aligned} $$

so that from the ellipticity condition on $a$, we have $u \partial_k v - v \partial_k u = 0$ a.e. on $\Omega$. On the other hand, from De Girogi-Nash-Moser regularity estimates (7.13) and (7.17), $f$ has Hölder regularity. In particular both functions $u$ and $v$ are continuous. Because $|f| \neq 0$, one of the two function is not identically vanishing, say for instance $v \neq 0$. There exists some points $x_0 \in \Omega$ such that $v(x_0) \neq 0$, say for instance $v(x_0) > 0$. Denoting by $\omega$ the connected component of the set $\{x \in \Omega; v(x) > 0\}$ containing $x_0$, we have $\nabla(u/v) = 0$ on $\omega$. Hence $u = \alpha v$ on $\omega$ for some $\alpha \in \mathbb{R}$, which implies that there exists $\sigma \in \mathbb{S}^1$ such that $f = \sigma|f|$ on $\omega$. If $\omega \neq \Omega$, we would have $|f| = 0$ on $\partial\omega \cap \Omega \neq \emptyset$, which would be a contradiction with the fact that $|f| > 0$. We conclude that $\omega = \Omega$ and thus that $f = \sigma|f|$, which is nothing but the reverse Kato's inequality condition (H5).

At this stage, we may use Theorem 5.16, in order to get the conclusion (C3) on the triviality of the principal point spectrum.

In order to go one step further and establish the asymptotic stability of $f_1$, we may use the two following approaches which are consequences respectively of Lemma 7.1 and Lemma 7.2.

**Lemma 7.1.** For any $R > 0$, the set

$$ \mathcal{K} := \{f \in D(\mathcal{L}); [f] \leq R, [\mathcal{L}f] \leq R\} $$

is strongly compact in $L_{\text{loc}}^1(\Omega)$, where $[g] := \|g\|_{L_{\phi_1}^1}$.

**Proof of Lemma 7.1.** Consider $f \in \mathcal{K}$ so that $f \in H_0^1(\Omega)$ and

$$ \partial_i (a_{ij} \partial_j f) + b_i \partial_i f + \partial_i (\beta_i f) + cf = g \in L^2(\Omega). $$

From the renormalization theory of elliptic equations and the GRE trick (see for instance [268] and the references therein) for any renormalizing function $H \in C^2(\mathbb{R})$, there holds

$$ H''(u)f_1\phi_1 a \nabla u \cdot \nabla u = \operatorname{div}(a\phi_1 \nabla(H(u)f_1)) - \operatorname{div}(f_1 H(u)a \nabla\phi_1) \\ + \operatorname{div}((b+\beta)H(u)f_1\phi_1) + gH'(u)f_1\phi_1, $$

with $u := f/f_1$. Considering $H \in W^{2,\infty}$ the even (and convex) function such that $H(0) = 0$ and $H'' := \mathbf{1}_{[n,n+1]}$, so that in particular $|H'(s)| \leq 1$, and integrating the previous equation, we deduce

$$ \nu \int_{|u| \in [n, n+1]} |\nabla u|^2 f_1 \phi_1 \leq \int |g| f_1 \phi_1 \leq \|f_1\|_{L^\infty} R. $$
Proof of Lemma 7.5. For $\varepsilon > 0$, we define

$$f_{\varepsilon}(x) := \frac{1}{\delta_{\varepsilon}(x)} \int_{B(x, \varepsilon)} f(y) \delta(y) dy, \quad \delta_{\varepsilon}(x) = \delta(B(x, \varepsilon)) := \int_{B(x, \varepsilon)} \delta(y) dy,$$

and $B(x, \varepsilon) := \{y \in \Omega; |x - y| < \varepsilon\}$. It is worth emphasizing that

$$(7.45) \qquad \varepsilon^{d+1} \lesssim \delta_{\varepsilon}(x) \lesssim \varepsilon^d, \quad \forall \varepsilon > 0.$$

For $f \in H^1(\delta)$, we compute

$$\begin{aligned} \|f - f_{\varepsilon}\|_{L^2(\delta)}^2 &= \int_{\Omega} \left| \frac{1}{\delta_{\varepsilon}(x)} \int_{B(x, \varepsilon)} (f(y) - f(x)) \delta(y) dy \right|^2 \delta(x) dx \\ &\le \int_{\Omega} \int_{\Omega} \mathbf{1}_{|y-x| \le \varepsilon} |f(y) - f(x)|^2 \frac{\delta(y)}{\delta_{\varepsilon}(x)} \delta(x) dxdy \\ &\le \varepsilon^2 \int_0^{1/2} \int_{\Omega} \int_{\Omega} |\nabla f((1-t)x + ty)|^2 \frac{\delta(y)}{\delta_{\varepsilon}(x)} \delta(x) dxdydt \\ &\quad + \varepsilon^2 \int_{1/2}^1 \int_{\Omega} \int_{\Omega} |\nabla f((1-t)x + ty)|^2 \frac{\delta(y)}{\delta_{\varepsilon}(x)} \delta(x) dxdydt \\ &\lesssim \varepsilon^2 \int_0^{1/2} \int_{\Omega} \int_{\Omega} |\nabla f(z)|^2 \frac{\delta(y)}{\varepsilon^{d+1}} dy 2 \delta(z) \frac{dz}{(1-t)^d} dt \\ &\quad + \varepsilon^2 \int_{1/2}^1 \int_{\Omega} \int_{\Omega} |\nabla f(z)|^2 \frac{2\delta(z)}{\varepsilon^{d+1}} \delta(x) \frac{dz}{td} dtdx \end{aligned}$$

where for the last inequality we have used the first inequality in (7.45), the fact that $\delta(x) \le 2\delta(z)$ when $0 < t < 1/2$ and the fact that $\delta(y) \le 2\delta(z)$ when $1/2 < t < 1$. Using the second inequality in (7.45), we straightforwardly obtain

$$\|f - f_{\varepsilon}\|_{L^2(\delta)}^2 \le C_1 \varepsilon \|\nabla f\|_{L^2(\delta)}^2, \quad \forall \varepsilon > 0,$$

for a constant $C_1 > 0$. On the other hand, we also observe that

$$\|f_{\varepsilon}\|_{L^{\infty}} \le \frac{C_2}{\varepsilon^{d+1}} \|f\|_{L_{\delta}^1}.$$

Writing now

$$f^2 = f(f - f_{\varepsilon}) + f f_{\varepsilon}$$

and using the above two estimates, we deduce

$$\begin{aligned} \|f\|_{L_{\delta}^2}^2 &\le \|f\|_{L_{\delta}^2} \|f - f_{\varepsilon}\|_{L_{\delta}^2} + \|f\|_{L_{\delta}^1} \|f_{\varepsilon}\|_{L^{\infty}} \\ &\le \|f\|_{L_{\delta}^2} C_1 \varepsilon^{1/2} \|\nabla f\|_{L_{\delta}^2} + C_2 \varepsilon^{-d-1} \|f\|_{L_{\delta}^1}^2 \\ &\le \frac{1}{2} \|f\|_{L_{\delta}^2}^2 + \frac{C_1}{2} \varepsilon \|\nabla f\|_{L_{\delta}^2}^2 + C_2 \varepsilon^{-d-1} \|f\|_{L_{\delta}^1}^2, \end{aligned}$$

and we obtain the weighted Nash inequality (7.46) by choosing $\varepsilon := (\|f\|_{L_{\delta}^1}^2 / \|\nabla f\|_{L_{\delta}^2}^2)^{1/(d+2)}$. $\square$

Defining

$$u := \int |f_t| \phi_0 dx e^{-2\kappa t}, \quad v := \int f_t^2 \phi_0 dx e^{-2\kappa t},$$

with $\kappa := \kappa_{1+}$, coming back to (7.43) and using (7.36), the Nash inequality (7.46) and the estimate (7.41), we get

$$\begin{aligned} v'(t) &\le -2\nu c_0 \int |\nabla f_t|^2 \delta e^{-2\kappa t} \\ &\le -2\nu c_0 C_N^{-2\frac{d+2}{d+1}} \frac{\left(\|f_t\|_{L^2(\delta)}^2 e^{-2\kappa t}\right)^{\frac{d+2}{d+1}}}{\left(\|f_t\|_{L^1(\delta)}^2 e^{-2\kappa t}\right)^{\frac{1}{d+1}}} \\ &\le -C \frac{v(t)^{1+\alpha}}{u(0)^{2\alpha}}, \end{aligned}$$
We proceed along the line of the proof of [62, Thm. 1]. For a fixed $\omega \subset \subset \Omega$, we define $B_n := \{x \in \omega; |u(x)| \in [n, n+1]\}$. Using that $f_1 > 0$ and $\phi_1 > 0$, there exists a constructive constant $C_{\omega,R} > 0$ such that

$$ \int_{B_n} |\nabla u|^2 \le C_{\omega}^2, \quad \forall n \ge 0. $$

From the Cauchy-Schwarz inequality, we have

$$ (7.29) \qquad \int_{B_n} |\nabla u| \le C_{\omega} \mathrm{meas}(B_n)^{1/2}, \quad \forall n \ge 0. $$

On the other hand, denoting by $1^* := d/(d-1)$ the Sobolev exponent, we have

$$ \int_{B_n} |\nabla u| \le C_{\omega,R} (n^{-1^*} \int_{B_n} |u|^{1^*})^{1/2}. $$

Summing up and using the Cauchy-Schwarz inequality again, we have

$$ \begin{aligned} \sum_{n \ge 1} \int_{B_n} |\nabla u| & \le C_{\omega,R} \left( \sum_{n \ge 1} n^{-1^*} \right)^{1/2} \left( \sum_{n \ge 1} \int_{B_n} |u|^{1^*} \right)^{1/2} \\ & \le C_{\omega,R} \left( \sum_{n \ge 1} n^{-1^*} \right)^{1/2} \|u\|_{L^{1^*}}^{1^*/2}. \end{aligned} $$

Together with (7.29) for $n=0$, we deduce

$$ \|\nabla u\|_{L^1(\omega)} \le C'_{\omega,R} (1 + \|\nabla u\|_{L^{1^*}(\omega)}^{1^*/2}). $$

Because $1^*/2 \le 3/4 < 1$ (recall that $d \ge 3$), we can kill the last term, and we obtain the estimate

$$ \|\nabla(f/f_1)\|_{L^1(\omega)} \le C'', \quad \forall f \in \mathcal{K}, $$

for some constant $C'' := C''_{\omega,R} > 0$. We classically conclude thanks to the Rellich-Kondrachov theorem. □

From the above lemma and Theorem 5.23, we deduce that $\tilde{S}(t)f \to \langle f, \phi_1 \rangle f_1$ in the $L^1_{\phi_1}$ norm sense as $t \to \infty$ for any $f \in L^2(\Omega)$. The alternative approach is based on the following result.

**Lemma 7.2.** *Setting $\kappa := \kappa_0 - 1$, there exist $A, \alpha, R > 0$ such that*

$$ (i) \sup_{z \in \Delta_{\kappa}} \langle y \rangle^{\alpha} \| \mathcal{R}_B(z) \|_{\mathcal{B}(L^2; H_0^1)} + \sup_{z \in \Delta_{\kappa} \setminus B_R} \| \mathcal{R}_{\mathcal{L}}(z) \|_{\mathcal{B}(L^2; H_0^1)} < \infty, $$

$$ (ii) \Sigma(\mathcal{L}) \cap \Delta_{\kappa} \subset \Sigma_d(\mathcal{L}) \cap B_R, $$

$$ \text{where } \mathcal{B} := \mathcal{L} - A \text{ and } z = x + iy, x, y \in \mathbb{R}. $$

*Proof of Lemma 7.2.* Let us consider an a priori solution to the stationary problem

$$ f \in H_0^1, \quad z = x + iy \in \Delta_{\kappa}, \quad (\mathcal{L} + z)f = g \in L^2. $$

This one satisfies

$$ |- \int (a \nabla f + \beta f) \cdot \nabla \bar{f} + \int b \cdot \nabla f \bar{f} + (c+z) |f|^2| = |- \int g \bar{f}|. $$

Using the elliptic condition, the Cauchy-Schwarz inequality and triangular inequalities, we get

$$ \begin{aligned} \left| \int g \bar{f} \right| & \ge \left| \int a \nabla f \nabla \bar{f} + ((c+x)_+ + iy) |f|^2 \right| - \left| \int b \cdot \nabla f \bar{f} - \beta f \cdot \nabla \bar{f} + (c+x)_- |f|^2 \right| \\ & \ge \frac{\nu}{2} \|\nabla f\|_{L^2}^2 + \left(\frac{|y|}{2} - x_-\right) \|f\|_{L^2}^2 - \|(b+|\beta|)f\|_{L^2} \|\nabla f\|_{L^2} - \|\sqrt{c_-} f\|_{L^2}^2. \end{aligned} $$

Using next similar arguments and those introduced in the paragraph dedicated to condition (H1) and with similar definition for the constant $M := M(b, \beta, c) > 0$, we deduce

$$ |- \int g \bar{f}| \ge \left( \frac{|y|}{2} - x_- - M \right) \|f\|_{L^2}^2 + \frac{\nu}{4} \|\nabla f\|_{L^2}^2. $$

Defining the sectorial set

$$ S := \{z = x + iy \in \mathbb{C}; |y| > 2x_- + M\}, $$
with $C := 2\nu C_N^{-2\frac{d+2}{d+1}} c_0^{1+2\frac{d+2}{d+1}} c_1^{-\frac{2}{d+1}}$ and $\alpha := 1/(d+1)$. Integrating in time, we deduce

$$v(t) \leq \frac{\alpha^{1/\alpha} u(0)^2}{C^{1/\alpha} t^{1/\alpha}}, \quad \forall t > 0.$$

We have thus established that there exist constructive constants $K > 0$ and $\kappa \ge 0$ such that

$$(7.46) \qquad \|S_C(t)f\|_{L^2(\phi_0)} \leq K \frac{e^{\kappa t}}{t^{(d+1)/2}} \|f\|_{L^1(\phi_0)}, \quad \forall f \in L^1(\phi_0).$$

From that last result, the estimates (7.36) and the properties of $\mathcal{A}$, we deduce that for $N \ge 1$ large enough

$$(7.47) \qquad (S_B \mathcal{A})^{(*N)} : L^1(\delta) \to L^2(\delta) \quad \text{with bound } \mathcal{O}(e^{\kappa t}).$$

We refer to [190, Prop. 3.9], [276, Prop. 2.5] and [231, Lem. 2.4] for details.

**- Step 4. Lyapunov condition.** We may next write

$$\tilde{S}_{\mathcal{L}} = V + W * \tilde{S}_{\mathcal{L}},$$

with

$$V := \tilde{S}_{\mathcal{B}} + \dots + (\tilde{S}_{\mathcal{B}} \mathcal{A})^{(*(N-1))}, \quad V := (\tilde{S}_{\mathcal{B}} \mathcal{A})^{(*N)}.$$

On the one hand, using that $\mathcal{A}: L^2 \to L^2$ is bounded and (7.40), we deduce that

$$V: L^2 \to L^2, \quad \text{with bound } \mathcal{O}(e^{\kappa t}),$$

for any $\kappa \in (\kappa_0 - \kappa_1, 0)$. On the other hand, using that $\mathcal{A}: L_{\delta}^2 \to L^2$ is bounded as well as (7.42) for $S_{\mathcal{L}}$, (7.47), (7.38), (7.36) and (7.40), we deduce that

$$W * \tilde{S}_{\mathcal{L}}: L_{\phi_1}^1 \to L^2, \quad \text{with bound } \mathcal{O}(e^{\kappa' t}),$$

for any $\kappa' > \kappa_1 - \kappa_0$. We may thus fix $t = T$ large enough such that the following Lyapunov inequality holds

$$(7.48) \qquad \|\tilde{S}_{\mathcal{L}}(T)f\|_{L^2} \leq \frac{1}{2} \|f\|_{L^2} + M_T \|f\|_{L_{\phi_1}^1},$$

which is nothing but (6.7) in the hypothesis of the Harris theorem.

**- Step 5. Harris condition** Let $A > 0$ and consider $0 \le f_0 \in L^2$ such that $\|f_0\|_2 \le A\langle f_0, \phi_0 \rangle$. We set $\tilde{f}_t := e^{-\lambda_1 t} S_{\mathcal{L}}(t) f_0$. From the first inequality in (7.23), we have

$$\frac{d}{dt} \langle \tilde{f}_t, \phi_0 \rangle = \langle \tilde{f}_t, (\mathcal{L}^* - \lambda_1) \phi_0 \rangle \geq -(\lambda_1 - \kappa_0) \langle \tilde{f}_t, \phi_0 \rangle,$$

and then, thanks to Gronwall lemma again, we obtain,

$$\langle \tilde{f}_t, \phi_0 \rangle \geq e^{-(\lambda_1 - \kappa_0)t} \langle f_0, \phi_0 \rangle.$$

This estimate, together with the previous step, shows that

$$\begin{align*}
\int_{\omega_{\varrho}} \tilde{f}_{t_0}(x) \phi_0 dx &= \int_{\Omega} \tilde{f}_{t_0}(x) \phi_0 dx - \int_{\omega_{\varrho}^c} \tilde{f}_{t_0}(x) \phi_0 dx \\
&\geq e^{-(\lambda_1 - \kappa_0)t_0} \langle f_0, \phi_0 \rangle - \|\tilde{f}_{t_0}\|_2 \|\phi_0\|_{\infty} |\omega_{\varrho}^c|^{1/2} \\
&\geq e^{-(\kappa_1 - \kappa_0)t_0} \langle f_0, \phi_0 \rangle - e^{(\kappa_1 - \kappa_0)t_0} \|f_0\|_2 \|\phi_0\|_{\infty} |\omega_{\varrho}^c|^{1/2} \\
&\geq \left( e^{-(\kappa_1 - \kappa_0)t_0} - A e^{(\kappa_1 - \kappa_0)t_0} \|\phi_0\|_{\infty} |\omega_{\varrho}^c|^{1/2} \right) \langle f_0, \phi_0 \rangle.
\end{align*}$$

Choosing $\varrho > 0$ small enough, we get

$$\int_{\omega_{\varrho}} \tilde{f}_{t_0}(x) \phi_0 dx \geq \gamma \langle f_0, \phi_0 \rangle, \quad \gamma := \frac{1}{2} e^{-(\lambda_1 - \kappa_0)t_0}.$$

As a consequence, there is $x_{t_0}^f \in \omega_{\varrho}$ such that

$$\tilde{f}_{t_0}(x_{t_0}^f) \geq \frac{1}{|\omega_{\varrho}|} \int_{\omega_{\varrho}} \tilde{f}_{t_0}(x) dx \geq \frac{1}{|\Omega| c_1 \varrho} \int_{\omega_{\varrho}} \tilde{f}_{t_0}(x) \phi_0 dx \geq \frac{\gamma}{|\Omega| c_1 \varrho} \langle f_0, \phi_0 \rangle.$$
we have established the a priori estimates

$$ \begin{aligned} \|f\|_{L^2} &\le \left(\frac{|y|}{2} - x_- - M\right)^{-1/2} \|g\|_{L^2}, \\ \|\nabla f\|_{L^2} &\le 2\nu^{-1/2} \left(\frac{|y|}{2} - x_- - M\right)^{-1/4} \|g\|_{L^2}, \end{aligned} $$

for any $z \in S$. We classically and immediately deduce that $\rho(\mathcal{L}) \supset S$ and the resolvent estimate $\|\mathcal{R}_{\mathcal{L}}(z)\|_{\mathcal{B}(L^2, H_1^1)} \lesssim (\frac{|y|}{2} - x_- - M)^{-1/2} + (\frac{|y|}{2} - x_- - M)^{-1/4}$ for any $z \in S$, and in particular the estimate (i) holds true.

On the other hand, because $\mathcal{L}$ has compact resolvent as established just above or during the proof of (H3) and using the Fredholm alternative, we have $\Sigma(\mathcal{L}) = \Sigma_d(\mathcal{L})$ and $\Sigma(\mathcal{L}) \cap \Delta_\kappa$ is finite for any $\kappa \in \mathbb{R}$, what is nothing but the property (ii). $\square$

From the above lemma and Theorem 5.30 or Theorem 5.32, we deduce that $\tilde{S}(t)f \to \langle f, \phi_1 \rangle f_1$ in the $L^2$ norm sense as $t \to \infty$ for any $f \in L^2(\Omega)$ with exponential rate.

We may summarize our analysis in the following result.

**Theorem 7.3.** Consider the elliptic operator (7.1) in a bounded domain and assume that the coefficients satisfy (7.2), (7.5) and (7.18). Then the conclusions (S1), (S2) and (S3$_3$) hold in $L^2$ as well as (E2) in $L_{\phi_1}^1$ norm and (E3$_1$) in $L^2$ with non constructive rate.

Under not optimal assumptions, the above result belongs to folklore. With the same kind of regularity assumptions the conclusion (S1)-(S2) is probably due to Chicco [105, 106] and to Lions [251, 8th course] for the constructive estimates. We believe that the ergodicity part (E2) and the convergence part (E3$_1$) are new in this framework. It is however worth emphasizing again that the above approach for the long time asymptotic behaviour is definitively not constructive. We propose now an alternative approach which is constructive.

**Quantitative estimate of stability.**

Using the Doblin-Harris type approach presented in Section 6, we are able to establish a rate of convergence to the principal dynamic, at least in a regular framework. We thus make some regularity assumptions on $\Omega$ and additional regularity assumptions on the coefficients.

- For the domain, we assume that there exists a constant $r_\Omega > 0$ such that for any $x \in \Omega$ there is $y \in \Omega$ such that $x \in B(y, r_\Omega) \subset \Omega$, in particular, for any $x \in \partial\Omega$ there is $y \in \Omega$ such that $x \in \partial B(y, r_\Omega)$, $B(y, r_\Omega) \subset \Omega$. We also assume that $\Omega$ is $C^{1,1}$.

- For the coefficients, we assume $a_{ij} \in C(\bar{\Omega})$, $\tilde{b}_i, \tilde{c} \in L^\infty(\Omega)$, where $\tilde{b}_i$ and $\tilde{c}$ are defined in (7.25).

**Theorem 7.4.** Consider the elliptic operator (7.1) in a bounded domain and assume that the assumptions of Theorem 7.3 hold together with the above additional regularity assumptions on the coefficients and the boundary. Then the conclusion (E3$_1$) holds with constructive exponential rate.

We are not aware of any such a result, apart from the recent work of Champagnat and Villemonais [103, 104], which demonstrates a similar result using a probabilistic method (also of Doblin-Harris type) under Holder regularity conditions. Although the constants involved in Theorem 7.4 are constructive, we do not attempt to make them explicit: because of the multiple steps required to establish the final estimate, tracing their dependency on the data would be tedious and probably of little interest. In particular, we do not claim that our method allows us to obtain optimal constants.

The proof of Theorem 7.4 follows from Theorem 6.4. We split the proof into several steps.

- **Step 1. Regularity estimates.** Thanks to De Giorgi-Nash-Moser regularity technique for parabolic equations developed for instance in [242] (in Russian), [345, Thm. 1.3, Thm. 2.2] as well as more recently in [229, Lem. 2.7] and [190, Thm. 1.1], there exists $\alpha = \alpha(a_{ij}) \in (0, 1)$ and for any $T_1 > T_0 > 0$ and any $\varrho \in (0, 1)$, there exist constructive constants $C_i = C_i(\|f\|_{L^\infty([T_0, T_1] \times \Omega)}$, $T, \tau, r)$ such that any solution $f \in L^\infty(0, \infty; L^2(\Omega))$ to the parabolic equation $\partial_t f = \mathcal{L}f$ satisfies

$$ (7.30) \qquad \|f\|_{L^\infty([T_0, T_1] \times \Omega)} \le C_1, \qquad \|f\|_{C^\alpha([T_0, T_1] \times \Omega_\varrho)} \le C_2, $$

with $\omega_r := \{x \in \Omega; d(x, \partial\Omega) > r\}$. More precisely, in order to establish the second estimate in (7.30) with constructive constant, one may observe that the proof of [190, Prop. 2.4] may be repeated in
On the other hand, from the Harnack inequality (7.33) established in Step 2, we know that for any $T > t_0$, there exists $C_H$ such that

$$ \tilde{f}_{t_0}(x_{t_0}^f) \leq \sup_{\omega_\varrho} \tilde{f}_{t_0} \leq C_H \inf_{\omega_\varrho} \tilde{f}_T. $$

The two last estimates together with (7.38) and (7.36) imply the Harris type estimate

$$ (7.49) \qquad \tilde{f}_T = \tilde{S}(T)f_0 \geq g_A\langle f_0, \phi_1 \rangle, $$

with $g_A := \frac{c_0\gamma}{C_H|\Omega|c_1^2\varrho}1_{w_\varrho}$, which is nothing but (6.8) in Harris theorem.

**- Step 6. Conclusion.** Because of the constructive estimates (7.39), (7.48) and (7.49), we may apply the Harris type Theorem 6.3, and we conclude to the exponential stability (E3) in the norm of $L^2(\Omega)$ with constructive constants.

**7.2. Diffusion in $\mathbb{R}^d$ with strong potential confinement.** We consider in this section the elliptic operator

$$ (7.50) \qquad \mathcal{L}f := \Delta f + b \cdot \nabla f + cf, \quad f \in H^1(\mathbb{R}^d), $$

with $b \in L_{\text{loc}}^{\infty}(\mathbb{R}^d)$, $c \in L_{\text{loc}}^2(\mathbb{R}^d)$ and a confinement condition that we impose through the properties of the potential function $c$, which is roughly speaking $c \to -\infty$ as $|x| \to \infty$. More precisely, we assume

$$ (7.51) \qquad \sigma_{i+} \in L^{d/2}, \quad \text{meas}\{\sigma_i \ge K\} < \infty, \quad \forall K < 0, $$

with either $\sigma_1 := c + |b|^2/\kappa$ for some constant $\kappa \in (0, 4)$ or either $\sigma_2 := c + \text{div}b/2$. When we assume that

$$ c \sim -|x|^\gamma \quad \text{and} \quad b \sim x|x|^{\beta-1} \quad \text{as} \quad |x| \to \infty, $$

the condition (7.51) for $\sigma_1$ is reached when $\gamma > \max(0, 2\beta)$ or $\gamma = 2\beta > 0$ and some conditions on the constants involved in the behavior of the coefficients. In that context, the condition (7.51) for $\sigma_2$ is more general since it is reached when $\gamma > \max(0, \beta - 1)$ or $\gamma = \beta - 1 > 0$ and some conditions on the constants involved in the behavior of the coefficients.

A similar framework is considered in [252] and for the reader convenance we just briefly check that it falls in the framework developed before by slightly modifying the arguments presented in the previous section. The integrability conditions on $b$ and $c$ may be probably weaken. For the sake of clarity we do not follow this line of research but rather focus on the new arguments which are necessary in order to deal with the unbounded domain $\Omega = \mathbb{R}^d$.

**Condition (H1).** The definition of the operator is still made through the formula (7.7). Under assumption (7.51) on $\sigma_1$, denoting $\theta_1 := 1 - \kappa/4$ and proceeding exactly as in the previous section during the proof of (7.8), for any $f \in H^1(\mathbb{R}^d)$ and $\lambda \in \mathbb{R}$, we have

$$ \begin{aligned} \langle (\lambda - \mathcal{L}) f, f \rangle &= \int_{\mathbb{R}^d} |\nabla f|^2 + \int_{\mathbb{R}^d} f b \cdot \nabla f + \int_{\mathbb{R}^d} (\lambda - c) f^2 \\ &\geq \theta_1 \int_{\mathbb{R}^d} |\nabla f|^2 + \int_{\mathbb{R}^d} (\lambda - \sigma_1) f^2, \end{aligned} $$

by using successively the Cauchy-Schwarz inequality and the Young inequality. On the other hand, under assumption (7.51) on $\sigma_2$, denoting $\theta_2 := 1$, for any $f \in H^1(\mathbb{R}^d)$ and $\lambda \in \mathbb{R}$, we write

$$ \langle (\lambda - \mathcal{L}) f, f \rangle = \theta_2 \int_{\mathbb{R}^d} |\nabla f|^2 + \int_{\mathbb{R}^d} (\lambda - \sigma_2) f^2, $$

by performing one integration by part in the previous equation. In both cases, for and any $M > 0$, proceeding again as in the previous section during the proof of (7.8), and denoting from now on $\sigma = \sigma_i$, $\theta = \theta_i$ we have

$$ \langle (\lambda - \mathcal{L}) f, f \rangle \geq \frac{\theta}{2} \|\nabla f\|_{L^2}^2 + \|\sqrt{\sigma_-} f\|_{L^2}^2 + (\lambda - M) \|f\|_{L^2}^2 + \left(\frac{\theta C_S}{2} - \|\sigma \mathbf{1}_{\sigma \ge M} \|_{L^{d/2}}\right) \|f\|_{L^{2*}}^2, $$

by using the Sobolev inequality (with associated constant $C_S$) and the Holder inequality. Taking $M > 0$ large enough, and next $\kappa_1 > 0$ large enough, we finally obtain

$$ (7.52) \qquad \langle (\lambda - \mathcal{L}) f, f \rangle \geq \frac{\theta}{2} \|\nabla f\|_{L^2}^2 + \|\sqrt{\sigma_-} f\|_{L^2}^2 + \|f\|_{L^2}^2, \quad \forall \lambda \geq \kappa_1. $$
order to get that solutions to the parabolic equation considered in the present framework fall into De Giorgi classes as defined in [190, Definition 2.3], and thus [190, Thm. 1.1] applies.

On the other hand, in this context and because of the regularity assumptions, we may establish a more accurate regularity estimate. More precisely, by gathering the Sobolev inequality and the Calderon-Zygmond estimate (7.26), we obtain the classical constructive regularity estimate

$$ (7.31) \qquad \|u\|_{C^{0,1}(\Omega)} \lesssim \|u\|_{W^{2,d+1}(\Omega)} \lesssim \|(\kappa_1 - \mathcal{L})u\|_{L^{d+1}(\Omega)}, $$

see for instance Theorem 7.10, Theorem 7.25 and Lemma 9.17 in [178]. Iterating the same kind of arguments, we get

$$ (7.32) \qquad \|u\|_{C^{0,1}(\Omega)} \le C \|(\kappa_1 - \mathcal{L})^k u\|_{L^2(\Omega)}, $$

with constructive constants C and k.

**- Step 2. Harnack estimate.** We claim that for any $T > t_0 > 0$ and $\varrho > 0$, there exist a constant $C_H > 0$ such that, for any $f_0 \in L^2$, the associated solution $f := S_{\mathcal{L}}f_0$ satisfies

$$ (7.33) \qquad \sup_{\omega_{\varrho}} f_{t_0} \le C_H \inf_{\omega_{\varrho}} f_T. $$

The proof mainly follows from Aronson-Serrin [25] (see also [291, 221, 223, 239, 222, 344, 345, 240] for similar results). First, we know from [25, Thm. 3] that

$$ (7.34) \qquad \max_{Q^*(\rho)} f \le C \min_{Q(\rho)} f, $$

for any $\rho > 0$, $t > 0$ such that $Q^*(3\rho) \subset (0, \infty) \times \Omega$, where $Q(\rho) := [t - \rho^2, t] \times \mathcal{C}(\rho)$, $Q^*(\rho) := [t - 8\rho^2, t - 7\rho^2] \times \mathcal{C}(\rho)$ and $\mathcal{C}(\rho)$ is a cube with length $\rho$. To avoid technical issues we assume that $w_\varrho$ is convex. In other case, the geometrical condition given above implies that there is $N \in \mathbb{Z}_+$ such that any two points $x, y \in \Omega$ can be connected by a polygonal path of at most $N$ segments, and we can argue as follows for any segment. We define $D := \sup_{a,b \in \Omega} d(a, b)$ the diameter of $\Omega$ and we choose $r' < \varrho/7$ such that

$$ 7(\lfloor \frac{D}{2r'} \rfloor + 1)(r')^2 < T - t_0. $$

For any $x, y \in \omega_\varrho$, we also define $N_c = \lfloor \frac{|x-y|}{r'} \rfloor$. Since $\omega_\varrho$ is convex, $r' < \varrho/7$, we have that the family of cubes $\{C(x_i, 2r')\}_{i=0, N_c}$ of center $x_i$ and length $2r'$ for $x_i = x + \frac{(x-y)i}{N_c}$ satisfy that $C(x_i, 6r') \subset \Omega$ and $C(x_i, 2r') \cap C(x_{i+1}, 2r') \neq \emptyset$ for any $i = 0, \dots, N_c$. As a consequence, we can apply Aronson-Serrin estimate (7.34) for each cube to obtain

$$ \max_{C(x_i, 2r')} f_{t_i} \le C_{2r'} \min_{C(x_i, 2r')} f_{t_{i+1}}, $$

with $t_i = t_0 + 7i(2r')^2$. Taking $y_i \in C(x_i, 2r') \cap C(x_{i+1}, 2r')$, we deduce

$$ \max_{C(x_i, 2r')} f_{t_i} \le C_{2r'} \min_{C(x_i, 2r')} f_{t_{i+1}} \le C_{2r'} f_{t_{i+1}}(y_i) \le C_{2r'} \max_{C(x_{i+1}, 2r')} f_{t_{i+1}} \le C_{2r'}^2 \min_{C(x_{i+1}, 2r')} f_{t_{i+2}}. $$

By induction, we obtain

$$ f_{t_0}(x) \le \max_{C(x_1, 2r')} f_{t_1} \le C_{2r'}^{N_c} \min_{C(x_{N_c}, 2r')} f_{t_{N_c}} \le C_{2r'}^{N_c} f_{t_{N_c}}(y), $$

with $t_{N_c} = t_0 + 7N_c r'^2 \le T$. Note that in any case the constant $C_{2r'}$ is the same since it only depends on the length $2r'$ and the coefficient of the equation. We have thus established (7.33) with $C_H := C_{2r'}^{\lfloor \frac{D}{2r'} \rfloor + 1}$.

On the other hand, we state an improved version of the already mentioned stationary Harnack inequality. Because of the interior ball condition the Hopf Lemma (see for instance the proof of [178, Lem. 3.4]) claims that for any $\varrho \in (0, r_\Omega/2]$ there exists a constructive constant $\alpha > 0$ such that if $u \in W^{2,p}(\Omega)$, $p > d$, is such that

$$ u \ge \mathbf{1}_{\omega_\varrho}, \quad (\kappa_1 - \mathcal{L}^*)u \ge 0, $$

then u satisfies

$$ (7.35) \qquad u \ge \chi(x) := e^{-\alpha(2\varrho - \delta(x))^2} - e^{-\alpha(2\varrho)^2} \quad \text{on } \omega_{\varrho}^c. $$
With the same arguments as in the previous section, we conclude that $\mathcal{L}$ is the generator in $L^2$ of a positive semigroup $S_{\mathcal{L}}$, so that (H1) holds.

**Condition (H2).** We may for instance use the third constructive argument (which is local) presented in section 7.1 and we establish

$$ \exists f_0 \in H_0^1 \setminus \{0\}, f_0 \ge 0, \exists \kappa_0 \in \mathbb{R}, \mathcal{L} f_0 \ge \kappa_0 f_0. $$

That is condition (ii) in Lemma 2.4, so that condition (H2) holds.

**Condition (H3).** We introduce again the splitting $\mathcal{L} = \mathcal{A} + \mathcal{B}$ with $\mathcal{A} := \kappa_1 - \kappa_0 + 1$, so that from (7.52), the operator $\lambda - \mathcal{B} = (\lambda - \kappa_0 + 1) + (\kappa_1 - \mathcal{L})$ is invertible for any $\lambda \ge \kappa_{\mathcal{B}} := \kappa_0 - 1$. We claim that the operator $(\lambda - \mathcal{B})^{-1}$ is compact for any $\lambda \ge \kappa_{\mathcal{B}}$. For that purpose, let us consider a sequence $(f_n)$ such that $(\lambda - \mathcal{B})f_n$ is bounded in $L^2$ and we have to prove that $(f_n)$ is relatively strongly compact. When condition (7.51) holds and because of the estimate (7.52) and the very definition of $\mathcal{B}$, we have

$$ (7.53) \qquad \frac{\theta}{2} \|\nabla f_n\|_{L^2}^2 + \|\sqrt{\sigma_-} f_n\|_{L^2}^2 + \|f_n\|_{L^2}^2 \le C, $$

for some constant $C \in \mathbb{R}_+$. Because of the Rellich-Kondrachov theorem, we just have to show that

$$ \lim_{R \to \infty} \sup_n \int_{B_R^c} f_n^2 = 0. $$

But that last convergence may be established using the assumption (7.51) in the following way. We write

$$ \begin{aligned} \int_{B_R^c} f_n^2 &= \int_{B_R^c \cap \{\sigma \ge K\}} f_n^2 + \int_{B_R^c \cap \{\sigma < K\}} f_n^2 \\ &\le \|f_n\|_{L^{2^*}}^{\frac{d-2}{d}} [\text{meas}(B_R^c \cap \{\sigma \ge K\})]^{\frac{2}{d}} + \frac{1}{|K|} \int \sigma_- f_n^2, \end{aligned} $$

for any $K < 0$, by using the Holder inequality. Using next the Sobolev inequality, the estimate (7.53) and the assumption (7.51), we deduce

$$ \limsup_{R \to \infty} \int_{B_R^c} f_n^2 \lesssim \limsup_{R \to \infty} \inf_{K < 0} \left\{ [\text{meas}(B_R^c \cap \{\sigma \ge K\})]^{\frac{2}{d}} + \frac{1}{|K|} \right\} = 0, $$

and the claim is proved. As a consequence, we may apply Lemma 2.8-(2) and we deduce that (H3) holds for both the primal and the dual problems.

**Condition (H4).** As in [231, Prop. 5.4], we establish the strong maximum principle by exhibiting a barrier function and using Lemma 4.11. An alternative argument should be to adapt the proof based on the Harnack inequality as presented in the previous section. Let us then consider $f \in D(\mathcal{L}^k) \cap X_+ \setminus \{0\}$ such that $(\lambda - \mathcal{L})f \ge 0$ with $k$ large enough ($k > d/2$ must be suitable) and $\lambda \ge \lambda_1$ large enough but fixed ($\lambda \ge \kappa_1$ is suitable). Using a very classically bootstrap argument based on iterated application of the Calderon-Zygmond elliptic regularity theorem and the Morrey estimate, we have $f \in C(\mathbb{R}^d)$. By assumption, there thus exist $x_0 \in \mathbb{R}^d$, and two constants $\tau, r > 0$ such that $f \ge \tau$ on $B(x_0, r)$ and we take choose $x_0 = 0$ in order to simplify the notations. We next fix $R > r$ and we observe that the function

$$ g(x) := \tau^*(g_0(|x|) - g_0(R)), \quad g_0(s) := \exp(\sigma r^2/2 - \sigma s^2/2), $$

satisfies

$$ \begin{aligned} (\tau^*)^{-1}(\lambda - \mathcal{L})g &= (\lambda - c)(g_0 - g_0(R)) + (d\sigma - \sigma b \cdot x - \sigma^2 r^2)g_0 \\ &\le [2(|\lambda| + \|c\|_{L^\infty(B_R)}) + \sigma(d + \|b \cdot x\|_{L^\infty(B_R)}) - \sigma^2 r^2]g_0 \le 0 \end{aligned} $$

on $\mathcal{O} := B(0, R) \setminus B(0, r)$ for $\sigma > 0$ large enough. We next fix $\tau^*$ such that $g = \tau$ on $\partial B(x_0, r)$. We also observe that from (7.52), $\lambda - \mathcal{L}$ is coercive on $\mathcal{O}$, in the sense that

$$ \forall h \in H_0^1(\mathcal{O}) \quad ((\lambda - \mathcal{L})h, h)_{L^2(\mathcal{O})} \ge \|h\|_{L^2(\mathcal{O})}. $$

In particular, $\lambda - \mathcal{L}$ satisfies the weak maximum principle as explained in the proof of (7.9). Arguing as in the proof of Lemma 4.11, we deduce that $f \ge g > 0$ on $\mathcal{O}$, what we also see directly by observing that $h := (g - f)_+ \in H_0^1(\mathcal{O})$, $(\lambda - \mathcal{L})h \le 0$ and using that the weak maximum
Let us give two applications of the above sharp regularity and positivity estimates. First, recalling (7.24) and using (7.31) and (7.35), we deduce that there exist two constructive constants $c_i \in (0, \infty)$ such that

$$ (7.36) \qquad c_0\delta \le \phi_0 \le c_1\delta \quad \text{on } \Omega. $$

Consider now $f_1 \in H_0^1(\Omega)$ the positive first eigenfunction with normalization $\|f_1\|_{L^2} = 1$. Using the estimate of regularity (7.32) on the iterated equation $(\kappa_1 - \mathcal{L})^k f_1 = (\kappa_1 - \lambda_1)^k f_1$, we have

$$ \|f_1\|_{L^\infty(\Omega)} \le \|f_1\|_{C^{0,1}(\Omega)} \le C_1, $$

for some constructive constant $C_1 \in (0, \infty)$. Next using the elementary inequality

$$ 1 = \int_{\Omega} f_1^2 \le \|f_1\|_{L^{\infty}} \|f_1\|_{L^1} \le C_1 \|f_1\|_{L^1}, $$

we deduce

$$ \begin{aligned} |\Omega| \sup_{\omega_{\varrho}} f_1 &\ge \int_{\omega_{\varrho}} f_1 = \|f_1\|_{L^1} - \int_{\omega_{\varrho}^c} f_1 \\ &\ge 1/C_1 - C_1|\omega_{\varrho}^c| \ge 1/(2C_1), \end{aligned} $$

by choosing $\varrho \in (0, r_\Omega/2)$ small enough. Then, from the Harnack inequality [178, Cor. 8.21], we deduce

$$ \inf_{\omega_{\varrho}} f_1 \ge C_H \sup_{\omega_{\varrho}} f_1 \ge C_H (2C_1|\Omega|)^{-1}. $$

Finally, from the above Hopf lemma and the above Lipschitz continuity, we have established

$$ (7.37) \qquad c_0\delta \le f_1 \le c_1\delta \quad \text{on } \Omega, $$

for two constructive constants $c_i \in (0, \infty)$. The same arguments on the normalized and positive first dual eigenfunction $\phi_1$ lead to the same estimate

$$ (7.38) \qquad c_0\delta \le \phi_1 \le c_1\delta \quad \text{on } \Omega. $$

In particular, for any such $\varrho \in (0, r_\Omega/2)$, we have

$$ (7.39) \qquad \langle \phi_1, \mathbf{1}_{\omega_{\varrho}} \rangle \ge r_{\varrho}, $$

with constructive constant $r_\varrho$, what is nothing but condition (6.10) in the Doblin-Harris theorem that we will use below.

**- Step 3. Splitting of $\mathcal{L}$**. We introduce the splitting $\mathcal{L} = \mathcal{A} + \mathcal{B}$, with $\mathcal{A}f = \mathcal{M}\mathbf{1}_{\omega_{\varrho}}f$, $\mathcal{M} \ge 0$ large enough and $\varrho > 0$ small enough that we fix just below. Using (7.8), we observe that

$$ \begin{aligned} (\mathcal{B}f, f)_{L^2} &= (\mathcal{L}f, f)_{L^2} - \mathcal{M}\|f\|_{L^2} + \mathcal{M} \int_{\omega_{\varrho}^c} f^2 \\ &\le -\frac{\nu}{4} \|\nabla f\|_{L^2}^2 + (\kappa_1 - \mathcal{M}) \|f\|_{L^2}^2 + \mathcal{M} |\omega_{\varrho}^c|^{4/d} \|f\|_{L^{2^*}}^2 \le \kappa_0 \|f\|_{L^2}^2, \end{aligned} $$

by choosing first $\mathcal{M} \ge \kappa_1 - \kappa_0$ and next $\varrho > 0$ small enough in order to be able to throw away the last term using the negative first term and the Sobolev inequality. We deduce

$$ (7.40) \qquad S_{\mathcal{B}}(t) : L^2 \to L^2 \quad \text{with bound} \quad \mathcal{O}(e^{\kappa_0 t}). $$

On the other hand, denoting $f_t := S_{\mathcal{L}}(t)f$ for $f \in L^2(\Omega)$ and recalling that $\phi_0$ defined by (7.22) satisfies (7.23), we have

$$ \frac{d}{dt} \int |f_t| \phi_0 \le \int \mathcal{L} |f_t| \phi_0 \le \int |f_t| \mathcal{L}^* \phi_0 \le \kappa_1 \int |f_t| \phi_0, $$

so that

$$ (7.41) \qquad \int |f_t| \phi_0 \le e^{\kappa_1 t} \int |f_0| \phi_0. $$

Arguing in the same way for $S_{\mathcal{B}}$ and using (7.36), we have established

$$ (7.42) \qquad S_{\mathcal{L}}(t), S_{\mathcal{B}}(t) : L_{\delta}^{1} \to L_{\delta}^{1} \quad \text{with bound} \quad \mathcal{O}(e^{\kappa_1 t}). $$
principle implies $h \le 0$, thus $h \equiv 0$ and finally $f \ge g$. Because $R > r$ can be chosen arbitrarily large, we conclude with $f > 0$ on $\mathbb{R}^d$.

**Condition (H5).** The reverse Kato's inequality condition is proved by using local arguments, so that it holds for the same reasons as in the previous section. Similarly, because the argument are local, the conclusion of Lemma 7.1 holds here.

As a consequence, using Theorem 2.21, Theorem 4.13, Theorem 5.16 and Theorem 5.23, we may summarize our analysis in the following result.

**Theorem 7.6.** Consider the elliptic operator (7.50) in the whole space and assume that the coefficients satisfy (7.51). Then the conclusions (C3) holds as well as (E2) in $L_{\phi_1}^1$.

We do not present an exponential constructive estimate, which we believe is possible to prove, but would require significantly more development.

**7.3. Diffusion in $\mathbb{R}^d$ with weak potential confinement.** We consider in this section the same elliptic operator (7.50) with now a weak confinement condition assuming that $c$ converges to a constant. With no loss of generality, we may assume $c \to 0$. More precisely, we consider the elliptic operator

$$ (7.54) \qquad \mathcal{L}f := \Delta f + b \cdot \nabla f + rcf, $$

with $c \in C_0(\mathbb{R}^d)$, $b \in C_0(\mathbb{R}^d)$ and $r \in \mathbb{R}_+$ a parameter. When not necessary in the discussion we will take $r = 1$. The associated first eigenvalue problem in such a situation has been studied in [252, 8th and 9th courses] to which we refer for more details. We define

$$ \lambda_1 = \lambda_1(r) := \inf\{\kappa \in \mathbb{R}; (\lambda - \mathcal{L})^{-1} \text{ well defined and positive for any } \lambda \ge \kappa\}. $$

Proceeding exactly as in the proof of (H1) in the preceding section, we see that the operator $\lambda - \mathcal{L}$ is invertible for any $\lambda > \|c_+\|_{L^\infty}$, and then its inverse is positive. Because the proof of (H2) in the preceding section also applies here, we deduce that the infimum $\lambda_1$ of the set $\mathcal{I}$ of real resolvent values is well defined with $\lambda_1 \in (\kappa_0, \kappa_1)$, for some constructive constants $\kappa_i \in \mathbb{R}$.

We split now the discussion into two cases.

**Case 1.** We start considering the case $b = 0$. In that case, $\mathcal{L}$ is self-adjoint so that $\lambda_1$ is also characterized by

$$ \lambda_1 = \sup_{\|f\|_{L^2}=1} \mathcal{E}(f), $$

with

$$ \mathcal{E}(f) := (\mathcal{L}f, f) = r \int c f^2 - \int |\nabla f|^2. $$

We make the following elementary observations :

* We claim that $\lambda_1 \ge 0$. Taking $f_n(x) := n^{-d/2}u(x/n)$ for some function $u \in H^1(\mathbb{R}^d)$, $\|u\|_{L^2} = 1$, we compute

$$ \begin{aligned} -\mathcal{E}(f_n) &= \int |\nabla f_n|^2 - \int_{B_R} rcf_n^2 - \int_{B_R^c} rcf_n^2 \\ &\le \frac{1}{n^2} \int |\nabla u|^2 + \|rc\|_{L^\infty(B_R)} \int_{B_{R/n}} u^2 + \|rc\|_{L^\infty(B_R^c)}, \end{aligned} $$

for any $R > 0$, so that

$$ -\lambda_1 \le \limsup(-\mathcal{E}(f_n)) \le 0. $$

* We claim that $\lambda_1 = 0$ when $c \le 0$. In that case, we have $\mathcal{E}(f) \le 0$ for any $f \in H^1(\mathbb{R}^d)$, and we deduce the reverse inequality $\lambda_1 \le 0$. In particular, as a function $\lambda_1 = \lambda_1(r)$ of $r \ge 0$, we have $\lambda_1(0) = 0$. We also claim that $\lambda_1(r) \to \infty$ as $r \to \infty$ when $c_+ \ne 0$. We may indeed fix $f \in H^1(\mathbb{R}^d)$, $\|f\|_{L^2} = 1$, $\supp f \subset \supp c_+$, and we compute

$$ \mathcal{E}(f) = r \int_{\mathbb{R}^d} c_+ f^2 - \int |\nabla f|^2 \to \infty, \quad \text{as } r \to \infty. $$

* We finally observe that $\lambda_1 : \mathbb{R}_+ \to \mathbb{R}_+$ is convex since it is defined as the supremum of linear functions $r \to \mathcal{E}(f)$ for any fixed $f \in H^1(\mathbb{R}^d)$. As a consequence, we have the following alternative:
For a solution to the evolution equation $\partial_t f = C f$, $C = \mathcal{L}$ or $C = \mathcal{B}$, we also classically compute

$$ \begin{aligned} \frac{d}{dt} \int f^2 \phi_0 &= 2 \int (C f) f \phi_0 \\ &= -2 \int (\nabla f \cdot a \nabla f) \phi_0 + \int f^2 C^* \phi_0. \end{aligned} $$

Thanks to (7.23) again, we have

$$ (7.43) \qquad \frac{d}{dt} \int f^2 \phi_0 \le -2\nu \int |\nabla f|^2 \phi_0 + \kappa_1 \int f^2 \phi_0, $$

from what we deduce

$$ S_{\mathcal{L}}(t), S_{\mathcal{B}}(t) : L^2(\delta) \to L^2(\delta) \text{ with bound } \mathcal{O}(e^{\kappa_1 t/2}). $$

In the sequel, we will need the following version of Nash inequality.

**Lemma 7.5** (weighted Nash inequality). There exists a constructive constant $C_N$ such that

$$ (7.44) \qquad \|f\|_{L^2(\delta)} \le C_N \|\nabla f\|_{L^2(\delta)}^{\frac{d+1}{d+2}} \|f\|_{L_\delta^1}^{\frac{1}{d+2}}, \quad \forall f \in H^1(\delta). $$

*Proof of Lemma 7.5.* For $\varepsilon > 0$, we define

$$ f_{\varepsilon}(x) := \frac{1}{\delta_{\varepsilon}(x)} \int_{B(x, \varepsilon)} f(y) \delta(y) dy, \quad \delta_{\varepsilon}(x) = \delta(B(x, \varepsilon)) := \int_{B(x, \varepsilon)} \delta(y) dy, $$

and $B(x, \varepsilon) := \{y \in \Omega; |x - y| < \varepsilon\}$. It is worth emphasizing that

$$ (7.45) \qquad \varepsilon^{d+1} \lesssim \delta_{\varepsilon}(x) \lesssim \varepsilon^d, \quad \forall \varepsilon > 0. $$

For $f \in H^1(\delta)$, we compute

$$ \begin{aligned} \|f - f_{\varepsilon}\|_{L^2(\delta)}^2 &= \left| \int_{\Omega} \frac{1}{\delta_{\varepsilon}(x)} \int_{B(x, \varepsilon)} (f(y) - f(x)) \delta(y) dy \right|^2 \delta(x) dx \\ &\le \int_{\Omega} \int_{\Omega} \mathbf{1}_{|y-x| \le \varepsilon} |f(y) - f(x)|^2 \frac{\delta(y)}{\delta_{\varepsilon}(x)} \delta(x) dxdy \\ &\le \varepsilon^2 \int_0^{1/2} \int_{\Omega} \int_{\Omega} |\nabla f((1-t)x + ty)|^2 \frac{\delta(y)}{\delta_{\varepsilon}(x)} \delta(x) dxdydt \\ &\quad + \varepsilon^2 \int_{1/2}^1 \int_{\Omega} \int_{\Omega} |\nabla f((1-t)x + ty)|^2 \frac{\delta(y)}{\delta_{\varepsilon}(x)} \delta(x) dxdydt \\ &\lesssim \varepsilon^2 \int_0^{1/2} \int_{\Omega} \int_{\Omega} |\nabla f(z)|^2 \frac{\delta(y)}{\varepsilon^{d+1}} dy 2 \delta(z) \frac{dz}{(1-t)^d} dt \\ &\quad + \varepsilon^2 \int_{1/2}^1 \int_{\Omega} \int_{\Omega} |\nabla f(z)|^2 \frac{2\delta(z)}{\varepsilon^{d+1}} \delta(x) \frac{dz}{td} dtdx \end{aligned} $$

where for the last inequality we have used the first inequality in (7.45), the fact that $\delta(x) \le 2\delta(z)$ when $0 < t < 1/2$ and the fact that $\delta(y) \le 2\delta(z)$ when $1/2 < t < 1$. Using the second inequality in (7.45), we straightforwardly obtain

$$ \|f - f_{\varepsilon}\|_{L^2(\delta)}^2 \le C_1 \varepsilon \|\nabla f\|_{L^2(\delta)}^2, \quad \forall \varepsilon > 0, $$

for a constant $C_1 > 0$. On the other hand, we also observe that

$$ \|f_{\varepsilon}\|_{L^{\infty}} \le \frac{C_2}{\varepsilon^{d+1}} \|f\|_{L_{\delta}^1}. $$

Writing now

$$ f^2 = f(f - f_{\varepsilon}) + ff_{\varepsilon} $$

and using the above two estimates, we deduce

$$ \begin{aligned} \|f\|_{L_{\delta}^2}^2 &\le \|f\|_{L_{\delta}^2} \|f - f_{\varepsilon}\|_{L_{\delta}^2} + \|f\|_{L_{\delta}^1} \|f_{\varepsilon}\|_{L^{\infty}} \\ &\le \|f\|_{L_{\delta}^2} C_1 \varepsilon^{1/2} \|\nabla f\|_{L_{\delta}^2} + C_2 \varepsilon^{-d-1} \|f\|_{L_{\delta}^1}^2 \\ &\le \frac{1}{2} \|f\|_{L_{\delta}^2}^2 + \frac{C_1}{2} \varepsilon \|\nabla f\|_{L_{\delta}^2}^2 + C_2 \varepsilon^{-d-1} \|f\|_{L_{\delta}^1}^2, \end{aligned} $$
- $\lambda_1 \equiv 0$;

- $\exists r_0 \in [0, \infty)$ such that $\lambda_1(r) = 0$ for $r \le r_0$ and $\lambda_1(r) > 0$ for $r > r_0$.

Concerning the value of $r_0$, it may happen that $r_0 > 0$, and that is the case when $c \in L^{d/2}$ because of the Sobolev inequality, or that $r_0 = 0$, and that is the case for instance when $c \ge 0$, $c(x) = |x|^{-m}$ for $x \in B_R^c$, $m \in (0, 2)$, $R > 0$. To prove that last claim, we may take the same sequence $(f_n)$ as above, and we compute

$$ 
\begin{aligned}
\mathcal{E}(f_n) & \ge \int_{B_R^c} r|x|^{-m} f_n^2 - \int |\nabla f_n|^2 dx \\
& = \frac{r}{n^m} \int_{B_{R/n}^c} |x|^{-m} u^2 - \frac{1}{n^2} \int |\nabla u|^2 dx > 0,
\end{aligned}
$$

for $n$ large enough (whatever is the value of $r > 0$).

**About condition (H3).** It is established in [252] that when $\lambda_1 = 0$, the condition (H3) is not satisfied and there does not exist a first eigenfunction $f_1 \in L^2(\mathbb{R}^d)$ to the operator $\mathcal{L}$ defined by (7.54). We refer to [252, 8th course] for a proof of that result. On the other hand, we claim that the condition (H3) is satisfied when $\lambda_1 > 0$. Consider indeed three sequences $(\lambda_n)$ of $\mathbb{R}$, $(f_n)$ of $H^1(\mathbb{R}^d)$ and $(\varepsilon_n)$ of $L^2(\mathbb{R}^d)$ such that $(\lambda_n - \mathcal{L})f_n = \varepsilon_n$, $\varepsilon_n$, $f_n \ge 0$, $\|f_n\|_{L^2} = 1$, for any $n \ge 1$, $\lambda_n \to \lambda_1$ and $\varepsilon_n \to 0$ in $L^2$ as $n \to \infty$. We then have

$$ \lambda_n - \mathcal{E}(f_n) = \langle (\lambda_n - \mathcal{L})f_n, f_n \rangle = \langle \varepsilon_n, f_n \rangle \to 0, $$

as $n \to \infty$. By definition of $\mathcal{E}$ and boundedness of $c$, we see that $(f_n)$ is bounded in $H^1$. As a consequence, up to the extraction of a subsequence, we have $f_n \to f_1 \ge 0$ in $L_{\text{loc}}^2$ and thus next $(\lambda_1 - \mathcal{L})f_1 = 0$ in the variational sense and

$$ \int c f_n^2 \to \int c f_1^2, \quad \|\nabla f_1\|_{L^2} \le \liminf \|\nabla f_n\|_{L^2}, $$

where we have used the dominated convergence theorem of Lebesgue and the fact that $c \to 0$ at infinity in order to get the first convergence. We finally deduce

$$ \mathcal{E}(f_1) \ge \limsup \mathcal{E}(f_n) = \lambda_1 > 0, $$

so that $f_1 \ne 0$, and the condition (H3) is verified.

As a conclusion, for a self-adjoint operator, condition (H3) is automatically fulfilled by its adjoint and the conditions (H4) and (H5) have been proved in a general situation, including the present framework. The same conclusions of existence, uniqueness and asymptotic stability of the first eigentriplet solution $(\lambda_1, f_1, \phi_1)$ as in section 7.2 hold true when $\lambda_1 > 0$.

**Case 2.** We consider the **general case** $b \in C_0(\mathbb{R}^d)$.

* We claim that $\lambda_1 \ge 0$. Adapting the second constructive argument in the proof of (H2) in Section 7.1, we consider $\chi \in C_c^1(\mathbb{R}_+) \cap W^{2,\infty}(\mathbb{R}_+)$ such that $\mathbf{1}_{[0,1/2]} \le \chi \le \mathbf{1}_{[0,1]}$, $\chi' \le 0$ on $[0, 1]$, $\chi(s) := (1-s)^2/2$ on $[\eta, 1]$ with $\eta \in (1/2, 1)$ large enough in such a way that

$$ (7.55) \qquad \chi''(s) + (d-1)\chi'(s)/s \ge 1/2, \quad \forall s \in (\eta, 1), $$

and define $f_0(x) := \chi(|x-x_0|/n)$ for $|x_0|$ large enough to be chosen later. We have $\text{supp } f_0 \subset B_n(x_0)$ for any $n \ge 1$ and we compute

$$ \mathcal{L}f_0(x) = \frac{1}{n^2} \{\chi''(r/n) + \frac{d-1}{r/n}\chi'(r/n)\} + \frac{1}{n}b(x) \cdot \hat{y}\chi'(r/n) + c(x)\chi(r/n) $$

where $y = x - x_0$ and $r = |y|$. On $B_{n\eta}(x_0)$, we have

$$ \mathcal{L}f_0(x) \ge -\frac{\|\chi''\|_{\infty}}{n^2} - \frac{d-1}{n^2} \left\| \frac{\chi'(r)}{r} \right\|_{\infty} - \frac{\|\chi'\|_{\infty}}{n} \sup_{B_n(x_0)} |b| - \|\chi\|_{\infty} \sup_{B_n(x_0)} |c|. $$

On $B_n(x_0) \setminus B_{n\eta}(x_0)$, thanks to (7.55), we have

$$ \mathcal{L}f_0(x) \ge \frac{1}{2n^2} - \frac{\|\chi'\|_{\infty}}{n} \sup_{B_n(x_0)} |b| - \|\chi\|_{\infty} \sup_{B_n(x_0)} |c|. $$
and we obtain the weighted Nash inequality (7.44) by choosing $\varepsilon := (\|f\|_{L_\delta^1}^2 / \|\nabla f\|_{L_\delta^2}^2)^{1/(d+2)}$. $\square$

Defining

$$u := \int |f_t| \phi_0 dx e^{-2\kappa t}, \quad v := \int f_t^2 \phi_0 dx e^{-2\kappa t},$$

with $\kappa := \kappa_{1+}$, coming back to (7.43) and using (7.36), the Nash inequality (7.44) and the estimate (7.41), we get

$$ \begin{aligned} v'(t) &\le -2\nu c_0 \int |\nabla f_t|^2 \delta e^{-2\kappa t} \\ &\le -2\nu c_0 C_N^{-2\frac{d+2}{d+1}} \frac{\left(\|f_t\|_{L^2(\delta)}^2 e^{-2\kappa t}\right)^{\frac{d+2}{d+1}}}{\left(\|f_t\|_{L^1(\delta)}^2 e^{-2\kappa t}\right)^{\frac{1}{d+1}}} \\ &\le -C \frac{v(t)^{1+\alpha}}{u(0)^{2\alpha}}, \end{aligned} $$

with $C := 2\nu C_N^{-2\frac{d+2}{d+1}} c_0^{1+2\frac{d+2}{d+1}} c_1^{-\frac{2}{d+1}}$ and $\alpha := 1/(d+1)$. Integrating in time, we deduce

$$v(t) \le \frac{\alpha^{1/\alpha} u(0)^2}{C^{1/\alpha} t^{1/\alpha}}, \quad \forall t > 0.$$

We have thus established that there exist constructive constants $K > 0$ and $\kappa \ge 0$ such that

$$(7.46) \qquad \|S_C(t)f\|_{L^2(\phi_0)} \le K \frac{e^{\kappa t}}{t^{(d+1)/2}} \|f\|_{L^1(\phi_0)}, \quad \forall f \in L^1(\phi_0).$$

From that last result, the estimates (7.36) and the properties of $\mathcal{A}$, we deduce that for $N \ge 1$ large enough

$$(7.47) \qquad (S_B \mathcal{A})^{(N)} : L^1(\delta) \to L^2(\delta) \quad \text{with bound } \mathcal{O}(e^{\kappa t}).$$

We refer to [189, Prop. 3.9], [275, Prop. 2.5] and [230, Lem. 2.4] for details.

**- Step 4. Lyapunov condition.** We may next write

$$\tilde{S}_{\mathcal{L}} = V + W * \tilde{S}_{\mathcal{L}},$$

with

$$V := \tilde{S}_B + \dots + (\tilde{S}_B \mathcal{A})^{(*(N-1))}, \quad V := (\tilde{S}_B \mathcal{A})^{(*N)}.$$

On the one hand, using that $\mathcal{A}: L^2 \to L^2$ is bounded and (7.40), we deduce that

$$V: L^2 \to L^2, \quad \text{with bound } \mathcal{O}(e^{\kappa t}),$$

for any $\kappa \in (\kappa_0 - \kappa_1, 0)$. On the other hand, using that $\mathcal{A}: L_\delta^2 \to L^2$ is bounded as well as (7.42) for $S_\mathcal{L}$, (7.47), (7.38), (7.36) and (7.40), we deduce that

$$W * \tilde{S}_{\mathcal{L}} : L_{\phi_1}^1 \to L^2, \quad \text{with bound } \mathcal{O}(e^{\kappa' t}),$$

for any $\kappa' > \kappa_1 - \kappa_0$. We may thus fix $t = T$ large enough such that the following Lyapunov inequality holds

$$(7.48) \qquad \|\tilde{S}_{\mathcal{L}}(T)f\|_{L^2} \le \frac{1}{2} \|f\|_{L^2} + M_T \|f\|_{L_{\phi_1}^1},$$

which is nothing but (6.8) in the hypothesis of the Doblin-Harris theorem.

**- Step 5. Doblin-Harris condition** Let $A > 0$ and consider $0 \le f_0 \in L^2$ such that $\|f_0\|_2 \le A \langle f_0, \phi_0 \rangle$. We set $\tilde{f}_t := e^{-\lambda_1 t} S_\mathcal{L}(t) f_0$. From the first inequality in (7.23), we have

$$\frac{d}{dt} \langle \tilde{f}_t, \phi_0 \rangle = \langle \tilde{f}_t, (\mathcal{L}^* - \lambda_1) \phi_0 \rangle \ge -(\lambda_1 - \kappa_0) \langle \tilde{f}_t, \phi_0 \rangle,$$

and then, thanks to Gronwall lemma again, we obtain,

$$\langle \tilde{f}_t, \phi_0 \rangle \ge e^{-(\lambda_1 - \kappa_0)t} \langle f_0, \phi_0 \rangle.$$
Let now fix $\varepsilon > 0$ and choose first $n$ large enough so that

$$ -\frac{\|\chi''\|_{\infty}}{n^2} - \frac{d-1}{n^2} \left\| \frac{\chi'(r)}{r} \right\|_{\infty} \geq -\frac{\varepsilon}{2} \inf_{(0,\eta)} \chi. $$

Then, using that $b, c \in C_0(\mathbb{R}^d)$, we can take $|x_0|$ large enough so that

$$ -\frac{\|\chi'\|_{\infty}}{n} \sup_{B_n(x_0)} |b| - \|\chi\|_{\infty} \sup_{B_n(x_0)} |c| \geq -\frac{\varepsilon}{2} \inf_{(0,\eta)} \chi $$

and

$$ \frac{\|\chi'\|_{\infty}}{n} \sup_{B_n(x_0)} |b| + \sup_{B_n(x_0)} |c| \leq \frac{1}{2n^2}. $$

Gathering the above inequalities, we obtain

$$ \mathcal{L}f_0 \geq -\varepsilon f_0, $$

and the condition (H2) is verified with $\kappa_0 = -\varepsilon$. Because $\varepsilon > 0$ can be choose arbitrarily small, we conclude with $\lambda_1 \geq 0$.

• We claim that $\lambda_1 = 0$ when $\sigma_2 \leq 0$. Indeed, we have already seen that

$$ \langle \mathcal{L}f, f \rangle = - \int_{\mathbb{R}^d} |\nabla f|^2 + \int_{\mathbb{R}^d} \sigma_2 f^2, $$

from which we deduce that

$$ \frac{d}{dt} \|S_t f\|^2 = 2 \langle \mathcal{L}f, f \rangle \leq 0. $$

This ensures that (H1) is verified with $\kappa_1 = 0$ and so $\lambda_1 \leq 0$.

• We claim that $\lambda_1 > 0$ when $c_+ \neq 0$ and $r > 0$ is large enough. For simplifying notations and up to translation and dilatation, we may reduce to the case $c \geq c_0 \mathbf{1}_{B(0,1)}$ with $c_0 > 0$. Adapting the second constructive argument in the proof of (H2) in Section 7.1, we consider $\chi \in C_c^1(\mathbb{R}_+) \cap W^{2,\infty}(\mathbb{R}_+)$, $\mathbf{1}_{[0,1/2]} \leq \chi \leq \mathbf{1}_{[0,1]}$, $\text{supp}\chi = [0,1]$, $\chi''(1) = 1$, $\chi' \leq 0$ on $[0,1]$ and we set $f_0(x) := \chi(|x|)$. We compute

$$ \mathcal{L}f_0 = \chi''(|x|) + \chi'(|x|)\left(\frac{d-1}{|x|} + b \cdot \hat{x}\right) + rc(x)\chi(|x|). $$

On the one hand, we fix $\eta \in (1/2, 1)$, $1 - \eta$ small enough, in such a way that

$$ \|\chi'\|_{L^{\infty}(\eta,1)} (2(d-1) + \|b\|_{L^{\infty}}) \leq 1/4, \quad 1/2 \leq \|\chi''\|_{L^{\infty}(\eta,1)}, $$

and thus

$$ \mathcal{L}f_0 \geq \frac{1}{4} \geq \frac{1}{4} f_0 \quad \text{on} \quad B(0, \eta)^c. $$

On the other hand, we fix $r > 0$, large enough, in such a way that

$$ \|\chi''\|_{L^{\infty}} + \|\chi'\|_{L^{\infty}} (2(d-1) + \|b\|_{L^{\infty}}) \leq \kappa(r) := \frac{1}{2} r c_0 \inf_{[0,\eta)} \chi, $$

and thus

$$ \mathcal{L}f_0 \geq \kappa(r) \geq \kappa(r)f_0 \quad \text{on} \quad B(0, \eta). $$

As a conclusion, we have established that condition (ii) in the statement Lemma 2.4 holds with $\kappa_0 := \min(1/4, \kappa(r))$, and that ends the constructive proof of condition (H2) by using Lemma 2.4. That implies in particular the claim since then $\lambda_1 \geq \kappa_0 > 0$.

• We finally claim that (H3) holds when $\lambda_1 > 0$. To see that, we consider again three sequences $(\lambda_n)$ of $\mathbb{R}$, $(f_n)$ of $H^1(\mathbb{R}^d)$ and $(\varepsilon_n)$ of $L^2(\mathbb{R}^d)$ such that $(\lambda_n - \mathcal{L})f_n = \varepsilon_n$, $\varepsilon_n$, $f_n \geq 0$, $\|f_n\|_{L^2} = 1$, for any $n \geq 1$, $\lambda_n \searrow \lambda_1$ and $\varepsilon_n \to 0$ in $L^2$ as $n \to \infty$. As a consequence, we have

$$ \lambda_n + \int |\nabla f_n|^2 - \int f_n b \cdot \nabla f_n - \int c f_n^2 = ((\lambda_n - \mathcal{L})f_n, f_n) = \langle \varepsilon_n, f_n \rangle \to 0, $$

as $n \to \infty$. Using the boundedness of $c, b$ and $\lambda_n$, we see that $(f_n)$ is bounded in $H^1$. As a consequence, up to the extraction of a subsequence, we have $f_n \to f_1 \geq 0$ in $L_{\text{loc}}^2$. We assume by contradiction that $f_1 \equiv 0$. We deduce that

$$ \int c f_n^2 \to 0, \quad \int f_n b \cdot \nabla f_n \to 0, $$
This estimate, together with the previous step, shows that

$$
\begin{align*}
\int_{\omega_{\varrho}} \tilde{f}_{t_0}(x) \phi_0 dx &= \int_{\Omega} \tilde{f}_{t_0}(x) \phi_0 dx - \int_{\omega_{\varrho}^c} \tilde{f}_{t_0}(x) \phi_0 dx \\
&\geq e^{-(\lambda_1 - \kappa_0)t_0} \langle f_0, \phi_0 \rangle - \|\tilde{f}_{t_0}\|_2 \|\phi_0\|_{\infty} |\omega_{\varrho}^c|^{1/2} \\
&\geq e^{-(\kappa_1 - \kappa_0)t_0} \langle f_0, \phi_0 \rangle - e^{(\kappa_1 - \kappa_0)t_0} \|f_0\|_2 \|\phi_0\|_{\infty} |\omega_{\varrho}^c|^{1/2} \\
&\geq \left( e^{-(\kappa_1 - \kappa_0)t_0} - A e^{(\kappa_1 - \kappa_0)t_0} \|\phi_0\|_{\infty} |\omega_{\varrho}^c|^{1/2} \right) \langle f_0, \phi_0 \rangle.
\end{align*}
$$

Choosing $\rho > 0$ small enough, we get

$$
\int_{\omega_{\rho}} \tilde{f}_{t_0}(x) \phi_0 dx \geq \gamma \langle f_0, \phi_0 \rangle, \quad \gamma := \frac{1}{2} e^{-(\lambda_1 - \kappa_0)t_0}.
$$

As a consequence, there is $x_{t_0}^f \in \omega_\rho$ such that

$$
\tilde{f}_{t_0}(x_{t_0}^f) \geq \frac{1}{|\omega_\rho|} \int_{\omega_\rho} \tilde{f}_{t_0}(x) dx \geq \frac{1}{|\Omega| c_1 \rho} \int_{\omega_\rho} \tilde{f}_{t_0}(x) \phi_0 dx \geq \frac{\gamma}{|\Omega| c_1 \rho} \langle f_0, \phi_0 \rangle.
$$

On the other hand, from the Harnack inequality (7.33) established in Step 2, we know that for any $T > t_0$, there exists $C_H$ such that

$$
\tilde{f}_{t_0}(x_{t_0}^f) \leq \sup_{\omega_\rho} \tilde{f}_{t_0} \leq C_H \inf_{\omega_\rho} \tilde{f}_T.
$$

The two last estimates together with (7.38) and (7.36) imply the Doblin-Harris type estimate

$$
(7.49) \qquad \tilde{f}_T = \tilde{S}(T) f_0 \geq g_A \langle f_0, \phi_1 \rangle,
$$

with $g_A := \frac{c_0 \gamma}{C_H |\Omega| c_1^2 \rho} \mathbf{1}_{w_\rho}$, which is nothing but (6.9) in Doblin-Harris theorem.

- Step 6. Conclusion. Because of the constructive estimates (7.39), (7.48) and (7.49), we may apply the Doblin-Harris type Theorem 6.4, and we conclude to the exponential stability (E3₁) in the norm of L²(Ω) with constructive constants.

7.2. **Diffusion in $\mathbb{R}^d$ with strong potential confinement.** We consider in this section the elliptic operator

$$
(7.50) \qquad \mathcal{L}f := \Delta f + b \cdot \nabla f + cf, \quad f \in H^1(\mathbb{R}^d),
$$

with $b \in L_{\text{loc}}^{\infty}(\mathbb{R}^d)$, $c \in L_{\text{loc}}^2(\mathbb{R}^d)$ and a confinement condition that we impose through the properties of the potential function $c$, which is roughly speaking $c \to -\infty$ as $|x| \to \infty$. More precisely, we assume

$$
(7.51) \qquad \sigma_{i+} \in L^{d/2}, \quad \text{meas}\{\sigma_i \ge K\} < \infty, \forall K < 0,
$$

with either $\sigma_1 := c + |b|^2/\kappa$ for some constant $\kappa \in (0, 4)$ or either $\sigma_2 := c + \text{div}b/2$. When we assume that

$$
c \sim -|x|^\gamma \quad \text{and} \quad b \sim x|x|^{\beta-1} \quad \text{as} \quad |x| \to \infty,
$$

the condition (7.51) for $\sigma_1$ is reached when $\gamma > \max(0, 2\beta)$ or $\gamma = 2\beta > 0$ and some conditions on the constants involved in the behavior of the coefficients. In that context, the condition (7.51) for $\sigma_2$ is more general since it is reached when $\gamma > \max(0, \beta - 1)$ or $\gamma = \beta - 1 > 0$ and some conditions on the constants involved in the behavior of the coefficients.

A similar framework is considered in [251] and for the reader convenance we just briefly check that it falls in the framework developed before by slightly modifying the arguments presented in the previous section. The integrability conditions on $b$ and $c$ may be probably weaken. For the sake of clarity we do not follow this line of research but rather focus on the new arguments which are necessary in order to deal with the unbounded domain $\Omega = \mathbb{R}^d$.

**Condition (H1).** The definition of the operator is still made through the formula (7.7). Under assumption (7.51) on $\sigma_1$, denoting $\theta_1 := 1 - \kappa/4$ and proceeding exactly as in the previous section
where we have used the dominated convergence theorem of Lebesgue and the fact that $b, c \to 0$ at infinity. We thus obtain

$$0 < \lambda_1 \le \lambda_n + \int |\nabla f_n|^2 = \int f_n b \cdot \nabla f_n + \int c f_n^2 + \langle \varepsilon_n, f_n \rangle \to 0,$$

and our contradiction. So that $f_1 \ne 0$, and the condition (H3) is verified.

For the dual problem, from the above analysis, we know that there exist two sequences $(\phi_n)$ of $H^1(\mathbb{R}^d)$, $(\varepsilon_n)$ of $L^2(\mathbb{R}^d)$ such that $(\lambda_n - \mathcal{L}^*)\phi_n = \varepsilon_n$, $\varepsilon_n$, $\phi_n \ge 0$ and $\|\phi_n\|_{L^2} = 1$, for any $n \ge 1$, and $\varepsilon_n \to 0$ in $L^2$ as $n \to \infty$. But we face the same situation as previously, since again

$$\lambda_n + \int |\nabla \phi_n|^2 - \int \phi_n b \cdot \nabla \phi_n - \int c \phi_n^2 = ((\lambda_n - \mathcal{L}^*) \phi_n, \phi_n) = (\varepsilon_n, \phi_n) \to 0,$$

and thus the same conclusion, namely $\phi_n \to \phi_1$, with $\phi_1 \in H^1(\mathbb{R}^d)$, $\phi_1 \ge 0$, $\phi_1 \ne 0$.

**Conclusion.** The conditions (H4) and (H5) have been proved in a general situation, including the present framework. The same conclusions as in section 7.2 hold true when $r > 0$ is large enough (and thus $\lambda_1 > 0$).

### 7.4. Diffusion in $\mathbb{R}^d$ with drift confinement.
We now consider the elliptic operator

$$\mathcal{L}f := \Delta f + b \cdot \nabla f + cf,$$

with a drift confinement as it is the case for the Fokker-Planck operator. More precisely, and for the sake of simplicity, we assume here

$$(7.56) \qquad b = \nabla U, \quad U(x) = \frac{1}{\gamma} \langle x \rangle^{\gamma}, \quad \gamma > 0.$$

When $\gamma = 2$ and $c = x$, that operator corresponds to the classical harmonic Fokker-Planck operator which is known to be related to the standard Poincaré inequality and to the standard log-Sobolev inequality, see [29, 26, 343] or more recently [27, 231] and the references therein. When $c = \text{div} b$, the operator $\mathcal{L}$ is on divergence form and $\mathcal{L}^*1 = 0$, so that $(0, 1) \in \mathbb{R} \times L^\infty(\mathbb{R}^d)$ is a solution to the dual first eigenvalue problem. Existence of stationary solution $f_1$ (which is also the first eigenfunction) and its stability have been widely studied. We refer for instance to [344, 332, 171, 28] as well as to [231, 274, 190] which techniques will be adapted here.

In the present situation, we impose that the contribution of $c$ has lower influence at the infinity that the drift term $b$ and we assume

$$(7.57) \qquad c \in L_{\text{loc}}^{\infty}(\mathbb{R}^d), \quad \exists C_0, R_0 > 0, \forall x \in B_{R_0}^c, |c(x)| = o(|x|^{2(\gamma-1)}).$$

We further assume that

$$(7.58) \qquad c \ge \text{div} b \quad \text{when} \quad \gamma \in (0, 1].$$

The action of the drift term will be revealed through the choice of a convenient “confining space”. More precisely, for a weight function $m: \mathbb{R}^d \to [1, \infty)$, we will work in a weighted Lebesgue space. Our analysis is based on the following elementary computation which can be readily adapted from [231, Lem. 2.1], [274, Lem. 3.8] and [190, Lem. 3.8].

**Lemma 7.7.** For any $p \in [1, \infty)$, any weight function $m$ and any smooth, rapidly decaying function $f$, we have

$$(7.59) \qquad \int (\mathcal{L}f) f |f|^{p-2} m^p = -(p-1) \int |\nabla f|^2 |f|^{p-2} m^p + \int |f|^p m^p \varphi_1,$$

with

$$(7.60) \qquad \varphi_1 := (p-1) \frac{|\nabla m|^2}{m^2} + \frac{\Delta m}{m} + \left(c - \frac{1}{p} \text{div } b\right) - b \cdot \frac{\nabla m}{m}$$

as well as

$$(7.61) \qquad \int (\mathcal{L}f) f |f|^{p-2} m^p = -(p-1) \int |\nabla (fm)|^2 |fm|^{p-2} + \int |f|^p m^p \varphi_2,$$

with

$$(7.62) \qquad \varphi_2 := 2(1 - \frac{1}{p}) \frac{|\nabla m|^2}{m^2} + (\frac{2}{p} - 1) \frac{\Delta m}{m} + \left(c - \frac{1}{p} \text{div } b\right) - b \cdot \frac{\nabla m}{m}.$$
during the proof of (7.8), for any $f \in H^1(\mathbb{R}^d)$ and $\lambda \in \mathbb{R}$, we have

$$ \begin{aligned} \langle (\lambda - \mathcal{L})f, f \rangle &= \int_{\mathbb{R}^d} |\nabla f|^2 + \int_{\mathbb{R}^d} fb \cdot \nabla f + \int_{\mathbb{R}^d} (\lambda - c)f^2 \\ &\geq \theta_1 \int_{\mathbb{R}^d} |\nabla f|^2 + \int_{\mathbb{R}^d} (\lambda - \sigma_1)f^2, \end{aligned} $$

by using successively the Cauchy-Schwarz inequality and the Young inequality. On the other hand, under assumption (7.51) on $\sigma_2$, denoting $\theta_2 := 1$, for any $f \in H^1(\mathbb{R}^d)$ and $\lambda \in \mathbb{R}$, we write

$$ \langle (\lambda - \mathcal{L})f, f \rangle = \theta_2 \int_{\mathbb{R}^d} |\nabla f|^2 + \int_{\mathbb{R}^d} (\lambda - \sigma_2)f^2, $$

by performing one integration by part in the previous equation. In both cases, for and any $M > 0$, proceeding again as in the previous section during the proof of (7.8), and denoting from now on $\sigma = \sigma_i$, $\theta = \theta_i$ we have

$$ \langle (\lambda - \mathcal{L})f, f \rangle \geq \frac{\theta}{2} \|\nabla f\|_{L^2}^2 + \|\sqrt{\sigma_-} f\|_{L^2}^2 + (\lambda - M) \|f\|_{L^2}^2 + \left(\frac{\theta C_S}{2} - \|\sigma \mathbf{1}_{\sigma \ge M}\|_{L^{d/2}}\right) \|f\|_{L^{2*}}^2, $$

by using the Sobolev inequality (with associated constant $C_S$) and the Holder inequality. Taking $M > 0$ large enough, and next $\kappa_1 > 0$ large enough, we finally obtain

$$ (7.52) \qquad \langle (\lambda - \mathcal{L})f, f \rangle \geq \frac{\theta}{2} \|\nabla f\|_{L^2}^2 + \|\sqrt{\sigma_-} f\|_{L^2}^2 + \|f\|_{L^2}^2, \quad \forall \lambda \geq \kappa_1. $$

With the same arguments as in the previous section, we conclude that $\mathcal{L}$ is the generator in $L^2$ of a positive semigroup $S_{\mathcal{L}}$, so that (H1) holds.

**Condition (H2).** We may for instance use the third constructive argument (which is local) presented in section 7.1 and we establish

$$ \exists f_0 \in H_0^1 \setminus \{0\}, \quad f_0 \ge 0, \quad \exists \kappa_0 \in \mathbb{R}, \quad \mathcal{L}f_0 \ge \kappa_0 f_0. $$

That is condition (ii) in Lemma 2.4, so that condition (H2) holds.

**Condition (H3).** We introduce again the splitting $\mathcal{L} = \mathcal{A} + \mathcal{B}$ with $\mathcal{A} := \kappa_1 - \kappa_0 + 1$, so that from (7.52), the operator $\lambda - \mathcal{B} = (\lambda - \kappa_0 + 1) + (\kappa_1 - \mathcal{L})$ is invertible for any $\lambda \ge \kappa_{\mathcal{B}} := \kappa_0 - 1$. We claim that the operator $(\lambda - \mathcal{B})^{-1}$ is compact for any $\lambda \ge \kappa_{\mathcal{B}}$. For that purpose, let us consider a sequence $(f_n)$ such that $(\lambda - \mathcal{B})f_n$ is bounded in $L^2$ and we have to prove that $(f_n)$ is relatively strongly compact. When condition (7.51) holds and because of the estimate (7.52) and the very definition of $\mathcal{B}$, we have

$$ (7.53) \qquad \frac{\theta}{2} \|\nabla f_n\|_{L^2}^2 + \|\sqrt{\sigma_-} f_n\|_{L^2}^2 + \|f_n\|_{L^2}^2 \le C, $$

for some constant $C \in \mathbb{R}_+$. Because of the Rellich-Kondrachov theorem, we just have to show that

$$ \lim_{R \to \infty} \sup_n \int_{B_R^c} f_n^2 = 0. $$

But that last convergence may be established using the assumption (7.51) in the following way. We write

$$ \begin{aligned} \int_{B_R^c} f_n^2 &= \int_{B_R^c \cap \{\sigma \ge K\}} f_n^2 + \int_{B_R^c \cap \{\sigma < K\}} f_n^2 \\ &\le \|f_n\|_{L^{2*}}^{\frac{d-2}{d}} [\mathrm{meas}(B_R^c \cap \{\sigma \ge K\})]^{\frac{2}{d}} + \frac{1}{|K|} \int_{|K|} \sigma_- f_n^2, \end{aligned} $$

for any $K < 0$, by using the Holder inequality. Using next the Sobolev inequality, the estimate (7.53) and the assumption (7.51), we deduce

$$ \limsup_{R \to \infty} \int_{B_R^c} f_n^2 \lesssim \limsup_{R \to \infty} \inf_{K < 0} \left\{ [\mathrm{meas}(B_R^c \cap \{\sigma \ge K\})]^{\frac{2}{d}} + \frac{1}{|K|} \right\} = 0, $$

and the claim is proved. As a consequence, we may apply Lemma 2.8-(2) and we deduce that (H3) holds for both the primal and the dual problems.

**Condition (H4).** As in [230, Prop. 5.4], we establish the strong maximum principle by exhibiting a barrier function and using Lemma 4.11. An alternative argument should be to adapt the proof