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# On the Krein-Rutman theorem and beyond

Claudia Fonte Sánchez, Pierre Gabriel, Stéphane Mischler

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Claudia Fonte Sánchez, Pierre Gabriel, Stéphane Mischler. On the Krein-Rutman theorem and beyond. 2024. hal-04093201v2

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# On the Krein-Rutman theorem and beyond

Claudia Fonte Sánchez, Pierre Gabriel, Stéphane Mischler

► To cite this version:

Claudia Fonte Sánchez, Pierre Gabriel, Stéphane Mischler. On the Krein-Rutman theorem and beyond. 2025. hal-04093201v3

HAL Id: hal-04093201

https://hal.science/hal-04093201v3

Preprint submitted on 20 Oct 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# ON THE KREIN-RUTMAN THEOREM AND BEYOND

CLAUDIA FONTE SANCHEZ, PIERRE GABRIEL, AND STÉPHANE MISCHLER

## CONTENTS

<table><tr><td>1. Introduction</td><td>2</td></tr><tr><td>1.1. Framework and the main result</td><td>2</td></tr><tr><td>1.2. Discussion about Theorem 1.1</td><td>6</td></tr><tr><td>1.3. Some examples of applications</td><td>11</td></tr><tr><td>1.4. Organization of the paper</td><td>15</td></tr><tr><td>2. Existence through a stationary problem approach</td><td>16</td></tr><tr><td>2.1. The Banach lattice framework</td><td>16</td></tr><tr><td>2.2. Existence part of the Krein-Rutman theorem</td><td>18</td></tr><tr><td>2.3. Discussion</td><td>28</td></tr><tr><td>3. Existence through a dynamical approach</td><td>31</td></tr><tr><td>3.1. About dissipativity</td><td>31</td></tr><tr><td>3.2. Existence in the dissipative case</td><td>32</td></tr><tr><td>3.3. About weak dissipativity</td><td>35</td></tr><tr><td>3.4. First existence result in the weakly dissipative case</td><td>37</td></tr><tr><td>3.5. Second existence result in the weakly dissipative case</td><td>39</td></tr><tr><td>4. Irreducibility and geometry of the first eigenvalue</td><td>41</td></tr><tr><td>4.1. More about positivity</td><td>42</td></tr><tr><td>4.2. Irreducibility and strong maximum principle</td><td>44</td></tr><tr><td>4.3. The geometry of the first eigenvalue problem</td><td>47</td></tr><tr><td>4.4. Mean ergodicity</td><td>49</td></tr><tr><td>5. The geometry of the boundary point spectrum</td><td>53</td></tr><tr><td>5.1. Complexification and the sign operator</td><td>53</td></tr><tr><td>5.2. On the subgroup and discrete structure of the boundary point spectrum</td><td>55</td></tr><tr><td>5.3. Stronger positivity</td><td>57</td></tr><tr><td>5.4. On the triviality of the boundary spectrum</td><td>61</td></tr><tr><td>5.5. Ergodicity</td><td>62</td></tr><tr><td>5.6. A word about spectral analysis argument</td><td>66</td></tr><tr><td>6. Quantitative stability</td><td>69</td></tr><tr><td>6.1. About quantified positivity conditions</td><td>69</td></tr><tr><td>6.2. Asymptotic stability under Doblín condition</td><td>70</td></tr><tr><td>6.3. Asymptotic stability under Harris condition</td><td>71</td></tr><tr><td>6.4. Quantified isolation of the first eigenvalue</td><td>73</td></tr><tr><td>6.5. The weak dissipativity case</td><td>74</td></tr><tr><td>7. Parabolic equations</td><td>76</td></tr><tr><td>7.1. Diffusion with rough coefficients in a bounded domain</td><td>77</td></tr><tr><td>7.2. Diffusion in ℝ<sup>d</sup> with strong potential confinement</td><td>89</td></tr><tr><td>7.3. Diffusion in ℝ<sup>d</sup> with weak potential confinement</td><td>91</td></tr><tr><td>7.4. Diffusion in ℝ<sup>d</sup> with drift confinement</td><td>94</td></tr><tr><td>8. Transport equations</td><td>100</td></tr><tr><td>8.1. The trace problem</td><td>101</td></tr><tr><td>8.2. Well-posedness for the transport equation with given inflow at the boundary</td><td>107</td></tr><tr><td>8.3. Optimal weighted trace theorem and transport equation with reflection at the boundary</td><td>114</td></tr><tr><td>8.4. Characteristics</td><td>120</td></tr><tr><td>8.5. On the Krein-Rutman theorem for the transport equation with kernel terms</td><td>127</td></tr></table>
# ON THE KREIN-RUTMAN THEOREM AND BEYOND

CLAUDIA FONTE SANCHEZ, PIERRE GABRIEL, AND STÉPHANE MISCHLER

## CONTENTS

1. Introduction 2

1.1. Framework and the main result 2

1.2. Discussion about Theorem 1.1 6

1.3. Some examples of applications 11

1.4. Organization of the paper 16

2. Existence through a stationary problem approach 16

2.1. The Banach lattice framework 16

2.2. Existence part of the Krein-Rutman theorem 19

2.3. Discussion 29

3. Existence through a dynamical approach 32

3.1. About dissipativity 32

3.2. Existence in the dissipative case 34

3.3. About weak dissipativity 36

3.4. First existence result in the weakly dissipative case 38

3.5. Second existence result in the weakly dissipative case 40

4. Irreducibility and geometry of the first eigenvalue 43

4.1. More about positivity 43

4.2. Irreducibility and strong maximum principle 45

4.3. The geometry of the first eigenvalue problem 48

4.4. Mean ergodicity 50

5. The geometry of the boundary point spectrum 55

5.1. Complexification and the sign operator 55

5.2. On the subgroup and discrete structure of the principal point spectrum 57

5.3. Stronger positivity 60

5.4. On the triviality of the principal point spectrum 63

5.5. Ergodicity 65

5.6. A word about spectral analysis argument 69

6. Quantitative stability 72

6.1. About quantified positivity conditions 73

6.2. Asymptotic stability under Doblin condition 73

6.3. Asymptotic stability under Doblin-Harris condition 75

6.4. Quantified isolation of the first eigenvalue 77

6.5. The weak dissipativity case 78

7. Parabolic equations 80

7.1. Diffusion with rough coefficients in a bounded domain 81

7.2. Diffusion in $\mathbb{R}^d$ with strong potential confinement 93

7.3. Diffusion in $\mathbb{R}^d$ with weak potential confinement 95

7.4. Diffusion in $\mathbb{R}^d$ with drift confinement 98

8. Transport equations 105

8.1. The trace problem 106

8.2. Well-posedness for the transport equation with given inflow at the boundary 112

8.3. Optimal weighted trace theorem and transport equation with reflection at the boundary 119

8.4. Characteristics 125

8.5. On the Krein-Rutman theorem for the transport equation with kernel terms 132
<table><tr><td>8.6.</td><td>A word about the renewal equation</td><td>131</td></tr><tr><td>9.</td><td>The growth-fragmentation equation</td><td>133</td></tr><tr><td>9.1.</td><td>The mitosis equation with mixing growth rate</td><td>134</td></tr><tr><td>9.2.</td><td>The mitosis equation with non-mixing growth rate</td><td>139</td></tr><tr><td>9.3.</td><td>The model with variability</td><td>143</td></tr><tr><td>10.</td><td>The kinetic linear Boltzmann equation</td><td>147</td></tr><tr><td>10.1.</td><td>The torus</td><td>148</td></tr><tr><td>10.2.</td><td>The whole space case</td><td>153</td></tr><tr><td>11.</td><td>The kinetic Fokker-Planck equation</td><td>156</td></tr><tr><td>11.1.</td><td>The trace problem</td><td>157</td></tr><tr><td>11.2.</td><td>Well-posedness problem with inflow term at the boundary</td><td>158</td></tr><tr><td>11.3.</td><td>Well-posedness problem with reflection condition at the boundary</td><td>162</td></tr><tr><td>11.4.</td><td>The first eigenvalue problem in a domain with reflection at the boundary</td><td>165</td></tr><tr><td>12.</td><td>A mutation-selection model</td><td>173</td></tr><tr><td>12.1.</td><td>Almost regular mutation kernel</td><td>173</td></tr><tr><td>12.2.</td><td>A singular mutation kernel</td><td>179</td></tr><tr><td colspan="2">References</td><td>183</td></tr></table>

# 1. INTRODUCTION

## 1.1. Framework and the main result.
In this work, we revisit the Krein-Rutman theory for semigroups of positive operators in a Banach lattice framework and we provide some very general, efficient and handy results with constructive estimates about

- the existence of a solution to the first eigentriplet problem;

- the geometry of the principal eigenvalue problem;

- the asymptotic stability of the first eigenvector with possible constructive rate of convergence.

This abstract theory is motivated and illustrated by several examples of differential, intro-differential and integral operators. In particular, we revisit the first eigenvalue problem and the asymptotic stability of the first eigenvector for

- some parabolic equations in a bounded domain and in the whole space;

- some transport equations in a bounded or unbounded domain, including some growth-fragmentation models and some kinetic models;

- the kinetic Fokker-Planck equation in bounded domain;

- some mutation-selection models.

The results we establish on these examples are more general and more accurate that what we can find in the literature. Our approach is in the same time able to tackle some critical cases, but also it is very natural and makes possible to bring out the main important properties for each example and to get rid of many technical issues.

The present work is motivated by new problems and ideas presented in the lectures on the Krein-Rutman theorem by P.-L. Lions at Collège de France [252] and by the recent contributions by Bansaye et al [35] and by Cañizo and Mischler [81] developing Harris techniques. Bringing and developing these ideas and techniques together with the more classical spectral analysis approach developed or synthesized in previous contributions by Krein and Rutman [238], by Arendt et al [15], by Mischler and Scher [278], by Bátkai et al [41] and many others, we are then able to significantly generalize and improve the Krein-Rutman theory for positive semigroups.

The abstract results are developed in the framework of a quite general Banach lattice $X$, that is a Banach space $(X, \| \cdot \|)$ endowed with a compatible order relation $\ge$ and thus with associated positive cone $X_+ := \{f \in X; f \ge 0\}$, which satisfies either $X = Y'$ or $X' = Y$ for another dual Banach lattice $Y$. The precise (and standard) framework will be presented in Section 2.1, and some additional properties will be added when needed (these ones always hold in usual Banach lattices used in PDE and stochastic processes theory). On the other hand, all the applications we will presented are made in the following examples of usual Banach lattices:
<table><tr><td>8.6.</td><td>A word about the renewal equation</td><td>136</td></tr><tr><td>9.</td><td>The growth-fragmentation equation</td><td>138</td></tr><tr><td>9.1.</td><td>The mitosis equation with mixing growth rate</td><td>139</td></tr><tr><td>9.2.</td><td>The mitosis equation with non-mixing growth rate</td><td>144</td></tr><tr><td>9.3.</td><td>The model with variability</td><td>148</td></tr><tr><td>10.</td><td>The kinetic linear Boltzmann equation</td><td>152</td></tr><tr><td>10.1.</td><td>The torus</td><td>153</td></tr><tr><td>10.2.</td><td>The whole space case</td><td>158</td></tr><tr><td>11.</td><td>The kinetic Fokker-Planck equation</td><td>161</td></tr><tr><td>11.1.</td><td>The trace problem</td><td>162</td></tr><tr><td>11.2.</td><td>Well-posedness problem with inflow term at the boundary</td><td>163</td></tr><tr><td>11.3.</td><td>Well-posedness problem with reflection condition at the boundary</td><td>167</td></tr><tr><td>11.4.</td><td>The first eigenvalue problem in a domain with reflection at the boundary</td><td>170</td></tr><tr><td>12.</td><td>A mutation-selection model</td><td>178</td></tr><tr><td>12.1.</td><td>Almost regular mutation kernel</td><td>178</td></tr><tr><td>12.2.</td><td>A singular mutation kernel</td><td>184</td></tr><tr><td colspan="2">References</td><td>188</td></tr></table>

# 1. INTRODUCTION

## 1.1. Framework and the main result.
In this work, we revisit the Krein-Rutman theory for semigroups of positive operators in a Banach lattice framework and we provide some very general, efficient and handy results with possible constructive estimates about

- the existence of a solution to the first eigentriplet problem;

- the geometry of the principal eigenvalue problem;

- the asymptotic stability of the first eigenvector.

This abstract theory is motivated and illustrated by several examples of (partial) differential, integro-differential and integral operators. In particular, we revisit the first eigenvalue problem and the asymptotic stability of the first eigenvector for

- some parabolic equations in a bounded domain and in the whole space;

- some transport equations in a bounded or unbounded domain, including some growth-fragmentation models and some kinetic models;

- the kinetic Fokker-Planck equation in a bounded domain;

- some mutation-selection models.

The results we establish on these examples are more general and more accurate that what we can find in the literature. Our approach is in the same time able to tackle some critical cases, but also it is very natural and makes possible to bring out the main important properties for each example and to get rid of many technical issues.

The present work is motivated by new problems and ideas presented in the lectures on the Krein-Rutman theorem by P.-L. Lions at Collège de France [251] and by the recent contributions by Bansaye et al [35] and by Cañizo and Mischler [80] developing Doblin-Harris techniques. Bringing and developing these ideas and techniques together with the more classical spectral analysis approach developed or synthesized in previous contributions by Krein and Rutman [237], by Arendt et al [15], by Mischler and Scher [277], by Bátkai et al [41] and many others, we are then able to significantly generalize and improve the Krein-Rutman theory for positive semigroups.

The abstract results are developed in a Banach lattice framework, that is in a Banach space $(X, \| \cdot \|)$ endowed with a compatible order relation $\ge$, and thus with an associated positive cone $X_+ := \{f \in X; f \ge 0\}$. We further assume either $X = Y'$ or $X' = Y$ for another dual Banach lattice $Y$ as well as, when needed, some additional structure properties. The (standard) Banach framework will be presented in Section 2.1 and the additional structure properties will be introduced in Sections 4.3 and 5.1. We are not seeking generality for its own sake, but rather in order to have a unified framework for all the applications we have in mind, in order to fit with the standard
• $X := C_0(E)$, the space of continuous functions which tend to 0 at infinity (when $E$ is not a compact set) endowed with the uniform norm, or $X := C_{0,m}(E)$ its weighted variant;

• $X := L^p(E) = L^p(E, \mathcal{E}, \mu)$, the Lebesgue space of functions associated to the Borel $\sigma$-algebra $\mathcal{E}$, a positive $\sigma$-finite measure $\mu$ and an exponent $p \in [1, \infty]$, or $X := L_m^p(E)$ its weighted variant;

• $X := M^1(E) = (C_0(E))'$, the space of Radon measures defined as the dual space of $C_0(E)$, or $X := M_m^1(E)$ its weighted variant.

In all the above examples, $E$ denotes a $\sigma$-compact metric space, and we write $E = \cup E_R$, with $E_R \subset E_{R+1}$, $E_R$ compact.

We next consider a positive one-parameter semigroup of operators $S = S_\mathcal{L}$ on $X$ (we will indifferently writes $S_t = S(t) = S_\mathcal{L}(t)$ for $t \ge 0$), and we denote by $\mathcal{L}$ its generator, by $D(\mathcal{L}) \subset X$ the domain of $\mathcal{L}$, by $\rho(\mathcal{L}) \subset \mathbb{C}$ the resolvent set of $\mathcal{L}$ and by $\Sigma(\mathcal{L}) = \mathbb{C}\backslash\rho(\mathcal{L})$ the spectrum of $\mathcal{L}$. We also denote by $S^*$ and $\mathcal{L}^*$ the corresponding semigroup and generator on the dual space $Y$, and we refer to Section 2.1 for more notations.

As announced, we may split the issue into several pieces concerning the stationary and the evolution associated problems.

• **Existence.** We are first interested in the existence part of the first or principal eigentriplet problem, namely we wish to bring out very general conditions under which

(S1) there exists a solution $(\lambda_1, f_1, \phi_1) \in \mathbb{R} \times X \times Y$ to the eigentriplet problem

$$ (1.1) \qquad \mathcal{L}f_1 = \lambda_1 f_1, \quad f_1 \ge 0, \quad f_1 \ne 0, $$

$$ (1.2) \qquad \mathcal{L}^*\phi_1 = \lambda_1\phi_1, \quad \phi_1 \ge 0, \quad \phi_1 \ne 0, $$

and furthermore $\lambda_1$ coincides with the spectral bound, namely

$$ (1.3) \qquad \lambda_1 = s(\mathcal{L}) := \sup\{\Re e\lambda; \lambda \in \Sigma(\mathcal{L})\} = \inf\{\kappa \in \mathbb{R}; \Delta_\kappa \subset \rho(\mathcal{L})\}, $$

where $\Delta_\alpha$ is the open half plan $\Delta_\alpha := \{z \in \mathbb{C}; \Re z > \alpha\}$.

We emphasize on the fact that this problem is named as the principal eigenvalue problem because

$$ \lambda_1 \in \Sigma(\mathcal{L}) \subset \{z \in \mathbb{C}, \Re e(z) \le \lambda_1\}. $$

• **Geometry.** A second issue is about an accurate analysis of the principal eigentriplet solution and of the geometry of the (principal part of the) spectrum.

On the one hand, concerning the eigentriplet solution, we investigate conditions such that

(S2) $f_1$ is strictly positive (we refer to Section 4.1 for a definition) and $f_1$ is the unique (up to normalization) positive eigenvector for $\mathcal{L}$, $\phi_1$ is strictly positive and $\phi_1$ is the unique (up to normalization) positive eigenvector for $\mathcal{L}^*$, and finally $\lambda_1$ is geometrically and algebraically simple for both $\mathcal{L}$ and $\mathcal{L}^*$. We then may make the (usual) normalization choice

$$ (1.4) \qquad (\|f_1\| = 1, \langle f_1, \phi_1 \rangle = 1) \quad \text{or} \quad (\|\phi_1\| = 1, \langle f_1, \phi_1 \rangle = 1). $$

We are next interested by describing the boundary point spectrum

$$ \Sigma_P^+(\mathcal{L}) := \Sigma_P(\mathcal{L}) \cap \Sigma_+(\mathcal{L}), $$

where we define the boundary spectrum $\Sigma_+(\mathcal{L}) := s(\mathcal{L}) + i\mathbb{R}$ and $\Sigma_P(\mathcal{L})$ as the point spectrum (or set of eigenvalues). More precisely, we exhibit some conditions such that

(S31) $\Sigma_P^+(\mathcal{L}) - \lambda_1$ is a (discrete) additive subgroup of $i\mathbb{R}$;

(S32) $\Sigma_P^+(\mathcal{L})$ is trivial, namely

$$ (1.5) \qquad \Sigma_P^+(\mathcal{L}) = \{\lambda_1\}; $$

or

(S33) $\Sigma_P^+(\mathcal{L})$ is trivial and $\Sigma(\mathcal{L})$ enjoys a spectral gap property (on its principal part), namely

$$ (1.6) \qquad \exists \kappa < \lambda_1; \quad \Sigma(\mathcal{L}) \cap \Delta_\kappa = \{\lambda_1\}. $$
framework and thus to be able to compare our results with the abundant literature, and finally in order to understand what is really needed at each step for proving our several results. We emphasize that all the above requirements hold in the case of the usual Banach lattices used in PDE and stochastic processes theory, in particular for the following examples of usual Banach lattices with which we will work in the applications sections:

• $X := C_0(E)$, the space of continuous functions which tend to 0 at infinity (when $E$ is not a compact set) endowed with the uniform norm, or $X := C_{0,m}(E)$ its weighted variant;

• $X := L^p(E) = L^p(E, \mathcal{E}, \mu)$, the Lebesgue space of functions associated to the Borel $\sigma$-algebra $\mathcal{E}$, a positive $\sigma$-finite measure $\mu$ and an exponent $p \in [1, \infty]$, or $X := L_m^p(E)$ its weighted variant;

• $X := M^1(E) = (C_0(E))'$, the space of Radon measures defined as the dual space of $C_0(E)$, or $X := M_m^1(E)$ its weighted variant.

In all the above examples, $E$ denotes a $\sigma$-compact metric space, and we write $E = \cup E_R$, with $E_R \subset E_{R+1}$, $E_R$ compact.

We next consider a positive one-parameter semigroup of operators $S = S_\mathcal{L}$ on $X$ (we will indifferently writes $S_t = S(t) = S_\mathcal{L}(t)$ for $t \ge 0$), and we denote by $\mathcal{L}$ its generator, by $D(\mathcal{L}) \subset X$ the domain of $\mathcal{L}$, by $\rho(\mathcal{L}) \subset \mathbb{C}$ the resolvent set of $\mathcal{L}$ and by $\Sigma(\mathcal{L}) = \mathbb{C}\setminus\rho(\mathcal{L})$ the spectrum of $\mathcal{L}$. We also denote by $S^*$ and $\mathcal{L}^*$ the corresponding semigroup and generator on the dual space $Y$, and we refer to Section 2.1 for more notations.

As announced, we may split the issue into several pieces concerning the stationary and the evolution associated problems.

• **Existence.** We are first interested in the existence part of the first or principal eigentriplet problem, namely we wish to bring out very general conditions under which

(S1) there exists a solution $(\lambda_1, f_1, \phi_1) \in \mathbb{R} \times X \times Y$ to the eigentriplet problem

$$ (1.1) \qquad \mathcal{L}f_1 = \lambda_1 f_1, \quad f_1 \ge 0, \quad f_1 \ne 0, $$

$$ (1.2) \qquad \mathcal{L}^*\phi_1 = \lambda_1\phi_1, \quad \phi_1 \ge 0, \quad \phi_1 \ne 0, $$

and furthermore $\lambda_1$ coincides with the spectral bound, namely

$$ (1.3) \qquad \lambda_1 = s(\mathcal{L}) := \sup\{\Re e(\lambda); \lambda \in \Sigma(\mathcal{L})\} = \inf\{\kappa \in \mathbb{R}; \Delta_\kappa \subset \rho(\mathcal{L})\}, $$

where $\Delta_\alpha$ is the open half plan $\Delta_\alpha := \{z \in \mathbb{C}; \Re e(z) > \alpha\}$.

Defining $\Sigma_P(\mathcal{L})$ as the point spectrum (or set of eigenvalues), we emphasize on the fact that this problem is named as the principal eigenvalue problem because

$$ \lambda_1 \in \Sigma_P(\mathcal{L}) \subset \{z \in \mathbb{C}, \Re e(z) \le \lambda_1\}. $$

• **Geometry.** A second issue is about an accurate analysis of the principal eigentriplet solution and of the geometry of the (principal part of the) spectrum.

On the one hand, concerning the eigentriplet solution, we investigate conditions such that

(S2) $f_1$ is strictly positive (we refer to Section 4.1 for a definition) and $f_1$ is the unique (up to normalization) positive eigenvector for $\mathcal{L}$, $\phi_1$ is strictly positive and $\phi_1$ is the unique (up to normalization) positive eigenvector for $\mathcal{L}^*$, and finally $\lambda_1$ is geometrically and algebraically simple for both $\mathcal{L}$ and $\mathcal{L}^*$. We then may make the (usual) normalization choice

$$ (1.4) \qquad (\|f_1\| = 1, \langle f_1, \phi_1 \rangle = 1) \quad \text{or} \quad (\|\phi_1\| = 1, \langle f_1, \phi_1 \rangle = 1). $$

We are next interested by describing the boundary point spectrum

$$ \Sigma_P^+(\mathcal{L}) := \Sigma_P(\mathcal{L}) \cap \Sigma_+(\mathcal{L}), $$

where we define the boundary spectrum $\Sigma_+(\mathcal{L}) := \Sigma(\mathcal{L}) \cap \{z \in \mathbb{C}; \Re z = s(\mathcal{L})\}$. More precisely, we exhibit some conditions such that

(S31) $\Sigma_P^+(\mathcal{L}) - \lambda_1$ is a (discrete) additive subgroup of $i\mathbb{R}$;

(S32) $\Sigma_P^+(\mathcal{L})$ is trivial, namely

$$ (1.5) \qquad \Sigma_P^+(\mathcal{L}) = \{\lambda_1\}; $$

or
In the last situation (1.6), a band separates the spectral value $\lambda_1$ to the remainder of the spectrum, while there is no spectral gap when (1.5) holds but (1.6) does not.

The importance of such a eigentriplet comes from the fact that we may associate the Malthusian function

$$F_1(t) := e^{\lambda_1 t} f_1,$$

which is a particular solution to the evolution equation (with maximal growth) and a natural candidate to capture the main asymptotic feature of generic semigroup flow.

• **Asymptotic stability.** In order to formulate our third main issue, namely the asymptotic stability of $F_1$, we introduce the rescaled operators $\tilde{\mathcal{L}} = \mathcal{L} - \lambda_1$ and $\tilde{\mathcal{L}}^* = \mathcal{L}^* - \lambda_1$, so that

$$\tilde{\mathcal{L}}f_1 = 0, \quad \tilde{\mathcal{L}}^*\phi_1 = 0,$$

or in other words, $f_1$ is a stationary state of the semigroup $\tilde{S} = S_{\tilde{\mathcal{L}}}$ and $\phi_1$ is a stationary state of the semigroup $\tilde{S}^* = S_{\tilde{\mathcal{L}}^*}$, and thus a conservation law for $\tilde{S}$:

$$\tilde{S}(t)f_1 = f_1, \quad \tilde{S}^*(t)\phi_1 = \phi_1, \quad \langle S(t)f, \phi_1 \rangle = \langle f, \phi_1 \rangle,$$

for any $t \ge 0$ and any $f \in X$. Because of the property of the eigentriplet and of the normalization assumption (1.4), we may reduce the issue to considering the case $f \in X$ satisfies $\langle \phi_1, f \rangle = 0$ when $(\mathbf{S}3_2)$ or $(\mathbf{S}3_3)$ holds and more generally $f \in Y_0^\perp$ when $(\mathbf{S}3_1)$ holds, where $Y_0$ stands for the eigenspace associated to the eigenvalues belonging to $\Sigma_P^+(\mathcal{L})$. Depending of the hypotheses we made on $\mathcal{L}$ and $S_\mathcal{L}$, we are able to establish some

(E1) **mean ergodic property**, namely

$$\frac{1}{T} \int_{0}^{T} \tilde{S}_{t} f dt \to 0 \quad \text{as} \quad T \to \infty;$$

(E2) **ergodic property**, namely

$$\tilde{S}_t f \to 0 \quad \text{as} \quad t \to \infty;$$

(E3) **quantitative asymptotic stability**, which may be geometric (or exponential) in the spectral gap (1.6) case, namely

$$(\mathbf{E}3_1) \quad \|\tilde{S}(t)f\| \le C e^{-\epsilon t} \|f\|, \quad \forall t \ge 0, \forall f \in X, \langle f, \phi_1 \rangle = 0,$$

for possible constructive constants $\epsilon > 0$ and $C \ge 1$, or under the weaker condition (1.5) only subgeometric, namely

$$(\mathbf{E}3_2) \quad \|\tilde{S}(t)f\|_1 \le \Theta(t) \|f\|_2, \quad \forall t \ge 0, \forall f \in X, \langle f, \phi_1 \rangle = 0,$$

where $\| \cdot \|_2 = \| \cdot \|_X$, $\| \cdot \|_1$ is a weaker norm and $\Theta : \mathbb{R}_+ \to \mathbb{R}_+$ is a constructive decay function satisfying $\Theta(t) \searrow 0$ when $t \nearrow \infty$.

We aim now to allude some general hypotheses on the semigroup $S_\mathcal{L}$ or its generator $\mathcal{L}$ such that the above three main issues may be tackled. Additionally to the yet mentioned fact that $S_\mathcal{L}$ is positive (which is almost equivalent to the fact that its resolvent is a positive operator, that $\mathcal{L}$ enjoys a weak maximum principle or that $\mathcal{L}$ enjoys Kato's inequality) our hypotheses are mainly of two kinds :

- strict positivity conditions;

- regularity conditions;

and these ones may be formulated at the stationary level directly on the generator $\mathcal{L}$ or its resolvent $\mathcal{R}_\mathcal{L}$ or they may be formulated at the evolution level on the semigroup of operators $S_\mathcal{L}$. Of course, in order to establish constructive results these hypotheses will have to be formulated in a quantitative way.

The strict positivity we will introduce and use are of different kinds:

- strong maximum principle on the generator, or equivalently irreducibility of the semigroup;

- reverse Kato's inequality for the generator or aperiodicity condition of the semigroup;

- Doblín-Harris condition on the semigroup, which may be formulated as

$$(1.7) \qquad S_T f \ge g_0(\psi_0, f), \quad \forall f \in X_+,$$
(S33) $\Sigma_P^+(\mathcal{L})$ is trivial and $\Sigma(\mathcal{L})$ enjoys a spectral gap property (on its principal part), namely
(1.6) \qquad $\exists \kappa < \lambda_1; \quad \Sigma(\mathcal{L}) \cap \Delta_\kappa = \{\lambda_1\}$.

In the last situation (1.6), a band separates the spectral value $\lambda_1$ to the remainder of the spectrum, while there is no spectral gap when (1.5) holds but (1.6) does not.

The importance of such a eigentriplet comes from the fact that we may associate the Malthusian function

$$F_1(t) := e^{\lambda_1 t} f_1,$$

which is a particular solution to the evolution equation (with maximal growth) and a natural candidate to capture the main asymptotic feature of generic trajectories.

• Asymptotic stability. In order to formulate our third main issue, namely the asymptotic stability of $F_1$, we introduce the rescaled operators $\tilde{\mathcal{L}} = \mathcal{L} - \lambda_1$ and $\tilde{\mathcal{L}}^* = \mathcal{L}^* - \lambda_1$, so that

$$\tilde{\mathcal{L}} f_1 = 0, \quad \tilde{\mathcal{L}}^* \phi_1 = 0,$$

or in other words, $f_1$ is a stationary state of the semigroup $\tilde{S} = S_{\tilde{\mathcal{L}}}$ and $\phi_1$ is a stationary state of the semigroup $\tilde{S}^* = S_{\tilde{\mathcal{L}}^*}$, and thus a conservation law for $\tilde{S}$:

$$\tilde{S}(t)f_1 = f_1, \quad \tilde{S}^*(t)\phi_1 = \phi_1, \quad \langle S(t)f, \phi_1 \rangle = \langle f, \phi_1 \rangle,$$

for any $t \ge 0$ and any $f \in X$. Because of the property of the eigentriplet and of the normalization assumption (1.4), we may reduce the issue to considering the case $f \in X$ satisfies $\langle \phi_1, f \rangle = 0$ when (S32) or (S33) holds and more generally $f \in Y_0^\perp$ when (S31) holds, where $Y_0 \subset Y$ stands for the eigenspace associated to the eigenvalues belonging to $\Sigma_P^+(\mathcal{L}^*)$. Depending of the hypotheses we made on $\mathcal{L}$ and $S_\mathcal{L}$, we are able to establish for such a $f$ some

(E1) mean ergodic property, namely

$$\frac{1}{T} \int_0^T \tilde{S}_t f dt \to 0 \quad \text{as} \quad T \to \infty;$$

(E2) ergodic property, namely

$$\tilde{S}_t f \to 0 \quad \text{as} \quad t \to \infty;$$

(E3) quantitative asymptotic stability, which may be geometric (or exponential) in the spectral gap (1.6) case, namely

$$\text{(E3_1)} \qquad \|\tilde{S}(t)f\| \le C e^{-\epsilon t} \|f\|, \quad \forall t \ge 0,$$

for possible constructive constants $\epsilon > 0$ and $C \ge 1$, or under the weaker condition (1.5) only subgeometric, namely

$$\text{(E3_2)} \qquad \|\tilde{S}(t)f\|_1 \le \Theta(t) \|f\|_2, \quad \forall t \ge 0,$$

where $\| \cdot \|_2 = \| \cdot \|_X$, $\| \cdot \|_1$ is a weaker norm and $\Theta : \mathbb{R}_+ \to \mathbb{R}_+$ is a constructive decay function satisfying $\Theta(t) \searrow 0$ when $t \nearrow \infty$.

We aim now to allude some general hypotheses on the semigroup $S_\mathcal{L}$ or its generator $\mathcal{L}$ such that the above three main issues may be tackled. Additionally to the yet mentioned fact that $S_\mathcal{L}$ is positive (which is almost equivalent to the fact that its resolvent is a positive operator, that $\mathcal{L}$ enjoys a weak maximum principle or that $\mathcal{L}$ enjoys Kato's inequality) our hypotheses are mainly of two kinds :

- strict positivity conditions;

- regularity conditions on the dominant part of the semigroup;

and these ones may be formulated at the stationary level directly on the generator $\mathcal{L}$ or its resolvent $\mathcal{R}_\mathcal{L}$ or they may be formulated at the evolution level on the semigroup of operators $S_\mathcal{L}$. Of course, in order to establish constructive results these hypotheses will have to be formulated in a quantitative way.

The strict positivity we will introduce and use are of different kinds:

- strong maximum principle on the generator, or equivalently irreducibility of the semigroup;

- reverse Kato's inequality for the generator or aperiodicity condition of the semigroup;
for some for some $T > 0$ and convenient $g_0 \in X_+ \setminus \{0\}$, $\psi_0 \in Y_+ \setminus \{0\}$.

Less systematically but in a crucial way, we will make use of somehow related
- barrier functions and positive subeigenfunctions, which for the last one typically writes

$$
(1.8) \qquad \exists \kappa_0 \in \mathbb{R}, \exists \phi_0 \in Y_+ \setminus \{0\}, \quad \mathcal{L}^* \phi_0 \ge \kappa_0 \phi_0.
$$

On the other hand, some regularity is needed on the dominant part of the semigroup. In order to briefly explain the issue we assume that $\mathcal{L} = \mathcal{A} + \mathcal{B}$ with $\mathcal{A} \in \mathcal{B}(X)$ and $\mathcal{B}$ is the generator of a positive semigroup $S_{\mathcal{B}}$. In such a context, we may write the resolvent factorization identity

$$
\mathcal{R}_{\mathcal{L}} = \mathcal{R}_{\mathcal{B}} + \mathcal{R}_{\mathcal{B}} \mathcal{A} \mathcal{R}_{\mathcal{L}}
$$

on the resolvent $\mathcal{R}_L$ of $\mathcal{L}$ and $\mathcal{R}_B$ of $\mathcal{B}$, and its iterated version

$$
(1.9) \qquad \mathcal{R}_L = \mathcal{V} + W \mathcal{R}_L, \quad \mathcal{V} := \mathcal{R}_B + \dots + \mathcal{R}_B (\mathcal{A} \mathcal{R}_B)^{N-1}, \quad W := (\mathcal{R}_B \mathcal{A})^N.
$$

At the level of the generator, our regularity assumption then typically writes

$$
(1.10) \qquad \sup_{z \in \Delta_{\kappa}} \|\mathcal{V}(z)\|_{\mathcal{B}(X)} < \infty, \quad \sup_{z \in \Delta_{\kappa}} \|\mathcal{W}(z)\|_{\mathcal{B}(X, X_1)} < \infty,
$$

for some $\kappa \in \mathbb{R}$ and $X_1 \subset X$, which is nothing but the classical Voigt's power compact condition when $X_1 \subset X$ with compact embedding. Similarly, at the level of the semigroup, we may write the associated Duhamel formula

$$
S_L = S_B + (S_B A) * S_L,
$$

(we refer to Section 3.1 for a precise definition) and its iterated version

$$
(1.11) \qquad S_L = V + W * S_L, \quad V := \sum_{\ell=0}^{N-1} S_B * (A S_B)^{(*\ell)}, \quad W := (S_B A)^{(*N)},
$$

with $N \ge 1$. At the level of the semigroup, our regularity assumption then typically writes

$$
(1.12) \quad \sup_{t \ge 0} \|V(t)e^{-\kappa t}\|_{\mathcal{B}(X)} < \infty, \quad \sup_{t \ge 0} \|W(t)e^{-\kappa t}\|_{\mathcal{B}(X, X_1)} < \infty,
$$

for some $\kappa \in \mathbb{R}$ and $X_1 \subset X$ in the dissipative framework and a variant of these estimates in a weak dissipative framework. The crucial information is $\kappa < \kappa_0$ (dissipative framework) or $\kappa = \kappa_0$ (more involved weak dissipative framework).

We are now in position to state in a very informal way our main result at the level of the abstract Banach lattice framework.

**Theorem 1.1** (rough version). Let us consider a Banach lattice $X$ picked up in the examples listed above and a positive semigroup $S_L$ on $X$ which enjoys the above splitting structure (1.9), (1.10), (1.11), (1.12).

(1) Conclusion (S1) holds under the localization of the principal spectrum assumption $\kappa < \kappa_0$ and a weak compactness assumption on the regular part $W$ or $W$ in the splitting.

(2) Under an additional strong maximum principle the conclusion (S2) holds. When $X \subset L_{\text{loc}}^1$, we additionally conclude that (S3$_1$) and (E1) hold.

In order to make one step further, we have the three next possibilities

(3) Under an additional inverse Kato's condition or an aperiodicity property, the conclusion (S3<sub>2</sub>) holds, as well as (E2) when X ⊂ L<sub>loc</sub><sup>1</sup>.

(4) Alternatively, under an additional strong compactness assumption on the regular part $W$ of the semigroup, the quantitative exponential asymptotic stability (E3<sub>1</sub>) holds without constructive constants, and thus also the spectral gap conclusion (S3<sub>3</sub>) holds (in a not constructive way).

(5) Alternatively, under the additional Doblin-Harris condition (1.7) and an appropriate regularity estimate on the regular part of the splitting $W$, the quantitative asymptotic stability (E3) holds for both the geometric and subgeometric framework with now constructive constants.

More general and precise statements will be presented in Sections 2, 3, 4, 5 and 6, where in
particular some variants in a weak dissipative framework ($\kappa = \kappa_0$) will be presented. It is worth
emphasizing that the assumptions in (4) and (5) may be optimal in the sense that reciprocal
implications are likely to be true. We do not follow that line of investigation but rather refer to
[278, 35] where such kind of results are established.
- Doblin-Harris condition on the semigroup, which may be formulated as

$$ (1.7) \qquad S_T f \ge g_0 \langle \psi_0, f \rangle, \quad \forall f \in X_+, $$

for some for some $T > 0$ and convenient $g_0 \in X_+ \setminus \{0\}$, $\psi_0 \in Y_+ \setminus \{0\}$.

Less systematically but in a crucial way, we will make use of somehow related

- barrier functions and positive subeigenfunctions, which for the last one typically writes

$$ (1.8) \qquad \exists \kappa_0 \in \mathbb{R}, \exists \phi_0 \in Y_+ \setminus \{0\}, \quad \mathcal{L}^* \phi_0 \ge \kappa_0 \phi_0. $$

On the other hand, some regularity is needed on the dominant part of the semigroup. In order to briefly explain the issue, we assume that $\mathcal{L} = \mathcal{A} + \mathcal{B}$ with $\mathcal{A} \in \mathcal{B}(X)$ and $\mathcal{B}$ is the generator of a semigroup $S_B$. In such a context, we may write the resolvent factorization identity

$$ \mathcal{R}_{\mathcal{L}} = \mathcal{R}_{\mathcal{B}} + \mathcal{R}_{\mathcal{B}} \mathcal{A} \mathcal{R}_{\mathcal{L}} $$

on the resolvent $\mathcal{R}_{\mathcal{L}}$ of $\mathcal{L}$ and $\mathcal{R}_{\mathcal{B}}$ of $\mathcal{B}$, and its iterated version

$$ (1.9) \qquad \mathcal{R}_{\mathcal{L}} = \mathcal{V} + \mathcal{W} \mathcal{R}_{\mathcal{L}}, \quad \mathcal{V} := \mathcal{R}_{\mathcal{B}} + \dots + \mathcal{R}_{\mathcal{B}} (\mathcal{A} \mathcal{R}_{\mathcal{B}})^{N-1}, \quad \mathcal{W} := (\mathcal{R}_{\mathcal{B}} \mathcal{A})^N. $$

At the level of the generator, our regularity assumption then typically writes

$$ (1.10) \qquad \sup_{z \in \Delta_{\kappa}} \|\mathcal{V}(z)\|_{\mathcal{B}(X)} < \infty, \quad \sup_{z \in \Delta_{\kappa}} \|\mathcal{W}(z)\|_{\mathcal{B}(X_0, X_1)} < \infty, $$

for some $\kappa \in \mathbb{R}$ and $X_1 \subset X \subset X_0$, which is nothing but the classical Voigt's power compact condition when $X_0 = X$ and $X_1 \subset X$ with compact embedding. Similarly, at the level of the semigroup, we may write the associated Duhamel formula

$$ S_{\mathcal{L}} = S_{\mathcal{B}} + (S_{\mathcal{B}} \mathcal{A}) * S_{\mathcal{L}}, $$

(we refer to Section 3.1 for a precise definition) and its iterated version

$$ (1.11) \qquad S_{\mathcal{L}} = V + W * S_{\mathcal{L}}, \quad V := \sum_{\ell=0}^{N-1} S_{\mathcal{B}} * (A S_{\mathcal{B}})^{(*\ell)}, \quad W := (S_{\mathcal{B}} \mathcal{A})^{(*N)}, $$

with $N \ge 1$. At the level of the semigroup, our regularity assumption then typically writes

$$ (1.12) \qquad \sup_{t \ge 0} \|V(t)e^{-\kappa t}\|_{\mathcal{B}(X)} < \infty, \quad \sup_{t \ge 0} \|W(t)e^{-\kappa t}\|_{\mathcal{B}(X_0, X_1)} < \infty, $$

for some $\kappa \in \mathbb{R}$ and $X_1 \subset X \subset X_0$ in the dissipative framework and a variant of these estimates in a weak dissipative framework. The crucial information is $\kappa < \kappa_0$ (dissipative framework) or $\kappa = \kappa_0$ (more involved weak dissipative framework).

We are now in position to state in a very informal way our main result in an abstract Banach lattice framework.

**Theorem 1.1** (rough version). Let us consider a Banach lattice $X$ picked up in the examples listed above and a positive semigroup $S_{\mathcal{L}}$ on $X$ which enjoys the above splitting structure (1.9), (1.10), (1.11), (1.12).

(1) Conclusion (S1) holds under the localization of the principal spectrum assumption $\kappa < \kappa_0$ and a weak compactness assumption on the regular part $W$ or $W$ in the splitting.

(2) Under an additional strong positivity assumption (irreducibility property) the conclusions (S2), (S31) and (E1) hold.

In order to make one step further, we have the three next possibilities

(3) Under an even stronger strict positivity condition (reverse Kato's inequality or aperiodicity property), the conclusion (S32) holds, as well as (E2) when $X \subset L_{\text{loc}}^1$.

(4) Alternatively, under an additional strong compactness assumption on the regular part $W$ of the splitting, the quantitative exponential asymptotic stability (E31) holds (without constructive constants), and thus also the spectral gap conclusion (S33) holds (in a not constructive way).

(5) Alternatively, under an additional Doblin-Harris-type condition similar to (1.7) and an appropriate regularity estimate on the regular part $W$ of the splitting, the quantitative asymptotic stability (E3) holds for both the geometric and subgeometric framework with now constructive constants.
## 1.2. Discussion about Theorem 1.1.
We discuss several works related to the main Theorem 1.1 as well as the hypotheses and the techniques used during the proof.

### 1.2.1. The Krein-Rutman work and related approaches.
For a strictly positive matrix in a finite dimensional space, Perron [311] and Frobenius [167] establish at the beginning of the 20th Century that the eigenvalue with largest real part is unique, real and simple. In their pioneer work, Krein and Rutman establish in [238] for the very first time possible infinite dimensional functional space versions of the Perron-Frobenius theorem.

**Theorem 1.2 (Krein-Rutman).** Consider a Banach lattice with positive cone $X_+$ and strictly positive cone $X_{++} := \text{int}X_+ \neq \emptyset$. Consider a linear and compact operator $\mathcal{R}: X \to X$ such that $\mathcal{R}: X_+ \to X_+$ and $\mathcal{R}: X_+ \setminus \{0\} \to X_{++}$. Then there exists a unique eigentriplet $(\mu_1, f_1, \phi_1)$ such that $\mu_1 > 0$, $f_1 \in X_{++}$, $f_1 = \mu_1 \mathcal{R}f_1$, $\phi_1 \in X'_{++}$, $\phi_1 = \mu_1 \mathcal{R}^*\phi_1$.

The non-emptiness of $X_{++}$ and the strict positivity assumption $\mathcal{R}: X_+ \setminus \{0\} \to X_{++}$ can be relaxed, to the price of losing the uniqueness and strict positivity properties of the eigenvectors. For a bounded operator $\mathcal{R}$ on $X$, we denote by $r(\mathcal{R})$ the spectral radius

$$r(\mathcal{R}) := \sup\{|\lambda|; \lambda \in \Sigma(\mathcal{R})\} \leq \|\mathcal{R}\|.$$

**Theorem 1.3 (Krein-Rutman).** Consider a Banach lattice with positive cone $X_+$ and a linear and compact operator $\mathcal{R}: X \to X$ such that $\mathcal{R}: X_+ \to X_+$ and $r(\mathcal{R}) > 0$. Then there exists an eigentriplet $(\mu_1, f_1, \phi_1)$ with $\mu_1 = r(\mathcal{R})$, $f_1 \in X_+ \setminus \{0\}$, $f_1 = \mu_1 \mathcal{R}f_1$, $\phi_1 \in X'_{++} \setminus \{0\}$, $\phi_1 = \mu_1 \mathcal{R}^*\phi_1$.

In Theorems 1.2 and 1.3, the operator $\mathcal{R}$ corresponds to a resolvent operator $\mathcal{R} := (\kappa - \mathcal{L})^{-1}$ for $\kappa > 0$ large enough, so that when it applies, we deduce in particular that the first eigenvalue problem (1.1)-(1.2) has a solution with $\lambda_1 = \kappa - \mu_1$. The two conditions $\text{int}X_+ \neq \emptyset$ and $\mathcal{R}: X_+ \setminus \{0\} \to X_{++}$ are very strong. The first one essentially imposes to work in the space of continuous function and the second one to work in a bounded domain. The result is however suitable and directly applicable (and somehow restricted) to an elliptic operator with smooth coefficients set in a bounded domain with suitable boundary conditions or to a Fredholm integral operator with positive kernel also set in a bounded domain. In the elliptic context, the property $\mathcal{R}: X_+ \setminus \{0\} \to X_{++}$ is nothing but the strong maximum principle while the compactness property of $\mathcal{R}$ comes from the elliptic regularity. We refer to Section 2.3 for further discussions. The weaker condition $r(\mathcal{R}) > 0$ is less restrictive and is in particular always satisfied for irreducible operators, by virtue of de Pagter's theorem [128]. In the same framework, Theorems 1.2 and 1.3 have been next slightly extended by Bonsall [65], Schaefer [336], Karlin [228] or Nussbaum [300] for instance. We also refer to the book by Dautray and Lions [125] for a clear and comprehensible presentation and several possible versions.

In his paper [58], G. Birkhoff derived the Perron-Frobenius theorem by proving a contraction principle in Hilbert's projective metric for positive matrices. His result actually applies to any "uniformly positive bounded" linear operators of a Banach lattice, such as integral operators with positive kernels, and also provides geometric stability estimates. A closely related result was proved by E. Hopf [216], and this Birkhoff-Hopf contraction theorem was subsequently generalized and sharpened, and its proof simplified, by several authors, see in particular [42, 73, 78, 155, 156, 233, 301, 305]. This approach of the Krein-Rutman theorem requires some "uniform positivity and boundedness" of the operator, which is quite restrictive, but it nevertheless allows to recover, through an approximation procedure, the standard result of Theorem 1.2, see [73, Thm. 6.18]. The contraction in Hilbert's projective metric has the advantage to be applicable in partially order linear vector spaces without any topological structure [156], and to nonlinear maps [301].

### 1.2.2. Spectral analysis approach.
In his paper [315], R.S. Phillips formalized the notion of positive semigroup acting on a Banach lattice paving the way to a new field of research. In the precursory work [351] by Vidav and next in a series of papers by Greiner and co-authors [187, 189, 15], Webb [359, 360] and Bürger [75] (see also [15, C-III, Cor. 2.12, Thm. 3.12], [152, Thm. VI.1.12, Cor. VI.1.13] or more recently Theorem 14.17 in the very pedagogical book [41]) significant generalizations of the Krein-Rutman theory were established leading to, roughly speaking, the following result.

**Theorem 1.4.** Consider a positive semigroup $S_{\mathcal{L}}$ on a (suitable) Banach lattice $X$ which is irreducible and such that $s(\mathcal{L}) > -\infty$ is a pole, then
In Section 2, we establish the conclusion (S1) about the existence of an eigentriplet (and in particular we establish Theorem 1.1-(1)) under a family of fundamental conditions denoted by (H1), (H2), (H3), and we discuss several handy conditions involving these ones, in particular some based on a splitting condition (HS1) related to (1.9)-(1.10). An alternative approach for proving at least part of the conclusion (S1) is presented in Section 3, it is based on a dynamical argument and some splitting conditions (HS2) and (HS3) related to (1.11)-(1.12). In Section 4, we establish conclusions (S2) and (E1) under an additional structure condition on X denoted by (X1) and an additional positivity condition denoted by (H4) (which is nothing but the irreducibility for a semigroup). In Section 5, we first introduce some additional structure conditions (X2) and (X3) on X and we establish (S31), thus concluding the proof of Theorem 1.1-(2). We next establish (S32) and (E2) (and thus Theorem 1.1-(3)) under two additional and somehow stronger positivity assumptions (H5) (reverse Kato's condition) and (H5') (aperiodicity condition). Introducing an additional alternative and more classical strong compactness assumption on the regular part W of the splitting, we prove (E31) and (S33) without constructive constants (and thus Theorem 1.1-(4)). In Section 6 finally, we introduce some relevant and somewhat stronger conditions and establish Theorem 1.1-(5). We emphasize that all the concrete Banach lattices previously introduced satisfy the structure assumptions (X1), (X2) and (X3). It is also worth mentioning that the assumptions in (4) and (5) may be optimal in the sense that reciprocal implications are likely to be true. We do not follow that line of investigation but rather refer to [277, 35] where such kind of results are established.

## 1.2. Discussion about Theorem 1.1.
We discuss several works related to the main Theorem 1.1 as well as the hypotheses and the techniques used during the proof.

### 1.2.1. The Krein-Rutman work and related approaches.
For a strictly positive matrix in a finite dimensional space, Perron [310] and Frobenius [166] establish at the beginning of the 20th Century that the eigenvalue with largest real part is unique, real and simple. In their pioneer work, Krein and Rutman establish in [237] for the very first time a Perron-Frobenius type theorem in an infinite dimensional Banach space.

**Theorem 1.2 (Krein-Rutman).** Consider a Banach lattice with positive cone $X_+$ and strictly positive cone $X_{++} := \text{int}X_+ \neq \emptyset$. Consider a linear and compact operator $\mathcal{R}: X \to X$ such that $\mathcal{R}: X_+ \to X_+$ and $\mathcal{R}: X_+ \setminus \{0\} \to X_{++}$. Then there exists a unique eigentriplet $(\mu_1, f_1, \phi_1)$ such that $\mu_1 > 0$, $f_1 \in X_{++}$, $f_1 = \mu_1 \mathcal{R}f_1$, $\phi_1 \in X'_+ \setminus \{0\}$, $\phi_1 = \mu_1 \mathcal{R}^*\phi_1$.

The non-emptiness of $X_{++}$ and the strict positivity assumption $\mathcal{R}: X_+ \setminus \{0\} \to X_{++}$ can be relaxed, to the price of losing the uniqueness and strict positivity properties of the eigenvectors. For a bounded operator $\mathcal{R}$ on $X$, we denote by $r(\mathcal{R})$ the spectral radius

$$r(\mathcal{R}) := \sup\{|\lambda|; \lambda \in \Sigma(\mathcal{R})\} \leq \|\mathcal{R}\|.$$

**Theorem 1.3 (Krein-Rutman).** Consider a Banach lattice with positive cone $X_+$ and a linear and compact operator $\mathcal{R}: X \to X$ such that $\mathcal{R}: X_+ \to X_+$ and $r(\mathcal{R}) > 0$. Then there exists an eigentriplet $(\mu_1, f_1, \phi_1)$ with $\mu_1 = r(\mathcal{R})$, $f_1 \in X_+ \setminus \{0\}$, $f_1 = \mu_1 \mathcal{R}f_1$, $\phi_1 \in X'_+ \setminus \{0\}$, $\phi_1 = \mu_1 \mathcal{R}^*\phi_1$.

In Theorems 1.2 and 1.3, the operator $\mathcal{R}$ corresponds to a resolvent operator $\mathcal{R} := (\kappa - \mathcal{L})^{-1}$ for $\kappa > 0$ large enough, so that when it applies, we deduce in particular that the first eigenvalue problem (1.1)-(1.2) has a solution with $\lambda_1 = \kappa - \mu_1$. The two conditions $\text{int}X_+ \neq \emptyset$ and $\mathcal{R}: X_+ \setminus \{0\} \to X_{++}$ are very strong. The first one essentially imposes to work in the space of continuous functions and the second one to work in a bounded domain. The result is however suitable and directly applicable (and somehow restricted) to an elliptic operator with smooth coefficients set in a bounded domain with suitable boundary conditions or to a Fredholm integral operator with positive kernel also set in a bounded domain. In the elliptic context, the property $\mathcal{R}: X_+ \setminus \{0\} \to X_{++}$ is nothing but the strong maximum principle while the compactness property of $\mathcal{R}$ comes from the elliptic regularity. We refer to Section 2.3 for further discussions. The weaker condition $r(\mathcal{R}) > 0$ is less restrictive and is in particular always satisfied for irreducible operators, by virtue of de Pagter's theorem [127]. In the same framework, Theorems 1.2 and 1.3 have been next slightly extended by Bonsall [65], Schaefer [335], Karlin [227] or Nussbaum [299] for instance. We also refer to the book by Dautray and Lions [124] for a clear and comprehensible presentation and several possible versions.
• $s(\mathcal{L})$ is a first-order pole with one-dimensional and strictly positive residue, so that in particular there exists a solution $(\lambda_1, f_1, \phi_1)$ to the eigentriplet problem;

• There exists $\alpha \in \mathbb{R}$ such that $\Sigma_+(\mathcal{L}) = \{s(\mathcal{L}) + i\alpha\mathbb{Z}\}$ consists of first-order poles with one-dimensional residue.

• A practical way for verifying that $s(\mathcal{L}) > -\infty$ is a pole consists in assuming that $\mathcal{L}$ enjoys the splitting structure $\mathcal{L} = \mathcal{A} + \mathcal{B}$, as described above, with $s(\mathcal{B}) < s(\mathcal{L})$ and $\mathcal{A}$ is $\mathcal{B}$ power compact, that is to say $\mathcal{W}$ is compact, on $\Delta_{s(\mathcal{B})}$.

Assuming furthermore that $\tilde{S}_{\mathcal{L}}$ is quasi-compact then

• $\tilde{S}_{\mathcal{L}}$ is exponential asymptotically stable in $\text{Span}\{\phi_1\}^{\perp}$ (without constructive constants).

The most important improvements here are the fact that the condition $\text{int}X_+ \neq \emptyset$ and the strong compactness of the resolvent operator $\mathcal{R}_{\mathcal{L}}$ are removed, and also that the exponential asymptotically stability is established. The hypotheses seem stronger to those stated in Theorem 1.1-(1), where only weak compactness is required what is not the case here. It is however worth emphasizing that in an AL-space and an AM-space (what includes the examples $C_0(E)$ and $L^1(E)$) a power weak compactness implies a power strong compactness (see [75, Rk. 2.1] and [337, Cor. 1 of Thm. II.9.9]). The hypotheses and conclusions are similar to those stated in Theorem 1.1-(4). The proof is based on the one hand on the Banach lattices theory as formalized for instance by Schaefer [337] (see also [11, 12, 13, 15] for significant developments) using notions as ideals and quasi-interior points. On the other hand, it takes advantage on the perturbation techniques initiated by Phillips in [314] and developed further by Jörgens [225], Vidav [331, 352] and Voigt [355] leading to the notions of power compact resolvent and quasi-compact semigroup, essential spectrum and Calkin algebra.

The above theorem in particular applies to a positive and irreducible semigroup which is eventually norm continuous and its generator has compact resolvent (see for instance Corollary VI.1.13 in [152] and for the definition of an eventually norm continuous). In that case indeed, one can show that $s(\mathcal{L}) > -\infty$, $\Sigma_+(\mathcal{L})$ is bounded and consists of poles, so that $\Sigma_+(\mathcal{L}) = \{s(\mathcal{L})\}$ and the essential growth bound $\omega_{\text{ess}}(S)$ associated to the essential spectrum (see for instance [41, Sec 14.1] for a definition) satisfies $\omega_{\text{ess}}(S) < \omega(S) = s(\mathcal{L})$. The theorem was motivated and successfully applied to Boltzmann like transport operator [351], cell division operator [134], age structured equation [360] and selection-mutation dynamics [75]. We also refer to [152, Ch. VI] and [41] for other numerous applications. Although very general and quite efficient, we formulate several criticisms about the above result.

- The exponential convergence result is definitively not constructive and that approach is not able to say anything about the weak dissipative case (a framework we will introduce latter, see in particular Section 3.3).

- We may observe that Theorem 1.4 is not so popular in the probability and the PDE communities and still many works in these domains refer to the original Krein and Rutman theorem even when some additional (approximation) arguments are needed rather than applying directly Theorem 1.4. By the way, we did not find in the literature where Theorem 1.4 is stated in such a handy way (the closer formulation is probably [152, Thm. VI.1.12] which is given without proof).

- The proof of Theorem 1.4 that we may find in the above quoted references is written in a very specific and abstract language which make it quite obscure.

In [278, 273], one of the authors proposes the following variant.

**Theorem 1.5.** Consider a positive semigroup $S_{\mathcal{L}}$ which satisfies (1.8) with $\kappa_0 \in \mathbb{R}$, it is irreducible and its generator enjoys the splitting structure (1.9)-(1.10) for some $\kappa < \kappa_0$ and $X_1 \subset X$ with compact imbedding. Assuming furthermore that

$$ (1.13) \qquad \exists \alpha > 0, \quad \sup_{z \in \Delta_{\kappa}} \langle z \rangle^{\alpha} \|W(z)\|_{\mathcal{B}(X)} < \infty, $$

the quantitative exponential asymptotic stability (E31) holds (without constructive constants).

The proof of Theorem 1.5 is based on a partial (but principal) spectral mapping and Weyl's theorem (in the spirit of Voigt [355]) coupled with a simple analysis of the first eigenelement problem based on the irreducibility of the semigroup, but which is really simpler than the deep result on irreducible semigroup stated in Theorem 1.4. On the other hand, that approach is unable to tackle the situation when $\Sigma_P^+(\mathcal{L})$ is not a singleton. One of the main features in Theorem 1.5
In his paper [58], G. Birkhoff derived the Perron-Frobenius theorem by proving a contraction principle in Hilbert's projective metric for positive matrices. His result actually applies to any "uniformly positive bounded" linear operators of a Banach lattice, such as integral operators with positive kernels, and also provides geometric stability estimates. A closely related result was proved by E. Hopf [215], and this Birkhoff-Hopf contraction theorem was subsequently generalized and sharpened, and its proof simplified, by several authors, see in particular [42, 72, 77, 154, 155, 232, 300, 304]. This approach of the Krein-Rutman theorem requires some "uniform positivity and boundedness" of the operator, which is quite restrictive, but it nevertheless allows to recover, through an approximation procedure, the standard result of Theorem 1.2, see [72, Thm. 6.18]. The contraction in Hilbert's projective metric has the advantage to be applicable in partially order linear vector spaces without any topological structure [155], and to nonlinear maps [300].

**1.2.2. Spectral analysis approach.** In his paper [314], R.S. Phillips formalized the notion of positive semigroup acting on a Banach lattice paving the way to a new field of research. In the precursory work [350] by Vidav and next in a series of papers by Greiner and co-authors [186, 188, 15], Webb [358, 359] and Bürger [74] (see also [15, C-III, Cor. 2.12, Thm. 3.12], [151, Thm. VI.1.12, Cor. VI.1.13] or more recently Theorem 14.17 in the very pedagogical book [41]) significant generalizations of the Krein-Rutman theory were established leading to, roughly speaking, the following result.

**Theorem 1.4.** Consider a positive semigroup $S_\mathcal{L}$ on a (suitable) Banach lattice $X$ which is irreducible and such that $s(\mathcal{L}) > -\infty$ is a pole, then

• $s(\mathcal{L})$ is a first-order pole with one-dimensional and strictly positive residue, so that in particular there exists a solution $(\lambda_1, f_1, \phi_1)$ to the eigentriplet problem;

• There exists $\alpha \in \mathbb{R}$ such that $\Sigma_+(\mathcal{L}) = \{s(\mathcal{L}) + i\alpha\mathbb{Z}\}$ consists of first-order poles with one-dimensional residue.

• A practical way for verifying that $s(\mathcal{L}) > -\infty$ is a pole consists in assuming that $\mathcal{L}$ enjoys the splitting structure $\mathcal{L} = \mathcal{A} + \mathcal{B}$, as described above, with $s(\mathcal{B}) < s(\mathcal{L})$ and $\mathcal{A}$ is $\mathcal{B}$ power compact, that is to say $W$ is compact, on $\Delta_s(\mathcal{B})$.

Assuming furthermore that $\tilde{S}_\mathcal{L}$ is quasi-compact then

• $\tilde{S}_\mathcal{L}$ is exponential asymptotically stable in $\text{Span}\{\phi_1\}^{\perp}$ (without constructive constants).

The most important improvements in the above result are the fact that the condition $\text{int}X_+ \neq \emptyset$ and the strong compactness of the resolvent operator $\mathcal{R}_\mathcal{L}$ are removed, and also that the exponential asymptotically stability is established. The hypotheses seem stronger to those stated in Theorem 1.1-(1), where only weak compactness is required what is not the case here. It is however worth emphasizing that in an AL-space and an AM-space (what includes the examples $C_0(E)$ and $L^1(E)$) a power weak compactness implies a power strong compactness (see [74, Rk. 2.1] and [336, Cor. 1 of Thm. II.9.9]). The hypotheses and conclusions are similar to those stated in Theorem 1.1-(4). The proof is based on the one hand on the Banach lattices theory as formalized for instance by Schaefer [336] (see also [11, 12, 13, 15] for significant developments) using notions as ideals and quasi-interior points. On the other hand, it takes advantage on the perturbation techniques initiated by Phillips in [313] and developed further by Jörgens [224], Vidav [330, 351] and Voigt [354] leading to the notions of power compact resolvent and quasi-compact semigroup, essential spectrum and Calkin algebra.

The above theorem in particular applies to a positive and irreducible semigroup which is eventually norm continuous and its generator has compact resolvent (see for instance Corollary VI.1.13 in [151]). In that case indeed, one can show that $s(\mathcal{L}) > -\infty$, $\Sigma_+(\mathcal{L})$ is bounded and consists of poles, so that $\Sigma_+(\mathcal{L}) = \{s(\mathcal{L})\}$ and the essential growth bound $\omega_{\text{ess}}(S)$ associated to the essential spectrum (see for instance [41, Sec 14.1] for a definition) satisfies $\omega_{\text{ess}}(S) < \omega(S) = s(\mathcal{L})$. The theorem was motivated and successfully applied to Boltzmann like transport operator [350], cell division operator [133], age structured equation [359] and selection-mutation dynamics [74]. We also refer to [151, Ch. VI] and [41] for other numerous applications. Although very general and quite efficient, we formulate several criticisms about the above result.

- The exponential convergence result is definitively not constructive and that approach is not able to say anything about the weak dissipative case (a framework we will introduce latter, see in particular Section 3.3).
and the other results established in [278, 273] is the clear identification of the simple localisation of the principal spectrum condition with (1.8).

### 1.2.3. Dynamical and probabilistic approach.

It is well known from the mean ergodicity theory of Von Neumann and Birkhoff introduced in the 1930s in [356, 60] that for a bounded semigroup a possible stationary state (and thus a first eigen-vector associated to the first eigenvalue $\lambda_1 = 0$) can be obtained through a dynamical approach by establishing that the Cesàro mean of the semigroup appropriately converges. A classical reference is [239], see also [152, Sec. V.4] for a short presentation.

The existence of invariant measures for Markov chains/processes can be derived through a contraction approach by using coupling arguments reminiscent from the ideas of Doeblin [140] and Harris [202]. This yields a simplified Krein-Rutman theorem in the Banach lattice of finite measures for Markov operators, providing the existence of $f_1$ whilst $\lambda_1 = 0$ and $\phi_1 = 1$ are known by definition. Doeblin's condition is a handy criterion which ensures contraction in total variation norm, and hence existence, uniqueness, and geometric stability of the invariant measure, see for instance [168, 81] for this very classical and easy result. It turns out that this contraction is related to the contraction in Hilbert's metric, see [174]. The drawback of Doeblin's condition is that it is quite demanding and typically requires the state space to be bounded. Harris's idea allows an extension to the unbounded setting by localizing Doeblin's condition in a "small set" which is visited infinitely often. The return to small sets is often obtained by using a Lyapunov function. When the Lyapunov function is strong enough for ensuring exponential return, contraction in weighted total variation norm can be established and geometric stability of the invariant measure is inferred [265, 268, 267, 268, 198, 81], leading to the following result (which is made constructive in the two last references).

**Theorem 1.6.** Consider a positive semigroup $S$ on the Banach space $X = M_m^1(E)$ for some weight function $m: E \to [1, \infty)$. Suppose that $S$ is conservative, in the sense that

(1) $S_t^*1 = 1$ for all $t \ge 0$, and assume that, for some subset $K \subset E$ on which $m$ is bounded and some time $T > 0$,

(2) $S_T^*m \le \alpha m + \theta 1_K$, for some $\alpha \in (0, 1)$ and $\theta > 0$;

(3) $S_T f \ge \langle f, 1_K \rangle g_0$, for all $f \in X_+$ and some $g_0 \in X_+$ such that $\langle g_0, 1_K \rangle > 0$.

Then there exists a unique probability measure $f_1 \in M_m^1$ such that $(\lambda_1 = 0, f_1, \phi_1 = 1)$ is solution to the first eigentriplet problem, and the quantitative exponential stability (E31) holds with constructive constants. Moreover, some reciprocal implication holds true.

When only a weak version of the above Lyapunov condition (2) is available, an extension of the theory to a weakly dissipative framework is possible and has been developed in [347, 145, 144, 197, 81] leading to existence, uniqueness, but only sub-geometric stability of the invariant measure. We also mention that ergodicity properties of Feynman-Kac semigroups were investigated in [130, 131] and [235, 236].

Using a condition proposed in [131, Condition $\mathcal{Z}$], the Doeblin-Harris method was extended to non-conservative semigroups in [35, 103, 105, 104, 109]. In [35] necessary and sufficient conditions for the geometric stability of $(\lambda_1, f_1, \phi_1)$ in weighted total variation norm are obtained. To our knowledge, no extension to the above mentioned weakly dissipative setting is available.

The following result is an immediate consequence of [35, Thm. 2.1].

**Theorem 1.7.** Consider the same situation as in Theorem 1.6 but relax the conservativeness assumption (1) by the assumption that there exists a function $\phi_0: E \to (0, \infty)$, bounded from above and below by positive constants on $K$, such that $\phi_0 \le m$ on $E$, and satisfying

(1a) $S_T^*\phi_0 \ge \beta\phi_0$, for some $\beta > 0$;

(1b) $1_K S_t^*\phi_0 \le C\langle g_0, 1_K S_t^*\phi_0 \rangle$, for all $t \ge 0$ and some $C > 0$;

and replace the condition $\alpha \in (0, 1)$ by $\alpha \in (0, \beta)$ in the assumption (2).

Then, there exists a unique solution $(\lambda_1, f_1, \phi_1)$ to the first eigentriplet problem and the quantitative exponential stability (E31) holds with constructive constants. Moreover, some reciprocal implication holds true.
- We may observe that Theorem 1.4 is not so popular in the probability and the PDE communities and still many works in these domains refer to the original Krein and Rutman theorem even when some additional (approximation) arguments are needed rather than applying directly Theorem 1.4. By the way, we did not find in the literature where Theorem 1.4 is stated in such an handy way (the closer formulation is probably [151, Thm. VI.1.12] which is given without proof).

- The proof of Theorem 1.4 that we may find in the above quoted references is written in a very specific and abstract language which make it quite obscure.

In [277, 272], one of the authors proposes the following variant.

**Theorem 1.5.** Consider a positive semigroup $S_L$ which satisfies (1.8) with $\kappa_0 \in \mathbb{R}$, it is irreducible and its generator enjoys the splitting structure (1.9)-(1.10) where $\kappa < \kappa_0$, $\mathcal{X}_0 = X$ and $\mathcal{X}_1 \subset X$ with compact imbedding. Assuming furthermore that

$$ (1.13) \qquad \exists \alpha > 0, \quad \sup_{z \in \Delta_{\kappa}} \langle z \rangle^{\alpha} \|W(z)\|_{\mathcal{B}(X)} < \infty, $$

the quantitative exponential asymptotic stability (E31) holds (without constructive constants).

The proof of Theorem 1.5 is based on a partial (but principal) spectral mapping and Weyl's theorem (in the spirit of Voigt [354]) coupled with a simple analysis of the first eigenelement problem based on the irreducibility of the semigroup, but which is really simpler than the deep result on irreducible semigroup stated in Theorem 1.4. On the other hand, that approach is unable to tackle the situation when $\Sigma_P^+(L)$ is not a singleton. One of the main features in Theorem 1.5 and the other results established in [277, 272] is the clear identification of the simple localization of the principal spectrum condition with (1.8).

### 1.2.3. Dynamical and probabilistic approach.

It is well known from the mean ergodicity theory of Von Neumann and Birkhoff introduced in the 1930s in [355, 60] that for a bounded semigroup a possible stationary state (and thus a first eigen-vector associated to the first eigenvalue $\lambda_1 = 0$) can be obtain through a dynamical approach by establishing that the Cesàro mean of the semigroup appropriately converges. A classical reference is [238], see also [151, Sec. V.4] for a short presentation.

The existence of invariant measures for Markov chains/processes can be derived through a contraction approach by using coupling arguments reminiscent from the ideas of Doblín [139] and Harris [201]. This yields a simplified Krein-Rutman theorem in the Banach lattice of finite measures for Markov operators, providing the existence of $f_1$ whilst $\lambda_1 = 0$ and $\phi_1 = 1$ are known by definition. Doblín's condition is a handy criterion which ensures contraction in total variation norm, and hence existence, uniqueness, and geometric stability of the invariant measure, see for instance [167, 80] for this very classical and easy result. It turns out that this contraction is related to the contraction in Hilbert's metric, see [173]. The drawback of Doblín's condition is that it is quite demanding and typically requires the state space to be bounded. Harris's idea allows an extension to the unbounded setting by localizing Doblín's condition in a "small set" which is visited infinitely often. The return to small sets can be obtained by using a Lyapunov function. When the Lyapunov function is strong enough for ensuring exponential return, contraction in weighted total variation norm can be established and geometric stability of the invariant measure is inferred [264, 265, 266, 267, 197, 80], leading to the following result (which is made constructive in the two last references).

**Theorem 1.6.** Consider a positive semigroup $S$ on the Banach space $X = M_m^1(E)$ for some weight function $m: E \to [1, \infty)$. Suppose that $S$ is conservative, in the sense that

(1) $S_t^*1 = 1$ for all $t \ge 0$, and assume that, for some subset $K \subset E$ on which $m$ is bounded and some time $T > 0$,

(2) $S_T^*m \le \alpha m + \theta 1_K$, for some $\alpha \in (0, 1)$ and $\theta > 0$;

(3) $S_T f \ge \langle f, 1_K \rangle g_0$, for all $f \in X_+$ and some $g_0 \in X_+$ such that $\langle g_0, 1_K \rangle > 0$.

Then there exists a unique probability measure $f_1 \in M_m^1$ such that $(\lambda_1 = 0, f_1, \phi_1 = 1)$ is solution to the first eigentriplet problem, and the quantitative exponential stability (E31) holds with constructive constants. Moreover, some reciprocal implication holds true.
Positivity conditions required for the Doeblin-Harris approach are less restrictive than for Birkhoff contraction. Conversely, unlike contraction in Hilbert's metric, Doeblin-Harris method strongly uses the linearity of the operators, and may thus not be easily extendable to nonlinear operator. However, since it is based on contraction arguments, it can be extended to time-inhomogeneous semigroups [34]. Finally, the existence of a first eigenmeasure in a non-conservative setting were established in [111, 112] through a Lyapunov function property, a suitable renormalization and a fixed point argument.

The key point in this approach is that it provides a constructive rate of convergence while its drawback is that it is somehow restricted to a $M_m^1$ (or $L_m^1$) framework and that some of the conditions (typically (1b) in Theorem 1.7) are not fully intuitive and may be hard to verify in the applications.

### 1.2.4. PDE approaches.

At least as far as the existence issue is concerned, one of the most common way in PDE papers in order to tackle the existence part of the first eigentriplet problem consists in approximating (by regularization of the coefficients, add of a small viscosity, discretization) the eigentriplet problem, then use the most classical Perron-Frobenius Theorem [311, 167] or Krein-Rutman Theorem [238, 125] and next to derive appropriate estimates and pass to the limit through a "stability argument".

Recently, in order to circumvent the above approximation step, a new abstract and general version of the existence part of the Krein-Rutman theory has been developed by Lions in [252] which, as for the early works [254, 253], is also adapted to nonlinear operators and it includes the following statement (in the linear operators framework).

**Theorem 1.8.** Consider a Banach lattice with positive cone $X_+$ and a linear and bounded operator $R: X \to X$ such that

(i) $R: X_+ \to X_+$;

(ii) $\exists g_2 \in X_+ \setminus \{0\}, \exists C_2 > 0$ such that $Rg_2 \le C_2g_2$, and set $K_2 := \{g \in X_+; \exists C, g \le Cg_2\}$;

(iii) $\mu_1 := \sup J < +\infty$, where

$$J := \{\mu \ge 0; \exists h \in K_2, h \ge \mu Rh + g_2\};$$

(iv) any sequence $(g^n)$ of almost first eigenvectors is relatively (possibly weakly) compact, where we say that $(g^n)$ is a sequence of almost first eigenvectors if $g^n = \mu^n R g^n + \varepsilon^n$, $(g^n)$ is bounded, $\mu^n \nearrow \mu_1$ and $\varepsilon^n \to 0$.

Then there exists $f_1 \in K_2$ such that $f_1 = \mu_1 R f_1$ and $\|f_1\| = 1$.

The statement and proof of Theorem 1.8 somehow generalize the existence part of the Krein-Rutman theorem presented in Theorem 1.4 because the required splitting structure and associated power compactness are replaced by the very natural stability principle (iv). Applications to elliptic operator with strong or critical confinement property in the whole space $\mathbb{R}^d$ setting are also presented in [252].

Let us also mention the huge literature on the characterization of the first eigenvalue by a min-max formula. As explained with more details below, this approach has first been introduced in the Courant-Fischer min-max theorem [160, 114, 115] providing a variational characterization of eigenvalues in an abstract Hilbert setting for self-adjoint elliptic operators. Inspired next by point-wise minmax formula established for simple self-adjoint operators [149, 323, 212] using a technique which goes back to Picard [316], it has been next generalized to non self-adjoint elliptic operators in [324, 47] among others. More recently, the same approach has been generalized to non elliptic operators, see for instance [118] and the references therein.

On the other hand, and beyond the eigentriplet problem, the convergence towards the first eigenfunction may be proved using the general relative entropy (GRE) method which has been applied to a large class of evolution PDE in [269] which principle is as follows. Assume that $(\lambda_1, f_1, \phi_1) \in \mathbb{R} \times X \times X'$ is a solution to the first eigenvalue problem, that $\lambda_1 = 0$ (a case to which one can always reduces from the general case by a mere change of operator and unknown), that $X, X' \subset L_{\text{loc}}^1(\mathcal{O})$ and then define the generalized relative entropy

$$\mathcal{J}(f) := \int_{\mathcal{O}} j(f/f_1) f_1 \phi_1 dz$$
When only a weak version of the above *Lyapunov condition (2)* is available, an extension of the theory to a *weakly dissipative* framework is possible and has been developed in [346, 144, 143, 196, 80] leading to existence, uniqueness, but only sub-geometric stability of the invariant measure. We also mention that ergodicity properties of Feynman-Kac semigroups were investigated in [129, 130] and [234, 235].

Using a condition proposed in [130, Condition $\mathcal{Z}$], the Doblin-Harris method was extended to non-conservative semigroups in [35, 102, 104, 103, 108]. In [35] necessary and sufficient conditions for the geometric stability of $(\lambda_1, f_1, \phi_1)$ in weighted total variation norm are obtained. To our knowledge, no extension to the above mentioned weakly dissipative setting is available.

The following result is an immediate consequence of [35, Thm. 2.1].

**Theorem 1.7.** Consider the same situation as in Theorem 1.6 but relax the conservativeness assumption (1) by the assumption that there exists a function $\phi_0 : E \to (0, \infty)$, bounded from above and below by positive constants on $K$, such that $\phi_0 \le m$ on $E$, and satisfying

(1a) $S_T^*\phi_0 \ge \beta\phi_0$, for some $\beta > 0$;

(1b) $\mathbf{1}_K S_t^* \phi_0 \le C(g_0, \mathbf{1}_K S_t^* \phi_0)$, for all $t \ge 0$ and some $C > 0$;

and replace the condition $\alpha \in (0, 1)$ by $\alpha \in (0, \beta)$ in the assumption (2).

Then, there exists a unique solution $(\lambda_1, f_1, \phi_1)$ to the first eigentriplet problem and the quantitative exponential stability ($\mathbf{E}3_1$) holds with constructive constants. Moreover, some reciprocal implication holds true.

Positivity conditions required for the Doblin-Harris approach are less restrictive than for Birkhoff contraction. Conversely, unlike contraction in Hilbert's metric, Doblin-Harris method strongly uses the linearity of the operators, and may thus not be easily extendable to nonlinear operator. However, since it is based on contraction arguments, it can be extended to time-inhomogeneous semigroups [34]. Finally, the existence of a first eigenmeasure in a non-conservative setting were established in [110, 111] through a Lyapunov function property, a suitable renormalization and a fixed point argument.

The key point in this approach is that it provides a constructive rate of convergence while its drawback is that it is somehow restricted to a $M_m^1$ (or $L_m^1$) framework and that some of the conditions (typically (1b) in Theorem 1.7) are not fully intuitive and may be hard to verify in the applications.

### 1.2.4. PDE approaches.

At least as far as the existence issue is concerned, one of the most common way in PDE papers in order to tackle the existence part of the first eigentriplet problem consists in approximating (by regularization of the coefficients, add of a small viscosity, discretization) the eigentriplet problem, then use the most classical Perron-Frobenius Theorem [310, 166] or Krein-Rutman Theorem [237, 124] and next to derive appropriate estimates and pass to the limit through a "stability argument".

Recently, in order to circumvent the above approximation step, a new abstract and general version of the existence part of the Krein-Rutman theory has been developed by Lions in [251] which, as for the early works [253, 252], is also adapted to nonlinear operators and it includes the following statement (in the linear operators framework).

**Theorem 1.8.** Consider a Banach lattice $X$ and a linear and bounded operator $\mathcal{R}: X \to X$ such that

(i) $\mathcal{R}: X_+ \to X_+$;

(ii) $\exists g_2 \in X_+ \setminus \{0\}, \exists C_2 > 0$ such that $\mathcal{R}g_2 \le C_2 g_2$;

(iii) $\mu_1 := \sup \mathcal{J} < +\infty$, where

$$ \mathcal{J} := \{\mu \ge 0; \exists h \in K_2, h \ge \mu R h + g_2\}, \quad K_2 := \{g \in X_+; \exists a, g \le ag_2\}; $$

(iv) any sequence $(g^n)$ of almost first eigenvectors is relatively (possibly weakly) compact, where we say that $(g^n)$ is a sequence of almost first eigenvectors if $g^n = \mu^n \mathcal{R}g^n + \varepsilon^n$, $(g^n)$ is bounded, $\mu^n \nearrow \mu_1$ and $\varepsilon^n \to 0$.

Then there exists $f_1 \in K_2$ such that $f_1 = \mu_1 \mathcal{R}f_1$ and $\|f_1\| = 1$.
for any given convex function $j: \mathbb{R} \to \mathbb{R}_+$. For any solution $f(t) \in X$ to the (appropriate) evolution PDE, one may establish (at least formal) the identity

$$ (1.14) \qquad \mathcal{J}(f(t)) + \int_0^t \mathcal{D}_{\mathcal{J}}(f(s)) ds = \mathcal{J}(f(0)), \quad \forall t \ge 0, $$

where $\mathcal{D}_{\mathcal{J}} \ge 0$ is the associated generalized dissipation of relative entropy, so that $\mathcal{J}$ is a Lyapunov functional (it is decreasing along the flow associated to the evolution PDE). Under suitable positivity hypothesis, one has $\mathcal{D}_{\mathcal{J}}(f) = 0$ if and only if $f \in \text{Vect}(f_1)$, and then one may deduce from (1.14) and some lower semicontinuity assumption on the operator $\mathcal{D}_{\mathcal{J}}$ that $f(t) \to cf_1$ as $t \to \infty$ (without rate and with $c \in \mathbb{R}$). The GRE method is of course connected to $j$-divergence in information theory and statistics [121, 122, 74, 262] and to $j$-entropy in probability and PDE theory [101, 176], where however here it is crucial to identify the associated operator $\mathcal{D}_{\mathcal{J}}$ and that this last one enjoys suitable properties.

### 1.2.5. Hypotheses and proof.

We now briefly discuss the strategy of the proof of Theorem 1.1 and how it is connected to the above material. Additional comments will be made in the corresponding Sections 2 to 6. As already said, the first eigenvalue problem is mainly split into three steps: existence, geometry and asymptotic stability. From a general point of view, our approach is more general than the initial Krein-Rutman theorem as well as less abstract than the usual semigroup school approach. We believe it is more intuitive and handy for the possible applications since it is presented as a series of estimates to be checked and the necessary assumptions are made clearer at each step.

• Concerning the existence of a solution to the first eigentriplet problem, our result improves the previous known results because (1) only weak compactness property is needed (while Theorems 1.4 & 1.5 require strong compactness assumptions), (2) it is more flexible than Theorems 1.4, 1.5, 1.7 & 1.8 (the two first ones being restricted to the generator of a strongly continuous semigroup, the third one being restricted to a $M_m^1$ framework and involving the tricky condition (1b) and the last one being somehow restricted to a weighted $L^\infty$ framework), (3) it applies to weakly dissipative cases (so that no spectral gap is needed). We present two different proofs: one based on a stationary problem approach and another one based on a dynamical problem approach (with which we are able to tackle the weakly dissipative case).

Our stationary problem approach mixes in a first step the (clearly formulated) approximation argument of [41, proof of Thm. 12.15] together with the stability argument of [252], where it is worth emphasizing that the condition $\kappa < \kappa_0$ in Theorem 1.1 is nothing but a practical (and possibly constructive) condition ensuring that assumption $s(B) < s(L)$ holds in Theorem 1.4. On a second step, we exhibit several practical situations where the required stability condition is fulfilled recovering as a particular case the existence part in Theorems 1.4 & 1.7. We would like to point out here that the splitting hypothesizes (1.11)-(1.12) on the semigroup is a generalization of the Lyapunov condition (2) in Theorem 1.6 on the semigroup which in turn generalizes the classical Lyapunov condition on the generator, namely for instance

$$ \mathcal{L}^*\psi_2 \le \kappa\psi_1 + K\psi_0 $$

with $\psi_i \in X'$, $\psi_1, \psi_2 \ge \psi_0$ together with $\psi_2 \le \psi_1$ (super Lyapunov condition), $\psi_2 = \psi_1$ (standard Lyapunov condition), $\psi_2 \ge \psi_1$ (weak Lyapunov condition). We refer to [81] and to Sections 2 and 3 for further discussions on that question.

On the other hand, our dynamical approach mixes the splitting method yet alluded above together with some argument picked up from Von Neumann & Birkoff mean ergodic theory in the spirit of but in a more elaborate way than in [81, Sec. 6].

• The proof about the geometry of the principal eigenvalue problem in Theorem 1.1 is a refinement of many arguments already developed in the literature. More precisely, the uniqueness of the first eigentriplet $(\lambda_1, f_1, \phi_1)$ and the strict positivity of the eigenvectors is established by taking up again in a more general setting some arguments developed in [313, 278, 231]. The subgroup structure of the boundary point spectrum $\Sigma_P^+(L)$ is next established under suitable (but not very restrictive) geometrical properties on the Banach lattice $X$, these ones being always true for the usual examples we have in mind and that we have already listed above. The proof mainly mimics the usual proof (as for instance presented in [41, Sec. 14.3]) but it is less abstract and more general.
The statement and proof of Theorem 1.8 somehow generalize the existence part of the Krein-Rutman theorem presented in Theorem 1.4 because the required splitting structure and associated power compactness are replaced by the very natural stability principle (iv). Applications to elliptic operator with strong or critical confinement property in the whole space $\mathbb{R}^d$ setting are also presented in [251].

Let us mention the huge literature on the characterization of the first eigenvalue by a minmax formula. As explained with more details below, this approach has first been introduced in the Courant-Fischer min-max theorem [159, 113, 114] providing a variational characterization of eigenvalues in an abstract Hilbert setting for self-adjoint elliptic operators. Inspired next by pointwise minmax formula established for simple self-adjoint operators [148, 322, 211] using a technique which goes back to Picard [315], it has been then generalized to non self-adjoint elliptic operators in [323, 47] among others. More recently, the same approach has been generalized to non elliptic operators, see for instance [117] and the references therein.

On the other hand, and beyond the eigentriplet problem, the convergence towards the first eigenfunction may be proved using the general relative entropy (GRE) method which has been applied to a large class of evolution PDE in [268] which principle is as follows. Assume that $(\lambda_1, f_1, \phi_1) \in \mathbb{R} \times X \times X'$ is a solution to the first eigenvalue problem, that $\lambda_1 = 0$ (a case to which one can always reduces from the general case by a mere change of operator and unknown), that $X, X' \subset L_{\text{loc}}^1(E)$ and then define the generalized relative entropy

$$ \mathcal{J}(f) := \int_E j(f/f_1) f_1 \phi_1 \, d\mu $$

for any given convex function $j: \mathbb{R} \to \mathbb{R}_+$. For any solution $f(t) \in X$ to the (appropriate) evolution PDE, one may establish (at least formal) the identity

$$ (1.14) \qquad \mathcal{J}(f(t)) + \int_0^t \mathcal{D}_{\mathcal{J}}(f(s)) \, ds = \mathcal{J}(f(0)), \quad \forall t \ge 0, $$

where $\mathcal{D}_{\mathcal{J}} \ge 0$ is the associated generalized dissipation of relative entropy, so that $\mathcal{J}$ is a Lyapunov functional (it is decreasing along the flow associated to the evolution PDE). Under suitable positivity hypothesis, one has $\mathcal{D}_{\mathcal{J}}(f) = 0$ if and only if $f \in \text{Vect}(f_1)$, and then one may deduce from (1.14) and some lower semicontinuity assumption on the operator $\mathcal{D}_{\mathcal{J}}$ that $f(t) \to cf_1$ as $t \to \infty$ (without rate and with $c \in \mathbb{R}$). The GRE method is of course connected to $j$-divergence in information theory and statistics [120, 121, 73, 261] and to $j$-entropy in probability and PDE theory [100, 175], where however here it is crucial to identify the associated operator $\mathcal{D}_{\mathcal{J}}$ and that this last one enjoys suitable properties.

### 1.2.5. Hypotheses and proof.

We now briefly discuss the strategy of the proof of Theorem 1.1 and how it is connected to the above material. Additional comments will be made in the corresponding Sections 2 to 6. As already said, the first eigenvalue problem is mainly split into three steps: existence, geometry and asymptotic stability. From a general point of view, our approach is more general than the initial Krein-Rutman theorem as well as less abstract than the usual semigroup school approach. We believe it is more intuitive and handy for the possible applications since it is presented as a series of estimates to be checked and the necessary assumptions are made clearer at each step.

• Concerning the existence of a solution to the first eigentriplet problem, our result improves the previous known results because (1) only weak compactness property is needed (while Theorems 1.4 & 1.5 require strong compactness assumptions), (2) it is more flexible than Theorems 1.4, 1.5, 1.7 & 1.8 (the two first ones being restricted to the generator of a strongly continuous semigroup, the third one being restricted to a $M_m^1$ framework and involving the tricky condition (1b) and the last one being somehow restricted to a weighted $L^\infty$ framework), (3) it applies to weakly dissipative cases (so that no spectral gap is needed). We present two different proofs: one based on a stationary problem approach and another one based on a dynamical problem approach (with which we are able to tackle the weakly dissipative case).

Our stationary problem approach mixes in a first step the (clearly formulated) approximation argument of [41, proof of Thm. 12.15] together with the stability argument of [251], where it
Especially, the proof does not refer to the notions of ideals, quasi-interior points or Calkin algebra nor uses the Kakutani lattice isomorphism theorem but rather uses the simpler notion of strict positivity (defined by duality) and some convenient structural properties of the signum operator. In order to go one step further and to prove the triviality property $\Sigma_P^+(\mathcal{L}) = \{\lambda_1\}$, we propose one quite original approach (which we believe to be new at this level of generality) based on an inverse Kato's inequality condition of $\mathcal{L}$ (by refining some arguments picked up from [278, 231]) and some more standard ones based on an aperiodicity condition on the semigroup $S_\mathcal{L}$, on a localisation of the point spectrum condition or on a quasi-compacteness condition on the semigroup $S_\mathcal{L}$.

• Finally, the proof on the asymptotic stability of the first eigenvector picks up and mixes some spectral analysis, dynamical system, entropy method and Doblin-Haris coupling arguments. On a first step, we mainly rewrite some very classical dynamical system results mixed together with some arguments coming from the General Relative Entropy method in order to get our mean ergodicity and ergodicity results which are really general and very little demanding about the trajectories. We also rewrite the most classical result about the exponential asymptotic stability (without constructive constants) of the first eigenfunction proposing a very simple (and self-contained) proof which does not make any references to abstract notions as Calkin algebra, essential spectrum or essential growth bound. Last, we adapt the Doblin-Harris approach as qualitatively formulated in [198, 81, 35] in order to get the quantitative asymptotic stability of the first eigenfunction with constructive constants.

### 1.3. Some examples of applications.

The abstract Krein-Rutman theory developed in these notes and alluded above have been cooked up in order to answer to the first eigenvalue problem for PDEs. We show its efficiency by applying it to several examples of evolution PDEs. These examples must be thus considered both as a motivation and an illustration of simultaneously developed abstract theory.

#### 1.3.1. Parabolic equations.
In Part 7, we are interested by parabolic equations in divergence form

$$ \partial_t f = \partial_i (a_{ij} \partial_j f) + \partial_i (\beta_i f) + b_j \partial_j f + cf \quad \text{in } (0, \infty) \times \Omega, $$

on the function $f = f(t, x)$, $t \ge 0$, $x \in \Omega$, with general conditions on the coefficients $a_{ij}$, $\beta_i$, $b_j$, $c$ and in both the case of a bounded domain $\Omega \subset \mathbb{R}^d$ (and we then complement the equation with a Dirichlet boundary condition) and the case $\Omega = \mathbb{R}^d$. The importance of parabolic equations for Physics, Chemistry, Biology and Economy modeling is well known and we do not discuss it here. We consider the four following cases.

• For a bounded domain $\Omega \subset \mathbb{R}^d$, we consider a general elliptic operator in divergence form

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

under the very general assumption about the regularity of the coefficients $a_{ij} \in L^\infty(\Omega)$, $a_{ij} \ge \nu\delta_{ij}$, for some $\nu > 0$, $b_i, \beta_j \in L^r(\Omega)$, $c \in L^{r/2}(\Omega)$, $r > d$.

• In the case when $\Omega = \mathbb{R}^d$, we focus first our analysis by considering

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

with drift $b \in L^\infty_{\text{loc}}(\mathbb{R}^d)$, potential $c \in L^2_{\text{loc}}(\mathbb{R}^d)$ and a confinement condition that (roughly speaking) we impose through the properties $c \to -\infty$ as $|x| \to \infty$ and $b$ is dominated by $c$ at the infinity. A typical case is given by $c \sim -|x|^\gamma$ and $b \sim x|x|^{\beta-1}$ as $|x| \to \infty$, with $\gamma > \max(0, \beta - 1)$.

• Still in the case when $\Omega = \mathbb{R}^d$, we next consider the similar problem

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

with now $c \in C_0(\mathbb{R}^d)$, $b \in C_0(\mathbb{R}^d)$ and $r \in \mathbb{R}_+$ a parameter. That hypotheses correspond to a critical confinement case and we further assume that $r > 0$ is large enough.

• In the case when $\Omega = \mathbb{R}^d$ again, we finally consider the elliptic operator

$$ \mathcal{L}f := \Delta f + b \cdot \nabla f + cf, $$

with the drift confinement

$$ b = \nabla U, \quad U(x) = \frac{1}{\beta} \langle x \rangle^\gamma, \quad \gamma > 0, $$
is worth emphasizing that the condition $\kappa < \kappa_0$ in Theorem 1.1 is nothing but a practical (and possibly constructive) condition ensuring that assumption $s(B) < s(L)$ holds in Theorem 1.4. On a second step, we exhibit several practical situations where the required stability condition is fulfilled recovering as a particular case the existence part in Theorems 1.4 & 1.7. We would like to point out here that the splitting hypotheses (1.11)-(1.12) on the semigroup is a generalization of the Lyapunov condition (2) in Theorem 1.6 on the semigroup which in turn generalizes the classical Lyapunov condition on the generator, namely for instance

$$ \mathcal{L}^*\psi_2 \leq \kappa\psi_1 + K\psi_0 $$

with $\psi_i \in X'$, $\psi_1, \psi_2 \geq \psi_0$ together with $\psi_2 \leq \psi_1$ (super Lyapunov condition), $\psi_2 = \psi_1$ (standard Lyapunov condition), $\psi_2 \geq \psi_1$ (weak Lyapunov condition). We refer to [80] and to Sections 2 and 3 for further discussions on that question.

On the other hand, our dynamical approach mixes the splitting method yet alluded above together with some argument picked up from Von Neumann & Birkoff mean ergodic theory in the spirit of but in a more elaborate way than in [80, Sec. 6].

* The proof about the geometry of the principal eigenvalue problem in Theorem 1.1 is a refinement of many arguments already developed in the literature. More precisely, the uniqueness of the first eigentriplet $(\lambda_1, f_1, \phi_1)$ and the strict positivity of the eigenvectors is established by taking up again in a more general setting some arguments developed in [312, 277, 230]. The subgroup structure of the boundary point spectrum $\Sigma_P^+(L)$ is next established under suitable (but not very restrictive) geometrical properties on the Banach lattice $X$, these ones being always true for the usual examples we have in mind and that we have already listed above. The proof mainly mimics the usual proof (as for instance presented in [41, Sec. 14.3]) but it is less abstract and more general. Especially, the proof does not refer to the notions of ideals, quasi-interior points or Calkin algebra nor uses the Kakutani lattice isomorphism theorem but rather uses the simpler notion of strict positivity (defined by duality) and some convenient structural properties of the signum operator. In order to go one step further and to prove the triviality property $\Sigma_P^+(L) = \{\lambda_1\}$, we propose one quite original approach (which we believe to be new at this level of generality) based on a reverse Kato's inequality condition of $L$ (by refining some arguments picked up from [277, 230]) and some more standard ones based on an aperiodicity condition on the semigroup $S_L$, on a localization of the point spectrum condition or on a quasi-compactness condition on the semigroup $S_L$.

* Finally, the proof on the asymptotic stability of the first eigenvector picks up and mixes some spectral analysis, dynamical system, entropy method and Doblin-Harris coupling arguments. On a first step, we mainly rewrite some very classical dynamical system results mixed together with some arguments coming from the General Relative Entropy method in order to get our mean ergodicity and ergodicity results which are really general and very little demanding about the trajectories. We also rewrite the most classical result about the exponential asymptotic stability (without constructive constants) of the first eigenfunction proposing a very simple (and self-contained) proof which does not make any references to abstract notions as Calkin algebra, essential spectrum or essential growth bound. Last, we adapt the Doblin-Harris approach as qualitatively formulated in [197, 80, 35] in order to get the quantitative asymptotic stability of the first eigenfunction with constructive constants.

## 1.3. Some examples of applications.

The abstract Krein-Rutman theory developed in these notes and alluded above have been cooked up in order to answer to the first eigenvalue problem for PDE. We show its efficiency by applying it to several examples of evolution PDE. These examples must be thus considered both as a motivation and an illustration of simultaneously developed abstract theory.

### 1.3.1. Parabolic equations.
In Part 7, we are interested in parabolic equations in divergence form

$$ \partial_t f = \partial_i(a_{ij}\partial_j f) + b_j\partial_j f + \partial_i(\beta_i f) + cf \quad \text{in } (0, \infty) \times \Omega, $$

on the function $f = f(t, x)$, $t \geq 0$, $x \in \Omega$, with general conditions on the coefficients $a_{ij}$, $b_j$, $\beta_i$, $c$ and in both the case of a bounded domain $\Omega \subset \mathbb{R}^d$ (and we then complement the equation with a Dirichlet boundary condition) and the case $\Omega = \mathbb{R}^d$. The importance of parabolic equations for
and with $c$ dominated by $b$ at the infinity. We further assume $c \ge \operatorname{div}b$ when $\gamma \in (0, 1]$. It is worth emphasizing that this corresponds to a perturbation of the classical Fokker-Planck operator associated to the potential $U$.

For each of these operators we are able to complete the existence, geometric and stability program as stated in Theorem 1.1, with constructive estimates on the first eigentriplet solution and more or less explicit rate of convergence to the first eigenfunction. Few suitable additional assumptions on the coefficients and on the regularity of $\Omega$ as well as the precise results will be discussed in the corresponding sections.

The first eigenvalue problem in the three first situations has been studied in [252, 8th and 9th courses] which inspired our study and to which we refer for motivations and possible extensions. Since mainly the existence issue is considered in [252], our results supplement the previous analysis by tackling the geometry of the principal spectrum and the exponential asymptotic stability of the first eigenfunction. On the other hand, the fourth situation in the conservative case ($c = \operatorname{div}b$) is very classical and we refer to [27, 28, 190, 274, 231] and the references therein. We believe that the extension to a non conservative case as considered here is new.

Of course, when the operator $\mathcal{L}$ is the Laplace operator or more generally is a self-adjoint elliptic operator, there exists a huge literature about the analysis of its spectrum and in particular about its first eigenvalue problem because among other things this is related to the ground state problem in quantum mechanic. We do not have the precise historical reference where similar results to the ones developed here are established for the first time. We may for instance refer to the contributions by Poincaré [319] and by Courant and Hilbert [114, 115]. We also refer to the textbook [179, Thm 8.38] for the quite general and modern proof which mixes minimisation technique, strong maximum principle and Hilbert structure arguments. It is worth mentioning that in earlier works, the Krein-Rutman theorem has been proved using elementary ODE method when considering the Sturm-Liouville operator (in dimension $d=1$), see for instance [63]. Still for a self-adjoint elliptic operator, the Courant-Fischer min-max theorem [160, 114] gives a variational characterization of eigenvalues through Rayleigh quotient [329] and the Weyl theorem [362, 363, 288, 312] provides some information about the distribution of the eigenvalues. More specifically, some constructive lower bound on the best constant in Poincaré inequality and thus on the first eigenvalue may be obtain through the Faber-Krahn [157, 237] isoperimetric inequality as presented in [92], see also Polya-Svzego [321, 320] and Payne-Weinberger [307, 308]. Other results on that direction but based on the Lyapunov condition are obtained in [27, 28] and we also refer to [30] and the references therein.

On the other hand, in the case of an elliptic operator which is not self-adjoint the first result on the principal eigenvalue problem seems to be Protter, Weinberger [324, Rk. 2] who consider the case of smooth domain and coefficients (without precise statement about the regularity) and use minmax formula and the Krein-Rutman Theorem 1.2, see also [310]. Next, Chicco [106, 107] establishes the existence, uniqueness and some monotony properties of the first eigenvalue-eigenfunction in the weak solutions framework of Stampacchia [339, 340] with mild regularity assumptions on the coefficients and which corresponds to the framework we will consider here (when we will consider the case of a bounded domain). These work has been followed by several papers by Donsker and Varadhan [143, 142] and next by the famous work of Berestycki, Nirenberg, Varadhan [47] opening a new field of research. These works are mainly based on strong maximum principle technique, see [325]. We also mention the recent works by Champagnat and Villemonais [104, 105] where similar results to ours for smooth enough coefficients are established using a variant of the probabilistic Doblin-Harris argument as already mentioned in Section 1.2.3. We also emphasize that in the conservative case, the long time behavior problem has been widely studied and some constructive estimates has been obtained in [29, 44, 215, 343, 344] by the mean of log-Sobolev inequality, in [27, 28, 332, 231] by the mean of Poincaré inequality and in [190, 231] by the mean of semigroup arguments.

1.3.2. Transport equation. In Part 8, we are interested in the general transport equation

$$ (1.15) \qquad \partial_t f + a \cdot \nabla_y f = \mathcal{H}[f] - Kf \quad \text{in } (0, \infty) \times \mathcal{O}, $$
Physics, Chemistry, Biology and Economy modeling is well known and we do not discuss it here. We consider the four following cases.

• For a bounded domain $\Omega \subset \mathbb{R}^d$, we consider a general elliptic operator in divergence form

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

under the very general assumption about the regularity of the coefficients $a_{ij} \in L^\infty(\Omega)$, $a_{ij} \ge \nu\delta_{ij}$, for some $\nu > 0$, $b_i, \beta_i \in L^r(\Omega)$, $c \in L^{r/2}(\Omega)$, $r > d$.

• In the case when $\Omega = \mathbb{R}^d$, we focus first our analysis by considering

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

with drift $b \in L_{\text{loc}}^{\infty}(\mathbb{R}^d)$, potential $c \in L_{\text{loc}}^2(\mathbb{R}^d)$ and a confinement condition that (roughly speaking) we impose through the properties $c \to -\infty$ as $|x| \to \infty$ and $b$ is dominated by $c$ at the infinity. A typical case is given by $c \sim -|x|^\gamma$ and $b \sim x|x|^{\beta-1}$ as $|x| \to \infty$, with $\gamma > \max(0, \beta - 1)$.

• Still in the case when $\Omega = \mathbb{R}^d$, we next consider the similar problem

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

with now $c \in C_0(\mathbb{R}^d)$, $b \in C_0(\mathbb{R}^d)$ and $r \in \mathbb{R}_+$ a parameter. That hypotheses correspond to a critical confinement case and we further assume that $r > 0$ is large enough.

• In the case when $\Omega = \mathbb{R}^d$ again, we finally consider the elliptic operator

$$ \mathcal{L}f := \Delta f + b \cdot \nabla f + cf, $$

with the drift confinement

$$ b = \nabla U, \quad U(x) = \frac{1}{\beta} \langle x \rangle^{\gamma}, \quad \gamma > 0, $$

and with $c$ dominated by $b$ at the infinity. We further assume $c \ge \text{div}b$ when $\gamma \in (0, 1]$. It is worth emphasizing that this corresponds to a perturbation of the classical Fokker-Planck operator associated to the potential $U$.

For each of these operators we are able to complete the existence, geometric and stability program as stated in Theorem 1.1, with constructive estimates on the first eigentriplet solution and more or less explicit rate of convergence to the first eigenfunction. Few suitable additional assumptions on the coefficients and on the regularity of $\Omega$ as well as the precise results will be discussed in the corresponding sections.

The first eigenvalue problem in the three first situations has been studied in [251, 8th and 9th courses] which inspired our study and to which we refer for motivations and possible extensions. Since mainly the existence issue is considered in [251], our results supplement the previous analysis by tackling the geometry of the principal spectrum and the exponential asymptotic stability of the first eigenfunction. On the other hand, the fourth situation in the conservative case (namely, when we additionally take $c := \text{div}b$) is very classical and we refer to [27, 28, 189, 273, 230] and the references therein. We believe that the extension to a non conservative case as considered here is new.

Of course, when the operator $\mathcal{L}$ is the Laplace operator or more generally is a self-adjoint elliptic operator, there exists a huge literature about the analysis of its spectrum and in particular about its first eigenvalue problem because among other things this is related to the ground state problem in quantum mechanic. We do not have the precise historical reference where similar results to the ones developed here are established for the first time. We may for instance refer to the contributions by Poincaré [318] and by Courant and Hilbert [113, 114]. We also refer to the textbook [178, Thm 8.38] for the quite general and modern proof which mixes minimization technique, strong maximum principle and Hilbert structure arguments. It is worth mentioning that in earlier works, the Krein-Rutman theorem has been proved using elementary ODE method when considering the Sturm-Liouville operator (in dimension $d = 1$), see for instance [63]. Still for a self-adjoint elliptic operator, the Courant-Fischer min-max theorem [159, 113] gives a variational characterization of eigenvalues through Rayleigh quotient [328] and the Weyl theorem [361, 362, 287, 311] provides some information about the distribution of the eigenvalues. More specifically, some constructive lower bound on the best constant in Poincaré inequality and thus on the first eigenvalue may be obtained through the Faber-Krahn [156, 236] isoperimetric inequality as presented in [91], see also
on the function $f = f(t, y)$, $t \ge 0$, $y \in \mathcal{O}$, with $\mathcal{O} \subset \mathbb{R}^D$, $D \ge 1$, a smooth open connected set. We assume that $a : \mathcal{O} \to \mathbb{R}^D$, $K : \mathcal{O} \to \mathbb{R}_+$, and that the collision operator $\mathcal{K}$ is linear and defined by

$$ (\mathcal{H}g)(y) := \int_{\mathcal{O}} k(y, y_{*}) g(y_{*}) dy_{*}, $$

for some kernel $k : \mathcal{O} \times \mathcal{O} \to \mathbb{R}_+$. When $\mathcal{O} \neq \mathbb{R}^D$, we complement the equation with a boundary condition on the incoming boundary $\Sigma_-$ which writes

$$ (\gamma_{-}f)(t, y) = \mathcal{R}_{\mathcal{O}}[f(t, \cdot)] + \mathcal{R}_{\Sigma}[\gamma_{+}f(t, \cdot)](y) \quad \text{on } (0, \infty) \times \Sigma_{-}, $$

where $\gamma_{\pm}f$ are the trace functions on the incoming and out going set $\Sigma_{\pm}$ and

$$ (\mathcal{R}_{\mathcal{O}}g)(y) := \int_{\mathcal{O}} r_{\mathcal{O}}(y, y_{*}) g(y_{*}) dy_{*}, \quad (\mathcal{R}_{\Sigma}h)(y) := \int_{\Sigma_{+}} r_{\Sigma}(y, y_{*}) h(y_{*}) d\sigma_{y_{*}}, $$

for some kernels $r_{\mathcal{O}} : \Sigma_{-} \times \mathcal{O} \to \mathbb{R}_+$, $r_{\Sigma} : \Sigma_{-} \times \Sigma_{+} \to \mathbb{R}_+$. All the (quite usual) notations will be explained at the begin of Part 8. It is worth emphasizing here that this framework in particular covers the cases of the renewal equation, the growth-fragmentation equation and the kinetic linear Boltzmann equation on which we will come back below. This framework is motivated by and generalizes the transport theory developed in [37, 43, 139, 70, 119].

In a first step, we consider a very general vector field $a$ by assuming that it satisfies the usual Sobolev regularity condition of DiPerna-Lions transport theory [139]. We also make general assumptions on $\mathcal{R}_{\mathcal{O}}$ and $\mathcal{R}_{\Sigma}$, but a very strong and somehow restrictive positivity condition on $\mathcal{H}$. Such an equation can be motivated by the abstract transport theory developed [43] as well as non-local reaction-diffusion models [109, 118, 249] and selection-mutation models in changing environment [162, 207]. Under these general conditions and additional ones we will detail later, we are able to solve the existence and geometrical part of the first eigenvalue problem and to prove an ergodicity result (without rate of convergence) generalizing some similar results obtained in [109, 118, 249].

Because of the strong positivity condition made on $\mathcal{H}$, the above mentioned result does not apply to the growth-fragmentation equation and the kinetic linear Boltzmann equation. We thus consider separately these important particular cases in the two next parts. Other singular jump kernels lacking strong positivity can appear in other models, for instance in neurosciences [150], and must also be treated through a specific study.

Another related model is the age structured (or renewal) equation

$$ \partial_t f + \partial_y f = -Kf \quad \text{in } (0, \infty) \times (0, \infty), $$

$$ f(t, 0) = (\mathcal{R}f(t, \cdot))(y) = \int_{0}^{\infty} r(y_{*})f(t, y_{*})dy_{*}. $$

It corresponds to the case $D = 1$, $\mathcal{O} = (0, \infty)$, $a = 1$, $\mathcal{R}_{\Sigma} = 0$, $\Sigma_{-} = \{0\}$ and $\mathcal{H} = 0$ in the transport equation (1.16). The age structured equation is very popular because it is useful for describing dynamic of populations [338, 19, 123, 359, 263] and simple neuronal dynamic [295, 306]. The long time behaviour can be analyzed though Laplace transform technique [158, 159, 221], relative entropy method [275, 269, 196], spectral analysis tool [358, 187, 278, 280, 277] and Doblín approach [34, 82, 168]. Because $\mathcal{H} = 0$, our previous result on the first eigenvalue problem does not apply. We just briefly observe that the method can be applied on the dual equation, thus guaranteeing the existence of $(\lambda_1, \phi_1)$, and then that the validity of Doblín's condition ensures the existence and uniqueness of the triplet $(\lambda_1, f_1, \phi_1)$, its positivity, and the exponential ergodicity.

### 1.3.3. Growth-fragmentation equation.
In Section 9, we consider the growth-fragmentation equation

$$ \partial_t f = \mathcal{L}f = \mathcal{G}f + \mathcal{F}f $$

posed on $\mathbb{R}_+$, with the growth operator $\mathcal{G}f = -\partial_x(af)$ and the fragmentation operator

$$ (\mathcal{F}f)(x) = \int_{x}^{\infty} k(y, x)f(y) dy - K(x)f(x), \quad K(x) := \int_{0}^{x} k(y, x) \frac{y}{x} dy. $$

Since the work of Diekmann, Heijmans and Thieme [134], many authors studied this equation by using various methods. We can mention, among many others, [269, 313, 246, 148, 84, 31, 50] for studies based on suitable weak distance, entropy and functional inequalities, [263, 317, 83, 33, 51,
Polya-Svzego [320, 319] and Payne-Weinberger [306, 307]. Other results on that direction but based on the Lyapunov condition are obtained in [27, 28] and we also refer to [30] and the references therein.

On the other hand, in the case of an elliptic operator which is not self-adjoint the first result on the principal eigenvalue problem seems to be Protter, Weinberger [323, Rk. 2] who consider the case of smooth domain and coefficients (without precise statement about the regularity) and use minmax formula and the Krein-Rutman Theorem 1.2, see also [309]. Next, Chicco [105, 106] establishes the existence, uniqueness and some monotony properties of the first eigenvalue-eigenfunction in the weak solutions framework of Stampacchia [338, 339] with mild regularity assumptions on the coefficients and which corresponds to the framework we will consider here (when we will consider the case of a bounded domain). These works have been followed by several papers by Donsker and Varadhan [142, 141] and next by the famous work of Berestycki, Nirenberg, Varadhan [47] opening a new field of research. These last works are mainly based on strong maximum principle technique, see [324]. We also mention the recent works by Champagnat and Villemonais [103, 104] where similar results to ours for smooth enough coefficients are established using a variant of the probabilistic Doblin-Harris argument as already mentioned in Section 1.2.3. We also emphasize that in the conservative case, the long time behavior problem has been widely studied and some constructive estimates has been obtained in [29, 44, 214, 342, 343] by the mean of log-Sobolev inequality, in [27, 28, 331, 230] by the mean of Poincaré inequality and in [189, 230] by the mean of semigroup arguments.

### 1.3.2. Transport equation.
In Part 8, we are interested in the general transport equation

$$ (1.15) \qquad \partial_t f + a \cdot \nabla_y f = \mathcal{K}[f] - Kf \quad \text{in } (0, \infty) \times \mathcal{O}, $$

on the function $f = f(t, y)$, $t \ge 0$, $y \in \mathcal{O}$, with $\mathcal{O} \subset \mathbb{R}^D$, $D \ge 1$, a smooth open connected set. We assume that $a: \mathcal{O} \to \mathbb{R}^D$, $K: \mathcal{O} \to \mathbb{R}_+$, and that the collision operator $\mathcal{K}$ is linear and defined by

$$ \mathcal{K}[g](y) := \int_{\mathcal{O}} k(y, y_*) g(y_*) dy_*, $$

for some kernel $k: \mathcal{O} \times \mathcal{O} \to \mathbb{R}_+$. When $\mathcal{O} \neq \mathbb{R}^D$, we complement the equation with a boundary condition on the incoming boundary $\Sigma_-$ which writes

$$ (\gamma_{-}f)(t, y) = \mathcal{R}_{\mathcal{O}}[f(t, \cdot)](y) + \mathcal{R}_{\Sigma}[\gamma_{+}f(t, \cdot)](y) \quad \text{on } (0, \infty) \times \Sigma_{-}, $$

where $\gamma_{\pm}f$ are the trace functions on the incoming and out going set $\Sigma_{\pm}$ and

$$ \mathcal{R}_{\mathcal{O}}[g](y) := \int_{\mathcal{O}} r_{\mathcal{O}}(y, y_*) g(y_*) dy_*, \quad \mathcal{R}_{\Sigma}[h](y) := \int_{\Sigma_+} r_{\Sigma}(y, y_*) h(y_*) d\sigma_{y_*}, $$

for some kernels $r_{\mathcal{O}}: \Sigma_- \times \mathcal{O} \to \mathbb{R}_+$, $r_{\Sigma}: \Sigma_- \times \Sigma_+ \to \mathbb{R}_+$. All the (quite usual) notations will be explained at the begin of Part 8. It is worth emphasizing here that this framework in particular covers the cases of the renewal equation, the growth-fragmentation equation and the kinetic linear Boltzmann equation on which we will come back below. This framework is motivated by and generalizes the transport theory developed in [37, 43, 138, 70, 118].

In a first step, we consider a very general vector field $a$ by assuming that it satisfies the usual Sobolev regularity condition of DiPerna-Lions transport theory [138]. We also make general assumptions on $\mathcal{R}_{\mathcal{O}}$ and $\mathcal{R}_{\Sigma}$, but a very strong and somehow restrictive positivity condition on $\mathcal{K}$. Such an equation can be motivated by the abstract transport theory developed [43] as well as non-local reaction-diffusion models [108, 117, 248] and selection-mutation models in changing environment [161, 206]. Under these general conditions and additional ones we will detail later, we are able to solve the existence and geometrical part of the first eigenvalue problem and to prove an ergodicity result (without rate of convergence) generalizing some similar results obtained in [108, 117, 248].

Because of the strong positivity condition made on $\mathcal{K}$, the above mentioned result does not apply to the growth-fragmentation equation and the kinetic linear Boltzmann equation. We thus consider separately these important particular cases in the two next parts. Other singular jump kernels lacking strong positivity can appear in other models, for instance in neurosciences [149], and must also be treated through a specific study.
52, 278, 287] in the framework of positive semigroups, [56, 57] for a probabilistic approach via the Feynman-Kac formula, [35, 80, 169, 354] for Harris's method, and also [163] for a recent new approach based on the reformulation of the equation as an abstract renewal problem. Our aim here is not to treat the most general cases of coefficients, but rather to illustrate the variety of the possible behaviors of the equation together with the efficiency and flexibility of the method developed in the first sections. We thus focus on a specific case of fragmentation operator, namely the equal mitosis kernel

$$k(x, y) = 2K(x)\delta_{x/2}(dy) = 4K(x)\delta_{2y}(dx),$$

so that the equation writes

$$\partial_t f(t, x) = -\partial_x(a(x)f(t, x)) - K(x)f(t, x) + 4K(2x)f(t, 2x).$$

In particular, we are interested in the case when the growth rate $a$ is such that $a(2x) = 2a(x)$ for all sizes $x$, for which the boundary spectrum is not trivial and the solutions then exhibit persistent oscillations in time. When this condition is not satisfied, we recover the more usual exponential convergence to the first eigenfunction.

We also aim at studying the variant of this equation where a variability $v$ is introduced as a growth speed parameter which is inherent to any individual, in the spirit of [275, 333] where such a variable is added in the renewal equation. More precisely we consider the growth-fragmentation equation with variability $v \in [1, 2]$ and the equal mitosis division kernel which reads

$$\partial_t f(t, x, v) = -v\partial_x(a(x)f(t, x, v)) - K(x)f(t, x, v) + 4 \int_1^2 K(2x)\wp(v, v_*)f(t, 2x, v_*)dv_*.$$

This model was introduced in [146], and then also considered in [304]. We prove that, unlike the case without variability, it exhibits exponential relaxation to the first eigenfunction even when $a(2x) = 2a(x)$ for all $x$.

### 1.3.4. Kinetic linear Boltzmann equation.
In Part 10, we are interested in another important subclass of transport equations, namely in the kinetic linear Boltzmann equation

$$(1.16) \qquad \partial_t f + v \cdot \nabla_x f - \nabla_x \Phi(x) \cdot \nabla_v f = \mathcal{K}[f] - Kf \quad \text{in } (0, \infty) \times \mathcal{O},$$

on the function $f = f(t, x, v)$, $t \ge 0$, $(x, v) \in \mathcal{O} = \Omega \times \mathcal{V}$, $\Omega \subset \mathbb{R}^d$, $\mathcal{V} \subset \mathbb{R}^d$, $d \ge 1$. We assume that $K: \mathcal{O} \to \mathbb{R}_+$, that $\mathcal{K}$ is a linear integral operator defined by

$$\mathcal{K}[g] := \int_{\mathbb{R}^d} rk(x, v, v_*) g(v_*) dv_*,$$

for some real number $r > 0$ and some kernel $k: \Omega \times \mathcal{V} \times \mathcal{V} \to \mathbb{R}_+$, and that $\Phi$ is a space confining potential $\Phi: \Omega \to \mathbb{R}$. We restrict our analysis to the case $\mathcal{V} := \mathbb{R}^d$ and $\Omega$ is either the torus $\Omega := \mathbb{T}^d$ (and we assume $\Phi = 0$) or it is the whole space $\Omega := \mathbb{R}^d$ (and we assume that $\Phi$ is a power function). This equation is very famous because it provides a model for neutron transport theory in nuclear reactors [93, 45] and for cells migration in a chemotactic gradient [7]. We refer to [40, 183, 39, 38, 282] for a mathematical analysis of the neutron transport equation and its diffusive approximation and to [214, 102] for the same concerning kinetic models for chemotaxis. Because the linear integral operator $\mathcal{K}$ is local in the position variable, this problem does not fall in the class of transport equation covered by the Krein-Rutman theorem established in Part 8 and a specific analysis is necessary. Under suitable positivity and regularity conditions on the kernel, we are able to complete the existence, geometric and stability program as stated in Theorem 1.1, with constructive estimates in the torus case, generalizing and improving previous works [59, 351, 352, 186, 257, 281, 283, 284, 285] where spectral analysis arguments are used and [103] based on a probability approach. It is worth emphasizing that these works are concerning the same equation in a bounded domain with no-flow boundary condition. Most of the literature is about the conservative case (when $\lambda_1 = 0$ and $\phi_1 = 1$) which has been tackled by the mean of spectral analysis method [53, 285, 286], of entropy method [132, 54], of geometric control method [200, 135], by hypocoercivity method [208, 141, 154] or by Harris coupling approach [79].
Another related model is the age-structured renewal equation

$$ \partial_t f + \partial_y f = -Kf \quad \text{in } (0, \infty) \times (0, \infty), \\ f(t, 0) = \mathcal{R}[f(t, \cdot)](y) = \int_0^\infty r(y_*)f(t, y_*)dy_* $$

It corresponds to the case $D = 1$, $O = (0, \infty)$, $a = 1$, $\mathcal{R}_\Sigma = 0$, $\Sigma_- = \{0\}$ and $\mathcal{K} = 0$ in the transport equation (1.15). The age structured equation is very popular because it is useful for describing dynamic of populations [337, 19, 122, 358, 262] and simple neuronal dynamic [294, 305]. The long time behaviour can be analyzed though Laplace transform technique [157, 158, 220], relative entropy method [274, 268, 195], spectral analysis tool [357, 186, 277, 279, 276] and Doblin approach [34, 81, 167]. Because $\mathcal{K} = 0$, our previous result on the first eigenvalue problem does not apply. We just briefly observe that the method can be applied on the dual equation, thus guaranteeing the existence of $(\lambda_1, \phi_1)$, and then that the validity of Doblin's condition ensures the existence and uniqueness of the triplet $(\lambda_1, f_1, \phi_1)$, its positivity, and the exponential ergodicity.

### 1.3.3. Growth-fragmentation equation.
In Section 9, we consider the growth-fragmentation equation

$$ \partial_t f = \mathcal{L}f = \mathcal{G}f + \mathcal{F}f \quad \text{in } (0, \infty) \times (0, \infty), $$

with the growth operator $\mathcal{G}f = -\partial_x(af)$ and the fragmentation operator

$$ (\mathcal{F}f)(x) = \int_x^\infty k(y, x)f(y)dy - K(x)f(x), \quad K(x) := \int_0^x k(y, x)\frac{y}{x}dy. $$

Since the work of Diekmann, Heijmans and Thieme [133], many authors studied this equation by using various methods. We can mention, among many others, [268, 312, 245, 147, 83, 31, 50] for studies based on suitable weak distance, entropy and functional inequalities, [262, 316, 82, 33, 51, 52, 277, 286] in the framework of positive semigroups, [56, 57] for a probabilistic approach via the Feynman-Kac formula, [35, 79, 168, 353] for Doblin-Harris's method, and also [162] for a recent new approach based on the reformulation of the equation as an abstract renewal problem. Our aim here is not to treat the most general cases of coefficients, but rather to illustrate the variety of the possible behaviors of the equation together with the efficiency and flexibility of the method developed in the first sections. We thus focus on a specific case of fragmentation operator, namely the equal mitosis kernel

$$ k(x, y) = 2K(x)\delta_{x/2}(dy) = 4K(x)\delta_{2y}(dx), $$

so that the equation writes

$$ \partial_t f(t, x) = -\partial_x(a(x)f(t, x)) - K(x)f(t, x) + 4K(2x)f(t, 2x). $$

In particular, we are interested in the case when the growth rate $a$ is such that $a(2x) = 2a(x)$ for all sizes $x > 0$, for which the boundary spectrum is not trivial and the solutions then exhibit persistent oscillations in time. When this condition is not satisfied, we recover the more usual exponential convergence to the first eigenfunction.

We also aim at studying the variant of this equation where a variability $v$ is introduced as a growth speed parameter which is inherent to any individual, in the spirit of [274, 332] where such a variable is added in the renewal equation. More precisely we consider the growth-fragmentation equation with variability $v \in [1, 2]$ and the equal mitosis division kernel which reads

$$ \partial_t f(t, x, v) = -v\partial_x(a(x)f(t, x, v)) - K(x)f(t, x, v) + 4 \int_1^2 K(2x)\wp(v, v_*)f(t, 2x, v_*)dv_* $$

This model was introduced in [145], and then also considered in [303]. We prove that, unlike the case without variability, it exhibits exponential relaxation to the first eigenfunction even when $a(2x) = 2a(x)$ for all $x > 0$.
1.3.5. Kinetic Fokker-Planck equation. In Part 11, we consider a kinetic Fokker-Planck equation

$$ (1.17) \qquad \partial_t f + v \cdot \nabla_y f = \Delta_v f + b \cdot \nabla_v f + c f \quad \text{in } (0, \infty) \times \mathcal{O}, $$

on the function $f = f(t, x, v)$, $t \ge 0$, $(x, v) \in \mathcal{O} := \Omega \times \mathbb{R}^d$, $\Omega \subset \mathbb{R}^d$ is a bounded domain, $b: \mathcal{O} \to \mathbb{R}^d$ is a given vector field and $c: \mathcal{O} \to \mathbb{R}$ is a given function. In contrast with the previous part, collisions are typically modeled by a Fokker-Planck operator $\Delta_v f + \text{div}_v(vf)$ which takes into account a thermal bath of (Gaussian) white-noise, see Kolmogorov [234], instead of the integral collisional operator $\mathcal{K}[f] - Kf$ in the linear Boltzmann equation (1.16). The above equation is complemented with the Maxwell boundary condition

$$ \gamma_{-}f = \alpha(x)\mathcal{D}_{x}\gamma_{+}f + \beta(x)\Gamma_{x}\gamma_{+}f, $$

where $\gamma_{\pm}f$ stand for the outgoing and incoming trace functions, $\alpha$ and $\beta$ are accommodation coefficients, $\mathcal{D}_x$ is a boundary diffusive operator and $\Gamma_x$ is the specular reflection operator. All these classical objects will be precisely defined in Part 11. We refer to [217, 298, 129, 66, 91, 67, 272, 353, 365] for a mathematical analysis of the kinetic Fokker-Planck equation or related problems. Under suitable boundedness and regularity conditions on the coefficients we are able to complete the existence, geometric and stability (without constructive estimates) program as stated in Theorem 1.1, generalizing the previous works [248, 193] (partially based on [330, 213, 247]) where similar results are established for the same kind of equation in a bounded domain with no-flow boundary condition. From a technical point of view, our proof is based on trace results as those developed in [257], boundary estimates picked up from [18, 257, 55] and regularity estimates recently obtained in [210, 181, 192]. We also emphasize that in the conservative case, many works have been done related to hypocoercivity and constructive rate of convergence to the steady state in [133, 195, 151, 209, 205, 353] or more recently in [141, 274, 87, 68, 5].

1.3.6. Mutation-selection equation. Last, in Section 12, we consider the mutation-selection evolution equation

$$ \partial_t f = \mathcal{L}f = J * f - W(x)f \quad \text{in } (0, \infty) \times \mathbb{R}^d, $$

This nonlocal-diffusion equation appears for instance in the modeling of genetic variability in evolutionary biology. In this context, $f = f(t, x)$ represents the density of a population, at time $t \ge 0$, of phenotypical trait $x$ on the multi-dimensional phenotypic trait space $\mathbb{R}^d$. The rate of change in $f$ per generation is given by the convolution term with kernel $J$ which models the mutations, and the fitness function $-W$ which stands for the difference between birth and death. This model has been widely used in the literature; we refer, for example, to the works of Kimura [232], Lande [245], Fleming [161] and Bürger [76] as examples of biological applications.

On the mathematical analysis point of view, the Krein-Rutman problem was investigated by Bürger in [75, 77] and more recently by Coville and co-authors [116, 249], as well as by Alfaro and co-authors in [6] where a quantified spectral gap is obtained for symmetric kernels $J$. A main difference of this equation compared to more classical “local” diffusion models, where the convolution is replaced by a Laplacian, is that the first eigenvector $f_1$ might be a measure with atoms [75, 77, 117]. Some conditions are then needed relating $W$ and $J$ for guaranteeing that the first eigenvector is an eigenfunction [6, 75, 249].

All the above mentioned results deal with kernels $J$ which are densities, namely absolutely continuous with respect to the Lebesgue measure. In our study, we allow the convolution kernel to have a singular part. In Section 12.1 we extend the results of the literature to the case of a small enough singular part. In Section 12.2 we consider a specific kernel which is purely singular, supported by the canonical axes of $\mathbb{R}^d$, and we extend the recent result of Velleret [350] to more general confining functions $W$.

1.4. Organization of the paper. The paper is organized in two main parts: the sections 2 to 6 are dedicated to the development of the abstract results about the Krein-Rutman problem, and the last sections 7 to 12 aim at illustrating the applicability of these results to various linear positivity preserving PDEs.

More precisely, with start with the existence part of the Krein-Rutman theorem, namely the conclusion (S1). This question is addressed through a stationary approach in Section 2 and through a dynamical approach in Section 3. Section 4 is devoted to the stronger conclusion of uniqueness of the first eigentriplet in the sense of (S2), as well as to the mean ergodic property (E1). In
### 1.3.4. Kinetic linear Boltzmann equation.
In Part 10, we are interested in another important subclass of transport equations, namely in the kinetic linear Boltzmann equation

$$ (1.16) \qquad \partial_t f + v \cdot \nabla_x f - \nabla_x \Phi(x) \cdot \nabla_v f = \mathcal{K}[f] - Kf \quad \text{in } (0, \infty) \times \mathcal{O}, $$

on the function $f = f(t, x, v)$, $t \ge 0$, $(x, v) \in \mathcal{O} = \Omega \times \mathcal{V}$, $\Omega \subset \mathbb{R}^d$, $\mathcal{V} \subset \mathbb{R}^d$, $d \ge 1$. We assume that $K: \mathcal{O} \to \mathbb{R}_+$, that $\mathcal{K}$ is a linear integral operator defined by

$$ \mathcal{K}[g] := \int_{\mathbb{R}^d} rk(x, v, v_*) g(v_*) dv_*, $$

for some real number $r > 0$ and some kernel $k: \Omega \times \mathcal{V} \times \mathcal{V} \to \mathbb{R}_+$, and that $\Phi$ is a space confining potential $\Phi: \Omega \to \mathbb{R}$. We restrict our analysis to the case $\mathcal{V} := \mathbb{R}^d$ and $\Omega$ is either the torus $\Omega := \mathbb{T}^d$ (and we assume $\Phi = 0$) or it is the whole space $\Omega := \mathbb{R}^d$ (and we assume that $\Phi$ is a power function). This equation is very famous because it provides a model for neutron transport theory in nuclear reactors [92, 45] and for cells migration in a chemotactic gradient [7]. We refer to [40, 182, 39, 38, 281] for a mathematical analysis of the neutron transport equation and its diffusive approximation and to [213, 101] for the same concerning kinetic models for chemotaxis. Because the linear integral operator $\mathcal{K}$ is local in the position variable, this problem does not fall in the class of transport equation covered by the Krein-Rutman theorem established in Part 8 and a specific analysis is necessary. Under suitable positivity and regularity conditions on the kernel, we are able to complete the existence, geometric and stability program as stated in Theorem 1.1, with constructive estimates in the torus case, generalizing and improving previous works [59, 350, 351, 185, 256, 280, 282, 283, 284] where spectral analysis arguments are used and [102] based on a probability approach. It is worth emphasizing that these works are concerning the same equation in a bounded domain with no-flow boundary condition. Most of the literature is about the conservative case (when $\lambda_1 = 0$ and $\phi_1 = 1$) which has been tackled by the mean of spectral analysis method [53, 284, 285], of entropy method [131, 54], of geometric control method [199, 134], of hypocoercivity method [207, 140, 153] or of Doblin-Harris coupling approach [78].

### 1.3.5. Kinetic Fokker-Planck equation.
In Part 11, we consider a kinetic Fokker-Planck equation

$$ (1.17) \qquad \partial_t f + v \cdot \nabla_y f = \Delta_v f + b \cdot \nabla_v f + cf \quad \text{in } (0, \infty) \times \mathcal{O}, $$

on the function $f = f(t, x, v)$, $t \ge 0$, $(x, v) \in \mathcal{O} := \Omega \times \mathbb{R}^d$, $\Omega \subset \mathbb{R}^d$ is a bounded domain, $b: \mathcal{O} \to \mathbb{R}^d$ is a given vector field and $c: \mathcal{O} \to \mathbb{R}$ is a given function. In contrast with the previous part, collisions are typically modeled by a Fokker-Planck operator $\Delta_v f + \text{div}_v(vf)$ which takes into account a thermal bath of (Gaussian) white-noise, see Kolmogorov [233], instead of the integral collisional operator $\mathcal{K}[f] - Kf$ in the linear Boltzmann equation (1.16). The above equation is complemented with the Maxwell boundary condition

$$ \gamma_{-}f = \alpha(x)\mathcal{D}_{x}\gamma_{+}f + \beta(x)\Gamma_{x}\gamma_{+}f, $$

where $\gamma_{\pm}f$ stand for the outgoing and incoming trace functions, $\alpha$ and $\beta$ are accommodation coefficients, $\mathcal{D}_x$ is a boundary diffusive operator and $\Gamma_x$ is the specular reflection operator. All these classical objects will be precisely defined in Part 11. We refer to [216, 297, 128, 66, 90, 67, 271, 352, 364] for a mathematical analysis of the kinetic Fokker-Planck equation or related problems. Under suitable boundedness and regularity conditions on the coefficients we are able to complete the existence, geometric and stability (without constructive estimates) program as stated in Theorem 1.1, generalizing the previous works [247, 192] (partially based on [329, 212, 246]) where similar results are established for the same kind of equation in a bounded domain with no-flow boundary condition. From a technical point of view, our proof is based on trace results as those developed in [271], boundary estimates picked up from [18, 271, 55] and regularity estimates recently obtained in [209, 180, 191]. We also emphasize that in the conservative case, many works have been done related to hypocoercivity and constructive rate of convergence to the steady state in [132, 194, 150, 208, 204, 352] or more recently in [140, 273, 86, 68, 5].

### 1.3.6. Mutation-selection equation.
Last, in Section 12, we consider the mutation-selection evolution equation

$$ \partial_t f = \mathcal{L}f = J * f - W(x)f \quad \text{in } (0, \infty) \times \mathbb{R}^d, $$
Section 5, we are interested in the geometry of the boundary point spectrum, deriving conditions that guarantee (S3₁), (S3₂) or (S3₃), as well as in the ergodic properties (E2) and (E3₁). Finally, in Section 6, we tackle the problem of quantifying the conclusions (S3₃) and (E3) by using constructive contraction arguments of the Doeblin-Harris type.

The purpose of the last six sections is to apply the theory developed in the first sections to the examples of PDEs presented in Section 1.3: some parabolic equations (Section 7), transport equations with integral terms (Section 8) and in particular growth-fragmentation equations (Section 9) and kinetic equations (Section 10), kinetic Fokker-Planck equations (Section 11), and purely integral mutation-selection equations (Section 12).

## 2. EXISTENCE THROUGH A STATIONARY PROBLEM APPROACH

In this part we provide a general existence result for the first eigentriplet problem by considering a family of approximating stationary problems and using a stability argument. We start by presenting the basic material about the Banach lattice framework and conclude with a comparison with several previous works.

### 2.1. The Banach lattice framework.
We start recalling the Banach lattice framework by stating (most of the time without proof) some well-known facts that one can find in reference monographs as [69, Chapitre II: Espaces de Riesz] or [337, 15, 32, 41].

**Banach lattice.** A real Banach lattice is a real Banach space $(X, \|\cdot\|)$ endowed with a partial order denoted by $\ge$ (or $\le$) such that the following holds:

(1) The set $X_+ := \{f \in X; f \ge 0\}$ is a nonempty convex cone (compatibility of the order with the vector space structure).

(2) For any $f \in X$, there exist some unique positive part $f_+ \in X_+$ and negative part $f_- \in X_+$ such that $f = f_+ - f_-$ which are minimal: $f = g - h$, $g, h \ge 0$ imply $g \ge f_+$ and $h \ge f_-$ (generation and properness of the positive cone). We set $|f| := f_+ + f_- \in X_+$ the absolute value of $f \in X$.

(3) For any $f, g \in X$, $|f| \le |g|$ implies $\|f\| \le \|g\|$ (compatibility of norm and order structures).

Under these assumptions, one can show that

- The convex cone $X_+$ is closed, pointed $X_+ \cap (-X_+) = \{0\}$ and generating $X = X_+ - X_+$.

- The lattice operations $f \mapsto f_+$, $f \mapsto f_-$ and $f \mapsto |f|$ are continuous (1-Lipschitz).

- The order intervals $\{h \in X; g \le h \le f\}$ are closed and bounded for any given $f, g \in X, f \ge g$.

It is worth emphasizing that one commonly defines the supremum and infimum operations by

$$f \vee g := g + (f - g)_+ \ge f, g, \quad f \wedge g := g - (g - f)_+ \le f, g,$$

for any $f, g \in X$, and these operations can be used as an alternative way for defining a Banach lattice (the lattice structure refers indeed to these supremum and infimum operations). We may note the following elementary formulas

$$(2.1) \qquad f_+ \wedge f_- = 0, \quad \|f\| = \|f\|, \quad \forall f \in X.$$

We write $f \perp g$ when $|f| \wedge |g| = 0$ or equivalently when $|f| + |g| = |f| \vee |g|$. In that case, we have

$$|f| + |g| = |f + g|.$$

**Dual Banach lattice.** On the dual space $X'$, we may naturally associate a dual order $\ge$ (or $\le$) by writing for $\varphi \in X'$

$$\varphi \ge 0 \text{ (or } \varphi \in X'_+) \quad \text{iff} \quad \forall f \in X_+ \quad \langle \varphi, f \rangle \ge 0.$$

For $\varphi \in X'$, there exist some unique $\varphi_{\pm} \in X'_+$ such that $\varphi = \varphi_+ - \varphi_-$ which also satisfy (and are defined by)

$$\forall f \in X_+, \quad \langle \varphi_{\pm}, f \rangle = \sup_{0 \le g \le f} \langle \pm \varphi, g \rangle.$$

One can show that the above conditions (1), (2) and (3) of a Banach lattice are fulfilled, and thus $X' = (X', \|\cdot\|_{X'}, \ge)$ is a Banach lattice. We observe that for any $f \in X_+$ there exists $f^* \in X'_+$ such that

$$(2.2) \qquad \langle f^*, f \rangle = \|f\|^2 = \|f^*\|_{X'}^2,$$
This nonlocal-diffusion equation appears for instance in the modeling of genetic variability in evolutionary biology. In this context, $f = f(t, x)$ represents the density of a population, at time $t \ge 0$, of phenotypical trait $x$ on the multi-dimensional phenotypic trait space $\mathbb{R}^d$. The rate of change in $f$ per generation is given by the convolution term with kernel $J$ which models the mutations, and the fitness function $-W$ which stands for the difference between birth and death. This model has been widely used in the literature; we refer, for example, to the works of Kimura [231], Lande [244], Fleming [160] and Bürger [75] as examples of biological applications.

On the mathematical analysis point of view, the Krein-Rutman problem was investigated by Bürger in [74, 76] and more recently by Coville and co-authors [115, 248], as well as by Alfaro et al. in [6] where a quantified spectral gap is obtained for symmetric kernels $J$. A main difference of this equation compared to more classical “local” diffusion models, where the convolution is replaced by a Laplacian, is that the first eigenvector $f_1$ might be a measure with atoms [74, 76, 116]. Some conditions are then needed relating $W$ and $J$ for guaranteeing that the first eigenvector is an eigenfunction [6, 74, 248].

All the above mentioned results deal with kernels $J$ which are densities, namely absolutely continuous with respect to the Lebesgue measure. In our study, we allow the convolution kernel to have a singular part. In Section 12.1 we extend the results of the literature to the case of a small enough singular part. In Section 12.2 we consider a specific kernel which is purely singular, supported by the canonical axes of $\mathbb{R}^d$, and we extend the recent result of Velleret [349] to more general confining functions $W$.

**1.4. Organization of the paper.** The paper is organized in two main parts: the sections 2 to 6 are dedicated to the development of the abstract results about the Krein-Rutman problem, and the last sections 7 to 12 aim at illustrating the applicability of these results to various linear positivity preserving PDE.

More precisely, we start with the existence part of the Krein-Rutman theorem, namely the conclusion (S1). This question is addressed through a stationary approach in Section 2 and through a dynamical approach in Section 3. Section 4 is devoted to the stronger conclusion of uniqueness of the first eigentriplet in the sense of (S2), as well as to the mean ergodic property (E1). In Section 5, we are interested in the geometry of the boundary point spectrum, deriving conditions that guarantee (S3₁), (S3₂) or (S3₃), as well as in the ergodic properties (E2) and (E3₁). Finally, in Section 6, we tackle the problem of quantifying the conclusions (S3₃) and (E3) by using constructive contraction arguments of the Doblin-Harris type.

The purpose of the last six sections is to apply the theory developed in the first sections to the examples of PDE presented in Section 1.3: parabolic equations (Section 7), transport equations with integral terms (Section 8) and in particular growth-fragmentation equations (Section 9) and kinetic equations (Section 10), kinetic Fokker-Planck equations (Section 11), and purely integral mutation-selection equations (Section 12).

## 2. EXISTENCE THROUGH A STATIONARY PROBLEM APPROACH

In this part we provide a general existence result for the first eigentriplet problem by considering a family of approximating stationary problems and using a stability argument, in particular we prove (S1). For that purpose we introduce some conditions (H1), (H2), (H3), as well as a splitting structure (HS1) useful for verifying (H3). We start by presenting the basic material about the Banach lattice framework and conclude with a comparison with several previous works.

**2.1. The Banach lattice framework.** We start recalling the Banach lattice framework by stating (most of the time without proof) some well-known facts that one can find in reference monographs as [69, Chapitre II: Espaces de Riesz] or [336, 15, 32, 41].

**Banach lattice.** A real Banach lattice is a real Banach space $(X, \| \cdot \|)$ endowed with a partial order denoted by $\ge$ (or $\le$) such that the following holds:

(1) The set $X_+ := \{f \in X; f \ge 0\}$ is a nonempty convex cone (compatibility of the order with the vector space structure).
as a classical corollary of the Hahn-Banach dominated extension theorem. Moreover, for any $f \in X$,
(2.3) \qquad $f \ge 0$ iff $\langle \varphi, f \rangle \ge 0, \forall \varphi \in X'_+$,

as an immediate application of the Hahn-Banach separation theorem. In other words, the restriction to $X$ of the dual order in $X''$ associated to the order defined (by duality) on $X'$ is nothing but the initial order, in particular the positive cone $X'_+$ is weakly * closed.

**The functional framework : The duality bracket.** We consider two Banach lattices $X, Y$ such that $X = Y'$ with $Y$ separable or such that $Y = X'$. We emphasize on the facts that

$$
(2.4) \qquad \text{for } f \in X : \quad f \in X_+ \text{ iff } \langle f, \varphi \rangle \ge 0, \forall \varphi \in Y_+,
$$

$$
(2.5) \qquad \text{for } \varphi \in Y : \quad \varphi \in Y_+ \text{ iff } \langle f, \varphi \rangle \ge 0, \forall f \in X_+,
$$

which are immediate consequences of (2.3) and of the definition of the dual order.

**Examples.** For the space $C_0(E)$, the order is defined by $f \ge 0$ iff $f(x) \ge 0$ for any $x \in E$. For a space $L^p(E, \mathcal{E}, \mu)$, $1 \le p \le \infty$, the order is defined by $f \ge 0$ iff $f(x) \ge 0$ for $\mu$-a.e. $x \in E$. For the space $M^1(E)$, the order is defined by $f \ge 0$ iff in the Hahn decomposition $f = f_+ - f_-$, there holds $f_- = 0$, or equivalently, by duality: $f \ge 0$ iff $\langle f, \varphi \rangle \ge 0$ for any $\varphi \in C_0(E)$, $\varphi \ge 0$.

Because confinement will play a major role in our analysis, we will use some weighted version of
the above space associated to a weight (continuous or Borel measurable) function $m : E \to (0, \infty)$
that we introduce now. We recall that $E$ always denotes a $\sigma$-compact metric space, and we write
$E = \cup E_R$, with $E_R \subset E_{R+1}$, $E_R$ compact. In that context, we write $x_n \to \infty$ if for any $R \ge 1$
there exists $n_R$ such that $x_n \notin E_R$ for any $n \ge n_R$.

• We denote by $C_{m,0}(E)$ the space

$$
C_{m,0}(E) := \{ \varphi \in C(E) ; |\varphi(x)|/m(x) \to 0 \text{ as } x \to \infty \}
$$

endowed with the norm $\|\varphi\|_{C_{m,0}} := \|\varphi/m\|_{C_0}$.

• We denote by $M_m^1(E) := (C_{m,0}(E))'$ the associated space of Radon measures.

• We denote by $L_m^p(E) = L_m^p(E, \mathcal{E}, \mu)$ the space

$$
L_m^p(E) := \{ f \in L_{\text{loc}}^1(E); \|f\|_{L_m^p} := \|f m\|_{L^p} < \infty \}.
$$

It is worth emphasizing that $L_m^p(E, \mathcal{E}, \mu) = L^p(E, \mathcal{E}, m^p \mu)$ when $p \in [1, \infty)$.

**Positive operator.** We denote by $\mathcal{B}(X)$ the set of linear and bounded operators on $X$. We also denote by $\mathcal{H}(X)$ the subspace of compact operators. We say that a bounded operator $A \in \mathcal{B}(X)$ is positive, and we write $A \ge 0$, if

$$
Af \in X_+, \quad \forall f \in X_+.
$$

We will also sometimes abuse notations by writing $A \in \mathcal{B}(X_+)$ for meaning that $A \ge 0$. For a positive operator $A \in \mathcal{B}(X)$, we have

$$
(2.6) \qquad |Af| \le A|f|, \quad \forall f \in X, \text{ and } \|A\| = \sup_{0 \le f \in B_X} \|Af\|,
$$

where $B_X$ is the unit closed ball. More generally, we have

$$
(2.7) \qquad (Af) \vee (Ag) \leq A(f \vee g), \quad \forall f, g \in X.
$$

For $X$ and $Y$ in duality, and $A \in \mathcal{B}(X)$ and $A^* \in \mathcal{B}(Y)$ in duality, in the sense that

$$
\langle Af, \phi \rangle = \langle f, A^* \phi \rangle, \quad \forall f \in X, \phi \in Y,
$$

there holds

$$
(2.8) \qquad A \ge 0 \quad \text{iff} \quad A^* \ge 0.
$$

Let us present the elementary and classical but instructive proof of the direct implication, the
reciprocal way being similar. We assume thus $A \ge 0$. We take $\varphi \in Y_+$ and we define $\psi := A^*\varphi$.
We then take $f \in X_+$ and we define $g := Af$, so that $g \ge 0$ by assumption. We compute

$$
\langle \psi, f \rangle = \langle A^* \varphi, f \rangle = \langle \varphi, Af \rangle = \langle \varphi, g \rangle \ge 0.
$$

Since $f \in X_+$ is arbitrary, we get $\psi \in Y_+$, and thus $A^* \ge 0$.
(2) For any $f \in X$, there exist some unique positive part $f_+ \in X_+$ and negative part $f_- \in X_+$ such that $f = f_+ - f_-$ which are minimal: $f = g - h$, $g, h \ge 0$ imply $g \ge f_+$ and $h \ge f_-$ (generation and properness of the positive cone). We set $|f| := f_+ + f_- \in X_+$ the absolute value of $f \in X$.

(3) For any $f, g \in X$, $|f| \le |g|$ implies $\|f\| \le \|g\|$ (compatibility of norm and order structures).

Under these assumptions, one can show that

- The convex cone $X_+$ is closed, pointed $X_+ \cap (-X_+) = \{0\}$ and generating $X = X_+ - X_+$.

- The lattice operations $f \mapsto f_+$, $f \mapsto f_-$ and $f \mapsto |f|$ are continuous (1-Lipschitz).

- The order intervals $\{h \in X; g \le h \le f\}$ are closed and bounded for any given $f, g \in X$, $f \ge g$.

It is worth emphasizing that one commonly defines the supremum and infimum operations by

$$f \vee g := g + (f - g)_+ \ge f, g, \quad f \wedge g := g - (g - f)_+ \le f, g,$$

for any $f, g \in X$, and these operations can be used as an alternative way for defining a Banach lattice (the lattice structure refers indeed to these supremum and infimum operations). We may note the following elementary formulas

$$(2.1) \qquad f_+ \wedge f_- = 0, \quad \|f\| = \|f\|, \quad \forall f \in X.$$

We write $f \perp g$ when $|f| \wedge |g| = 0$ or equivalently when $|f| + |g| = |f| \vee |g|$.

**Dual Banach lattice.** On the dual space $X'$, we may naturally associate a dual order $\ge$ (or $\le$) by writing for $\varphi \in X'$

$$\varphi \ge 0 \text{ (or } \varphi \in X'_+) \quad \text{iff} \quad \forall f \in X_+ \quad \langle \varphi, f \rangle \ge 0.$$

For $\varphi \in X'$, there exist some unique $\varphi_\pm \in X'_+$ such that $\varphi = \varphi_+ - \varphi_-$ which also satisfy (and are defined by)

$$\forall f \in X_+, \quad \langle \varphi_\pm, f \rangle = \sup_{0 \le g \le f} \langle \pm\varphi, g \rangle.$$

One can show that the above conditions (1), (2) and (3) of a Banach lattice are fulfilled, and thus $X' = (X', \| \cdot \|_{X'} \ge)$ is a Banach lattice. We observe that for any $f \in X_+$ there exists $f^* \in X'_+$ such that

$$(2.2) \qquad \langle f^*, f \rangle = \|f\|^2 = \|f^*\|_{X'}^2,$$

as a classical corollary of the Hahn-Banach dominated extension theorem. Moreover, for any $f \in X$,

$$(2.3) \qquad f \ge 0 \quad \text{iff} \quad \langle \varphi, f \rangle \ge 0, \forall \varphi \in X'_+,$$

as an immediate application of the Hahn-Banach separation theorem. In other words, the restriction to $X$ of the dual order in $X''$ associated to the order defined (by duality) on $X'$ is nothing but the initial order, in particular the positive cone $X'_+$ is weakly * closed.

**The functional framework : The duality bracket.** We consider two Banach lattices $X, Y$ such that $X = Y'$ with $Y$ separable or such that $Y = X'$. We emphasize on the facts that

$$(2.4) \qquad \text{for } f \in X: \quad f \in X_+ \text{ iff } \langle f, \varphi \rangle \ge 0, \forall \varphi \in Y_+,$$

$$(2.5) \qquad \text{for } \varphi \in Y: \quad \varphi \in Y_+ \text{ iff } \langle f, \varphi \rangle \ge 0, \forall f \in X_+,$$

which are immediate consequences of (2.3) and of the definition of the dual order.

**Examples.** For the space $C_0(E)$, the order is defined by $f \ge 0$ iff $f(x) \ge 0$ for any $x \in E$. For a space $L^p(E, \mathcal{E}, \mu)$, $1 \le p \le \infty$, the order is defined by $f \ge 0$ iff $f(x) \ge 0$ for $\mu$-a.e. $x \in E$. For the space $M^1(E)$, the order is defined by $f \ge 0$ iff in the Hahn decomposition $f = f_+ - f_-$, there holds $f_- = 0$, or equivalently, by duality: $f \ge 0$ iff $\langle f, \varphi \rangle \ge 0$ for any $\varphi \in C_0(E)$, $\varphi \ge 0$.

Because confinement will play a major role in our analysis, we will use some weighted version of the above space associated to a weight (continuous or Borel measurable) function $m: E \to (0, \infty)$ that we introduce now. We recall that $E$ always denotes a $\sigma$-compact metric space, and we write $E = \cup E_R$, with $E_R \subset E_{R+1}$, $E_R$ compact. In that context, we write $x_n \to \infty$ if for any $R \ge 1$ there exists $n_R$ such that $x_n \notin E_R$ for any $n \ge n_R$.

• We denote by $C_{m,0}(E)$ the space

$$C_{m,0}(E) := \{\varphi \in C(E); |\varphi(x)|m(x) \to 0 \text{ as } x \to \infty\}$$

endowed with the norm $\|\varphi\|_{C_{m,0}} := \|\varphi m\|_{L^\infty}$.
**Semigroup, generator and spectrum.** In this work, a semigroup $S = S(t) = (S_t)$ on $X$ will always denote a semigroup of linear and bounded operators on a Banach lattice $X$ which trajectories are

- either strongly continuous, namely, the mapping $t \mapsto S_t f$ is continuous for the norm of $X$ for any fixed $f \in X$;

- either weakly * continuous, namely $X = Y'$ for some separable Banach lattice $Y$ such that $\forall f \in X, \forall \phi \in Y, t \mapsto \langle S_t f, \phi \rangle_{X,Y}$ is continuous and $\forall t \ge 0, \forall \phi \in Y, f \mapsto \langle S_t f, \phi \rangle_{X,Y}$ is continuous. That is in particular the case when there exists a strongly continuous semigroup $P$ on $Y$ such that $S_t = P_t^*$ for any $t \ge 0$.

For a semigroup $S$, we denote by $\mathcal{L}$ its generator and $D(\mathcal{L})$ the associated domain, and thus we sometimes write $S = S_{\mathcal{L}}$. We also denote the iterated domain defined recursively by $D(\mathcal{L}^k) := \{f \in D(\mathcal{L}^{k-1}), \mathcal{L}f \in D(\mathcal{L}^{k-1})\}$ for any $k \ge 2$ and $D(\mathcal{L}^\infty) := \bigcap_{k \ge 1} D(\mathcal{L}^k)$. We recall that $D(\mathcal{L})$ is dense in $X$ and the graph of $\mathcal{L}$ is closed in $X \times X$. We define the growth bound

$$ (2.9) \qquad \omega = \omega(S) := \limsup_{t \to \infty} \frac{1}{t} \log \|S(t)\| \in \mathbb{R} \cup \{-\infty\}, $$

so that

$$ (2.10) \qquad \forall \omega' > \omega, \quad \exists M \ge 1, \quad \|S(t)\|_{\mathcal{B}(X)} \le Me^{\omega't}, \quad \forall t \ge 0, $$

and $\omega$ is the infimum of $\omega' \in \mathbb{R}$ such that (2.10) holds. We say that $S$ is a semigroup of contractions when $S$ satisfies (2.10) with $M = 1$ and $\omega' = 0$.

The resolvent set $\rho(\mathcal{L})$ is the set of $z \in \mathbb{C}$ such that if $z - \mathcal{L} : D(\mathcal{L}) \to X$ is bijective and its inverse belongs to $\mathcal{B}(X)$. We define the resolvent operator by

$$ (2.11) \qquad \mathcal{R}(z) = \mathcal{R}_{\mathcal{L}}(z) := (z - \mathcal{L})^{-1}, \quad \forall z \in \rho(\mathcal{L}), $$

and the spectrum by $\Sigma(\mathcal{L}) := \mathbb{C} \setminus \rho(\mathcal{L})$. Denoting the half complex plane of abscissa $\alpha \in \mathbb{R}$ by

$$ (2.12) \qquad \Delta_{\alpha} := \{z \in \mathbb{C}; \Re(e(z)) > \alpha\}, $$

we have $\rho(\mathcal{L}) \supset \Delta_{\omega}$ and, for any $z \in \Delta_{\omega}$, there holds

$$ (2.13) \qquad \mathcal{R}(z) = \int_{0}^{\infty} S(t)e^{-zt} dt. $$

**Positive semigroup.** We say that a semigroup $(S_t)$ on a Banach lattice $X$ is positive if
$$ S_t \ge 0, \quad \forall t \ge 0. $$

**Lemma 2.1.** For a semigroup $S$ on a Banach lattice $X$, there is equivalence between

(a) $S$ is positive;

(b) the associate resolvent operator $\mathcal{R}$ is positive: $\mathcal{R}(\kappa) \ge 0$ for all $\kappa > \omega$ (or for all sufficiently large $\kappa$).

It is immediate from Hille's identity (2.13) that (a) implies (b). The reciprocal implication comes from the relation $S(t) = \lim_{n \to \infty} [n/t \mathcal{R}(n/t)]^n$ at the foundation of the Hille-Yosida theory, see for instance [309, Thm. I.8.3].

2.2. **Existence part of the Krein-Rutman theorem.** From now on in this section, we consider a Banach lattice $X$ and an operator $\mathcal{L}$ with dense domain and closed graph. Our goal is mainly to prove the existence part for the primal problem in the Krein-Rutman theorem, namely

$$ (2.14) \qquad \exists \lambda_1 \in \mathbb{R}, \exists f_1 \in X_+ \setminus \{0\}, \quad \mathcal{L}f_1 = \lambda_1 f_1. $$

We will also discuss the existence part for the dual problem at the end of the section.

We first assume

(H1) $\exists \kappa_1 \in \mathbb{R}$ such that $\lambda - \mathcal{L}$ is invertible and $(\lambda - \mathcal{L})^{-1} : X_+ \to X_+$ for any $\lambda \ge \kappa_1$.

Note that an operator $\mathcal{L}$ satisfying (H1) is sometimes called a *resolvent positive operator* after the paper of Arendt [14].

We then set

$$ (2.15) \qquad \mathcal{I} := \{\kappa \in \mathbb{R}; \lambda - \mathcal{L} \text{ is invertible, } (\lambda - \mathcal{L})^{-1} \ge 0 \text{ for any } \lambda \ge \kappa\}, $$
• We denote by $M_m^1(E) := (C_{m-1,0}(E))'$ the associated space of Radon measures.

• We denote by $L_m^p(E) = L_m^p(E, \mathcal{E}, \mu)$ the space

$$L_m^p(E) := \{f \in L_{\text{loc}}^1(E); \|f\|_{L_m^p} := \|f m\|_{L^p} < \infty\}.$$

It is worth emphasizing that $L_m^p(E, \mathcal{E}, \mu) = L^p(E, \mathcal{E}, m^p\mu)$ when $p \in [1, \infty)$.

**Positive operator.** We denote by $\mathcal{B}(X)$ the set of linear and bounded operators on $X$. We also denote by $\mathcal{K}(X)$ the subspace of compact operators. We say that a bounded operator $A \in \mathcal{B}(X)$ is positive, and we write $A \ge 0$, if

$$Af \in X_+, \quad \forall f \in X_+.$$

We will also sometimes abuse notations by writing $A \in \mathcal{B}(X_+)$ for meaning that $A \ge 0$. For a positive operator $A \in \mathcal{B}(X)$, we have

$$(2.6) \qquad |Af| \le A|f|, \quad \forall f \in X, \quad \text{and} \quad \|A\| = \sup_{0 \le f \in B_X} \|Af\|,$$

where $B_X$ is the unit closed ball. More generally, we have

$$(2.7) \qquad (Af) \vee (Ag) \le A(f \vee g), \quad \forall f, g \in X.$$

For $X$ and $Y$ in duality, and $A \in \mathcal{B}(X)$ and $A^* \in \mathcal{B}(Y)$ in duality, in the sense that

$$\langle Af, \phi \rangle = \langle f, A^* \phi \rangle, \quad \forall f \in X, \quad \phi \in Y,$$

there holds

$$(2.8) \qquad A \ge 0 \quad \text{iff} \quad A^* \ge 0.$$

Let us present the elementary and classical but instructive proof of the direct implication, the reciprocal way being similar. We assume thus $A \ge 0$. We take $\varphi \in Y_+$ and we define $\psi := A^*\varphi$. We then take $f \in X_+$ and we define $g := Af$, so that $g \ge 0$ by assumption. We compute

$$\langle \psi, f \rangle = \langle A^* \varphi, f \rangle = \langle \varphi, Af \rangle = \langle \varphi, g \rangle \ge 0.$$

Since $f \in X_+$ is arbitrary, we get $\psi \in Y_+$, and thus $A^* \ge 0$.

**Semigroup, generator and spectrum.** In this work, a semigroup $S = S(t) = (S_t)$ on $X$ will always denote a semigroup of linear and bounded operators on a Banach lattice $X$ which trajectories are

- either strongly continuous, namely, the mapping $t \mapsto S_t f$ is continuous for the norm of $X$ for any fixed $f \in X$;

- either weakly $*$ continuous, namely $X = Y'$ for some separable Banach lattice $Y$ such that $\forall f \in X, \forall \phi \in Y, t \mapsto \langle S_t f, \phi \rangle_{X,Y}$ is continuous and $\forall t \ge 0, \forall \phi \in Y, f \mapsto \langle S_t f, \phi \rangle_{X,Y}$ is continuous. That is in particular the case when there exists a strongly continuous semigroup $P$ on $Y$ such that $S_t = P_t^*$ for any $t \ge 0$.

For a semigroup $S$, we denote by $\mathcal{L}$ its generator and $D(\mathcal{L})$ the associated domain, and thus we sometimes write $S = S_{\mathcal{L}}$. We also denote the iterated domain defined recursively by $D(\mathcal{L}^k) := \{f \in D(\mathcal{L}^{k-1}), \mathcal{L}f \in D(\mathcal{L}^{k-1})\}$ for any $k \ge 2$ and $D(\mathcal{L}^\infty) := \bigcap_{k \ge 1} D(\mathcal{L}^k)$. We recall that $D(\mathcal{L})$ is dense in $X$ and the graph of $\mathcal{L}$ is closed in $X \times X$. We define the growth bound

$$(2.9) \qquad \omega = \omega(S) := \limsup_{t \to \infty} \frac{1}{t} \log \|S(t)\| \in \mathbb{R} \cup \{-\infty\},$$

so that

$$(2.10) \qquad \forall \omega' > \omega, \quad \exists M \ge 1, \quad \|S(t)\|_{\mathcal{B}(X)} \le Me^{\omega't}, \quad \forall t \ge 0,$$

and $\omega$ is the infimum of $\omega' \in \mathbb{R}$ such that (2.10) holds. We say that $S$ is a semigroup of contractions when $S$ satisfies (2.10) with $M = 1$ and $\omega' = 0$.

The resolvent set $\rho(\mathcal{L})$ is the set of $z \in \mathbb{C}$ such that if $z - \mathcal{L} : D(\mathcal{L}) \to X$ is bijective and its inverse belongs to $\mathcal{B}(X)$. We define the resolvent operator by

$$(2.11) \qquad \mathcal{R}(z) = \mathcal{R}_{\mathcal{L}}(z) := (z - \mathcal{L})^{-1}, \quad \forall z \in \rho(\mathcal{L}),$$

and the spectrum by $\Sigma(\mathcal{L}) := \mathbb{C} \setminus \rho(\mathcal{L})$. Denoting the half complex plane of abscissa $\alpha \in \mathbb{R}$ by

$$(2.12) \qquad \Delta_{\alpha} := \{z \in \mathbb{C}; \Re e(z) > \alpha\},$$
which is a non empty and non upper bounded interval due to (H1). We finally set

$$ (2.16) \qquad \lambda_1 := \inf \mathcal{I} \in [-\infty, \kappa_1]. $$

For the sake of completeness, we recall now some general facts about $\mathcal{I}$ and $\lambda_1$ when $\mathcal{L}$ is the generator of a positive semigroup. We also refer to [152, Sec. 1.b, Chap. VI] or [41, Chapter 12] and the references therein for more details.

**Lemma 2.2.** When $\mathcal{L}$ is the generator of a positive semigroup $S = S_{\mathcal{L}}$, then

(i) (H1) automatically holds with any $\kappa_1 > \omega(S)$, so that $\lambda_1 \le \omega(S)$;

(ii) $\Sigma(\mathcal{L}) \cap \Delta_{\lambda_1} = \emptyset$ and the representation formula (2.13) holds true for any $z \in \Delta_{\lambda_1}$;

(iii) it may happen that $\lambda_1 = -\infty$.

The important property (ii) is probably due to [189].

*Proof of Lemma 2.2.* The claim (i) is an immediate consequence of the representation formula (2.13) for any $\kappa_1 > \omega(S)$ and the positivity of $S(t)$ for any $t \ge 0$ (that is nothing but Lemma 2.1). We prove (ii). Take $\lambda > \lambda_1$. From the classical identity

$$ S(t)e^{-\lambda t} - I = (\mathcal{L} - \lambda) \int_0^t S(s)e^{-\lambda s} ds, \quad \forall t \ge 0, $$

and the positivity property of $S$, we have

$$ 0 \le V(t) := \int_0^t S(s)e^{-\lambda s} ds = \mathcal{R}(\lambda) - \mathcal{R}(\lambda)S(t)e^{-\lambda t} \le \mathcal{R}(\lambda), $$

for any $t \ge 0$. From that estimate, we get $\|V(t)\| \le \|\mathcal{R}(\lambda)\|$. For any $z \in \Delta_{\lambda}$, an integration by part yields

$$ \int_0^t e^{-zs} S(s) ds = e^{-(z-\lambda)t} V(t) + (z-\lambda) \int_0^t e^{-(z-\lambda)s} V(s) ds. $$

The estimate on $V$ makes possible to pass to the limit $t \to \infty$ in the above identity, and we deduce

$$ \mathcal{U}(z) := \int_0^\infty e^{-zs} S(s) ds = (z - \lambda) \int_0^\infty e^{-(z-\lambda)s} V(s) ds \in \mathcal{B}(X). $$

In that situation, one classically knows that $z \in \rho(\mathcal{L})$ and $(z - \mathcal{L})^{-1} = \mathcal{U}(z)$. We have thus established $\Sigma(\mathcal{L}) \cap \Delta_{\lambda} = \emptyset$ and we conclude the proof of (ii) by observing that (2.13) is then nothing but the above formula.

(iii) On $L^p(0,1)$, $1 \le p < \infty$, the translation semigroup defined for $a > 0$ by

$$ S(t)f(x) := f(x+at)\mathbf{1}_{x+at \le 1}, \quad \forall t \ge 0, x \in (0,1), $$

is strongly continuous and positive. Since $S(t) \equiv 0$ for any $t \ge 1/a$, we have $\omega(S) = -\infty$, and thus $\lambda_1 = -\infty$ because of (i). $\square$

For further discussion, we give some probably classical results about the condition (H1) and some equivalent definitions of the set $\mathcal{I}$.

**Lemma 2.3.** The operator $\mathcal{L}$ satisfies (H1) if and only if the operator $\mathcal{L}^*$ satisfies (H1). Furthermore, under condition (H1) for $\mathcal{L}$ (or $\mathcal{L}^*$), we have

$$ (2.17) \qquad \mathcal{I} = \mathcal{I}_i, \quad \forall i = 2, 3, 4, $$

with

$$ \mathcal{I}_2 := \{\kappa \in \mathbb{R}; \lambda - \mathcal{L} \text{ is invertible for any } \lambda \ge \kappa\}, $$

$$ \mathcal{I}_3 := \{\kappa \in \mathbb{R}; \lambda - \mathcal{L}^* \text{ is invertible, } (\lambda - \mathcal{L}^*)^{-1} \ge 0 \text{ for any } \lambda \ge \kappa\}, $$

$$ \mathcal{I}_4 := \{\kappa \in \mathbb{R}; \lambda - \mathcal{L}^* \text{ is invertible for any } \lambda \ge \kappa\}. $$

*Proof of Lemma 2.3.* The equivalence of condition (H1) for the operators $\mathcal{L}$ and $\mathcal{L}^*$ is an immediate consequence of the identity $\rho(\mathcal{L}) = \rho(\mathcal{L}^*)$ (see for instance [229, Thm. III.6.22]) and the fact that $(\lambda - \mathcal{L})^{-1} : X_+ \to X_+$ iff $(\lambda - \mathcal{L}^*)^{-1} : Y_+ \to Y_+$ as recalled in (2.8). As a consequence, we have $\mathcal{I} = \mathcal{I}_3$ and $\mathcal{I}_2 = \mathcal{I}_4$.
we have $\rho(\mathcal{L}) \supset \Delta_\omega$ and, for any $z \in \Delta_\omega$, there holds

$$ (2.13) \qquad \mathcal{R}(z) = \int_0^\infty S(t)e^{-zt} dt. $$

**Positive semigroup.** We say that a semigroup $(S_t)$ on a Banach lattice $X$ is positive if

$$ S_t \ge 0, \quad \forall t \ge 0. $$

**Lemma 2.1.** For a semigroup $S$ on a Banach lattice $X$, there is equivalence between

(a) $S$ is positive;

(b) the associate resolvent operator $\mathcal{R}$ is positive: $\mathcal{R}(\kappa) \ge 0$ for all $\kappa > \omega$ (or for all sufficiently large $\kappa$).

It is immediate from Hille's identity (2.13) that (a) implies (b). The reciprocal implication comes from the relation $S(t) = \lim_{n\to\infty}[n/t\mathcal{R}(n/t)]^n$ at the foundation of the Hille-Yosida theory, see for instance [308, Thm. I.8.3].

**2.2. Existence part of the Krein-Rutman theorem.** From now on in this section, we consider a Banach lattice $X$ and an operator $\mathcal{L}$ with dense domain and closed graph. Our goal is mainly to prove the existence part for the primal problem in the Krein-Rutman theorem, namely

$$ (2.14) \qquad \exists \lambda_1 \in \mathbb{R}, \exists f_1 \in X_+ \setminus \{0\}, \quad \mathcal{L}f_1 = \lambda_1 f_1. $$

We will also discuss the existence part for the dual problem at the end of the section.

We first assume

(H1) $\exists \kappa_1 \in \mathbb{R}$ such that $\lambda - \mathcal{L}$ is invertible and $(\lambda - \mathcal{L})^{-1} : X_+ \to X_+$ for any $\lambda \ge \kappa_1$.

Note that an operator $\mathcal{L}$ satisfying (H1) is sometimes called a *resolvent positive operator* after the paper of Arendt [14].

We then set

$$ (2.15) \qquad \mathcal{I} := \{\kappa \in \mathbb{R}; \lambda - \mathcal{L} \text{ is invertible, } (\lambda - \mathcal{L})^{-1} \ge 0 \text{ for any } \lambda \ge \kappa\}, $$

which is a non empty and non upper bounded interval due to (H1). We finally set

$$ (2.16) \qquad \lambda_1 := \inf \mathcal{I} \in [-\infty, \kappa_1]. $$

For the sake of completeness, we recall now some general facts about $\mathcal{I}$ and $\lambda_1$ when $\mathcal{L}$ is the generator of a positive semigroup. We also refer to [151, Sec. 1.b, Chap. VI] or [41, Chapter 12] and the references therein for more details.

**Lemma 2.2.** When $\mathcal{L}$ is the generator of a positive semigroup $S = S_\mathcal{L}$, then

(i) (H1) automatically holds with any $\kappa_1 > \omega(S)$, so that $\lambda_1 \le \omega(S)$;

(ii) $\Sigma(\mathcal{L}) \cap \Delta_{\lambda_1} = \emptyset$ and the representation formula (2.13) holds true for any $z \in \Delta_{\lambda_1}$;

(iii) it may happen that $\lambda_1 = -\infty$.

The important property (ii) is probably due to [188].

**Proof of Lemma 2.2.** The claim (i) is an immediate consequence of the representation formula (2.13) for any $\kappa_1 > \omega(S)$ and the positivity of $S(t)$ for any $t \ge 0$ (that is nothing but Lemma 2.1). We prove (ii). Take $\lambda > \lambda_1$. From the classical identity

$$ S(t)e^{-\lambda t} - I = (\mathcal{L} - \lambda) \int_0^t S(s)e^{-\lambda s} ds, \quad \forall t \ge 0, $$

and the positivity property of $S$, we have

$$ 0 \le V(t) := \int_0^t S(s)e^{-\lambda s} ds = \mathcal{R}(\lambda) - \mathcal{R}(\lambda)S(t)e^{-\lambda t} \le \mathcal{R}(\lambda), $$

for any $t \ge 0$. From that estimate, we get $\|V(t)\| \le \|\mathcal{R}(\lambda)\|$. For any $z \in \Delta_\lambda$, an integration by part yields

$$ \int_0^t e^{-zs} S(s) ds = e^{-(z-\lambda)t} V(t) + (z-\lambda) \int_0^t e^{-(z-\lambda)s} V(s) ds. $$
We obviously have $\mathcal{I}_2 \subset \mathcal{I}$ and let us show the reverse inclusion. We denote $\mathcal{R} = \mathcal{R}_{\mathcal{L}}$. On the one hand, for any $z_0 \in \rho(\mathcal{L})$ and any $z \in \mathbb{C}$, $|z - z_0| < \|\mathcal{R}(z_0)\|^{-1}$, we know that

$$ (2.18) \qquad \mathcal{R}(z) = \mathcal{R}(z_0) \sum_{k=0}^{\infty} (z_0 - z)^k \mathcal{R}(z_0)^k, $$

which gives a proof of the fact that resolvent set $\rho(\mathcal{L})$ is open and that $\mathcal{R}$ is an holomorphic function on $\rho(\mathcal{L})$. Formula (2.18) also ensures that for $\lambda_0, \lambda \in \mathbb{R}$, the condition $\mathcal{R}(\lambda_0) \ge 0$ implies that $\mathcal{R}(\lambda) \ge 0$ provided that $\lambda_0 - \lambda > 0$ is small enough and thus $\mathcal{R}(\lambda) \ge 0$ for any $\lambda$ in the non upper bounded connected component of the set $\rho(\mathcal{L}) \cap \mathbb{R}$ thanks to a continuation argument. In particular, $\mathcal{I}$ is an open set and $\mathcal{I} = \mathcal{I}_2$. $\square$

We next assume

(H2) $\exists \kappa_0 \in \mathbb{R}$ such that $\inf \mathcal{I} \ge \kappa_0$.

We do not further consider in these notes the case when $\inf \mathcal{I} = -\infty$ and moreover we will particularly focus on the possibility to exhibit constructive lower bound $\kappa_0$.

We point out several conditions under which (H2) is satisfied.

**Lemma 2.4.** Condition (H2) holds under one of the four following conditions

(i) $\exists \kappa_0 \in \mathbb{R}, \exists \phi_0 \in Y_+ \setminus \{0\}$ such that $\mathcal{L}^* \phi_0 \ge \kappa_0 \phi_0$, which means

$$ \forall f \in D(\mathcal{L}) \cap X_+, \quad \langle \phi_0, (\kappa_0 - \mathcal{L})f \rangle \le 0; $$

(ii) $\exists \kappa_0 \in \mathbb{R}, \exists f_0 \in X_+ \setminus \{0\}$ such that $\mathcal{L} f_0 \ge \kappa_0 f_0$, which means

$$ \forall \phi \in D(\mathcal{L}^*) \cap Y_+, \quad \langle (\kappa_0 - \mathcal{L}^*)\phi, f_0 \rangle \le 0; $$

(iii) $\mathcal{L}^*$ is the generator of a positive semigroup $S^* = (S_t^*)$ and

$$ \exists \kappa_0 \in \mathbb{R}, \exists \phi_0 \in Y_+ \setminus \{0\}, \exists T > 0 \text{ such that } S_T^* \phi_0 \ge e^{\kappa_0 T} \phi_0; $$

(iv) $\mathcal{L}$ is the generator of a positive semigroup $S = (S_t)$ and

$$ \exists \kappa_0 \in \mathbb{R}, \exists f_0 \in X_+ \setminus \{0\}, \exists T > 0 \text{ such that } S_T f_0 \ge e^{\kappa_0 T} f_0. $$

**Proof of Lemma 2.4.** In the three cases, we claim that $\kappa_0 \notin \mathcal{I}$ and thus $\inf \mathcal{I} \ge \kappa_0$. We argue by contradiction, assuming $\lambda_1 < \kappa_0$, so that $\kappa_0 \in \mathcal{I} = \mathcal{I}_i$ for any $i = 2, 3, 4$.

We assume (i). For any $g \in X_+$, we define $f := (\kappa_0 - \mathcal{L})^{-1}g \in X_+$ and we compute

$$ 0 \le \langle \phi_0, g \rangle = \langle \phi_0, (\kappa_0 - \mathcal{L})f \rangle \le 0. $$

That implies $\langle \phi_0, g \rangle = 0$ for any $g \ge 0$, so that $\phi_0 = 0$ and a contradiction.

We assume (ii). For any $\psi \in Y_+$, we define $\phi := (\kappa_0 - \mathcal{L}^*)^{-1}\psi \in Y_+$ and we compute

$$ 0 \le \langle \psi, f_0 \rangle = \langle (\kappa_0 - \mathcal{L}^*)\phi, f_0 \rangle \le 0. $$

That implies $\langle \psi, f_0 \rangle = 0$ for any $\psi \ge 0$, so that $f_0 = 0$ and a contradiction.

We assume first that (iii) holds for any $T > 0$. For any $f \in D(\mathcal{L}) \cap X_+ \setminus \{0\}$, we compute

$$ \langle \phi_0, (\kappa_0 - \mathcal{L})f \rangle = -\frac{d}{dt} \langle \phi_0, e^{-\kappa_0 t} S_t f \rangle \le 0, $$

which is precisely (i). We assume now that (iii) holds. If $\kappa_0 \in \mathcal{I}$, for any $g \in X_+$, we may define $f = (\kappa_0 - \mathcal{L})^{-1}g \in X_+ \cap D(\mathcal{L})$ and from condition (iii), we have

$$ 0 \le \langle e^{-\kappa_0 T} S_{nT} f - f, \phi_0 \rangle = \langle (\mathcal{L} - \kappa_0) \int_0^{nT} e^{-\kappa_0 t} S_t f dt, \phi_0 \rangle, $$

for any $n \in \mathbb{N}$. From the very definition of $f$, we also have

$$ (\mathcal{L} - \kappa_0) \int_0^{nT} e^{-\kappa_0 t} S_t f dt = \int_0^{nT} e^{-\kappa_0 t} S_t (\mathcal{L} - \kappa_0) f dt = - \int_0^{nT} e^{-\kappa_0 t} S_t g dt \le 0. $$

The two pieces of information together imply

$$ \langle \int_0^{nT} e^{-\kappa_0 t} S_t g dt, \phi_0 \rangle = 0. $$
The estimate on $V$ makes possible to pass to the limit $t \to \infty$ in the above identity, and we deduce

$$ \mathcal{U}(z) := \int_{0}^{\infty} e^{-zs} S(s) \, ds = (z - \lambda) \int_{0}^{\infty} e^{-(z-\lambda)s} V(s) \, ds \in \mathcal{B}(X). $$

In that situation, one classically knows that $z \in \rho(\mathcal{L})$ and $(z - \mathcal{L})^{-1} = \mathcal{U}(z)$. We have thus established $\Sigma(\mathcal{L}) \cap \Delta_{\lambda} = \emptyset$ and we conclude the proof of (ii) by observing that (2.13) is then nothing but the above formula.

(iii) On $L^p(0, 1)$, $1 \le p < \infty$, the translation semigroup defined for $a > 0$ by

$$ S(t)f(x) := f(x + at)\mathbf{1}_{x+at \le 1}, \quad \forall t \ge 0, x \in (0, 1), $$

is strongly continuous and positive. Since $S(t) \equiv 0$ for any $t \ge 1/a$, we have $\omega(S) = -\infty$, and thus $\lambda_1 = -\infty$ because of (i). $\square$

For further discussion, we give some probably classical results about the condition (H1) and some equivalent definitions of the set $\mathcal{I}$.

**Lemma 2.3.** The operator $\mathcal{L}$ satisfies (H1) if and only if the operator $\mathcal{L}^*$ satisfies (H1). Furthermore, under condition (H1) for $\mathcal{L}$ (or $\mathcal{L}^*$), we have

$$ (2.17) \qquad \mathcal{I} = \mathcal{I}_i, \quad \forall i = 2, 3, 4, $$

with

$$ \mathcal{I}_2 := \{ \kappa \in \mathbb{R} ; \lambda - \mathcal{L} \text{ is invertible for any } \lambda \ge \kappa \}, $$

$$ \mathcal{I}_3 := \{ \kappa \in \mathbb{R} ; \lambda - \mathcal{L}^* \text{ is invertible}, (\lambda - \mathcal{L}^*)^{-1} \ge 0 \text{ for any } \lambda \ge \kappa \}, $$

$$ \mathcal{I}_4 := \{ \kappa \in \mathbb{R} ; \lambda - \mathcal{L}^* \text{ is invertible for any } \lambda \ge \kappa \}. $$

*Proof of Lemma 2.3.* The equivalence of condition (H1) for the operators $\mathcal{L}$ and $\mathcal{L}^*$ is an immediate consequence of the identity $\rho(\mathcal{L}) = \rho(\mathcal{L}^*)$ (see for instance [228, Thm. III.6.22]) and the fact that $(\lambda - \mathcal{L})^{-1} : X_+ \to X_+$ iff $(\lambda - \mathcal{L}^*)^{-1} : Y_+ \to Y_+$ as recalled in (2.8). As a consequence, we have $\mathcal{I} = \mathcal{I}_3$ and $\mathcal{I}_2 = \mathcal{I}_4$.

We obviously have $\mathcal{I}_2 \subset \mathcal{I}$ and let us show the reverse inclusion. We denote $\mathcal{R} = \mathcal{R}_{\mathcal{L}}$. On the one hand, for any $z_0 \in \rho(\mathcal{L})$ and any $z \in \mathbb{C}$, $|z - z_0| < \|\mathcal{R}(z_0)\|^{-1}$, we know that

$$ (2.18) \qquad \mathcal{R}(z) = \mathcal{R}(z_0) \sum_{k=0}^{\infty} (z_0 - z)^k \mathcal{R}(z_0)^k, $$

which gives a proof of the fact that resolvent set $\rho(\mathcal{L})$ is open and that $\mathcal{R}$ is an holomorphic function on $\rho(\mathcal{L})$. Formula (2.18) also ensures that for $\lambda_0, \lambda \in \mathbb{R}$, the condition $\mathcal{R}(\lambda_0) \ge 0$ implies that $\mathcal{R}(\lambda) \ge 0$ provided that $\lambda_0 - \lambda > 0$ is small enough and thus $\mathcal{R}(\lambda) \ge 0$ for any $\lambda$ in the non upper bounded connected component of the set $\rho(\mathcal{L}) \cap \mathbb{R}$ thanks to a continuation argument. In particular, $\mathcal{I}$ is an open set and $\mathcal{I} = \mathcal{I}_2$. $\square$

We next assume

$$ (H2) \quad \exists \kappa_0 \in \mathbb{R} \text{ such that } \inf \mathcal{I} \ge \kappa_0. $$

We do not further consider in these notes the case when $\inf \mathcal{I} = -\infty$ and moreover we will particularly focus on the possibility to exhibit constructive lower bound $\kappa_0$.

We point out several conditions under which (H2) is satisfied. In the applications part and depending of the case, we will be able to establish (mostly constructively) one of these different criteria.

**Lemma 2.4.** Condition (H2) holds under one of the four following conditions

(i) $\exists \kappa_0 \in \mathbb{R}, \exists \phi_0 \in Y_+ \setminus \{0\}$ such that $\mathcal{L}^*\phi_0 \ge \kappa_0\phi_0$, which means

$$ \forall f \in D(\mathcal{L}) \cap X_+, \quad \langle \phi_0, (\kappa_0 - \mathcal{L})f \rangle \le 0; $$

(ii) $\exists \kappa_0 \in \mathbb{R}, \exists f_0 \in X_+ \setminus \{0\}$ such that $\mathcal{L}f_0 \ge \kappa_0 f_0$, which means

$$ \forall \phi \in D(\mathcal{L}^*) \cap Y_+, \quad \langle (\kappa_0 - \mathcal{L}^*)\phi, f_0 \rangle \le 0; $$

(iii) $\mathcal{L}^*$ is the generator of a positive semigroup $S^* = (S_t^*)$ and

$$ \exists \kappa_0 \in \mathbb{R}, \exists \phi_0 \in Y_+ \setminus \{0\}, \exists T > 0 \text{ such that } S_T^*\phi_0 \ge e^{\kappa_0 T}\phi_0; $$
Passing to the limit $n \to \infty$ thanks to Lemma 2.2-(ii) and using (2.11)-(2.13), we obtain

$$0 = \left\langle \int_{0}^{\infty} e^{-\kappa_{0}t} S_{t} g dt, \phi_{0} \right\rangle = \langle f, \phi_{0} \rangle = \langle g, (\kappa_{0} - \mathcal{L}^{*})^{-1} \phi_{0} \rangle.$$

That implies $(\kappa_0 - \mathcal{L}^*)^{-1}\phi_0 = 0$ since $g$ is arbitrary, what is not possible since $\phi_0 \neq 0$. The proof of (H2) under assumption (iv) is similar and thus skipped. $\square$

**Remark 2.5.** (1) In practice, we may build $f_0$ or $\phi_0$ through an explicit computation or use a barrier fonction and strong maximum principle techniques. We refer to Lemma 4.12 for a possible general result in that direction.

(2) When (ii) holds with $f_0 \in X_+ \setminus \{0\} \cap D(\mathcal{L})$ and $\mathcal{L}$ is the generator of a positive semigroup $S$, then (iv) holds for any $T > 0$. In that case, we may indeed compute

$$S_{T}e^{-\kappa_{0}T}f_{0} - f_{0} = \int_{0}^{T} S_{t}e^{-\kappa_{0}t}(\mathcal{L} - \kappa_{0})f_{0}ds \geq 0.$$

**Lemma 2.6.** Under conditions (H1) and (H2), there hold

$$(2.19) \qquad \lambda_1 \in [\kappa_0, \kappa_1]$$

and

$$(2.20) \qquad \exists \lambda_n \searrow \lambda_1, \exists \hat{f}_n \in D(\mathcal{L}) \cap X_+, \varepsilon_n := \lambda_n \hat{f}_n - \mathcal{L} \hat{f}_n \ge 0, \|\hat{f}_n\| = 1, \|\varepsilon_n\| \to 0.$$

*Proof of Lemma 2.6.* We obviously have $\lambda_1 \le \kappa_1$ from assumption (H1) and $\lambda_1 \ge \kappa_0$ by assumption (H2), so that (2.19) is proved.

Consider now a sequence $(\lambda_n)_{n \ge 2}$ such that $\lambda_n \searrow \lambda_1$ as $n \to \infty$. We eventually have $\|\mathcal{R}(\lambda_n)\| \to \infty$ as $n \to \infty$, where we denote by $\mathcal{R} = \mathcal{R}_\mathcal{L}$ the resolvent of $\mathcal{L}$. On the contrary, we would have $\|\mathcal{R}(\lambda_{n'})\| \le M$ for some subsequence $\lambda_{n'} \searrow \lambda_1$ and some constant $M > 0$. Because of (2.18) this implies that $(\lambda_{n'} - \varepsilon, \lambda_{n'}) \subset \mathcal{I}$ for any $n'$ and some $\varepsilon > 0$, and this is in contradiction with the definition of $\lambda_1$. The blow up $\|\mathcal{R}(\lambda_n)\| \to \infty$ means that

$$\exists f_n \in D(\mathcal{L}), \exists g_n \in X, \mathcal{R}(\lambda_n)g_n = f_n, \|f_n\| \to \infty, \|g_n\| \le 1.$$

By splitting $g_n = g_n^+ - g_n^-$, we get

$$f_n = \mathcal{R}(\lambda_n)g_n^+ - \mathcal{R}(\lambda_n)g_n^-$$

with

$$\|g_n^\pm\| \le 1 \quad \text{and} \quad (\|\mathcal{R}(\lambda_n)g_n^+\| \to \infty \text{ or } \|\mathcal{R}(\lambda_n)g_n^-\| \to \infty).$$

Changing notations, we have thus

$$\exists f_n \ge 0, \exists g_n \ge 0, \mathcal{R}(\lambda_n)g_n = f_n, \|f_n\| \to \infty, \|g_n\| \le 1.$$

We get (2.20) by defining $\hat{f}_n := f_n/\|f_n\|$ and $\varepsilon_n := g_n/\|f_n\|$.

$\square$

We learn a very similar proof in [252], from which our own proof is adapted. The same type of arguments can also be found in [41, proof of Theorem 12.15].

We finally assume that

(H3) for any sequence $(\hat{f}_n)$ of $X$ such that (2.20) holds, there exist $f_1 \in X_+ \setminus \{0\}$ and a subsequence $(\hat{f}_{n'})$ such that $\hat{f}_{n'} \to f_1$ for the weak convergence or the weak * convergence.

We discuss several situations in which assumption (H3) holds. We start with a very classical framework formalized for instance by Voigt [355], see also Karlin [228, Cor. 1] or Sasser [335] for earlier similar situations and results, which is however somehow restrictive since it is based on a strong compactness property assumed at the level of the associated semigroup of operators.

**Lemma 2.7.** We assume that $\mathcal{L}$ generates a positive semigroup $S$, that (H2) holds for a constant $\kappa_0 \in \mathbb{R}$ and that there exists $T > 0$ such that the splitting

$$(2.21) \qquad S_T = V_T + K_T,$$

holds with $\|V_T\|_{\mathcal{B}(X)} \le e^{\kappa T}$, $\kappa < \kappa_0$, and $K_T \in \mathcal{K}(X)$. Then condition (H3) holds for the primal and the dual problems.
(iv) $\mathcal{L}$ is the generator of a positive semigroup $S = (S_t)$ and

$$ \exists \kappa_0 \in \mathbb{R}, \exists f_0 \in X_+ \setminus \{0\}, \exists T > 0 \text{ such that } S_T f_0 \ge e^{\kappa_0 T} f_0. $$

*Proof of Lemma 2.4.* In the four cases, we claim that $\kappa_0 \notin \mathcal{I}$ and thus $\inf \mathcal{I} \ge \kappa_0$. We argue by contradiction, assuming $\lambda_1 < \kappa_0$, so that $\kappa_0 \in \mathcal{I} = \mathcal{I}_i$ for any $i = 2, 3, 4$.

We assume (i). For any $g \in X_+$, we define $f := (\kappa_0 - \mathcal{L})^{-1}g \in X_+$ and we compute

$$ 0 \le \langle \phi_0, g \rangle = \langle \phi_0, (\kappa_0 - \mathcal{L})f \rangle \le 0. $$

That implies $\langle \phi_0, g \rangle = 0$ for any $g \ge 0$, so that $\phi_0 = 0$ and a contradiction.

We assume first that (iii) holds for any $T > 0$. For any $f \in D(\mathcal{L}) \cap X_+ \setminus \{0\}$, we compute

$$ \langle \phi_0, (\kappa_0 - \mathcal{L})f \rangle = -\frac{d}{dt} \langle \phi_0, e^{-\kappa_0 t} S_t f \rangle \le 0, $$

which is precisely (i). We assume now that (iii) holds. If $\kappa_0 \in \mathcal{I}$, for any $g \in X_+$, we may define $f = (\kappa_0 - \mathcal{L})^{-1}g \in X_+ \cap D(\mathcal{L})$ and from condition (iii), we have

$$ 0 \le \langle e^{-n\kappa_0 T} S_{nT} f - f, \phi_0 \rangle = \langle (\mathcal{L} - \kappa_0) \int_0^{nT} e^{-\kappa_0 t} S_t f dt, \phi_0 \rangle, $$

for any $n \in \mathbb{N}$. From the very definition of $f$, we also have

$$ (\mathcal{L} - \kappa_0) \int_0^{nT} e^{-\kappa_0 t} S_t f dt = \int_0^{nT} e^{-\kappa_0 t} S_t (\mathcal{L} - \kappa_0) f dt = - \int_0^{nT} e^{-\kappa_0 t} S_t g dt \le 0. $$

The two pieces of information together imply

$$ \langle \int_0^{nT} e^{-\kappa_0 t} S_t g dt, \phi_0 \rangle = 0. $$

Passing to the limit $n \to \infty$ thanks to Lemma 2.2-(ii) and using (2.11)-(2.13), we obtain

$$ 0 = \langle \int_0^\infty e^{-\kappa_0 t} S_t g dt, \phi_0 \rangle = \langle f, \phi_0 \rangle = \langle g, (\kappa_0 - \mathcal{L}^*)^{-1} \phi_0 \rangle. $$

That implies $(\kappa_0 - \mathcal{L}^*)^{-1}\phi_0 = 0$ since $g$ is arbitrary, what is not possible since $\phi_0 \ne 0$. The proofs of (H2) under assumption (ii) or (iv) are similar to the previous ones and are thus skipped. $\square$

**Remark 2.5.** (1) In practice, we may build $f_0$ or $\phi_0$ through an explicit computation or use a barrier function and strong maximum principle techniques. We refer to Lemma 4.12 for a possible general result in that direction. More precisely, in all the applications we will use the criteria (i) or (ii), except in Section 8.5 where the criteria (iv) is used.

(2) When (ii) holds with $f_0 \in X_+ \setminus \{0\} \cap D(\mathcal{L})$ and $\mathcal{L}$ is the generator of a positive semigroup $S$, then (iv) holds for any $T > 0$. In that case, we may indeed compute

$$ e^{-\kappa_0 T} S_T f_0 - f_0 = \int_0^T e^{-\kappa_0 t} S_t (\mathcal{L} - \kappa_0) f_0 ds \ge 0. $$

**Lemma 2.6.** Under conditions (H1) and (H2), there hold

$$ (2.19) \qquad \lambda_1 \in [\kappa_0, \kappa_1] $$

and

$$ (2.20) \qquad \exists \lambda_n \searrow \lambda_1, \exists \hat{f}_n \in D(\mathcal{L}) \cap X_+, \varepsilon_n := \lambda_n \hat{f}_n - \mathcal{L} \hat{f}_n \ge 0, \|\hat{f}_n\| = 1, \|\varepsilon_n\| \to 0. $$

*Proof of Lemma 2.6.* We obviously have $\lambda_1 \le \kappa_1$ from assumption (H1) and $\lambda_1 \ge \kappa_0$ by assumption (H2), so that (2.19) is proved.

Consider now a sequence $(\lambda_n)_{n \ge 2}$ such that $\lambda_n \searrow \lambda_1$ as $n \to \infty$. We eventually have $\|\mathcal{R}(\lambda_n)\| \to \infty$ as $n \to \infty$, where we denote by $\mathcal{R} = \mathcal{R}_\mathcal{L}$ the resolvent of $\mathcal{L}$. On the contrary, we would have $\|\mathcal{R}(\lambda_{n'})\| \le M$ for some subsequence $\lambda_{n'} \searrow \lambda_1$ and some constant $M > 0$. Because of (2.18) this implies that $(\lambda_{n'} - \varepsilon, \lambda_{n'}) \subset \mathcal{I}$ for any $n'$ and some $\varepsilon > 0$, and this is in contradiction with the definition of $\lambda_1$. The blow up $\|\mathcal{R}(\lambda_n)\| \to \infty$ means that

$$ \exists f_n \in D(\mathcal{L}), \exists g_n \in X, \mathcal{R}(\lambda_n)g_n = f_n, \|f_n\| \to \infty, \|g_n\| \le 1. $$
Proof of Lemma 2.7. The condition (H1) holds because of Lemma 2.2-(i). Let us then consider three sequences $(\lambda_n)$, $(\hat{f}_n)$ and $(\varepsilon_n)$ satisfying (2.20). Integrating along the rescaled flow, this yields

$$ 
\begin{aligned}
e^{-\lambda_n T} S_T \hat{f}_n - \hat{f}_n &= \int_0^T e^{-\lambda_n t} S_t (\mathcal{L} - \lambda_n) \hat{f}_n dt \\
&= -\int_0^T e^{-\lambda_n t} S_t \varepsilon_n dt :=: \tilde{\varepsilon}_n,
\end{aligned}
$$

which also reads

$$ V \hat{f}_n + K \hat{f}_n - e^{\lambda_n T} \hat{f}_n = e^{\lambda_n T} \tilde{\varepsilon}_n. $$

Since $e^{\lambda_n T} \ge e^{\kappa_0 T} > e^{\kappa T}$, the operator $e^{\lambda_n T} - V_T$ is invertible with inverse $Q(\lambda_n) := (e^{\lambda_n T} - V_T)^{-1}$ uniformly bounded and converging in $\mathcal{B}(X)$ to $Q(\lambda_1) = (e^{\lambda_1 T} - V_T)^{-1}$. We thus have

$$ \hat{f}_n = w_n + v_n, \quad w_n := Q(\lambda_n) K_T \hat{f}_n, \quad v_n := -Q(\lambda_n) e^{\lambda_n T} \tilde{\varepsilon}_n, $$

with $\|v_n\|_X \to 0$ and $(w_n)$ relatively compact in $X$. There exist thus a subsequence $(\hat{f}_{n_k})$ and $g \in X$ such that $K_T \hat{f}_{n_k} \to g$ and next

$$ w_{n_k} - Q(\lambda_1)g = (Q(\lambda_{n_k}) - Q(\lambda_1))K_T \hat{f}_{n_k} + Q(\lambda_1)(K_T \hat{f}_{n_k} - g) \to 0. $$

We deduce that $\hat{f}_{n_k} \to f_1$ strongly in $X$. Because of the positivity and normalized properties of $\hat{f}_n$, we get $f_1 \in X_+$, $\|f_1\|_X = 1$, and we conclude that (H3) holds for the primal problem

Observing that the dual semigroup $S^*$ satisfies $S_T^* = V_T^* + K_T^*$ with $\|V_T^*\|_{\mathcal{B}(Y)} \le e^{\kappa T}$ and $K_T^* \in \mathcal{K}(Y)$, the same proof implies that (H3) holds for the dual problem. □

In the six next lemmas, we will assume that (H1)-(H2) holds associated to some constants $\kappa_i \in \mathbb{R}$, $\kappa_0 < \kappa_1$, and we always make the following splitting structure hypothesis

(HS1) there exists a splitting $\mathcal{L} = \mathcal{A} + \mathcal{B}$ such that $\mathcal{B} - \alpha$ is invertible for any $\alpha \ge \kappa_0$ and

$$ (2.22) \qquad \mathcal{V}(\alpha) := \sum_{i=0}^{N-1} (\mathcal{R}_B(\alpha)\mathcal{A})^i \mathcal{R}_B(\alpha), \quad \mathcal{W}(\alpha) := (\mathcal{R}_B(\alpha)\mathcal{A})^N, $$

are bounded in $\mathcal{B}(X)$ uniformly with respect to $\alpha \ge \kappa_0$ and for some $N \ge 1$, where we recall that $\mathcal{R}_B(\alpha) := (\alpha - \mathcal{B})^{-1}$ is the resolvent of $\mathcal{B}$.

We first present a result also based on a strong compactness property which is assumed to hold however at the level of the resolvent operator. We will be able to use that result in most of the applications.

**Lemma 2.8.** (1) We assume (H1)-(H2)-(HS1) and there exists $N \ge 1$ such that

$$ (2.23) \qquad \mathcal{W}(\alpha) \text{ is strongly compact locally uniformly on } \alpha \ge \kappa_0, $$

in the sense that if $\alpha_n \to \alpha$, $\alpha_n \ge \kappa_0$, and $(g_n)$ is a bounded sequence in $X$, then there exist $f \in X$ and a subsequence $(g_{n_k})$ such that $\mathcal{W}(\alpha_{n_k})g_{n_k} \to f$ strongly in $X$. Then condition (H3) holds.

(2) We assume (H1)-(H2) and (HS1) where $\mathcal{R}_B(\alpha)$ is bounded uniformly in $\alpha \ge \kappa_0$, $\mathcal{A} \in \mathcal{B}(X)$ and $\mathcal{W}(\alpha) \in \mathcal{K}(X)$ for any fixed $\alpha \ge \kappa_0$ and some $N \ge 1$. Then condition (H3) holds both for the primal and the dual problems.

**Remark 2.9.** (1) The property (2.23) holds if we assume $\mathcal{W}(\alpha) : X \to \mathcal{X}_1$ is bounded uniformly in $\alpha \ge \kappa_0$ and $\mathcal{X}_1 \subset X$ with strong compact embedding.

(2) The property (2.23) holds if we assume (H1)-(H2)-(HS1) together with the facts that $\mathcal{R}_B(\alpha)$ and $\mathcal{R}_B(\alpha)\mathcal{A}$ are bounded uniformly in $\alpha \ge \kappa_0$ and $\mathcal{W}(\alpha) \in \mathcal{K}(X)$ for any fixed $\alpha \ge \kappa_0$. Consider indeed $\alpha_n \to \alpha$, $\alpha_n \ge \kappa_0$, and $(g_n)$ a bounded sequence in $X$. On the one hand, there exist $f \in X$ and a subsequence $(g_{n_k})$ such that $\mathcal{W}(\alpha)g_{n_k} \to f$ strongly in $X$, because $\mathcal{W}(\alpha) \in \mathcal{K}(X)$. On the other hand, using the resolvent identity $\mathcal{R}_B(\lambda) - \mathcal{R}_B(\mu) = (\mu - \lambda)\mathcal{R}_B(\lambda)\mathcal{R}_B(\mu)$, we have

$$ \mathcal{W}(\alpha) - \mathcal{W}(\alpha_n) = (\alpha_n - \alpha) \sum_{j=1}^{N} (\mathcal{R}_B(\alpha)\mathcal{A})^{N-j} \mathcal{R}_B(\alpha) (\mathcal{R}_B(\alpha_n)\mathcal{A})^j \to 0, $$

so that $\mathcal{W}(\alpha_{n_k})g_{n_k} \to f$ strongly in $X$, and (2.23) holds true.
By splitting $g_n = g_n^+ - g_n^-$, we get

$$f_n = \mathcal{R}(\lambda_n)g_n^+ - \mathcal{R}(\lambda_n)g_n^-$$

with

$$\|g_n^\pm\| \le 1 \quad \text{and} \quad (\|\mathcal{R}(\lambda_n)g_n^+\| \to \infty \text{ or } \|\mathcal{R}(\lambda_n)g_n^-\| \to \infty).$$

Changing notations, we have thus

$$\exists f_n \ge 0, \exists g_n \ge 0, \mathcal{R}(\lambda_n)g_n = f_n, \|f_n\| \to \infty, \|g_n\| \le 1.$$

We get (2.20) by defining $\hat{f}_n := f_n/\|f_n\|$ and $\varepsilon_n := g_n/\|f_n\|$.

□

We learn a very similar proof in [251], from which our own proof is adapted. The same type of arguments can also be found in [41, proof of Thm 12.15].

We finally assume that

(H3) for any sequence $(\hat{f}_n)$ of $X$ such that (2.20) holds, there exist $f_1 \in X_+ \setminus \{0\}$ and a subsequence $(\hat{f}_{n'})$ such that $\hat{f}_{n'} \to f_1$ for the weak convergence or the weak * convergence.

We discuss several situations in which assumption (H3) holds. We start with a very classical framework formalized for instance by Voigt [354], see also Karlin [227, Cor. 1] or Sasser [334] for earlier similar situations and results, which is however somehow restrictive since it is based on a strong compactness property assumed at the level of the associated semigroup of operators.

**Lemma 2.7.** We assume that $\mathcal{L}$ generates a positive semigroup $S$, that (H2) holds for a constant $\kappa_0 \in \mathbb{R}$ and that there exists $T > 0$ such that the splitting

$$S_T = V_T + K_T$$

holds with $\|V_T\|_{\mathcal{B}(X)} \le e^{\kappa T}$, $\kappa < \kappa_0$, and $K_T \in \mathcal{K}(X)$. Then condition (H3) holds for the primal and the dual problems.

**Proof of Lemma 2.7.** The condition (H1) holds because of Lemma 2.2-(i). Let us then consider three sequences $(\lambda_n)$, $(\hat{f}_n)$ and $(\varepsilon_n)$ satisfying (2.20). Integrating along the rescaled flow, this yields

$$\begin{aligned} e^{-\lambda_n T} S_T \hat{f}_n - \hat{f}_n &= \int_0^T e^{-\lambda_n t} S_t (\mathcal{L} - \lambda_n) \hat{f}_n dt \\ &= - \int_0^T e^{-\lambda_n t} S_t \varepsilon_n dt :=: \tilde{\varepsilon}_n, \end{aligned}$$

which also reads

$$V_T \hat{f}_n + K_T \hat{f}_n - e^{\lambda_n T} \hat{f}_n = e^{\lambda_n T} \tilde{\varepsilon}_n.$$

Since $e^{\lambda_n T} \ge e^{\kappa_0 T} > e^{\kappa T}$, the operator $e^{\lambda_n T} - V_T$ is invertible with inverse $Q(\lambda_n) := (e^{\lambda_n T} - V_T)^{-1}$ uniformly bounded and converging in $\mathcal{B}(X)$ to $Q(\lambda_1) = (e^{\lambda_1 T} - V_T)^{-1}$. We thus have

$$\hat{f}_n = w_n + v_n, \quad w_n := Q(\lambda_n) K_T \hat{f}_n, \quad v_n := -Q(\lambda_n) e^{\lambda_n T} \tilde{\varepsilon}_n,$$

with $\|v_n\|_X \to 0$ and $(w_n)$ relatively compact in $X$. There exist thus a subsequence $(\hat{f}_{n_k})$ and $g \in X$ such that $K_T \hat{f}_{n_k} \to g$ and next

$$w_{n_k} - Q(\lambda_1)g = (Q(\lambda_{n_k}) - Q(\lambda_1))K_T \hat{f}_{n_k} + Q(\lambda_1)(K_T \hat{f}_{n_k} - g) \to 0.$$

We deduce that $\hat{f}_{n_k} \to f_1 := Q(\lambda_1)g$ strongly in $X$. Because of the positivity and normalized properties of $\hat{f}_n$, we get $f_1 \in X_+$, $\|f_1\|_X = 1$, and we conclude that (H3) holds for the primal problem. Observing that the dual semigroup $S^*$ satisfies $S_T^* = V_T^* + K_T^*$ with $\|V_T^*\|_{\mathcal{B}(Y)} \le e^{\kappa T}$ and $K_T^* \in \mathcal{K}(Y)$, the same proof implies that (H3) holds for the dual problem. □

Although this classic result can probably be used in many of the applications to PDE that we present in the second part of this paper, we develop alternative arguments below because: (1) some of the applications presented in the sequel certainly do not fall within the scope of the above result (we are thinking in particular to the parabolic equation under weakly dissipative conditions in Section 7.4 and to the mitosis equation with non-mixing growth rate in Section 9.2) and (2) the framework developed below is much simpler and direct to use (it requires no information on the semigroup itself and little information about the properties of the operator). In the six next
**Proof of Lemma 2.8.** We first assume (1). Taking advantage of the splitting structure (HS1), we write equation (2.20) as

$$ (2.24) \qquad (\lambda_n - \mathcal{B})\hat{f}_n = \mathcal{A}\hat{f}_n + \varepsilon_n, $$

or equivalently

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

Iterating that last identity and using the notations (2.22), we get

$$ (2.25) \qquad \hat{f}_n = w_n + v_n, \quad w_n := \mathcal{W}(\lambda_n)\hat{f}_n, \quad v_n := \mathcal{V}(\lambda_n)\varepsilon_n. $$

We observe that $(w_n)$ is strongly relatively compact from (2.23) and $\|\hat{f}_n\|_X = 1$, so that there exist a subsequence $(w_{n_k})$ and $f_1 \in X$ such that $w_{n_k} \to f_1$ strongly in $X$. Since $v_n \to 0$ strongly in $X$, we deduce that $\hat{f}_{n_k} \to f_1$ strongly in $X$. We conclude that condition (H3) holds as in the proof of Lemma 2.7.

We next assume (2). As observed in Remark 2.9-(2), the property (2.23) holds and thus also the condition (H3) for the primal problem. We claim that the same locally uniform strong compactness property (2.23) holds for the dual problem at order $N+1$ and thus condition (H3) holds for the dual problem. We may indeed use Remark 2.9-(2) since then $\mathcal{R}_B^*(\alpha)$ and $\mathcal{A}^*\mathcal{R}_B^*(\alpha)$ are bounded uniformly in $\alpha \ge \kappa_0$ and

$$ (\mathcal{A}^*\mathcal{R}_B^*(\alpha))^{N+1} = \mathcal{A}^*\mathcal{W}(\alpha)^*\mathcal{R}_B^*(\alpha) \in \mathcal{H}(Y), \quad \forall \alpha \ge \kappa_0, $$

as a product of two bounded operator with a compact operator. $\square$

**Remark 2.10.** Instead of (HS1) in Lemma 2.8, one can assume that there exists a splitting $\mathcal{L} = \mathcal{A} + \mathcal{B}$ and $N \ge 1$ such that $\mathcal{B} - \alpha$ is invertible for any $\alpha \ge \kappa_0$ and

$$ \mathcal{R}_B(\alpha) := (\alpha - \mathcal{B})^{-1}, \quad \check{\mathcal{V}}(\alpha) := \sum_{i=0}^{N-1} (\mathcal{A}\mathcal{R}_B(\alpha))^i, \quad \check{\mathcal{W}}(\alpha) := (\mathcal{A}\mathcal{R}_B(\alpha))^N $$

are respectively bounded in $\mathcal{B}(X)$ uniformly with respect to $\alpha \ge \kappa_0$ and strongly compact locally uniformly on $\alpha \ge \kappa_0$. Starting indeed again from (2.24) and defining $h_n := (\lambda_n - \mathcal{B})\hat{f}_n$, we may write

$$ h_n = \mathcal{A}\mathcal{R}_B(\lambda_n)h_n + \varepsilon_n. $$

Observing that $\|h_n\|_X \ge \|\mathcal{R}_B(\lambda_n)\|_{\mathcal{B}(X)}^{-1} \ge c > 0$ by assumption, we deduce that $\hat{h}_n := h_n/\|h_n\|_X$ satisfies

$$ \hat{h}_n = \tilde{w}_n + \tilde{v}_n, \quad \tilde{w}_n := \check{\mathcal{W}}(\lambda_n)\hat{h}_n, \quad \tilde{v}_n := \check{\mathcal{V}}(\lambda_n)\hat{\varepsilon}_n, $$

with $\|\hat{h}_n\| = 1$ and $\hat{\varepsilon}_n := \varepsilon_n/\|h_n\|_X \to 0$. Similarly as in the proof of Lemma 2.8, we conclude to the existence of subsequence $(\hat{h}_{n_k})$ and $h_1 \in X_+ \setminus \{0\}$ such that $\hat{h}_{n_k} \to h_1$ strongly in $X$. Defining $f_1 := \mathcal{R}_B(\lambda_1)h_1/\|\mathcal{R}_B(\lambda_1)h_1\|$, we have again $\hat{f}_{n_k} \to f_1$ strongly in $X$ and next that condition (H3) holds.

As we see now, strong compactness is not really necessary.

**Lemma 2.11.** We assume (H1)-(H2)-(HS1) and there exists $N \ge 1$ such that

$$ \mathcal{W}(\alpha) : X \to \mathcal{X}_1 \subset X \quad \text{is positive and uniformly bounded in } \alpha \ge \kappa_0 $$

and, denoting $\mathcal{X}_0 := X$, we assume that for any $R_1 \ge R_0 > 0$ the set

$$ (2.26) \qquad C = C_{R_0, R_1} := \{g \in X_+; \|g\|_{\mathcal{X}_0} \ge R_0, \|g\|_{\mathcal{X}_1} \le R_1\} $$

is relatively sequentially compact for the weak topology $\sigma(X, Y)$ and $0 \notin \bar{C}$, where the closure is taken in the sense of the weak topology $\sigma(X, Y)$. Then condition (H3) holds.

**Remark 2.12.** When $\mathcal{X}_1 \subset \mathcal{X}_0$ with strongly compact embedding the above set $C$ clearly satisfies the required conditions. In particular, Lemma 2.11 generalizes the result stated in Remark 2.9-(1).