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On the Krein-Rutman theorem and beyond

Claudia Fonte Sanchez, Pierre Gabriel, Stéphane Mischler

TL;DR

The work extends Krein-Rutman theory to semigroups of positive operators on Banach lattices, presenting a unified framework that yields existence, geometric structure, and quantitative stability for the principal eigenpair with constructive estimates. It develops both stationary and dynamical approaches to secure the eigentriplet under weak compactness and dissipativity assumptions, and proves strict positivity and simplicity of the principal eigenvalue under irreducibility and related maximum-principle conditions. The authors demonstrate the theory on a broad class of PDEs and integro-differential models, including parabolic, transport, growth-fragmentation, kinetic Boltzmann/Fokker-Planck, and mutation-selection equations, with explicit rates and spectral-gap insights in several cases. This provides a versatile, constructive toolkit for spectral analysis of positive operators in PDEs and kinetic theories, enabling rigorous long-time behavior descriptions and principled guidance for applications.

Abstract

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, 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 the torus and in the whole space; some mutation-selection models.

On the Krein-Rutman theorem and beyond

TL;DR

The work extends Krein-Rutman theory to semigroups of positive operators on Banach lattices, presenting a unified framework that yields existence, geometric structure, and quantitative stability for the principal eigenpair with constructive estimates. It develops both stationary and dynamical approaches to secure the eigentriplet under weak compactness and dissipativity assumptions, and proves strict positivity and simplicity of the principal eigenvalue under irreducibility and related maximum-principle conditions. The authors demonstrate the theory on a broad class of PDEs and integro-differential models, including parabolic, transport, growth-fragmentation, kinetic Boltzmann/Fokker-Planck, and mutation-selection equations, with explicit rates and spectral-gap insights in several cases. This provides a versatile, constructive toolkit for spectral analysis of positive operators in PDEs and kinetic theories, enabling rigorous long-time behavior descriptions and principled guidance for applications.

Abstract

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, 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 the torus and in the whole space; some mutation-selection models.
Paper Structure (71 sections, 147 theorems, 1509 equations)

This paper contains 71 sections, 147 theorems, 1509 equations.

Table of Contents

  1. Introduction
  2. Framework and the main result
  3. Discussion about Theorem \ref{['theo:main-intro']}
  4. The Krein-Rutman work and related approaches
  5. Spectral analysis approach
  6. Dynamical and probabilistic approach.
  7. PDE approaches
  8. Hypotheses and proof
  9. Some examples of applications
  10. Parabolic equations
  11. Transport equation
  12. Growth-fragmentation equation
  13. Kinetic linear Boltzmann equation
  14. Kinetic Fokker-Planck equation
  15. Mutation-selection equation
  16. ...and 56 more sections

Key Result

Theorem 1.1

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 eq:intro:RLRB-RLARBN, eq:intro-regRL, eq:intro:stopDyson, eq:intro-regSL. (1) Conclusion S1 holds under the localization of the pri

Theorems & Definitions (312)

  • Theorem 1.1: rough version
  • Theorem 1.2: Krein-Rutman
  • Theorem 1.3: Krein-Rutman
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Lemma 2.1
  • Lemma 2.2
  • ...and 302 more