A Lyapunov approach to stability of positive semigroups: an overview with illustrations

• Published in: Stochastic analysis and applications Vol. 42; no. 1; pp. 121 - 200
• Main Authors: Arnaudon, Marc, Del Moral, Pierre, Ouhabaz, El Maati
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 02.01.2024; Taylor & Francis Ltd; Taylor & Francis: STM, Behavioural Science and Public Health Titles
• Subjects: Applications of mathematics; Boundary conditions; boundary problems; Branching (mathematics); Dirichlet problem; Harmonic oscillators; Hyperspaces; hypersurfaces; Integral operators; Langevin diffusions; Liapunov functions; Lyapunov function; Markov and Sub-Markov semigroups; Markov processes; Mathematics; Molecular composites; Neumann problem; Primary 47D08; Probability; secondary 47B65; Semigroups; shape matrices; Stability analysis; Stochastic processes; Integral operators; Markov and Sub-Markov semigroups; hypersurfaces; harmonic oscillators; Langevin diffusions; semigroups; shape matrices; Lyapunov function
• ISSN: 0736-2994; 1532-9356
• DOI: 10.1080/07362994.2023.2206880

The stability analysis of possibly time varying positive semigroups on non-necessarily compact state spaces, including Neumann and Dirichlet boundary conditions is a notoriously difficult subject. These crucial questions arise in a variety of areas of applied mathematics, including nonlinear filtering, rare event analysis, branching processes, physics and molecular chemistry. This article presents an overview of some recent Lyapunov-based approaches, focusing principally on practical and powerful tools for designing Lyapunov functions. These techniques include semigroup comparisons as well as conjugacy principles on non-necessarily bounded manifolds with locally Lipschitz boundaries. All the Lyapunov methodologies discussed in the article are illustrated in a variety of situations, ranging from conventional Markov semigroups on general state spaces to more sophisticated conditional stochastic processes possibly restricted to some non-necessarily bounded domains, including locally Lipschitz and smooth hypersurface boundaries, Langevin diffusions as well as coupled harmonic oscillators.