A Lyapunov approach to stability of positive semigroups: An overview with illustrations - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Preprints, Working Papers, ... Year : 2023

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

Abstract

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.
Fichier principal
Vignette du fichier
Lyapunov.positive.semigroups-arxiv.pdf (547.37 Ko) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

hal-03931007 , version 1 (09-01-2023)

Identifiers

Cite

Marc Arnaudon, Pierre Del Moral, El Maati Ouhabaz. A Lyapunov approach to stability of positive semigroups: An overview with illustrations. 2023. ⟨hal-03931007⟩
51 View
55 Download

Altmetric

Share

Gmail Facebook X LinkedIn More