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

TL;DR

This paper reviews Lyapunov-based methods for analyzing the stability of positive semigroups in complex, possibly unbounded, state spaces, with applications across various fields including physics and chemistry.

Contribution

It provides a comprehensive overview of recent Lyapunov techniques, including comparison and conjugacy principles, for stability analysis of diverse positive semigroups.

Findings

Lyapunov methods effectively analyze stability in non-compact spaces

Comparison and conjugacy principles are practical tools

Applications include Markov semigroups, Langevin diffusions, and coupled oscillators

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.