2023/01/09 by Marc Arnaudon, Pierre Del Moral, Arnaudon, Marc +3 · 1 citation
Biochemistry, Genetics and Molecular Biology · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Diffusion and Search Dynamics #FOS: Mathematics #Probability (math.PR) #stochastic dynamics and bifurcation
paper · pdf · doi:10.48550/arxiv.2301.03484
openalex publication_date 2023/01/09 · openalex created_date 2023/02/13 · openalex updated_date 2026/07/28
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.