  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.
