On the Stability of Positive Semigroups

Pierre del Moral (1) , Emma Horton (1) , Ajay Jasra (2)

1 ASTRAL - Méthodes avancées d’apprentissage statistique et de contrôle
2 KAUST - King Abdullah University of Science and Technology [Saudi Arabia]

The stability and contraction properties of positive integral semigroups on Polish spaces are investigated. Our novel analysis is based on the extension of V-norm contraction methods, associated to functionally weighted Banach spaces for Markov semigroups, to positive semigroups. This methodology is applied to a general class of positive and possibly time-inhomogeneous bounded integral semigroups and their normalised versions. The spectral theorems that we develop are an extension of Perron-Frobenius and Krein-Rutman theorems for positive operators to a class of time-varying positive semigroups. In the context of time-homogeneous models, the regularity conditions discussed in the present article appear to be necessary and sufficient condition for the existence of leading eigenvalues. We review and illustrate the impact of these results in the context of positive semigroups arising in transport theory, physics, mathematical biology and signal processing.

Mots clés: Positive semigroups; Boltzmann-Gibbs transformations; Contraction inequalities; Dobrushin's ergodic coefficient; Spectral theorems; Foster-Lyapunov conditions

Domaines: Probabilités [math.PR]

Pierre del Moral, Emma Horton, Ajay Jasra. On the Stability of Positive Semigroups. The Annals of Applied Probability, inPress. ⟨hal-03468416v6⟩