On the Stability of Positive Semigroups

Abstract

The stability and contraction properties of positive integral semigroups on Polish spaces are investigated. We provide a novel analysis based on an extension of V -norm, Dobrushin-type, contraction techniques on functionally weighted Banach spaces for Markov operators. These are applied to a general class of positive and possibly time-inhomogeneous bounded integral semigroups and their normalised versions. Under mild regularity conditions, the Lipschitz-type contraction analysis presented in this article simplifies and extends several exponential estimates developed in the literature. The spectral-type theorems that we develop can also be seen as an extension of Perron-Frobenius and Krein-Rutman theorems for positive operators to time-varying positive semigroups. Incidentally, in the context of time-homogeneous models, the regularity conditions discussed in the present article appears to be a necessary and sufficient condition for the existence of leading eigenvalues. We review and illustrate in detail the impact of these results in the context of positive semigroups arising in transport theory, physics, mathematical biology and advanced signal processing.

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

“…The proof of the above assertion can be found in [21] (see also [22] in the context of time homogeneous models). The next exponential contraction theorem is a direct consequence of the operator norm estimates (12) and it is valid on abstract measurable spaces as well as for any function V ě 1.…”

“…Their use in the context of positive semigroup arising in discrete time nonlinear filtering goes back to the pioneering articles [27,64], based on coupling techniques developed in [41,42]. The extension of Foster-Lyapunov criterion to discrete or continuous time varying positive semigroups and their normalized versions on general state spaces were further developed in [21], extending Dobrushin's ergodic coefficient techniques introduced in [11,12] and further developed in [15,16,17,14,22] to unbounded state space models.…”

“…The exhausting sequence of compact subsets needs to be equivalent to the sub-level sets of some sufficiently strong Lyapunov function near entrance boundaries. For a more thorough discussion on this subject we refer to Section 2 and the article [21], see also the series of Riccati-type diffusions discussed in Section 5.…”

“…The main subject of this article is to find practical ways to design these Lyapunov functions for various classes of positive semigroups that have been discussed in the literature, including conditional diffusions on manifolds with Neumann and Dirichlet boundaries. We did our best to cover the subject as broadly as possible, we also refer to the article [21] for additional historical and reference pointers. Due to the vast literature on this subject we apologize for possible omissions of some important contributions due to the lack of knowledge.…”

Uniform in time propagation of chaos for a Moran model

“…See for example the work of Del Moral and Miclo [30], the articles of Champagnat and Villemonais on exponential convergence towards the quasi-stationary distribution, specifically [31,32], and also the works of Bansaye et al [33,34]. See also the recent work of Del Moral et al [35] on the stability of positive semigroups. In Section 1.2, we provide several examples of process where Assumption (C2) holds.…”