Quasi-stationary distributions and diffusion models in population dynamics

By Patrick Cattiaux, Pierre Collet, Amaury Lambert, Servet Martínez, S. M'el'eard, Jaime San Martín. 2007

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In this paper, we study quasi-stationarity for a large class of Kolmogorov diffusions. The main novelty here is that we allow the drift to go to $- \infty$ at the origin, and the diffusion to have an entrance boundary at $+\infty$. These diffusions arise as images, by a deterministic map, of generalized Feller diffusions, which themselves are obtained as limits of rescaled birth--death processes. Generalized Feller diffusions take nonnegative values and are absorbed at zero in finite time with probability $1$. An important example is the logistic Feller diffusion. We give sufficient conditions on the drift near $0$ and near $+ \infty$ for the existence of quasi-stationary distributions, as well as rate of convergence in the Yaglom limit and existence of the $Q$-process. We also show that under these conditions, there is exactly one quasi-stationary distribution, and that this distribution attracts all initial distributions under the conditional evolution, if and only if $+\infty$ is an entrance boundary. In particular this gives a sufficient condition for the uniqueness of quasi-stationary distributions. In the proofs spectral theory plays an important role on $L^2$ of the reference measure for the killed process.

Subfields: Management Information Systems, Mathematical Physics, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/73f21d45c25faecf5c6ce82021545e5500016479

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Citations in the corpus

Poincaré inequalities and hitting times (2010, background): “Conversely, [11], Theorem 7.3, tells us that (3) implies the condition (called (H5) therein) z (cid:12)→ F(z)/z is thus non-increasing and we have x a e V(y) d y = 1 .”
General criteria for the study of quasi-stationarity (2017, methodology): “Also, the case considered in [10] is restricted to (transformations of) Kolmogorov diffusions where the drift derives from a potential (b = ∇V ), which allow the authors to use a spectral theoretic approach as in the one-dimensional case [9].”
Quasilimiting behavior for one-dimensional diffusions with killing (2010, background): “…of the vast array of applications of killed Markov processes to biology [Seneta and Vere-Jones (1966), Högnäs (1997), Haridas and Tuljapurkar (2005), Cattiaux et al. (2009), demography: Steinsaltz and Evans (2004), Le Bras (1976), Li and Anderson (2009), medicine: Manton and Stallard (1988),…”
Population Dynamics and Random Genealogies (2008, background): “The goal of Ref.[20] is to display sufficient conditions on h to get the existence of QSDs.”
Exponential convergence to quasi-stationary distribution and Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q$$\end{ (2014, unknown): “When E = (0 , + ∞ ) or N and ∂ = 0, this is implied by the fact that the process X comes down from inﬁnity [4] (see Section 4.1 for the discrete case).”
Quasi-stationary distributions and population processes (2011, background): “[13] Assume that X0, r and c are positive.”
Quasi-Stationary Distributions and the Continuous-State Branching Process Conditioned to Be Never Extinct (2007, background): “More interestingly, it is proved in (5) that if competition intensity increases rapidly enough with population size, then the diffusion ‘comes down from infinity’ and has a unique QSD.”