Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Galton-Watson processes in dynamical environments

Thomas Morand

Abstract

We define a model of Galton Watson processes in dynamical environments where the environment evolves according to a dynamical system (X, T). Three behaviours are possible: uniformly subcritical, critical, and uniformly supercritical. We study the extinction probability q in the uniformly supercritical case. In particular, we investigate the regularity of this application as a function of the environment x. In the critical case, we study the set of bad environments N (where the probability of extinction is one), which is T -invariant. We give its Hausdorff dimension in some cases.

Galton-Watson processes in dynamical environments

Abstract

We define a model of Galton Watson processes in dynamical environments where the environment evolves according to a dynamical system (X, T). Three behaviours are possible: uniformly subcritical, critical, and uniformly supercritical. We study the extinction probability q in the uniformly supercritical case. In particular, we investigate the regularity of this application as a function of the environment x. In the critical case, we study the set of bad environments N (where the probability of extinction is one), which is T -invariant. We give its Hausdorff dimension in some cases.

Paper Structure

This paper contains 30 sections, 38 theorems, 100 equations, 2 figures.

Table of Contents

  1. Model and results
  2. Galton-Watson processes: classical case and random environments
  3. Galton-Watson processes
  4. Galton-Watson processes in varying environments
  5. Galton-Watson processes in random environments
  6. Galton-Watson processes: dynamical environments
  7. Presentation
  8. Extinction and generating function
  9. Main results
  10. The set of bad environments
  11. Measure of the set of bad environments
  12. Dimension of the set of bad environments
  13. Regularity of the invariant graph
  14. Continuity in the supercritical case
  15. Hölder continuity in the supercritical case
  16. ...and 15 more sections

Key Result

Theorem 1.1.1

The probability of extinction $q$ is the smallest $s\in[0,1]$ such that $\varphi_{\mu}(s)=s$. In particular, if $m\in[0,1]$ and $\mu\neq\delta_1$, then $q=1$, and if $m \in(1,+\infty]$, then $q<1$.

Figures (2)

  • Figure 1: Probability-generating function for the Poisson distribution with parameter $a$, and the probability of extinction $q$ of the associated Galton-Watson process.
  • Figure 2: Plot of the probability of extinction $q_\lambda$ for some $\lambda\in\mathbb{R}$.

Theorems & Definitions (86)

  • Theorem 1.1.1
  • Example 1.2.1
  • Example 1.2.2
  • Definition 1.2.3
  • Proposition 1.2.4
  • Definition 1.2.5
  • Proposition 1.2.6
  • Definition 1.2.7
  • Definition 1.2.8
  • Theorem 1.3.1
  • ...and 76 more