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Remaining-lifetime age-structured branching processes

Ziling Cheng, Zenghu Li

Abstract

We study age-structured branching models with reproduction law depending on the remaining lifetime of the parent. The lifespan of an individual is decided at its birth and its remaining lifetime decreases at the unit speed. The models without or with immigration are constructed as measure-valued processes by pathwise unique solutions of stochastic equations driven by time-space Poisson random measures. In the subcritical branching case, we give a sufficient condition for the ergodicity of the process with immigration. Two large number laws and a central limit theorem of the occupation times are proved.

Remaining-lifetime age-structured branching processes

Abstract

We study age-structured branching models with reproduction law depending on the remaining lifetime of the parent. The lifespan of an individual is decided at its birth and its remaining lifetime decreases at the unit speed. The models without or with immigration are constructed as measure-valued processes by pathwise unique solutions of stochastic equations driven by time-space Poisson random measures. In the subcritical branching case, we give a sufficient condition for the ergodicity of the process with immigration. Two large number laws and a central limit theorem of the occupation times are proved.
Paper Structure (6 sections, 44 theorems, 179 equations)

This paper contains 6 sections, 44 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. An age-structured branching process
  3. Stochastic equations and martingale problems
  4. The branching process with immigration
  5. The occupation time processes
  6. Limit Theorems

Key Result

proposition 1

For any $f\in B(0,\infty)^{+}$, the map $(t,x)\mapsto u_t f(x)$ is the unique solution to which is equivalent to the evolution equation (2.4).

Theorems & Definitions (69)

  • proposition 1
  • proof
  • proposition 2
  • proof
  • proposition 3
  • proof
  • proposition 4
  • proof
  • proposition 5
  • proof
  • ...and 59 more