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Markov branching processes with disasters: extinction, survival and duality to p-jump processes

F. Hermann, P. Pfaffelhuber

Abstract

A $p$-jump process is a piecewise deterministic Markov process with jumps by a factor of $p$. We prove a limit theorem for such processes on the unit interval. Via duality with respect to probability generating functions, we deduce limiting results for the survival probabilities of time-homogeneous branching processes with arbitrary offspring distributions, underlying binomial disasters. Extending this method, we obtain corresponding results for time-inhomogeneous birth-death processes underlying time-dependent binomial disasters and continuous state branching processes with $p$-jumps.

Markov branching processes with disasters: extinction, survival and duality to p-jump processes

Abstract

A -jump process is a piecewise deterministic Markov process with jumps by a factor of . We prove a limit theorem for such processes on the unit interval. Via duality with respect to probability generating functions, we deduce limiting results for the survival probabilities of time-homogeneous branching processes with arbitrary offspring distributions, underlying binomial disasters. Extending this method, we obtain corresponding results for time-inhomogeneous birth-death processes underlying time-dependent binomial disasters and continuous state branching processes with -jumps.

Paper Structure

This paper contains 13 sections, 14 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Results
  3. $p$-jump Processes
  4. Branching Processes with Binomial Disasters
  5. Continuous State Branching Processes with Binomial Disasters
  6. Piecewise Deterministic Markov Processes
  7. Proof of Theorem \ref{['thm:convergence-$p$-jump-pros']}
  8. Proof of Corollary \ref{['cor:p-jump-concave']}
  9. Branching processes with binomial disasters
  10. Proof of Theorem \ref{['thm:bp-arb.off.dis']}
  11. Preparation: Regular Variation
  12. Proof of Theorem \ref{['thm:inhom-bdp']}
  13. Large Deviations

Key Result

Theorem 1

Let $I$ and $\alpha$ be as in Definition def:p-jump-process, $p\in(0,1)$ and $X_0\in I$. Also, suppose if $I=\mathbb R^+$ that $s_\alpha:=\sup\{x:\alpha(x)>0\}<\infty$. Furthermore, let $\alpha'_0:=\lim_{x\to0}\frac{1}{x}\alpha(x)$ and assume that $\alpha$ satisfies one of the following: Then, there is a $p$-jump process $\mathcal{X}$ with drift $\alpha$ on $I$ starting in $X_0$, such that, letti

Theorems & Definitions (44)

  • Definition 2.1
  • Remark 2.2
  • Theorem 1: Convergence of $p$-jump-processes
  • Remark 2.3
  • Corollary 2.4
  • Definition 2.5
  • Theorem 2
  • Remark 2.6
  • Corollary 2.7
  • proof
  • ...and 34 more