[Paper Review] Markov branching processes with disasters: extinction, survival and duality to p-jump processes

Felix Hermann, Peter Pfaffelhuber|arXiv (Cornell University)|Jul 31, 2018

Stochastic processes and statistical mechanics32 references3 citations

TL;DR

This paper establishes a duality between Markov branching processes with binomial disasters and p-jump processes—piecewise deterministic Markov processes on [0,1] with multiplicative jumps by factor p. Using this duality and limit theorems for p-jump processes, the authors derive exact asymptotic expressions for extinction and survival probabilities in time-homogeneous and time-inhomogeneous branching processes with disasters, revealing phase transitions in decay rates and conditions for almost sure extinction or survival.

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.

Motivation & Objective

• To analyze the long-time behavior of Markov branching processes subject to binomial disasters, where each individual survives with probability p after a disaster.
• To extend existing results on extinction and survival probabilities by incorporating time-inhomogeneous rates and continuous-state branching processes.
• To develop a duality framework using probability generating functions to translate properties of p-jump processes into results for branching processes.
• To derive precise asymptotic expressions for survival and extinction probabilities under general offspring and disaster dynamics.
• To characterize phase transitions in the decay rate of survival probabilities based on the interplay between branching, disaster, and survival parameters.

Proposed method

• Introduce p-jump processes as piecewise deterministic Markov processes on [0,1], where the process jumps by a factor p at disaster times.
• Establish duality between the branching process and the p-jump process via probability generating functions, linking the generating function of the branching process to the solution of a stochastic differential equation.
• Derive a key SDE for the dual process: $ \dot{X} = -\lambda(X - h(X)) $, modified to include jumps by factor p at disaster rate $ \kappa $.
• Use large deviations for Poisson processes (from Dembo and Zeitouni, 1998) to analyze the tail behavior of the cumulative jump intensity and survival probability.
• Apply limit theorems for p-jump processes under concavity and growth conditions on the jump rate and branching intensity.
• Use the duality relation $ \mathbb{E}[x^{Z_t} \mid Z_0 = z] = X_t^z $ to translate convergence of $ X_t $ into extinction/survival probabilities of $ Z_t $.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of the survival probability in a time-homogeneous branching process with binomial disasters and arbitrary offspring distribution?
• RQ2 How do time-inhomogeneous branching and disaster rates affect the long-term survival or extinction of a population?
• RQ3 What conditions lead to almost sure extinction or non-extinction in branching processes with disasters, and how does the decay rate of survival probability change?
• RQ4 Can the duality between p-jump processes and branching processes be extended to continuous-state branching processes with disasters?
• RQ5 What are the precise convergence rates of the survival probability conditioned on the history of disaster times?

Key findings

• For time-homogeneous branching processes with disasters, the survival probability decays exponentially, and the decay rate exhibits a phase transition depending on the balance between branching, disaster, and survival parameters.
• If the integral $ \int_0^t \log(1/p_s) \kappa_s \, ds $ grows faster than the branching rate, extinction is almost sure; otherwise, survival occurs with positive probability.
• In the time-inhomogeneous case, the survival probability conditioned on disaster times converges to a limit determined by the dual p-jump process, with convergence rates expressible via $ \log \mathbb{P}(Z_t > 0 \mid \mathcal{D}_\infty) \sim \max\{ -L_t, \log \int_0^t e^{-L_s} b_s \, ds \} / h(t) $.
• When the cumulative drift $ \ell(t) = \int_0^t (b_s - d_s - \kappa_s \log(1/p_s)) \, ds $ converges to a constant, the process almost surely has only finitely many births and converges to a non-degenerate limit distribution.
• For continuous-state branching processes with disasters, the extinction probability is characterized via the same duality, and the limit behavior depends on the asymptotic growth of the dual process.
• In cases where the integral $ \int_0^t e^{-L_s} b_s \, ds $ diverges, the survival probability decays slower than exponential, while convergence of the integral implies faster decay or extinction with probability one.