[Paper Review] On time scales and quasi-stationary distributions for multitype birth-and-death processes

Jean-René Chazottes, Pierre Collet+1 more|arXiv (Cornell University)|Feb 17, 2017

Stochastic processes and statistical mechanics12 references5 citations

TL;DR

This paper establishes the existence of a unique quasi-stationary distribution (qsd) for multitype birth-and-death processes with a unique interior attractive fixed point in the associated deterministic ODE, proving that the process conditioned on non-extinction converges exponentially fast to the qsd. For large $ K $, the mean time to extinction is exponentially large in $ K $, and the total variation distance between the conditioned process and the qsd decays exponentially in time $ t \gg \log K $, quantifying the quasi-stationary regime.

ABSTRACT

We consider a class of birth-and-death processes describing a population made of $d$ sub-populations of different types which interact with one another. The state space is $\mathbb{Z}_+^d$ (unbounded). We assume that the population goes almost surely to extinction, so that the unique stationary distribution is the Dirac measure at the origin. These processes are parametrized by a scaling parameter $K$ which can be thought as the order of magnitude of the total size of the population at time $0$. For any fixed finite time span, it is well-known that such processes, when renormalized by $K$, are close, in the limit $K o+\infty$, to the solutions of a certain differential equation in $\mathbb{R}_+^d$ whose vector field is determined by the birth and death rates. We consider the case where there is a unique attractive fixed point (off the boundary of the positive orthant) for the vector field (while the origin is repulsive). What is expected is that, for $K$ large, the process will stay in the vicinity of the fixed point for a very long time before being absorbed at the origin. To precisely describe this behavior, we prove the existence of a quasi-stationary distribution (qsd). In fact, we establish a bound for the total variation distance between the process conditioned to non-extinction before time $t$ and the qsd. This bound is exponentially small in $t$, for $t\gg \log K$. As a by-product, we obtain an estimate for the mean time to extinction in the qsd. We also quantify how close is the law of the process (not conditioned to non-extinction) either to the Dirac measure at the origin or to the qsd, for times much larger than $\log K$ and much smaller than the mean time to extinction, which is exponentially large as a function of $K$. Let us stress that we are interested in what happens for finite $K$. We obtain results much beyond what large deviation techniques could provide.

Motivation & Objective

• To describe the long-term behavior of multitype birth-and-death processes with finite populations that almost surely go extinct.
• To establish the existence of a unique quasi-stationary distribution (qsd) for such processes when the associated deterministic ODE has a unique interior attractive fixed point.
• To quantify the convergence rate of the process conditioned on non-extinction to the qsd, especially for large $ K $.
• To estimate the mean time to extinction under the qsd and characterize the law of the process before absorption.

Proposed method

• Analyzes a class of Markov jump processes on $ \mathbb{Z}^d_+ \setminus \{0\} $ with birth and death rates scaled by $ K $, leading to a deterministic ODE in the limit $ K \to \infty $.
• Uses a Lyapunov function to control the dynamics near the fixed point $ \mathbf{x}^* $, ensuring stability and guiding the construction of the qsd.
• Applies a 'lemma of four domains' to decompose the state space and control the process's behavior in different regions: near the fixed point, near the origin, and in the transient zones.
• Establishes exponential bounds on the total variation distance between the conditioned process and the qsd, showing decay as $ \exp(-c t) $ for $ t \gg \log K $.
• Derives upper and lower bounds on the mean time to extinction under the qsd, showing it is exponentially large in $ K $.
• Uses duality and detailed balance arguments to analyze the existence of invariant measures, though the main approach avoids relying on detailed balance due to its restrictive nature in higher dimensions.

Experimental results

Research questions

• RQ1 Does a quasi-stationary distribution exist for multitype birth-and-death processes with a unique interior attractive fixed point in the associated ODE?
• RQ2 How fast does the process conditioned on non-extinction converge to the quasi-stationary distribution?
• RQ3 What is the asymptotic behavior of the mean time to extinction for large $ K $?
• RQ4 How close is the unconditioned process to the Dirac measure at extinction or to the qsd at times much larger than $ \log K $ but much smaller than the mean extinction time?

Key findings

• A unique quasi-stationary distribution exists for the multitype birth-and-death process under the assumption of a unique interior attractive fixed point for the associated ODE.
• The total variation distance between the process conditioned on non-extinction and the qsd decays exponentially in time $ t $, specifically as $ \exp(-c t) $, for $ t \gg \log K $.
• The mean time to extinction under the qsd is exponentially large in $ K $, with the rate depending on the spectral gap of the generator near the fixed point.
• For times $ t \gg \log K $ but $ t \ll \mathbb{E}[T_{\text{ext}}] $, the law of the unconditioned process is exponentially close to the qsd, while for $ t \gg \mathbb{E}[T_{\text{ext}}] $, it is close to the Dirac measure at the origin.
• The results are non-asymptotic in $ K $, providing sharp quantitative estimates beyond what large deviation theory can achieve.
• The method avoids relying on detailed balance, making it applicable to a broader class of models, including those with inter-specific competition, where detailed balance fails.