Large population limit of the spectrum of killed birth-and-death processes

TL;DR

This paper analyzes the asymptotic spectral properties of large population birth-and-death processes killed at zero, revealing a superposition of spectra from Ornstein-Uhlenbeck and branching processes.

Contribution

It provides a detailed characterization of the spectrum of the generator in the large population limit, combining different process types and overcoming technical challenges.

Findings

Spectral gap converges to the minimum of two key derivatives.

Spectrum asymptotically decomposes into two parts from different processes.

Results extend understanding of population extinction dynamics.

Abstract

We consider a general class of birth-and-death processes with state space $\{0,1,2,3,\ldots\}$ which describes the size of a population going eventually to extinction with probability one. We obtain the complete spectrum of the generator of the process killed at $0$ in the large population limit, that is, we scale the process by a parameter $K$, and take the limit $K\to+\infty$. We assume that the differential equation $\mathrm{d} x/\mathrm{d} t=b(x)-d(x)$ describing the infinite population limit (in any finite-time interval) has a repulsive fixed point at $0$, and an attractive fixed point $x_*>0$. We prove that, asymptotically, the spectrum is the superposition of two spectra. One is the spectrum of the generator of an Ornstein-Uhlenbeck process, which is $n(b'(x_*)-d'(x_*))$, $n\ge 0$. The other one is the spectrum of a continuous-time binary branching process conditioned on non-extinction, and is given by $n(d'(0)-b'(0))$, $n\ge 1$. A major difficulty is that different scales and function spaces are involved. We work at the level of the eigenfunctions that we split over different regions, and study their asymptotic dependence on $K$ in each region. In particular, we prove that the spectral gap goes to $\min\big\{b'(0)-d'(0),\,d'(x_*)-b'(x_*)\big\}$. This work complements a previous work of ours in which we studied the approximation of the quasi-stationary distribution and of the mean time to extinction.