论文标题
杀死的出生和死亡过程范围的较大人口限制
Large population limit of the spectrum of killed birth-and-death processes
论文作者
论文摘要
我们考虑使用状态空间$ \ {0,1,2,3，\ ldots \} $的一般性出生和死亡过程，该过程描述了最终将以一个概率灭绝的人群的大小。我们获得了该过程的完整频谱，即在较大的人口限制中以$ 0 $杀死的流程，也就是说，我们按参数$ k $进行扩展，并将限制$ k \ to限制为+\ \ \ infty $。我们假设微分方程$ \ mathrm {d} x/\ mathrm {d} t = b（x）-d（x）$描述了无限群体限制（在任何有限的时间间隔中）具有$ 0 $的排斥固定点，并且有吸引力的固定点$ x _*> 0 $。我们渐近地证明，光谱是两个光谱的叠加。一个是Ornstein-uhlenbeck过程的生成器的频谱，即$ n（b'（x _*）-d'（x _*））$，$ n \ ge 0 $。另一个是连续的二进制分支过程的频谱，该过程以非伸入为条件，由$ n（d'（d'（0）-b'（0））$，$ n \ ge 1 $给出。一个主要的困难是涉及不同的尺度和功能空间。我们在不同地区分裂的本征函数级别工作，并研究了他们对每个地区$ k $的渐近依赖性。特别是，我们证明频谱差距为$ \ min \ big \ {b'（0）-d'（0），\，d'（x _*） - b'（x _*）\ big \} $。这项工作补充了我们先前的工作，其中我们研究了准平台分布的近似值和平均灭绝时间的近似值。
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.
