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

We consider a general class of birth-and-death processes with state space {0,1,2,3,…} which describes the size population going eventually to extinction probability one. obtain complete spectrum generator process killed at 0 in large limit, that is, we scale by parameter K, and take limit K→+∞. assume differential equation dx/dt=b(x)−d(x) describing infinite (in any finite-time interval) has repulsive fixed point 0, an attractive x⁎>0. prove that, asymptotically, is superposition two spectra. One Ornstein-Uhlenbeck process, n(b′(x⁎)−d′(x⁎)), n≥0. The other one continuous-time binary branching conditioned on non-extinction, given n(d′(0)−b′(0)), n≥1. A major difficulty different scales function spaces are involved. work level eigenfunctions split over regions, study their asymptotic dependence K each region. In particular, spectral gap goes min⁡{b′(0)−d′(0),d′(x⁎)−b′(x⁎)}. This complements previous ours studied approximation quasi-stationary distribution mean time extinction.