Large Deviations and Quasi-stationarity for Density-dependent Birth-death Processes

Consider a density-dependent birth-death process X N on a finite state space of size N . Let PN be the law (on D.[0; T ]/ where T > 0 is arbitrary) of the density process XN =N and let 5N be the unique stationary distribution (on [0,1]) of X N =N , if it exists. Typically, these distributions converge weakly to a degenerate distribution as N ! 1, so the probability of sets not containing the degenerate point will tend to 0; large deviations is concerned with obtaining the exponential decay rate of these probabilities. Friedlin-Wentzel theory is used to establish the large deviations behaviour (as N !1) of PN . In the one-dimensional case, a large deviations principle for the stationary distribution 5N is obtained by elementary explicit computations. However, when the birth-death process has an absorbing state at 0 (so 5N no longer exists), the same elementary computations are still applicable to the quasi-stationary distribution, and we show that the quasi-stationary distributions obey the same large deviations principle as in the recurrent case. In addition, we address some questions related to the estimated time to absorption and obtain a large deviations principle for the invariant distribution in higher dimensions by studying a quasi-potential.