Analytic Form of the Quasi-stationary Distribution of a Simple Birth-Death Process

I consider a simple birth-death model with an absorbing state, where the stable fixed point of the corresponding deterministic mean-field dynamics turns into a transient peak of the probability distribution due to the presence of a tiny fluctuation. The model satisfies the detailed-balance condition, enabling one not only to obtain the analytic form of a quasi-stationary distribution, but also to obtain the analytic form of the escape time under the assumption of quasi-stationarity. I argue that the quasi-steady distribution with exponentially decaying normalization is an excellent approximation of the dynamics at late times, especially for small fluctuations. The analytic expressions for the quasi-stationary distribution and the escape time are expected to be more accurate, hence more useful, for systems with larger sizes.