Quasi-stationary concentration and extinction-rate conjecture

Let ${\bf X}^{(N)}$ be the multinomial Wright–Fisher Markov chain, let $\mu_N$ be its quasi-stationary distribution on the interior state space, let $\lambda_N$ satisfy $\mu_NP_{\circ,N}=\lambda_N\mu_N$, and let ${\bm\chi_{eq}}$ be the equilibrium from Theorem~. Let $\alpha$ be the parameter appearing in that theorem. Assume the conditions of Theorem~.

Quasi-stationary distribution conjecture. Then:

(i) $\mu_N$ converges as $N\to\infty$ to the degenerate distribution supported at ${\bm\chi_{eq}}$; equivalently,

$$\lim_{N\to\infty}\mu_N(\mathcal N)=1$$

for every open neighborhood $\mathcal N$ of ${\bm\chi_{eq}}$; and

(ii)

$$\lim_{N\to\infty}\frac{\alpha^N}{1-\lambda_N}=1.$$

This conjecture concerns concentration near the stable deterministic equilibrium and the sharp asymptotic extinction rate from the quasi-stationary regime. The paper explains the first part as natural for stochastic perturbations of asymptotically stable deterministic systems, while the second is motivated by comparison with the extinction calculation in Theorem~; neither part is proved there.