Exact subcritical quasi-stationary density

Determine an exact expression for the limiting quasi-stationary density of the subcritical multi-type branching diffusion with \alpha<0, including its dependence on the mutation-rate matrix \gamma.

Background

For subcritical branching diffusions, the population becomes extinct almost surely, so the relevant long-time object is the distribution conditioned on survival. Unlike the supercritical and critical cases discussed in the paper, the subcritical quasi-stationary density does not collapse onto a one-dimensional ray and is independent of the initial condition.

The paper notes that only the asymptotic mean is known explicitly, namely a vector proportional to the stationary left eigenvector of the mutation-rate matrix. The authors instead use a small-mutation-rate perturbation to obtain moments and do not determine the full exact density.

References

Although an exact solution of the limiting asymptotic density of the diffusion is not known, the asymptotic mean of the population is known to be $(2|\alpha|){-1}\bm{\pi}$, where $\bm{\pi}$ is the left-stationary eigenvector of the rate matrix $\gamma$ normalised so that $\sum_{i = 1}d \pi_i = 1$ \citep[Lemma~2]{BurdenGriffiths23}.

— Multi-type branching diffusions with small mutation rates  (2610.05804 - Burden, 5 Oct 2026) in Section 1, Introduction