  In this paper we study the asymptotic behavior of the normalized weighted
empirical occupation measures of a diffusion process on a compact manifold
which is killed at a smooth rate and then regenerated at a random location,
distributed according to the weighted empirical occupation measure. We show
that the weighted occupation measures almost surely comprise an asymptotic
pseudo-trajectory for a certain deterministic measure-valued semiflow, after
suitably rescaling the time, and that with probability one they converge to the
quasi-stationary distribution of the killed diffusion. These results provide
theoretical justification for a scalable quasi-stationary Monte Carlo method
for sampling from Bayesian posterior distributions.
