An approximation scheme for quasi-stationary distributions of killed diffusions

TL;DR

This paper analyzes the asymptotic behavior of empirical measures of killed diffusions on manifolds, establishing convergence to quasi-stationary distributions and supporting scalable Monte Carlo sampling methods.

Contribution

It introduces a theoretical framework showing empirical measures converge to quasi-stationary distributions, justifying a new Monte Carlo sampling approach.

Findings

Empirical measures form an asymptotic pseudo-trajectory.

Convergence to quasi-stationary distribution is almost sure.

Supports scalable Bayesian posterior sampling.

Abstract

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.