Convergence of a particle approximation for the quasi-stationary distribution of a diffusion process: uniform estimates in a compact soft case

TL;DR

This paper proves the convergence of a particle approximation scheme for the quasi-stationary distribution of a diffusion process on a torus, providing uniform bounds in simulation time, particle number, and timestep.

Contribution

It introduces a Moran/Fleming-Viot type particle scheme with uniform convergence bounds for approximating quasi-stationary distributions of diffusions.

Findings

Convergence of the particle scheme is established in multiple parameters.

Quantitative bounds are independent across simulation time, particle number, and timestep.

The results apply to diffusions killed at a smooth rate on a compact domain.

Abstract

We establish the convergences (with respect to the simulation time $t$; the number of particles $N$; the timestep $\gamma$) of a Moran/Fleming-Viot type particle scheme toward the quasi-stationary distribution of a diffusion on the $d$-dimensional torus, killed at a smooth rate. In these conditions, quantitative bounds are obtained that, for each parameter ($t\rightarrow \infty$, $N\rightarrow \infty$ or $\gamma\rightarrow 0$) are independent from the two others.