Abstract
We establish the convergences (with respect to the simulation time t; the number of particles N; the timestep γ) 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 →∞, N →∞ or γ → 0) are independent from the two others.
Highlights
The initial motivation of the present study was to test the general strategy of the proof in a first simple case, with the goal of extending it later on to the metastable hard case by combining it with some Lyapunov arguments to control the variations of the killing rate near the boundary
The use of particle systems with death and re-birth to approximate the quasi-stationary distribution (QSD) of a Markov process has been introduced in [7], for two-dimensional Brownian motions killed at the boundary of a box
We don’t expect this to happen for the Fleming-Viot particle system in our context where the QSD is unique and the long-time convergence of the limit process and the uniform in time propagation of chaos have been established in non-perturbative cases
Summary
As will be seen below, as far as the long-time behaviour of the process is concerned we will work in a perturbative regime, namely we will assume that the variations of λ are small with respect to the mixing time of the diffusion (1.1) (while λ ∞ itself is not required to be small) These very restrictive conditions, which rule out many cases of practical interest, have to be considered in light of our very strong results The initial motivation of the present study was to test the general strategy of the proof (via coupling aguments) in a first simple case, with the goal of extending it later on to the metastable hard case by combining it with some Lyapunov arguments to control the variations of the killing rate near the boundary
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