Quasistationary Distributions and Fleming-Viot Processes in Finite Spaces

Abstract

Consider a continuous-time Markov process with transition rates matrix Q in the state space ∪ {0}. In the associated Fleming-Viot process N particles evolve independently in with transition rates matrix Q until one of them attempts to jump to state 0. At this moment the particle jumps to one of the positions of the other particles, chosen uniformly at random. When is finite, we show that the empirical distribution of the particles at a fixed time converges as N → ∞ to the distribution of a single particle at the same time conditioned on not touching {0}. Furthermore, the empirical profile of the unique invariant measure for the Fleming-Viot process with N particles converges as N → ∞ to the unique quasistationary distribution of the one-particle motion. A key element of the approach is to show that the two-particle correlations are of order 1/N .

Fleming–Viot selects the minimal quasi-stationary distribution: The Galton–Watson case

“…For a signed measure µ in N we will need to work with the ℓ 2 norm given by µ 2 = x∈N (µ(x)) 2 . The paper [1] proves this proposition for processes with bounded rates, but the extension to our case is straightforward. Thus, we obtain (8.6).…”

“…The arguments are similar to those used in [17,1]. The key is a control of the correlations that we state below.…”

“…The main conjecture in [11,12] is that assuming ξ has distribution λ N , the law of the random measure m(., ξ) converges to the law concentrated on the constant ν qs . This was proven for diffusion processes on a bounded domain of R d , killed at the boundary [5,19,20,26], for jump processes under a Doeblin condition [17] and for finite state jump processes [1].…”

“…The issue here is a uniform bound for the correlations of the empirical distribution of Fleming-Viot at sites x, y ∈ N at fixed time t. This was carried out in [1]. To show (1.4), it suffices to bound the three terms in the bottom line of (2.6).…”

Cheeger inequalities for absorbing Markov chains

“…Yaglom limits for continuous-time Markov processes have been analysed in [9], [16], [29], [30]. The Yaglom limit has also been studied via Fleming-Viot processes [18] in the continuous-time Markov process setting [2], [14]. Here, a group of N particles evolve independently according to the underlying Markov process, and when a particle enters the absorbing state it chooses uniformly one of the other particles and immediately jumps to its location.…”