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

Abstract

Consider N particles moving independently, each one according to a subcritical continuous-time Galton-Watson process unless it hits 0, at which time it jumps instantaneously to the position of one of the other particles chosen uniformly at random. The resulting dynamics is called Fleming-Viot process. We show that for each N there exists a unique invariant measure for the Fleming-Viot process, and that its stationary empirical distribution converges, as N goes to infinity, to the minimal quasi-stationary distribution of the Galton-Watson process conditioned on non-extinction.

Convergence of the Fleming-Viot process toward the minimal quasi-stationary distribution

“…Before proving the above proposition, we state a criterion implying that the continuous time Galton-Watson process satisfies Assumption H. Note that, since m > 1−p(0), the assumption on α implies that α > 1. Thanks to Theorem 1.1 in [2], the ergodicity of the Fleming-Viot process and its weak convergence toward the quasi-stationary distribution were already known with the condition that p admits an exponential moment. It is thus improved here by considering reproduction laws admitting a polynomial moment of explicit order and by providing convergence in a stronger sense.…”

“…This Fleming-Viot type system has been introduced by Burdzy, Holyst, Ingermann and March in [5] and studied in [6], [13], [21], [14] for multi-dimensional diffusion processes. The study of this system when the underlying Markov process X is a continuous time Markov chain in a countable state space has been initiated in [12] and followed by [1], [2], [16], [3] and [10]. We also refer the reader to [15], where general considerations on the link between the study of such systems and front propagation problems are considered and to [7,11] where CLTs for this Fleming-Viot type process have been proved.…”

On laws of large numbers in $L^{2}$ for supercritical branching Markov processes beyond $\lambda $-positivity

“…Indeed, there exists a large body of literature studying empirical measures of population models with mutations and selection [10,9,11,6,25,4,39,16,26], for which hydrodynamic limits were obtained on finite time windows [44,3,6,16]. However, it is still an open problem in many of these systems to obtain scaling limits of the empirical measure in its stationary regime, see [4,26,39].…”

“…Different couplings with branching Markov processes have been proposed to study this problem, but typically the resulting branching process is not λ-positive, see [4,39]. Thus, this highlights the need of a convergence theory for empirical measures of branching Markov processes going past this assumption of λ-positivity.…”

Yaglom limit for stable processes in cones

“…Let Γ ⊂ R d be an arbitrary Lipschitz cone with vertex at the origin 0. We define (1) τ Γ = inf{t > 0 : X t / ∈ Γ} , the time of the first exit of X from Γ. The following measure µ will be called the Yaglom limit for X and Γ.…”