Existence, uniqueness and ergodicity for the centered Fleming-Viot process

TL;DR

This paper investigates the centered Fleming-Viot process, establishing existence, uniqueness under certain conditions, and ergodicity with exponential convergence, thereby advancing understanding of its long-term behavior.

Contribution

It characterizes the centered Fleming-Viot process via a martingale problem, proving existence, partial uniqueness, and ergodicity with exponential convergence.

Findings

Existence of solutions via asymptotic expansions.

Uniqueness for initial conditions with finite moments.

Exponential ergodicity and invariant measure characterization.

Abstract

Motivated by questions of ergodicity for shift invariant Fleming-Viot process, we consider the centered Fleming-Viot process $\left(Z_{t} \right)_{t\geqslant 0}$ defined by $Z_{t} := \tau_{-\left\langle {\rm id}, Y_{t} \right\rangle} \sharp \, Y_{t}$, where $\left(Y_{t}\right)_{t\geqslant 0}$ is the original Fleming-Viot process. Our goal is to characterise the centered Fleming-Viot process with a martingale problem. To establish the existence of a solution to this martingale problem, we exploit the original Fleming-Viot martingale problem and asymptotic expansions. The proof of uniqueness is based on a weakened version of the duality method, allowing us to prove uniqueness for initial conditions admitting finite moments. We also provide counter examples showing that our approach based on the duality method cannot be expected to give uniqueness for more general initial conditions. Finally, we establish ergodicity properties with exponential convergence in total variation for the centered Fleming-Viot process and characterise the invariant measure.