Fleming-Viot couples live forever

TL;DR

This paper proves non-extinction and describes invariant measures for Fleming-Viot systems with two particles driven by symmetric Hunt processes, highlighting conditions for ergodicity and stationarity.

Contribution

It establishes non-extinction results and characterizes invariant and stationary measures for Fleming-Viot systems with symmetric Hunt process dynamics.

Findings

Non-extinction of two-particle Fleming-Viot systems under finite reference measure.

Existence of an invariant measure for the system.

Reference measure is stationary for the embedded Markov chain.

Abstract

We prove a non-extinction result for Fleming-Viot-type systems of two particles with dynamics described by an arbitrary symmetric Hunt process under the assumption that the reference measure is finite. Additionally, we describe an invariant measure for the system, we discuss its ergodicity, and we prove that the reference measure is a stationary measure for the embedded Markov chain of positions of the surviving particle at successive branching times.