Fleming–Viot couples live forever

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.