Correction to: Branching-coalescing particle systems

TL;DR

This paper corrects a previously published theorem about branching-coalescing particle systems, showing that the original claim was only valid for almost every time, and provides a revised, weaker version suitable for applications.

Contribution

It identifies an error in a prior theorem and offers a corrected, weaker statement that maintains usefulness for related probabilistic models.

Findings

Counterexample demonstrates the original theorem's inaccuracy.

Theorem holds only for almost every time, not all times.

A revised, weaker theorem is established for practical use.

Abstract

In the article titled "Branching-Coalescing Particle Systems" published in Probability Theory and Related Fields 131(3), pages 376-414, (2005), Theorem 7 as stated there is incorrect. Indeed, we show by counterexample that the equality that we claimed there to hold for all time, in general holds only for almost every time with respect to Lebesgue measure. We prove a weaker version of the theorem that is still sufficient for our applications in the mentioned paper.