On the coming down from infinity of coalescing Brownian motions

TL;DR

This paper investigates the conditions under which an infinite system of coalescing Brownian particles on the real line reduces to finitely many over time, providing criteria and rates for this phenomenon.

Contribution

It establishes a necessary and sufficient condition for the coming down from infinity in coalescing Brownian motions and determines the rate of this reduction for various initial setups.

Findings

Derived a precise criterion for coming down from infinity.

Identified the rate at which particles coalesce over time.

Applied results to different initial configurations.

Abstract

Consider a system of Brownian particles on the real line where each pair of particles coalesces at a certain rate according to their intersection local time. Assume that there are infinitely many initial particles in the system. We give a necessary and sufficient condition for the number of particles to come down from infinity. We also identify the rate of this coming down from infinity for different initial configurations.