On Number of Particles in Coalescing-Fragmentating Wasserstein Dynamics

TL;DR

This paper studies a modified coalescing-fragmentating Wasserstein particle system, revealing conditions under which the system can have infinitely many particles on dense time subsets, depending on the splitting function.

Contribution

It characterizes when the particle system admits infinitely many particles, extending understanding of its dynamics based on the splitting function’s properties.

Findings

The system has finitely many particles for almost all times.

It can have infinitely many particles on dense subsets of time.

Infinite particle count occurs if the splitting function takes infinitely many values.

Abstract

We consider the system of sticky-reflected Brownian particles on the real line proposed in [arXiv:1711.03011]. The model is a modification of the Howitt-Warren flow but now the diffusion rate of particles is inversely proportional to the mass which they transfer. It is known that the system consists of a finite number of distinct particles for almost all times. In this paper, we show that the system also admits an infinite number of distinct particles on a dense subset of the time interval if and only if the function responsible for the splitting of particles takes an infinite number of values.