On asymptotic behavior of the modified Arratia flow

TL;DR

This paper investigates the long-term behavior of a generalized Arratia flow, a system of interacting diffusion particles that coalesce and change their diffusion rates based on their mass, linking asymptotics to initial mass distribution properties.

Contribution

It constructs the system for initial mass distributions with finite second moments as an $L_2$-valued martingale and relates its asymptotic behavior to initial local mass properties.

Findings

Constructed the system as an $L_2$-valued martingale.

Established the relationship between asymptotics and initial mass distribution.

Analyzed the influence of initial local properties on long-term behavior.

Abstract

We study asymptotic properties of the system of interacting diffusion particles on the real line which transfer a mass [arXiv:1408.0628]. The system is a natural generalization of the coalescing Brownian motions. The main difference is that diffusion particles coalesce summing their mass and changing their diffusion rate inversely proportional to the mass. First we construct the system in the case where the initial mass distribution has the moment of the order greater then two as an $L_2$-valued martingale with a suitable quadratic variation. Then we find the relationship between the asymptotic behavior of the particles and local properties of the mass distribution at the initial time.