A system of coalescing heavy diffusion particles on the real line

TL;DR

This paper constructs a modified Arratia flow with mass conservation, where particles' diffusion inversely depends on their mass, and introduces a stochastic integral related to the flow, providing new insights into particle systems with conserved quantities.

Contribution

It presents a novel construction of a mass-conserving Arratia flow with inverse proportional diffusion and develops a stochastic integral framework for analyzing the flow.

Findings

Existence of the flow under uniform initial mass distribution

Introduction of a stochastic integral with respect to the flow

Total local time as the density of occupation measure

Abstract

We construct a modified Arratia flow with mass and energy conservation. We suppose that particles have a mass obeying the conservation law, and their diffusion is inversely proportional to the mass. Our main result asserts that such a system exists under the assumption of the uniform mass distribution on an interval at the starting moment. We introduce a stochastic integral with respect to such a flow and obtain the total local time as the density of the occupation measure for all particles.