Coalescing-fragmentating Wasserstein dynamics: particle approach

TL;DR

This paper introduces a novel particle system model with mass-dependent diffusion and sticky-reflecting interactions, extending the Howitt-Warren flow and providing a new infinite-dimensional equation framework.

Contribution

It develops a new family of semimartingales modeling particle systems with mass-dependent diffusion and sticky interactions, advancing the mathematical understanding of such systems.

Findings

Constructed semimartingales for particle systems with sticky-reflecting interactions.

Extended the Howitt-Warren flow to include mass and diffusion rate dependence.

Linked the particle model to a corrected Dean-Kawasaki equation.

Abstract

We construct a family of semimartingales that describes the behavior of a particle system with sticky-reflecting interaction. The model is a physical improvement of the Howitt-Warren flow, an infinite system of diffusion particles on the real line that sticky-reflect from each other. But now particles have masses obeying the conservation law and the diffusion rate of each particle depends on its mass. The equation which describes the evolution of the particle system is a new type of equations in infinite-dimensional space and can be interpreted as an infinite-dimensional analog of the equation for sticky-reflected Brownian motion. The particle model appears as a particular solution to the corrected version of the Dean-Kawasaki equation.