Mean-field limit of collective dynamics with time-varying weights

TL;DR

This paper establishes the mean-field limit for a collective dynamics model with time-varying weights, resulting in a transport equation with source that captures both position and weight redistribution among agents.

Contribution

It derives the mean-field limit for a model with weight dynamics preserving total mass and agent indistinguishability, including existence, uniqueness, and convergence results.

Findings

Proved existence and uniqueness of solutions for microscopic and macroscopic models.

Established convergence of microscopic to macroscopic models in Wasserstein and Lipschitz topologies.

Introduced a new empirical measure accounting for agent weights.

Abstract

In this paper, we derive the mean-field limit of a collective dynamics model with time-varying weights, for weight dynamics that preserve the total mass of the system as well as indistinguishability of the agents. The limit equation is a transport equation with source, where the (non-local) transport term corresponds to the position dynamics, and the (non-local) source term comes from the weight redistribution among the agents. We show existence and uniqueness of the solution for both microscopic and macroscopic models and introduce a new empirical measure taking into account the weights. We obtain the convergence of the microscopic model to the macroscopic one by showing continuity of the macroscopic solution with respect to the initial data, in the Wasserstein and Bounded Lipschitz topologies.