# Recursive tree processes and the mean-field limit of stochastic flows

## TL;DR

This paper develops a continuous-time theory for recursive tree processes related to mean-field limits of interacting particle systems, illustrating it with a cooperative branching example that is not endogenous.

## Contribution

It introduces a continuous-time analogue for recursive tree processes and analyzes their behavior in the mean-field limit for coupled systems.

## Key findings

- Developed a continuous-time recursive tree process theory.
- Connected recursive tree processes with mean-field limits of particle systems.
- Provided an example of a non-endogenous recursive tree process.

## Abstract

Interacting particle systems can often be constructed from a graphical representation, by applying local maps at the times of associated Poisson processes. This leads to a natural coupling of systems started in different initial states. We consider interacting particle systems on the complete graph in the mean-field limit, i.e., as the number of vertices tends to infinity. We are not only interested in the mean-field limit of a single process, but mainly in how several coupled processes behave in the limit. This turns out to be closely related to recursive tree processes as studied by Aldous and Bandyopadyay in discrete time. We here develop an analogue theory for recursive tree processes in continuous time. We illustrate the abstract theory on an example of a particle system with cooperative branching. This yields an interesting new example of a recursive tree process that is not endogenous.