An example driven introduction to probabilistic rough paths

TL;DR

This paper introduces a new regularity structure called Lions trees for solving rough mean-field equations, extending previous ideas to better understand collective interactions in mean-field dynamics.

Contribution

It presents a novel regularity structure based on Lions trees, generalizing prior work to analyze mean-field equations with complex interactions.

Findings

Lions trees effectively capture mean-field interactions.

The new structure generalizes existing frameworks.

Potential for deeper insights into collective dynamics.

Abstract

In these notes, we provide an introduction to a new regularity structure used for solving rough mean-field equations. The index set of this regularity structure is described a collection of novel objects which we refer to as Lions trees. These objects arise in Taylor expansions involving the Lions derivative and capture many of the desirable properties of mean-field dynamics. This work represents a comprehensive generalisation of the ideas first introduced in [BCD20] that promise powerful insights into how interactions with a collective determine the dynamics of an individual within this collective.