An example driven introduction to probabilistic rough paths

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.