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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 Bailleul et al. (Electron J Probab 25:Paper No. 21, 51, 2020) that promise powerful insights into how interactions with a collective determine the dynamics of an individual within this collective.

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An Example Driven Introduction to Probabilistic Rough Paths

  • 章节

  • 发表日期：

    2025年

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  • François Delarue,
  • William Salkeld

摘要

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 Bailleul et al. (Electron J Probab 25:Paper No. 21, 51, 2020) that promise powerful insights into how interactions with a collective determine the dynamics of an individual within this collective.