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Coupled bialgebras and Lions trees

William Salkeld

TL;DR

This work develops an algebraic framework for higher-order Lions-Taylor expansions by introducing a coupled tensor product that governs interactions among Lions derivatives. It formalizes coupled bialgebras built from Lions words and Lions forests, including a deconcatenation-type coupled coproduct Δ, a twisted interaction with a shuffle product, and a two-parameter grading that yields finite, well-structured components. By connecting partition-based indexing with hypergraphic Lions forests, the authors extend the combinatorial toolkit for probabilistic rough paths and mean-field regularity structures, providing coassociativity and counital properties alongside explicit grading. The results lay a robust foundation for analyzing coupled interactions in McKean-Vlasov-type systems and pave the way for applications to regularity structures and rough path theory in mean-field settings.

Abstract

A Bialgebra is a module over a ring that is both an associative algebra and a co-associative coalgebra with the product and coproduct additionally satisfying an appropriate commutative relationship. One application of Bialgebras is in the study of Taylor expansions where the product describes how for any increment, two terms of the Taylor expansion can be combined together to obtain a higher order term, while the coproduct describes how a term can be decomposed into lower order terms over two adjacent increments. Our motivation is the study of higher order Lions-Taylor expansions and more generally to probabilistic rough paths. The application of iterative Lions derivatives generates a collection of free variables. When terms from a Lions-Taylor expansion are themselves Lions-Taylor expanded, each of these free variables are viewed as tagged and so need Taylor expanding on which greatly increases the number of variables that one needs to keep track of. This additional structure typically takes the form of couplings between terms that arise through the application of the coproduct. The purpose of this work is to document the combinatorial properties of Lions forests (which are central to many key interpretations of probabilistic rough paths) and more formally describe the coupled Bialgebra structure.

Coupled bialgebras and Lions trees

TL;DR

This work develops an algebraic framework for higher-order Lions-Taylor expansions by introducing a coupled tensor product that governs interactions among Lions derivatives. It formalizes coupled bialgebras built from Lions words and Lions forests, including a deconcatenation-type coupled coproduct Δ, a twisted interaction with a shuffle product, and a two-parameter grading that yields finite, well-structured components. By connecting partition-based indexing with hypergraphic Lions forests, the authors extend the combinatorial toolkit for probabilistic rough paths and mean-field regularity structures, providing coassociativity and counital properties alongside explicit grading. The results lay a robust foundation for analyzing coupled interactions in McKean-Vlasov-type systems and pave the way for applications to regularity structures and rough path theory in mean-field settings.

Abstract

A Bialgebra is a module over a ring that is both an associative algebra and a co-associative coalgebra with the product and coproduct additionally satisfying an appropriate commutative relationship. One application of Bialgebras is in the study of Taylor expansions where the product describes how for any increment, two terms of the Taylor expansion can be combined together to obtain a higher order term, while the coproduct describes how a term can be decomposed into lower order terms over two adjacent increments. Our motivation is the study of higher order Lions-Taylor expansions and more generally to probabilistic rough paths. The application of iterative Lions derivatives generates a collection of free variables. When terms from a Lions-Taylor expansion are themselves Lions-Taylor expanded, each of these free variables are viewed as tagged and so need Taylor expanding on which greatly increases the number of variables that one needs to keep track of. This additional structure typically takes the form of couplings between terms that arise through the application of the coproduct. The purpose of this work is to document the combinatorial properties of Lions forests (which are central to many key interpretations of probabilistic rough paths) and more formally describe the coupled Bialgebra structure.
Paper Structure (21 sections, 28 theorems, 293 equations, 1 figure)

This paper contains 21 sections, 28 theorems, 293 equations, 1 figure.

Table of Contents

  1. Contribution
  2. Partitions instead of labels?
  3. Lions-Taylor expansions
  4. Lions-Taylor expansions and coupled structures
  5. Couplings between Lions derivatives
  6. A motivation from probabilistic rough paths
  7. The coupled tensor algebra and Lions words
  8. Lions words and the shuffle product
  9. Couplings for Lions words and coupled coproduct
  10. Coupled coproduct
  11. Bialgebra identity on Lions words
  12. Grading on Lions words
  13. Lions forests and the associated coupled bialgebra
  14. Lions forests and their properties
  15. Partition sequences and Lions forests
  16. ...and 6 more sections

Key Result

Proposition 1.2

Let $\mathscr{N}$ be a finite set and let $I$ be a finite index set (which for clarity should not be confused with $\mathscr{N}$). For each $\iota\in I$, let $p_\iota \subseteq \mathscr{N}$ such that the collection of subsets $p_I = (p_\iota)_{\iota\in I}$ are mutually disjoint. Let $P\in \mathscr{P We denote $\mathfrak{m} :\mathscr{P}(\mathscr{N})[I]\to \mathbb{N}_0$ to be the function such that

Figures (1)

  • Figure 1: Two examples of ancestral lines that are included in the same hyperedge: $(i)$ (Left pane) The two nodes (Node 1 and Node 2) have ancestral lines in the same hyperedge up to the roots; $(ii)$ The two nodes (Node 1 and Node 2) have ancestral lines in the same hyperedge up to a common ancestor, but the common ancestor may be in another hyperdege (hence a different color on the graph).

Theorems & Definitions (86)

  • Definition 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Theorem 1.4
  • Theorem 1.5: Schwarz Theorem for Lions Derivatives
  • Corollary 1.6
  • Remark 1.7
  • Remark 1.9
  • Proposition 1.10
  • proof
  • ...and 76 more