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Hopf algebra structures for the backward error analysis of ergodic stochastic differential equations

Eugen Bronasco, Adrien Laurent

TL;DR

The paper develops a comprehensive Hopf-algebraic framework for backward error analysis in ergodic stochastic differential equations with additive noise, addressing weak-sense accuracy and invariant-measure sampling. Central to the approach are exotic aromatic S-series and the novel clumping construction, which together yield explicit expressions for modified vector fields as exotic aromatic B-series and enable high-order order conditions via composition and substitution laws. The authors introduce decorated aromatic forests and their D-algebra, alongside Grossman–Larson Hopf algebroids, and extend these ideas to exotic forests, linking decorated and clumped representations through CEM coactions and IBP techniques. This algebraic foundation supports rigorous, high-order stochastic numerical analysis on manifolds and in Euclidean spaces, with potential extensions to rough paths and intrinsic geometric discretizations for sampling invariant measures.

Abstract

While backward error analysis does not generalise straightforwardly to the strong and weak approximation of stochastic differential equations, it extends for the sampling of ergodic dynamics. The calculation of the modified equation relies on tedious calculations and there is no expression of the modified vector field, in opposition to the deterministic setting. We uncover in this paper the Hopf algebra structures associated to the laws of composition and substitution of exotic aromatic S-series, relying on the new idea of clumping. We use these algebraic structures to provide the algebraic foundations of stochastic numerical analysis with S-series, as well as an explicit expression of the modified vector field as an exotic aromatic B-series.

Hopf algebra structures for the backward error analysis of ergodic stochastic differential equations

TL;DR

The paper develops a comprehensive Hopf-algebraic framework for backward error analysis in ergodic stochastic differential equations with additive noise, addressing weak-sense accuracy and invariant-measure sampling. Central to the approach are exotic aromatic S-series and the novel clumping construction, which together yield explicit expressions for modified vector fields as exotic aromatic B-series and enable high-order order conditions via composition and substitution laws. The authors introduce decorated aromatic forests and their D-algebra, alongside Grossman–Larson Hopf algebroids, and extend these ideas to exotic forests, linking decorated and clumped representations through CEM coactions and IBP techniques. This algebraic foundation supports rigorous, high-order stochastic numerical analysis on manifolds and in Euclidean spaces, with potential extensions to rough paths and intrinsic geometric discretizations for sampling invariant measures.

Abstract

While backward error analysis does not generalise straightforwardly to the strong and weak approximation of stochastic differential equations, it extends for the sampling of ergodic dynamics. The calculation of the modified equation relies on tedious calculations and there is no expression of the modified vector field, in opposition to the deterministic setting. We uncover in this paper the Hopf algebra structures associated to the laws of composition and substitution of exotic aromatic S-series, relying on the new idea of clumping. We use these algebraic structures to provide the algebraic foundations of stochastic numerical analysis with S-series, as well as an explicit expression of the modified vector field as an exotic aromatic B-series.
Paper Structure (15 sections, 25 theorems, 135 equations)

This paper contains 15 sections, 25 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Exotic aromatic and clumped forests
  3. Decorated aromatic S-series
  4. Exotic aromatic S-series
  5. Backward order analysis and modified equations for stochastic differential equations
  6. Stochastic order theory with exotic aromatic S-series
  7. The composition law in stochastic numerical analysis
  8. The substitution law in stochastic numerical analysis
  9. Algebraic structure of decorated and exotic aromatic forests
  10. D-algebra of decorated aromatic forests
  11. Decorated clumped forests
  12. Substitution law for decorated aromatic S-series
  13. Algebraic structures of exotic aromatic forests
  14. Conclusion
  15. Examples for forests of small order

Key Result

Theorem 2.5

Bronasco22ebs Let $S(a)$ and $S(b)$ be two exotic aromatic S-series and let $\phi$ be a test function. Then, with $\Delta_{BCK}$ being the Butcher-Connes-Kreimer coproduct over exotic aromatic forests.

Theorems & Definitions (47)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Theorem 2.5: Composition law
  • Definition 2.6
  • Example 2.7
  • Theorem 2.8: Substitution law
  • Definition 3.1
  • Example 3.2
  • ...and 37 more