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

TL;DR

This paper explores the algebraic structures, specifically Hopf algebras, underlying the backward error analysis of ergodic stochastic differential equations, enabling explicit expressions of modified vector fields in stochastic numerical methods.

Contribution

It introduces Hopf algebra structures for exotic aromatic S-series, facilitating the algebraic foundation and explicit computation of modified vector fields in stochastic numerical analysis.

Findings

Uncovered Hopf algebra structures for exotic aromatic S-series.

Provided explicit expressions for modified vector fields as exotic aromatic B-series.

Established algebraic foundations for stochastic numerical analysis.

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.