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

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.

47 pages

References in corpus (4)

• The aromatic bicomplex for the description of divergence-free aromatic forms and volume-preserving integrators
• A Survey on the Munthe-Kaas-Wright Hopf Algebra
• The Lie derivative and Noether's theorem on the aromatic bicomplex for the study of volume-preserving numerical integrators
• The universal equivariance properties of exotic aromatic B-series

Cited by in corpus (1)

• The universal equivariance properties of exotic aromatic B-series