遍历随机微分方程向后误差分析的Hopf代数结构
Hopf algebra structures for the backward error analysis of ergodic stochastic differential equations
AI总结：
针对遍历随机微分方程向后误差分析中修正向量场无显式表达、计算繁琐的问题，本文提出聚集思路，揭示相关Hopf代数结构，奠定随机数值分析的代数基础并给出修正向量场的显式表达式。
摘要：
尽管向后误差分析无法直接推广到随机微分方程的强逼近与弱逼近，但它可拓展至遍历动力学的采样场景。修正方程的计算依赖繁琐的运算，且与确定性情形不同，目前尚无修正向量场的表达式。本文依托“聚集（clumping）”这一新思路，揭示了与异域芳香S-级数的合成及替换法则相关的Hopf代数结构。我们利用这些代数结构，为基于S-级数的随机数值分析奠定了代数基础，同时给出了修正向量场作为异域芳香B-级数的显式表达式。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
