AI 中文总结
本文利用自然丛理论，开发了由全分支粗糙路径驱动的流形上的粗糙微分方程新框架，证明了解的存在唯一性，给出子流形不变的充要条件，无需额外结构。
AI 中文摘要
本文利用自然丛理论，开发了由分支粗糙路径驱动的有限维光滑流形上的粗糙微分方程的新框架。向量场的作用由某些伴随纤维丛的截面承担。解以广义Davie意义通过局部逼近定义，使得解在坐标变换下不变。本文证明了解的存在性与唯一性，并给出了子流形对解不变的充要条件。该方法可处理由全分支粗糙路径驱动的流形上的粗糙微分方程，无需直接依赖洗牌乘积公式、括号扩展或Connes-Kreimer Hopf代数，也无需在流形上施加额外结构。
英文摘要
In the article, a novel framework for rough differential equations on finite-dimensional smooth manifolds driven by branched rough paths is developed utilizing the theory of natural bundles. The role of vector fields is played by sections of certain associated fiber bundles. The solutions are defined in a generalized Davie sense via a local approximation in such a way that they are invariant under changes of coordinates. Existence and uniqueness of the solutions is proved and a necessary and sufficient condition for the invariance of a submanifold for the solution is given. The approach allows the treatment of rough differential equations driven by fully branched rough paths on manifolds without directly relying on a shuffle product formula, bracket extension, or the Connes-Kreimer Hopf algebra and without imposing additional structure on the manifold.
Comments33 pages
