论文标题
受控的粗糙路径的几何形状
The geometry of controlled rough paths
论文作者
论文摘要
我们证明，被控制（分支）任意顺序的粗糙路径的空间形成了Banach空间的连续场。该结构与（无限维）矢量束具有许多相似之处，并允许在总空间上定义拓扑，这是所有受控路径空间的收集，在几何情况下，该空间被证明是抛光的。该结构是固有的，并且基于受控粗糙路径的新近似结果。该框架将众所周知的地图（例如粗糙的集成图和Itô-Laneons映射）变成连续的（结构保存）映射。此外，它与对粗糙整合的稳定性理论的先前感兴趣构建兼容。
We prove that the spaces of controlled (branched) rough paths of arbitrary order form a continuous field of Banach spaces. This structure has many similarities to an (infinite-dimensional) vector bundle and allows to define a topology on the total space, the collection of all controlled path spaces, which turns out to be Polish in the geometric case. The construction is intrinsic and based on a new approximation result for controlled rough paths. This framework turns well-known maps such as the rough integration map and the Itô-Lyons map into continuous (structure preserving) mappings. Moreover, it is compatible with previous constructions of interest in the stability theory for rough integration.
