论文标题
概率粗糙路径II狮子taylor膨胀和随机控制的粗糙路径
Probabilistic rough paths II Lions-Taylor expansions and Random controlled rough paths
论文作者
论文摘要
与先前的贡献中引入的概率粗糙路径的概念\ cite {salkeld2021-培育}，我们解决了相应的随机控制粗糙路径（首先在\ cite {2019Arxiv18020205882.2b}中引入），该路径的结构由雄鹿森林索引。这些是通过基础概率粗糙路径和剩余项上射流组合所描述的路径空间的统计分布。后者的规律性促进了粗糙积分的定义。 我们在随机控制的粗糙路径上建立了两个关键运算符的闭合度和稳定性：通过在Wasserstein空间上的平滑函数进行粗糙整合和组成。这些是针对仍处于妊娠中的粗糙McKean-Vlasov方程的完整理论的重要结果。证明经历了我们严格阐述的狮子衍生物的高阶泰勒膨胀。 耦合的Hopf代数结构（请参见\ Cite {Salkeld2021-蛋白稳态}）和Lions-Taylor的扩展（在\第\ ref Ref {部分：Taylorexpansions}中建立）引入了许多其他挑战，这意味着这些结果不仅是简单的经典理论扩展。我们将这项工作致力于追求这些细节。
In line with the notion of probabilistic rough paths introduced in the previous contribution \cite{salkeld2021Probabilistic}, we address corresponding random controlled rough paths (first introduced in \cite{2019arXiv180205882.2B}), the structure of which is indexed by Lions forests. These are statistical distributions over the space of paths described by the combination of a jet on the underlying probabilistic rough path and a remainder term. The regularity of the latter facilitates the definition of a rough integral. We establish closedness and stability of two key operators on random controlled rough paths: rough integration and composition by a smooth function on the Wasserstein space. These are important results towards a complete theory of rough McKean-Vlasov equations that is still in gestation. The proof goes through a higher-order Taylor expansion for the Lions derivative which we rigorously expound. The coupled Hopf algebra structure (see \cite{salkeld2021Probabilistic}) and the Lions-Taylor expansion (established in Section \ref{section:TaylorExpansions}) introduce a number of additional challenges which mean these results are not simply a natural extension of classical theory. We dedicate this work to pursuing these details.
