AI 中文总结
该研究针对流形上任意低正则性路径发展内禀微积分，将Föllmer逐段伊藤微积分扩展到流形值路径，适用于不规则随机过程，以分数布朗运动提升为例说明其应用。
AI 中文摘要
我们针对光滑流形上任意低正则性的路径的光滑函数发展了一种内禀微积分，并定义了沿此类路径的恰当微分形式的逐段积分。路径的正则性由沿一系列划分的p阶变分定义，其中p为任意整数。我们将路径沿一系列划分的p阶变分张量定义为沿路径的局部对称张量测度，并推导了具有有限p阶变分的路径的光滑函数的变量替换公式。当p=2时，我们的结果将H. Föllmer的逐段伊藤微积分扩展到了流形值路径。我们的构造仅需要流形上的一个仿射联络，可视为L. Schwartz二阶微分几何的高阶类似物。该联络将高阶切向量分解为对称张量分量，并导出了几何转移原理：变量替换公式定义了测试函数的约化p-jet的一种内禀、与联络无关的泛函，其规范最高阶分量由p阶变分张量确定。尽管我们的结果是纯几何的，但它们适用于具有高度不规则路径的流形值随机过程，并为此类过程产生了高阶伊藤型微积分。我们以分数布朗运动的指数提升到黎曼流形和李群为例说明了该微积分。
英文摘要
We develop an intrinsic calculus for smooth functions of paths of arbitrarily low regularity on smooth manifolds. The regularity of paths is defined in terms of a $p$-th variation tensor along a sequence of partitions, for an arbitrary integer $p$; this tensor is constructed as a local symmetric tensor measure along the path. We define pathwise integrals of closed one-forms along paths with finite $p$-th variation and derive a change of variable formula for smooth functions of such paths. For $p=2$, our results give a manifold version of H. Föllmer's pathwise Itô calculus. Our construction only requires an affine connection on the manifold and may be viewed as a higher-order analogue of L. Schwartz's second-order differential geometry. The connection provides a splitting of higher-order tangent vectors into symmetric tensor components and leads to a geometric transfer principle: the change-of-variable formula defines an intrinsic, connection-independent functional of the reduced $p$-jet of the test function, whose canonical highest-order component is determined by the $p$-th variation tensor. Although our results are purely geometric, they apply to manifold-valued stochastic processes with highly irregular paths and yield a higher-order Itô-type calculus for such processes. We illustrate this calculus for exponential lifts of fractional Brownian motions to Riemannian manifolds and Lie groups.
Comments46 pages, 1 figure
