论文标题
迈向粗略差异形式的几何整合
Towards geometric integration of rough differential forms
论文作者
论文摘要
我们提供了粗略差异形式的几何整合理论的草案，这些形式是对具有非常低规律性的类似物体的经典（平滑）差异形式的概括，例如，涉及Hölder连续函数，可能无处可区分。从粗糙的路径理论中借用思想，我们表明可以构建这种几何整合，以更一般的渐近扩张来代替适当的差异。这可以将其视为几何整合的基础，类似于几何测量理论中使用的几何整合，但没有任何潜在的可区分结构，因此可以允许Lipschitz函数和可重新定义的集合被替换为较少的规则对象（例如，Hölder函数及其图像及其图像，这可能纯粹是无效的）。我们的构建包括R.Züst最近推出的一维的年轻积分和多维积分。为了简化说明，我们将自己限制为不超过两个的尺寸形式的整合。
We provide a draft of a theory of geometric integration of rough differential forms which are generalizations of classical (smooth) differential forms to similar objects with very low regularity, for instance, involving Hölder continuous functions that may be nowhere differentiable. Borrowing ideas from the theory of rough paths, we show that such a geometric integration can be constructed substituting appropriately differentials with more general asymptotic expansions. This can be seen as the basis of geometric integration similar to that used in geometric measure theory, but without any underlying differentiable structure, thus allowing Lipschitz functions and rectifiable sets to be substituted by far less regular objects (e.g. Hölder functions and their images which may be purely unrectifiable). Our construction includes both the one-dimensional Young integral and multidimensional integrals introduced recently by R. Züst. To simplify the exposition, we limit ourselves to integration of forms of dimensions not exceeding two.
