Stochastic Jacobi fields and vector fields induced by varying area on path spaces
Stochastic Jacobi fields and vector fields induced by varying area on path spaces
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路径空间上变化面积引起的随机雅可比场和矢量场
DOI：
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发表时间：
1997
期刊：
影响因子：
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通讯作者：
Z. Qian
中科院分区：
文献类型：
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作者：
Terry Lyons;Z. Qian
摘要研究了闭流形上路空间上两类具有Wiener黎曼测度的向量场。采用杨-米尔斯场论的观点，研究了由变化的度量联络定义的向量场。考虑到布朗运动在正交群作用下是不变的，我们证明了这样得到的向量场满足不同于经典的Jacobi场方程，因而它是连续路径空间上的几何向量场，并在路径空间上导出了一个拟不变解流.本文的第二个研究对象是由变面积得到的向量场。这里我们遵循这样的思想：连续半鞅确实是一条粗糙路，它不仅包含经典意义上的路，而且包含它的Lévy区域。证明了在初始切空间中通过联络平行平移一条曲线所得到的向量场，就是在Malliavin演算意义下，在Cameron-Martin空间中沿沿着方向平移路径，同时适当改变其Lévy面积所得到的向量场.这就导出了路径空间上分部积分公式的一个新的推导过程。
Summary. We study two classes of vector fields on the path space over a closed manifold with a Wiener Riemannian measure. By adopting the viewpoint of Yang-Mills field theory, we study a vector field defined by varying a metric connection. We prove that the vector field obtained in this way satisfies a Jacobi field equation which is different from that of classical one by taking in account that a Brownian motion is invariant under the orthogonal group action, so that it is a geometric vector field on the space of continuous paths, and induces a quasi-invariant solution flow on the path space. The second object of this paper is vector fields obtained by varying area. Here we follow the idea that a continuous semimartingale is indeed a rough path consisting of not only the path in the classical sense, but also its Lévy area. We prove that the vector field obtained by parallel translating a curve in the initial tangent space via a connection is just the vector field generated by translating the path along a direction in the Cameron-Martin space in the Malliavin calculus sense, and at the same time changing its Lévy area in an appropriate way. This leads to a new derivation of the integration by parts formula on the path space.