Stochastic Jacobi fields and vector fields induced by varying area on path spaces

Journal article

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 denned 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.

Authors: Lyons, T; Qian, Z

• Journal: Probability Theory and Related Fields
• Volume: 109
• Issue: 4
• Pages: 539-570
• Publication date: 1997-12-01
• DOI: 10.1007/s004400050141
• ISSN: 0178-8051
• Language: English
• Keywords: Driver's flow; Path space; Itô map; Lévy area; Affine connection; Covariant equation; Vector field; Rough path