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.
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Received: 8 August 1996 / In revised form: 8 January 1997
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Lyons, T., Qian, Z. Stochastic Jacobi fields and vector fields induced by varying area on path spaces. Probab Theory Relat Fields 109, 539–570 (1997). https://doi.org/10.1007/s004400050141
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DOI: https://doi.org/10.1007/s004400050141
- AMS Subject Classification (1991): Primary 60D05
- 28D05
- 58D02
- 58D25