 
Summary: The Annals of Probability
2003, Vol. 31, No. 2, 769790
© Institute of Mathematical Statistics, 2003
YANGMILLS FIELDS AND RANDOM HOLONOMY
ALONG BROWNIAN BRIDGES
BY MARC ARNAUDON AND ANTON THALMAIER1
Université de Poitiers and Universität Bonn
We characterize YangMills connections in vector bundles in terms
of covariant derivatives of stochastic parallel transport along variations of
Brownian bridges on the base manifold. In particular, we prove that a
connection in a vector bundle E is YangMills if and only if the covariant
derivative of parallel transport along Brownian bridges (in the direction of
their drift) is a local martingale, when transported back to the starting point.
We present a Taylor expansion up to order 3 for stochastic parallel transport
in E along small rescaled Brownian bridges and prove that the connection
in E is YangMills if and only if all drift terms in the expansion (up to order 3)
vanish or, equivalently, if and only if the average rotation of parallel transport
along small bridges and loops is of order 4.
1. Introduction. This article is concerned with the characterization of Yang
Mills connections in a vector bundle E over a compact Riemannian manifold M
