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April 2003 Yang--Mills fields and random holonomy along Brownian bridges
Marc Arnaudon, Anton Thalmaier
Ann. Probab. 31(2): 769-790 (April 2003). DOI: 10.1214/aop/1048516535


We characterize Yang--Mills 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 Yang--Mills 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 Yang--Mills 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 $4$.


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Marc Arnaudon. Anton Thalmaier. "Yang--Mills fields and random holonomy along Brownian bridges." Ann. Probab. 31 (2) 769 - 790, April 2003.


Published: April 2003
First available in Project Euclid: 24 March 2003

zbMATH: 1029.58025
MathSciNet: MR1964948
Digital Object Identifier: 10.1214/aop/1048516535

Primary: 58J65
Secondary: 60H30

Keywords: Brownian bridge , random holonomy , stochastic calculus of variation , stochastic parallel transport , Yang--Mills connection

Rights: Copyright © 2003 Institute of Mathematical Statistics


Vol.31 • No. 2 • April 2003
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