The Annals of Probability

Yang--Mills fields and random holonomy along Brownian bridges

Marc Arnaudon and Anton Thalmaier

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Abstract

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 $4$.

Article information

Source
Ann. Probab., Volume 31, Number 2 (2003), 769-790.

Dates
First available in Project Euclid: 24 March 2003

Permanent link to this document
https://projecteuclid.org/euclid.aop/1048516535

Digital Object Identifier
doi:10.1214/aop/1048516535

Mathematical Reviews number (MathSciNet)
MR1964948

Zentralblatt MATH identifier
1029.58025

Subjects
Primary: 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60]
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.)

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

Citation

Arnaudon, Marc; Thalmaier, Anton. Yang--Mills fields and random holonomy along Brownian bridges. Ann. Probab. 31 (2003), no. 2, 769--790. doi:10.1214/aop/1048516535. https://projecteuclid.org/euclid.aop/1048516535


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References

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