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 $4$.
Publié le : 2003-04-14
Classification:
Yang--Mills connection,
Brownian bridge,
stochastic parallel transport,
random holonomy,
stochastic calculus of variation,
58J65,
60H30
@article{1048516535,
author = {Arnaudon, Marc and Thalmaier, Anton},
title = {Yang--Mills fields and random holonomy along Brownian bridges},
journal = {Ann. Probab.},
volume = {31},
number = {1},
year = {2003},
pages = { 769-790},
language = {en},
url = {http://dml.mathdoc.fr/item/1048516535}
}
Arnaudon, Marc; Thalmaier, Anton. Yang--Mills fields and random holonomy along Brownian bridges. Ann. Probab., Tome 31 (2003) no. 1, pp. 769-790. http://gdmltest.u-ga.fr/item/1048516535/