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A probabilistic approach to the Yang-Mills heat equation

by Marc Arnaudon, Robert O. Bauer and Anton Thalmaier

We construct a parallel transport *U* in a vector bundle *E*, along the
paths of a Brownian motion in the underlying manifold, with respect
to a time dependent covariant derivative
on *E*, and consider the covariant derivative
_{0}*U*
of the parallel transport with respect to perturbations of the Brownian motion.
We show that the vertical part
*U*^{-1}_{0}*U*
of this covariant derivative has quadratic variation twice the
Yang-Mills energy density (i.e. the square norm of the curvature 2-form)
integrated along the Brownian motion, and
that the drift of such processes vanishes if and only if
_{0}*U*
solves the Yang-Mills heat equation.
A monotonicity property for the quadratic variation of
*U*^{-1}_{0}*U*
is given, both in terms of change of time and in terms of scaling of
*U*^{-1}_{0}*U*.
This allows us to find a priori energy bounds
for solutions to the Yang-Mills heat equation, as well
as criteria for non-explosion given in terms of this
quadratic variation.
*J. Math. Pures Appl. ***81** (2002) 143-166

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