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
0U
of the parallel transport with respect to perturbations of the Brownian motion.
We show that the vertical part
U-10U
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
0U
solves the Yang-Mills heat equation.
A monotonicity property for the quadratic variation of
U-10U
is given, both in terms of change of time and in terms of scaling of
U-10U.
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
The paper is available here:
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