 
Summary: J. Math. Pures Appl. 81 (2002) 143166
A probabilistic approach to the YangMills heat equation
Marc Arnaudon a
, Robert O. Bauer b
, Anton Thalmaier c
a Département de Mathématiques, Université de Poitiers, Téléport 2, BP 30179, F86962 Futuroscope Chasseneuil cedex, France
b Department of Mathematics, University of Illinois at UrbanaChampaign, Urbana, IL 618012975, USA
c Institut für Angewandte Mathematik, Universität Bonn, Wegelerstraße 6, D53115 Bonn, Germany
Received 1 December 2001
Abstract
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 U10U of this covariant derivative
has quadratic variation twice the YangMills energy density (i.e., the square norm of the curvature 2form) integrated along
the Brownian motion, and that the drift of such processes vanishes if and only if solves the YangMills heat equation.
A monotonicity property for the quadratic variation of U10U is given, both in terms of change of time and in terms of
scaling of U10U. This allows us to find a priori energy bounds for solutions to the YangMills heat equation, as well as
criteria for nonexplosion given in terms of this quadratic variation. 2002 Éditions scientifiques et médicales Elsevier SAS.
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1. Introduction, notations
