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, F-86962 Futuroscope Chasseneuil cedex, France
b Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801-2975, USA
c Institut für Angewandte Mathematik, Universität Bonn, Wegelerstraße 6, D-53115 Bonn, Germany
Received 1 December 2001
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 YangMills 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 solves the YangMills 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 YangMills heat equation, as well as
criteria for non-explosion given in terms of this quadratic variation. 2002 Éditions scientifiques et médicales Elsevier SAS.
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1. Introduction, notations