Affordable Access

Publisher Website

A probabilistic approach to the Yang–Mills heat equation

Authors
Journal
Journal de Mathématiques Pures et Appliquées
0021-7824
Publisher
Elsevier
Publication Date
Volume
81
Issue
2
Identifiers
DOI: 10.1016/s0021-7824(02)01254-0

Abstract

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 ∇ 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 ∇ 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.

There are no comments yet on this publication. Be the first to share your thoughts.