## 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.

Original language | English (US) |
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Pages (from-to) | 143-166 |

Number of pages | 24 |

Journal | Journal des Mathematiques Pures et Appliquees |

Volume | 81 |

Issue number | 2 |

DOIs | |

State | Published - 2002 |

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics