A probabilistic approach to the Yang-Mills heat equation

Marc Arnaudon, Robert O. Bauer, Anton Thalmaier

    Research output: Contribution to journalArticlepeer-review


    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 ∇ 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.

    Original languageEnglish (US)
    Pages (from-to)143-166
    Number of pages24
    JournalJournal des Mathematiques Pures et Appliquees
    Issue number2
    StatePublished - 2002

    ASJC Scopus subject areas

    • General Mathematics
    • Applied Mathematics


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