A probabilistic approach to the Yang-Mills heat equation
by Marc Arnaudon, Robert O. Bauer and Anton Thalmaier

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 $\nabla $on E, and consider the covariant derivative $\nabla$0U of the parallel transport with respect to perturbations of the Brownian motion. We show that the vertical part U-1$\nabla $0U 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 $\nabla$0U solves the Yang-Mills heat equation. A monotonicity property for the quadratic variation of U-1$\nabla $0U is given, both in terms of change of time and in terms of scaling of U-1$\nabla$0U. 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.

J. Math. Pures Appl. 81 (2002) 143-166   [ORIGINAL ARTICLE]

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Marc Arnaudon
Robert Bauer
Anton Thalmaier

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