A probabilistic approach to the Yang-Mills heat equation

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^-1 ∇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 ∇ solves the Yang-Mills heat equation. A monotonicity property for the quadratic variation of U^-1∇₀U is given, both in terms of change of time and in terms of scaling of U^-1∇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.