Yang–Mills fields and stochastic parallel transport in small geodesic balls

We develop a new method to obtain stochastic characterizations of Yang–Mills fields. Our main tool is the Itô-equation for the stochastic parallel transport. We estimate the drift terms in a small ball of radius ε and find that for a general connection the average rotation is of order ε3 but that for a Yang–Mills connections the average rotation is of order ε4. Using a Doob h-transform we give a new proof of the stochastic characterization of Yang–Mills fields by S. Stafford. Varying the starting point of the Brownian motion we obtain an unconditioned version of this result. By considering the horizontal Laplace equation we then apply our result to obtain a new analytic characterization of Yang–Mills fields.