Characterizing Yang–Mills Fields by Stochastic Parallel Transport

Abstract

We give a new stochastic approach to Yang–Mills fields on vector bundles by studying the derivative of the stochastic parallel transport in the vector bundle under a variation of connection. We establish martingale criterions for variations transversal to the gauge orbit and for Yang–Mills fields using variations induced by the flow of vector fields of gradient type on the base manifold. In dimension four we link the local Yang–Mills action to the quadratic variation of the derivative process defined by scalings of the driving Brownian motion in a coordinate chart.