Random holonomy for Yang-Mills fields: Long-time asymptotics

We study weak and strong convergence of the stochastic parallel transport for time t --> infinity on Euclidean space. We show that the asymptotic behavior can be controlled by the Yang-Mills action and the Yang-Mills equations. For open paths we show that under appropriate curvature conditions there exits a gauge in which the stochastic parallel transport converges almost surely. For closed paths we show that there exists a gauge invariant notion of a weak limit of the random holonomy and we give conditions that insure the existence of such a limit. Finally, we study the asymptotic behavior of the average of the random holonomy in the case of t'Hooft's 1-instanton.