Random Holonomy for Yang–Mills Fields: Long-Time Asymptotics

We study weak and strong convergence of the stochastic parallel transport for time t→∞ 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.