A Basic Estimate for Two-Dimensional Stochastic Holonomy Along Brownian Bridges

We consider R2-valued Gaussian random fields over R2, realizing the free electromagnetic field. We associate to them stochastic holonomy operators, by integrating along compositions of C1 curves from an initial point x (time 0) in R2 to a final point y (time t), and Brownian bridges from y to x. We show that after an infinite renormalization the stochastic holonomy is well defined in Lp. For the proof we use new tools of stochastic analysis, in particular fractional Sobolev spaces in Malliavin calculus and martingale methods. The results have applications to the representation of Higgs fields in terms of Brownian bridges (an extension to Higgs fields of Symanzik′s "polymer representation" of quantum fields).