Renormalised Amperean Area of Brownian Motions and Symanzik Representation of the 2D Abelian Yang--Mills--Higgs Field

Abstract: We construct and study the renormalised Amperean area of a Brownian motion. First studied by W.Werner, the Amperean area is related to L\'evy area and stochastic integrals in a way akin to the relation between self-intersection measure and occupation measure. As we explain, it plays a central role in the Symanzik's polymer representation of the continuous Abelian Yang--Mills--Higgs field in 2 dimensions and allows to study this field using classical stochastic calculus and martingale theory.