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

TL;DR

This paper introduces a renormalised Amperean area for Brownian motion, linking it to stochastic integrals and enabling analysis of the 2D Abelian Yang--Mills--Higgs field via classical stochastic calculus.

Contribution

It constructs and studies a new renormalised geometric quantity for Brownian motion, connecting stochastic calculus with quantum field theory representations.

Findings

Defines the renormalised Amperean area for Brownian motion.

Establishes its relation to stochastic integrals and Lévý area.

Facilitates analysis 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.