A para-controlled approach to the stochastic Yang-Mills equation in two dimensions

TL;DR

This paper revisits the stochastic Yang-Mills heat equation in two dimensions, employing para-controlled calculus to establish local well-posedness and gauge-covariance, offering a new coordinate-invariant viewpoint on stochastic objects.

Contribution

It introduces a para-controlled calculus approach to the stochastic Yang-Mills equation, providing an alternative to regularity structures and emphasizing coordinate invariance.

Findings

Established local well-posedness using para-controlled calculus

Proved gauge-covariance of the stochastic Yang-Mills model

Developed a new coordinate-invariant framework for stochastic vector-valued objects

Abstract

We consider the stochastic Yang-Mills heat equation on the two-dimensional torus. Using regularity structures, Chandra, Chevyrev, Hairer, and Shen previously proved both the local well-posedness and gauge-covariance of this model. In this article, we revisit their results using para-controlled calculus. One of the main ingredients is a new coordinate-invariant perspective on vector-valued stochastic objects.