The Yang-Mills heat flow with random distributional initial data

TL;DR

This paper develops a method to construct local solutions to the Yang-Mills heat flow starting from highly irregular, random initial data, including the 3D Gaussian free field, by decomposing the solution and applying probabilistic techniques.

Contribution

It introduces a novel approach to handle distributional initial data for Yang-Mills heat flow, extending the analysis to include random fields like the Gaussian free field.

Findings

Successfully constructed local solutions for random distributional initial data.

Extended the analysis to include the 3D Gaussian free field.

Provided a framework for future construction of 3D Yang-Mills measures.

Abstract

We construct local solutions to the Yang-Mills heat flow (in the DeTurck gauge) for a certain class of random distributional initial data, which includes the 3D Gaussian free field. The main idea, which goes back to work of Bourgain as well as work of Da Prato-Debussche, is to decompose the solution into a rougher linear part and a smoother nonlinear part, and to control the latter by probabilistic arguments. In a companion work, we use the main results of this paper to propose a way towards the construction of 3D Yang-Mills measures.