A tree-free approach to 3D Yang-Mills Langevin dynamic. Analytic estimates and the existence of a model for a regularity structure

TL;DR

This paper constructs a regularity structure and model for the 3D Euclidean Yang-Mills Langevin equation using a multi-index approach, providing global estimates and convergence results.

Contribution

It develops a novel multi-index approach for systems with vector-valued white noises, enabling the construction of a model for the 3D Yang-Mills Langevin equation.

Findings

Constructed a regularity structure and model for the stochastic 3D Yang-Mills equation.

Established global stochastic and pointwise weighted Besov estimates for the model.

Proved convergence of smooth models to the constructed model as mollification is removed.

Abstract

Using the multi-index approach to regularity structures due to F. Otto et al., we construct a regularity structure and a model for it associated to the stochastic Langevin equation for the 3D Euclidean Yang-Mills functional. For the model we also obtain global stochastic and global pointwise weighted Besov type estimates which hold almost surely. The model is defined as a limit of a sequence of smooth models introduced with the help of a mollified noise. When the mollification is removed the sequence converges in a certain topology defined with the help of the stochastic estimates. To obtain these results we develop the multi-index approach for systems of equations with vector-valued white noises. This project is motivated by the problem for constructing 3D Euclidean Yang-Mills measure and by the earlier results of the author on the related problem of canonical quantization of the Yang-Mills field on the Minkowski space.