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

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.

arXiv comment128 pages; misprints corrected; more details in proofs have been added; remarks on symmetries of the model have been added in Section 2.4; remarks on the independence of the structure group maps on the choice of the basis have been added in Sections 2.5, 2.6 and in Lemma 2.5.2; some unused auxiliary statements in Section 3.2 have been removed