Abstract
We define a natural state space and Markov process associated to the stochastic YangMills heat flow in two dimensions. To accomplish this we first introduce a space of distributional connections for which holonomies along sufficiently regular curves (Wilson loop observables) and the action of an associated group of gauge transformations are both welldefined and satisfy good continuity properties. The desired state space is obtained as the corresponding space of orbits under this group action and is shown to be a Polish space when equipped with a natural Hausdorff metric. To construct the Markov process we show that the stochastic YangMills heat flow takes values in our space of connections and use the "DeTurck trick" of introducing a time dependent gauge transformation to show invariance, in law, of the solution under gauge transformations. Our main tool for solving for the YangMills heat flow is the theory of regularity structures and along the way we also develop a "basisfree" framework for applying the theory of regularity structures in the context of vectorvalued noise  this provides a conceptual framework for interpreting several previous constructions and we expect this framework to be of independent interest.
Original language  English 

Publisher  ArXiv 
Number of pages  120 
Publication status  Published  8 Jun 2020 
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Ilya Chevyrev
 School of Mathematics  Reader in Probability and Stochastic Analysis
Person: Academic: Research Active (Teaching)