  We define a natural state space and Markov process associated to the
stochastic Yang-Mills 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 well-defined 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 Yang-Mills 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 Yang-Mills heat
flow is the theory of regularity structures and along the way we also develop a
"basis-free" framework for applying the theory of regularity structures in the
context of vector-valued noise - this provides a conceptual framework for
interpreting several previous constructions and we expect this framework to be
of independent interest.
