Description

This text defines and studies a class of stochastic processes
indexed by curves drawn on a compact surface and taking
their values in a compact Lie group. We call these processes
two-dimensional Markovian holonomy fields. The proto-type
of these processes, and the only one to have been constructed
before the present work, is the canonical process
under the Yang-Mills measure, first defined by Ambar Sengupta
and later by the author. The Yang-Mills measure sits
in the class of Markovian holonomy fields very much like the
Brownian motion in the class of Lévy processes. We prove
that every regular Markovian holonomy field determines a
Lévy process of a certain class on the Lie group in which
it takes its values, and we construct, for each Lévy process
in this class, a Markovian holonomy field to which it is associated.
When the Lie group is in fact a finite group, we
give an alternative construction of this Markovian holonomy
field as the monodromy of a random ramified principal
bundle. Heuristically, this agrees with the physical origin
of the Yang-Mills measure as the holonomy of a random
connection on a principal bundle.