  We define a notion of Markov process indexed by curves drawn on a compact
surface and taking its values in a compact Lie group. We call such a process a
two-dimensional Markovian holonomy field. The prototype of this class of
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
Levy processes. We prove that every regular Markovian holonomy field determines
a Levy process of a certain class on the Lie group in which it takes its
values, and construct, for each Levy 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.
