Two-Dimensional Markovian Holonomy Fields

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. The author calls these processes two-dimensional Markovian holonomy fields. The prototype 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...
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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. The author calls these processes two-dimensional Markovian holonomy fields. The prototype 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 Levy processes. The author proves 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 he constructs, 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, the author gives 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.