Astérisque 2010; 172 pp; softcover Number: 329 ISBN10: 2856292836 ISBN13: 9782856292839 List Price: US$52 Member Price: US$41.60 Order Code: AST/329
 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 twodimensional 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 YangMills measure, first defined by Ambar Sengupta and later by the author. The YangMills measure sits in the class of Markovian holonomy fields very much like the Brownian motion in the class of Lévy processes. The author proves 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 he constructs, 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, 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 YangMills measure as the holonomy of a random connection on a principal bundle. A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Readership Graduate students and research mathematicians interested in probability. Table of Contents  Introduction
 Surfaces and graphs
 Multiplicative processes indexed by paths
 Markovian holonomy fields
 Lévy processes and Markovian holonomy fields
 Random ramified coverings
 Index
 Bibliography
