We consider the class of stationary, zero-mean Gaussian processes, indexed by the circle, satisfying a two-point Markov property and taking values in a vector bundle over the circle with given holonomy. We establish, subject to certain additional symmetry properties, a classification of all such processes. We then propose a construction of a Brownian motion of loops, in which these processes provide the infinitesimal increments.
"Ornstein-Uhlenbeck processes indexed by the circle." Ann. Probab. 26 (2) 465 - 478, April 1998. https://doi.org/10.1214/aop/1022855640