Brownian motion is introduced as a tool in Riemannian geometry to show how useful it is in the function theory of manifolds, as well as the study of maps between manifolds. As applications, a generalization of Picard's little theorem, and a version of it for Riemann surfaces of large genus are given.
S. I. Goldberg. C. Mueller. "Brownian Motion, Geometry, and Generalizations of Picard's Little Theorem." Ann. Probab. 11 (4) 833 - 846, November, 1983. https://doi.org/10.1214/aop/1176993435