Self-similarity of Brownian motion and a large deviation principle for random fields on a binary tree

Summary Using self-similarity of Brownian motion and its representation as a product measure on a binary tree, we construct a random sequence of probability measures which converges to the distribution of the Brownian bridge. We establish a large deviation principle for random fields on a binary tree. This leads to a class of probability measures with a certain self-similarity property. The same construction can be carried out forC0, 1]-valued processes and we can describe, for instance, aC0, 1]-valued Ornstein-Uhlenbeck process as a large deviation of Brownian sheet.