Couplings of Brownian Motions of deterministic distance in the Euclidean space and on the sphere

Mihai N. Pascu, Ionel Popescu

Published 2015-07-18Version 1

We obtain a characterization of all co-adapted couplings of Brownian motions in the Euclidean space and on the unit spheres in Euclidean spaces whose distance process is deterministic. More precisely, we show that there exists a co-adapted coupling of Brownian motions on these spaces for which the distance between the processes is a deterministic function if and only if the given function is continuous and satisfies almost everywhere a certain differential inequality.