Probability, Convexity, and Harmonic Maps. II. Smoothness via Probabilistic Gradient Inequalities

A conformal change of metric is used to construct a coupling of two time-changed Riemannian Brownian motions, so that there is an upper bound on the probability of the processes not coupling before either process leaves a small geodesic ball, with the bound being linear in the distance between the two starting points if these two points are close enough to the centre of the ball. This coupling is an alternative to that introduced by Cranston (J. Funct. Anal.99 (1991), 110-124) and is more geometric (and probabilistically less technical) than Cranston′s coupling. It is used to provide a probabilistic approach to the regularity of harmonic maps, thus completing the probabilistic construction of solutions to the small-image harmonic map Dirichlet problems described in Kendall (Proc. London Math. Soc. (3) 61 (1990), 371-406). The approach includes a mild generalization to the case of harmonic maps defined using non-Riemannian connections in the target manifold and smooth strictly elliptic second-order differential operators in the domain.