Manifold-valued martingales, changes of probabilities, and smoothness of finely harmonic maps
by Marc Arnaudon, Xue-Mei Li and Anton Thalmaier

This paper is concerned with regularity results for starting points of continuous manifold-valued martingales with fixed terminal value under a possibly singular change of probability. In particular, if the martingales live in a small neighbourhood of a point and if the stochastic logarithm M of the change of probability varies in some Hardy space Hr for sufficiently large r < 2, then the starting point is differentiable at M = 0. As an application, our results imply that continuous finely harmonic maps between manifolds are smooth, and the differentials have stochastic representations not involving derivatives. This gives a probabilistic alternative to the coupling technique used by Kendall (1994) to prove smoothness of finely harmonic maps.

Ann. Inst. H. Poincaré Probab. Statist.  35 (1999)  765-792

The paper is available here:

Marc Arnaudon
Xue-Mei Li
Anton Thalmaier

Back to Homepage