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:
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