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Probability, convexity, and harmonic maps with small image. I: Uniqueness and fine existence. (English) Zbl 0675.58042

This paper uses probability theory to provide an approach to the Dirichlet problem for harmonic maps. The probabilistic tools used are those of manifold-valued Brownian motion and \(\Gamma\)-martingales. Probabilistic proofs are given of uniqueness, continuous dependence on data, and existence of generalized solutions (finely harmonic maps), for target domains which have convex geometry. (This class of domains includes all regular geodesic balls.) Links are made with new results in other fields: proofs of new theorems for the \(\Gamma\)-martingale Dirichlet problem, Riemannian centres of mass (herein termed “Karcher means”), and certain convex functions. Future prospects include the development of the probabilistic approach in order to formulate the notion of a harmonic map defined on a fractal and to attack the corresponding Dirichlet problem.
Reviewer: W.S.Kendall

MSC:

58J65 Diffusion processes and stochastic analysis on manifolds
60J65 Brownian motion
58E20 Harmonic maps, etc.
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