STOCHASTIC CHARACTERIZATION OF HARMONIC MAPS ON RIEMANNIAN POLYHEDRA.

Abstract. The aim of this paper is to relate the theory of harmonic-ity, in the sense of Korevaar-Schoen and Eells-Fuglede, to the notion of Brownian motion in a Riemannian polyhedron, achieved by the second author. We define an exponential map at some singular points. Under the assumption that these exponential maps are totally geodesic (for in-stance in dimension one), we find the infinitesimal generator of a Brow-nian motion in a Riemannian polyhedron. We prove that it is uniquely defined on some Banach space. Finally, we show that harmonic maps, in the sense of Eells-Fuglede, with target smooth Riemannian manifolds, are characterized by mapping Brownian motions in Riemannian polyhe-dra into martingales, while harmonic morphisms are exactly the maps which are Brownian preserving paths. 1. Introduction. Brownian motions in Riemannian manifolds are intimately related to har-monic functions, maps and morphisms. The origin of this relationship is th