Fisher information geometry, Poisson kernel and asymptotical harmonicity

摘要

Let (X,g) be an Hadamard manifold with ideal boundary ? X. We can then define the map φ:X~→P(?X) associated with Poisson kernel on X, where P(?X) is the space of probability measures on ? X, together with the Fisher information metric G. We make geometrical investigation of homothetic property and minimality of this map with respect to the metrics g and G. The map φ is shown to be a minimal homothetic embedding for a rank one symmetric space of noncompact type as well as for a nonsymmetric Damek-Ricci space. The following is also obtained. If φ is assumed to be homothetic and minimal, then, (X,g) turns out to be an asymptotically harmonic, visibility manifold with the Poisson kernel being expressed in terms of the Busemann function.