Fisher information geometry, Poisson kernel and asymptotical harmonicity

Abstract

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.