Fisher information geometry, Poisson kernel and asymptotical harmonicity

Let (X,g)(X,g) be an Hadamard manifold with ideal boundary ∂X  . We can then define the map φ:X→P(∂X)φ:X→P(∂X) associated with Poisson kernel on X  , where P(∂X)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)(X,g) turns out to be an asymptotically harmonic, visibility manifold with the Poisson kernel being expressed in terms of the Busemann function.