Information Geometry of Poisson Kernels on Damek-Ricci Spaces

Abstract

For Damek-Ricci spaces $(X,g)$ we compute the exact form of the Busemann function which is needed to represent the Poisson kernel of $(X,g)$ in exponential form in terms of the Busemann function and the volume entropy. From this fact, we show that the Poisson kernel map $\varphi: (X,g) \rightarrow (\mathcal{P}(\partial X),G)$ is a homothetic embedding. Here $\mathcal{P}(\partial X)$ is the space of probability measures having positive density function on the ideal boundary $\partial X$ of $X$, and $G$ is the Fisher information metric on $\mathcal{P}(\partial X)$.