Riemannian Geometry and Information Geometry of Poisson kernels and heat kernels

我们研究了从Hadamard流形X到空间P（∂X）的泊松核映射φ的黎曼几何和Fisher信息几何，该空间P（∂X）是正密度函数的概率测度，相对于黎曼体积绝对连续，定义在X的理想边界上。我们得到以下结果：如果映射φ是齐次最小的，则泊松核用Busemann函数表示，使得X是渐近调和的，满足可见性公理，并且我们可以进一步证明这个齐次常数与X的体积熵一致。

We investigated Riemannian geometry and Fisher information geometry of the Poisson kernel map φ from an Hadamard manifold X into the space P(∂X) of probability measures of positive density function, absolutely continous with respect to the Riemannian volume, defined on the ideal boundary of X. We obtained the following results : if the mapφ is homothetic and minimal, then the Poisson kernel is represented in terms of the Busemann function so that X turns out to be asymptotically harmonic and satisfies the axiom of visibility and furthermore we can show that this homothetic constant coincides with the volume entropy of X.