Riemannian Distance and Diameter of the Space of Probability Measures and the Parametrix

Abstract

Information geometry of the space ${\mathcal {P}}(M)$ of probability measures defined on a compact smooth Riemannian manifold M and equipped with Fisher metric G, is investigated from the viewpoint of Riemannian geometry. The function $\ell \,:\, {\mathcal {P}}(M)\times {\mathcal {P}}(M)\rightarrow [0,\pi )$ associated to the geometric mean of two probability measures is introduced. From the formulae of Levi-Civita geodesics the Riemannian distance $d(\cdot ,\cdot )$ of $({\mathcal {P}}(M),G)$ is exactly given by $\ell (\cdot ,\cdot )$. By applying the parametrix $H(x,x_0;t)$ of the heat kernel of M it is shown that the diameter D satisfies $D= \pi$.