Edgewise Pullback Geometry from Maas Transport and Heat Kernels

This paper develops a heat-kernel pullback theory for the Maas transport geometry of finite irreducible reversible Markov chains. The Maas transport geometry constructs a Wasserstein-type geometry on the probability simplex and studies the heat flow as an entropy gradient flow; the present work addresses the complementary problem of extracting vertex-level and edge-level geometry on the underlying graph from this simplex-level structure. Each vertex is embedded into the probability simplexby its heat-kernel density, and the Maas geometry is pulled back along this embedding. The resulting pairwise midpoint energy, defined through a mobility-weighted Poisson equation, is shown to be well posed and to admit equivalent gradient, dual, and bilinear formulations. It also controls the corresponding pathwise pullback action from both sides, with explicit distortion factors that become sharp in the near-equilibrium regime. The induced pullback quantities satisfy spectral-gap decay and exact tensorization for weighted product chains. This structure further yields a dynamic Ollivier-type curvature, obtained by normalizing the Maas-type pathwise action between heat-kernel measures by the base distance between vertices. Comparison with static heat-kernel $W_1$ transport connects this curvature to classical coarse Ricci geometry, while a rescaled and calibrated graph-to-manifold limit recovers fixed-time heat-kernel Ollivier curvature on the continuum space.