  Based on a study of the coupling by reflection of diffusion processes, a new
monotonicity in time of a time-dependent transportation cost between heat
distribution is shown under Bakry-Emery's curvature-dimension condition on a
Riemannian manifold. The cost function comes from the total variation between
heat distributions on spaceforms. As a corollary, we obtain a comparison
theorem for the total variation between heat distributions. In addition, we
show that our monotonicity is stable under the Gromov-Hausdorff convergence of
the underlying space under a uniform curvature-dimension and diameter bound.
