Monotonicity of time-dependent transportation costs and coupling by reflection

TL;DR

This paper demonstrates a new monotonicity property of time-dependent transportation costs between heat distributions on Riemannian manifolds, using coupling by reflection under curvature conditions, with stability results under space convergence.

Contribution

It introduces a novel monotonicity result for transportation costs based on reflection coupling under Bakry-Emery conditions, and establishes stability under Gromov-Hausdorff convergence.

Findings

Monotonicity of transportation cost under curvature-dimension condition

Comparison theorem for total variation between heat distributions

Stability of monotonicity under Gromov-Hausdorff convergence

Abstract

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.