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Eulerian calculus for the displacement convexity in the Wasserstein distance
Sara Daneri, Giuseppe Savare
TL;DR
The paper addresses how to prove displacement convexity on compact Riemannian manifolds without relying on regular optimal-transport maps. It uses an Eulerian gradient-flow and metric EVI approach, and proves strong displacement convexity under lower Ricci curvature bounds.
Problem
A proof of displacement convexity is needed that does not depend on existence and smoothness results for optimal transport maps and Wasserstein geodesics on Riemannian manifolds.
Method
The paper combines the Eulerian approach, the Riemannian characterization of W2, nonlinear diffusion gradient flows, and Evolution Variational Inequalities.
Results
The method proves strong displacement convexity for the integral functional under nonnegative Ricci curvature and strong displacement λ-convexity for relative entropy when Ricci curvature is bounded below by λ.
Takeaways & Limitations
Displacement convexity can be reduced to differential estimates along smooth positive solutions of the functional’s gradient flow.
Takeaways & Limitations
The argument is developed for compact, connected Riemannian manifolds without boundary and uses the stated assumptions on e and U.
Abstract
from arXiv · showhide
In this paper we give a new proof of the (strong) displacement convexity of a class of integral functionals defined on a compact Riemannian manifold satisfying a lower Ricci curvature bound. Our approach does not rely on existence and regularity results for optimal transport maps on Riemannian manifolds, but it is based on the Eulerian point of view recently introduced by Otto-Westdickenberg and on the metric characterization of the gradient flows generated by the functionals in the Wasserstein space.
1 Introduction
The paper proves strong displacement convexity for integral functionals on compact Riemannian manifolds using an Eulerian gradient-flow strategy rather than regular optimal-transport maps. The approach connects differential estimates for nonlinear diffusion flows and Evolution Variational Inequalities to metric displacement convexity.
- Background: The paper’s motivation is the established link between displacement λ-convexity of entropy functionals and lower Ricci curvature bounds.Earlier approaches relied on optimal-transport characterization and Jacobian estimates for the exponential map.
- Main results: The main result establishes strong displacement convexity of the integral functional E when the manifold has nonnegative Ricci curvature and e satisfies the stated McCann conditions.The functional includes an absolutely continuous density term and a singular-measure contribution under the paper’s assumptions.
- Main results: With Ricci curvature bounded below by λ, the relative entropy functional is strongly displacement λ-convex.The relative entropy case uses e(ρ)=ρ log ρ.
- Approach: The proof avoids existence and smoothness results for optimal transport maps and Wasserstein geodesics by working from the Eulerian viewpoint and gradient flows.This addresses regularity difficulties caused by nonsmooth Wasserstein geodesics and the cut locus on Riemannian manifolds.
- Approach: The strategy works on smooth positive densities, uses the Riemannian Benamou–Brenier characterization of W2, and identifies nonlinear diffusion equations as gradient flows of E.The logarithmic entropy gives the heat equation, while another choice of U gives the porous medium equation.
- Approach: Solutions of the diffusion flow satisfy an Evolution Variational Inequality, and this metric property implies strong displacement convexity from a dense class of initial data.Thus geodesic behavior of E is reduced to differential estimates along smooth positive gradient-flow solutions.
2 Gradient ows and geodesic convexity in a smooth setting
The paper develops a smooth-setting strategy that derives geodesic convexity from differential estimates for a semigroup’s action along perturbed curves. A more precise Evolution Variational Inequality overcomes the limitations of contraction-based arguments and yields strong geodesic λ-convexity.
- A metric derivation of convexity: Contraction alone does not retain all information linking S to F, so it cannot by itself establish geodesic convexity.The smooth differential proof also depends on regular geodesics and covariant differentiation, which may be difficult to extend to nonsmooth settings.
- A metric derivation of convexity: Theorem 2.1 shows that a continuous semigroup satisfying the Evolution Variational Inequality implies strong geodesic λ-convexity of F.The result applies along every minimal constant-speed geodesic.
- E.V.I. through action-differential estimates: The action-based strategy keeps one endpoint fixed, evolves the other through S, and studies a family of perturbed curves.This differs from contraction estimates, which evolve both endpoints.
- E.V.I. through action-differential estimates: Theorem 2.2 reduces the E.V.I. to a differential inequality for the perturbed action, and checking it at t = 0 is sufficient.The resulting condition also identifies S as the gradient flow of F and implies geodesic λ-convexity.
- E.V.I. through action-differential estimates: Theorem 2.3 gives a simpler criterion: combining the action estimate with the contraction differential inequality yields the E.V.I. and convexity.Under these conditions, S is the gradient flow of F.
3 Gradient ows and geodesic convexity in a metric setting
The metric-setting framework characterizes gradient flows through Evolution Variational Inequalities and derives strong geodesic λ-convexity without requiring smooth geodesics. It also extends flows from dense subsets and applies the strategy to entropy-driven diffusion on manifolds.
- Metric characterization: The E.V.I. provides an integral characterization of the flow and implies its λ-contraction property.The contraction estimate is expressed through a differential inequality for squared distances.
- Metric characterization: A λ-flow satisfies uniform regularization and continuity estimates, including an energy bound involving Eλ(t) and the squared distance.These estimates support the metric convexity argument.
- Metric characterization: Theorem 3.2 shows that a λ-flow satisfying the metric flow inequalities makes F strongly geodesically λ-convex along geodesics.The theorem first treats Lipschitz curves with an approximation error, then specializes to geodesics.
- Extension from dense subsets: If the flow is initially defined on a dense subset satisfying the approximation property, completeness allows a unique extension to the whole space.The extension preserves the semigroup, flow, and contraction properties.
- Application to nonlinear diffusion: For Ric(M) ≥ 0 and McCann-admissible e, the entropy functional is strongly displacement convex along every Wasserstein geodesic.The conclusion is stated as E(µs) ≤ (1 − s)E(µ0) + sE(µ1).
5 The Heat equation and the displacement λ-convexity of the logarithmic Entropy
The heat semigroup is analyzed as the Wasserstein gradient flow of logarithmic entropy on a manifold with a lower Ricci bound. The resulting E.V.I. yields strong displacement λ-convexity of the entropy.
- Heat flow as a gradient flow: The heat flow satisfies the E.V.I. involving W2(ν, µt), λW2(ν, µt)^2, and the entropy difference E(ν) − E(µt).This inequality is the metric characterization used to obtain convexity.
- Displacement convexity conclusion: The logarithmic entropy functional is strongly displacement λ-convex along every Wasserstein geodesic between µ0 and µ1.The conclusion follows from the established λ-flow property.
- Proof strategy: The proof obtains the E.V.I. by applying the Eulerian action-differential strategy to smooth solutions of the heat equation.The argument uses smooth continuity-equation curves and reparametrization estimates.