Curvature bounds for configuration space
Calculus of Variations
We show that the configuration space Υ over a manifold M inherits many curvature properties of the manifold. For instance, we show that a lower Ricci curvature bound on M implies a lower Ricci curvature bound on Υ in the sense of Lott–Sturm–Villani, the Bochner inequality, gradient estimates and Wasserstein contraction. Moreover, we show that the heat flow on Υ can be identified as the gradient flow of the entropy.
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