Curvature bounds for configuration spaces

We show that the configuration space over a manifold M inherits many curvature properties of the manifold. For instance, we show that a lower Ricci curvature bound on M implies for the configuration space a lower Ricci curvature bound in the sense of Lott-Sturm-Villani, the Bochner inequality, gradient estimates and Wasserstein contraction. Moreover, we show that the heat flow on the configuration space, or the infinite independent particle process, can be identified as the gradient flow of the entropy.

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