Gradient flow identification in metric measure spaces

In this talk, we outline a proof of the identification of the gradient flow of the Renyi entropy in the p-Wasserstein space with the flow of the q-heat flow, i.e. the parabolic q-Laplace equation. We work in the purely non-smooth setting of metric measure spaces and only require some growth bounds of the reference measure (for q2, none if q>2) and lower semicontinuity of the upper gradient of the Renyi entropy which holds under a curvature assumption shown in a previous talk. Those spaces include smooth Riemannian/Finsler manifolds with lower bound on the Ricci curvature.

Instead of following Otto’s idea of using the p-Wasserstein space to construct the q-heat flow, we follow Ambrosio-Gigli-Savaré (Inv. Math. 2013) and show existence, mass preservation and other properties of the q-heat flow even in the non-compact setting and use this to show that it “solves" the gradient flow of the (3-p)-Renyi entropy in the p-Wasserstein space. Uniqueness will follow from some convexity assumptions of the upper gradient of the Renyi entropy.