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Talk

Gradient flow identification in metric measure spaces

  • Martin Kell (MPI MiS, Leipzig)
A3 01 (Sophus-Lie room)

Abstract

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.

Katja Heid

MPI for Mathematics in the Sciences Contact via Mail

Upcoming Events of This Seminar

  • Mar 11, 2024 tba with Carlos Román Parra
  • Mar 15, 2024 tba with Esther Bou Dagher
  • Mar 27, 2024 tba with Christian Wagner
  • May 21, 2024 tba with Immanuel Zachhuber