Gradient flows for semiconvex functions on metric measure spaces – existence, uniqueness, and Lipschitz continuity

Abstract

Given any continuous, lower bounded, and κ \kappa -convex function V V on a metric measure space ( X , d , m ) (X,d,m) which is infinitesimally Hilbertian and satisfies some synthetic lower bound for the Ricci curvature in the sense of Lott-Sturm-Villani, we prove existence and uniqueness for the (downward) gradient flow for V V . Moreover, we prove Lipschitz continuity of the flow w.r.t. the starting point d ( x t , x t ′ ) ≤ e − κ t d ( x 0 , x 0 ′ ) . \begin{equation*} d(x_t,x’_t)\le e^{-\kappa \, t} d(x_0,x_0’). \end{equation*}