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

Abstract:

Given any continuous, lower bounded, and $\kappa$-convex function $V$ on a metric measure space $(X,d,m)$ 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 $V$. Moreover, we prove Lipschitz continuity of the flow w.r.t. the starting point \begin{equation*} d(x_t,x’_t)\le e^{-\kappa t} d(x_0,x_0’). \end{equation*}