Gradient Flows for Semiconvex Functions on Metric Measure Spaces - Existence, Uniqueness and Lipschitz Continuity

TL;DR

This paper establishes the existence, uniqueness, and Lipschitz continuity of gradient flows for semiconvex functions on certain metric measure spaces with Ricci curvature bounds, extending analysis in non-smooth geometric contexts.

Contribution

It proves the well-posedness and stability of gradient flows for semiconvex functions on infinitesimally Hilbertian metric measure spaces with Ricci curvature bounds.

Findings

Existence and uniqueness of gradient flows for semiconvex functions.

Lipschitz continuity of the flow with exponential contraction.

Applicability to spaces satisfying synthetic Ricci curvature bounds.

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 $d(x_t,x'_t)\le e^{-\kappa\, t} d(x_0,x_0').$