Heat Flows on Time-dependent Metric Measure Spaces

Abstract

We consider time-dependent metric measure spaces evolving as a super Ricci flow in the sense of Sturm. We haracterize these by means of the (time-dependent) heat flow. We establish equivalence of super Ricci flows, gradient and transport estimates as well as a dynamic Bochner inequality. This can be seen as a dynamic analogue to the results by Ambrosio, Gigli and Savaré. For this we first define heat flows on time-dependent metric measure spaces. Due to the lack of symmetry we need to consider the heat equation and its adjoint. The solution to the heat equation gives rise to the heat flow on functions, whereas the solution to the adjoint gives rise to the adjoint heat flow on measures. Existence and uniqueness are guaranteed by the general theory of time-dependent elliptic operators. Moreover the two can be retained as gradient flows; the heat flow is the gradient flow of the Cheeger's energy and the dual heat flow is the gradient flow of the relative entropy. Both, functional and space, vary in time, so we first need to make sense of gradient flows in a dynamic setting. We define Brownian motions on time-dependent metric measure spaces. We prove that under a super Ricci flow there exists a coupling of Brownian motions with pathwise contraction of their trajectories.