Heat Flow on Time‐Dependent Metric Measure Spaces and Super‐Ricci Flows

Abstract

We study the heat equation on time-dependent metric measure spaces (as well as the dual and the adjoint heat equation) and prove existence, uniqueness, and regularity. Of particular interest are properties that characterize the underlying space as a super-Ricci flow as previously introduced by the second author [51]. Our main result yields the equivalence of F dynamic convexity of the Boltzmann entropy on the (time-dependent) L 2 -Wasserstein space, F monotonicity of L 2 -Kantorovich-Wasserstein distances under the dual heat flow acting on probability measures (backward in time), F gradient estimates for the heat flow acting on functions (forward in time), and F a Bochner inequality involving the time-derivative of the metric. Moreover, we characterize the heat flow on functions as the unique forward EVI flow for the (time-dependent) energy in L 2 -Hilbert space and the dual heat flow on probability measures as the unique backward EVI flow for the (timedependent) Boltzmann entropy in L 2 -Wasserstein space.

Super-Ricci flows and improved gradient and transport estimates

“…In [18] the authors show existence and uniqueness of the heat flow (P t,s ) t≥s on (X, d t , m t ) t∈I , i.e. P t,s u solves ∂ t u t = ∆ t u t , for t > s and u s = u, and moreover, that super-Ricci flows are characterized by the time-dependent gradient estimate Γ t (P t,s u) ≤ P t,s (Γ s (u)), or equivalently, by the L 2 -Kantorovich transport estimate…”

“…In the same setting as in [18] we will introduce backward Brownian motions (see Definition 3.1) and prove their existence and uniqueness (Proposition 3.2). Moreover we construct couplings of backward Brownian motions satisfying a pathwise contraction estimate assuming that stronger L p -transport estimates hold (Theorem 3.8).…”

“…For the precise statement see Theorem 4.12. Note that the authors in [18] derive a dynamic version of Bochner's inequality, where the test functions u appear as heat flows P t,s u s and the estimate holds in an almost everywhere sense. The pointwise dynamic Bochner inequality is an essential modification, since from this we obtain the improved gradient estimates i) in Theorem 1.2.…”

Diffusion, Optimal Transport and Ricci Curvature for Metric Measure Space

“…We conclude this brief review by noting that there have been some recent striking applications to time-dependent metric measure spaces and Ricci flows [43,33].…”

Hamilton differential Harnack inequality and W-entropy for Witten Laplacian on Riemannian manifolds

“…We would like to point out that the notion of super Ricci flows has been also independently introduced by K.-T. Sturm on time-dependent metric measure spaces [24]. See also Kopfer-Sturm [7]. In [9], the Li-Yau Harnack inequality (2) has been extended to positive solutions of the heat equation of the Witten Laplacian…”