Super-Ricci flows and improved gradient and transport estimates

Abstract

We introduce Brownian motions on time-dependent metric measure spaces, proving their existence and uniqueness. We prove contraction estimates for their trajectories assuming that the time-dependent heat flow satisfies transport estimates with respect to every L p -Kantorovich distance, p ∈ [1, ∞]. These transport estimates turn out to characterize super-Ricci flows, introduced by Sturm in [31].where the last two inequalities follow from i) of Lemma 3.5 and the tightness ofand similarly for e 2 .Proposition 3.7. There exists a projective family {Q ν J |J finite ⊂ D} of probability measures and a subsequence (n l ) l∈N such that for each finiteProof. Lemma 3.6 yields for each fixed J the existence of a weakly converging subsequence Q (n l ) J by virtue of Prokhorov's theorem. By a diagonal argument we may choose a subsequence such that Q (n l ) J weakly converges for all finite J ⊂ D. Note that ( e 1 ) # Q (n l ) J = P ν1 J , ( e 2 ) # Q (n l ) J = P ν2 J

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

“…log d t instead of Lipschitz continuity-in a subsequent work of the first author [29] a refined version of the dynamic Bochner inequality (IV N ) will be deduced with estimate (1.11) for every r and all u r ; g r in respective domains, without requiring that they are solutions to heat and adjoint heat equations, respectively. a r / a denotes the W r -geodesic connecting y P r;t and y P r;t .…”

“…The current paper, together with the previous paper by the second author [51], will lay the foundations for a broad systematic study of (super-)Ricci flows in the context of mm-spaces with various subsequent publications in preparation that, among others, will address the following challenges: time-discrete gradient flow scheme à la Jordan-Kinderlehrer-Otto for the heat equation and its dual as gradient flows of energy and entropy, respectively, [28]; improved dynamic Bochner inequality; L p -gradient and L q -transport estimates; construction and optimal coupling of Brownian motions on timedependent mm-spaces [29]; geometric functional inequalities on time-dependent mm-spaces-in particular, local Poincaré, logarithmic Sobolev, and dimension-free Harnack inequalities-and characterization of super-Ricci flows in terms of them [30]; synthetic approaches to upper Ricci bounds [52] and rigidity results for Ricci flat metric cones [18].…”

“…(c) Under slightly more restrictive assumptions on .X; d t ; m t /-namely, C 1dependence of t 7 ! log d t instead of Lipschitz continuity-in a subsequent work of the first author [29] a refined version of the dynamic Bochner inequality (IV N ) will be deduced with estimate (1.11) for every r and all u r ; g r in respective domains, without requiring that they are solutions to heat and adjoint heat equations, respectively. (d) Note that the regularity assumption (1.7) in our formulation of the dynamic Bochner inequality is not really a restriction.…”

Super-Ricci flows for metric measure spaces

“…• construction and detailed analysis of the heat flow and Brownian motion on timedependent mm-spaces [KoS1], [Ko2]; Jordan-Kinderlehrer-Otto gradient flow scheme for the entropy [Ko1]; • geometric functional inequalities on mm-spaces -in particular, local Poincaré, logarithmic Sobolev and dimension-free Harnack inequalities -and characterization of super-Ricci flows in terms of them [KoS2]; • synthetic approaches to upper Ricci bounds [St6] and rigidity results for Ricci flat metric cones [ErS].…”

Functional inequalities for the heat flow on time‐dependent metric measure spaces

“…This self-improvement strategy in the timedependent case requires additional time regularity of the involved quantities. It was carried out by the first author in [13] and can be reformulated with the notation from the current paper as follows. Theorem 1.2 [13].…”