We introduce the notions of `super-Ricci flows' and `Ricci flows' for time-dependent families of metric measure spaces (X,dt,mt)t∈I. The former property is proven to be stable under suitable space-time versions of mGH-convergence. Uniformly bounded families of super-Ricci flows are compact. In the spirit of the synthetic lower Ricci bounds of Lott-Sturm-Villani for static metric measure spaces, the defining property for super-Ricci flows is the `dynamic convexity' of the Boltzmann entropy
Nearby in the stack
Ent(.∣mt)
regarded as a functions on the time-dependent geodesic space
(P(X),Wt)t∈I
. For Ricci flows, in addition a nearly dynamic concavity of the Boltzmann entropy is requested. Alternatively, super-Ricci flows will be studied in the framework of the
Γ
-calculus of Bakry-Émery-Ledoux and equivalence to gradient estimates will be derived. For both notions of super-Ricci flows, also enforced versions involving an `upper dimension bound'