Karl-Theodor Sturm: Heat flow on time-dependent metric measure spaces and super Ricci flows

We study the heat equation on time-dependent metric measure spaces
(being a dynamic forward gradient flow for the energy) and its dual
(being a dynamic backward gradient flow for the Boltzmann
entropy). Monotonicity estimates for transportation distances and for
gradients will be shown to be equivalent to the so-called dynamical
convexity of the Boltzmann entropy on the Wasserstein space. For
time-dependent families of Riemannian manifolds the latter is
equivalent to be a super-Ricci flow.  This includes all static
manifolds of nonnegative Ricci curvature as well as all solutions to
the Ricci flow equation.  The latter will also be characterized in
terms of coupled pairs of Brownian motions.