Ricci limit flows and weak solutions

In this paper we reconcile several different approaches to Ricci flow through singularities that have been proposed over the last few years by Kleiner–Lott, Haslhofer–Naber and Bamler. Specifically, we prove that every noncollapsed limit of Ricci flows, as provided by Bamler’s precompactness theorem, as well as every singular Ricci flow of Kleiner–Lott, is a weak solution in the sense of Haslhofer–Naber. We also generalize all path-space estimates of Haslhofer–Naber to the setting of noncollapsed Ricci limit flows. The key step to establish these results is a new hitting estimate for Brownian motion. A fundamental difficulty, in stark contrast to all prior hitting estimates in the literature, is the lack of lower heat kernel bounds under Ricci flow. To overcome this, we introduce a novel approach to hitting estimates that compensates for the lack of lower heat kernel bounds by making use of the heat kernel geometry of space-time.