Hitting estimates on Einstein manifolds and applications

We generalize the Benjamini-Pemantle-Peres estimate relating hitting probability and Martin capacity to the setting of manifolds with Ricci curvature bounded below. As applications we obtain: (1) a sharp estimate for the probability that Brownian motion comes close to the high curvature part of a Ricci-flat manifold, (2) a proof of an unpublished theorem of Naber that every noncollapsed limit of Ricci-flat manifolds is a weak solution of the Einstein equations, (3) an effective intersection estimate for two independent Brownian motions on manifolds with nonnegative Ricci curvature and positive asymptotic volume ratio. We also obtain generalizations of (1) and (2) for the manifolds with two-sided Ricci bounds and Einstein manifolds with nonzero Einstein constant.

Automated summary

This paper generalizes the Benjamini-Pemantle-Peres estimate to manifolds with Ricci curvature bounded below and explores its applications in various aspects. The applications include a sharp estimate for the probability of Brownian motion approaching the high curvature part of a Ricci-flat manifold, a proof of an unpublished theorem by Naber regarding noncollapsed limits of Ricci-flat manifolds as weak solutions of the Einstein equations, and an effective intersection estimate for two independent Brownian motions on manifolds with nonnegative Ricci curvature and positive asymptotic volume ratio.