Hitting estimates on Einstein manifolds and applications

TL;DR

This paper extends hitting probability estimates to Einstein manifolds with Ricci curvature bounds, providing new bounds and applications in geometric analysis and stochastic processes on such manifolds.

Contribution

It generalizes the Benjamini-Pemantle-Peres estimate to manifolds with Ricci curvature bounds and proves new results on Brownian motion and Einstein manifolds.

Findings

Sharp hitting probability estimates near high curvature regions

Proof that limits of Ricci-flat manifolds satisfy Einstein equations

Effective intersection bounds for independent Brownian motions

Abstract

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the probability that Brownian motion comes close to the high curvature part of a Ricci-flat manifold, (2) a proof of an unpublished theorem of Naber that every noncollapsed limit of Ricci-flat manifolds is a weak solution of the Einstein equations, (3) an effective intersection estimate for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 non-negative 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.