  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a Ricci-flat manifold, (2) a proof of an unpublished theorem of Naber that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.
