The surface limit of Brownian motion in tubular neighborhoods of an embedded Riemannian manifold

Abstract

We construct the surface measure on the space C([0,1], M) of paths in a compact Riemannian manifold M without boundary embedded into R n which is induced by the usual flat Wiener measure on C([0,1], R n) conditioned to the event that the Brownian particle does not leave the tubular ε-neighborhood of M up to time 1. We prove that the limit as ε→0 exists, the limit measure is equivalent to the Wiener measure on C([0,1], M), and we compute the corresponding density explicitly in terms of scalar and mean curvature.