The Surface Limit of Brownian Motion in Tubular Neighbourhoods of an embedded Riemannian Manifold

We construct the surface measure on the space C([0, 1],M) of paths in a compact Riemannian manifold M without boundary embedded into Rn which is induced by the usual flat Wiener measure on C([0, 1], Rn) 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. Preprint submitted to Elsevier Science 12 August 2004