A Convergence Result for Dirichlet Semigroups on Tubular Neighbourhoods and the Marginals of Conditional Brownian Motion

Abstract: We investigate yet another approach to understand the limit behaviour of Brownian motion conditioned to stay within a tubular neighbourhood around a closed and connected submanifold of a Riemannian manifold. In this context, we identify a second order generator subject to Dirichlet conditions on the boundary of the tube and study its associated semigroups. After a suitable rescaling and renormalization procedure, we obtain convergence of these semigroups, both in $L^2$ and in Sobolev spaces of arbitrarily large index, to a limit semigroup, as the tube diameter tends to zero. As a byproduct, we conclude that the conditional Brownian motion converges in finite dimensional distributions to a limit process supported by the path space of the submanifold.