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

## TL;DR

This paper studies the limiting behavior of Brownian motion conditioned to stay within shrinking tubular neighborhoods around a submanifold, showing convergence of associated semigroups and the process itself to a limit supported on the submanifold.

## Contribution

It introduces a second order generator with Dirichlet boundary conditions and proves the convergence of the semigroups and conditioned Brownian motions as the tube diameter approaches zero.

## Key findings

- Semigroups converge in $L^2$ and Sobolev spaces
- Conditional Brownian motion converges in finite-dimensional distributions
- Limit process supported on the submanifold

## 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.