A Sub-Gaussian estimate for Dirichlet Heat Kernels on Tubular Neighbourhoods and Tightness of Conditional Brownian Motion

Abstract

We prove tightness of a family of path measures ${\nu }_{\epsilon }$ on tubes $L(\epsilon )$ of small diameters around a closed and connected submanifold $L$ of another Riemannian manifold $M$. Together with a convergence result for Dirichlet semigroups on tubular neighbourhoods, that implies weak convergence of the measures as the tube radius $\epsilon$ tends to zero to a measure supported by the path space of the submanifold. As a consequence, we obtain weak convergence of the measures obtained by conditioning Brownian motion to stay within the tubes $L(\epsilon )$ up to a finite time $T>0$, and we identify the limit measure.

Cited by in corpus (1)

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