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

TL;DR

This paper establishes the weak convergence of conditioned Brownian motion measures within tubular neighborhoods of a submanifold, using sub-Gaussian estimates for Dirichlet heat kernels to analyze the limiting behavior.

Contribution

It provides a new sub-Gaussian estimate for Dirichlet heat kernels on tubular neighborhoods and demonstrates the tightness and convergence of conditioned Brownian motion measures.

Findings

Weak convergence of measures conditioned on staying within tubes

Identification of the limit measure supported on the submanifold

Establishment of tightness for path measures in tubular neighborhoods

Abstract

We prove tightness of a family of path measures $\nu_{\varepsilon}$ on tubes $L(\varepsilon)$ 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 $\varepsilon$ 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 $L(\varepsilon)$ up to a finite time $T>0$, and we identify the limit measure.