Brownian bridges to submanifolds

TL;DR

This paper develops a mathematical framework for Brownian bridges to submanifolds, providing formulas and estimates that extend the analysis of path measures in Riemannian geometry.

Contribution

It introduces a general integral formula for the minimal heat kernel over submanifolds and applies it to derive bounds, asymptotics, and derivative estimates, connecting to hypersurface local time.

Findings

Derived a general integral formula for heat kernels on submanifolds

Established lower bounds and asymptotic relations for heat kernel integrals

Connected Brownian bridges to hypersurface local time

Abstract

We introduce and study Brownian bridges to submanifolds. Our method involves proving a general formula for the integral over a submanifold of the minimal heat kernel on a complete Riemannian manifold. We use the formula to derive lower bounds, an asymptotic relation and derivative estimates. We also see a connection to hypersurface local time. This work is motivated by the desire to extend the analysis of path and loop spaces to measures on paths which terminate on a submanifold.