On the semimartingale property of Brownian bridges on complete manifolds

TL;DR

This paper proves that all adapted Brownian bridges on complete Riemannian manifolds are semimartingales, allowing for their horizontal lift to principal fiber bundles, using a localized gradient estimate.

Contribution

It establishes the semimartingale property of Brownian bridges on complete manifolds without additional geometric assumptions.

Findings

Brownian bridges are semimartingales on complete manifolds

Terminal time behavior is included in the semimartingale property

Enables horizontal lifting of Brownian bridges to fiber bundles

Abstract

I prove that every adapted Brownian bridge on a geodesically complete connected Riemannian manifold is a semimartingale including its terminal time, without any further assumptions on the geometry. In particular, it follows that every such process can be horizontally lifted to a smooth principal fiber bundle with connection, including its terminal time. The proof is based on a localized Hamilton-type gradient estimate by Arnaudon/Thalmaier.