Brownian motion and the distance to a submanifold

TL;DR

This paper investigates the behavior of Brownian motion relative to submanifolds in Riemannian manifolds, providing inequalities, local time characterizations, and concentration bounds with broad applicability.

Contribution

It introduces new inequalities and estimates for the distance function and local time of Brownian motion near submanifolds, extending existing theoretical frameworks.

Findings

Laplacian inequality for the distance function

Characterization and formula for local time on hypersurfaces

Exit time estimates and concentration inequalities for tubular neighborhoods

Abstract

We present a study of the distance between a Brownian motion and a submanifold of a complete Riemannian manifold. We include a variety of results, including an inequality for the Laplacian of the distance function derived from a Jacobian comparison theorem, a characterization of local time on a hypersurface which includes a formula for the mean local time, an exit time estimate for tubular neighbourhoods and a concentration inequality. We derive the concentration inequality using moment estimates to obtain an exponential bound, which holds under fairly general assumptions and which is sufficiently sharp to imply a comparison theorem. We provide numerous examples throughout. Further applications will feature in a subsequent article, where we see how the main results and methods presented here can be applied to certain study objects which appear naturally in the theory of submanifold bridge processes.