Brownian Motion and the Distance to a Submanifold

Authors: James Thompson

Publish Date: 2016/04/02

Volume: 45, Issue: 3, Pages: 485-508

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Abstract

This is a study of the distance between a Brownian motion and a submanifold of a complete Riemannian manifold It contains 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 The concentration inequality is derived 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 processesOpen AccessThis article is distributed under the terms of the Creative Commons Attribution 40 International License http//creativecommonsorg/licenses/by/40/ which permits unrestricted use distribution and reproduction in any medium provided you give appropriate credit to the original authors and the source provide a link to the Creative Commons license and indicate if changes were made

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