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 processes.
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Thompson, J. Brownian Motion and the Distance to a Submanifold. Potential Anal 45, 485–508 (2016). https://doi.org/10.1007/s11118-016-9553-2
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DOI: https://doi.org/10.1007/s11118-016-9553-2