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Uniqueness and regularity for a system of interacting Bessel processes via the Muckenhoupt condition

Authors: Sebastian Andres and Max-K. von Renesse
Journal: Trans. Amer. Math. Soc. 364 (2012), 1413-1426
MSC (2010): Primary 60J60, 42B37
Published electronically: October 11, 2011
MathSciNet review: 2869181
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Abstract: We study the regularity of a diffusion on a simplex with singular drift and reflecting boundary condition which describes a finite system of particles on an interval with Coulomb interaction and reflection between nearest neighbors.

As our main result we establish the strong Feller property for the process in both cases of repulsion and attraction. In particular, the system can be started from any initial state, including multiple point configurations. Moreover, we show that the process is a Euclidean semi-martingale if and only if the interaction is repulsive. Hence, contrary to classical results about reflecting Brownian motion in smooth domains, in the attractive regime a construction via a system of Skorokhod SDEs is impossible. Finally, we establish exponential heat kernel gradient estimates in the repulsive regime.

The main proof for the attractive case is based on potential theory in Sobolev spaces with Muckenhoupt weights.

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Additional Information

Sebastian Andres
Affiliation: Department of Mathematics, Technische Universität Berlin, Strasse des 17, Juni 136, 10623 Berlin, Germany

Max-K. von Renesse
Affiliation: Department of Mathematics, Technische Universität Berlin, Strasse des 17, Juni 136, 10623 Berlin, Germany

Keywords: Bessel process, reflecting boundary condition, Coulomb interaction, Feller property, Muckenhoupt weights
Received by editor(s): November 20, 2009
Received by editor(s) in revised form: May 21, 2010
Published electronically: October 11, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.