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

Andres, Sebastian; von Renesse, Max-K.

doi:10.1090/S0002-9947-2011-05457-7

Uniqueness and regularity for a system of interacting Bessel processes via the Muckenhoupt condition
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by Sebastian Andres and Max-K. von Renesse PDF

Trans. Amer. Math. Soc. 364 (2012), 1413-1426 Request permission

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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