  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 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 spaceswith Muckenhoupt weights.
