  We consider a system of particles undergoing correlated diffusion with
elastic boundary conditions on the half-line. By taking the large particle
limit we establish existence and uniqueness for the limiting empirical measure
valued process for the surviving particles. This process can be viewed as the
weak form for an SPDE with a noisy Robin boundary condition satisfied by the
particle density. We establish results on the $L^2$-regularity properties of
this density process, showing that it is well behaved in the interior of the
domain but may exhibit singularities on the boundary at a dense set of times.
We also show existence of limit points for the empirical measure in the
non-linear case where the particles have a measure dependent drift. We make
connections for our linear problem to the corresponding absorbing and
reflecting SPDEs, as the elastic parameter takes its extreme values.
