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We consider a system of particles undergoing correlated diffusion withelastic boundary conditions on the half-line. By taking the large particlelimit we establish existence and uniqueness for the limiting empirical measurevalued process for the surviving particles. This process can be viewed as theweak form for an SPDE with a noisy Robin boundary condition satisfied by theparticle density. We establish results on the $L^2$-regularity properties ofthis density process, showing that it is well behaved in the interior of thedomain 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 thenon-linear case where the particles have a measure dependent drift. We makeconnections for our linear problem to the corresponding absorbing andreflecting SPDEs, as the elastic parameter takes its extreme values.

An SPDE with Robin-type boundary for a system of elastically killed diffusions on the positive half-line