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

TL;DR

This paper studies a stochastic partial differential equation with Robin boundary conditions arising from a system of correlated diffusions with elastic boundaries, establishing existence, uniqueness, and regularity properties of the solution.

Contribution

It introduces a novel SPDE model with Robin boundary conditions for elastic diffusions and proves key properties including existence, uniqueness, and boundary regularity.

Findings

Existence and uniqueness of the limiting empirical measure process.

The density is regular inside the domain but may have boundary singularities.

Connections to absorbing and reflecting SPDEs at extreme elastic parameters.

Abstract

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.