Hydrodynamic limit for interacting Ornstein–Uhlenbeck particles

We consider a system of interacting Ornstein–Uhlenbeck particles moving in a d-dimensional torus. The interaction between particles is given by a short-range superstable pair potential V. We prove that, in a diffusive scaling limit, the density of particles satisfies a non-linear partial differential equation. This generalizes to higher dimensions a result of Olla and Varadhan (cf. (Comm. Math. Phys. 125 (1993) 523)).