It has been shown that for a system of Brownian motions with local interaction considered in a diffusive scaling, under some regularity assumptions on the initial profile, the tagged particle process converges to a diffusion. We provide a sufficient condition for granting both the existence and the uniqueness of the tagged particle process for an arbitrary initial profile.
Ilie Grigorescu. "Uniqueness of the Tagged Particle Process in a System with Local Interactions." Ann. Probab. 27 (3) 1268 - 1282, July 1999. https://doi.org/10.1214/aop/1022677446