Uniqueness for a weak nonlinear evolution equation and large deviations for diffusing particles with electrostatic repulsion

We use hydrodynamics techniques to study the large deviations properties of the McKean–Vlasov model with singular interactions introduced by Cépa and Lépingle (Probab. Theory Related Fields 107 (1997) 429). In a general framework, we prove upper bounds and exponential tightness, and study the action functional. The study of lower bounds is much harder and requires a uniqueness result for a class of nonlinear evolution equations. In the case of interacting Ornstein–Uhlenbeck particles, we prove a general uniqueness statement by extending techniques of Cabannal-Duvillard and Guionnet (Ann. Probab. 29 (2001) 1205). Using this result we deduce some lower bounds for interacting particles with constant diffusion coefficient and general drift terms.