Abstract

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. © 2004 Elsevier B.V. All rights reserved.