Large deviation properties of weakly interacting processes via weak convergence methods

We study large deviation properties of systems of weakly interacting particles modeled by Itô stochastic di erential equations (SDEs). It is known under certain conditions that the corresponding sequence of empirical measures converges, as the number of particles tends to in nity, to the weak solution of an associated McKean-Vlasov equation. We derive a large deviation principle via the weak convergence approach. The proof, which avoids discretization arguments, is based on a representation theorem, weak convergence and ideas from stochastic optimal control. The method works under rather mild assumptions and also for models described by SDEs not of di usion type. To illustrate this, we treat the case of SDEs with delay. 2000 AMS subject classi cations: primary 60F10, 60K35; secondary 60B10, 60H10, 34K50, 93E20. ∗Department of Statistics and Operations Research, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA. Research supported in part by the Army Research O ce (Grant W911NF-0-1-0080). †Lefschetz Center for Dynamical Systems, Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. Research supported in part by the National Science Foundation (DMS-0706003), the Army Research O ce (W911NF-09-1-0155), and the Air Force O ce of Scienti c Research (FA9550-09-1-0378). ‡Lefschetz Center for Dynamical Systems, Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. Research supported by the German Research Foundation (DFG research fellowship) and the National Science Foundation (DMS0706003).