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Consider a system of $n$ weakly interacting particles driven by independentBrownian motions. In many instances, it is well known that the empiricalmeasure converges to the solution of a partial differential equation, usuallycalled McKean-Vlasov or Fokker-Planck equation, as $n$ tends to infinity. Wepropose a relatively new approach to show this convergence by directly studyingthe stochastic partial differential equation that the empirical measuresatisfies for each fixed $n$. Under a suitable control on the noise term, whichappears due to the finiteness of the system, we are able to prove that thestochastic perturbation goes to zero, showing that the limiting measure is asolution to the classical McKean-Vlasov equation. In contrast with knownresults, we do not require any independence or finite moment assumption on theinitial condition, but the only weak convergence. The evolution of theempirical measure is studied in a suitable class of Hilbert spaces where thenoise term is controlled using two distinct but complementary techniques: roughpaths theory and maximal inequalities for self-normalized processes.

A Law of Large Numbers for interacting diffusions via a mild formulation