  Consider a system of $n$ weakly interacting particles driven by independent
Brownian motions. In many instances, it is well known that the empirical
measure converges to the solution of a partial differential equation, usually
called McKean-Vlasov or Fokker-Planck equation, as $n$ tends to infinity. We
propose a relatively new approach to show this convergence by directly studying
the stochastic partial differential equation that the empirical measure
satisfies for each fixed $n$. Under a suitable control on the noise term, which
appears due to the finiteness of the system, we are able to prove that the
stochastic perturbation goes to zero, showing that the limiting measure is a
solution to the classical McKean-Vlasov equation. In contrast with known
results, we do not require any independence or finite moment assumption on the
initial condition, but the only weak convergence. The evolution of the
empirical measure is studied in a suitable class of Hilbert spaces where the
noise term is controlled using two distinct but complementary techniques: rough
paths theory and maximal inequalities for self-normalized processes.
