  This paper is devoted to the problem of approximating non-linear Stochastic
Partial Differential Equations (SPDEs) via interacting particle systems. In
particular, we consider the Stochastic McKean-Vlasov equation, which is the
McKean-Vlasov (MKV) PDE, perturbed by additive trace class noise. As is
well-known, the MKV PDE can be obtained as mean field limit of the empirical
measure of a stochastic system of interacting particles, where particles are
subject to independent sources of noise. There is now a natural question, which
is the one we consider and answer in this paper: can we obtain the SMKV
equation, i.e. additive perturbations of the MKV PDE, as limit of interacting
particle systems? It turns out that, in order to obtain the SMKV equation, one
needs to study weighted empirical measures of particles, where the particles
evolve according to a system of SDEs with independent noise, while the weights
are time evolving and subject to common noise. The work of this manuscript
therefore complements and contributes to various streams of literature, in
particular: i) much attention in the community is currently devoted to
obtaining SPDEs as scaling limits of appropriate dynamics; this paper
contributes to a complementary stream, which is devoted to obtaining
representations of SPDE through limits of empirical measures of interacting
particle systems; ii) since the literature on limits of weighted empirical
measures is often constrained to the case of static (random or deterministic)
weights, this paper contributes to further expanding this line of research to
the case of time-evolving weights.
