  We consider conditional McKean-Vlasov stochastic differential equations
(SDEs), such as the ones arising in the large-system limit of mean field games
and particle systems with mean field interactions when common noise is present.
The conditional time-marginals of the solutions to these SDEs satisfy
non-linear stochastic partial differential equations (SPDEs) of the second
order, whereas the laws of the conditional time-marginals follow Fokker-Planck
equations on the space of probability measures. We prove two superposition
principles: The first establishes that any solution of the SPDE can be lifted
to a solution of the conditional McKean-Vlasov SDE, and the second guarantees
that any solution of the Fokker-Planck equation on the space of probability
measures can be lifted to a solution of the SPDE. We use these results to
obtain a mimicking theorem which shows that the conditional time-marginals of
an Ito process can be emulated by those of a solution to a conditional
McKean-Vlasov SDE with Markovian coefficients. This yields, in particular, a
tool for converting open-loop controls into Markovian ones in the context of
controlled McKean-Vlasov dynamics.
