  This paper is composed of two parts. In the first part we consider
McKean-Vlasov Partial Differential Equations (PDEs), obtained as thermodynamic
limits of interacting particle systems (i.e. in the limit $N\to\infty$, where N
is the number of particles). It is well-known that, even when the particle
system has a unique invariant measure (stationary solution), the limiting PDE
very often displays a phase transition: for certain choices of (coefficients
and) parameter values, the PDE has a unique stationary solution, but as the
value of the parameter varies multiple stationary states appear. In the first
part of this paper, we add to this stream of literature and consider a specific
instance of a McKean-Vlasov type equation, namely the Kuramoto model on the
torus perturbed by a symmetric double-well potential, and show that this PDE
undergoes the type of phase transition just described, as the diffusion
coefficient is varied. In the second part of the paper, we consider a rather
general class of McKean-Vlasov PDEs on the torus (which includes both the
original Kuramoto model and the Kuramoto model in double well potential of part
one) perturbed by (strong enough) infinite-dimensional additive noise. To the
best of our knowledge, the resulting Stochastic PDE, which we refer to as the
Stochastic McKean-Vlasov equation, has not been studied before, so we first
study its well-posedness. We then show that the addition of noise to the PDE
has the effect of restoring uniqueness of the stationary state in the sense
that, irrespective of the choice of coefficients and parameter values in the
McKean-Vlasov PDE, the Stochastic McKean-Vlasov PDE always admits at most one
invariant measure.
