  This paper presents an investigation into the long-term behaviour of
solutions to a nonlinear stochastic McKean-Vlasov equation with common noise.
The equation arises naturally in themean-field limit of systems composed of
interacting particles subject to both idiosyncratic and common noise.
Initially, we demonstrate that the addition of common diffusion in each
particle's dynamics does not disrupt the established stability results observed
in the absence of common noise.However, our main objective is to understand how
the presence of common noise can restore theuniqueness of equilibria.
Specifically, in a non-convex landscape, we establish uniqueness and
convergence towards equilibria under two specific conditions: (1) when the
dimension of the ambientspace equals 1, and (2) in the absence of idiosyncratic
noise in the system.
