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

A note on the Long-Time behaviour of Stochastic McKean-Vlasov Equations with common noise