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

This paper focuses on the long-term behavior of solutions to nonlinear stochastic Fokker-Planck equations driven by common noise, where the drift term has a linear dependence on the measure. These equations, which describe the evolution of probability distributions, naturally arise in the mean-field limit of interacting particle systems driven by both idiosyncratic and common noises. After proving the existence of an invariant measure under some mild conditions, we first consider the case where the confinement potential is uniformly convex. In this setting, we establish a result of uniform-in-time conditional propagation of chaos for the associated particle system. This result directly implies the uniqueness of the long-term behavior for solutions of the nonlinear stochastic Fokker-Planck equation. Then, we highlight a more surprising phenomenon of uniqueness recovery induced by the addition of common noise in the non-convex case, albeit under more restrictive structural assumptions. Specifically, we show that the presence of common noise leads to uniqueness and exponential convergence towards equilibrium in the absence of idiosyncratic noise. This result emphasizes the stabilizing role of common noise in systems where non-convex potentials would typically allow for multiple invariant measures.