Metastable confinement of particle systems near linearly stable McKean-Vlasov equilibria

Source abstract

We establish a general metastability principle showing that the local stability of a McKean-Vlasov equilibrium implies exponentially long confinement of the associated particle system. Consider interacting diffusions on $\mathbb{R}^d$ with a general smooth and bounded interaction kernel and a globally Lipschitz drift that is dissipative at large distances. Assume the McKean-Vlasov equation has at least one equilibrium which is locally exponentially stable. We prove that the probability that the empirical measure leaves a sufficiently small fixed Wasserstein-1 neighborhood of the selected equilibrium before time $T$ is bounded by $C(1+T)e^{-cN}$, where $N$ is the number of particles and $C,c>0$ are independent of $N$ and $T$. Consequently, confinement holds up to times $e^{γN}$, for every $0<γ<c$, with exponentially small exit probability. The proof combines an explicit Volterra construction of the linearized Fokker-Planck semigroup, a mild representation of empirical fluctuations, and exponential maximal estimates for stochastic convolutions. Our approach does not require a gradient flow structure or reversibility of the equations.