Abstract We show that the empirical process associated with a system of weakly interacting diffusion processes exhibits a form of noise-induced metastability. The result is based on an analysis of the associated McKean–Vlasov free energy, which, for suitable attractive interaction potentials, has at least two distinct global minimisers at the critical parameter value β = β c . On the torus, one of these states is the spatially homogeneous constant state, and the other is a clustered state. We show that a third critical point exists at this value. As a result, we obtain that the probability of transition of the empirical process from the constant state scales like exp ⁡ ( − N Δ ) , with Δ the energy gap at β = β c . The proof is based on a version of the mountain pass theorem for lower semicontinuous and λ-geodesically convex functionals on the space of probability measures P 2 ( M ) equipped with the 2-Wasserstein metric, where M is a complete, connected, and smooth Riemannian manifold.

中文翻译：

---

概率测度空间中通过山口定理对麦基恩-弗拉索夫能量的障碍

摘要 我们表明，与弱相互作用扩散过程系统相关的经验过程表现出一种噪声引起的亚稳态。结果基于对相关 McKean-Vlasov 自由能的分析，对于合适的吸引力相互作用势，其在临界参数值 β = β c 处至少具有两个不同的全局最小值。在环面上，这些状态之一是空间均匀的恒定状态，另一个是聚集状态。我们表明在该值处存在第三个临界点。结果，我们从恒定状态尺度如 exp ⁡ ( − N Δ ) 获得经验过程的转变概率，其中 Δ 是 β = β c 处的能隙。