论文标题
关于McKean-Vlasov路径的长期行为
On the long-time behaviour of McKean-Vlasov paths
论文作者
论文摘要
众所周知，在某个参数制度中，所谓的McKean-Vlasov Evolution Evolution $（μ_t）_ {t \ in [0，\ infty）} $完全允许三个固定状态。在本文中，我们研究了此制度中流量$（μ_t）_ {t \ in [0，\ infty）} $的长期行为。主要结果是，对于任何初始度量$μ_0$，流量$（μ_t）_ {t \ in [0，\ infty）} $收敛到固定状态，为$ t \ rightarrow \ rightarrow \ infty $。此外，我们表明，如果初始度量的能量低于某些临界阈值，则可以识别限制固定态。最后，我们还展示了麦基 - 维拉索夫进化吸引吸引力的盆地的一些拓扑特性。证明是基于$（μ_t）_ {t \ in [0，\ infty）} $的表示形式作为Wasserstein梯度流。 本文的一些结果并不是全新的。这里的主要贡献是表明Wasserstein框架为这些结果提供了简短而优雅的证明。但是，根据作者的最大知识，关于吸引力盆地拓扑特性的说法是一个新的结果。
It is well-known that, in a certain parameter regime, the so-called McKean-Vlasov evolution $ (μ_t)_{t\in [0,\infty)} $ admits exactly three stationary states. In this paper we study the long-time behaviour of the flow $ (μ_t)_{t\in [0,\infty)} $ in this regime. The main result is that, for any initial measure $ μ_0 $, the flow $ (μ_t)_{t\in [0,\infty)} $ converges to a stationary state as $ t\rightarrow \infty $. Moreover, we show that if the energy of the initial measure is below some critical threshold, then the limiting stationary state can be identified. Finally, we also show some topological properties of the basins of attraction of the McKean-Vlasov evolution. The proofs are based on the representation of $ (μ_t)_{t\in [0,\infty)} $ as a Wasserstein gradient flow. Some results of this paper are not entirely new. The main contribution here is to show that the Wasserstein framework provides short and elegant proofs for these results. However, up to the author's best knowledge, the statement on the topological properties of the basins of attraction is a new result.
