On the long-time behaviour of McKean-Vlasov paths

It is well-known that, in a certain parameter regime, the so-called McKean-Vlasov evolution (mu(t))(t)(is an element of[0,infinity)) admits exactly three stationary states. In this paper we study the long-time behaviour of the flow (mu(t))(t)(is an element of[0,infinity)) in this regime. The main result is that, for any initial measure mu(0), the flow (mu(t))(t)(is an element of[0,infinity) )converges to a stationary state as t -> infinity (see Theorem 1.2). Moreover, we show that if the energy of the initial measure is below some critical threshold, then the limiting stationary state can be identified (see Proposition 1.3). Finally, we also show some topological properties of the basins of attraction of the McKean-Vlasov evolution (see Proposition 1.4). The proofs are based on the representation of (mu(t))(t)(is an element of[0,infinity)) 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 (Proposition 1.4) is a new result.