On the long-time behaviour of McKean-Vlasov paths. (English) Zbl 1448.35040

Summary: It is well-known that, in a certain parameter regime, the so-called McKean-Vlasov evolution \((\mu_t)_{t\in [0,\infty )}\) admits exactly three stationary states. In this paper we study the long-time behaviour of the flow \((\mu_t)_{t\in [0,\infty )}\) in this regime. The main result is that, for any initial measure \(\mu_0\), the flow \((\mu_t)_{t\in [0,\infty )}\) converges to a stationary state as \(t\rightarrow \infty\) (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 \((\mu_t)_{t\in [0,\infty)}\) 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 (Proposition 1.4) is a new result.


35B40 Asymptotic behavior of solutions to PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35K55 Nonlinear parabolic equations
35R60 PDEs with randomness, stochastic partial differential equations
49J40 Variational inequalities
60G10 Stationary stochastic processes
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