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 ( µ t ) t ∈ [0 , ∞ ) admits exactly three stationary states . In this paper we study the long-time behaviour of the ﬂow ( µ t ) t ∈ [0 , ∞ ) in this regime. The main result is that, for any initial measure µ 0 , the ﬂow ( µ t ) t ∈ [0 , ∞ ) converges to a stationary state as t → ∞ (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 ( µ t ) t ∈ [0 , ∞ ) 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.


Introduction
In this paper we study the ergodicity and the energy landscape of the flow (µ t ) t∈[0,∞) of marginal laws associated to the stochastic differential equation given by (1.1) Here, the single-site potential Ψ : R → R and the interaction strength J ∈ R satisfy Assumption 1.1 below, and B is a one-dimensional Brownian motion. This flow (µ t ) t∈[0,∞) is often called McKean-Vlasov evolution in the literature. In order to understand the main motivation for this paper, we recall five well-known facts.
(iii) Already in the paper [7] it was conjectured that the process (L N (t)) t∈[0,∞) exhibits metastable behaviour 1 . It is a long outstanding problem to verify this conjecture rigorously. Although some progress in this direction was established in the paper [4], there are still many open and challenging questions.
(iv) It is well-known that, in order to analyse the metastable behaviour of a stochastic system, it is essential to have deep knowledge on the underlying energy landscape of the system and its ergodicity, i.e., its possible convergence towards stationary measures.
(v) In order to study curves and other objects that belong to the infinite-dimensional space of probability measures, the Wasserstein formalism provides a natural and convenient framework. Indeed, ever since the seminal papers [12] and [15], it is known that the Wasserstein formalism provides the structure of a Riemannian manifold on the space of probability measures. We refer to [2, p. 421] or [4, Section 1.4] for more arguments that speak in favour of the Wasserstein formalism.
We now formulate the main motivation for this paper. Combining the facts (ii), (iii) and (iv), we see that, in order to understand the metastable behaviour of (L N (t)) t∈[0,∞) , it is essential to study the ergodicity and the energy landscape of the McKean-Vlasov evolution. Moreover, from fact (i) we see that the energy landscape associated to (µ t ) t∈[0,∞) is determined by the functional F and its basins of attraction; see Proposition 1.4 for the precise definition of the latter. This is the main motivation why we study the ergodicity of (µ t ) t∈[0,∞) and the basins of attraction of F. Finally, fact (v) explains why we use the Wasserstein setting as the framework for this paper. We make the following assumptions throughout this paper.
In particular, Assumption 1.1 is fulfilled if Ψ is a polynomial of degree 2 for some ∈ N ∩ [2, ∞) such that Assumption 1.1 (4) and Assumption 1.1 (5) are satisfied. In Section 6 we briefly discuss the assumptions we make in this paper.
An important observation in Lemma 2.5 is that, as an immediate consequence of Assumption 1.1, the system (1.1) admits exactly three stationary points at some measures µ − , µ 0 , µ + ∈ P 2 (R), which are defined in (2.18); see Lemma 2.5 for more details. We also mention here that, as we will see in Lemma 2.4, the measures µ − and µ + are the global minimizers of the functional F.
We now formulate the main result of this paper in the following theorem.  Proof. The proof is postponed to Section 5.
As a by-product of the proof of Theorem 1.2, we obtain the following two propositions, which are interesting on their own. The first one shows that inside the valleys of the set {µ ∈ P 2 (R) | F(µ) ≤ F(µ 0 )} the convergence of the gradient flows for F is determined by the sign of the mean of the initial value.   Proof. The proof is postponed to Section 3.
The second by-product is the following proposition, which provides useful informations on the energy landscape determined by F.
Then, B − and B + are open subsets of the metric space 2 (P 2 (R), W 2 ), and B 0 is a closed subset of (P 2 (R), W 2 ).
Proof. The claim follows from Proposition 4.1 and Corollary 5.1 below.
The results of this paper are not completely new. Indeed, Theorem 1.2 and Proposition 1.3 have already been obtained in the paper [16] 3 . The proofs in [16] are based on methods from the theory of partial differential equations. The main contributions of this paper are that we use the Wasserstein framework to prove these results (which provides shorter proofs than in [16]), and that the results hold in the stronger topology of the Wasserstein distance (whereas the results in [16] are formulated in terms of the weak topology). However, to our knowledge, Proposition 1.4 is a new result. It is expected that this proposition will become useful in the study of the metastable behaviour of the system (1.3) via the Wasserstein framework. The latter is left for future research.
This paper is organized as follows. First, we recall some elements of the construction of Wasserstein gradient flows in Section 2.1. Then, in Section 2.2, we compare F with the functionalH, which appeared in [4]. In Section 2. briefly discuss the assumptions we make in this paper.

Wasserstein gradient flows
In this section, we briefly recall some elements of the construction of Wasserstein gradient flows. For simplicity, we restrict all definitions to the functional F from (1.2).
For more general functionals and for the details, we refer to [1].
Let P 2 (R) denote the space of all probability measures on R, whose second moment is finite. We equip P 2 (R) with the Wasserstein distance W 2 , which, for µ, ν ∈ P 2 (R) is defined by where Cpl(µ, ν) denotes the space of all probability measures on R 2 that have µ and ν as marginals.
We denote the set of all absolutely continuous curves in (P 2 (R), W 2 ) by AC((0, ∞); P 2 (R)). On We are now in the position to define the notion of Wasserstein gradient flows for F.
There are several different and equivalent ways to do this; some of them are listed in [1,Chapter 11]. In this paper, we choose the definition as a curve of maximal slope (cf. [1, 1.3.2]).
We conclude this section with some useful properties of F and Wasserstein gradient flows for F, which we use many times in this paper. (i) (Lower bound on F) There exists c > 0 such that (iii) (Existence) For each µ ∈ P 2 (R), there exists a gradient flow (S[µ](t)) t∈[0,∞) for F.
(iv) (Energy identity) Let µ ∈ D(F). Then, for all t ∈ (0, ∞), (vi) (contraction and semigroup property) Let µ, ν ∈ P 2 (R). Then, It remains to show part (i). Let µ ∈ D(F), since otherwise the claim is trivial. In the following let C > 0 denote a constant which does not depend on µ, and may change from 4 Note that there is a typo in [1, (4 (2.10) Note that, as a consequence of the classic Young inequalities, for all x,x ∈ R and all α > 0, |xx| ≤ |x| 2 /2 + |x| 2 /2 and |x| 2+ ≥ α|x| 2 − C α for some constant C α > 0 (which only depends on α and ). Then, by choosing α large enough, we can show that the last term on the right-hand side of (2.10) is greater or equal to −C α c 4 . This concludes the proof.

Macroscopic Hamiltonians
In this section we first introduce and recall some facts about the functionH : R → R, which was the object of investigation in the paper [4]. Then, in Lemma 2.4, we show the relation between F andH, and infer from that useful analytic facts about F.
Let the function ϕ * : R → R be defined by  where, for σ ∈ R, the probability measure µ σ ∈ P 2 (R) is defined by (2.14) Finally, we define the functionH : R → R bȳ In the following let m[µ] = R zdµ(z) denote the mean of a probability measure µ ∈ P 2 (R). We have the following relation 5    Then, F admits exactly two global minima, one at µ − and one at µ + , and we have that Proof. If F(µ) = ∞, then (2.16) is trivially satisfied. So we assume that F(µ) < ∞. In the following let H(·|·) denote the relative entropy functional (see e.g. [1, 9.4.1]), and let m[µ] = m. Then, by using (2.13) and by denoting the Lebesgue density of µ by ρ,

Stationary points of the McKean-Vlasov evolution
In this section we characterize the stationary points of the McKean-Vlasov evolution 6 , where we say that µ ∈ P 2 (R) is stationary if  (i) µ is stationary.

Convergence in the valleys
In this chapter we first show some compactness property of the McKean-Vlasov paths in Lemma 3.1. Then, we use this result to prove Proposition 1.3.  Proof. In the following let µ t = S[µ](t). We prove this lemma in three steps.
With this compactness result in hand, we are able to prove Proposition 1.3.