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The exit time and the exit location of a non-Markovian diffusion is analyzed. More particularly, we focus on the so-called self-stabilizing process. The question has been studied by Herrmann, Imkeller and Peithmann (in 2008) with results similar to those by Freidlin and Wentzell. We aim to provide the same results by a more intuitive approach and without reconstructing the proofs of Freidlin and Wentzell. Our arguments are as follows. In one hand, we establish a strong version of the propagation of chaos which allows to link the exit time of the McKean-Vlasov diffusion and the one of a particle in a mean-field system. In the other hand, we apply the Freidlin-Wentzell theory to the associated mean field system, which is a Markovian diffusion.


Introduction
The questions that we address in this paper are about the pathwise asymptotic behavior of a particular class of inhomogeneous diffusions: We study here the so-called self-stabilizing process.The term "self-stabilizing" comes from the fact that each trajectory is attracted by the whole set of trajectories in the following sense: b (t, x) := ∇V (x) + E {∇F (x − X t )} .What is the exit time?What is the exit location?In the small-noise limit, the questions become: 1. What is the exit time τ ( ) for going to 0? 2. What is the exit location X τ ( ) for going to 0?
The subject of this article is to study these questions.They have been solved by Freidlin and Wentzell for homogeneous difffusions.See [5,4] for a complete review.Let us briefly present their results.We study the diffusion U is a C ∞ -continuous function from R k (k ≥ 1) to R and β is a Brownian motion in R k .a 0 is a minimizer of U and G is a domain which contains a 0 .Under easy to check assumptions (which are detailed in Appendix A), for all δ > 0, the following Kramers' type law holds: The star in the previous line corresponds to a convolution and u t is the own law of the diffusion X at time t.Let us point out that E {∇W t (X t )} is not equal to E {∇V (X t )}.
It is equal to E {∇V (X t ) + ∇F (X t − Y t )} where Y is an independent version of X .
Since the own law of the process intervenes in the drift, this equation is nonlinear, in the sense of McKean.Three terms generate the dynamic.The first one is a Brownian motion B in R d with intensity 2 d.It allows X to visit the whole space.The second force describes the attraction between one trajectory t → X t (ω 0 ) and the whole set of trajectories.Indeed, we notice: ∇F * u t (X t (ω 0 )) = ω∈Ω ∇F (X t (ω 0 ) − X t (ω)) dP (ω) where (Ω, F, P) is the underlying measurable space.Consequently, we say that F is the interaction potential.The last term is V , the so-called confining potential.It forces the diffusion to move to the minimizers of V .These three forces are concurrent.
As a first observation, we note that the future of the couple (X ; u ) is independent of its past if its present is known.However, the diffusion X is not Markovian since the past intervenes in the drift ∇W t through the law u t .This kind of processes were introduced by McKean.The reader is referred to [14].X corresponds to the hydrodynamic limit of the interacting particle system: s ds for all 1 ≤ i ≤ N .The N Brownian motions are supposed independent and B 1 = B.
Each particle is attracted by the whole set of particles.We call this a mean-field system.
The drift which intervenes in each diffusion Z ,i,N can be written similarly to the one of the self-stabilizing diffusion (0.1): Heuristically, the empirical law δ Z ,i,N t of the system converges to u t as N tends to 0. This phenomenon is called propagation of chaos.Under some hypotheses on V and F , the self-stabilizing diffusion X corresponds to the limit for large N of the first particle Z ,1,N in the following sense: [15,1,12,13,3].Proofs of the classical results on propagation of chaos are in Appendix B. The mean-field system is Markovian.Indeed, by denoting where the potential Υ N is defined by Then we can apply Freidlin-Wentzell results to the homogeneous diffusion Z ,N .We note that the exit problem of Z ,1,N from D is equivalent to the one of Z ,N from D × R d(N −1) .A strong version of the propagation of chaos allows then to link the exit time of X from D and the one of Z ,1,N from D.
Let us briefly recall some previous results on McKean-Vlasov diffusions.The existence and the uniqueness of a strong solution X on R + for equation (0.1) has been proved in [6] (Theorem 2.13).The asymptotic behavior of the law has been studied in [3,2] (for the convex case) and in [16,18] in the non-convex case by using the results in [7,8,9] about the non-uniqueness of the stationary measures and their small-noise behavior.
The exit problem of self-stabilizing processes has already been solved if both V and F are uniformly strictly convex, see [6].The authors follow and extend the method of Freidlin and Wentzell.The difficulty is the lack of Markov property.Indeed, in inhomogeneous diffusions, the first exit time and the second exit time can not be identified up to a shift.However, if V and F are uniformly strictly convex that is to say if Hess F (x) ≥ α > 0, they prove a Kramers' type law.The exit time τ ( ) of X from a domain D satisfies the limit: for all δ > 0. Here, H := inf ) where a 0 is the unique minimizer of V .They also provide a result on the exit location which is similar to the one of Freidlin-Wentzell.They also give an example of the influence of self-stabilizing term on the exit location.This paper proposes a new simpler and more intuitive approach of the problem.The article is organized as follows.First, we present the assumptions on the potentials and the definitions.Then, the uniform boundedness of the moments is established.This justifies the assumptions on the domain D. The main results are written in the end of Section 1.In the second section, the exit problem of the particle Z ,1,N is addressed by applying classical Freidlin-Wentzell theory.The third section deals with a new version of the propagation of chaos.Finally, the main results are proved.
The article contains also two appendixes.One deals with the results and the hypotheses of the Freidlin-Wentzell theory and the other one presents the classical results on the propagation of chaos, including the proofs.

Preliminaries and main results
First, let us denote by || .|| the euclidian norm on R d : ||x|| We assume the following properties on the confining potential V : V is uniformly strictly convex: Hess V ≥ θ > 0.
We would like to point out that the aim of the hypothesis (V-3) is just to simplify the writing.Indeed, if the point of the global minimum is a 0 = 0, it is sufficient to consider the diffusion X := X − a 0 and the potential V := V (. + a 0 ) − V (a 0 ).An immediate consequence of (V-1)-(V-3) is the following inequality: x Let us now present the assumptions on the interaction potential F : (F-1) There exists a function G from R + to itself such that F (x) = G (||x||).
Let us note that (F-1)-(F-4) imply Since the initial law is a Dirac measure, we know that there exists a unique strong solution X to the equation (0.1), see Theorem 2.13 in [6] for a proof.Moreover: for all p ∈ N * .We immediately deduce the tightness of the family (u t ) t∈R+ .
We now present some notations concerning the space R d N =: R dN .

3.
Finally, for all x ∈ R d , the vector (x, • • • , x) ∈ R dN is denoted by x.
We remark that |||x||| = ||x|| for all x ∈ R d .In order to simplify the writing, we use the following terminology in the whole article: Let G be a subset of R k and let U be a C ∞ -continuous function from R k to R. For all x ∈ R k , we consider the dynamical system We say that the domain G is stable by −∇U if the orbit {ψ t (x) ; t ∈ R + } is included in G for all x ∈ G.
We now establish an important result about the moments of X .Indeed, since these moments intervene in the drift, the asymptotic behavior (deterministic) of the law u t is related to the asymptotic behavior (probabilistic) of the trajectories.Moreover, it allows to understand what are the relevant sets from which we shoud study the exit problem.Proposition 1.3.1.The 2n-moment is uniformly bounded: . (1.3)

2.
For all κ > 0 and > 0, we introduce the deterministic time For < κ 2 θ 2n−1 , we have the inequality: (1.4) By definition, the second term b (t) can be written as where Y is a solution of (0.1) independent from X .We can exchange X and Y .

3.
Finally, for all T > 0, (1.5) implies sup Then, for all t ≥ T κ ( ), This means that the self-stabilizing process tends to be trapped in a ball with center 0. This result concerns the law u t and not the trajectories t → X t (ω).But it points out the importance of δ 0 in the study.Indeed, Proposition 1.3 implies Consequently, the relevant sets for the exit problem of the McKean-Vlasov diffusions are the ones which contain the attractive point 0.
Remark 1.4.In Proposition 1.3, we established the uniform boundedness of the moment of degree 2n.We would like to point out that we can prove for all p ∈ N.
We now give the assumptions on the domain D.
Assumption 1.5.We consider the dynamical system where X 0 is introduced in (0.1).There exists T 0 ≥ 0 such that {ϕ T0+t ; t > 0} is included in D and the orbit {ϕ t ; 0 We point out that the domain D is not necessary stable by −∇V .
In order to heuristically understand this assumption, let us consider the dynamical system where Υ N is defined in (0.4).We remark that Z N t is equal to ϕ t for all t ≥ 0.Then, by Assumption 1.5, the orbit Let us note that this assumption is weaker than Assumption 4.1.i)in [6].We now present the other hypothesis: This hypothesis is natural according to Proposition 1.3.Indeed, the law u t is as close as we want to δ 0 .Consequently, the drift ∇V +∇F * u t is close to ∇V +∇F * δ 0 = ∇V +∇F .
Next, we define the exit cost.
We now give an example of a domain satisfying both Assumptions 1.5-1.6.Lemma 1.8.For all H > 0, the domain K Let us prove the first hypothesis.We take any x ∈ R d and we consider the dynamical system Since V is convex, ϕ t (x) converges to 0 so there exists T 0 ≥ 0 such that the orbit This finishes the proof.
Before giving the main results of the paper, we recall a simple fact.
Lemma 1.9.Υ N admits exactly one critical point: 0.Moreover, it is the point of the global minimum.
The proof is similar -up to some details due to the dimension d -to the one of Proposition 2.1 in [19].Thereby, it is left to the reader.
Let us now provide the two main results.
Theorem: We consider a function V which satisfies (V-1)-(V-3), a function F which satisfies (F-1)-(F-4).Under Assumptions 1.5-1.6,for all ξ > 0, we have the limit: Let us note that this result is stronger than the one in [6] since we do not assume that the domain D is stable by −∇V .Theorem: We consider a function V which satisfies (V-1)-(V-3), a function F which satisfies (F-1)-(F-4).Let H and ρ be two positive real numbers.For all δ > 0, there exist N δ ∈ N * and δ > 0 such that: sup This result establishes that -in the small-noise limit -the particle Z ,1,N is a good approximation of the McKean-Vlasov diffusion, even in the long-time.

Exit problem of the first particle
In this section, we study the exit problem of the diffusion Z ,1,N from the domain D with large N and small .We recall the equation satisfied by each particle And the whole system where the potential Υ N is defined in (0.4).We observe that the exit problem of Z ,1,N from D is equivalent to the one of Z ,N from D × R d(N −1) .Furthermore, the diffusion Z ,N is homogeneous.The domain D satisfies Assumptions 1.5-1.6.However, nothing ensures us that the domain D × R d(N −1) satisfies Assumption A.1, described in the appendix.Assumption A.2 is obvious since the potential Υ N is convex due to the convexity of both V and F .It is then necessary and sufficient to prove the stability of D × R d(N −1) by −N ∇Υ N for applying the Freidlin-Wentzell theory.We recall that the notion of "stable by" has been introduced in Definition 1.2.
As remarked previously, the drift term −∇V − ∇F * u s is close to −∇V − ∇F * δ 0 for s sufficiently large.The propagation of chaos implies that −∇V − ∇F * 1 N N j=1 δ Z ,j,N s tends also to −∇V − ∇F * δ 0 .Heuristically, since D is stable by −∇V − ∇F , we can imagine that it is stable by −∇V − ∇F * ν for all the measures ν sufficiently close to δ 0 .This would imply that D × R d(N −1) B N κ is stable by −N ∇Υ N for κ sufficiently small.
Of course, this does not have any reason to be true.Consequently, we consider two sequences of sets which frame the domain and which satisfy Assumption 1.6.Let us consider κ > 0. We recall that 2n = deg(G), see (F-1)-(F-2).2. For all the measures µ, W µ is equal to V + F * µ.
3. For all ν ∈ (B ∞ κ ) R+ =: M ∞ κ and for all x ∈ R d , we also introduce the dynamical system: 4. Let r be an increasing function from R + to itself such that r(0) = 0.This function is chosen subsequently, see Section 3.For all κ > 0, we introduce the following two domains: and D e,κ := Proof.
Step 1.Let µ be a measure in B ∞ κ .We note that, by applying Lemma 1.1 in [17], the drift ∇F * µ is the product of x with a polynomial function of degree 2n − 2 of ||x|| and with a finite number of parameters of the form: x ; e i li ||x|| l0 µ(dx) Thereby, for any compact set K which contains D, there exists f (κ) which tends to 0 when κ goes to 0 such that Step 2. Let x 0 be an element of D. Let us prove that x 0 ∈ D i,κ when κ is small enough.We introduce the dynamical system ψ(x 0 ): We remark that ψ t (x 0 ) ∈ K for all t ≥ 0. We recall From now on, we take ) for all t ≥ 0 and for all ν ∈ M ∞ κ .This means that x 0 ∈ D i,κ for κ small enough.
Step 3. We now prove lim κ→0 sup z∈De,κ d (z ; D) = 0. Let x 0 be a point in R d satisfying d (x 0 ; D) ≤ r(κ).There exists y 0 ∈ D such that d(x 0 , y 0 ) ≤ r(2κ).We study the two dynamical systems: Since Hess W ≥ θ, the function t → d(ψ t (x 0 ), ψ t (y 0 )) is nonincreasing.This means d(ψ t (x 0 ), ψ t (y 0 )) ≤ r(2κ) for all t ≥ 0. By proceeding like in Step 2, the distance d(ψ t (x 0 ), θ .Hence: We deduce that We define the two domains to which we will apply Freidlin-Wentzell theory: First, let us prove that the ball B N κ is stable by −N ∇Υ N .It is not an obvious consequence of the convexity of Υ N because the norm ||| .||| does not derive from a scalar product.

Lemma 2.3. The open domain B N
κ is stable by −N ∇Υ N .Moreover, its exit cost goes to infinity when N goes to infinity: ∈ R dN and we consider the deterministic dynamical system already introduced in (1.6) Step 1 in the proof of Proposition 1.3, we can prove: . Consequently, the ball B N κ is stable by −N ∇Υ N .
Step 2. We now compute the exit cost.Hypotheses (V-2) and (F-1) imply Lemma 2.4.Let O be a bounded domain which contains 0. We have: Proof.We study the function ξ z from R d(N −1) to R: and the unique minimizer is This implies the existence of a continuous function f N 1 satisfying lim Simple computations imply κ .However, by definition, 0 .Since D i,κ is stable by −∇V − ∇F * µ N t for all t ≥ 0, we deduce that Z 1 t ∈ D i,κ for all t ≥ 0. This finishes to prove the stability of D (N ) i,κ by −N ∇Υ N .We proceed in the same way with D (N ) e,κ .
We now define the exit times that we use.We recall that Assumption 1.5 is assumed.
Consequently, nothing forbides X 0 to be an element of D c .In this case, we introduce the first hitting time.
We already know that these times are less than a deterministic time with high probability for going to 0: Lemma 2.7.For all κ > 0, we have the limit where T 0 has been defined in Assumption 1.5.
Since 0 ∈ D, this result is an obvious consequence of Assumption 1.5, Proposition A.4 and Proposition 2.2.The proof is left to the reader.
We can now define the exit times.
Definition 2.8.We denote by the first exit time of the diffusion Z ,N defined in and τ 1,N e,κ ( ) : the first exit time of the diffusion Z ,N from D e,κ × R d(N −1) .
We recall that we can not apply Freidlin-Wentzell theory directly to the two domains 1) and D e,κ × R d(N −1) .Consequently, we introduce two other exit times.Definition 2.9.We denote by the first exit time of the diffusion Z ,N from D the first exit time of the diffusion Z ,N from D We have all the ingredients in order to obtain the exit times.
Furthermore, we have information on the exit location.Indeed, for all N ⊂ ∂D e,κ such that inf z∈N W (z) > inf z∈∂De,κ W (z), we have: for κ small enough and N large enough.
Proof.Outline First, we prove that the whole system Z ,N enters with high probability before a time T κ (finite, independent of N , independent of and deterministic) in the domain B N κ .Next, we prove that the system does not exit from D e,κ × R d(N −1) before this time T κ with probability close to 1.
The set D e,κ is stable by −N ∇Υ N .We apply Freidlin-Wentzell theory.Finally, we prove that the diffusion Z ,N exits from the domain D e,κ × R d(N −1) before exiting from B N κ .
Step 1.We recall the dynamical system introduced in (1.6): As Z N 0 = X 0 , we deduce that for all t ≥ 0, Hypotheses (V-2) and (V-3) imply the convergence of Z N to 0 and there exists T κ , deterministic and independent from N such that We assume without any loss of generality that T κ ≥ T 0 + 1 where T 0 is defined in Lemma 2.7.Proposition A.4 and Lemma 2.7 allow to obtain the following limits: (2.6) Step 2. From now on, we consider the new exit time: The limits in (2.6) and in (2.7) imply Step 3. We now compute the exit cost H N κ .By definition, Υ N (Z) ; N inf Lemmas 2.3 and 2.4 imply that H N κ converges to inf z∈∂De,κ W (z) when N goes to infinity.
Finally, the continuity of the function W and Proposition 2.2 imply the convergence of inf z∈∂De,κ W (z) to H when κ tends to 0. By taking κ sufficiently small, then N sufficiently large, we obtain H N κ − H < δ 2 which ends the proof of (2.4).
Step 4. We now prove that the two exit times T N e,κ ( ) and τ 1,N e,κ ( ) are equal with probability close to 1 for N large enough and small enough.We just remark that inf for N large enough, and we apply (A.3) of Proposition A.3.
Step 5.By applying Lemma 2.4, we have An analogous result holds with D i,κ .We do not give the proof since it is similar to the previous one.Proposition 2.11.For all δ > 0, there exists κ 0 such that for all 0 < κ < κ 0 and for all N large enough, the following limit holds: W (z), we have: if κ is small enough and if N is sufficiently large.
Proposition 2.10 and Proposition 2.11 allow to obtain the results on D.
Corollary 2.12.By τ 1,N ( ), we denote the exit time of the diffusion Z ,1,N from the domain D. For all ρ > 0, there exists N 0 ≥ 2 such that for all N ≥ N 0 , we have the following limit: where H is like in in Definition 1.7: Furthermore, for all N ⊂ ∂D such that inf z∈N W (z) > inf z∈∂D W (z), there exists N 1 ≥ 2 such that for all N ≥ N 1 , we have: (2.9) Proof.
Step 1.For all κ > 0, Z ,1,N needs to exit from D i,κ before exiting from D.
Consequently, for all ρ > 0, we have: We apply Proposition 2.11 by taking κ sufficiently small and N large enough.This implies the convergence of P τ 1,N i,κ ( ) ≤ exp 2 (H − ρ) to 0 when goes to 0 ; if N is large enough.
Step 2. If Z ,1,N does not exit from D, it does not exit from D e,κ .We apply Proposition 2.10 by taking κ sufficiently small and N large enough.It implies for N large enough.
Step 3. By definition of N , there exists ξ > 0 such that inf z∈N W (z) = H + 3ξ.In order to prove (2.9), we introduce the set By Lemma 1.8, the domain K H+2ξ satisfies Assumptions 1.5-1.6.Then, we can apply (2.8) to K H+2ξ .We denote τ 1,N ξ ( ) the first exit time of Z ,1,N from K H+2ξ .We immediately have: for all ρ > 0 and for N large enough.By construction of K H+2ξ , we have N ⊂ K c H+2ξ .This implies: The limit (2.10) with ρ = ξ implies the convergence to 0 of the first term as going to 0. By applying (2.8), the second term goes to 0 when tends to 0.

Strong propagation of chaos
It is well known that the two diffusions X and Z ,1,N , defined by (0.1) and (0.2), are close.Indeed, propagation of chaos holds: there exist K > 0 and M > 0 such that Exit problem See Appendix B for the proof of the first statement.These two inequalities have strong restrictions.In the first one, the supremum is not under the expectation.Consequently, if τ is a (not necessary bounded) stopping time, nothing tells us that the quantity E X τ − Z ,1,N τ 2 tends to 0 when N goes to infinity.
Note that this cannot be deduced from the second inequality since the supremum is restricted to a fixed and finite interval.
However, by Proposition A.4, we know that the exit time of X from a domain D which satisfies both Assumptions 1.5-1.6 goes to infinity when tends to 0.
From now on, we consider a compact convex set K ⊂ R d which contains 0 and X 0 .
We introduce the following exit times.
Definition 3.1.By τ ( ) (resp.by τ 1,N ( )), we denote the first exit time of the diffusion X (resp.Z ,1,N ) from the compact set K. The first exit time of the whole system Z ,N from the ball B N κ is denoted by τ N κ ( ), where κ > 0.
We now introduce From now on, we use the function r: .
By Lemma 1.1 in [17], ∇F * µ is the product of x with a polynomial function of degree 2n − 2 of ||x|| and with a finite number of parameters of the form: x ; e i li ||x|| l0 µ(dx) , for some constant C > 0. Consequently, the quantity r(κ) goes to 0 when κ tends to 0.
The following result tells us that the propagation of chaos is uniform on 0 ; T N κ ( ) .
Step 1.By Proposition 1.3, there exist 1 > 0 and a time T κ which is deterministic and independent from N and such that for all t ≥ 0 and < 1 .Furthermore, by Proposition B.3, there exists 2 > 0 such that sup for N large enough.Note that (3.2) holds in the small-noise case, uniformly with respect to N .Also, (3.3) is true for N large enough, uniformly with respect to .
Step 2. We denote µ ,N t . Recall that W µ := V + F * µ for all the measures µ and that B ∞ κ denotes the set of all the measures µ such that ||x|| 2n µ(dx) < κ 2n .The assumptions on V and F imply Hess W µ ≥ θ > 0. From now on, we put ξ N (t) := Tκ ∈ K and Z ,N Tκ ∈ B N κ then, for all T κ ≤ t ≤ T N κ ( ), we have: where we have set Inequality (3.4) directly implies: which together with (3.3) yields

E sup
Tκ≤t≤T N κ ( ) (3.6) The claim thus follows from the Markov inequality.
This theorem links the exit time of X with the one of Z ,1,N .It also shows that the McKean-Vlasov diffusion is a good approximation (even in the long time) of the first particle in a mean-field system in the small-noise limit.Let us point out that the only use of the convexity was in the inequality E ||X t || 2n ≤ κ 2n for all t ≥ T κ .

Exit problem of the McKean-Vlasov diffusion
In this section, we provide our main results: the exit time and the exit location of the McKean-Vlasov diffusion.
Let us consider a domain D ⊂ R d satisfying Assumptions 1.5-1.6.By τ ( ), we denote the first exit time of the diffusion (0.1) from the domain D. Let K be a compact set which contains D and such that d (D ; K c ) ≥ 1.Let us recall that τ 1,N i,κ ( ) (resp.τ 1,N e,κ ( )) is the first exit time of Z ,1,N from D i,κ (resp.D e,κ ).Set τ N κ ( ) to be the exit time of the diffusion Z ,N from the domain B N κ .Finally, we denote T N κ ( ) := min τ ( ) ; τ 1,N e,κ ( ) ; τ N κ ( ) .
Step 2. We prove here the upper bound: Step 2.1.By Proposition 2.10, there exists κ 1 > 0 such that for all 0 < κ < κ 1 and N large enough Therefore, the first term a N ( ) tends to 0 as goes to 0.
Step 2.2.Let us look at the third term c N ( ).For κ sufficiently small, we have D e,κ ⊂ K.
On the event where δ(κ) denotes the distance between D and D c e,κ .By construction, we have δ(κ) ≥ r(κ).According to Theorem 3.2, there exist N 0 ≥ 2 and 0 > 0 such that for all N ≥ N 0 and < 0 . Step As an immediate application of Theorem 3.2 and Theorem 4.1, we obtain a good approximation of the self-stabilizing process on unbounded family of intervals: Corollary 4.2.Let H and ρ be two positive real numbers.For all δ > 0, there exist N δ ∈ N * and δ > 0 such that sup It satisfies Assumptions 1.5-1.6 by Lemma 1.8.For κ > 0 suffficiently small, Inequality (3.6) gives the existence of N 0 ∈ N * and 0 > 0 such that for all N ≥ N 0 and < 0 , where T N κ ( ) := inf τ ( ) ; τ 1,N ( ) ; τ N κ ( ) .Here τ ( ) (resp.τ 1,N ( )) is the first exit time of the diffusion X (resp.Z ,1,N ) from K H 2 +1 and τ N κ ( ) is the first exit time of the whole system Z ,N from the ball B N κ .For N large enough, Lemma 2.3 implies The domain K H 2 +1 satisfies Assumptions 1.5-1.6 so we can apply Theorem 4.1 and Corollary 2.12 to deduce that for all ξ > 0,  This ends the proof.
We now provide the result on the exit location.Proof.The proof is similar to the Step 3 of the proof of Corollary 2.12.
By definition of N , there exists ξ > 0 such that inf z∈N W (z) = H + 3ξ.We introduce By Lemma 1.8, the domain K H+2ξ satisfies Assumptions 1.5-1.6.Then, we can apply Remark 4.4.Note that we have not used convexity of V in the whole space R d .We have used the convexity in a compact set which contains the point of the global minimum 0 and the captivity of the law u t in a small ball which contains δ 0 .This means that it is possible to characterize the exit time and the exit location even if V is not convex by using the new approach of this paper.

A Freidlin-Wentzell Theory
Here we present the main results of the Freidlin-Wentzell theory.We restrict ourselves to a simple case in R k , k ≥ 1.We consider a homogeneous diffusion x : with ρ j (i) := ∇F X ,i t − X ,j t − ∇F * u t Z ,i,N t .
Taking the conditional expectation with respect to Z ,i,N t and then to Z ,j,N t , we obtain E ρ j (i)ρ k (i) = 0 for j = k.Since ξ i (0) = 0, This inequality holds uniformly with respect to 0 < < 0 .
Let us note that this uniform propagation of chaos would not hold if V was not convex.But it is true even if V is not uniformly strictly convex which means that the Hessian of V is not necessary definite positive.Then, there exists K > 0 such that We can also remark that the supremum is not under the expectation.However, such a result is available on a finite interval (even if V is not convex): The model is detailed subsequently.Let us present what we denote by exit problem.We consider a domain D ⊂ R d and we introduce S( ) := inf {t ≥ 0 | X t ∈ D} Exit problem the first hitting time of X in the domain D.Then, we define τ ( ) := inf {t ≥ S( ) | X t / ∈ D} the first exit time of X from the domain D. The exit problem consists of two questions.

1 n κ 2
which converges to infinity when N goes to infinity.Before looking at the sets D (N ) i,κ and D (N ) e,κ , we compute the exit cost of a set of the form O × R d(N −1) .

Let us note that lim N →∞ sup z∈∂O f N 2 Lemma 2 . 5 .
(z) = 0 since ∂O is bounded.This ends the proof.Now we study the two sets D (N ) i,κ and D (N ) e,κ .The two domains D (N ) i,κ and D (N ) e,κ are stable by −N ∇Υ N .Proof.Let Z 1 0 , • • • , Z N 0 be an element of D (N ) i,κ .By definition, it is in B N κ .The stability of the ball B

.
is stable by −N ∇Υ N according to Proposition 2.5.We apply Proposition A.3 to D (N ) e,κ and we obtain

W
(z)  for N large enough.Applying Proposition A.3 for N large enough leads to (2.5).

x t = x 0 + √ B t − t 0 ∇U
(x s ) ds ,(A.1)To deal with the last term, we apply the Cauchy-Schwarz inequality:
2.3.Let us look at the second term b N ( ).By Lemma 2.3, Let ξ > 0. By taking first κ small enough and then N large enough, we obtain Step 2.4.