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Electron. J. Probab. 17 (2012), no. 76, 1–26.
ISSN: 1083-6489 DOI: 10.1214/EJP.v17-1914
Exit problem of McKean-Vlasov diffusions
in convex landscapes∗
Julian Tugaut†
Abstract
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 [6] 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.
Keywords: Self-stabilizing diffusion ; Exit time ; Exit location ; Large deviations ; Interacting
particle systems ; Propagation of chaos ; Granular media equation.
AMS MSC 2010: Primary 60F10, Secondary 60J60 ; 60H10 ; 82C22.
Submitted to EJP on March 30, 2012, final version accepted on August 20, 2012.
Introduction
The questions that we address in this paper are about the pathwise asymptotic behavior of a particular class of inhomogeneous diffusions:
Xt = X0 +
√
Z
Bt −
t
b (s, Xs ) ds .
0
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 − Xt )} .
The model is detailed subsequently. Let us present what we denote by exit problem. We
consider a domain D ⊂ Rd and we introduce
S() := inf {t ≥ 0 | Xt ∈ D}
∗ Supported by the DFG-funded CRC 701, Spectral Structures and Topological Methods in Mathematics.
† University of Bielefeld, Germany E-mail: jtugaut@math.uni-bielefeld.de
Exit problem
the first hitting time of X in the domain D . Then, we define
τ () := inf {t ≥ S() | Xt ∈
/ D}
the first exit time of X from the domain D . The exit problem consists of two questions.
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 Xτ () 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
xt
= x0 +
√
Z
βt −
t
∇U (xs ) ds .
0
U is a C ∞ -continuous function from Rk (k ≥ 1) to R and β is a Brownian motion in
Rk . a0 is a minimizer of U and G is a domain which contains a0 . Under easy to check
assumptions (which are detailed in Appendix A), for all δ > 0, the following Kramers’
type law holds:
2
2
lim P exp
(H − δ) < τ () < exp
(H + δ)
= 1.
→0
Here, the exit cost is H :=
inf U (z) − U (a0 ). We immediately remark that H =
z∈∂G
1
lim log {τ ()}. Moreover, the exit location is near the points of the boundary which
2 →0
minimize U . Indeed,
n
o
lim P xτ () ∈ N = 0
→0
if N ⊂ ∂D is such that inf U (z) > H .
z∈N
Let us note that we also have results if we replace Brownian motion by a Lévy process, see [11, 10].
Let us present more precisely the model studied in the article. Let X0 be an element
of Rd , d ≥ 1. We consider the McKean-Vlasov diffusion
Rt
√
Xt = X0 + Bt − 0 ∇Ws (Xs ) ds
.
Wt := V + F ∗ ut := V + F ∗ L (Xt )
(0.1)
The star in the previous line corresponds to a convolution and ut is the own law of the
diffusion X at time t. Let us point out that E {∇Wt (Xt )} is not equal to E {∇V (Xt )}.
It is equal to E {∇V (Xt ) + ∇F (Xt − Yt )} 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 Rd with intensity 2 d. It allows X to visit the whole space. The second
force describes the attraction between one trajectory t 7→ Xt (ω0 ) and the whole set of
R
trajectories. Indeed, we notice: ∇F ∗ ut (Xt (ω0 )) = ω∈Ω ∇F (Xt (ω0 ) − Xt (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
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Exit problem
past intervenes in the drift ∇Wt through the law ut . 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:
Zt,i,N
= X0 +
√
Bti
Z
t
−
Zs,i,N
∇V
0
N Z
1 X t
∇F Zs,i,N − Zs,j,N ds
ds −
N j=1 0
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):
Wt,N
Heuristically, the empirical law
:= V + F ∗
N
1 X
δ ,i,N
N i=1 Zt
!
.
(0.2)
N
1 X
δ ,i,N of the system converges to ut as N tends
N i=1 Zt
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:
(
lim E
N →∞
2
sup Xt − Zt,1,N )
=0
t∈[0;T ]
for all T ∈ R+ . See [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
Z ,N := Z ,1,N , · · · , Z ,N,N , B N := B 1 , · · · , B N and Z0N := (X0 , · · · , X0 ), equation
(0.2) can be rewritten
Zt,N = Z0N +
√
BtN − N
Z
t
∇ΥN Zs,N ds
(0.3)
0
where the potential ΥN is defined by
N
N N
1 X
1 XX
Υ (Z) :=
V (Zj ) +
F (Zi − Zj )
N j=1
2N 2 i=1 j=1
N
(0.4)
N
for all Z := (Z1 , · · · , ZN ) ∈ Rd . 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 × Rd(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
EJP 17 (2012), paper 76.
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Exit problem
inf Hess V (x) ≥ θ > 0 and inf Hess F (x) ≥ α > 0, they prove a Kramers’ type law. The
x∈Rd
x∈Rd
exit time τ () of X from a domain D satisfies the limit:
2
2
lim P exp
(H − δ) ≤ τ () ≤ exp
(H + δ)
=1
→0
for all δ > 0. Here, H := inf (V + F ∗ δa0 ) − V (a0 ) where a0 is the unique minimizer of
∂D
V . They also provide a result on the exit location which is similar to the one of FreidlinWentzell. 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.
1
Preliminaries and main results
2
Pd
First, let us denote by || . || the euclidian norm on Rd : ||x|| :=
(x1 , · · · , xd ) ∈ Rd . The associated distance is d.
We assume the following properties on the confining potential V :
r=1
x2r for all x =
(V-1) V is a smooth function from Rd to R.
(V-2) V is uniformly strictly convex: Hess V ≥ θ > 0.
(V-3) The unique minimizer of V is 0 and V (0) = 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 a0 6= 0, it is sufficient to consider
e := X − a0 and the potential Ve := V (. + a0 ) − V (a0 ). An immediate
the diffusion X
consequence of (V-1)–(V-3) is the following inequality:
2
hx ; ∇V (x)i ≥ θ ||x||
for all x ∈ Rd .
(1.1)
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||).
(F-2) G is an even polynomial function such that deg(G) =: 2n ≥ 2.
(F-3) G is convex.
(F-4) G(0) = 0.
Let us note that (F-1)–(F-4) imply
∇F (x) = x
n
X
G0 (||x||)
G(2k) (0)
2k−2
=x
||x||
.
||x||
(2k − 1)!
k=1
EJP 17 (2012), paper 76.
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Exit problem
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:
n
o
2p
sup E ||Xt ||
<∞
(1.2)
t∈R+
for all p ∈ N∗ . We immediately deduce the tightness of the family (ut )t∈R+ .
We now present some notations concerning the space Rd
N
=: RdN .
Definition 1.1. 1. For all Z = (Z1 , · · · , ZN ) ∈ RdN , we define the following norm:
(
|||Z||| :=
N
1 X
2n
||Zi ||
N i=1
1
) 2n
.
2. For all κ > 0, we introduce the ball:
o
n
dN BN
:=
Z
∈
R
|||Z||| < κ .
κ
3. Finally, for all x ∈ Rd , the vector (x, · · · , x) ∈ RdN is denoted by x.
We remark that |||x||| = ||x|| for all x ∈ Rd . In order to simplify the writing, we use
the following terminology in the whole article:
Definition 1.2. Let G be a subset of Rk and let U be a C ∞ -continuous function from Rk
to R. For all x ∈ Rk , we consider the dynamical system
Z
ψt (x) = x −
t
∇U (ψs (x)) ds .
0
We say that the domain G is stable by −∇U if the orbit {ψt (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 ut 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:
n n
o
2n − 1
2n
2n
sup E ||Xt ||
≤ max ||X0 || ;
n .
2θ
t∈R+
(1.3)
2. For all κ > 0 and > 0, we introduce the deterministic time
n
n
o
o
2n
Tκ () := min t ≥ 0 E ||Xt ||
≤ κ2n .
For <
κ2 θ
2n−1 ,
we have the inequality:
1
2n
||X0 || .
nθκ2n
n
o
2n
3. Moreover, for all t ≥ Tκ (), E ||Xt ||
≤ κ2n .
Tκ () ≤
EJP 17 (2012), paper 76.
(1.4)
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Exit problem
Proof. 1. After applying the Itô formula and integrating, we obtain
2n
||Xt ||
√
2n
Z
t
2n−2
||Xs ||
hXt ; dBs i
= ||X0 || + 2n 0
Z t
n
o
2n−2
||Xs ||
hXt ; ∇V (Xs )i + hXs ; ∇F ∗ us (Xs )i ds
− 2n
0
Z t
2n−2
||Xs ||
ds .
+ n(2n − 1)
0
n
o
2n
We put ξ (t) := E ||Xt ||
. The previous equality implies:
n
o
2n−2
ξ0 (t) = − 2nE ||Xt ||
hXt ; ∇V (Xt )i
n
o
n
o
2n−2
2n−2
− 2nE ||Xt ||
hXt ; ∇F ∗ ut (Xt )i + n(2n − 1)E ||Xt ||
=: a (t) + b (t) + c (t) .
By definition, the second term b (t) can be written as
h
i
2n−2
b (t) = E ||Xt ||
hXt ; ∇F (Xt − Yt )i
where Y is a solution of (0.1) independent from X . We can exchange X and Y .
Thereby, by using (F-1)–(F-4), we get:
E
G0 (||Xt − Yt ||) D 2n−2 ||X
||
X
;
X
−
Y
t
t
t
t
||X − Y ||
0 t t D
E
1
G (||Xt − Yt ||)
2n−2
2n−2
X
||X
||
−
Y
||Y
||
;
X
−
Y
= E
.
t
t
t
t
t
t
2
||Xt − Yt ||
b (t) = E
This last term is nonnegative. Indeed, the Cauchy-Schwarz inequality implies
D
2n−2
x ||x||
2n−2
− y ||y||
E 2n−1
2n−1
; x − y ≥ ||x||
− ||y||
(||x|| − ||y||) ≥ 0
n
2n−1
for all x, y ∈ Rd . Therefore, we obtain b (t) = E ||Xt ||
o
∇F ∗ ut (Xt ) ≥ 0.
Moreover, inequality (1.1) implies
n
i
n
o
2n−2
2n
a (t) = E ||Xt ||
hXt ; ∇V (Xt )i ≥ θE ||Xt ||
= θξ (t) .
1
Hence, by using Jensen inequality, we deduce c (t) ≤ n(2n − 1)ξ (t)1− 2n . By combining
results on a (t), b (t) and c (t), we obtain
1
ξ0 (t) ≤ −2nθξ (t) + n(2n − 1)ξ (t)1− n
1
1
(2n − 1)
1− n
n
≤ −2nθξ (t)
ξ (t) −
.
2θ
Inequality (1.3) is an obvious consequence of (1.5).
κ2 θ
2. From now on, we take < 2n−1
. This implies
t < Tκ (), we have
1
ξ (t) n ≥ κ2 ≥
κ2
2
>
(2n−1)
.
2θ
(1.5)
Consequently, for all
(2n − 1)
.
θ
We obtain from (1.5):
−ξ0 (t) ≥ nθκ2n .
EJP 17 (2012), paper 76.
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Exit problem
n
2n
o
Z
Tκ ()
By definition, if ξ (0) = E ||X0 ||
Z
Tκ ()
nθκ2n dt ≤
0
≥ κ2n :
2n
−ξ0 (t)dt = ξ (0) − κ2n ≤ ||X0 ||
.
0
(1.4) immediately holds.
3. Finally, for all T > 0, (1.5) implies sup ξ (t) ≤ max
ξ (T ) ;
t≥T
(2n − 1)
2θ
n . Then, for
all t ≥ Tκ (),
n n
o
(2n − 1)
2n
E ||Xt ||
≤ max ξ (Tκ ()) ;
≤ κ2n .
2θ
This means that the self-stabilizing process tends to be trapped in a ball with center
0. This result concerns the law ut and not the trajectories t 7→ Xt (ω). But it points out
the importance of δ0 in the study. Indeed, Proposition 1.3 implies
n
o
2n
lim lim sup E ||Xt ||
= 0.
→0
t→∞
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
p n
o
2p − 1
2p
2p
p
sup E ||Xt ||
≤ max ||X0 || ;
2θ
t∈R+
for all p ∈ N.
We now give the assumptions on the domain D .
Assumption 1.5. We consider the dynamical system
t
Z
ϕt = X0 −
∇V (ϕs ) ds
0
where X0 is introduced in (0.1). There exists T0 ≥ 0 such that {ϕT0 +t ; t > 0} is included
in D and the orbit {ϕt ; 0 < t < T0 } is included in D c .
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
ZtN
t
Z
∇ΥN ZsN ds
= X0 − N
(1.6)
0
N
where ΥN is defined in (0.4).
t ≥ 0. Then, by
NWe remarkthat Zt is equal Nto ϕt for all
Assumption 1.5, the orbit ZT0 +t ; t ∈ R+ is included in D ⊂ D × Rd(N −1) .
Let us note that this assumption is weaker than Assumption 4.1.i) in [6]. We now
present the other hypothesis:
Assumption 1.6. The open domain D is stable by −∇V − ∇F =: −∇W .
This hypothesis is natural according to Proposition 1.3. Indeed, the law ut is as close
as we want to δ0 . Consequently, the drift ∇V +∇F ∗ut is close to ∇V +∇F ∗δ0 = ∇V +∇F .
Next, we define the exit cost.
EJP 17 (2012), paper 76.
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Exit problem
Definition 1.7. The exit cost of a bounded domain D which contains 0 is
H := inf W (z)
z∈∂D
with W (z) := V (z) + F (z).
We now give an example of a domain satisfying both Assumptions 1.5–1.6.
Lemma 1.8. For all H > 0, the domain KH := x ∈ Rd | V (x) + F (x) < H
Assumptions 1.5–1.6. Moreover, its exit cost is H .
satisfies
Proof. Assumption 1.6 is obviously verified since KH is a level set of the potential V + F
and its exit cost is H by definition.
Let us prove the first hypothesis. We take any x ∈ Rd and we consider the dynamical
system
Z
t
∇V (ϕs (x)) ds .
ϕt (x) = x −
0
Since V is convex, ϕt (x) converges to 0 so there exists T0 ≥ 0 such that the orbit
c
{ϕt (x) ; t < T0 } is included in KH
. Let us show {ϕt (x) ; t > T0 } ⊂ KH . For this, it is
now sufficient to establish that KH is stable by −∇V :
d
W (ϕt (x)) = − h∇V (ϕt (x)) ; ∇V (ϕt (x))i − h∇V (ϕt (x)) ; ∇F (ϕt (x))i
dt
G0 (||ϕt (x)||)
2
= − ||∇V (ϕt (x))|| − ∇V (ϕt (x)) ; ϕt (x)
||ϕt (x)||
2
≤ − ||∇V (ϕt (x))|| − θG0 (||ϕt (x)||) ||ϕt (x)|| < 0 .
This finishes the proof.
Before giving the main results of the paper, we recall a simple fact.
Lemma 1.9. ΥN 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:
2
2
lim P exp
(H − ξ) < τ () < exp
(H + ξ)
=1
→0
with H := inf W (z) where the potential W is defined as W (z) := V (z) + F (z). Let N
z∈∂D
be a subset of ∂D such that inf W (z) > inf W (z). Then:
z∈N
z∈∂D
n
o
lim P Xτ () ∈ N = 0 .
→0
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
Nδ ∈ N∗ and δ > 0 such that:




,1,N sup sup P
sup
Xt − Zt
≥ ρ ≤ δ .
0≤t≤exp[ H ]

N ≥Nδ <δ
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.
EJP 17 (2012), paper 76.
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Exit problem
2
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
Zt,i,N =X0 +
√
Bti −
Z
t
∇V Zs,i,N ds
0
N Z
1 X t
∇F Zs,i,N − Zs,j,N ds .
−
N j=1 0
And the whole system Z ,N := Z ,1,N , · · · , Z ,N,N verifies
Zt,N
= X0 +
√
BtN − N
t
Z
∇ΥN Zs,N ds
0
where the potential ΥN 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 × Rd(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 × Rd(N −1) satisfies Assumption A.1, described in the appendix. Assumption
A.2 is obvious since the potential ΥN is convex due to the convexity of both V and F .
It is then necessary and sufficient to prove the stability of D × Rd(N −1) by −N ∇ΥN 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 ∗ us is close to −∇V
− ∇F ∗ δ0 for s
sufficiently large. The propagation of chaos implies that −∇V − ∇F ∗
1
N
PN
j=1 δZs,j,N
tends also to −∇V − ∇F ∗ δ0 . Heuristically, since D is stable by −∇V − ∇F , we can
imagine that it is stable by −∇V − ∇F
T ∗ νN for all the measuresNν sufficiently close to δ0 .
d(N −1)
This would imply that D × R
Bκ is stable by −N ∇Υ 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).
d
Definition 2.1. 1. B∞
κ denotes the set of all the probability measures µ on R satisfying
R
2n
2n
||x|| µ(dx) ≤ κ .
Rd
2. For all the measures µ, Wµ is equal to V + F ∗ µ.
3. For all ν ∈ (B∞
κ )
system:
R+
d
=: M∞
κ and for all x ∈ R , we also introduce the dynamical
ψtν (x) = x −
Z
t
∇Wνs (ψsν (x)) ds .
0
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:
Di,κ :=
and
De,κ
d (ψtν (x) ; Dc )
x ∈ D | inf∞ inf
> r(κ)
ν∈Mκ t∈R+
n
o
:= ψtν (x) | t ≥ 0, ν ∈ M∞
,
d
(x
,
D)
<
r(κ)
.
κ
(2.1)
(2.2)
Obviously, for all κ > 0, and for all µ ∈ B∞
κ , the two sets Di,κ and De,κ are stable by
−∇Wµ = −∇V − ∇F ∗ µ. Moreover, we have the inclusions
Di,κ2 ⊂ Di,κ1 ⊂ D ⊂ De,κ1 ⊂ De,κ2 ,
EJP 17 (2012), paper 76.
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Exit problem
for all 0 < κ1 < κ2 . More precisely:
c
d (Di,κ ; Dc ) ≥ r(κ) and d D ; De,κ
≥ r(κ) .
Now we justify why the two sets frame the open D .
Proposition 2.2. The following limits hold:
lim
sup d (z ; Dc ) = lim
κ→0 z∈∂Di,κ
sup d (z ; D) = 0 .
κ→0 z∈∂De,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:
d
Y
Z
C(l0 , l1 , · · · , ld ) :=
Rd
l
l
hx ; ei i i ||x|| 0 µ(dx)
i=1
Pd
2n
≤ 2n. The definition of B∞
for all
κ implies C(l0 , l1 , · · · , ld ) ≤ κ
l0 , · · · , ld ≥ 0 such that l0 + · · · + ld ≤ 2n. Thereby, for any compact set K which contains
D, there exists f (κ) which tends to 0 when κ goes to 0 such that
where l0 +
i=1 li
sup sup ||∇F ∗ µ(x) − ∇F (x)|| ≤ f (κ) .
µ∈B∞
κ x∈K
Moreover, (V-2) and (F-3) imply inf
inf Hess Wµ (x) ≥ θ for any compact set K as
x∈K µ∈B∞
κ
above.
Step 2. Let x0 be an element of D . Let us prove that x0 ∈ Di,κ when κ is small enough.
We introduce the dynamical system ψ(x0 ):
t
Z
ψt (x0 ) = x0 −
t
Z
∇V (ψs (x0 ))ds −
∇F (ψs (x0 ))ds .
0
0
We remark that ψt (x0 ) ∈ K for all t ≥ 0. We recall
ψtν (x0 ) = x0 −
Z
t
∇V (ψsν (x0 ))ds −
0
Z
t
∇F ∗ νs (ψsν (x0 ))ds .
0
Assumption 1.6 implies that δ(x0 ) := inf d (ψt (x0 ) ; D c ) > 0. From now on, we take
t≥0
ν
r(κ) < δ(x40 ) . We introduce ξt (x0 ) := ||ψtν (x0 ) − ψt (x0 )||. Then, for all ν ∈ M∞
κ , if ψt (x0 ) ∈
K , we get
d
ξt (x0 )2 = − 2 h∇Wνt (ψtν (x0 )) − ∇W (ψt (x0 )) ; ψtν (x0 ) − ψt (x0 )i
dt
= − 2 h∇Wνt (ψtν (x0 )) − ∇Wνt (ψt (x0 )) ; ψtν (x0 ) − ψt (x0 )i
− 2 h∇Wνt (ψt (x0 )) − ∇W (ψt (x0 )) ; ψtν (x0 ) − ψt (x0 )i
≤ − 2θξt (x0 )2 + 2ξt (x0 ) sup sup ||∇F ∗ µ(x) − ∇F (x)||
µ∈B∞
κ x∈K
≤2ξt (x0 ) {f (κ) − θξt (x0 )} .
By taking κ sufficiently small, d (ψtν (x0 ) ; ψt (x0 )) ≤
(2.3)
δ(x0 )
2
for all 0 ≤ t ≤ τK with τK :=
δ(x0 )
for all t ≤ τK . This
2
ν
c
= ∞ and inf d (ψt (x0 ) ; D ) ≥ r(2κ) for all t ≥ 0 and for all ν ∈ M∞
κ . This
inf {t ≥ 0 | ψtν (x0 ) ∈
/ K}. We deduce: inf d (ψtν (x0 ) ; Dc ) ≥
t≥0
implies τK
t≥0
means that x0 ∈ Di,κ for κ small enough.
EJP 17 (2012), paper 76.
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Exit problem
Step 3. We now prove lim sup d (z ; D) = 0. Let x0 be a point in Rd satisfying
κ→0 z∈De,κ
d (x0 ; D) ≤ r(κ). There exists y0 ∈ D such that d(x0 , y0 ) ≤ r(2κ). We study the two
dynamical systems:
Z
t
ψt (x0 ) = x0 −
t
Z
∇W (ψs (x0 ))ds and ψt (y0 ) = y0 −
0
∇W (ψs (y0 ))ds .
0
Since Hess W ≥ θ , the function t 7→ d(ψt (x0 ), ψt (y0 )) is nonincreasing. This means
d(ψt (x0 ), ψt (y0 )) ≤ r(2κ) for all t ≥ 0. By proceeding like in Step 2, the distance
d(ψt (x0 ), ψtν (x0 )) is less than f (κ)
θ . Hence:
d (ψtν (x0 ), ψt (y0 )) ≤ d (ψtν (x0 ), ψt (x0 )) + d (ψt (x0 ), ψt (y0 )) ≤
sup d (ψtν (x0 ) ; D) → 0 as κ goes to 0. It implies the conver-
sup
We deduce that
f (κ)
+ r(2κ) .
θ
d(x ; D)≤r(κ) t∈R+
gence of sup d (z ; D) to 0 when κ tends to 0.
z∈De,κ
We define the two domains to which we will apply Freidlin-Wentzell theory:
\
(N )
Di,κ := Di,κ × Rd(N −1)
BN
κ
\
(N )
and De,κ
:= De,κ × Rd(N −1)
BN
κ .
N
First, let us prove that the ball BN
κ is stable by −N ∇Υ . It is not an obvious conseN
quence of the convexity of Υ because the norm ||| . ||| does not derive from a scalar
product.
N
Lemma 2.3. The open domain BN
κ is stable by −N ∇Υ . Moreover, its exit cost goes
to infinity when N goes to infinity:
lim
inf
N →+∞ Z∈∂BN
κ
N ΥN (Z) = +∞ .
Proof. Step 1. We take Z0N := Z01 , · · · , Z0N ∈ RdN and we consider the deterministic
dynamical system already introduced in (1.6)
ZtN
=
Z0N
Z
−N
t
∇ΥN ZsN ds =: Zt1 , · · · , ZtN .
0
We recall that ΥN (Z1 , · · · , ZN ) =
2n
tion, ZtN :=
1
N
1
N
PN
i=1
V (Zi ) +
1
2N 2
PN PN
i=1
j=1
F (Zi − Zj ). By defini-
PN i 2n
. Then:
i=1 Zt
N
d N 2n
2n X i 2n−2 i
Zt
=−
Zt
Zt ; ∇V Zti
dt
N i=1
−
N N
E
2n X X i 2n−2 D i
Zt
Zt ; ∇F Zti − Ztj
N i=1 j=1
N
2n X i 2n−2 i
Zt
Zt ; ∇V Zti
N i=1
N N n X X i 2n−2 i j 2n−2 j
j
i
Zt ; ∇F Zt − Zt
−
Zt
Zt − Zt N i=1 j=1
=−
EJP 17 (2012), paper 76.
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Exit problem
Like in Step 1 in the proof of Proposition 1.3, we can prove:
N X
N X
i 2n−2 i j 2n−2 j
j
i
Zt Zt ; ∇F Zt − Zt
≥ 0.
Zt − Zt i=1 j=1
2n
2n
d N Hypothesis (V-2) implies dt
≤ −2nθ ZtN . Consequently, the ball BN
Zt
κ is
N
stable by −N ∇Υ .
Step 2. We now compute the exit cost. Hypotheses (V-2) and (F-1) imply
N
θX
θ 1
2
||Zi || ≥ N n
N ΥN (Z1 , · · · , ZN ) ≥
2 i=1
2
Consequently,
inf
Z∈∂BN
κ
N ΥN (Z) ≥
N
1 X
2n
||Zi ||
N i=1
! n1
.
θ 1 2
N n κ which converges to infinity when N goes to
2
infinity.
(N )
(N )
Before looking at the sets Di,κ and De,κ , we compute the exit cost of a set of the
form O × Rd(N −1) .
Lemma 2.4. Let O be a bounded domain which contains 0. We have:
lim
inf
N →∞ Z∈∂O×Rd(N −1)
N ΥN (Z) = inf (V (z) + F (z)) .
z∈∂O
Proof. We study the function ξz from Rd(N −1) to R:
ξz (x2 , · · · , xN ) := ΥN (z, x2 , · · · , xN ) .
N
d(N −1)
ξz is convex on Rd(N −1) and the unique minimizer is xN
where
0 (z), · · · , x0 (z) ∈ R
N
x0 (z) satisfies
1
∇F xN
∇V xN
0 (z) − z = 0 .
0 (z) +
N
This implies the existence of a continuous function f1N satisfying lim f1N (z) = 0 for all
N →∞
z ∈ Rd such that
xN
0 (z) =
f N (z)
1
−1
(Hess V (0)) ∇F (z) + 1
.
N
N
Simple computations imply
1
f N (z)
N
ΥN z, xN
{V (z) + F (z} + 2
0 (z), · · · , x0 (z) =
N
N
where f2N is a continuous function satisfying lim f2N (z) = 0 for all z ∈ Rd . Then:
N →∞
N
N
N ΥN (z, xN
0 (z), · · · , x0 (z)) = W (z) + f2 (z) .
Let us note that lim
sup f2N (z) = 0 since ∂O is bounded. This ends the proof.
N →∞ z∈∂O
(N )
(N )
Now we study the two sets Di,κ and De,κ .
(N )
(N )
Lemma 2.5. The two domains Di,κ and De,κ are stable by −N ∇ΥN .
EJP 17 (2012), paper 76.
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Exit problem
(N )
Proof. Let Z01 , · · · , Z0N be an element of Di,κ . By definition, it is in BN
κ . The stability
1
N
of the ball BN
∈ BN
κ proved in Lemma 2.3 implies Zt , · · · , Zt
κ for all t ≥ 0. Then,
P
N
1
∞
N
∞
N
µt := N j=1 δZ j ∈ Bκ for all t ≥ 0 which means µ ∈ Mκ . However, by definition,
t
N
Zt1 = ψtµ Z01 . Since Di,κ is stable by −∇V − ∇F ∗ µN
t for all t ≥ 0, we deduce that
(N )
1
Zt ∈ Di,κ for all t ≥ 0. This finishes to prove the stability of Di,κ by −N ∇ΥN . We
(N )
proceed in the same way with De,κ .
We now define the exit times that we use. We recall that Assumption 1.5 is assumed.
Consequently, nothing forbides X0 to be an element of D c . In this case, we introduce
the first hitting time.
1,N
1,N
()), we denote the first hitting time of the
Definition 2.6. By Si,κ () (resp. by Se,κ
diffusion Z ,N defined in (0.3)–(0.4) on the domain Di,κ ×Rd(N −1) (resp. De,κ ×Rd(N −1) ).
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
n
o
1,N
1,N
lim P Si,κ
() ≤ T0 + 1 = lim P Se,κ
() ≤ T0 + 1 = 1
→0
→0
where T0 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
n
o
1,N
1,N
τi,κ
() := inf t ≥ Si,κ
() | Zt,N ∈
/ Di,κ × Rd(N −1)
the first exit time of the diffusion Z ,N defined in (0.3)–(0.4) from Di,κ × Rd(N −1)
n
o
1,N
1,N
τe,κ
() := inf t ≥ Se,κ
() | Zt,N ∈
/ De,κ × Rd(N −1)
and
the first exit time of the diffusion Z ,N from De,κ × Rd(N −1) .
1,N
1,N
We remark that τi,κ () (resp. τe,κ
()) is the exit time of the diffusion Z ,1,N defined
in (0.2) from the domain Di,κ (resp. from the domain De,κ ).
We recall that we can not apply Freidlin-Wentzell theory directly to the two domains
Di,κ × Rd(N −1) and De,κ × Rd(N −1) . Consequently, we introduce two other exit times.
Definition 2.9. We denote by
n
o
(N )
1,N
N
Ti,κ
() := inf t ≥ Si,κ
() | Zt,N ∈
/ Di,κ
(N )
the first exit time of the diffusion Z ,N from Di,κ = Di,κ × Rd(N −1)
and
T
BN
κ
n
o
N
1,N
(N )
Te,κ
() := inf t ≥ Se,κ
() | Zt,N ∈
/ De,κ
(N )
the first exit time of the diffusion Z ,N from De,κ = De,κ × Rd(N −1)
T
BN
κ .
We have all the ingredients in order to obtain the exit times.
EJP 17 (2012), paper 76.
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Exit problem
Proposition 2.10. For all δ > 0, there exists κ0 such that for all 0 < κ < κ0 and for all
N large enough, the following limit holds:
2
2
1,N
lim P exp
(H − δ) < τe,κ () < exp
(H + δ)
=1
→0
with H := inf W (z) and
(2.4)
W (z) = V (z) + F (z) .
z∈∂D
Furthermore, we have information on the exit location. Indeed, for all N ⊂ ∂De,κ such
that inf W (z) > inf W (z), we have:
z∈N
z∈∂De,κ
n
o
lim P Zτ,1,N
∈
N
=0
1,N
()
→0
(2.5)
e,κ
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 Tκ (finite, independent of N , independent of and deterministic) in the
d(N −1)
domain BN
before
κ . Next, we prove that the system does not exit from De,κ × R
this time Tκ with probability close to 1.
(N )
The set De,κ is stable by −N ∇ΥN . We apply Freidlin-Wentzell theory. Finally, we
prove that the diffusion Z ,N exits from the domain De,κ × Rd(N −1) before exiting from
BN
κ .
Step 1. We recall the dynamical system introduced in (1.6):
ZtN
Z
t
= X0 − N
∇ΥN ZsN ds .
0
As Z0N = X0 , we deduce that for all t ≥ 0, ZtN = ψt (X0 ) with
Z
ψt (X0 ) = X0 −
t
∇V (ψs (X0 ))ds .
0
Hypotheses (V-2) and (V-3) imply the convergence of Z N to 0 and there exists Tκ , deterministic and independent from N such that
ZTNκ ∈ BN
κ .
We assume without any loss of generality that Tκ ≥ T0 + 1 where T0 is defined in Lemma
2.7. Proposition A.4 and Lemma 2.7 allow to obtain the following limits:
N
lim P Te,κ
() ≤ Tκ = 0 and
→0
n
o
N
lim P ZT,N
∈
B
= 1.
κ
κ
→0
(2.6)
Step 2. From now on, we consider the new exit time:
n
o
N
(N )
ηe,κ
() := inf t ≥ Tκ | Zt,N ∈
/ De,κ
.
(N )
The domain De,κ is stable by −N ∇ΥN according to Proposition 2.5. We apply Proposition A.3 to
(N )
De,κ
and we obtain
2
δ
2
δ
N
N
N
lim P exp
Hκ −
< ηe,κ () < exp
Hκ +
=1
→0
2
2
with
HκN := N
(2.7)
ΥN (Z) .
inf
(N )
Z∈∂De,κ
EJP 17 (2012), paper 76.
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Exit problem
The limits in (2.6) and in (2.7) imply
2
δ
2
δ
N
lim P exp
HκN −
< Te,κ
() < exp
HκN +
= 1.
→0
2
2
Step 3. We now compute the exit cost HκN . By definition,
HκN :=N
ΥN (Z)
inf
(N )
Z∈∂De,κ
= inf N
inf
Z∈∂De,κ ×Rd(N −1)
ΥN (Z) ; N
Lemmas 2.3 and 2.4 imply that HκN converges to
inf
inf
Z∈∂BN
κ
z∈∂De,κ
ΥN (Z)
.
W (z) when N goes to infinity.
Finally, the continuity of the function W and Proposition 2.2 imply the convergence of
inf W (z) to H when κ tends to 0. By taking κ sufficiently small, then N sufficiently
z∈∂De,κ
large, we obtain HκN − H < 2δ which ends the proof of (2.4).
N
1,N
Step 4. We now prove that the two exit times Te,κ
() and τe,κ
() are equal with probability close to 1 for N large enough and small enough. We just remark that
inf
N ΥN (Z) <
(N )
Z∈∂De,κ
inf
Z∈∂BN
κ
N ΥN (Z)
for N large enough, and we apply (A.3) of Proposition A.3.
Step 5. By applying Lemma 2.4, we have
inf
Z∈N ×Rd(N −1)
N ΥN (Z) >
(N )
if N ⊂ ∂De,κ such that inf W (z) >
z∈N
N ΥN (Z)
inf
Z∈∂De,κ
inf
z∈∂De,κ
W (z) for N large enough. Applying Proposi-
tion A.3 for N large enough leads to (2.5).
An analogous result holds with Di,κ . We do not give the proof since it is similar to
the previous one.
Proposition 2.11. For all δ > 0, there exists κ0 such that for all 0 < κ < κ0 and for all
N large enough, the following limit holds:
2
2
1,N
(H − δ) < τi,κ () < exp
(H + δ)
=1
lim P exp
→0
with H := inf W (z) and
W (z) := V (z) + F (z) .
z∈∂D
Furthermore, for all N ⊂ ∂Di,κ such that inf W (z) >
z∈N
lim P
→0
Zτ,1,N
1,N
i,κ ()
inf
z∈∂Di,κ
W (z), we have:
∈N
=0
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 N0 ≥ 2 such that for all N ≥ N0 , we have the
following limit:
2
2
lim P exp
(H − ρ) < τ 1,N () < exp
(H + ρ)
=1
→0
EJP 17 (2012), paper 76.
(2.8)
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Exit problem
where H is like in in Definition 1.7: H := inf (V (z) + F (z)).
z∈∂D
Furthermore, for all N ⊂ ∂D such that inf W (z) > inf W (z), there exists N1 ≥ 2
z∈N
z∈∂D
such that for all N ≥ N1 , we have:
n
o
lim P Zτ,1,N
= 0.
1,N () ∈ N
(2.9)
→0
Proof. Step 1. For all κ > 0, Z ,1,N needs to exit from Di,κ before exiting from D .
Consequently, for all ρ > 0, we have:
P τ
1,N
2
2
1,N
() ≤ exp
(H − ρ)
≤ P τi,κ () ≤ exp
(H − ρ)
.
We apply Proposition 2.11 byn taking κ sufficiently small
o and N large enough. This
1,N
implies the convergence of P τi,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 De,κ . We apply Proposition
2.10 by taking κ sufficiently small and N large enough. It implies
lim P τ
1,N
→0
2
() ≥ exp
(H + ρ)
=0
for N large enough.
Step 3. By definition of N , there exists ξ > 0 such that inf W (z) = H + 3ξ . In order to
z∈N
prove (2.9), we introduce the set
KH+2ξ := x ∈ Rd | W (x) < H + 2ξ .
By Lemma 1.8, the domain KH+2ξ satisfies Assumptions 1.5–1.6. Then, we can apply
1,N
(2.8) to KH+2ξ . We denote τξ () the first exit time of Z ,1,N from KH+2ξ . We immediately have:
2
2
lim P exp
(H + 2ξ − ρ) < τξ1,N () < exp
(H + 2ξ + ρ)
=1
→0
(2.10)
c
for all ρ > 0 and for N large enough. By construction of KH+2ξ , we have N ⊂ KH+2ξ
.
This implies:
o
n
o
n
/ KH+2ξ
≤P Zτ,1,N
P Zτ,1,N
1,N () ∈
1,N () ∈ N
n
o
≤P τξ1,N () ≤ τ 1,N ()
H +ξ
1,N
≤P τξ () ≤ exp
H +ξ
1,N
+ P exp
≤τ
() .
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.
3
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
2 K
,1,N sup E Xt − Zt
≤
N
t∈R+
(
)
2
MT
and E
sup Xt − Zt,1,N ≤
N
t∈[0;T ]
EJP 17 (2012), paper 76.
for all T > 0 .
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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,
necessary bounded) stopping time,
n if τ is a (not o
2
nothing tells us that the quantity E Xτ − Zτ,1,N 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 ⊂ Rd which contains 0 and X0 .
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
N
from the ball BN
κ is denoted by τκ (), where κ > 0.
We now introduce
TκN () := inf τ () ; τ 1,N () ; τκN () .
(3.1)
From now on, we use the function r :
(
r(κ) :=
) 13
2
sup sup ||∇F ∗ µ1 (x) − ∇F ∗ µ2 (x)||
.
θ µ1 ,µ2 ∈B∞
κ x∈K
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:
Z
C(l0 , l1 , · · · , ld ) :=
Rd
d
Y
l
l
hx ; ei i i ||x|| 0 µ(dx) ,
i=1
Pd
2n
where l0 + i=1 li ≤ 2n. If µ is in B∞
for some constant C > 0.
κ , |C(l0 , l1 , · · · , ld )| ≤ Cκ
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 ; TκN () .
Theorem 3.2. There exists κ0 such that for all κ < κ0 , there exists N0 (κ) ∈ N∗ and
0 (κ) > 0 such that
(
P
sup
0≤t≤TκN ()
)
,1,N
Xt − Zt
≥ r(κ) ≤ r(κ) ,
for all N ≥ N0 (κ) and for all < 0 (κ).
Proof. Step 1. By Proposition 1.3, there exist 1 > 0 and a time Tκ which is deterministic and independent from N and such that
n
2n o
E XT κ +t < κ2n
(3.2)
for all t ≥ 0 and < 1 . Furthermore, by Proposition B.3, there exists 2 > 0 such that
2 r(κ)3
,1,N sup E
sup Xt − Zt
≤
2
0<<2
0≤t≤Tκ
(3.3)
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 .
EJP 17 (2012), paper 76.
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Exit problem
,N
Step 2. We denote µt
:=
PN
1
N
i=1 δZt,i,N .
Recall that Wµ := V + F ∗ µ for all the
B∞
κ
R
2n
measures µ and that
denotes the set of all the measures µ such that ||x|| µ(dx) <
κ2n .
on V and F imply Hess Wµ ≥ θ > 0. From now on, we put ξN (t) :=
The assumptions
,1,N
,N
,1,N N
. If XT κ , ZTκ ∈ K and ZTκ ∈ BN
Xt − Zt
κ then, for all Tκ ≤ t ≤ Tκ (), we have:
D
E
d N 2
ξ (t) = − 2 ∇Wut (Xt ) − ∇Wµ,N Zt,1,N ; Xt − Zt,1,N
t
dt
E
D
= − 2 ∇Wut (Xt ) − ∇Wut Zt,1,N ; Xt − Zt,1,N
D
E
Zt,1,N ; Xt − Zt,1,N
− 2 ∇F ∗ ut Zt,1,N − ∇F ∗ µ,N
t
2
≤ − 2θ ξN (t) + 2ξN (t)fK (κ) ,
(3.4)
where we have set
fK (κ) :=
sup ||∇F ∗ µ1 (x) − ∇F ∗ µ2 (x)|| =
sup
µ1 ,µ2 ∈B∞
κ x∈K
θ
r(κ)3 .
2
Inequality (3.4) directly implies:
2
2 r(κ)3 ,1,N ,1,N ,
Xt − Zt
≤ max XT κ − ZTκ ;
2
Tκ ≤t≤TκN ()
sup
which together with (3.3) yields
(
E
sup
Tκ ≤t≤TκN ()
)
2
,1,N Xt − Zt
≤
r(κ)3
.
2
(3.5)
From (3.3), (3.5) and the inequality max{a, b} ≤ a + b for all a, b ∈ R+ , we obtain
(
E
sup
0≤t≤TκN ()
2
,1,N Xt − Zt
)
≤ r(κ)3 .
(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.
n
o Let us point out that the only
2n
use of the convexity was in the inequality E ||Xt ||
4
≤ κ2n for all t ≥ Tκ .
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 ⊂ Rd 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 ; Kc ) ≥ 1.
Theorem 4.1. For all ξ > 0
2
2
(H − ξ) < τ () < exp
(H + ξ)
=1
lim P exp
→0
where H := inf W (z) with W (z) := V (z) + F (z).
z∈∂D
EJP 17 (2012), paper 76.
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Exit problem
Proof. Step 1. Let κ > 0. According to Definition 2.1 and Proposition 2.2, there exist
two families of domains (Di,κ )κ>0 and (De,κ )κ>0 such that
• Di,κ ⊂ D ⊂ De,κ .
• Di,κ and De,κ are stable by −∇V − ∇F ∗ µ for all µ ∈ B∞
κ . The terminology “stable
by” has been introduced in Definition 1.2.
• sup d (z ; D c ) + sup d (z ; D) tends to 0 when κ goes to 0.
z∈∂Di,κ
z∈∂De,κ
1,N
1,N
Let us recall that τi,κ () (resp. τe,κ
()) is the first exit time of Z ,1,N from Di,κ (resp.
N
De,κ ). Set τκ () to be the
exit time of the diffusion
Z ,N from the domain BN
κ . Finally,
1,N
we denote TκN () := min τ () ; τe,κ
() ; τκN () .
Step 2. We prove here the upper bound:
H +ξ
P τ () ≥ exp
=
H +ξ
H +ξ
1,N
P τ () ≥ exp
; τe,κ () ≥ exp
H +ξ
H +ξ
1,N
N
1,N
; τe,κ () < exp
; τκ () ≤ τe,κ ()
+ P τ () ≥ exp
H +ξ
H +ξ
1,N
N
1,N
+ P τ () ≥ exp
; τe,κ () < exp
; τκ () > τe,κ ()
H +ξ
H +ξ
N
N
+ P τκ () ≤ exp
≤P τe,κ () ≥ exp
H +ξ
H +ξ
1,N
N
1,N
+ P τ () ≥ exp
; τe,κ () < exp
; τκ () > τe,κ ()
=: aN () + bN () + cN () .
Step 2.1. By Proposition 2.10, there exists κ1 > 0 such that for all 0 < κ < κ1 and N
large enough
1,N
lim P τe,κ
() < exp
→0
2
(H + ξ)
=0
Therefore, the first term aN () tends to 0 as goes to 0.
Step 2.2. Let us look at the third term cN (). For κ sufficiently small, we have De,κ ⊂ K.
On the event
τ () ≥ exp
H +ξ
1,N
; τe,κ
() ≤ exp
H +ξ
1,N
; τκN () > τe,κ
()
,
1,N
1,N
1,N
() ≤ TκN (). We deduce
() ≤ τ () and τe,κ
() ≤ τκN (). This implies τe,κ
we have τe,κ
that
H +ξ
H +ξ
1,N
N
1,N
P τ () ≥ exp
; τe,κ () ≤ exp
; τκ () > τe,κ ()
n
o
1,N
N
≤ P Xτ 1,N () − Zτ,1,N
≥
δ(κ)
;
τ
()
≤
T
()
1,N
e,κ
κ
e,κ
e,κ ()
(
)
,1,N ≤P
sup
Xt − Zt
≥ δ(κ) ,
0≤t≤TκN ()
c
where δ(κ) denotes the distance between D and De,κ
. By construction, we have δ(κ) ≥
r(κ). According to Theorem 3.2, there exist N0 ≥ 2 and 0 > 0 such that
(
P
sup
0≤t≤TκN ()
)
,1,N Xt − Zt
≥ r(κ) ≤ r(κ) ,
EJP 17 (2012), paper 76.
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Exit problem
for all N ≥ N0 and < 0 .
Step 2.3. Let us look at the second term bN (). By Lemma 2.3,
lim
inf
N →+∞ Z∈∂BN
κ
N ΥN (Z) = +∞ .
Consequently, for N large enough, we obtain
lim P
→0
τκN ()
H +ξ
≤ exp
= 0.
Step 2.4. Let ξ > 0. By taking first κ small enough and then N large enough, we obtain
the upper bound
lim P τ () ≥ exp
→0
H +ξ
= 0.
Step 3. Analogous arguments with Proposition 2.11 instead of Proposition 2.10 show
that
H −ξ
lim P τ () ≤ exp
= 0.
→0
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
Nδ ∈ N∗ and δ > 0 such that


,1,N sup sup P
sup
Xt − Zt
≥ ρ ≤ δ .

0≤t≤exp[ H ]
N ≥Nδ <δ


Proof. We introduce the set K H +1 := x ∈ Rd | V (x) + F (x) < H
+ 1 . It satisfies As2
2
sumptions 1.5–1.6 by Lemma 1.8. For κ > 0 suffficiently small, Inequality (3.6) gives
the existence of N0 ∈ N∗ and 0 > 0 such that for all N ≥ N0 and < 0 ,
(
)
2
δ
,1,N E
sup
≤ ,
Xt − Xt
2
0≤t≤TκN ()
where TκN () := inf τ () ; τ 1,N () ; τκN () . Here τ () (resp. τ 1,N ()) is the first exit time
of the diffusion X (resp. Z ,1,N ) from K H +1 and τκN () is the first exit time of the whole
2
system Z ,N from the ball BN
κ . For N large enough, Lemma 2.3 implies
H
lim P τκN () < exp
= 0.
→0
(4.1)
The domain K H +1 satisfies Assumptions 1.5–1.6 so we can apply Theorem 4.1 and Corol2
lary 2.12 to deduce that for all ξ > 0,
and
2 H
lim P τ () < exp
+1−ξ
=0
→0
2
2 H
+1−ξ
= 0.
lim P τ 1,N () < exp
→0
2
(4.2)
(4.3)
In particular, for ξ = 1, (4.1), (4.2) and (4.3) imply
H
δ
P TκN () < exp
<
2
EJP 17 (2012), paper 76.
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Exit problem
for small enough. Finally,




,1,N P
sup
≥ ρ
Xt − Zt

0≤t≤exp[ H ]
)
(
H
,1,N ≤P
sup
≤ δ.
≥ ρ + P TκN () < exp
Xt − Zt
N
0≤t≤Tκ ()
This ends the proof.
We now provide the result on the exit location.
Theorem 4.3. Let N be a subset of ∂D satisfying
inf W (z) > inf W (z) .
z∈N
z∈∂D
Then
n
o
lim P Xτ () ∈ N = 0 .
→0
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 W (z) = H + 3ξ . We introduce
z∈N
KH+2ξ := x ∈ Rd | W (x) < H + 2ξ .
By Lemma 1.8, the domain KH+2ξ satisfies Assumptions 1.5–1.6. Then, we can apply
(2.8) to KH+2ξ . If we denote by τξ () the first exit time of X from KH+2ξ , then we obtain
2
2
(H + 2ξ − ρ) < τξ () < exp
(H + 2ξ + ρ)
=1
lim P exp
→0
(4.4)
c
for all ρ > 0. By construction of KH+2ξ , N ⊂ KH+2ξ
, which implies
n
o
n
o
P Xτ () ∈ N ≤P Xτ () ∈
/ KH+2ξ
≤P {τξ () ≤ τ ()}
H +ξ
H +ξ
≤P τξ () ≤ exp
+ P exp
≤ τ () .
Applying (4.4) with ρ := ξ to the first term and Theorem 4.1 to the second one, we
obtain the result.
Remark 4.4. Note that we have not used convexity of V in the whole space Rd . We have
used the convexity in a compact set which contains the point of the global minimum 0
and the captivity of the law ut in a small ball which contains δ0 . 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 Rk , k ≥ 1. We consider a homogeneous diffusion x :
xt = x0 +
√
Z
Bt −
t
∇U (xs ) ds ,
(A.1)
0
EJP 17 (2012), paper 76.
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Exit problem
where x0 ∈ Rk , B is a Brownian motion and potential U ∈ C ∞ Rk . For a more general
setting and the proofs, the reader is referred to [4].
Let a0 be a minimizer of the potential U . Let G be an open domain which contains
x0 and a0 . τ () denotes the first exit time of the diffusion xt from the domain G . Let us
introduce the exit cost H :
H := inf U (z) − U (a0 ) .
z∈∂G
Define the deterministic dynamical system
Z
t
ϕt (x) = x −
∇U (ϕs (x)) ds .
0
We need two assumptions.
Assumption A.1. The unique critical point of U in the domain G is a0 . Moreover, for
all x ∈ G , ϕt (x) ∈ G for all t > 0 and lim ϕt (x) = a0 .
t→∞
Note that this asssumption is about the domain G and it is always true if G is the
basin of attraction of a0 .
Assumption A.2. All the trajectories of the deterministic system ϕt (x) with x ∈ ∂G
converge to a0 as t → ∞.
If U is convex on G then Asssumption A.2 is satisfied.
Assume that Assumptions A.1 and A.2 hold.
Proposition A.3. For all δ > 0, the following limit holds:
2
2
(H − δ) < τ () < exp
(H + δ)
= 1.
lim P exp
→0
(A.2)
Moreover, for each subset N ⊂ ∂G satisfying inf U (z) > inf U (z), we have:
z∈N
z∈∂G
n
o
lim P xτ () ∈ N = 0 .
(A.3)
→0
At the end, we recall a classical result of the theory of large deviations.
Proposition A.4. If
Fδ :=
d
z ∈ R | inf d (z ; ϕt (x0 )) ≤ δ
t≥0
for δ > 0 and τδ () denotes the first exit time of the diffusion x from the domain Fδ ,
then
lim P {τδ () < T } = 0 ,
→0
for all δ > 0 and T > 0.
By using Proposition A.4, we can improve the results of Proposition A.3 with domains
which do not satisfy Hypotheses A.1 and A.2.
Proposition A.5. Let us consider a domain G which satisfies Assumptions A.1 and A.2
and let x0 be a point in Rk such that x0 ∈
/ G . Also assume that ϕt (x0 ) converges to a0 as
t goes to infinity. Let T0 be the hitting time of G for the dynamical system ϕ (x0 ). If we
denote by
S() := inf {t ≥ 0 | xt ∈ G}
the first hitting time in G of the diffusion x and by
τ () := inf {t ≥ S() | xt ∈
/ G}
the first exit time, then (A.2) holds for τ ().
The proof is left to the reader.
EJP 17 (2012), paper 76.
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Exit problem
B
Propagation of chaos
The aim of this appendix is to present the classical results of the propagation of
chaos and the proofs. We recall the mean-field system (0.2):
Zt,i,N = X0 +
√
Bti −
Z
t
∇V Zs,i,N ds
(B.1)
0
N Z
1 X t
−
∇F Zs,i,N − Zs,j,N ds .
N j=1 0
Also, we consider a system of N independent self-stabilizing diffusions:
Xt,i = X0 +
√
Bti −
Z
t
∇V Xs,i ds −
Z
0
t
∇F ∗ us Xs,i ds .
(B.2)
0
The two diffusions X ,i and Z ,i,N are close when N is large enough.
Proposition B.1. There exists K > 0 and 0 > 0 such that, for all N ≥ 1:
2 K
.
sup E Xt,1 − Zt,1,N ≤
N
0<<0 t∈R+
sup
,i
Proof. We apply the Itô formula to Xt
2
ξi (t) := Xt,i − Zt,i,N , we obtain
N
X
N N
2 XX (∆2 (i, j, t) + ∆3 (i, j, t)) dt
N
i=1
i=1
i=1 j=1
D
E
,i
,i,N
with ∆1 (i, t) := Xt − Zt
; ∇V (Xt,i ) − ∇V Zt,i,N
,
D
E
∆2 (i, j, t) := Xt,i − Zt,i,N ; ∇F (Xt,i − Xt,j ) − ∇F Zt,i,N − Zt,j,N
and
D
E
∆3 (i, j, t) := Xt,i − Zt,i,N ; ∇F Zt,i,N − Zt,j,N − ∇F ∗ ut Zt,i,N
.
d
N
X
− Zt,i,N and the function x 7→ x2 . By denoting
ξi (t) = −2
∆1 (i, t)dt −
,i,j
The convexity of F implies ∆2 (i, j, t) + ∆2 (j, i, t) ≥ 0. Indeed, by writing ηt
Xt,j and ζt,i,j,N := Zt,i,N − Zt,j,N , we have:
:= Xt,i −
∆2 (i, j, t) + ∆2 (j, i, t)
D
E
= ∇F (ηt,i,j ) − ∇F (ζt,i,j,N ) ; Xt,i − Zt,i,N − Xt,j − Zt,j,N
2
D
E
= ∇F (ηt,i,j ) − ∇F (ζt,i,j,N ) ; ηt,i,j − ζt,i,j,N ≥ α ηt,i,j − ζt,i,j,N ≥ 0 ,
where α ≥ 0 depends on F . Consequently
E

N X
N
X

i=1 j=1




∆2 (i, j, t) = E


N
X
1≤i<j≤N


∆2 (i, j, t) + ∆2 (j, i, t)
≥ 0.

(B.3)
Inequality (1.1) implies
−2
N
X
i=1
∆1 (i, t) ≤ −2θ
N
X
ξi (t) .
(B.4)
i=1
EJP 17 (2012), paper 76.
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Exit problem
To deal with the last term, we apply the Cauchy-Schwarz inequality:



 12
 q
N
N X
N
X
X

∆3 (i, j, t) ≤ ξi (t) ×
E E ρj (i) ; ρk (i)




j=1
j=1 k=1
,i
,j
with ρj (i) := ∇F Xt − Xt
− ∇F ∗ ut Zt,i,N .
,i,N
Taking
the conditional
expectation with respect to Zt
E ρj (i)ρk (i) = 0 for j 6= k . Therefore
,j,N
and then to Zt
, we obtain


 r
N
X
h
i
2
∆3 (i, j, t) ≤ N E [ξi (t)] E ||∇F (Xt − Yt ) − ∇F ∗ ut (Xt )||
E 

j=1
where Xt and Yt are two independent random variables with law ut . We know by
Lemma 1.1 in [17] that ∇F is the product of x with a polynomial function of degree
2n − 2 of ||x||. By Proposition 1.3 and Remark 1.4, there exist 0 and C > 0 such that



N
X
p
≤ C N E {ξi (t)} ,
∆3 (i, j, t)
E 

j=1
(B.5)
for every < 0 . By combining (B.3), (B.4) and (B.5), we obtain
N
N q
X
d X
C
E {ξi (t)} ≤ 2
−θE {ξi (t)} + √
E {ξi (t)} .
dt i=1
N
i=1
The particles are exchangeable and so
d
2C
E {ξ1 (t)} ≤ −2θE {ξ1 (t)} + √
dt
N
q
E {ξ1 (t)} .
Since ξi (0) = 0,
E {ξ1 (t)} ≤
C2
.
θ2 N
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.
Remark B.2. Instead of (V-2), let us assume that there exist ζ ≥ 2 and λ > 0 such that
ζ
h∇V (x) − ∇V (y) ; x − yi ≥ λ ||x − y|| .
Then, there exists K > 0 such that
2 1
,1
,1,N sup sup E Xt − Zt
≤ KN − ζ−1 .
0<<1 t≥0
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):
EJP 17 (2012), paper 76.
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Exit problem
Proposition B.3. There exists 0 > 0 and M > 0 such that for all T > 0 and for all
N ≥ 1,
sup E
0<<0
2
sup Xt,1 − Zt,1,N 0≤t≤T
≤
MT
.
N
By putting the supremum under the expectation, we lose the uniformity with respect
,i
,i,N
to the time. However, the position of the two particles Xt and Zt
was not used in
the previous proofs. By doing it, we obtained a stronger result in this paper, see Section
3.
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[16] J. Tugaut: Convergence to the equilibria for self-stabilizing processes in double well landscape. To appear in The Annals of Probability, 2010.
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Exit problem
[18] J. Tugaut: Self-stabilizing processes in multi-wells landscape in Rd - Convergence. Available
on http://hal.archives-ouvertes.fr/hal-00628086/fr/, 2011
[19] J. Tugaut: Captivity of mean-field systems. Available on http://hal.archives-ouvertes.
fr/hal-00573047/fr/, 2011.
Acknowledgments. I would like to thank Christophe Bahadoran for having given me
the idea of using particles system to find the exit time of the self-stabilizing process on
Wednesday 23th April 2008. I would also like to thank the colleagues who helped me
to correct the English language of this paper. Philippe Gambette: Merci. Ante Mimica:
Hvala ti. Diana Putan: Mulţumesc. Finalement, un très grand merci à Manue et à
Sandra pour tout.
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