Non-Equilibrium Steady States for Networks of Oscillators
Noé Cuneo Jean-Pierre Eckmann Martin Hairer Luc Rey-Bellet
math-ph cond-mat.stat-mech math.MP



Département de mathématiques, Université de Cergy-Pontoise, CNRS UMR 8088
Département de Physique Théorique and Section de Mathématiques, University of Geneva
Department of Mathematics, Imperial College London
Department of Mathematics and Statistics, University of Massachusetts Amherst




Non-Equilibrium Steady States for Networks of Oscillators
    Noé Cuneo1, Jean-Pierre Eckmann2, Martin Hairer3, Luc Rey-Bellet4
    December 30, 2023
=====================================================================




Non-equilibrium steady states for chains of oscillators (masses) connected by harmonic and anharmonic springs and interacting with heat baths at different temperatures have been the subject of several studies.  In this paper, we show how some of the
results extend to more complicated networks.
We establish the existence and uniqueness of the non-equilibrium steady state, and show that the system converges to it at an exponential rate. The arguments are based on controllability and conditions on the potentials at infinity.




§ INTRODUCTION
 sec:intro
 
The aim of this paper is to state and prove an extension of the results
of <cit.> to the multidimensional case. 
We consider a network of masses connected with springs (interaction
potentials), where some of
the masses interact with stochastic heat baths which can have different
temperatures. We also let each mass interact with a substrate
through some pinning potential.
 We will show that under conditions spelled out in
this paper, any such system has a unique non-equilibrium stationary state (invariant measure). We show, moreover,
that the convergence to the steady state is exponential.
The proof follows in principle the ideas of <cit.>, but 
the controllability argument uses the more general conditions of <cit.>,
and the compactness part relies on a Lyapunov function argument similar to <cit.>.

The new aspects of this paper are twofold: First, we deal with networks of 
springs connecting the masses, and not just with 
1-dimensional chains.
Second, we correct an oversight of <cit.> (see rem:liglpnoncompact)
by a careful analysis of the interplay between the coupling potentials, which hold the 
system together, and the pinning potentials, which prevent it from “flying 
away”. This will require decomposing the phase space into two 
regions, depending on whether the pinning forces or the interaction forces dominate.
In the process, we also obtain sharper estimates on the rate of energy dissipation (see rem:sharper).


The conditions on the system come in the following flavors:

  
  
  
  
  C1: The masses are sufficiently connected to the heat baths.
     
  C2: The interaction potentials are non-degenerate.
       
  C3: The potentials are homogeneous at infinity and coercive.
       
  C4: The limiting interaction forces are locally injective.
	    
  C5: The interaction potentials grow at least as fast as the pinning potentials.




We will make C1–C5 precise in the next section. 
C1–C2 will be required to show the uniqueness of the steady state, and actually C1 will have to be more specific
than what common sense would seem to dictate.
C3–C5 will be further required for existence and exponential convergence.

As was shown in <cit.>, it is useful to assume that all
potentials are quadratic (at least at infinity). These
results have been extended 
in <cit.> to potentials of polynomial growth subject to C5.

Without C5, decoupling phenomena (related to “breathers”) may lead to subexponential convergence
to the invariant measure (and much more difficult proofs). In fact, the existence of the invariant measure
when the pinning potentials grow faster than the interaction potentials
has only been obtained for a chain of 3 masses so far (see the extensive
discussion in <cit.>), and for some closely related chains of 
rotors, which correspond to the “infinite pinning” limit in a sense (see <cit.>).



The paper is organized as follows. In sec:setup we give the precise definitions of the conditions C1–C5 above and state the main result about existence and uniqueness of the invariant measure, and exponential convergence. The proof relies on two main ingredients: (1) Hörmander's bracket condition, and  (2) a Lyapunov condition on the energy. 
In sec:genHam we prove that these two ingredients lead to the desired result, and there we consider more general thermalized Hamiltonian systems (of which networks of oscillators are a special case). Finally, we check that under  C1–C5, networks of oscillators indeed satisfy Hörmander's condition (s:hypo) and the Lyapunov property (sec:Lyapunov).

While the discussion in sec:genHam is rather standard, the proofs in s:hypo and sec:Lyapunov are quite specific to our setup; the main technical difficulty there is that the heat baths do not act on all the oscillators directly, so that propagation within the network has to be carefully studied. This difficulty was already present, to a lesser extent, in the works on chains of oscillators mentioned above.

Finally, although we restrict ourselves to smooth potentials here, we mention that systems of particles with singular interactions (but with heat baths acting on all particles) have attracted significant attention lately (see for example <cit.>). 





§ SETUP AND RESULTS


We consider a finite set  of masses.
We denote by q_v ∈^n
and  p_v ∈^n the position and momentum of each mass v∈ (we assume n≥ 1).
The phase space is then Ω≡^2|| n, and we write z = (p,q) = ((p_v)_v∈, (q_v)_v∈).

We then introduce a set ⊂×
of edges representing the springs, and consider Hamiltonians of the form

 H(p,q) = ∑_v ∈ (p_v^2/2 + U_v(q_v)) +
 ∑_e ∈ V_e(δq_e) ,

where the functions U_v are pinning potentials, the functions V_e
are interaction potentials, and where for e=(v,v') ∈
we write δ q_e = q_v'-q_v ∈^n.

We view (, ) as an undirected graph with
no loop (, no edge of the kind (v,v)).
Since the edges e = (v,v') and e̅ = (v',v) are identified, we also adopt the convention that V_e(q_v'-q_v) and V_e̅(q_v-q_v') are equal and both express just one interaction, which appears only once in e:hamil.

We now choose a subset ⊂ of vertices where thermal baths act,
and for every b∈ we assume that some temperature T_b>0 and
some coupling constant γ_b>0 are given. For v∉ we
set, for convenience, γ_v = T_v =0.
With this notation, our model is described by the system of stochastic differential
equations (one equation per v∈):
[e:SDE]
dq_v = p_v dt ,    dp_v = -∇_q_v H(p,q) dt - γ_v p_v dt + √(2T_v γ_v) dW_v(t) ,

where the W_v are mutually independent standard n-dimensional Wiener processes. 
Note that for v∉, the last two terms in e:SDE are absent. We denote by z_t = (p_t, q_t) the solution of e:SDE.
For each fixed initial condition z∈Ω, we denote by _z the probability distribution of the solutions to (<ref>), and by _z the corresponding expectation. We also introduce the transition kernels P_t(z, ) defined for all z∈Ω, t≥ 0, and all Borel sets A⊂Ω by

P_t(z, A) = _z{z_t ∈A} .	


The Langevin heat baths used in  (<ref>) are slightly simpler than those in <cit.>. There, the oscillators interact with some classical field theories which are initially Gibbs-distributed, and the (linear) coupling between the oscillators and the fields is chosen so that the latter can be integrated out. The resulting dynamics is similar to (<ref>), but instead of directly acting on the momenta as in (<ref>), the noise and dissipation act on some auxiliary variables which in turn interact with the momenta. The choice of Langevin heat baths (also made in <cit.>) is only for convenience, and the present analysis is easily transposed to the setup of <cit.>. 

We now make C1–C5 precise. We start with C1 in s:control, which is in particular satisfied if the network is a chain with heat baths at both ends. In s:potential and s:nearly, we discuss C2–C5. An example of potentials satisfying C2–C5 that the reader might want to have in mind is[Throughout the paper,  denotes the Euclidean norm.] V_e=(1+^2)^ℓ_i/2 and U_v = (1+^2)^ℓ_p/2, where ℓ_i, ℓ_p ∈ satisfy ℓ_i ≥ℓ_p ≥ 2. (If ℓ_i and ℓ_p are even numbers subject to the same condition, then one may also take V_e=^ℓ_i and U_v = ^ℓ_p.)




 §.§ Controllability through the springs



The following definition is useful: Let B be a subset of . 
We say that B is nicely connected to v∈∖ B if 
there exists a vertex b ∈ B and an edge of the form (b,v)∈,
and there is no other edge from b to
  ∖ B. We define B as the union of B with its
nicely connected vertices in ∖ B (see fig:nicelyconnected).
We denote by ^2 , ^3 , … the iterates of this construction.








Let (,) be as above. We say that ⊂ controls (, )
if there exists k ≥ 1 such that
^k  =.


This allows us to make C1 precise as[It was brought to our attention that the same condition appears in <cit.>.]
C1 The graph is connected and  controls (, ).



Note that connectedness is a trivial restriction, 
for if the graph is not connected the results apply to each connected component separately. Chains with heat baths at both ends (or even at just one end) obviously satisfy C1. So do some finite pieces of regular lattices, see fig:slabs.
As some examples in fig:slabs illustrate, controllability is,  unfortunately, not a monotone property in : Adding edges, ,
  more springs, will sometimes improve controllability, and sometimes destroy it.
On the other hand, given (,), controllability is a monotone property in the set 
 of “initially controlled” nodes.










r:1d
One always has the inequality |^k+1| ≤ |^k| + ||.
Indeed, let B_k be the set of vertices in ^k that are connected to
at least one other vertex in ∖^k.
It is then clear from the definition of  that |^k+1| ≤ |^k| + |B_k|.
On the other hand, it follows from the definition of  that,
for every “newly added” vertex v in ^k+1∖^k,
there must be at least one vertex w in B_k such that v is the only element in ∖^k
that is connected to w. As a consequence, |B_k+1| ≤ |B_k| ≤ || for every k, from which the claim
follows at once.

In a way, this remark says that the system is
effectively almost 1-dimensional with respect to the propagation of
information.
No point in  and no point in ^k will ever
control more than one new point 
as one iterates from k to k+1 above.



Another criterion for the controllability of networks of interacting
oscillators was introduced in <cit.>. While the results
in <cit.> allow in some cases to control networks with more
general topologies, in particular some which do not satisfy <ref>,
they only apply to strictly anharmonic
polynomial potentials in 1D (n=1).




 §.§ Non-degenerate potentials


We now discuss the conditions on the potentials V_e. The attentive reader
will note that, in fact, the non-degeneracy conditions below
are not necessary on all the
links, but only on those which are needed for <ref> to hold.
This means, for example, that in
fig:slabs,
the potentials associated with the “vertical” springs may be degenerate.
We will not deal with this any further, and make the assumptions on
all V_e.

Given a multi-index α =(α _1,…,α_n)  of
non-negative integers, we set |α |=∑_i=1^n α _i,
and define D^α as the differential operator with α _i
derivatives in the i^ th direction of ^n.
Given a potential V^n →, (, any of the V_e) we
introduce the following notion of non-degeneracy <cit.>. The idea is that the V_e do not have “infinitely flat” pieces. 



  A smooth potential V:^n→ is non-degenerate if there
  exists an ℓ<∞ such that the set of derivatives

    { D^α∇ V (x) :  1 ≤ |α| ≤ℓ}

spans ^n for every x∈^n.


We now have the following precise version of C2:
C2
The interaction potentials V_e are non-degenerate.




Any potential of the form V(x)=x^r with r=2, 4, 6, … is non-degenerate. The same is true of V(x) = (1+x^2)^r/2  with any real number r > 0. On the contrary, if x is replaced by |x_1| here, then the resulting potential is degenerate (unless n=1). 



	The condition in d:nondegenerate allows for controllability in
the following sense:
Consider a given continuous trajectory
 q̅ [0,1] →^n and the problem
[e:control]
ṗ_f(t) = -∇V(q̅(t) + f(t)) 

with p_f(0) = p_*.
If V is non-degenerate, then the set of solutions p_f(1) of
e:control at time 1,  as f is varied over all smooth
functions with sup_t≤1 |f(t)|≤ 1,
contains an open (and in particular “full-dimensional”) set. 	






 §.§ Nearly homogeneous potentials


One of the difficulties with models of the type e:hamil, e:SDE is to show the existence
of a non-equilibrium steady state.
As was demonstrated in <cit.>, this can be highly non-trivial, and even with “nice” potentials, there are
situations where the convergence to the steady state can be arbitrarily slow.

For the purpose of proving the existence of the steady state, a convenient class of interactions is given by potentials that behave at infinity like homogeneous functions.
We say that a function Ψ^n → is homogeneous of degree[The degree r is not assumed to be an integer. The restriction r≥ 2 is required for some of the results below.] r ≥ 2 if
Ψ(λ x) = λ^r Ψ(x) for every λ > 0 and every x ∈^n ∖{0}.
With this notion at hand, we give the following definition, which is slightly weaker than the one
in <cit.>:


A smooth function V^n → is said to be nearly homogeneous of degree r if
there exists a homogeneous (of degree r), differentiable function V_∞ :^n → such that ∇ V_∞ is locally Lipschitz, and such that for all 0≤ |α| ≤ 1,

lim_λ→∞ sup_x = 1	|(D^αV)(λx)/λ^r-|α|  - D^αV_∞(x)| = 0 .




If V(x) = x^r with r = 2, 4, 6, …, then V is nearly homogeneous. Moreover, for any real number r≥ 2, the potential V(x) = (1+x^2)^r/2 is nearly homogeneous. In both cases, V_∞(x) = x^r.




It is easy to see that nearly homogeneous functions (of degree
r≥2) also satisfy some derived properties, for 0≤|α |≤1:


  (i)
lim_x→∞x^|α|-r(D^α V(x)-D^α V_∞(x))= 0.

  (ii)
|D^α V(x)|	≤ C_V (1+x^r-|α|),
for some C_V>0.


  (iii)
lim_λ→∞sup_x∈ K( λ^|α|-r(D^α
  V)(λ x) -D^α V_∞(x))=
0, for every compact set K.



  (iv)If inf_x = 1 V_∞(x)>0, then 
V(x)	≥ C'_V x^r,
for some C'_V>0 when x is large enough.






We can now define C3–C5 properly as follows.

C3The potentials U_v are nearly homogeneous of degree ℓ_p ≥ 2 with limiting functions U_v, ∞, and the potentials V_e are nearly homogeneous of degree ℓ_i ≥ 2 with limiting functions V_e, ∞. Moreover, the limiting potentials are coercive, ,  inf_x=1 V_e, ∞(x) > 0 and inf_x=1 U_v,∞(x) > 0. 


C4 The limiting
  interaction forces -∇ V_e, ∞ are locally injective in
  the sense that for each  e∈ and each x∈^n, we have
  ∇  V_e, ∞(x')≠∇  V_e, ∞(x) for all x' in
  a neighborhood of x.



C5The interaction and pinning powers satisfy ℓ_i ≥ℓ_p.


Note that Conditions <ref> and <ref> are not comparable: the former guarantees that the forces -∇ V_e are locally surjective in a sense, and the latter guarantees that the limiting forces -∇ V_e, ∞ are locally injective.

For example, consider a smooth homogeneous function V: ^3 → given by x^4/4 + y^2z^2/2 on the set M = {(x, y, z)∈^3 : z^2+y^2 ≤  x^2/10}. Then ∇ V(x,y,z) = (x^3, yz^2, y^2z), and obviously { D^α∇ V (x) :  1 ≤ |α| ≤ 3 } spans ^3 for all (x,y,z)∈ M. However, ∇ V(x,y,0) = (x^3, 0, 0), and thus ∇ V is not locally injective.

There are specific systems for which <ref> is not actually required, and
others for which it is, as we
illustrate in Remarks <ref> and <ref>.

 <ref> holds for example if the V_e, ∞ are strictly convex. In particular if n=1, then the V_e, ∞ are automatically strictly convex, since they are homogeneous of degree ℓ_i ≥ 2 and coercive.



The requirement that all interaction potentials have the same degree ℓ_i is crucial. Indeed, if one of the interactions in the bulk of the network (, involving two oscillators in ∖) has a higher degree than the others, the system may find itself in a regime where the two corresponding oscillators oscillate in phase opposition and with a frequency much higher than the other natural frequencies of the system, leading to a decoupling phenomenon comparable to the situation in <cit.>. This is again expected to lead to subgeometric convergence to the invariant measure and much more involved proofs.


As will be clear from the proofs in sec:Lyapunov, it is actually not necessary for all the limiting pinning potentials U_v,∞ to be coercive (or even to be non-zero). In fact, we only need the quantity defined in (<ref>) to be coercive.



Without loss of generality, we also assume that the potentials U_v and V_e are non-negative (by the coercivity condition above,
this is always achievable by adding a constant).




 §.§ Main result


Given the definitions of s:control–<ref>,
we can now state the main result.  In order to emphasize the role of each assumption,
we introduce the following (very weak) auxiliary condition.
CAThe Hamiltonian H has compact level sets (, the set {z: H(z) ≤ K} is compact for each K>0), and there exists some β> 0 such that the function exp(-β H) is integrable on Ω.



<ref> follows immediately from Condition <ref> (one can choose any β > 0). 



The following holds.


  * Under Conditions <ref>, <ref> and  <ref>, the system e:SDE admits at most one invariant measure, and if it exists, it has a smooth density with respect to Lebesgue measure.

  * Under Conditions <ref>, <ref>, <ref> and <ref>, the system  e:SDE admits a least one invariant measure, and e^θ H is integrable with respect to it for all 0 < θ < 1/T_ max, with T_ max = max{T_b: b∈}.

  * Finally, assuming Conditions <ref>–<ref>, the system e:SDE admits a unique invariant measure μ_⋆. Moreover, for all 0 < θ < T_ max, there are constants C,c>0 such that for every initial condition z = (p,q) ∈Ω and all t≥ 0,
[eq:convergencemu]
sup_f∈C(Ω) : |f| ≤e^θH|_z f(z_t) - ∫f d μ_⋆| ≤C e^θH(z)-c t .










This theorem is a special case of theo:maingeneral below, as we will show.




§ A GENERAL RESULT ABOUT THERMALIZED HAMILTONIAN SYSTEMS
 


In this section, we prove a version of  theo:main which applies to more general thermalized Hamiltonian systems subject to two assumptions H1 and H2 (see below). As we show in s:hypo and sec:Lyapunov, these assumptions follow from Conditions <ref>–<ref>. Although the material discussed in this section is mostly standard (see for example <cit.>), we provide a complete exposition relying on the version of Harris' ergodic theorem proved in <cit.>. We hope that by considering more general Hamiltonian systems and conditions in this section, the proofs will be both easier to read and useful beyond the scope of this paper.


 The setup is as in sec:setup, except that we do not assume that  the set of masses  has the structure of a graph and that the Hamiltonian has the form (<ref>). More precisely, we study the SDE
[e:SDEgen]
dq_v = p_v dt ,    dp_v = -∇_q_v H(p,q) dt - γ_v p_v dt + √(2T_v γ_v) dW_v(t) ,

where the friction constants γ_v, the temperatures T_v and the set  ⊂ are as in sec:setup, and where the Hamiltonian is given by

 H(p,q) = ∑_v ∈ p_v^2/2 + U(q) ,

for some arbitrary smooth, non-negative potential U on ^n||.

We also assume throughout this section that <ref> holds, , that H has compact level sets and that exp(-β H) is integrable on Ω for some β > 0.

We define the semigroup (^t)_t≥ 0 acting on the space of bounded measurable functions on Ω by ^t f(z) = _z f(z_t) = ∫_Ω f(z') P_t(z, dz'). 
We also fix 0 < θ < 1/T_ max, with T_ max  = max{T_b: b∈} as in theo:main.  We let moreover 
    V = e^ϑ H .


The solutions to e:SDEgen form a Markov process whose generator  is
[e:defL]
= X_0 + ∑_b ∈ ∑_i=1, …, nX_b,i^2 ,

where X_b,i = √(T_b γ_b)_̣p^i_b and 

X_0 = ∑_v∈ (p_v ·∇_q_v - ∇_q_v U(q) ·∇_p_v - γ_v p_v ·∇_p_v)  .

From now on, we will view X_0 and the X_b,i interchangeably as first-order differential operators and as vector fields on Ω. 



With C_* = θ∑_ b∈γ_b T_b, we obtain
[eq:lyapunovplus]
V = ∑_b∈ θγ_b ([θT_b-1]p_b^2 + T_b)e^θH ≤C_*V .

 Since H, and hence V, have compact level sets by assumption, the process admits strong solutions that are continuous and defined for all t≥ 0 (almost surely),  the strong Markov property is satisfied, and for all t≥ 0 we have
[eq:lyapunovplusint]
^t V ≤e^C_*tV

 (see for example <cit.>, <cit.>, and <cit.> for the strong Markov claim).  

We now introduce  Hör­mander's celebrated “Lie bracket condition” <cit.>. Define a family of vector fields _0 by _0 = {X_b,i :  b∈, i=1, …, n} and then, recursively,

_k+1 = _k ∪{[X,Y] :  X ∈_k , Y ∈_0 ∪{X_0}} ,

where [X,Y] denotes the Lie bracket (commutator) of X and Y.
With this notation at hand, we formulate



H1 The operator  defined in e:defL satisfies Hörmander's bracket condition, , for every z ∈Ω,
there exists an integer k>0 such that the linear span of {Y(z): Y ∈_k} is all of Ω.




<ref> is sufficient (and “almost necessary”) for ∂_t - to be hypoelliptic, so that the semigroup associated to e:SDEgen has
a smoothing effect (see prop:regularite below). We note that the requirement in <ref> is made for all z∈Ω; see for example <cit.> for an argument which only requires Hörmander's condition to hold at one point, but which is specific to quasi-harmonic systems whose harmonic part is subject to Kalman's controllability condition.

Next, we introduce a Lyapunov condition, which will be crucial in order to obtain the existence of an invariant measure and the exponential convergence (<ref>).

H2 There exists t_* > 0 and κ∈ (0,1) such that[Here and below, _K denotes the characteristic function of the set K.]

^t_* V ≤κV+ c_K ,

where c>0 is a constant and K is a compact set. 





In s:hypo, we show that for the original system (<ref>), Conditions <ref> and <ref>  imply <ref>, and in sec:Lyapunov we show that Conditions <ref>, <ref>, <ref> and <ref> imply <ref>. With this in mind, theo:main is a special case of


The following holds (recall that <ref> is assumed throughout this section).


  * Under <ref>, the system e:SDEgen admits at most one invariant measure, and if it exists, it has a smooth density with respect to Lebesgue measure.

  * Assuming <ref>, the system  e:SDEgen admits a least one invariant measure, and V is integrable with respect to it.

  * Finally, assuming Conditions <ref> and <ref>, the system e:SDEgen admits a unique invariant measure μ_⋆, and the exponential convergence in (<ref>) holds.



The three parts of the theorem are proved in Propositions <ref>, <ref> and <ref> below. 	














 §.§ Controllability and uniqueness



The following consequence of Hörmander's condition is well known <cit.> (see <cit.>, <cit.>, and <cit.> for introductions).


	Assume <ref>. Then the transition kernel in (<ref>) can be written as P_t(z, dz') = p_t(z,z')dz', where the map (t, z, z')↦ p_t(z,z') is smooth on (0,∞) ×Ω×Ω. In particular, the process is strong Feller. Finally, every invariant measure has a smooth density with respect to Lebesgue measure on Ω.



We now prove the following “accessibility” result (see also <cit.> for another variant of this argument).

 Assume <ref>. Then the system e:SDEgen admits at most one invariant measure, and for every non-empty open set ⊂Ω and all z∈Ω, we have sup_t> 0P_t(z, ) > 0.



The argument follows the same lines as the reasoning first given in <cit.>, see also
<cit.>.
Take β > 0 as in <ref> and consider instead
of e:SDEgen the modified equation
[e:SDE2]
dq_v = p_v dt ,    dp_v = -∇_q_v H(p,q) dt - γ_v p_v dt+ √(2γ_v β^-1) dW_v(t) .

The only difference is that all the temperatures have been replaced by 1/β (we still have γ_v = 0 for all v∉). By the same argument as above, the solutions to (<ref>) almost surely exist for all times.
It is well known that the measure 
    dμ_β = 1/Ze^-β H(p,q) dp dq
 is invariant for e:SDE2, and by <ref>, one can choose Z>0 so that μ_β is a probability measure. (The invariance of μ_β can be seen by checking that ^* e^-β H = 0, where ^* is the formal adjoint of the generator of (<ref>).)

We  next show that μ_β is the only invariant probability measure for e:SDE2. It is easy to show that, as a consequence of prop:regularite, the map z↦P_t(z, ) is continuous in the total variation topology, where P_t denotes the transition probabilities for (<ref>). Since distinct ergodic invariant probability measures for e:SDE2
are mutually singular by Birkhoff's ergodic theorem, this immediately implies that if ν is an ergodic invariant
measure for e:SDE2 and z ∈ν, then there exists a neighborhood _z of z such that 
_z ∩ν̅= ∅ for every other ergodic invariant measure ν̅. 

As a consequence, let us choose some (there exists at least one) ergodic invariant measure ν  of e:SDE2.
Assuming by contradiction that ν is not unique, we have ν≠Ω.
As a consequence, setting = ⋃_z ∈ν (_z ∖ν), we have
constructed a non-empty open set  such that ∩ν̅= ∅
for every ergodic invariant measure ν̅ of e:SDE2 and therefore, by the ergodic representation theorem,
for every invariant measure ν̅. (We must have ≠∅ for otherwise ν would be both open and
closed, which cannot be.) However, μ_β  = Ω, thus yielding a contradiction.

Returning to our main line of argument, since μ_β is the unique invariant probability measure for e:SDE2,
it must be ergodic. Since μ_β has full support, it then follows from Birkhoff's ergodic theorem that for every open set 
and Lebesgue-almost every initial condition z∈Ω, we have sup_t> 0P_t(z, ) > 0. An easy application 
of the Chapman-Kolmogorov equation, using the smoothness of the transition probabilities, shows that this actually holds for every z∈Ω.[One can also use that the strong Feller property 
implies that Birkhoff's ergodic theorem holds for every initial
condition in the support of the invariant measure <cit.>.]
The conclusion of the proposition thus holds for (<ref>). We now return to (<ref>).

The key is that for each z∈Ω and t≥ 0, the transition probabilities P_t(z, ) for (<ref>) and P_t(z, ) for (<ref>) are equivalent, since the two stochastic differential equations differ only by the scaling of the Brownian motions. Thus, we indeed have sup_t> 0P_t(z, ) > 0 for all z∈Ω and every non-empty open set ⊂Ω. Assume now by contradiction that (<ref>) admits more than one invariant probability measure. Then by the ergodic decomposition theorem there exist two distinct ergodic measures, which then have distinct supports S_1 and S_2. By smoothness, there exists a non-empty open set ⊂ S_2, and by taking z∈ S_1 we find sup_t> 0P_t(z, ) = 0, which is a contradiction.



Although this will not be needed, we state the following corollary, which follows from the Stroock–Varadhan support theorem (see <cit.>, and <cit.> for an extension to case of unbounded coefficients).

Assume <ref>.
Then, for any starting point 
z_0 ∈Ω and any non-empty open set ⊂Ω, there exists a time t > 0 and smooth
controls u_b  [0,t] →^n for b∈ such that the solution at time t to[The same is true without the dissipative terms - γ_v p_v, since they can be absorbed into the controls u_v (recall that γ_v = 0 when v∉).]

q̇_v = p_v ,    ṗ_v =  -∇_q_v H(p,q) - γ_v p_v + _(v)u_v(t) ,     v∈ ,

with initial condition z_0, lies in . 



In the case of chains of oscillators, a stronger controllability argument is used in <cit.>. The argument given above is “softer”. As a consequence, it applies to a larger class of Langevin equations, at the expense of
having less explicit control. The argument in <cit.> actually implies that, in the statement of prop:uniq, the quantity P_t(z, ) is positive for all t>0. 





 §.§ Minorization


The next proposition shows that every compact set is small in the terminology of <cit.>. In fact, we show that for each given compact set C, the minorization condition holds for all large enough t. In the proof, p_t(,) is as in prop:regularite.

Assume Condition <ref>. Then, for every compact set C, there exists a time t_C such that for all t≥ t_C, there exists a non-negative and non-trivial measure ν (which may depend on t) such that P_t(z, ) ≥ν for all z∈ C.


We start by showing that there exists z_* ∈Ω such that for all z ∈Ω, there are t_♯(z) and δ_z > 0 satisfying

	p_t(z', z_*) > 0     for all t≥ t_♯(z) and all z' ∈ B(z, δ_z) .


First, pick any z_0 ∈Ω. We now fix any z_* such that p_1(z_0,z_*) > 0. By continuity, there exists δ > 0 such that inf_z∈ B(z_0, δ)p_1(z,z_*) > 0. By prop:uniq, there exists for each z∈Ω some t_0(z) such that P_t_0(z)(z, B(z_0, δ)) > 0. It then follows from the semigroup property 
that p_t_1(z)(z, z_*) > 0 with t_1(z) = t_0(z) +1. Using continuity again, we can choose δ_z > 0 so that
[eq:part1lemferry]
	p_t_1(z)(z', z_*) > 0     for all  z'∈B(z, δ_z) .

We now show that there exists t_2 > 0 such that
[eq:part2lemferry]
	p_t(z_*, z_*) > 0     for all t≥ t_2 .

Since p_t_1(z_*)(z_*, z_*) > 0, continuity with respect to time implies that for some Δ > 0 small enough, we have p_t(z_*, z_*) > 0 for all t∈ [t_1(z_*), t_1(z_*) +Δ]. But then the same holds for all t∈ [nt_1(z_*), nt_1(z_*) + nΔ], n ∈. Thus eq:part2lemferry holds with t_2 = n_*t_1(z_*) for any integer n_* ≥  t_1(z_*)/Δ. Using eq:part1lemferry, eq:part2lemferry and the semigroup property yields  (<ref>) with t_♯(z) = t_1(z) + t_2.

We now prove the main claim. Let C be a compact set. The balls {B(z, δ_z): z∈ C} form an open cover of C, and by compactness we can extract a finite subcover, yielding a maximum time t_C such that p_t(z, z_*) > 0 for all z∈ C and all t≥ t_C. For any such t, since p_t(,) is continuous on Ω^2 and C is compact, the result follows with dν = ε_B(z_*, r)dz for small enough ε, r > 0.







 §.§ Existence of an invariant measure and exponential convergence


As an elementary consequence of <ref>, we find that 

	^nt_* V  ≤κ^nV + c∑_i=0^∞κ^i  for all n∈ .

From this and (<ref>), we obtain that

	^t V  ≤c_1 +  c_2 ρ^tV   for all t≥0 ,

with c_1, c_2  > 0 and ρ = κ^1/t_*∈ (0,1). 
In particular, since V has compact level sets, this implies that for any z∈Ω, the family of probability measures (P_t(z, ))_t≥ 0 is tight. Since the process is Feller, the standard Krylov–Bogolyubov construction then implies that for some sequence t_k increasing to infinity, 1/t_k∫_0^t_kP_s(z, )ds converges weakly to some measure which is invariant, and with respect to which V is integrable. We thus obtain

Under <ref>, the process admits an invariant measure μ_⋆, and V is integrable with respect to μ_*.



Assuming in addition <ref> implies that μ_⋆ is unique (prop:uniq), and we now prove exponential convergence.

	Under Conditions <ref> and <ref>, the exponential convergence in (<ref>) holds.


	We will apply the main result of <cit.> to the discrete-time semigroup (^nt_0)_n=0,1,2, …, for some large enough t_0>0.  Let first R  = 2c_1/(1-ρ). Here c_1, c_2 and ρ are as in (<ref>). We then define the compact set C = {z: V(z) ≤ R}. We choose now t_0 ≥ t_C with the t_C from prop:minorization, and large enough so that c_2 ρ^t_0 < ρ. It follows that R > 2c_1/(1-c_2 ρ^t_0), so that by (<ref>) the main result of <cit.> applies to (^nt_0)_n=0,1,2, …. We obtain[Another way to obtain (<ref>) with t_0 = t_* is to use <cit.>. Indeed, (<ref>)  implies that the process is aperiodic, <ref> provides the required drift condition, and by prop:minorization the compact set K in (<ref>) is small (and hence petite). An alternative proof of convergence
using quasi-compactness of the semigroup can be found in <cit.>.] that for some C_0, c_0 > 0 and all z∈Ω,

sup_f∈C(Ω)  :  |f| ≤V|_z f(z_nt_0) - ∫f d μ_⋆| ≤C_0 V(z)e^-c_0 nt_0    for all n∈ .

For |f| ≤ V, we define g(z,t) = _z f(z_t)-∫ f d μ_⋆. Decomposing t = n t_0 + r with n∈ and r∈ [0,t_0), we obtain from the Markov property that

|g(z,t)|     = |_z g(z_r,nt_0)|≤C_0e^-c_0 nt_0 _z V(z_r) ≤C_0e^C_*t_0-c_0 nt_0  V(z) ,

where we have also used (<ref>). This immediately implies (<ref>) for some C,c>0, and thus the proof is complete.











§ HYPOELLIPTICITY


In this section, we prove 
 Under Conditions <ref> and <ref>, the system  e:SDE satisfies <ref>.



For the system e:SDE, the vector field X_0 in the decomposition e:defL reads

X_0    = ∑_v∈ (p_v ·∇_q_v - ∇U_v(q_v)·∇_p_v - γ_v p_v ·∇_p_v)

        - ∑_(u,v) ∈ ∇V_(u,v)(q_v - q_u) ·(∇_p_v-∇_p_u) .


We will actually prove the following statement, which implies <ref>. 
Let X̅_0 = _̣t - X_0 and set 
_0 = {X̅_0}∪{X_b,i :  b ∈, i=1, …, n}, which we view as a family of smooth vector fields on ^1+2n||. 
Denote by  the smallest set of vector fields containing _0 that is closed under Lie brackets and multiplication
by smooth functions.

We will show that _̣t, as well as ∇_p_v and ∇_q_v for every v ∈, all belong to . Since X̅_0 ∈, it is sufficient to prove the claim about the ∇_p_v and ∇_q_v.  (Here and below, what we mean by ∇_p_v∈ is that ∂_p_v^i∈ for all i=1, …, n, and similarly for ∇_q_v.)

Note first that, by the definition of X_b,i and _0, we have
∇_p_b∈ for all b ∈. Furthermore,
since

 [_̣p_v^i, X̅_0] = -_̣q_v^i + γ_v _̣p_v^i

for all v∈, it follows that one has the implication

∇_p_v ∈  ⇒  ∇_q_v ∈ .

By the definition of the notion of  controlling , the claim
now follows if we can show that, for any set '⊂, one
has the implication

∇_q_b,∇_p_b ∈ for all  b ∈'   ⇒  ∇_p_v ∈ for all  v ∈' .

Assume therefore that ' is such that ∇_q_b,∇_p_b
are in  for all b ∈'.
Note that, for all i ∈{1,…,n} and every b∈',
[e:comm]
[_̣q_b^i, X̅_0] = (_̣i ∇U_b)(q_b)·∇_p_b - ∑_e=(b,v) ∈_b (_̣i ∇V_e)(δq_e) ·(∇_p_v-∇_p_b) ,

where we denote by _b the subset of those edges in  that are of the form (b,v) for some v ∈.
Fix now v ∈'∖'. By the definition of ', there exists
then b ∈' such that (b,v) ∈_b and, for every other 
w for which  (b,w) is in _b, one has w ∈'. 
For such a b∈', we conclude that in e:comm all the terms but (_̣i∇ V_(b,v))(q_v - q_b) ·∇_p_v are of the form f_u(z)·∇_p_u for some u∈', so that 

(_̣i ∇V_(b,v))(q_v - q_b)·∇_p_v ∈ .

By the definition of ', this holds for every v ∈'∖'. We now get rid of the potential term in
e:step1. Repeatedly taking Lie brackets with _̣q_b^j, e:step1 implies that, for every non-zero multi-index α, we have

(D^α∇V_(b,v))(q_v - q_b)·∇_p_v ∈ .

Let now ℓ be the value appearing in the non-degeneracy
assumption for V_(b,v) and let M be the n× n matrix-valued function
 whose elements are given by

M_ij(x) = ∑_1≤|α| ≤ℓ(D^α_̣i V_(b,v))(x)   (D^α_̣j V_(b,v))(x) ,   x∈^n .

It follows from the non-degeneracy assumption that M is invertible for every x∈^n, so that M^-1_ij(x) is a smooth function.
An explicit calculation shows, furthermore, that one has the identity 

_̣p_v^j = ∑_i=1^n ∑_1 ≤|α| ≤ℓ M^-1_ij(q_v - q_b) (D^α_̣i V_(b,v))(q_v - q_b)  (D^α∇V_(b,v))(q_v - q_b) ·∇_p_v  .

From e:M and the fact that M^-1_ij(q_v - q_b) (D^α_̣i V_(b,v))(q_v - q_b) is a smooth function,
we deduce that we indeed have ∇_p_v∈, thus completing the proof. 





§ LYAPUNOV CONDITION


In this section, we show that Conditions <ref>, <ref>, <ref> and <ref> imply that the system e:SDE satisfies <ref> above, , that V = e^θ H satisfies the Lyapunov property if  θ is small enough.

The proof follows the lines of the argument that can be found in <cit.>. Unfortunately,
these works both contained a gap in the argument, which we presently correct (see rem:liglpnoncompact).


We fix t_* > 0 and θ < 1/T_ max with T_ max= max{T_b: b∈}.  The main result of this section is

Under Conditions <ref>, <ref>, <ref> and <ref>, there is a constant C_1> 0 such that for all z_0 such that H(z_0) is large enough, we have
[eq:boundC1C2]
_z_0 e^θH(z_t_*)- θH(z_0)	≤e^- C_1 H(z_0) .



By the coercivity of H, the theorem above implies that there exist constants  κ∈ (0,1) and c>0, and a compact set K such that

_z e^θH(z_t_*) ≤κe^θH(z) + c_K(z) ,

which is the usual Lyapunov condition used in <ref>.



For the remainder of the paper, we assume that Conditions <ref>, <ref>, <ref> and <ref> are satisfied.

The central role in the proof of pr:lyap will be played by the
dissipation integral

Γ(t) = ∑_b∈ γ_b ∫_0^t p_b^2(s) ds .

In a nutshell, we will prove (<ref>) by showing that if H(z_0) is large enough, then with very high probability the main contribution to the energy difference H(z_t_*) - H(z_0) comes from (minus) the dissipation integral Γ(t_*), which, also with very high probability, scales like H(z_0).

In order to do this, we start by partitioning, for each initial condition z_0 ∈Ω, the probability space into the following three events:

A_1    = {H(z_s) ∈[H(z_0)/2, 2H(z_0)]  ∀s∈[0,t_*]} ,

A_2    = {inf_s∈[0,t_*] H(z_s) < H(z_0)/2} ,

A_3    = {sup_s∈[0,t_*] H(z_s) > 2H(z_0)} .


The event A_1 will be the center of most of our analysis. 
The event A_2 will be of no trouble, since after getting as low as H(z_0)/2, it is unlikely that the energy will increase again to a large value. Finally, the event A_3 will be of negligible probability at high energy. 

When the event A_1 is realized, we will cut the time interval [0,t_*] into subintervals. The length of each subinterval will depend on the distribution of energy between the interaction and center of mass degrees of freedom as follows.

We introduce the center of mass coordinates
[eq:defCDMcoord]
Q    = 1/||∑_v∈ q_v ,     P = ∑_v∈ p_v ,

and split the Hamiltonian according to
[eq:separationHams]
H    = H_c+ H_i ,

where

H_c    =  P^2/2|| + ∑_v∈ U_v(q_v) ,

H_i     = 1/2 ∑_v∈ (p_v -P/||)^2 + ∑_e∈ V_e(δq_e) .


We then let
[eq:defstoptx]
τ(z) =   λH(z)^1/ℓ_i - 1/2     if  H_i(z) ≥H(z)/2 ,

λH(z)^1/ℓ_p - 1/2     if  H_c(z) > H(z)/2 ,


where λ > 0 is arbitrary if ℓ_p > 2, and subject to the condition 0 < λ≤ t_*/2 if ℓ_p = 2.
Note that τ(z) is not random when z is fixed.

The rationale behind eq:defstoptx is simple: when the system is
dominated by the “internal” dynamics, the natural time scale is
H(z)^ 1/ℓ_i - 1/2. In the opposite case, the time
scale H(z)^ 1/ℓ_p - 1/2 of the pinning potentials is
relevant. When ℓ_i = ℓ_p, this distinction of time scales
obviously vanishes.


The following proposition, which we will prove in sec:interacDom and sec:pinningdom, says that with a very large probability, the average dissipation rate over the time interval [0, τ(z_0)] is at least some fraction of the initial energy.


Let

Ã = {H(z_s) ≤4H(z_0) ∀s∈[0,τ(z_0)]} .

Then there exist ε, B > 0 such that for all  z_0 with H(z_0) large enough,
[eq:resultptauxU]
_z_0(Ã ∩{Γ(τ(z_0))< εH(z_0)τ(z_0) }) ≤e^-B H(z_0) .



For the remainder of this section, we assume that ε, B  are fixed as in p:tauxU. 
We start with a corollary of p:tauxU, which says that one can basically apply p:tauxU to successive time intervals in order to obtain estimates on Γ(t_*).

There exists B'>0 such that for all z_0 with H(z_0) large enough,
[eq:toshowcor54]
_z_0(A_1 ∩{Γ(t_*) < εt_* /4 H(z_0) }) ≤e^- B'H(z_0) .




Fix z_0 and let E =  H(z_0).
Consider the sequence of stopping times
[eq:stoppingtimes]
τ_0 =0 ,     τ_j+1= τ_j + τ(z_τ_j) ,

with τ(z) for z∈Ω as in eq:defstoptx. We now introduce the random variable

J = sup{ j:  τ_j ≤t_*} .


On A_1, we have for all t≤ t_* that

λ(2E)^1/ℓ_i - 1/2 ≤τ(z_t) ≤λ(E/2)^1/ℓ_p - 1/2 ,

and hence that

J ≤Ĵ ≡⌊t_*(2E)^1/2 - 1/ℓ_iλ^-1 ⌋ .

Moreover, if E is large enough (and in the case ℓ_p = 2, using that λ≤ t_*/2), we have on A_1 that J > 0 and that

τ_J > t_* - τ(z_τ_J)  ≥t_*/2 .



Consider next the events

G_j    =   {J > j}∩{∑_b∈ γ_b ∫_τ_j^τ_j+1 p_b^2(s) ds < ετ(z_τ_j) H(z_τ_j) } ,

G    = ⋃_j≥0  G_j =  { ∃j< J : ∑_b∈ γ_b ∫_τ_j^τ_j+1 p_b^2(s) ds < ετ(z_τ_j) H(z_τ_j) } .

We observe that the event A_1 ∩{J > j} is a subset of

	Ã_j ≡{H(z_τ_j) ≥E/2  and   H(z_t) ≤4H(z_τ_j)  ∀t∈[τ_j, τ_j+1]} .

Thus, if E is large enough, we find by p:tauxU and the strong Markov property that
for all j≥ 0,

_z_0(A_1 ∩G_j) ≤e^-BE/2 ,

so that
[eq:probGG]
_z_0(A_1 ∩G) ≤∑_j=0^Ĵ-1 _z_0(A_1 ∩G_j) ≤Ĵ e^-BE/2 ≤e^-B' E

if B' > 0 is small enough and E large enough. 

We observe next that on A_1 ∩ G^c and for all E large enough,
[eq:decompdiss]
Γ(t_*)    ≥∑_j=0^J-1∑_b∈ γ_b ∫_τ_j^τ_j+1 p_b^2(s) ds ≥∑_j=0^J-1ετ(z_τ_j) H(z_τ_j) 

    ≥εE/2  ∑_j=0^J-1 τ(z_τ_j) =  εE/2  τ_J ≥εE/4  t_* .

Thus, the left-hand side of (<ref>) is bounded by _z_0(A_1 ∩ G), which by  eq:probGG completes the proof.





There are constants ρ, q >0 such that for every initial condition z_0, every event A, and all t>0,
[eq:rel1sq]
_z_0 (e^θH(z_t) - θH(z_0)_A)	     ≤e^C_* t(_z (e^-ρΓ(t) _A) )^1/q ≤e^C_* t  ,

with again C_* = θ∑_ b∈γ_b T_b.



	This proof is as in <cit.>. By applying the Itô formula to H(z_t), we find

_z_0 (e^θH(z_t) - θH(z_0)_A)	     = e^C_* t _z_0 (e^-θΓ(t) + θM_t  _A) ,

where

M_t     =  ∫_0^t ∑_b∈ √(2γ_b T_b) p_b(s) dW_b(s) .

The quadratic variation of M_t satisfies
[eq:quadvarMt]
 [M]_t =  2∫_0^t ∑_b∈ γ_b T_b p^2_b(s) ds ≤2 T_max Γ(t) .

Let p> 1 be such that pθ  < 1/T_ max and let q be such that 1/q + 1/p = 1. By Hölder's inequality,

   _z_0 (e^-θΓ(t) + θM_t  _A) = _z_0 ( e^-θΓ(t) + pθ^2/2 [M]_t _A e^ θM_t -  pθ^2/2 [M]_t)

        ≤(_z_0 e^-θq Γ(t) + qpθ^2/2 [M]_t _A )^1/q  (_z_0 e^ p θM_t -  p^2θ^2/2 [M]_t )^1/p .

The expectation in the second bracket in the last line is ≤ 1, since the exponential there is a  Doléans–Dade exponential, and thus a supermartingale. Finally, by (<ref>) we obtain eq:rel1sq with ρ = θ q(1-pθ T_ max) > 0.





There exists c>0 such that for all z_0 with H(z_0) large enough,
[eq:2ndbound]
_z_0(A_3)≤e^-c H(z_0) .



This is a classical result (see for example <cit.> or the proof of Theorem 3.5 in <cit.>). Observe that by eq:lyapunovplus,

(∂_t + )(e^θH-C_*t) = (-C_*) e^θH-C_*t≤0 .

Consider the stopping time σ = min(t_*, inf{t≥ 0: H(z_t) >  2H(z_0)}) (with the convention inf∅ = +∞).	 Then,  σ is a bounded stopping time, and we have by Dynkin's formula

_z_0 e^θH(z_ σ)-C_* t_*     ≤_z_0  e^θH(z_ σ)-C_* σ 

    =  e^θH(z_0) + _z_0 ∫_0^σ ((∂_s + )(e^θH-C_*s))(z_s) d s .

As the expectation in the last line is non-positive, we find _z_0  e^θ H(z_σ)≤ 	 e^C_* t_* +θ H(z_0), and thus

_z_0(A_3)     = _z_0{σ<  t_*} ≤e^-2θH(z_0) _z_0 (e^θH(z_ σ) _σ< t_*)≤e^C_* t_* - θH(z_0) ,

where the last inequality uses eq:rel1sq.
Thus, choosing c small enough completes the proof.




We can now give the


First, we have by lem:bornerhoUt and <Ref> that if H(z_0) is large enough,
[eq:pieceA1]
   _z_0 (e^θH(z_t_*) - θH(z_0)_A_1)	  ≤e^C_* t(_z_0 (e^-ρΓ(t_*) _A_1) )^1/q 

        ≤e^C_* t(e^- B'H(z_0) + e^-ρεt_* H(z_0)/4  )^1/q ≤e^- cH(z_0)

for some small enough c>0.
We next work on A_2. Consider the stopping time σ = min(t_*, inf{t≥ 0: H(z_t) <  H(z_0)/2}) (again with inf∅ = +∞). We have A_2 = {σ <  t_*} and

	_z_0 (e^θH(z_t_*)_A_2)    ≤e^θH(z_0)/2_z_0 (e^θH(z_t_*)- θH(z_σ)_A_2)  ≤e^θH(z_0)/2 + C_* t_* ,

where we have used the strong Markov property, eq:rel1sq, and the fact that t_*-σ≤ t_*. But then,
[eq:pieceA2]
_z_0 (e^θH(z_t_*)- θH(z_0)_A_2) ≤e^C_*t_*- θH(z_0)/2 ≤e^-c H(z_0) ,

if c>0 is small enough and H(z_0) is large enough.


Finally, by lem:bornerhoUt and lem:apriori, we have
[eq:pieceA3]
_z_0 (e^θH(z_t_*)- θH(z_0)_A_3)     ≤e^C_* t_* (_z_0 (A_3) )^1/q ≤e^- cH(z_0) ,

which has the desired form again. Summing eq:pieceA1, eq:pieceA2 and eq:pieceA3 completes the proof.




 Above, we split the time interval [0,t_*] into many subintervals, and apply p:tauxU to each of them. This is what allows us to obtain eq:boundC1C2, which is very natural from the dimensional point of view. In comparison,  <cit.>	 use the same Lyapunov function, but obtain weaker estimates (but still sufficient to obtain exponential convergence in eq:convergencemu): the bound obtained in <cit.> is _z_0 e^θ H(z_t_*)- θ H(z_0)≤ e^- C_1 H^r(z_0) with r∈ (0,1), and in <cit.> it is only shown that lim_z_0→∞_z_0 e^θ H(z_t_*)- θ H(z_0) = 0.




It now remains to prove p:tauxU. In order to do so, we start with some technical lemmas.


Let r≥ 1 and let f:^r →^r be a locally Lipschitz function.
For T>0, let V ∈([0,T], ^r) and consider

d x_t = f(x_t)dt + dV(t),    d y_t = f(y_t)dt

with initial conditions x_0 = y_0 ∈^r.
Then, provided that both x and y exist up to time T,

sup_t∈[0,T]x_t-y_t  ≤e^k_*T   sup_t ∈[0,T]V(t)  ,    k_* = sup_t ∈[0,T]f(x_t)-f(y_t)/x_t-y_t ,

with the convention 0/0 = 0.


Setting Δ_s = x_s-y_s, we have
Δ_t ≤∫_0^t k_* Δ_s ds  	+  V(t)
and the result follows from Gronwall's inequality.


rem:afterlemmecontinuite
We will later use lem:lemmecontinuite to show that, after adequate rescaling, e:SDE (or a component thereof) converges to a deterministic dynamics at high energy.




As a consequence of the definition of H, <ref> and replace510 (iv), we immediately
obtain

There is a constant C>0 such that for all z∈Ω, v∈ and e∈,
[eq:scalingvar1]
q_v ≤C (1+H^1/ℓ_p(z)) ,  δq_e ≤C(1+H^1/ℓ_i(z)) ,  p_v ≤C H^1/2(z) .

	


We are now ready to prove p:tauxU. We treat the case where
H_i(z_0) ≥ H(z_0)/2 in sec:interacDom and the case where H_c(z_0) > H(z_0)/2 in sec:pinningdom.
When ℓ_i = ℓ_p, such a distinction is not necessary and only the analysis in sec:interacDom is required.




 §.§ When the interactions dominate

In this subsection, we make
 If ℓ_i > ℓ_p, we assume that z_0 ∈Ω is such that H_i(z_0) ≥ H(z_0)/2.
(If ℓ_i = ℓ_p, we make no such restriction.)


We write E =  H(z_0). Consider the rescaled time σ = E^1/2 - 1/ℓ_it and the variables
[eq:rescaletilde1]
p̃_v(σ)    = E^-1/2p_v(E^1/ℓ_i - 1/2σ) ,

q̃_v(σ)    = E^-1/ℓ_iq_v(E^1/ℓ_i - 1/2σ) .


We write z̃ = (p̃, q̃) and z̃_0 for the rescaled initial condition. We consider times t∈ [0,τ(z_0)] = [0,λ E^1/ℓ_i -  1/2], or equivalently σ∈ [0, λ].
Observe that in terms of the rescaled time and variables,  eq:resultptauxU reads
[eq:toshowinteracdom]
_z_0(Ã ∩{∫_0^λ∑_b∈ γ_b p̃_b^2(σ) dσ< ελ} )  ≤e^-BE .

In the remainder of this section, we show that eq:toshowinteracdom holds provided E is large enough and z_0 satisfies ass:interdom.

Introducing

    H̃(p,q) = ∑_v∈p_v^2/2 +  ∑_v∈ E^-1 U_v(E^1/ℓ_iq_v)+ ∑_e∈E^-1V_e(E^1/ℓ_iδ q_e) ,

it is easy to see that
[eq:tildesysinterac]
d q̃_v   = p̃_v d σ ,

d p̃_v    = -(∇_q_v H̃)(p̃, q̃)dσ- E^1/ℓ_i - 1/2  γ_v p̃_v d σ

        + E^1/2 ℓ_i - 3/4 √(2 T_v γ_v) d W̃_v(σ)  ,

where W̃_v(σ) = E^-1/2 ℓ_i + 1/4 W_v(E^1/ℓ_i - 1/2σ) is again an n-dimensional Brownian motion. Clearly, in eq:tildesysinterac, the stochastic term vanishes in the limit E →∞, and so does the dissipative term, except when ℓ_i = 2.


Observe that when E→∞, the Hamiltonian H̃ converges pointwise to

    Ĥ(p,q) = ∑_v∈p_v^2/2 +  δ_ℓ_i, ℓ_p∑_v∈ U_v, ∞(q_v)+ ∑_e∈V_e,∞(δ q_e) ,

where U_v, ∞ and V_e,∞ are defined in <ref>.


Moreover, by construction,

 	H(z) = E H̃(z̃) ,

and in particular, 
[eq:initenergy1]
H̃(z̃_0) = 1 .



We introduce the set

K̃_E = {z̃: H̃(z̃) ≤4} .

On the event Ã, we have H(z_t) ≤ 4E for all t∈ [0, τ(z_0)], and hence also

z̃_σ∈K̃_E,   0≤σ≤λ .



By eq:scalingvar1, there exists C̃ > 0 such that if E is large enough, we have that for all z̃∈K̃_E,
[eq:scalingvartilde1]
q̃_v ≤C̃ E^1/ℓ_p - 1/ℓ_i ,  δq̃_e ≤C̃  ,  p̃_v ≤C̃  .



Note that if ℓ_i > ℓ_p, then q̃_v may become arbitrarily large when E is large, so that the set K̃_E is not bounded uniformly in E. In fact, when ℓ_i > ℓ_p, it is not true that sup_z̃∈K̃_E | H̃(z̃) -    Ĥ(z̃)| goes to zero when E→∞. Indeed, for all E one can find z̃∈K̃_E such that all the energy is in the pinning potential, so that Ĥ(z̃) = 0 but H̃(z̃) = 1. This explains why we have to restrict ourselves to initial conditions such that H_i(z_0) ≥ H(z_0)/2 (which will guarantee that Ĥ (z̃_0) is not too small), and then treat the opposite case separately in sec:pinningdom. This distinction is missing from the proofs in <cit.>.




For all 0≤ |α | ≤ 1 and e∈, we have
[eq:limInteracTilde]
	lim_E→∞sup_z̃ ∈K̃_E |D^α_z̃(E^-1V_e(E^1/ℓ_iδq̃_e) - V_e,∞(δq̃_e)) |    = 0 .


Let v∈.
If ℓ_i = ℓ_p and 0≤ |α | ≤ 1 (case 1) or if ℓ_i >ℓ_p
and |α |=1 (case 2), then:

lim_E→∞sup_z̃ ∈K̃_E
        |D^α_z̃(E^-1 U_v(E^1/ℓ_iq̃_v) - δ_ℓ_i, ℓ_p U_v,∞(q̃_v))|    = 0 ,

	lim_E→∞sup_z̃ ∈K̃_E |(D^αH̃)(z̃) -   (D^αĤ)(z̃)|    = 0 .
	





The first identity follows immediately from <ref>, replace510 (iii) and eq:scalingvartilde1.
Assume now we are in case 1. By <ref> and  replace510 (iii),

	lim_E→∞sup_x ≤C̃   |(E^|α|/ℓ_i-1 (D^αU_v)(E^1/ℓ_ix) - (D^αU_v,∞)(x))| = 0 .

This together with eq:scalingvartilde1 proves eq:limPinningTilde. 

Assume now we are in case 2. By eq:scalingvartilde1, in order to prove eq:limPinningTilde, it is enough to show that when  |α| = 1,
[eq:equivboundlime]
	lim_E→∞sup_x ≤C̃ E^1/ℓ_p - 1/ℓ_i  |E^|α|/ℓ_i-1 (D^αU_v)(E^1/ℓ_ix) | = 0 .

By replace510 (ii), there exists a constant c such that |(D^α U_v)(E^1/ℓ_ix)| ≤ c(1+E^ℓ_p/ℓ_i - |α|/ℓ_ix^ℓ_p - |α|). From this, we obtain that for some c'>0, the supremum in eq:equivboundlime is bounded above by

	 c'(E^|α|/ℓ_i-1 + E^   |α|/ℓ_i -|α|/ℓ_p) .

Clearly, since  |α | = 1 < 2 ≤ℓ_p < ℓ_i, the above vanishes when E→∞, which proves eq:equivboundlime and hence eq:limPinningTilde.

Finally, in both cases, combining eq:limInteracTilde and eq:limPinningTilde yields eq:limHTildeequa (recalling that the kinetic parts in Ĥ and H̃ are identical).


We now observe that for all E large enough,
[eq:hatHTildex0]
Ĥ(z̃_0) ∈[1/4, 2] .
 
Indeed, if ℓ_i = ℓ_p, this follows from eq:initenergy1 and eq:limHTildeequa. If ℓ_i > ℓ_p, then  ass:interdom ensures that |∑_v∈ E^-1 U_v(E^ 1/ℓ_iq_v)| ≤  1/2, so that eq:initenergy1 and eq:limInteracTilde indeed imply eq:hatHTildex0 for E large enough. 


Next, eq:tildesysinterac can be rewritten as
[eq:tildesysinteracR]
d q̃_v   = p̃_v d σ ,

d p̃_v    = -(∇_q_v Ĥ)(p̃, q̃)dσ+ R̃_v(q̃)d σ

        - E^1/ℓ_i - 1/2  γ_v p̃_v d σ+ E^1/2 ℓ_i - 3/4 √(2 T_v γ_v) d W̃_v(σ)  ,

where R̃_v(q̃) =  -∇_q̃_v (H̃(z̃) - Ĥ(z̃)), which by lem:convergenceTildeInteracDom satisfies, regardless of whether ℓ_i > ℓ_p or ℓ_i = ℓ_p, 
[eq:tildeRtozero]
lim_E→∞	sup_z̃∈K̃_E R̃_v(q̃_v)  →0 .


Consider now the deterministic limiting system
[eq:limitsysteminterac]
d q̂_v   = p̂_v d σ ,

d p̂_v    = -(∇_q_v Ĥ)(ẑ)dσ- δ_ℓ_i, 2γ_v p̂_v d σ ,

with initial condition ẑ_0 = z̃_0.






There is a constant C>0 such that for every initial condition ẑ_0 such that Ĥ(ẑ_0) ∈ [1/4, 2], the solution of eq:limitsysteminterac satisfies
[eq:int0tauhat]
∫_0^λ∑_b∈ γ_b p̂_b^2(σ) dσ≥C .



We first show that 
[eq:int0tauhatavant]
∫_0^λ∑_b∈ γ_b p̂_b^2(σ) dσ> 0   if   Ĥ(ẑ_0)> 0 .

Indeed, assume  the left-hand side of eq:int0tauhatavant is
zero. Then, for all b∈, we have p̂_b(σ) ≡  0
on [0, λ]. Take now v ∈∖. There exists then b∈ such that b is linked only to v
and possibly some vertices in . Now, since the masses in  do not move,
all forces among them are constant (this applies to both the
interaction forces -∇ V_e, ∞ with e∈× and, if ℓ_i = ℓ_p, to the pinning forces -∇ U_b', ∞ with b'∈). Thus, since the total
force on b is identically zero, we must have that ∇ V_(b,v),
  ∞(q̂_b(σ)-q̂_v(σ)) is constant. But then,
by <ref>, this means that actually q̂_b(σ) -q̂_v(σ) is constant, and hence that so is q̂_v(σ). We have thus shown that p̂_v(σ) ≡ 0 for all v∈. Proceeding in the same way, we obtain inductively that the same
holds for all v in ^2, ^3, etc. Thus, by
<ref>, we eventually obtain that 
no mass moves during
the time interval [0, λ]. But then we
have p̂_v(0)=0 and ∇_q_vĤ(ẑ_0) = 0 for all v∈,
which is only possible if Ĥ (ẑ_0) = 0, so that (<ref>) holds.

We now complete the proof of the proposition using a compactness argument and the fact that the solution of eq:limitsysteminterac depends continuously on the initial condition ẑ_0. In order to do so, there are two cases to consider.

	
  * ℓ_i = ℓ_p. Then, the set {ẑ: Ĥ(ẑ) ∈ [1/4,2]} is compact, and hence eq:int0tauhat holds for some C>0.

  * ℓ_i > ℓ_p. Then, the set {ẑ: Ĥ(ẑ) ∈ [1/4,2]} is not compact, since it is invariant under global translations q_v ↦ q_v + ρ, where ρ is any vector in ^n independent of v. But when ℓ_i > ℓ_p, both the dynamics eq:limitsysteminterac and the left-hand side of eq:int0tauhat are invariant under such translations. Since the set {ẑ: Ĥ(ẑ) ∈ [1/4,2]} is compact modulo such translations, we obtain eq:int0tauhat for some C>0.

This completes the proof.


 Note that
  <ref>
is only used to prove
  eq:int0tauhatavant.
In fact, there are systems for which
  eq:int0tauhatavant, and hence all the results in the present
  paper, hold without <ref>. For example,
  consider a chain of N oscillators with heat baths at both ends, , = {1, …, N},
  = {1, N} and = {(1, 2), (2, 3), …, (N-1,
  N)}. Let ℓ_i > ℓ_p, so that the limiting system only involves
the interaction potentials. Assume the left-hand side of eq:int0tauhatavant is
  zero. Then, on the time interval [0,λ], we have p̂_1(σ) ≡ 0. But then, we must have ∇ V_(1,2),
    ∞(q̂_2(σ)-q̂_1(σ)) ≡ 0 (unlike in
  the general case, we know here that the constant is zero, since no
  other force acts on the first oscillator). As a consequence, since
  the only stationary point of V_(1,2), ∞ is at the origin
  (this is true of any coercive, homogeneous function without the need
  for <ref>), we must have q̂_2(σ)
  ≡q̂_1(σ). But then we also have p_2(σ) ≡
  0. Continuing like this along the chain, we eventually obtain that
  all the masses stand still, and conclude as above that Ĥ(ẑ_0) = 0. 





 <ref> cannot be waived in general. We give here a counterexample in three[Everything in this example happens in the Oxy-plane. The third dimension is necessary only to ensure that the interaction potential is non-degenerate.] dimensions consisting of two oscillators 1 and 2, the first of which is coupled to a heat bath (see fig:C4matters). We start with both oscillators at rest at position  (0,1,0) and  (4,2,0) respectively. We assume that V_21(x,y,z) = y^4/4 + x^2z^2/2 when (x,y,z)≈ (4,1,0), and that U_1 = U_2 = U, where U(x,y,z) = x^4+y^4+z^4/4 when (x,y,z)≈(0,1,0) and U(x,y,z) =  x^4/64-y^4/32 + z^4/4  when (x,y,z)≈(4,2,0). The potentials above can be extended to non-degenerate, coercive, homogeneous functions of degree 4 (note in particular that Ĥ = H). Moreover, <ref> is not satisfied, as ∇ V_21(4 + ε, 1, 0) = (0, 1, 0) for all small enough |ε|. In this setup, the initial energy of the system is non-zero, and there exists a finite time interval during which the following happens: oscillator 1 does not move at all (so that (<ref>) fails to hold if λ is small enough), and the position of oscillator 2 is (x(t), 2, 0), for some decreasing x(t). The interaction force f_21 (see fig:C4matters) remains equal to (0,1,0), and the pinning force f_1 acting on oscillator 1 remains equal to (0,-1, 0). During the same time, the pinning force f_2 acting on oscillator 2 is equal to (-x^3(t)/16, 1, 0), consistently with the motion described above.







Returning to the proof of (<ref>), we compare now the systems (p̃, q̃) and (p̂, q̂).
There exist a constant c>0, a family of constants (G_E)_E > 0 satisfying lim_E→∞ G_E = 0, 
and a family of non-negative random variables (η_E)_E > 0 satisfying
[eq:Rtails]
{η_E ≥s}	≤e^-s^2/2 ,

such that if E is large enough,

_Ãsup_σ∈[0, λ] z̃_σ- ẑ_σ ≤G_E + c E^1/2 ℓ_i - 3/4 η_E .



The result immediately follows from  lem:lemmecontinuite and eq:tildeRtozero, provided we can show that there exists an absolute constant k>0 such that on the event Ã, we have (∇Ĥ)(z̃_σ) - (∇Ĥ)(ẑ_σ) ≤ k z̃_σ - ẑ_σ for all 0≤σ≤λ (we need not worry about the other terms in eq:limitsysteminterac, as they are globally Lipschitz).
	As mentioned above, on the event Ã, we have z̃_σ∈K̃_E for all 0≤σ≤λ. Moreover, since d/d σĤ(ẑ_σ) = - δ_ℓ_i,2∑_b∈γ_b p̂_b^2 ≤ 0, we have by eq:hatHTildex0 that Ĥ(ẑ_σ) ≤ 2 for all 0≤σ≤λ. We consider again two cases separately.


	
  * ℓ_i = ℓ_p. Then, there exists R>0 such that for all E large enough, z̃_σ≤ R and ẑ_σ≤ R for all 0≤σ≤λ. Since ∇Ĥ is locally Lipschitz (by <ref>), the proof is complete.

  * ℓ_i > ℓ_p. Then, one can find R>0 such that δq̃_e(σ), δq̂_e(σ) , p̃_v(σ) and p̂_v(σ) are bounded by R for all   0≤σ≤λ. Since ∇Ĥ  is locally Lipschitz and depends only the δ q_e and p_v, the proof is complete.


Note that by  lem:lemmecontinuite, the random variable η_E can be chosen as a constant times sup_σ∈ [0,λ]W̃(σ), where W̃ = (W̃_b)_b∈ is an n||-dimensional Brownian motion. While W̃ depends on E pathwise, its distribution does not. Moreover, W̃ does not depend on  z_0 for a given energy E, and thus the same is true of η_E.


Using lem:convInteracDom, prop:dissipGCInteracDom and the inequality x^2  ≥y^2/2 -	(x-y)^2, we obtain that there exist c, c'>0 such that on Ã and if E is large enough,

∫_0^λ∑_b∈ γ_b p̃_b^2(σ) dσ   ≥c -  c'(G_E + E^1/2 ℓ_i - 3/4 η_E)^2  ≥c -  2c'G_E^2 - 2 c'E^1/ ℓ_i - 3/2 η_E^2 .

Since G_E→ 0, we find for E large enough that

∫_0^λ∑_b∈ γ_b p̃_b^2(σ) dσ≥c/2 - 2c' E^1/ ℓ_i - 3/2 η_E^2 ,

so that

_z_0(Ã∩{∫_0^λ∑_b∈ γ_b p̃_b^2(σ) dσ< c/4}) ≤{η_E >E^ 3/4- 1/2 ℓ_i √( c/  8c')} .

Using now eq:Rtails and the fact that 1/ℓ_i - 3/2≤ -1 completes the proof of eq:toshowinteracdom (for an adequate choice of ε and B).

Thus, if ℓ_i = ℓ_p, the proof of p:tauxU is complete. If now ℓ_i > ℓ_p, then because of ass:interdom, the conclusion of p:tauxU is proved only in the case where H_i(z_0) ≥ H(z_0)/2, and the next subsection is required.



 §.§ When the pinning dominates


Recalling the decomposition of H introduced in eq:separationHams, we now make the following assumption.

 We assume that ℓ_i > ℓ_p and that
the initial condition z_0 ∈Ω satisfies H_c(z_0) > H(z_0)/2.



We start by rescaling the system in much the same way as in sec:interacDom, except that we now choose the natural scaling of the pinning.
More precisely, we introduce the rescaled time σ = E^1/2 -  1 /ℓ_pt and the variables

p̃_v(σ)    = E^-1/2p_v(E^1/ℓ_p - 1/2σ) ,

q̃_v(σ)    = E^-1/ℓ_pq_v(E^1/ℓ_p - 1/2σ) .

 We consider times t∈ [0,τ(z_0)] = [0,λ E^1/ℓ_p - 1/2], or equivalently σ∈ [0, λ]. As in sec:interacDom, the analogue of eq:resultptauxU in terms of the rescaled variables and time is
[eq:toshowpinningdom]
_z_0(Ã ∩{∫_0^λ∑_b∈ γ_b p̃_b^2(σ) dσ< ελ} )  ≤e^-BE .




We let now

H̃(p,q) = ∑_v∈p_v^2/2 +  ∑_v∈ E^-1 U_v(E^1/ℓ_pq_v)+ ∑_e∈E^-1V_e(E^1/ℓ_pδq_e) ,

and obtain
[eq:tildesysintpin]
d q̃_v   = p̃_v d σ ,

d p̃_v    = -(∇_q_v H̃)(p̃, q̃)dσ- E^1/ℓ_p - 1/2  γ_v p̃_v d σ+ E^1/2 ℓ_p - 3/4 √(2 T_v γ_v) d W̃_v(σ)  ,  
where W̃_v(σ) = E^-1/2 ℓ_p + 1/4 W_v(E^1/ℓ_p - 1/2σ) is again an n-dimensional Brownian motion.
We define, as in sec:interacDom,

K̃_E = {z̃: H̃(z̃) ≤4} ,

and obtain that on the event Ã, we have z̃_σ∈K̃_E for all 0≤σ≤λ.

By eq:scalingvar1, there is some C̃ such that if E is large enough, then for all z̃∈K̃_E,
[eq:scalingvartildepin]
q̃_v ≤C̃ ,  δq̃_e ≤C̃  E^1/ℓ_i - 1/ℓ_p  ,  p̃_v ≤C̃  .


Note that unlike in sec:interacDom, the collection of sets (K̃_E)_E>0 is uniformly bounded.
In fact, the maximum allowed value of δq̃_e becomes very small at high energy. 


	The difficulty is that the dynamics eq:tildesysintpin does not converge to a nice limit when E is large. Indeed, we have for any edge e=(v,v') that

	∇_q̃_v (E^-1 V_e(E^1/ℓ_pδq̃_e)) ∼E^ℓ_i/ℓ_p-1 δ̃q_e^ℓ_i-1 ,

which diverges pointwise when E→∞ if δq̃_e ≠ 0. The supremum of this quantity over K̃_E diverges like 
E^1 /ℓ_p - 1/ ℓ_i (as can be seen by the scaling in eq:scalingvartildepin).
The interpretation is that at high energy and under ass:pindom, while the rescaled system behaves like a  “tight molecule” 
with vanishing relative distance δ̃q_e between the masses, the dynamics is still dominated by
the fast oscillations of the internal degrees of freedom. The way around this is to consider the center of mass coordinates.


The center of mass coordinates in eq:defCDMcoord are expressed, after rescaling, as

P̃(σ)    = E^-1/2P(E^1/ℓ_p - 1/2σ) = E^-1/2 ∑_v∈ p_v(E^1/ℓ_p - 1/2σ) ,

Q̃(σ)    = E^-1/ℓ_pQ(E^1/ℓ_p - 1/2σ) = 1/|| E^-1/ℓ_p ∑_v∈ q_v(E^1/ℓ_p - 1/2σ) .

We denote by (P̃_0, Q̃_0) the rescaled initial condition.
As the interaction forces cancel out, the dynamics we obtain is 
[eq:PtQtpinfirst]
d Q̃    =  1/|| P̃ d σ ,

d P̃     =-∑_v∈E^1/ℓ_p-1∇U_v(E^1/ℓ_pq̃_v) dσ

            - E^1/ℓ_p -1/2 ∑_b∈ γ_b p̃_b d σ+  E^1/2 ℓ_p - 3/4 ∑_b∈√(2 T_b γ_b) d W̃_b(σ)  .



Moreover, since the graph (, ) is connected by <ref>, we have for all z̃∈K̃_E that
[eq:qtildemqtilde]
max_v∈Q̃-q̃_v	≤max_(v, v') ∈^2 q̃_v - q̃_v'≤∑_e∈ δq̃_e ≤||C̃ E^1/ℓ_i - 1/ℓ_p   .


Defining now

U_∞(Q̃) = ∑_v∈U_v, ∞(Q̃) ,

we have

For all 0≤ |α| ≤ 1,

lim_E→∞sup_z̃ ∈K̃_E| ∑_v∈E^|α|/ℓ_p-1(D^αU_v)(E^1/ℓ_pq̃_v)-(D^αU_∞) (Q̃)| = 0 .



By <ref>, eq:scalingvartildepin and replace510 (iii) we have, for all v∈,
[eq:premierbornelem217]
lim_E→∞sup_z̃ ∈K̃_E|E^|α|/ℓ_p-1(D^αU_v)(E^1/ℓ_pq̃_v) - (D^αU_v,∞)(q̃_v)| = 0 .

Moreover, by <ref>, there exists c>0 such that D^α U_v, ∞ is c-Lipschitz on the ball B(0, C̃)⊂^n, which  by eq:scalingvartildepin contains q̃_v and Q̃ for all z̃∈K̃_E. Thus,

sup_z̃ ∈K̃_E|(D^αU_v,∞)(q̃_v)-(D^αU_v,∞)(Q̃)| ≤csup_z̃ ∈K̃_EQ̃ - q̃_v  .

By eq:qtildemqtilde, the right-hand side vanishes when E→∞. This and eq:premierbornelem217 imply that

lim_E→∞sup_z̃ ∈K̃_E|E^|α|/ℓ_p-1(D^αU_v)(E^1/ℓ_pq̃_v) - (D^αU_v,∞)(Q̃)| = 0 .

By the definition of U_∞ and the triangle inequality, the proof is complete.


It is then natural to consider the limiting system
[eq:limitsystempinning]
d Q̂    = 1/||P̂ d σ ,

d P̂     =-∇U_∞(Q̂)d σ ,

which corresponds to the Hamiltonian

Ĥ(P̂,Q̂) = P̂^2/2||   + U_∞(Q̂) .



We can rewrite eq:PtQtpinfirst as
[eq:prelimitsystempinning]
d Q̃    =  1/||P̃ d σ ,

d P̃     =-∇U_∞(Q̃)d σ+ R̃(z̃) dσ- E^1/ℓ_p -1/2 ∑_b∈ γ_b p̃_b d σ

        +  E^1/2 ℓ_p - 3/4 ∑_b∈√(2 T_b γ_b) d W̃_b(σ) ,

where

R̃(z̃) = ∇U_∞(Q̃)- E^1/ℓ_p-1 ∑_v∈ (∇U_v)(E^1/ℓ_pq̃_v) ,

which by lem:convergencepotpindom satisfies
[eq:tildeRtozero pinning]
lim_E→∞	sup_z̃∈K̃_E R̃(z̃)  →0 .



Note that the dynamics eq:prelimitsystempinning does not converge to eq:limitsystempinning when ℓ_p = 2, as the dissipative terms in eq:prelimitsystempinning remain in the limit. This will complicate the argument slightly (see the proof of Lemma <ref>).

As a consequence of lem:convergencepotpindom, and since H_c(z_0) > H(z_0)/2, we have when E is large enough that

Ĥ(P̃_0, Q̃_0) ∈[1/4, 2] .



There is a constant C>0 such that for every initial condition (P̂_0,Q̂_0) with Ĥ(P̂_0, Q̂_0) ∈ [1/4, 2], the solution of eq:limitsystempinning satisfies
[eq:int0tauhatpin]
sup_σ∈[0, λ] Q̂(σ)-Q̂(0)≥C .



The left-hand side of eq:int0tauhatpin is obviously strictly positive provided that Ĥ(P̂_0, Q̂_0) > 0. Moreover, the map (P̂_0, Q̂_0)↦sup_σ∈ [0, λ]Q̂(σ)-Q̂(0) is lower semicontinuous (as the supremum of a family of continuous functions). Thus, since the set {(P̂_0, Q̂_0) ∈^2n :Ĥ(P̂_0, Q̂_0) ∈ [1/4, 2]} is compact, the proof is complete.




We introduce the random variable

X = sup_b∈ ∫_0^λp̃_b^2 d σ .



There exist constants B,ε > 0 such that if E is large enough,

_z_0(Ã ∩{X  < ε}) ≤e^-BE .
	


In this proof, the constant c>0 may be different each time it appears, and is not allowed to depend on E, provided E is large enough.
First, observe that for all b∈, Hölder's inequality implies that
[eq:byholder]
	 ∫_0^λp̃_bdσ≤c√(X) .


Next, by eq:scalingvartildepin and the fact that Ĥ is conserved by eq:limitsystempinning, we have that on the event Ã, there is some R>0 such that Q̂(σ), Q̃(σ), P̂(σ) and P̃(σ) are bounded by R for all 0≤σ≤λ. As ∇Ĥ is locally Lipschitz by <ref>, there exists k>0 such that on the event Ã, we have for all 0≤σ≤λ that

(∇Ĥ)(P̂(σ), Q̂(σ)) - (∇Ĥ)( P̃(σ), Q̃(σ))  ≤k (P̂(σ), Q̂(σ))- (P̃(σ), Q̃(σ))	 .

As a consequence, we can apply lem:lemmecontinuite to  (P̂, Q̂) and (P̃, Q̃) to obtain that
[eq:resultatdiffpetitere]
_Ãsup_σ∈[0, λ] Q̃(σ) - Q̂(σ)  ≤c G_E + cE^1/ℓ_p -1/2 √(X)+ c E^1/2 ℓ_p - 3/4 η_E ,

where lim_E→∞G_E = 0 and where η_E is a non-negative random variable satisfying eq:Rtails. 

Pick now any b∈. By eq:tildesysintpin and eq:byholder, we have
[eq:supsqtv]
	sup_σ∈[0,λ]q̃_b(σ) - q̃_b(0)  ≤c ∫_0^λp̃_bdσ≤c√(X) .

Moreover, by eq:qtildemqtilde, we also find that on Ã,
[eq:supqtildeQtildeA1]
	sup_σ∈[0,λ]q̃_b(σ) - Q̃(σ)  ≤c E^1/ℓ_i-1/ℓ_p  .

From eq:supsqtv and eq:supqtildeQtildeA1 we deduce that
[eq:surQtilde0sigma]
	sup_σ∈[0,λ]Q̃(σ) - Q̃(0)  ≤c E^1/ℓ_i-1/ℓ_p + c √(X)  .

This together with eq:resultatdiffpetitere implies that on Ã,

sup_σ∈[0, λ] Q̂(σ)-Q̂(0)≤c E^1/ℓ_i-1/ℓ_p +c G_E + c√(X)(1+E^1/ℓ_p -1/2 )+ c E^1/2 ℓ_p - 3/4 η_E .

But by prop:limitDomBoundedBelow, the left-hand side is bounded below by C>0. Thus,

cE^1/2 ℓ_p - 3/4 η_E ≥C 	-c E^1/ℓ_i-1/ℓ_p -c G_E -c√(X)(1+E^1/ℓ_p -1/2 ) .

We next choose ε > 0 small enough so that for all E large enough,

c E^1/ℓ_i-1/ℓ_p +c G_E + c√(ε)(1+E^1/ℓ_p -1/2 ) ≤C/2 .

Then, on Ã and for E large enough,  X<ε implies cE^1/2 ℓ_p - 3/4η_E > C/2, so that

_z_0(Ã ∩{X < ε}) ≤_z_0{cE^1/2 ℓ_p - 3/4 η_E > C/2} ≤e^-cE^3/2 - 1/ℓ_p .

	Since 3/2 - 1/ℓ_p≥ 1, the proof is complete.


By lem:BvarepsilonX, and since

 ∑_b∈ ∫_0^λγ_b  p̃_b^2 d σ≥Xinf_b∈γ_b  ,

we obtain eq:toshowpinningdom (for some ε>0 possibly smaller than that of lem:BvarepsilonX). Thus, the proof of p:tauxU is complete.


This work was initiated in the pleasant atmosphere of the Fields
Institute, Toronto.
Financial support was kindly provided by the ERC Advanced grant  290843:BRIDGES (JPE),
the Swiss National Science Foundation Grant 165057 (NC),  the National Science Foundation (NSF)  grant DMS-1515712 (LRB), 
the Leverhulme Trust (MH), as well as the ERC Consolidator grant 615897:CRITICAL (MH). 








Martin




