Path-space moderate deviations for a Curie-Weiss model of self-organized criticality
Francesca Collet Matthias Gorny Richard Clemens Kraaij
math.PR 60F10, 60K35


Calibration for Weak Variance-Alpha-Gamma Processes[This
research was partially supported by ARC grant DP160104037.]
    Boris BuchmannResearch School of Finance, Actuarial Studies & Statistics,
Australian National University,
ACT 0200,
Australia.
Email: boris.buchmann@anu.edu.au

Kevin W. LuMathematical Sciences Institute,
	Australian National University,
	ACT 0200,
	Australia.
	Email: kevin.lu@anu.edu.au
Dilip B. MadanRobert H. Smith School of Business, University of Maryland, College Park, MD. 20742, USA, Email: dbm@rhsmith.umd.edu

    December 30, 2023
========================================================================================================================================================================================================================================================================================================================================================================================================================================




	
	The dynamical Curie-Weiss model of self-organized criticality (SOC) was introduced in <cit.> and it is derived from the classical generalized Curie-Weiss by imposing a microscopic Markovian evolution having the distribution of the Curie-Weiss model of SOC <cit.> as unique invariant measure. In the case of Gaussian single-spin distribution, we analyze the dynamics of moderate fluctuations for the magnetization. We obtain a path-space moderate deviation principle via a general analytic approach based on convergence of non-linear generators and uniqueness of viscosity solutions for associated Hamilton-Jacobi equations. Our result shows that, under a peculiar moderate space-time scaling and without tuning external parameters, the typical behavior of the magnetization is critical. 

	
	Keywords: moderate deviations ·  interacting particle systems · mean-field interaction · self-organized criticality · Hamilton–Jacobi equation · perturbation theory for Markov processes








§ INTRODUCTION



In their very well-known article <cit.>, Bak, Tang and Wiesenfeld showed that certain large dynamical systems have the tendency to organize themselves into a critical state, without any external intervention. The amplification of small internal fluctuations can lead to a critical state and cause a chain reaction leading to a radical change of the system behavior. These systems exhibit the phenomenon of self-organized criticality (SOC) that since its introduction has been successfully applied to describe quite a number of
natural phenomena (e.g., forest fires, earthquakes, species evolution). Indeed, it has been conjectured that living systems self-organize by putting themselves in a state which is close to criticality.
In general, features of SOC have been observed empirically or simulated on a computer in various models; however, the mathematical analysis turns out to be extremely difficult, even for models whose definition is very simple  <cit.>. Self-organized criticality has been reviewed in recent works <cit.>.

The simplest models exhibiting SOC are obtained by forcing standard critical transitions into a self-organized state <cit.>. The idea is to start with a model presenting a phase transition and to create a feedback from the configuration to the control parameters in order to converge towards a critical state. Following this guideline, Cerf and Gorny designed an interacting particle system exhibiting self-organized criticality that is as simple as possible and is amenable to a rigorous mathematical analysis: a Curie-Weiss model of SOC <cit.>. They modified the equilibrium distribution associated to the generalized Curie-Weiss model (i.e., with real-valued spins <cit.>) by implementing an automatic control of the inverse temperature that, in the limit as the size n goes to infinity, drives the system into criticality without tuning any external parameter. Under an exponential moment condition and a symmetry assumption on the spin distribution, they proved that the magnetization behaves as in the generalized Curie-Weiss model when posed at the critical point: the fluctuations are of order n^3/4 and have limiting law ν(x) ∝exp( -x^4/12)   x. 

More recently, Gorny approached the problem from a non-equilibrium viewpoint and constructed a dynamical Curie-Weiss model of SOC <cit.>. He considered a Markov process whose unique invariant distribution is the Curie-Weiss model of SOC and proved, in the case of Gaussian spins, that the fluctuations evolve on a peculiar space-time scale (orders n^3/4, √(n)  t) and their limit is the solution of a “critical” SDE having ν as invariant measure.

The advantage of dealing with Gaussian spins is that it is possible to find a finite-dimensional order parameter to describe the system. In particular, the problem can be reduced to a bi-dimensional problem: the Langevin spin dynamics induce a Markovian evolution on the pair ((n^-1S_n(t),n^-1T_n(t)), t ≥ 0), with S_n := ∑_i=1^n X_i and T_n := ∑_i=1^n X_i^2, X_i's being the spin values. Therefore it suffices to analyze the behaviour of the latter observable. 



Our purpose is to characterize path-space moderate deviations for the dynamical model of SOC with Gaussian spins introduced in <cit.>. A moderate deviation principle is technically a large deviation principle and consists in a refinement of a central limit theorem, in the sense that it characterizes the exponential decay of the probability of deviations from the average on a smaller scale. 

We apply the approach to large deviations by Feng-Kurtz <cit.> to characterize the most likely behavior for the trajectories of fluctuations. The techniques are based on the convergence of Hamiltonians and well-posedness of a class of Hamilton-Jacobi equations corresponding to a limiting Hamiltonian H. These techniques have been recently exploited to analyze moderate fluctuations from equilibrium in the various regimes in the standard <cit.> and the random-field version <cit.> of the Curie-Weiss model. 
The major difference in comparison to these papers is that now we are dealing with unbounded spin state space. Nevertheless, we can implement the same strategy as in <cit.>. We use the perturbation theory for Markov processes <cit.> to formally identify a limiting operator H and we relax our definition of limiting operator to allow for unbounded functions in the domain. More precisely, we follow <cit.> and introduce two Hamiltonians H_† and H_, that are limiting upper and lower bounds for the sequence of Hamiltonians H_n, respectively. We then characterize H by matching the upper and lower bound. 

From a qualitative viewpoint, we derive a projected large deviation principle. Indeed, there is a natural time-scale separation for the evolutions of the two processes (n^-1S_n(t), t ≥ 0) and : n^-1T_n is fast and converges exponentially quickly to σ^2, the variance of the single-spin distribution, while n^-1S_n is slow and its limiting behavior can be determined after suitably “averaging out” the dynamics of n^-1T_n. Corresponding to this observation, we need to prove a large deviation principle for the component n^-1S_n only. 
Our main result shows that self-organized criticality is reflected by moderate deviations, since the rate function for the path-space moderate deviation principle retains the features of the “critical” evolution derived in <cit.>.


The outline of the paper is as follows: in Section <ref> we formally introduce the dynamical version of the Curie-Weiss model of SOC and we state the large deviation principle. The proof is given in Section <ref>. Appendix <ref> contains the mathematical tools needed to derive our large deviation principle via solving a class of associated Hamilton-Jacobi equations and it is included to make the paper self-contained. A similar version of the appendix appears also in <cit.>. 








§ MODEL AND MAIN RESULT







 §.§ Notation and definitions
 


Before starting with the main contents of the paper, we introduce some notation. We start with the definition of good rate-function and of large deviation principle for a sequence of random variables. 
	

Let (X_n)_n ∈ℕ^* be a sequence of random variables on a Polish space 𝒳. Furthermore, consider a function I : 𝒳→ [0,∞] and a sequence (r_n)_n ∈ℕ^* of positive numbers such that r_n ↑∞. We say that


  * the function I is a good rate-function if the set {x   |   I(x) ≤ c} is compact for every c ≥ 0.


  * the sequence (X_n)_n ∈ℕ^* is exponentially tight at speed r_n if, for every a ≥ 0, there exists a compact set K_a ⊆𝒳 such that lim sup_n r_n^-1log [X_n ∉ K_a] ≤ - a.


  * the sequence (X_n)_n ∈ℕ^* satisfies the large deviation principle with speed r_n and good rate-function I, denoted by 

    [X_n ≈ a] ≍ e^-r_n I(a),

if, for every closed set A ⊆𝒳, we have 

    lim sup_n ↑∞  r_n^-1log[X_n ∈ A] ≤ - inf_x ∈ A I(x),

and, for every open set U ⊆𝒳, 

    lim inf_n ↑∞  r_n^-1log[X_n ∈ U] ≥ - inf_x ∈ U I(x).




 
A curve γ: [0,T] →ℝ is absolutely continuous if there exists a function  such that for t ∈ [0,T] we have γ(t) = γ(0) + ∫_0^t g(s)  s. We write g = γ̇. A curve γ: ^+ →ℝ is absolutely continuous if the restriction to [0,T] is absolutely continuous for every T ≥ 0. Throughout the whole paper  will denote the set of absolutely continuous curves in .


To conclude we fix notation for some collections of function-spaces. 


Let k ≥ 1 and E a closed subset of ℝ^d. We will denote by 


  * C_l^k(E) (resp. C_u^k(E)) the set of functions that are bounded from below (resp. above) in E and are k times differentiable on a neighborhood of E in ℝ^d.


  * C_c^k(E) the set of functions that are constant outside some compact set in E and are k times continuously differentiable on a neighborhood of E in ℝ^d. Finally, we set 









 §.§ Description of the model and main result





Let ρ be a symmetric probability measure on ℝ, with variance σ^2, and such that we have ∫_ℝexp(az^2) ρ(z) < ∞, for every a ≥ 0. The generalized Curie-Weiss model associated with ρ and inverse temperature β >0 is an infinite triangular array of real-valued spin random variables (X_n^k)_1 ≤ k ≤ n having joint distribution


    μ^CW_n,ρ,β (z_1, …, z_n) = 1/Z_n(β)exp( β/2(z_1 + ⋯ + z_n)^2/n) ∏_i=1^n ρ(z_i),


where Z_n(β) is a normalizing constant. For any n ≥ 1, set S_n := X_n^1+⋯+X_n^n. We have the following results for the asymptotics of S_n (cf. <cit.>):

	
  * If β < 1/σ^2, then the fluctuations of S_n are of order √(n) and, in particular, S_n/√(n) converges in law to a centered Gaussian random variable with variance σ^2/1-βσ^2.
	
	
  * The point β = 1/σ^2 is the critical point for the system. The fluctuations of S_n become of higher order and their limit is no more Gaussian. Indeed, there exist k ∈ℕ∖{0,1} and λ > 0  (both depending on ρ), such that
	
	
    S_n/n^1-1/2k   C_k,λexp( -λs^2k/(2k)!)   s,

	
	where C_k,λ is a normalizing constant.
 

In <cit.> the authors modified the distribution (<ref>) so as to build a system of interacting random variables that exhibits a phenomenon of self-organized criticality. In other words, they constructed a spin system converging to the critical state of (<ref>) (corresponding to β=1/σ^2) without tuning any external parameter. Based on the observation that if the spins were independent the quantity n (z_1^2+⋯+z_n^2)^-1 would be a good estimator for 1/σ^2 by strong law of large numbers, they decided to replace the inverse temperature β in (<ref>) with n (z_1^2+⋯+z_n^2)^-1, obtaining


    μ^SOC_n,ρ (z_1, …, z_n) =  1/Z_nexp( 1/2(z_1 + ⋯ + z_n)^2/z_1^2+⋯+z_n^2) ∏_i=1^n ρ(z_i),


where Z_n is a normalizing constant. An infinite triangular array of real-valued spins  (X_n^k)_1 ≤ k ≤ n having joint distribution (<ref>) is a Curie-Weiss model of self-organized criticality  and it indeed evolves spontaneously towards criticality. The fluctuations of S_n, under (<ref>), have the same asymptotics as the critical generalized Curie-Weiss model, in the sense that they obey the same result as (<ref>) with a universal exponent k=2.

In <cit.> a dynamical version of the Curie-Weiss model of SOC was introduced. It consists in a Markov process, defined through a system of n interacting Langevin diffusions, whose unique invariant distribution is 


    μ̃^SOC_n,ρ (z_1, …, z_n) = 1/Z_nexp( 1/2(z_1 + ⋯ + z_n)^2/z_1^2+⋯+z_n^2+1) ∏_i=1^n ρ(z_i),


where Z_n is a normalizing constant. Observe that (<ref>) is a slight modification of (<ref>) aimed at avoiding technical difficulties due to ill-definition of the distribution at the origin and to the non-Lipschitzianity of the coefficients of the associated Langevin diffusions. Nevertheless the distributions (<ref>) and (<ref>) provide two equivalent formulations for a Curie-Weiss model of SOC (see <cit.> and references therein for further details). Now we come to the description of the dynamics we are interested in.

Let φ: ℝ→ℝ be an even function of class C^2 such that exp(2φ) is integrable over . Moreover, suppose that there exists a positive constant c such that, for any z ∈ℝ, zφ'(z)≤ c(1+z^2). We define ρ to be the probability measure having density


    ρ(z) = exp(2φ(z)) (∫_exp(2φ(w))  w)^-1,


with respect to the Lebesgue measure on . The dynamical counterpart of the Curie-Weiss model of SOC (<ref>) is an infinite triangular array of stochastic processes (X_n^k(t), t≥ 0)_1≤ k ≤ n such that, for all n ≥ 1, ((X^1_n(t),…,X^n_n(t)), t≥ 0) is the unique solution of the following system of stochastic differential equations:


    X_n^j(t)= 1/2[ 2 φ' ( X_n^j(t) ) + S_n(t)/T_n(t)+1 - X_n^j(t)( S_n(t)/T_n(t)+1)^2 ]  t +  B_j(t),    (j=1,…,n)


where

	
  * for every t ≥ 0, 
	
	
    S_n(t) := X_n^1(t)+…+X_n^n(t)      T_n(t)=(X_n^1(t))^2+…+(X_n^n(t))^2;

	
	
  * the process ((B_1(t),…,B_n(t)), t ≥ 0) is a standard n-dimensional Brownian motion.

The solution ((X^1_n(t),…,X^n_n(t)), t≥ 0) of (<ref>) is a Markov diffusion process on ℝ^n. For any f∈ C^2(^n) and z ∈^n, it evolves with infinitesimal generator


    L_nf(z)=1/2∑_j=1^n∂^2 f(z)/∂ z_j^2+ 1/2∑_j=1^n[ 2φ'(z_j) + S_n[z]/T_n[z]+1-z_j(S_n[z]/T_n[z]+1)^2]∂ f(z)/∂ z_j,


with S_n[z] := ∑_i=1^n z_i and T_n[z] := ∑_i=1^n z_i^2. We recall once more that the measure (<ref>) is the unique invariant distribution for L_n.


Our main aim is to describe the limiting behavior of moderate fluctuations for the evolution (<ref>); the technical difficulties arising have not allowed us to obtained the desired results under the present assumptions, in particular with no requirements on the function φ (except evenness and exponential integrability). Thus we find it preferable to make the following assumption at this point:


	
  (A) φ (z) = - z^24 σ^2, for some σ>0.


Assumption (A) corresponds to choosing the Gaussian probability density as reference measure ρ for the spin variables. Under assumption (A), the process ( (n^-1 S_n(t), n^-1T_n(t)), t ≥ 0 ) is a sufficient statistics for our model. Indeed, the dynamics (<ref>) on the configurations induce a Markovian dynamics on ℝ^2 for the process ( (n^-1 S_n(t), n^-1T_n(t)-σ^2), t ≥ 0 )  that evolves with generator 


    A_n f (x,y) = 1/2n∂^2 f/∂ x^2(x,y) +2x/n∂^2 f/∂ x ∂ y(x,y) + 2(y+σ^2)/n∂^2 f/∂ y^2(x,y)
    
    + 1/2[ -n^2 x^3/(ny+nσ^2+1)^2 + nx/ny+nσ^2+1 - x/σ^2] ∂ f/∂ x(x,y) 
    
    + [ nx^2/(ny+nσ^2+1)^2 - y/σ^2] ∂ f/∂ y(x,y).


The derivation  of the previous formula from (<ref>) is omitted, since it is tedious and rather standard. We refer to <cit.> for the detailed derivation of a similar result (the main difference being the space-time scaling the process is subject to).
	
As a consequence of (<ref>), the task of characterizing the time-evolution of the fluctuation flow


    ( 1/n∑_k=1^n δ_X_n^k(t) - ρ(z) )_t≥0


turns into analyzing the path-space deviations of ((n^-1 S_n(t), n^-1T_n(t) - σ^2), t ≥ 0 ). From being infinite dimensional, the problem reduces to a two dimensional problem.




First consider a standard central limit theorem setting and therefore consider the two dimensional process classically rescaled by √(n). Computing the formal limit of (<ref>) for functions of the variable x (resp. y) only, we find that, as n ↑∞, the process (n^-1/2 S_n(t), t ≥ 0 ) converges weakly to a standard Brownian motion, whereas ( √(n)(n^-1T_n(t) - σ^2), t ≥ 0 ) to the Ornstein-Uhlenbeck process solution of
	
    Y(t) = - Y(t)/σ^2 t + 2 σ  B_1(t),
 
	where B_1 is a standard Brownian motion.





Thus, the second component of the pair ((n^-1/2 S_n(t), √(n)(n^-1T_n(t) - σ^2 )), t ≥ 0 ) has a confined process as a limit, whereas the first one fluctuates homogeneously in space. Indeed, in this last case, as shown in <cit.>, a further rescaling  allows one to see that the process (n^-3/4 S_n(√(n)t), t ≥ 0 ) converges weakly to the solution of 
	
    X(t) = - X^3(t)/2 σ^4 t +  B(t),
 
with B(t) standard Brownian motion.
Under this critical space-time rescaling the process, ( n^-1T_n(t) - σ^2, t ≥ 0 ) collapses: at times of order √(n)t the process (√(n)(n^-1T_n(√(n) t) - σ^2), t ≥ 0 ) equilibrates at a Gaussian measure and therefore, when refining the space rescaling, the process ( n^1/4(n^-1T_n(√(n) t) - σ^2) ) equilibrates at δ_0. This was proven in <cit.>.
	
We complement the analysis by considering the moderate deviations of n^-1 S_n(t) around  equilibrium, under the microscopic dynamics (<ref>). As in the weak convergence setting mentioned above, corresponding to the separation of time-scales for the evolutions of the two processes, we need to prove a projected path-space large deviation principle, in other words for the component n^-1S_n only. More precisely, we get the following statement.  


	Let (b_n)_n ∈ℕ^* be a sequence of positive real numbers such that b_n ↑∞ and  . Suppose that b_n n^-1 S_n(0) satisfies a large deviation principle with speed nb_n^-4 on ℝ and rate function I_0. Then, the trajectories ( b_n n^-1 S_n(b_n^2 t), t ≥ 0 ) satisfy the large deviation principle
	
	
    ℙ[ ( b_n n^-1 S_n(b_n^2 t), t ≥ 0 ) ≈ (γ(t), t ≥ 0) ] ≍ e^-nb_n^-4 I(γ)

	
	on C_ℝ(ℝ^+), with good rate function
	
	
    I(γ) = 
    	
    	I_0(γ(0)) + ∫_0^+∞(γ(s),γ̇(s))  s     if γ∈𝒜𝒞,
    ∞    otherwise,

	
	where
	
	
    ℒ(x,v) := 1/2| v + x^3/2σ^4|^2.

	


By choosing the sequence b_n = n^α, with α > 0, we can rephrase Theorem <ref> in terms of more familiar moderate scalings involving powers of the system-size. We therefore get estimates for the probability of a typical trajectory on a scale that is between a law of large numbers and a central limit theorem.  

This result extends our understanding of the path-space fluctuations for the Curie-Weiss model of self-organized criticality, in the case of Gaussian spins. We have stated this result, in combination with the non-standard central limit theorem in <cit.> in Table <ref>.
The displayed conclusions are drawn under the assumption that in each case the initial condition satisfies a large deviation principle at the correct speed. Observe that self-organized criticality is reflected by moderate deviations, since the rate function retains the features of the “critical” evolution (<ref>).  

To conclude, it is worth to mention that the methods of the papers <cit.> are not sufficient to obtain a path-space large deviation principle for the process ((n^-1S_n(t),n^-1T_n(t), t ≥ 0) by the Feng-Kurtz approach. Indeed, the Hamiltonian is not of the standard type dealt with in <cit.> and it is not  immediately clear how the comparison principle can be treated.











§ PROOF




We aim at studying moderate deviations by following the methods in <cit.>. The techniques are based on the convergence of Hamiltonians and well-posedness of a class of Hamilton-Jacobi equations corresponding to a limiting Hamiltonian. These techniques have been applied also in <cit.>.  
In particular, in <cit.> moderate deviation principles for projected processes are proved by combining the perturbation theory for Markov processes with a sophisticated notion of convergence of Hamiltonians, based on limiting upper and lower bounds.  Here we apply those same techniques, as they allow to take care of unbounded spin state space. We summarize the notions needed for our result and the abstract machinery used for the proof of a large deviation principle via well-posedness of Hamilton-Jacobi equations in Appendix <ref>. We rely on Theorem <ref> for which we must check the following conditions:



  * The processes ( (b_n n^-1 S_n(b_n^2t), b_n ( n^-1 T_n(b_n^2 t) - σ^2 ) ), t ≥ 0 )  satisfy an appropriate exponential compact containment condition. See Section <ref>.


  * There exist two Hamiltonians H_†⊆ C_l(ℝ^2) × C_b(ℝ^2) and H_⊆ C_u(ℝ^2) × C_b(ℝ^2) such that H_†⊆ ex-_n H_n and H_⊆ ex-_n H_n. This extension allows for unbounded functions in the domain. See Section <ref>. Moreover, we refer to Definition <ref> for the notions of  and .


  * There is an operator H ⊆ C_b() × C_b() such that, for all λ > 0 and h ∈ C_b(ℝ), every viscosity subsolution to f- λ H_† f = h is a viscosity subsolution to f - λ Hf = h and  every viscosity supersolution to f - λ H_ f = h is a viscosity supersolution to f - λ H f = h. The operators H_† and H_ should be thought of as upper and lower bounds for the “true” limiting H of the sequence H_n. See Section <ref>. 


  * The comparison principle holds for the Hamilton-Jacobi equation f - λ H f = h for all h ∈ C_b(ℝ) and all λ>0. The proof of this statement is immediate, since the operator H we will be dealing with is of the type considered in <cit.>.


For the verification of all the open conditions we use the limiting behaviour of the sequence of Hamiltonians H_n. We then start by deriving an expansion for the Hamiltonians associated to the re-scaled fluctuation process.






 §.§ Expansion of the Hamiltonian



Let (b_n)_n ∈ℕ^* be a sequence of positive real numbers such that b_n ↑∞ and b_n^4 n^-1↓ 0. The fluctuation process ( (b_n n^-1 S_n(b_n^2t), b_n ( n^-1 T_n(b_n^2 t) - σ^2 ) ), t ≥ 0 ) has Markovian  evolution on state space  and its generator G_n can be deduced from (<ref>). 


Let n∈^*. The Markov process ( (b_n n^-1 S_n(b_n^2t), b_n ( n^-1 T_n(b_n^2 t) - σ^2 ) ), t ≥ 0 )  has infinitesimal generator _n that, for any f ∈ C^2_c(E_n), satisfies 


    _nf(x,y)= 1/2(xb_n^2σ^2(h_n(y)-1)-x^3σ^4h_n^2(y))∂ f/∂ x(x,y)+(b_nx^2nσ^4h_n^2(y)-b_n^2yσ^2)∂ f/∂ y(x,y)
    
    +b_n^42n∂^2 f/∂ x^2(x,y) +2 b_n^3xn∂^2 f/∂ x∂ y(x,y)+2b_n^4n(yb_n+σ^2)
    ∂^2 f/∂ y^2(x,y),


where the function h_n:   (-σ^2 b_n,+∞)  →ℝ is defined by h_n(y) = ( 1+ y/b_nσ^2+1/nσ^2)^-1.


By applying the chain rule to the function exp{ nb_n^-4f(x,y) }, for f ∈ C^2_c(E_n), it is easy to see that, at speed nb_n^-4, the Hamiltonian


    H_n f (x,y) = b_n^4 n^-1 e^-nb_n^-4 f(x,y) G_n ( e^nb_n^-4 f) (x,y)


results in


    _nf(x,y)=_nf(x,y)+12(∂ f∂ x(x,y))^2+2σ^2(∂ f∂ y(x,y))^2
    
    
    +2 x b_n∂ f∂ x(x,y)∂ f∂ y(x,y)+2y b_n(∂ f/∂ y(x,y))^2,


with G_n given by (<ref>). We Taylor expand the function h_n(y) appearing in the definition of G_n up to second order:


    h_n(y)=1-y/b_nσ^2+y^2/b_n^2σ^4+1/b_n^2ε_n(y),


where the sequence of functions (ε_n)_n∈^* converges to zero, uniformly in n, on compact sets of . 


In what follows we will require a more accurate control on the reminder ε_n(y). For this reason we give here the following lemma.

  
Set K_n = [-σ^2 log^1/2 b_n^1/2,σ^2 log^1/2 b_n^1/2]. There exists a positive constant c, independent of n, such that we have


    sup_y ∈ K_n |ϵ_n(y)| ≤ c   b_n^-1log^3/2 b_n^1/2.


 


We Taylor expand the function h_n(y) up to second order and we express the reminder in Lagrange's form. Taking out the highest order terms to obtain (<ref>), we find


    ϵ_n(y) = - b_n^2/1 + nσ^2 - b_n y/σ^2[( 1 - 1/1 + nσ^2)^2 - 1 ] +  y^2/σ^4[( 1 - 1/1 + nσ^2)^3 - 1 ] 
    
    - y^3/b_n σ^6(1 + ζ/b_n σ^2 + 1/nσ^2)^-4,


with |ζ| < |y|. Note that the first three terms on the right-hand side are at most of order b_n^-2. The final term is of order b_n^-1log^3/2 b_n^1/2, as the fraction that is taken to the fourth power is asymptotically converging to 1. 	


Turning back to the expansion of G_n in (<ref>), by (<ref>) we get


    _nf(x,y) = 1/2(-b_n xy/σ^4 + x y^2/σ^6 - x^3σ^4)∂ f/∂ x(x,y) -b_n^2yσ^2∂ f/∂ y(x,y) 
    
    +12(∂ f∂ x(x,y))^2+2σ^2(∂ f∂ y(x,y))^2 + R^f_n(x,y)


and the remainder


    R^f_n(x,y) = (x ε_n(y)2σ^2-x^32σ^4( h_n^2(y)-1 ) )∂ f/∂ x(x,y) + b_n x^2 h_n^2(y)nσ^4 ∂ f/∂ y(x,y) 
    
    + b_n^42n∂^2 f/∂ x^2(x,y)+ 2 (b_n^4 σ^2/n + b_n^3y/n) ∂^2 f/∂ y^2(x,y) + 2b_n^3x/n∂^2 f/∂ x∂ y(x,y)
    
    +2 x b_n∂ f∂ x(x,y)∂ f∂ y(x,y)+2y b_n(∂ f/∂ y(x,y))^2


converges to zero, uniformly in n, on compact sets of ℝ^2. 








 §.§ Perturbative approach and approximating Hamiltonians





Observe that the expansion (<ref>) is diverging and, more precisely, is diverging through terms containing the y variable, thus relative to the time-evolution of the process (b_n ( n^-1T_n(b_n^2 t) - σ^2 ), t ≥ 0 ). Indeed, the two components of ((b_n n^-1 S_n(b_n^2t), b_n ( n^-1T_n(b_n^2 t) - σ^2 )), t ≥ 0 ) live on two different time-scales and the asymptotic behavior of  can be determined after having averaged out the evolution of (b_n ( n^-1T_n(b_n^2 t) - σ^2 ), t ≥ 0 ). The “averaging” is obtained through a perturbative approach leading to a projected large deviation principle. This argument takes inspiration from the perturbation theory for Markov processes applied in <cit.> and it was also used to study path-space moderate deviations for the Curie-Weiss model with random field in <cit.>.



In the present section we will first give some heuristics about the perturbative method, since it will provide  guideline for getting the approximating Hamiltonians H_†, H_, and then we will make it rigorous.





 



  
Heuristics on perturbation. In the expansion (<ref>) the leading term is of order b_n^2 and thus explodes as n ↑∞. We think of b_n^-1 as a perturbative parameter and we use a second order perturbation F_n,f of f to introduce some negligible (in the infinite volume limit) terms providing that the whole expansion does not diverge. 

More precisely, given two arbitrary functions _f, _f: ℝ^2 →ℝ, we define the perturbation of f as


    F_n,f:(x,y)⟼ f(x)+ b_n^-1 _f(x,y)+ b_n^-2 _f(x,y)


and then we choose _f and _f so that


    _n F_n,f(x,y) =   f(x) + ,


where H f(x) is of order 1 with respect to b_n and the remainder contains smaller order terms. We assume that  _f and _f are at least of class C^2 and we compute _n F_n,f. Using (<ref>) yields


    _nF_n,f(x,y)=-b_n xy/2σ^4f'(x)+(x y^2/2σ^6-x^32σ^4)f'(x)+12(f'(x))^2
    
    -yb_nσ^2∂_f/∂ y(x,y)-xy/2σ^4∂_f/∂ x(x,y)-yσ^2∂_f/∂ y(x,y)+.


To eliminate the terms of order b_n and of order 1 in the variable y, the functions _f and _f must necessarily verify


    ∀ (x,y)∈^2
        {[                             -yσ^2∂_f∂ y(x,y)-xy2σ^4f'(x)=0; -yσ^2∂_f∂ y(x,y) -xy2σ^4∂_f∂ x(x,y) + x y^22σ^6 f'(x) = 0. ].


If we take


    _f:(x,y)⟼-xy/2σ^2f'(x)      _f:(x,y)⟼xy^2/8σ^4(3f'(x)+xf”(x)),


then the conditions (<ref>) are satisfied and we obtain


    H_n F_n,f (x,y) = -x^32σ^4f'(x)+12(f'(x))^2 + .


Provided we can control the remainder, for any function f in a suitable regularity class, we formally get the following candidate limiting operator


    f(x)=-x^32σ^4f'(x)+12(f'(x))^2.


To rigorously conclude that the Hamiltonian H is the limit of the sequence (H_n)_n ∈ℕ^*, with H_n given in (<ref>), we should prove that H ⊆_n H_n (see Definition <ref>). The proof of the latter assertion would consist in showing that, for every f ∈ C_c^4(ℝ), we have _n F_n,f = f and _n H_n F_n,f = Hf. Recall that in our setting (x,y) ∈ E_n = ℝ× (-σ^2b_n,+∞). Therefore, the functions Γ_f and Λ_f in (<ref>)  are unbounded in E_n, implying in turn that also F_n,f is unbounded in E_n. Due to this unboundedness, even if f ∈ C_c^4(ℝ), we can not guarantee sup_n  F_n,f < ∞ and thus we can not prove _n F_n,f = f. 

We apply the same techniques as in <cit.>. To circumvent the problem and allow for unbounded functions in the domain, we relax our definition of limiting operator. In particular, we introduce two limiting Hamiltonians H_† and H_, approximating H from above and below respectively, and then we characterize H by matching  upper and lower bound. 








  
Approximating Hamiltonians and domain extensions.

We have seen that the natural perturbations of our functions f are unbounded. We repair this unboundedness by cutting off the functions. To this purpose, we introduce a collection of smooth increasing functions χ_n : → such that


    χ_n(z) = 
    
     -σ^2 log b_n^1/2 + 1     if  z ≤ - σ^2 log b_n^1/2
    
    z     if  -σ^2 log b_n^1/2 + 2 ≤ z ≤σ^2 log b_n^1/2 - 2 
    σ^2 log b_n^1/2 - 1      if  z ≥σ^2 log b_n^1/2.


To make sure that the cut-off acts only outside a compact set, we first perturb our function f by a Lyapunov function ε (y^2 + log(1+x^2)). The latter function will indeed play a special role in establishing the exponential compact containment condition in Section <ref> below.


Let ε∈ (0,1) and f ∈ C_c^4(ℝ). Consider the cut-off (<ref>) and define the functions


    χ_n ( F_n,f(x,y) ±ε (y^2 + F_n,g(x,y))),


with F_n,∙ as in (<ref>), (<ref>) and g(x) = log(1+x^2). Then, 


  * 
For any C > 0 there is an N = N(C) such that, for any n ≥ N, we have

    χ_n ( F_n,f(x,y) ±ε (y^2 + F_n,g(x,y)) ) = F_n,f(x,y) ±ε (y^2 + F_n,g(x,y))

on the set K_1 = K_1(C) := {(x,y) ∈^2  |  ε (y^2 + log(1+x^2))≤ C }.


  * 



Let C be the positive constant defined in (<ref>) and set N_1 := sup{n ∈ℕ | ε≤ 6 C b_n^-2}. Then, for any n > N_1, the function χ_n ( F_n,f(x,y) ±ε (y^2 + F_n,g(x,y))) is constant outside the compact set K_2,n := {(x,y) ∈ℝ^2  |  ε (y^2 + log (1+x^2)) ≤ 2 σ^2 log b_n^1/2 + 6 C}.







We start by proving (<ref>). Recall that f ∈ C_c^4(ℝ), so there exists a positive constant M such that the derivatives of f (and, as a consequence, Γ_f and Λ_f) vanish at x ∉ [-M,M]. Therefore, it yields 





    | f(x) | + b_n^-1|Γ_f (x, y) | + b_n^-2|Λ_f (x, y) |≤f + M/2σ^2f' |y| + M/8σ^4(3f' + Mf”) y^2,
 

where · denotes the L^∞-norm. Moreover, since |xg'(x)| ≤ 1 and |x(3g'(x) + xg”(x))| ≤ 8  we also get the bound







    ε( b_n^-1|Γ_g (x, y) | + b_n^-2|Λ_g (x, y) |) ≤| y |/σ^2 + y^2/σ^4.


Setting 


    C :=  max{f,  1/2σ^2(M f'+2),  1/8σ^4(3 M f' + M^2 f” + 8) }


and putting the two previous estimates together we obtain


    | F_n,f(x,y) ±ε( b_n^-1Γ_g (x, y) + b_n^-2Λ_g (x, y) ) |≤C( 1+ b_n^-1| y | + b_n^-2 y^2 ) ≤ 3 C( 1 +  b_n^-2 y^2 ),




where the last inequality follows from |ab| ≤ 2(a^2+b^2), with a,b ∈ℝ.



Consider an arbitrary C > 0. By (<ref>), we find that (x,y) ↦ F_n,f(x,y) ±ε (y^2 + F_n,g(x,y)) is bounded uniformly in n on the set K_1. To conclude, simply observe that, since the cut-off is moving to infinity, for sufficiently large n, we obtain χ_n ≡id on K_1.

We proceed with the proof of (b). For any n > N_1 and  any (x,y) ∉ K_2,n, we obtain 

    F_n,f(x,y)    + ε ( y^2 + F_n,g(x,y) ) 
         =   F_n,f(x,y) + ε( b_n^-1Γ_g (x, y) + b_n^-2Λ_g (x, y) ) + ε(y^2 + log(1+x^2)) 
       ≥ - 3C( 1+ b_n^-2 y^2 ) + ε/2 y^2 + ε/2(y^2 + 2log(1+x^2)) 
       ≥ - 3 C +  ε/2 (y^2+2log(1+x^2)) 
        > σ^2 log b_n^1/2.

The definition (<ref>) of the cut-off leads then to the conclusion. The proof for the function F_n,f(x,y) - ε ( y^2+F_n,g(x,y)) follows similarly.



Before stating the next lemma, we want to make a remark on the notation N_⋆ used therein. This index is an explicit positive integer larger than N_1 introduced in Lemma <ref>(<ref>) and it will be defined precisely in (<ref>) at the end of this section.



Let ε∈ (0,1) and f ∈ C_c^4(ℝ). Consider the cut-off (<ref>) and  define the functions


    f_n^ε,± (x,y) :=
    
    0     if  n ≤ N_⋆
    χ_n ( F_n,f(x,y) ±ε (y^2 + F_n,g(x,y)))     if  n > N_⋆


and


    f^ε,± (x,y) := f(x) ±ε( y^2 + g(x) ),


with F_n,∙ as in (<ref>), (<ref>) and g(x) = log(1+x^2). Then, for every ε∈ (0,1), the following properties are satisfied:


  * 
f_n^ε,±∈(H_n).


  * 
f^ε,+∈ C_l(^2) and f^ε,-∈ C_u(^2).


  * 
We have 


    inf_n  inf_(x,y) ∈ E_n f_n^ε,+(x,y) > - ∞    sup_n sup_(x,y) ∈ E_n f_n ^ε,-(x,y) < ∞.



  * 
For every compact set K ⊆^2, there exists a positive integer N = N(K) such that, for n ≥ N and (x,y) ∈ K, we have 


    f_n^ε,±(x,y) =  F_n,f(x,y) ±ε( y^2 + F_n,g(x,y) ).



  *  
For every c ∈, we have


    _n ↑∞  f_n^ε,+∧ c  = f^ε,+∧ c     _n ↑∞  f_n^ε,-∨ c = f^ε,-∨ c.



Moreover, it holds 


  *  
For every c ∈, we have 

    lim_ε↓ 0f^ε,+∧ c - f ∧ c + f^ε,-∨ c - f ∨ c = 0.
 


	

If n < N_⋆ all the statements are trivial. We assume n ≥ N_⋆ and we prove all the properties for the `+' superscript case, the other being similar. 


  * 
It is clear from the definition (<ref>) of N_⋆ that N_⋆≥ N_1. Then, as the cut-off (<ref>) is smooth, Lemma <ref>(<ref>) yields f_n^ε,±∈ C_c^∞(^2). In addition, the location of the cut-off and Lemma <ref>(<ref>) make sure that f_n^ε,± is constant outside a compact set K ⊂ E_n, implying f_n^ε,±∈(G_n) and, as a consequence, f_n^ε,±∈(H_n). See equations (<ref>) and (<ref>) for the definitions of G_n and H_n respectively.


  * This is immediate from the definitions of f^ε,±.


  * From the estimate (<ref>), we deduce (keeping the same notation)




    inf_(x,y) ∈ℝ^2 F_n,f(x,y) + ε( y^2 + F_n,g(x,y) ) ≥ - 3 C (1 + b_n^-2 y^2) + ε( y^2 + log(1+x^2) ),


which is bounded from below uniformly in n > N_1. The conclusion follows as N_⋆≥ N_1 (cf. equation (<ref>)). 




  * This follows immediately by Lemma <ref>(<ref>).


  * Fix ε > 0 and c ∈. By (<ref>), the sequence ( f_n^ε,+∧ c )_n ∈ℕ^* is uniformly bounded from below and then, we obviously get sup_n ∈ℕ^*f_n^ε,+∧ c < ∞. Thus, it suffices to prove uniform convergence on compact sets. Let us consider an arbitrary sequence (x_n,y_n) converging to (x,y) and prove lim_n f_n^ε,+(x_n,y_n) = f^ε,+(x,y). As a converging sequence is bounded, it follows from (<ref>) that, for sufficiently large n, we have 


    f_n^ε,+(x_n,y_n) = F_n,f (x_n,y_n) + ε( y_n^2 + F_n,g(x_n,y_n) ),


which indeed converges to f^ε,+(x,y) as n ↑∞.


  * This follows similarly as in the proof of (<ref>).

		



Let H ⊆ C_b() × C_b(), with domain (H) = C_c^∞(), be defined as 


    Hf(x) = - x^3/2σ^4 f'(x) + 1/2( f'(x) )^2.


We define the approximating Hamiltonians H_†⊆ C_l(^2) × C_b(^2) and H_⊆ C_u(^2) × C_b(^2) as


    H_†    := {(  f(x) + ε( y^2 + log(1+x^2) ),   Hf(x) + ε2f' + ε^2 )  |   f ∈ C_c^∞(ℝ), ε∈ (0,1) }, 
    
    H_    := {(  f(x) - ε( y^2 + log(1+x^2) ),   Hf(x) - ε2f' - ε^2 )  |   f ∈ C_c^∞(ℝ), ε∈ (0,1) }.


	
	
 
Consider notation as in Definitions <ref> and <ref>. We have H_†⊆ ex-_n  H_n and H_⊆ ex-_n  H_n.




We prove only the first statement, i.e. H_†⊆ ex-_n  H_n. Fix ε > 0 and f ∈ C_c^4(). Set f_n := f_n^ε,+ as in Lemma <ref>.  We show that (f(x) + ε(y^2 + log(1+x^2)),Hf(x)+ε2f' + ε^2) is approximated by (f_n,H_nf_n) as in Definition <ref>(a). Since (<ref>) was proved in Lemma <ref>(<ref>), we are left to check conditions (<ref>) and (<ref>).


  (<ref>)
We start by showing that we can get a uniform (in n) upper bound for the function H_n f_n^ε,+. To avoid trivialities, we consider the sequence for n ≥ N_⋆.


  * If | F_n,f(x,y) +ε(y^2 + F_n,g(x,y) )|≥σ^2 log b_n^1/2, then the function f_n^ε,+ is constant and therefore H_n f_n^ε,+≡ 0. 


  * If | F_n,f(x,y) +ε(y^2 + F_n,g(x,y) )| < σ^2 log b_n^1/2, the variables x and y are at most of order b_n^1/4 and log^1/2 b_n^1/2 respectively and we can characterize H_n f_n^ε,+ by means of (<ref>), since we can control the remainder term. Indeed,


  * by Lemma <ref>, we control ε_n(y) up to y's of order log^1/2 b_n^1/2;


  * the function f is constant outside a compact set and thus has zero derivatives outside such a compact set;


  * by smoothness of the cut-off (<ref>), the derivatives χ_n' and χ_n” are bounded.


We thus find


    H_n f_n^ε,+ (x,y) = [ -x^3/2σ^4 f'(x) - ε( x^4/σ^4(1+x^2) + 2 b_n^2 y^2/σ^2) ] χ_n'(-) 
    
    + 1/2[ ( f'(x))^2 + 4 ε^2 x^2/(1+x^2)^2] ( χ_n'(-) )^2 
    
    + 8 ε^2 σ^2 y^2 [ b_n^4/nχ_n”(-) + ( χ_n'(-) )^2 ] +  Q_n(x,y)


and sup_(x,y)| Q_n(x,y) |≤ c_0, for a suitable positive constant c_0, independent of n and ε. Observe that the remainder term Q_n(x,y) collects all the smaller order contributions coming from F_n,f(x,y), F_n,g(x,y) and y^2. 

We want to show that (<ref>) is uniformly bounded from above. The terms involving f are ok, since f ∈ C_c^4(ℝ) implies that there exists a positive constant M such that f' vanishes at . The function -x^4/σ^4(1+x^2) is non-positive and the term 2x^2/(1+x^2)^2 is bounded from above by 2. Moreover, if we set


    N_2 := sup{ n ∈ℕ|  -2b_n^2/σ^2 + 8 σ^2 [ b_n^4/nχ_n”(-) + ( χ_n'(-) )^2 ] > 0 .},


we obtain that -2 b_n^2 y^2/σ^2 + 8 σ^2 [ b_n^4/nχ_n”(-) + ( χ_n'(-) )^2 ] y^2 is uniformly bounded from above, for all n > N_2. By definition (<ref>), N_⋆≥ N_2. Therefore, we can find a positive constant c_1 (dependent on M and σ, but not on n and ε) such that .



To conclude, observe that, since there exists a positive constant c_2 (independent of n) such that | H_n f_n^ε,+|≤ c_2   b_n^2 log b_n + c_0 (cf. equation (<ref>)), choosing the sequence v_n := b_n leads to sup_n v_n^-1log H_n f_n^ε,+ < ∞.


  (<ref>)
Let K be a compact set. Consider an arbitrary converging sequence (x_n, y_n) ∈ K and let (x,y) ∈ K be its limit. We want to show lim sup_n H_n f_n^ε,+(x_n,y_n) ≤ Hf(x).

As a converging sequence is bounded, by Lemma <ref>(<ref>) we can find a sufficiently large N=N(K) ∈ℕ such that, for all n ≥ N, we have 


    f_n^ε,+ (x_n,y_n) = F_n,f(x_n,y_n) + ε( y_n^2 + F_n,g(x_n,y_n) ).


Thus, for any n ≥ N, it yields 


    H_n f_n^ε,+(x_n,y_n) ≤ -x_n^3/2σ^4 f'(x_n) + 1/2( f'(x_n) )^2
    
    
    
    + ε( x_n/1+x_n^2 f'(x_n) -x_n^4/σ^4(1+x_n^2) - 2 b_n^2 y_n^2/σ^2)  
    
    
    
    + ε^2 ( 2x_n^2/(1+x_n^2)^2 + 8 σ^2 y_n^2 ) + Q_n(x_n,y_n).


Using that x(1+x^2)^-1≤ 1/2, we find

    H_n f_n^ε,+(x_n,y_n) ≤ Hf(x_n) +  ε/2f' + ε^2 + ε y_n^2 [ 8 εσ^2 - 2 b_n^2/σ^2] + Q_n(x_n,y_n),

where the remainder term Q_n converges to zero uniformly on compact sets. Since b_n ↑∞, the conclusion follows.



At this point we are ready to complete the definition of the sequences {f_n^ε,±}_n ∈ℕ^* by defining the index N_⋆. We set

    N_⋆ := N_1 ∨ N_2,

with N_1 and N_2 given respectively in Lemma <ref>(<ref>) and in (<ref>). To conclude this section we obtain the Hamiltonian extensions.


 
Consider notation as in Definition <ref>. Moreover, set Ĥ_† := H_†∪ H and Ĥ_ := H_∪ H. Then Ĥ_† is a sub-extension of H_† and Ĥ_ is a super-extension of H_.



We prove only that Ĥ_† is a sub-extension of H_†. We use the first statement of Lemma <ref>. Let f ∈(H). We must show that (f,Hf) is appropriately approximated  by elements in the graph of H_†.
 
For any n ∈ℕ^*, set ε(n) = n^-1 and consider the function f_n (x,y) = f(x) + ε(n) ( y^2 + log(1+x^2) ), with H_† f_n = Hf+ ε/2f'+ε^2. From Lemma <ref>(<ref>)  we obtain that f_n ∧ c - f ∧ c→ 0, for every c ∈ℝ. In addition, as Hf ∈ C_b(), we have H_† f_n - Hf→ 0. This concludes the proof.


	






 §.§ Exponential compact containment




The last open question we must address consists in verifying exponential compact containment for the fluctuation process. The validity of the compactness condition will be shown in Proposition <ref> below.
We start with an informal discussion on the validity of this property. 

Recall that the sequence of processes (√(n)(n^-1T_n(t) - σ^2), t ≥ 0 ) converges weakly to the solution of (<ref>). Thus, speeding up time by a factor b_n^2, we find that the process √(n)(n^-1T_n(t) - σ^2) has roughly equilibrated as a centered normal random variable with variance 2 σ^4. This implies that, for any a > 0, the tail probability [b_n (n^-1T_n(t) - σ^2) ≥ a] scales like
	
    ∫_a^∞1/2σ^2√(π)√(n)/b_n e^- n/b_n^2y^2/4 σ^4 y.

	By Lemma 2 in <cit.>, (<ref>) is bounded above by
	
    1/2aσ^2 √(π)b_n/√(n)exp{- n/b_n^2a^2/4σ^4},

which is indeed decaying on an exponential scale that is faster than n b_n^-4. As a consequence, it is the dynamics of the process ( b_n n^-1 S_n(b_n^2 t), t ≥ 0 ) that needs to be properly controlled, as well as the interplay between the two processes.
	
To do so, we use a Lyapunov argument based on <cit.> (included for completeness as Lemma <ref>). We start by proving an auxiliary lemma showing that the function (x,y) ↦1/2(y^2 + log (1 +x^2)) is appropriate for this purpose, whenever we carry out the appropriate perturbation and cut-off as in the previous section.



Let G ⊆ℝ^2 be a relatively compact open set. Consider the cut-off introduced in (<ref>) and define

    Υ_n (x,y) = χ_n [ 1/2( y^2 + F_n, g (x,y) ) ],


with F_n,∙ as in (<ref>), (<ref>) and  g(x) = log(1+x^2). Then, we have


    lim sup_n ↑∞sup_(x,y) ∈ G ∩ E_n H_n Υ_n (x,y) ≤1/4.





This follows immediately from the statement H_†⊆ ex-_n H_n proved in Proposition <ref>. Namely, one can consider f ≡ 0 and ε = 1/2. 


 
Assume that the sequence (b_n n^-1 S_n(0), b_n (n^-1 T_n(0) - σ^2)) is exponentially tight at speed nb_n^-4. Then, the processes 


    ((X_n(t),Y_n(t)),t ≥ 0) := ((b_n n^-1 S_n(b_n^2t), b_n ( n^-1T_n(b_n^2 t) - σ^2 )), t ≥ 0 )
 

satisfy the exponential compact containment condition at speed nb_n^-4. In other words, for every compact set K ⊆^2, every constant a ≥ 0 and time T ≥ 0, there exists a compact set  such that


    lim sup_n ↑∞sup_(x,y) ∈ K ∩ E_n b_n^4 n^-1log[ (X_n(t),Y_n(t)) ∉ K'  for some  t ≤ T   |   (X_n(0),Y_n(0)) = (x,y) ] ≤ - a.





The statement follows from Lemmas <ref> and <ref> by choosing f_n ≡Υ_n on a fixed, sufficiently large, compact set of ℝ^2. For similar proofs see e.g. <cit.>, <cit.>.









 §.§ Proof of Theorem <ref>




We check the assumptions of Theorem <ref>. We use operators H_†, H_ as in Definition <ref> and limiting Hamiltonian H ⊆ C_b() × C_b(), with domain C_c^∞(), of the form Hf(x) = H(x,f'(x)) where


    H(x,p) = - x^3/2σ^4p + 1/2p^2.


We first verify Condition <ref>: (<ref>) follows from Proposition <ref>, (<ref>) is satisfied by definition and (<ref>) follows from Proposition <ref>.

The comparison principle for f- λ Hf = h for h ∈ C_b() and λ > 0 has been verified in e.g. <cit.>. Two things should be noted. The statement of the latter proposition is valid for f ∈ C_c^2(), but the result generalizes straightforwardly to class C_c^∞() as the penalization and containment functions used in the proof are C^∞. In addition, the proposition was stated for strong viscosity solutions, but the proof of <cit.> works for our notion of viscosity solutions as well. See the discussion following <cit.> on the difference of the two notions of solutions.

Finally, the exponential compact containment condition follows from Proposition <ref>.












§ APPENDIX: PATH-SPACE LARGE DEVIATIONS FOR A PROJECTED PROCESS
 


We turn to the derivation of the large deviation principle. We first introduce our setting.

 
Assume that, for each n ∈ℕ^*, we have a Polish subset E_n ⊆^2 such that for each x ∈^2 there are x_n ∈ E_n with x_n → x.  Let A_n ⊆ C_b(E_n) × C_b(E_n) and existence and uniqueness holds for the D_E_n(^+) martingale problem for (A_n,μ) for each initial distribution μ∈(E_n). Letting _z^n ∈(D_E_n(^+)) be the solution to (A_n,δ_z), the mapping z ↦_z^n is measurable for the weak topology on (D_E_n(^+)). Let Z_n be the solution to the martingale problem for A_n and set

    H_n f = 1/r_n e^-r_n fA_n e^r_n f     e^r_n f∈(A_n),

for some sequence of speeds (r_n)_n ∈ℕ^*, with lim_n ↑∞ r_n = ∞.


Following the strategy of <cit.>, the convergence of Hamiltonians (H_n)_n ∈ℕ^* is a major component in the proof of the large deviation principle. We postpone the discussion on how determining a limiting Hamiltonian H due to the difficulties that taking the n ↑∞ limit introduces in our particular context. We first focus on exponential tightness, an equally important aspect. 




 §.§ Compact containment condition



Given the convergence of the Hamiltonians, to have exponential tightness it suffices to establish an exponential compact containment condition.

 
	We say that a sequence of processes (Z_n(t),t ≥ 0) on E_n ⊆^2 satisfies the exponential compact containment condition at speed (r_n)_n ∈ℕ^*, with lim_n ↑∞ r_n = ∞, if for all compact sets K⊆^2,  constants a ≥ 0 and times T > 0, there is a compact set K' ⊆^2 with the property that
	
    lim sup_n ↑∞sup_z ∈ K1/r_nlog[Z_n(t) ∉ K'  for some  t ≤ T  |   Z_n(0) = z] ≤ - a.




The exponential compact containment condition can be verified by using approximate Lyapunov functions and martingale methods. This is summarized in the following lemma. Note that exponential compact containment can be obtained by taking deterministic initial conditions.


 
Suppose Assumption <ref> is satisfied. Let Z_n(t) be a solution of the martingale problem for A_n and assume that (Z_n(0))_n ∈ℕ^* is exponentially tight with speed (r_n)_n ∈ℕ^*. Consider the compact set K = [a,b] × [c,d] and let G ⊆ℝ^2 be open and such that [a,b]× [c,d] ⊆ G. For each n, suppose we have (f_n,g_n) ∈ H_n. Define
	
    β(q,G)    := lim inf_n ↑∞( inf_(x,y) ∈ G^c f_n(x,y) - sup_(x,y) ∈ K f_n(x,y)), 
    γ(G)     := lim sup_n ↑∞sup_(x,y) ∈ G g_n(x,y).

	Then
	
    lim sup_n ↑∞1/r_nlog[Z_n(t) ∉ G  for some  t ≤ T  ] 
    ≤max{-β(q,G) + T γ(G), lim sup_n ↑∞[Z_n(0) ∉ [a,b]×[c,d] ] }.







 §.§ Operator convergence for a projected process



In the papers <cit.> one of the main steps in proving the large deviation principle was proving directly the existence of an operator H such that H ⊆_n H_n; in other words, verifying that, for all (f,g) ∈ H, there are f_n ∈ H_n such that _n f_n = f and _n H_n f_n = g (the notion of  is introduced in Definition <ref>). Here it is hard to follow a similar strategy. We rather proceed as done in <cit.>.

We are dealing with functions 


    f_n(x,y) = f(x) + b_n^-1 f_1(x,y) + b_n^-2 f_2(x,y)


given in a perturbative fashion and satisfying intuitively f_n → f and H_n f_n → Hf with Hamiltonian H ⊆ C_b() × C_b() of the form (<ref>). The unboundedness of the state space E_n causes that for most functions f ∈ C_c^4(ℝ): sup_n f_n = ∞, implying we do not have f_n = f. To circumvent this issue, we relax our definition of limiting operator.

In particular, we will work with two Hamiltonians H_† and H_, that are limiting upper and lower bounds for the sequence of Hamiltonians H_n, respectively, and thus serve as natural upper and lower bounds for H. This extension allows us to consider unbounded functions in the domain and to argue with inequalities rather than equalities.


For f_n ∈ C_b(E_n) and f ∈ C_b(^2), we will write f_n = f if sup_n f_n < ∞ and, for all compact sets K ⊆^2,


    lim_n ↑∞sup_(x,y) ∈ K ∩ E_n|f_n(x,y) - f(x,y) | = 0.






 
Suppose that for each n we have an operator H_n⊆ C_b(E_n) × C_b(E_n). 
Let (v_n)_n ∈ℕ^* be a sequence of real numbers such that v_n ↑∞.


	
  * The extended sub-limit, denoted by ex-_n H_n, is defined by the collection (f,g) ∈ C_l(^2)× C_b() for which there exist (f_n,g_n) ∈ H_n such that
	
    f_n ∧ c = f ∧ c,     ∀  c ∈, 
    sup_n 1/v_nlogg_n < ∞,     sup_nsup_x ∈^2 g_n(x) < ∞,

	and that, for every compact set K ⊆^2 and every sequence z_n ∈ K satisfying lim_n z_n = z and lim_n f_n(z_n) = f(z) < ∞, 
	
    lim sup_n ↑∞g_n(z_n) ≤ g(z).

	
  * The extended super-limit, denoted by ex-_n H_n, is defined by the collection (f,g) ∈ C_u(^2)× C_b() for which there exist (f_n,g_n) ∈ H_n such that
	
    f_n ∨ c = f ∨ c,     ∀  c ∈, 
    sup_n 1/v_nlogg_n < ∞,     inf_ninf_x ∈^2 g_n(x) > - ∞,

	and that, for every compact set K ⊆^2 and every sequence z_n ∈ K satisfying lim_n z_n = z and lim_n f_n(z_n) = f(z) > - ∞,
	
    lim inf_n ↑∞g_n(z_n) ≥ g(z).




For completeness, we also give the definition of the extended limit.


Suppose that for each n we have an operator H_n⊆ C_b(E_n) × C_b(E_n). We write ex- H_n for the set of (f,g) ∈ C_b(^2) × C_b(^2) for which there exist (f_n,g_n) ∈ H_n such that f =  f_n and g =  g_n.



Let H_†⊆ C_l(^2) × C_b(^2) and H_⊆ C_u(^2) × C_b(^2) and let λ > 0 and h ∈ C_b(^2). Consider the Hamilton-Jacobi equations

    f - λ H_† f     = h, 
    
    f - λ H_ f     = h.

We say that u is a (viscosity) subsolution of equation (<ref>) if u is bounded, upper semi-continuous and if, for every f ∈(H_†) such that sup_x u(x) - f(x) < ∞ there exists a sequence  x_n ∈ℝ^2 such that

    lim_n ↑∞ u(x_n) - f(x_n)  = sup_x u(x) - f(x),

and 

    lim_n ↑∞ u(x_n) - λ H_† f(x_n) - h(x_n) ≤ 0.

We say that v is a (viscosity) supersolution of equation (<ref>) if v is bounded, lower semi-continuous and if, for every f ∈(H_) such that inf_x v(x) - f(x) > - ∞ there exists a sequence x_n ∈ℝ^2 such that

    lim_n ↑∞ v(x_n) - f(x_n)  = inf_x v(x) - f(x),

and 

    lim_n ↑∞ v(x_n) - λ H_ f(x_n) - h(x_n) ≥ 0.

We say that u is a (viscosity) solution of equations (<ref>) and (<ref>) if it is both a subsolution to (<ref>) and a supersolution to (<ref>).

We say that (<ref>) and (<ref>) satisfy the comparison principle if for every subsolution u to (<ref>) and supersolution v to (<ref>), we have u ≤ v.


Note that the comparison principle implies uniqueness of viscosity solutions. This in turn implies that a new Hamiltonian can be constructed based on the set of viscosity solutions.


 
Suppose we are in the setting of Assumption <ref>. Suppose there are operators H_†⊆ C_l(^2) × C_b(^2), H_⊆ C_u(^2) × C_b(^2) and H ⊆ C_b() × C_b() with the following properties:


  *  
H_†⊆ ex-_n H_n and H_⊆ ex-_n H_n (recall Definition <ref>).


  *  
The domain (H) contains C^∞_c(ℝ) and, for f ∈ C_c^∞(), we have Hf(x) = H(x,∇ f(x)).


  * 
For all λ > 0 and h ∈ C_b(), every subsolution to f - λ H_† f = h is a subsolution to f - λ H f = h and every supersolution to f - λ H_ f = h is a supersolution to f - λ H f = h.



Now we are ready to state the main result of this appendix: the large deviation principle for the projected process. We denote by η_n: E_n →ℝ the projection map η_n(x,y)=x.

 
Suppose we are in the setting of Assumption <ref> and Condition <ref> is satisfied. Suppose that for all λ > 0 and h ∈ C_b(ℝ) the comparison principle holds for .   

Let Z_n(t) be the solution to the martingale problem for A_n. Suppose that the large deviation principle at speed (r_n)_n ∈ℕ^* holds for η_n(Z_n(0)) on  with good rate-function I_0. Additionally suppose that the exponential compact containment condition holds at speed (r_n)_n ∈ℕ^* for the processes Z_n(t).

Then the large deviation principle holds with speed (r_n)_n ∈ℕ^*  for (η_n(Z_n(t)))_n ∈ℕ^* on D_ℝ(^+) with good rate function I. Additionally, suppose that the map p ↦ H(x,p) is convex and differentiable for every x and that the map (x,p) ↦/ p H(x,p) is continuous. Then the rate function I is given by

    I(γ) = 
    I_0(γ(0)) + ∫_0^∞(γ(s),γ̇(s))  s    if γ∈, 
    ∞   otherwise,

where : ℝ^2 → is defined by (x,v) = sup_p {pv - H(x,p)}.



The large deviation result follows by <cit.> with H_† and H_ as in the present paper and _† = _ = H. The verification of the conditions for <cit.> corresponding to a Hamiltonian of this type have been carried out in e.g. <cit.> or <cit.>.






 §.§ Relating two sets of Hamiltonians



For Condition <ref>, we need to relate the Hamiltonians H_†⊆ C_l(^2) × C_b(^2) and H_⊆ C_u(^2) × C_b(^2) to H ⊆ C_b() × C_b(). 


Let H_†⊆ C_l(^2) × C_b(^2) and H_⊆ C_u(^2) × C_b(^2). We say that Ĥ_†⊆  C_l(^2) × C_b(^2) is a viscosity sub-extension of H_† if H_†⊆Ĥ_† and if for every λ >0 and h ∈ C_b(^2) a viscosity subsolution to f-λ H_† f = h is also a viscosity subsolution to f - λĤ_† f = h. Similarly, we define a viscosity super-extension Ĥ_ of H_.


The following lemma allows us to obtain viscosity extensions.

 
Let H_†⊆Ĥ_†⊆ C_l(^2) × C_b(^2) and H_⊆Ĥ_⊆ C_u(^2) × C_b(^2).



Suppose that for each (f,g) ∈Ĥ_† there exist (f_n,g_n) ∈ H_† such that, for every c,d ∈, we have

    lim_n ↑∞f_n ∧ c - f ∧ c = 0

and 

    lim sup_n ↑∞sup_z :  f(γ(z)) ∨ f_n(γ(z)) ≤ c g_n(z) ∨ d - g(z) ∨ d ≤ 0.

Then Ĥ_† is a sub-extension of H_†.



Suppose that for each (f,g) ∈Ĥ_ there exist (f_n,g_n) ∈ H_ such that, for every c,d ∈, we have

    lim_n ↑∞f_n ∨ c - f ∨ c = 0

and 

    lim inf_n ↑∞inf_z :  f(γ(z)) ∧ f_n(γ(z)) ≥ c g_n(z) ∧ d - g(z) ∧ d ≥ 0.

Then Ĥ_ is a super-extension of H_.






Acknowledgements
The authors are grateful to the anonymous referees who pointed out important points which led to an improvement of the paper.
FC was supported by The Netherlands Organisation for Scientific Research (NWO) via TOP-1 grant 613.001.552 and by the French foundation Fondation Sciences Ma­thé­ma­ti­ques de Paris. Part of this work was done during FC's stay at the Institut Henri Poincaré – Centre Émile Borel during the trimester Stochastic Dynamics Out of Equilibrium. FC thanks this institution for hospitality. 

MG was supported by the French foundation Fondation Mathématique Jacques Hadamard and partially by the Dutch cluster Stochastics – Theoretical and Applied Research (STAR). 

RK was supported by the Deutsche Forschungsgemeinschaft (DFG) via RTG 2131 High-dimensional Phenomena in Probability – Fluctuations and Discontinuity.







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