Path-space moderate deviation principles for the random field Curie-Weiss model

We analyze the dynamics of moderate fluctuations for macroscopic observables of the random field Curie Weiss model (i.e., standard Curie-Weiss model embedded in a site dependent, i.i.d. random environment). We obtain path space large deviation principles via a general analytic approach based on convergence of non-linear generators and uniqueness of viscosity solutions for associated Hamilton Jacobi equations. The moderate asymptotics depend crucially on the phase we consider and moreover, the space-time scale range for which fluctuations can be proven is restricted by the addition of the disorder.


Introduction
The study of the normalized sum of random variables and its asymptotic behavior plays a central role in probability and statistical mechanics. Whenever the variables are independent and have finite variance, the central limit theorem ensures that the sum with square-root normalization converges to a Gaussian distribution. The generalization of this result to dependent variables is particularly interesting in statistical mechanics where the random variables are correlated through an interaction Hamiltonian. For explicitly solvable models many properties are well understood. In this category fall the so-called Curie-Weiss models for which one can explicitly explain important phenomena such as multiple phases, metastable states and, particularly, how macroscopic observables fluctuate around their mean values when close to or at critical temperatures. Ellis and Newman characterized the distribution of the normalized sum of spins (empirical magnetization) for a wide class of mean-field Hamiltonian of Curie-Weiss type [EN78a,EN78b,ENR80]. They found conditions, in terms of thermodynamic properties, that lead in the infinite volume limit to a Gaussian behavior and those which lead to a higher order exponential probability distribution. Equilibrium large deviation principles have been established in [Ell85], wheras path-space counterparts have been derived in [Com87]. Static and dynamical moderate deviations have been obtained in [EL04,CK17] respectively. We are interested in the fluctuations of the magnetization for the random field Curie-Weiss model, which is derived from the standard Curie-Weiss by replacing the constant external magnetic field by local and random fields which interact with each spin of the system. The random field Curie-Weiss model has the advantage that, while still being analytically tractable, it has a very rich phase-structure. The phase diagram exhibits interesting critical points: a critical curve where the transition from paramagnetism to ferromagnetism is second-order, a first-order boundary line and moreover, depending on the distribution of the randomness, a tri-critical point may exist [SW85]. As a consequence, the model has been used as a playground to test new ideas. We refer to [APZ92] for the characterization of infinite volume Gibbs states; [KLN07] for Gibbs/non-Gibbs transitions; [Kül97,IK10,FKR12] for the study of metastates; [MP98,FMP00,BBI09] for the metastability analysis; and references therein. From a static viewpoint, the behavior of the fluctuations for this system is clear. In [AP91], a central limit theorem is proved and some remarkable new features as compared to the usual non-random model are shown. In particular, depending on temperature, fluctuations may have Gaussian or non-Gaussian limit; in both cases, however, such a limit depends on the realization of the local random external fields, implying that fluctuations are non-self-averaging. Large and moderate deviations with respect to the corresponding (disorder dependent) Gibbs measure have been studied as well. An almost sure large deviation principle can be obtained from [Com89] if the external fields are bounded and from [LMT13] if they are unbounded or dependent. Almost sure moderate deviations are characterized in [LM12] under mild assumptions on the randomness. As already mentioned, all the results recalled so far have been derived at equilibrium; on the contrary, we are interested in describing the time evolution of fluctuations, obtaining non-equilibrium properties. Fluctuations for the random field Curie-Weiss model were studied on the level of a path-space large deviation principle in [DPdH96] and on the level of a path-space (standard and non-standard) central limit theorem in [CDP12]. The purpose of the present paper is to study dynamical moderate deviations of a suitable macroscopic observable. In the random field Curie-Weiss model we are considering, the disorder comes from a site-dependent magnetic field which is η i = ±1. The single spin-flip dynamics induces a Markovian evolution on a bi-dimensional magnetization. The first component is the usual empirical average of the spin values: m n = n −1 n i=1 σ i . The second component is q n = n −1 n i=1 σ i η i and measures the relative alignment between the spins and their local random fields. The observable we are interested in is therefore the pair (m n , q n ) and we aim at analyzing its path-space moderate fluctuations. A moderate deviation principle is technically a large deviation principle and consists in a refinement of a (standard or non-standard) central limit theorem, in the sense that it characterizes the exponential decay of deviations from the average on a smaller scale. We apply the generator convergence approach to large deviations by Feng-Kurtz [FK06] to characterize the most likely behavior for the trajectories of fluctuations around the stationary solution(s) in the various regimes. Our findings highlight the following distinctive aspects: • The moderate asymptotics depend crucially on the phase we are considering.
The physical phase transition is reflected at this level via a sudden change in the speed and rate function of the moderate deviation principle. In particular, our findings indicate that fluctuations are Gaussian-like in the sub-and supercritical regimes, while they are not at criticalities. Moreover, if the inverse temperature and the magnetic field intensity are sizedependent and approach a critical threshold, the rate function retains the features of the phases traversed by the sequence of parameters and is a mixture of the rate functions corresponding to the visited regimes.
• In the sub-and supercritical regimes, the processes m n and q n evolve on the same time-scale and we characterize deviations from the average of the pair (m n , q n ). For the proof we will refer to the large deviation principle in [CK17,Appendix A]. On the contrary, at criticality, we have a natural timescale separation for the evolutions of our processes: q n is fast and converges exponentially quickly to zero, whereas m n is slow and its limiting behavior can be determined after suitably "averaging out" the dynamics of q n . Corresponding to this observation, we need to prove a path-space large deviation principle for a projected process, in other words for the component m n only.
The projection on a one-dimensional subspace relies on the synergy between the convergence of the Hamiltonians [FK06] and the perturbation theory for Markov processes [PSV77]. The method exploits a technique known for (linear) infinitesimal generators in the context of non-linear generators and, to the best of our knowledge, is original. Moreover, due to the fact that the perturbed functions we are considering do not allow for a uniform bound for the sequence of Hamiltonians, in the present case we need a more sophisticated notion of convergence of Hamiltonians than the one used in [CK17]. To circumvent this unboundedness problem, we relax our definition of limiting operator. More precisely, we follow [FK06] 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. The same techniques have been recently applied in [CGK] to tackle path-space moderate deviations for a system of interacting particles with unbounded state space.
• The fluctuations are considerably affected by the addition of quenched disorder: the range of space-time scalings for which moderate deviation principles can be proven is restricted by the necessity of controlling the fluctuations of the field.
• In [CDP12], at second or higher order criticalities, the contribution to fluctuations coming from the random field is enhanced so as to completely offset the contribution coming from thermal fluctuations. The moderate scaling allows to go beyond this picture and to characterize the thermal fluctuations at the critical line and at the tri-critical point.
It is worth to mention that our statements are in agreement with the static results found in [LM12]. The paper is organized as follows.
Outline: Appendix A is devoted to the derivation of a large deviation principle via solution of Hamilton-Jacobi equation and it is included to make the paper as much selfcontained as possible.

Notation and definitions
Before entering the 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.
Definition 2.1. Let {X n } n 1 be a sequence of random variables on a Polish space X. Furthermore, consider a function I : X → [0, ∞] and a sequence {r n } n 1 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 1 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 −1 n log P[X n / ∈ K a ] −a.
• the sequence {X n } n 1 satisfies the large deviation principle with speed r n and good rate-function I, denoted by if, for every closed set A ⊆ X, we have lim sup and, for every open set U ⊆ X, Throughout the whole paper AC will denote the set of absolutely continuous curves in R d . For the sake of completeness, we recall the definition of absolute continuity.
A curve γ : R + → R d is absolutely continuous if the restriction to [0, T ] is absolutely continuous for every T 0.
An important and non-standard definition that we will often use is the notion of o(1) for a sequence of functions. Definition 2.3. Let {g n } n 1 be a sequence of real functions. We say that if sup n 1 sup x |g n (x)| < ∞ and lim n→∞ sup x∈K |g n (x) − g(x)| = 0, for all compact sets K.
To conclude we fix notation for a collection of function-spaces.
Definition 2.4. Let k 1 and E a closed subset of R d . We will denote by ) 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 k c (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,

Microscopic and macroscopic description of the model
For a given realization of η, {σ(t)} t 0 evolves as a Markov process on {−1, +1} n , with infinitesimal generator where σ i is the configuration obtained from σ by flipping the i-th spin; β and B are positive parameters representing the inverse temperature and the coupling strength of the external magnetic field, and m n = 1 n n i=1 σ i . The two terms in the rates of (2.1) have different effects: the first one tends to align the spins, while the second one tends to point each of them in the direction of its local field. In addition to the usual empirical magnetization, we define also the empirical averages Let E n be the image of {−1,1} n × {−1,1} n under the map (σ, η) → (m n , q n ). The Glauber dynamics on the configurations, corresponding to the generator (2.1), induce Markovian dynamics on E n for the process {(m n (t), q n (t))} t 0 , that in turn evolves with generator For later convenience, let us introduce the functions 3) We start with a large deviation principle for the trajectory of {(m n (t), q n (t))} t 0 . Note that which implies that given η, (m n + q n , m n − q n ) is a pair of variables taking their value in discrete subsets of the square Proposition 2.5 (Large deviations, Theorem 1 in [Kra16]). Suppose that (m n (0), q n (0)) satisfies a large deviation principle with speed n on R 2 with a good rate function I such that {(x, y) | I(x, y) < ∞} ⊆ E 0 . Then, µ-almost surely, the trajectories {(m n (t), q n (t))} t 0 satisfy the large deviation principle on D R 2 (R + ), with good rate function I that is finite only for trajectories in E 0 and Proof. Arguing for the pair (m n + q n , m n − q n ), we can use Theorem 1 in [Kra16]. We obtain our result by undoing the coordinate transformation.
We recall that a large deviation principle in the trajectory space can also be derived via contraction of a large deviation principle for the non-interacting particle system; see [DPdH96] for details. Moreover, a static quenched large deviation principle for the empirical magnetization has been proved in [LMT13]. In both the aforementioned papers, the large deviation principle is obtained under assumptions that cover more general disorder than dichotomous.
The path-space large deviation principle in Proposition 2.5 allows to derive the infinite volume dynamics for our model: if (m n (0), q n (0)) converges weakly to the constant (m 0 , q 0 ), then the empirical process (m n (t), q n (t)) t 0 converges weakly, as n → ∞, to the solution oḟ with initial condition (m 0 , q 0 ). The phase portrait of system (2.5) is known; for instance, see [APZ92,DPdH95]. We briefly recall the analysis of equilibria. First of all, observe that any stationary solution of (2.5) is of the form (2.6) and that (0, tanh(βB)) satisfies (2.6) for all the values of the parameters. Solutions with m = 0 are called paramagnetic, those with m = 0 ferromagnetic. On the phase space (β, B) we get the following: (I) If β 1, then (0, tanh(βB)) is the unique fixed point for (2.5) and it is globally stable.
We refer to Figure 1 for a visualization of the previous assertions.

Main results
We consider the moderate deviations of the microscopic dynamics (2.2) around their stationary macroscopic limit in the various regimes. The first of our statements is mainly of interest in the paramagnetic phase, but is indeed valid for all values of the parameters. Theorem 2.6 (Moderate deviations around (0, tanh(βB))). Let {b n } n 1 be a sequence of positive real numbers such that b n → ∞ and b 2 n n −1 log log n → 0.
Suppose that (b n m n (0), b n (q n (0) − tanh(βB))) satisfies a large deviation principle with speed nb −2 n on R 2 and rate function I 0 . Then, µ-almost surely, the trajectories {(b n m n (t), b n (q n (t) − tanh(βB)))} t 0 Each colored region represents a phase with as many ferromagnetic solutions of (2.6) as indicated by the numerical label. The thick red and blue separation lines are g 1 and g 2 respectively. The thin dashed blue line is the coexistence line relevant for metastability (cf. II(iv)).
satisfy the large deviation principle on D R×R (R + ), with good rate function (2.8) Observe that the growth condition b 2 n n −1 log log n → 0 is necessary to ensure that b n η n (re-scaled empirical average of the local fields) converges to zero almost surely as n → +∞. A similar effect is also known in moderate deviation principles for the overlap in the Hopfield model, see [EL04]. The peculiar scaling is prescribed by the law of iterated logarithm, that provides the scaling factor where the limits of the weak and strong law of large numbers become different, cf. [Kal02, Corollary 14.8]. Analogous requirements will appear also in the following statements.
Theorem 2.7 (Moderate deviations: super-critical regime β > 1, B < g 2 (β)). Let (m, q) be a solution of (2.6) with m = 0. Moreover, let {b n } n 1 be a sequence of positive real numbers such that b n → ∞ and b 2 n n −1 log log n → 0.
Suppose that (b n (m n (0) − m), b n (q n (0) − q)) satisfies a large deviation principle with speed nb −2 n on R 2 and rate function I 0 . Then, µ-almost surely, the trajectories satisfy the large deviation principle on D R×R (R + ), with good rate function We see that the Lagrangian (2.8) trivializes in the x coordinate if β = cosh 2 (βB). The latter equation corresponds to (β, B) lying on the critical curve B = g 1 (β). This fact can be seen as the dynamical counterpart of the bifurcation occurring at the stationary point as B varies for fixed β: (0, tanh(βB)) is turning unstable from being a stable equilibrium. At the critical curve, the fluctuations of m n (t) behave homogeneously in the distance from the stationary point, whereas the fluctuations of q n (t) are confined around 0.
To further study the fluctuations of m n (t), we speed up time to capture higher order effects of the microscopic dynamics. Speeding up time implies that the probability of deviations from q n (t) decays faster than exponentially.
Suppose that b n m n (0) satisfies a large deviation principle with speed nb −4 n on R and rate function I 0 . Then, µ-almost surely, the trajectories b n m n (b 2 n t) t 0 satisfy the large deviation principle on D R (R + ), with good rate function At the tri-critical point, again the Lagrangian trivializes, and a further speed-up of time is possible.
Theorem 2.9 (Moderate deviations: tri-critical point β = 3 2 and B = g 1 ( 3 2 )). Let {b n } n 1 be a sequence of positive real numbers such that b n → ∞ and b 10 n n −1 log log n → 0.
Suppose that b n m n (0) satisfies a large deviation principle with speed nb −6 n on R and rate function I 0 . Then, µ-almost surely, the trajectories b n m n (b 4 n t) t 0 satisfy the large deviation principle on D R (R + ), with good rate function We want to conclude the analysis by considering moderate deviations for volumedependent temperature and magnetic field approaching the critical curve first and the tri-critical point afterwards. In the sequel let {m β,B n (t)} t 0 denote the process evolving at temperature β and subject to a random field of strength B.
Theorem 2.10 (Moderate deviations: critical curve 1 < β 3 2 , B = g 1 (β), temperature and field rescaling). Let {b n } n 1 be a sequence of positive real numbers such that b n → ∞ and b 6 n n −1 log log n → 0. Let {κ n } n 1 , {θ n } n 1 be two sequences of real numbers such that κ n b 2 n → κ and θ n b 2 n → θ. where I is the good rate function and For approximations of the tri-critical point, we consider two scenarios. The first considers an approximation along the critical curve, whereas the second scenario considers approximation from an arbitrary direction.
Theorem 2.11 (Moderate deviations: tri-critical point β = 3 2 , B = g 1 ( 3 2 ), temperature and field rescaling on the critical curve). Let {b n } n 1 be a sequence of positive real numbers such that b n → ∞ and b 10 n n −1 log log n → 0. Let {κ n } n 1 , {θ n } n 1 be two sequences of real numbers such that Set β n := β tc +κ n and B n := B tc +θ n , where (β tc , B tc ) = ( 3 2 , 2 3 arccosh( 3 2 )). Assume that β n = cosh 2 (β n B n ) for all n ∈ N. Moreover, suppose that b n m βn,Bn n (0) satisfies the large deviation principle with speed nb −6 n on R with rate function I 0 . Then, µ-almost surely, the trajectories b n m βn,Bn where I is the good rate function and Remark. To ensure that (β n , B n ) approximate (β tc , B tc ) over the critical curve, κ and θ must satisfy (2.17) Theorem 2.12 (Moderate deviations: tri-critical point β = 3 2 , B = g 1 ( 3 2 ), temperature and field rescaling). Let {b n } n 1 be a sequence of positive real numbers such that b n → ∞ and b 10 n n −1 log log n → 0. Let {κ n } n 1 , {θ n } n 1 be two sequences of real numbers such that κ n b 4 n → κ and θ n b 4 n → θ.
Set β n := β tc + κ n and B n : where I is the good rate function and By choosing the sequence b n = n α , with α > 0, we can rephrase Theorems 2.6-2.12 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. These results, together with the central limit theorem and the study of fluctuations at β = cosh 2 (βB) in [CDP12, Prop. 2.7 and Thm. 2.12], give a clear picture of the behaviour of fluctuations for the random field Curie-Weiss model. We summarize these facts in Tables 1 and 2. The displayed conclusions are drawn under the assumption that in each case either the initial condition satisfies a large deviation principle at the correct speed or the initial measure converges weakly. Observe that not all scales can be covered. Indeed, to control disorder fluctuations and avoid they are too large, the range of allowed spatial scalings becomes quite limited.
To conclude, it is worth to mention that our results are consistent with the moderate deviation principles obtained in [LM12] for the random field Curie-Weiss model at equilibrium. Indeed, as prescribed by Thm. 5.4.3 in [FW98], in each of the cases above, the rate function of the stationary measure satisfies LDP at speed n with rate function (2.4) LDP at speed n 1−2α with rate function (2.12) α = 1 2 all β B > g 1 (β) n 1/2 m n (t), n 1/2 (q n (t) − tanh(βB)) CLT weak convergence to the unique solution of a linear diffusion equation (see [CDP12,Prop. 2.7]) β > 1 B < g 2 (β) n 1/2 (m n (t) − m), n 1/2 (q n (t) − q)

Expansion of the Hamiltonian and moderate deviations in the sub-and supercritical regimes
Following the methods of [FK06], the authors have studied large and moderate deviations for the Curie-Weiss model based on the convergence of Hamiltonians and well-posedness of a class of Hamilton-Jacobi equations corresponding to a limiting Hamiltonian in [Kra16,CK17]. For the results in Theorems 2.6 and 2.7, considering moderate deviations for the pair (m n (t), q n (t)), we will follow a similar strategy and we will refer to the large deviation principle in [CK17, Appendix A]. For the results in Theorems 2.8-2.12 stated for the process m n (t) only, we need a more sophisticated large deviation result, which is based on the abstract framework introduced in [FK06]. We will recall the notions needed for these results in Appendix A.
n α m n n 2α t LDP at speed n 1−4α with rate function (2.13) n α m n n 4α t LDP at speed n 1−6α with rate function (2.14) β n = 3 2 + κ n , B n = g 1 ( 3 2 ) + θ n where β n = cosh 2 (β n B n ), ∀n ∈ N κ n n 2α → κ, θ n n 2α → θ n α m n n 4α t LDP at speed n 1−6α with rate function (2.16) β n = 3 2 + κ n , B n = g 1 ( 3 2 ) + θ n where κ n n 4α → κ, θ n n 4α → θ n α m n n 4α t LDP at speed n 1−6α with rate function (2.18) In both settings, however, a main ingredient is the convergence of Hamiltonians. Therefore, we start by deriving an expansion for the Hamiltonian associated to a generic time-space scaling of the fluctuation process. We will then use such an expansion to obtain the results stated in Theorems 2.6 and 2.7. For Theorems 2.8-2.12, we need additional methods to obtain a limiting Hamiltonian, that will be introduced in Sections 4 and 5 below.

Expansion of the Hamiltonian
Let (m, q) be a stationary solution of equation (2.5). We consider the fluctuation process b n (m n (b ν n t) − m) , b n (q n (b ν n t) − q) . Its generator A n can be deduced from (2.2) and is given by Therefore, at speed nb −δ n , the Hamiltonian Let ∇f(x, y) = (f x (x, y), f y (x, y)) be the gradient of f. Moreover, denote We Taylor expand the exponential functions containing f up to second order: To write down the intermediate result after Taylor-expansion, we introduce the functions (3.2) and the matrix , which are the finite-volume analogues of (2.3) and (2.9). In what follows, not to clutter our formulas we will drop subscripts highlighting the dependence on the inverse temperature β and the magnetic field B. A tedious but straightforward calculation yields To have an interesting reminder in the limit, we need δ = ν + 2. This gives In the sequel we will Taylor expand G ± n,1 and G ± n,2 around (m, q). Therefore we need the derivatives of the G's. By direct computation, we get the following lemma.
Lemma 3.1. Let G ± n,1 and G ± n,2 be defined by (3.2). Then, we obtain Note that the functions G, whenever differentiated twice in the y direction, equal zero. For the sake of readability, we again put the terms of the Taylor expansions of G ± n,1 and G ± n,2 in matrix form. For k ∈ N, k 1, let us denote , where G ± n,2 are defined in (3.2) and G ± 2 in (2.3). Moreover, set We obtain the following expansion.
, ∇f(x, y) (3.5) where the remainder terms converge to zero uniformly on compact sets.
Observe that o(1) + o(b ν−4 n ) (cf. lines (3.7) and (3.9)) includes all the remainder terms coming from first Taylor expanding the exponentials, and then the functions . For any f ∈ C 3 c (R 2 ), let us denote by R exp n,f and R G n,f these two contributions. In what follows we will need a more accurate control on these remainders. For this reason we state the following lemma.
There exists a positive constant C (dependent on the sup-norms of the first to third order partial derivatives of f, but not on n) such that we have (3.10) Proof. We study the Taylor expansion of the exponential functions first. We treat explicitly only the case of the others being analogous. We denote by R exp,+ n,f the remainder terms coming from Taylor expanding such a function. To shorten our next formula, we set x = (x, y) and ξ = (ξ 1 , ξ 2 ) . By Lagrange's form of the Taylor expansion, there is some ξ ∈ R 2 with ξ 1 ∈ (x, x + 2b n n −1 ) and ξ 2 ∈ (y, y + 2b n n −1 ) and Observe that, by the mean-value theorem, we can control the exponential. Indeed, there exists a point z, on the line-segment connecting ξ and x, for which we have Therefore, we can find positive constants c 1 and c 2 (depending on sup-norms of the first, second and third order partial derivatives of f, but not on n), such that Analogously, we get the same control for the other three exponential terms. We conclude sup We focus now on the remainder terms relative to the expansion of the G's function.
Turning back to the expansion of H n in Lemma 3.2, we analyze now the terms containing η n that appear in (3.5) and (3.7). As cosh is a positive function and b n → ∞, any contribution by η n is dominated by the one in (3.5). To make sure that this term vanishes, we apply the Law of Iterated Logarithm. As an immediate corollary, we obtain conditions ensuring that b ν+1 n η n converges to zero almost surely. Note that condition (3.11) corresponds to the growth assumption in Theorems 2.6 and 2.7 for ν = 0, in Theorems 2.8 and 2.10 for ν = 2 and in Theorems 2.9, 2.11 and 2.12 for ν = 4. The result of Lemma 3.2, combined with the corollary, yields a preliminary expansion for which is obtained from the generic Hamiltonian (3.1) after the choice δ = ν + 2 we made to get a non-trivial expansion with controlled remainder.
Then, µ-almost surely, we have (3.14) In the setting of our main theorems ν ∈ {0,2,4} and (m, q) is a stationary point. This implies that all contributions on the right hand side of (3.12) vanish almost surely and uniformly on compact sets as n → ∞. Furthermore, the expression in (3.14) is constant and we do not need to consider this expression any further. Thus, the analysis for our main results focuses on the expressions in (3.13). The next lemma gives expressions for the matrices D k G 2 (m, q).
Lemma 3.7. Let k ∈ N, k 1. ) is a stationary point, then

Proof of Theorems 2.6 and 2.7
The setting of Theorems 2.6 and 2.7 corresponds to that of Proposition 3.6 with ν = 0. Having chosen a stationary point (m, q) and applying Lemma 3.7(b), we find where the matricesĜ 1 and B are defined in (2.10) and (2.11) respectively. The remainder o(1) is uniform on compact sets. Therefore, for f ∈ C 2 c (R 2 ), H n f converges uniformly to Hf(x) = H(x, ∇f(x)), where The large deviation results follow by Theorem A.14, Lemma 3.4 and Proposition 3.5 in [CK17]. The Lagrangian is found by taking the Legendre-Fenchel transform of H and is given by Observe that, in the case when (m, q) = (0, tanh(βB)), we get This concludes the proof.

Projection on a one-dimension subspace and moderate deviations at criticality
For the proofs of Theorems 2.8 and 2.9, we consider the stationary point (m, q) = (0, tanh(βB)). Recall that, given the correct assumptions on the sequence {b n } n 1 , the expression for the Hamiltonian in Proposition 3.6 reduces µ-a.s. to x k kx k−1 y , ∇f(x, y) If ν ∈ {2,4}, the term corresponding to D 1 G 2 (0, tanh(βB)) is diverging and, more precisely, is diverging through a term containing the y variable (see Lemma 3.7(d)).
We have a natural time-scale separation for the evolutions of our variables: y is fast and converges very quickly to zero, whereas x is slow and its limiting behavior can be characterized after suitably "averaging out" the dynamics of y. Corresponding to this observation, our aim is to prove that the sequence {H n } n 1 admits a limiting operator H and, additionally, the graph of this limit depends only on the x variable. In other words, we want to prove a path-space large deviation principle for a projected process.
The projection on a one-dimensional subspace relies on the formal and recursive calculus explained in the next section (an analogous approach will be implemented also in Section 5.1 to achieve the large deviation principles of Theorems 2.10-2.12). We want to mention that the results presented in Sections 4.1 and 5.1 take inspiration from the perturbation theory for Markov processes introduced in [PSV77].

Formal calculus with operators and a recursive structure
We start by introducing a formal structure allowing to write the drift component in (4.1) in abstract form. Afterwards, we introduce a method based on this abstract structure to perturb a function ψ depending on the only variable x to a function F n,ψ depending on (x, y), so that the perturbation exactly cancels out the contributions of the drift operators to the y variable.
Consider the vector spaces of functions and Moreover, for notational convenience, we will denote Next we define a collection of operators on V. Let a ∈ R and g : R 2 → R a differentiable function. We consider the operators for even k In particular, note that all operators map V into V. Furthermore, the operators Q 1 k , with odd k, have V 0 as a kernel. We will see that Q 1 1 plays a special role.
Assumption 4.2. Assume there exist real constants (a + k ) k 1 , (a − k ) k 1 if k is even and a 0 1 = 0, (a 0 k ) k>1 , (a 1 k ) k 1 if k is odd, for which, given a continuously differentiable function g : R 2 → R, we can write • for even k, • for odd k, Observe that the drift term in (4.1) is of the form We aim at abstractly showing that, for any function ψ ∈ V 0 and sequence b n → ∞, we can find a perturbation F n,ψ ∈ V of ψ for which there existsψ ∈ V 0 such that We will construct the perturbation in an inductive fashion. We start by motivating the first step of the construction. Let ψ ∈ V 0 , i.e. a function only depending on x. Then: (1) Q 1 ψ = 0, but Q 2 ψ = 0 and moreover Q 2 ψ ∈ V 1 because of the action of Q + 2 .
(2) The leading order term in Q (n) ψ is given by b ν−1 n Q 2 ψ.
Note that we can only assure that the sum is in V 0 . This is due to the specific structure of the operators that we will discuss in Lemma 4.6. Now there are two possibilities (a) Q 3 ψ + Q 2 ψ[1] + Q 1 ψ[2] = 0. In this case ν = 2 is the maximal ν that we can use for this particular problem. In addition, the outcome of this sum will be in the form cx 3 ψ (x) and hence determine the limiting drift in the Hamiltonian.
In this case, ν = 2 is a possible option. However, we can use a larger ν and proceed with perturbing ψ with even higher order terms.
Proof. By direct computation, we get from which the conclusion follows.
Starting from ψ = ψ[0] ∈ V 0 , we define recursively  , which is exactly the result that we aimed to find in steps (4) and (5) above.
Next, we evaluate the action of Q (n) applied to our perturbation of ψ.

Proposition 4.5. Fix ν 2 an even natural number and suppose that Assumption 4.2 holds true for this ν. Consider the operator
where o(1) is meant according to Definition 2.3.
Proof. We aim at determining the leading order term of The remainder o(1) contains lower order terms in the expansion and it is small as b −l n ψ[l] is uniformly bounded on the state space E n for any n ∈ N (see Lemma 4.10). We re-arrange the first sum by changing indices r = k + l − 1. It yields Observe that the term corresponding to r = 0 vanishes as Q 1 ψ[0] = 0. By (4.5) and the properties stated in Remark 4.4, we get For the cases we are interested in, we will use ν ∈ {2,4}. Thus, to conclude, we need to consider the action of P 0 on the functions φ[r], for r = 1, . . . , 4. This is the content of the next two statements. The functions ψ[r], φ[r] belong to the vector spaces V i according to the classification given in the next lemma.
Proof. As all operators Q k map V i k into V i k+1 and the projection P maps V 0 to {0}, it suffices to prove that, for any r ∈ N, ψ[2r] ∈ V even and ψ[2r + 1] ∈ V odd . We proceed by induction. Clearly the result holds true for r = 0. We are left to show the inductive step. Suppose the claim is valid for all positive integers less than r, we must prove that, if r is odd (resp. even) and ψ[r] ∈ V odd (resp. V even ), then ψ[r + 1] ∈ V even (resp. V odd ). We stick on the odd r case, the other being similar. By definition, we have Let us analyze the sum on the right-hand side of the previous formula. It is composed of terms of two types: either l is even or it is odd.
• If l even, then by inductive hypothesis ψ[l] ∈ V even . Additionally, r + 2 − l is odd, so that by Lemma 4.1, the operator Q k maps V even to V even and therefore, Q r+2−l ψ[l] ∈ V even .
• If l odd, then by inductive hypothesis ψ[l] ∈ V odd . Additionally, r + 2 − l is even, so that by Lemma 4.1, the operator Q k maps V odd to V even and therefore, Q r+2−l ψ[l] ∈ V even .
This yields that ψ[r + 1] ∈ V even , giving the induction step.
To evaluate the limiting drift from the expression in Proposition 4.5, we need to evaluate P 0 in the functions φ[i], with i ∈ N, i ν. From Lemma 4.6, we have φ[2i + 1] ∈ V odd which is in the kernel of P 0 . An explicit calculation for the even i is done in the next lemma. (4.7) Proof. By Lemma 4.6, if l is odd, then φ[l] ∈ V odd and, as a consequence, P 0 φ[l] = 0.
We are left to understand how the even terms contribute to V 0 . We exploit the recursive structure of the functions ψ[l] and φ[l].
For k = 2, we find Q 3 ψ + Q − 2 ψ[1], as Q + 2 always maps into the kernel of P 0 . For k = 4 , as ψ[2] has no V 0 component and Q 3 maps V i into V i for all i. Thus, we obtain A straightforward computation yields the following result that will be useful for the computation of the constants involved in the operators in the previous lemma.
Lemma 4.8. Given ψ ∈ V, it holds We see in (4.7) that P 0 φ[4] contains a part resembling P 0 φ[2]. On the one hand, if P 0 φ[2] = 0, then P 0 φ[4] has a much simpler structure. On the other, whenever is not needed as in (3.13) there are terms of higher order that dominate. As a consequence, we will only ever work with the simplified result for P 0 φ[4]. Similar observations involving higher order recursions can be made for any arbitrary P 0 φ[2l] with l ∈ N, l 3. By combining these remarks with the type of calculations made for getting the expressions presented in Lemma 4.8, we conjecture the following general structure. We neither prove nor use (4.8), but we believe it is of interest from a structural point of view and deserves to be stated.

Proofs of Theorems 2.8 and 2.9
The formal calculus we developed in Section 4.1 is used to formally identify the limiting operator H of the sequence H n given in (4.1). However, it is not possible to show directly H ⊆ LIM n H n as in the proof of Theorems 2.6 and 2.7, since the most functions ψ ∈ C ∞ c (R) cause sup n H n F n,ψ ≮ ∞ and thus we can not prove LIM n H n F n,ψ = Hψ.
To circumvent the problem, 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. 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 A. We rely on Theorem A.9 for which we must check the following conditions: (a) The processes ((b n m n (b ν n t), b n (q n (b ν n t) − tanh(βB))) , t 0) satisfy an appropriate exponential compact containment condition.
such that H † ⊆ ex − subLIM n H n and H ‡ ⊆ ex − superLIM n H n .
(c) There is an operator H ⊆ C b (R) × C b (R) such that every viscosity subsolution to f − λH † f = h is a viscosity subsolution to f − λHf = h and such that every supersolution to f − λH ‡ f = h is a viscosity supersolution to f − λHf. The operators H † and H ‡ should be thought of as upper and lower bounds for the "true" limiting H of the sequence H n .
(d) The comparison principle holds for the Hamilton-Jacobi equation f − λHf = 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 [CK17].
We will start with the verification of (b)+(c), which are based on the expansion in Proposition 3.6 and the formal calculus in Section 4.1. Afterwards, we proceed with the verification of (a), for which we will use the result of (b). Finally, the form of the operator H is of the type considered in e.g. [CK17] or [FK06, Section 10.3.3] and thus, the establishment of (d) is immediate.
Consider the statement of Proposition 3.6. We want to extract the limiting behavior of the operators H n presented there. If (m, q) = (0, tanh(βB)) the term in (3.12) vanishes, whereas the term in (3.14) converges if ∇f n (x, y) n→∞ − −−− → ∇f(x) uniformly on compact sets (see Lemma 4.10 below). For the term in (3.13), we use the results from Section 4.1.
The next lemma proves uniform convergence for the sequence of perturbation functions F n,ψ and for the sequence of the gradients.
for all rectangles R ⊆ R 2 . The second part of this limiting statement establishes (4.15). To show LIM F n,ψ = ψ, we need the first part of this limiting statement and uniform boundedness of the sequence F n,ψ . This final property follows since E n ⊆ R × [−2b n ,2b n ], implying that b −l n ψ[l] is bounded for each l.
We start by calculating the terms in (3.13) that contribute to the limit via Proposition 4.5 and Lemma 4.7.
We prove our first claim. The term Q 0 3 f is given in (4.11), whereas Lemma 4.8 yields Combining these two results, we get By using the fundamental identity cosh 2 − sinh 2 = 1 for hyperbolic functions and the fact we are on the critical curve, simple algebraic manipulations lead to the first conclusion.
We proceed with proving (4.17). The term Q 0 5 is defined in (4.13). By Lemma 4.8 we find and

Adding the contributions above gives
Plugging the value β = β tc = 3 2 yields the result. Approximating Hamiltonians and domain extensions. The natural perturbations of our functions ψ do not allow for uniform bounds of H n F n,ψ . We repair this lack by cutting off the functions. To this purpose, we introduce a collection of smooth increasing functions χ n : R → R such that 0 χ n 1 and (4.18) Lemma 4.12. Suppose we are either in the setting of Theorem 2.8 with ν = 2 or in the setting of Theorem 2.9 with ν = 4. Let ε ∈ (0,1) and ψ ∈ C ∞ c (R). Consider the cut-off (4.18) and define the functions with F n,ψ as in (4.4)+(4.5). Then, (a) For any C > 0 there is an N = N(C) such that, for any n N, we have χ n ≡ id on the compact set K 1 = K 1 (C) := (x, y) ∈ R 2 ε log(1 + x 2 + y 2 ) C .
(b) Let C be a positive constant providing a uniform bound for the sequence (F n,ψ ) n 1 (cf. Lemma 4.10) and consider the compact set K 2,n := (x, y) ∈ R 2 ε 2 log(1 + x 2 + y 2 ) max{C, 2 log log b n } .
The function (4.19) is constant outside K 2,n .
Proof. We start by proving (a). The function F n,ψ is uniformly bounded by Lemma 4.10.
Consider an arbitrary C > 0. The mapping (x, y) → F n,ψ (x, y) ± ε log(1 + x 2 + y 2 ) is thus 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 (x, y) / ∈ K 2,n , we obtain The definition (4.18) of the cut-off leads then to the conclusion. The proof for the function F n,ψ (x, y) − ε log(1 + x 2 + y 2 ) follows similarly.
(e) For every c ∈ R, we have

Moreover, it holds
Proof. We prove all the properties for the '+' superscript case, the other being similar.
(a) As the cut-off (4.18) is smooth, it yields ψ ε,± n ∈ C ∞ (R 2 ). In addition, the location of the cut-off and Lemma 4.12(b) make sure that ψ ε,± n is constant outside a compact set K ⊂ E n , implying ψ ε,± n ∈ D(H n ).
(b) This is immediate from the definitions of ψ ε,± .
(d) This follows immediately by Lemma 4.12(a).
(f) This follows similarly as in the proof of (e).
Definition 4.14. Suppose we are either in the setting of Theorem 2.8 with ν = 2 or in the setting of Theorem 2.9 with ν = 4.
We have H † ⊆ ex − subLIM n H n and H ‡ ⊆ ex − superLIM n H n .
-If |F n,ψ (x, y) + ε log(1 + x 2 + y 2 )| < log log b n , the variables x and y are at most of order log 1/2 b n and we can characterize H n ψ ε,+ n by means of (4.1), since we can control the remainder term. Indeed, * the first, second and third order partial derivatives of ψ ∈ C ∞ c (R 2 ) and log(1 + x 2 + y 2 ) are bounded, therefore by means of (3.10) we get control of the remainder up to order log 1/2 b n variables x and y; * the function ψ is constant outside a compact set and thus has zero derivatives outside such a compact set; * by smoothness of the cut-off (4.18), the derivative χ n is bounded between 0 and 1. We want to show that (4.23) is uniformly bounded from above. We start by analyzing the terms in Ξ n (x, y). By completing the square, we can write
To conclude, observe that, since there exist positive constants c 1 and c 2 (independent of n) such that H n ψ ε,+ n c 1 b ν n log b n + c 2 (cf. equation (4.23)), choosing the sequence v n := b n leads to sup n v −1 n log H n ψ ε,+ n < +∞.
(A.3) 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 ψ ε,+ n (x n , y n ) Hψ(x).
As a converging sequence is bounded, by Lemma 4.13(d) we can find a sufficiently large N = N(K) ∈ N such that, for all n N, we have ψ ε,+ n (x n , y n ) = F n,ψ (x n , y n ) + ε log(1 + x 2 n + y 2 n ).
Thus, for any n N, equation (4.27) yields H n ψ ε,+ n (x n , y n ) Hψ(x) + 8 where the remainder terms converge to zero uniformly on compact sets. Since b n ↑ ∞, the conclusion follows.
To conclude this section we obtain the Hamiltonian extensions.

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 4.18. We start by proving an auxiliary lemma. H(x, p) = 2 cosh(βB) • in the setting of Theorem 2.9 with ν = 4: H(x, p) = 2 2 3 p 2 − 9 10 3 2 x 5 p.
We first verify Condition A.8: (a) follows from Proposition 4.15, (b) is satisfied by definition and (c) follows from Proposition 4.16.
The comparison principle for f − λHf = h for h ∈ C b (R) and λ > 0 has been verified in e.g. [CK17,Prop. 3.5]. Note that the statement of the latter proposition is valid for f ∈ C 2 c (R), but the result generalizes straightforwardly to class C ∞ c (R) as the penalization and containment functions used in the proof are C ∞ . Finally, the exponential compact containment condition follows from Proposition 4.18.

Variations in the external parameters
Suppose we are either in the setting of Theorem 2.10 with ν = 2 or in the setting of Theorems 2.11 and 2.12 with ν = 4. The major difference of these theorems compared to Theorems 2.8 and 2.9 is the variation in the parameters β and B as the system size increases. The inverse temperature and the magnetic field are respectively β n := β + κ n and B n := B + θ n , where {κ n } n 1 and {θ n } n 1 are real sequences converging to zero. In this more general framework, due to an extra Taylor expansion in β and B, the Hamiltonian (4.1) changes into x k kx k−1 y , ∇f(x, y) and determining the terms that contribute to the limiting operator is trickier than before. In the linear part of (4.1) the operators appearing with a factor b k n introduce terms with x k ; whereas, now this is no longer the case. Operators with pre-factor b k n may introduce terms with x m for m k (the power m depends on the order of κ n and θ n with respect to b n ). Therefore, we need to extend the method presented in Section 4.1 appropriately.

Extending the formal calculus of operators
Let V and V i , with i ∈ N, be the vector spaces of functions introduced at the beginning of Section 4.1. We are going to define an alternative set of operators on V. Let a ∈ R and g : R 2 → R be a differentiable function. Fix k ∈ N and consider the array of operators We have the direct analogue of Lemma 4.1.
Lemma 5.1. For all a ∈ R and k, i ∈ N, we have Notice that also in this extended setting the operators with superscript 1 (i.e., Q 1 k,m with odd m and m k) have the peculiarity of admitting V 0 as a kernel.
Assumption 5.2. Assume there exist real constants a + k,m , a − k,m if m is even and 1 m k and a 0 1,1 = 0, a 1 1,1 , a 0 k,m , a 1 k,m if m is odd and 1 < m k, for which, given a continuously differentiable function g : R 2 → R, we can write for even m and m k and Q 0 k,m g(x, y) : for odd m and m k.
Observe that we recover Assumption 4.2 if a z k,m = 0 whenever k = m and set Q z k,k := Q z k for appropriate z ∈ {+, −,1,0}. Using our new definitions, Lemma 4.3 and the recursion relationships (4.5) are unchanged and furthermore, the result of Proposition 4.5 is still valid. The main modification is that we need to re-evalute the functions P 0 φ[i] as the Q k are defined by using a larger set of operators. We get the following two statements.
Proposition 5.3. Fix ν 2 an even natural number and suppose that Assumption 5.2 holds true for this ν. Consider the operator where o(1) is meant according to Definition 2.3.
We can evaluate the functions P 0 φ[i] as we did in Lemma 4.7. Under the more general Assumption 5.2, more terms survive the infinite volume limit. We calculate the outcomes only for the cases we will need below.
(5.6) Following Conjecture 4.9, we can make a similar conjecture in this extended setting as well.

Preliminaries for the proofs of Theorems 2.10-2.12
As we did before, we now connect the discussion of Section 5.1 with the proofs of Theorems 2.10-2.12 via Theorem A.9. Recall that our purpose is to find an operator In other words, for f ∈ D(H), we need to determine f n ∈ H n such that LIM f n = f and LIM H n f n = Hf. Consider the statement of Proposition 3.6. We want to find the limit of the operator H n presented there. We analyze term by term. If (m, q) = (0, tanh(βB)) the term in (3.12) vanishes. The very same proof as the one of Lemma 4.10 gives the next lemma implying that the term in (3.14) converges as a consequence of the uniform convergence of the gradients.
Lemma 5.6. Suppose we are either in the setting of Theorem 2.10 with ν = 2 or in the setting of Theorems 2.11 and 2.12 with ν = 4. For ψ ∈ C ∞ c (R), define the approximation For the term in (3.13), we use the results from Section 5.1. At this point the proofs of Theorem 2.10 and 2.11 differ from the proof of Theorem 2.12 in the sense that in the first case κ n , θ n are of order b −2 n , whereas in the latter κ n , θ n are of order b −4 n . Therefore, the connection between the linear part in (5.1) and the operators in Assumption 5.2 changes. To give an explicit example: Q 0 5,1 is a different operator in the two settings. Using (5.6) of Lemma 5.4, we calculate the drift of the limiting Hamiltonians by considering the relevant operators in the two sections below.

Proof of Theorems 2.10 and 2.11
In the setting of Theorems 2.10 and 2.11, κ n and θ n are of order b −2 n . We identify the relevant operators for Assumption 5.2 from the linear part in the expansion in (5.1). First of all, there are the operators that do not involve derivations in the κ and θ directions. These are the operators we also considered for Theorems 2.8 and 2.9. Turning to our extended notation, we find for k ∈ {1, . . . ,5} and appropriate z ∈ {+, −,0,1}, the operators Q z k,k = Q z k as defined in (4.9)-(4.13). Additional operators are being introduced by the differentiations in the θ, κ directions. In particular, the relevant operators are (a) Q 0 3,1 and Q 1 3,1 arising from the first and second coordinate of (∂ κ + ∂ θ )D 1 G 2 (0, tanh(βB)) x y ; (b) Q + 4,2 and Q − 4,2 arising from the first and second coordinate of x 2 2xy ; (c) Q 0 5,3 arising from the first coordinate of 1 6 (∂ κ + ∂ θ )D 3 G 2 (0, tanh(βB)) x 3 3x 2 y ; (d) Q 0 5,1 arising from the first coordinate of x 2 2xy .

A.1 Compact containment condition
Given the convergence of the Hamiltonians, to have exponential tightness it suffices to establish an exponential compact containment condition.
Definition A.2. We say that a sequence of processes (Z n (t), t 0) on E n ⊆ R 2 satisfies the exponential compact containment condition at speed (r n ) n∈N * , with lim n↑∞ r n = ∞, if for all compact sets K ⊆ R 2 , constants a 0 and times T > 0, there is a compact set K ⊆ R 2 with the property that lim sup n↑∞ sup z∈K 1 r n log P [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.

A.2 Operator convergence for a projected process
In the papers [Kra16, CK17, DFL11], one of the main steps in proving the large deviation principle was proving directly the existence of an operator H such that H ⊆ LIM n H n ; in other words by verifying that, for all (f, g) ∈ H, there are f n ∈ H n such that LIM n f n = f and LIM n H n f n = g (the notion of LIM is introduced in Definition A.4). Here it is hard to follow a similar strategy. We are dealing with functions f n (x, y) = f(x) + b −1 n f 1 (x, y) + b −2 n f 2 (x, y) (for suitably chosen f 1 and f 2 ) given in a perturbative fashion and satisfying intuitively f n → f and H n f n → Hf with Hamiltonian H ⊆ C b (R) × C b (R). In contrast to the setting of [CK17], even if F n,f ∈ C ∞ c (R 2 ), we can not guarantee sup n H n F n,ψ < ∞, implying we do not have LIM H n f n = Hf. 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.
Definition A.5 (Condition 7.11 in [FK06]). 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 ↑ ∞.
(a) The extended sub-limit, denoted by ex−subLIM n H n , is defined by the collection (f, g) ∈ C l (R 2 ) × C b (R) for which there exist (f n , g n ) ∈ H n such that and that, for every compact set K ⊆ R 2 and every sequence z n ∈ K satisfying lim n z n = z and lim n f n (z n ) = f(z) < ∞, and that, for every compact set K ⊆ R 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). (A.6) For completeness, we also give the definition of the extended limit.
Definition A.6. Suppose that for each n we have an operator H n ⊆ C b (E n ) × C b (E n ). We write ex − LIM H n for the set of (f, g) ∈ C b (R 2 ) × C b (R 2 ) for which there exist (f n , g n ) ∈ H n such that f = LIM f n and g = LIM g n .
with the following properties: (c) For all λ > 0 and h ∈ C b (R), every subsolution to f − λH † f = h is a subsolution to f − λHf = h and every supersolution to f − λH ‡ f = h is a supersolution to 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.
Theorem A.9 (Large deviation principle). Suppose we are in the setting of Assumption A.1 and Condition A.8 is satisfied. Suppose that for all λ > 0 and h ∈ C b (R) the comparison principle holds for f − λHf = h. 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 R with good ratefunction 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 (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) → d dp H(x, p) is continuous. Then the rate function I is given by I(γ) = I 0 (γ(0)) +