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Large deviations for interacting multiscale particle systems

We consider a collection of weakly interacting diffusion processes moving in a two-scale locally periodic environment. We study the large deviations principle of the empirical distribution of the particles' positions in the combined limit as the number of particles grow to infinity and the time-scale separation parameter goes to zero simultaneously. We make use of weak convergence methods providing a convenient representation for the large deviations rate function, which allow us to characterize the effective controlled mean field dynamics. In addition, we obtain equivalent representations for the large deviations rate function of the form of Dawson-G\"artner which hold even in the case where the diffusion matrix depends on the empirical measure and when the particles undergo averaging in addition to the propagation of chaos.


Introduction
The goal of this article is to obtain the large deviations principle (LDP) for interacting particle systems of diffusion type in multiscale environments. We use methods from weak convergence and stochastic control, [21], ultimately making connections with mean field stochastic control problems [12].
Our goal is to obtain the large deviations principle for the measure-valued process µ N t , t ∈ [0, 1] N ∈N in the combined limit N → ∞ and δ ↓ 0. Here, δ is the time scale separation parameter. One can regard X i,N as the slow i th component and Y i,N = X i,N /δ as the fast i th component.
Systems of interacting diffusions arise in many areas of science, finance and engineering, see for example [6,32,33,38,44,46] to name just a few. On the other hand, diffusions in multiscale environments are also common in many applications ranging from chemical physics to finance and climate modeling, see for example [1, 7, 23-25, 37, 45, 64] for a representative, but by no means complete, list. Our goal in this paper is to study the combined effect of weak mean field interactions in a fast oscillating multiscale environment from the point of view of large deviations for the empirical measure of particles.
In the case δ = 1, i.e. in the absence of multiple scales, the limiting problem of N → ∞ has been very well studied in the literature. Typical behavior, fluctuations as well as large deviations have been obtained, see for example [11,15,16] for related classical works. Analogously, if N = 1, i.e., in the single particle case, the limiting behavior as δ ↓ 0 has also been extensively studied in the literature under various modeling assumptions, see for example [2,9,20,26,31,48,50,51,[56][57][58][59][60] and the references therein. In this paper, we study the combined limit as N → ∞ and δ(N ) ↓ 0. The main result of the paper is Theorem 3.4 that gives the large deviations principle of the empirical distribution of the particles in the combined limit N → ∞ and δ ↓ 0. As a byproduct we also obtain in Theorem 3.7 the typical behavior, i.e. the law of large numbers. We use weak convergence methods of [21] which leads to the study of related mean field stochastic control problems [12,29,42]. In addition, in Section 4, we connect, in a simplified regime, the variational form of the action functional that we obtain in Theorem 3.4 with the expected "dual" form based on the classical work of Dawson and Gärtner [16] in the δ = 1 case and [20] in the N = 1 case. As pointed out in [29], while the limit problems here are not mean-field games, but rather optimal control problems of McKean-Vlasov-type, the modern advances in control theory for mean-field models and McKean-Vlasov equations resulting from mean-field games' current popularity may provide a rigorous means of proving the formal results provided in Section 4. This analysis will appear elsewhere.
As an example, we consider in Section 5 a system of interacting processes in dimension one in a multiscale confining potential, i.e. a Curie-Weiss type of interaction. This example in motivated by the work of Dawson in [15]. We verify that this system satisfies the required assumptions for an LDP to hold, derive the large deviations principle and discuss equivalent formulations. We provide an alternative variational form of the rate function for this particular system in Theorem 5.2.
To our knowledge, this is the first large deviations result for the combined δ ↓ 0 and N → ∞ limit. Some similar results include the proof of an averaging principal for slow-fast McKean-Vlasov SDEs found in [54], i.e. the δ ↓ 0 limit for a system of the type we get after N → ∞. There is also the result of [3], in which the authors establish an averaging principal for the empirical mean a system of mean field multiscale diffusions at the level of large deviations. What is meant by this is that they study the known large deviations rate functions for the N → ∞ limit of the empirical measures with fixed δ, J δ , and are able to prove Γ-convergence of the sequence {J δ } δ>0 to a functional J as δ ↓ 0. Lastly, in [17], a result similar to Theorem 3.7 appears (only typical behavior, not LDP). A key difference between the regime of [17] and the regime of our paper is that rather than depending on the slow process X i,N t , the fast process X i,N t /δ, and the empirical measure µ N t , their coefficients depend on the fast process X i,N t /δ and the "fast empirical measure" µ N,δ t := 1 N N i=1 δ X i,N t /δ . As a result, the invariant measure π (see Equation 3) depends not on the parameter µ = L(X · ) in the limit, as in our regime, but implicitly on itself as µ = π. Consequently, in [17], multiple steady states can exist, potentially affecting the way in which the limits δ → 0 and N → ∞ interact. We discuss this further in our conclusion section 11.
The rest of the paper is organized as follows. In Section 2, we lay out notation and main assumptions in regards to the model (1). In addition, we introduce the corresponding controlled particle system and controlled McKean-Vlasov process which will be crucial components of the large deviations analysis. In Section 3 we present our main result on large deviations for the measure-valued process µ N t , t ∈ [0, 1] N ∈N in the combined limit N → ∞ and δ ↓ 0. Section 4 connects the obtained Laplace principle with other classical works in the literature, i.e. the LDP in the δ = 1 case of [16] and the LDP in the N = 1 case of [20]. Section 5 includes an example with a bistable confining potential motivated by the classical work of Dawson in [15] and an alternative variational form of the rate function. In Section 6 we discuss the limiting behavior of the controlled particle system, proving tightness and identifying the limit. In Sections 7 and 8 we prove the Laplace's principle (which is equivalent to the large deviations principle) lower and upper bounds respectively. Compactness of the level sets of the rate function is proven in Section 9. In Section 10 we discuss how the assumptions on the coefficients can be relaxed. In Appendices A,C, and D we discuss technical preliminary results that are used in various places of the paper. For purposes of self containment and for the reader's convenience, Appendix B reviews the necessary material from Lions differentiation. Lastly, an equivalent variational representation for the rate function offered in Theorem 3.4 is discussed in Appendix E. Section 11 has our conclusions and directions for future work.

Notation, Assumptions, and the Controlled McKean-Vlasov process
Consider a probability space (Ω, F , P) with filtration F t that satisfies the usual conditions. For S a Polish space, we will use C([0, 1]; S) to denote the space of continuous functions from [0, 1] to S, equipped with the topology of uniform convergence and the sup norm. We will use D S [0, 1] to denote B : [0, 1] → S such that B is right-continuous and has left-hand limits, equipped with the Skorohod topology. A useful fact is that both D R d [0, 1] and C([0, 1]; R d ) with the previously described topologies are Polish spaces (see [21] Theorem A.6.5). We will use C b (S) to denote the space of continuous, bounded functions B : S → R, and let B ∞ := sup x∈S |B(x)|. We use C k b (S) to denote the space of continuous, bounded functions B : S → R with k continuous, bounded derivatives. We use C ∞ c (S) to denote the space of continuous, infinitely differentiable functions B : S → R with compact support. L 2 (S, µ; R k ), where µ is a measure on S will denote the class of functions B : S → R k such that B L 2 (µ) := S |B(x)| 2 µ(dx) 1/2 < ∞. We may omit the codomain in this notation when convenient. P(S) will denote the space of probability measures on the Borel σ-field B(S), where open sets are induced by the metric on S. P(S) is given the topology of weak convergence and the Prokhorov metric, and is itself a Polish space ( [22] Theorem 3.1.7). P 2 (E) ⊂ P(E) will denote the set of square integrable measures on E. It is given the L 2 -Wasserstein distance (see Definition B.1) as its metric and is also a Polish space ( [12] p.360). Given a random variable η, L(η) will denote the distribution of η. Assume the following: (A1) For g = f, b, σ, g,∇ x g, ∇ y g, ∇ x ∇ x g, and ∇ x ∇ y g exist, are uniformly bounded, and are continuous in R d × T d × P(R d ). (A2) For A = σσ ⊤ there exists γ > 0 such that uniformly in x ∈ R d , y ∈ T d , µ ∈ P 2 , ξ ⊤ A(x, y, µ)ξ ≥ γ|ξ| 2 , ∀ξ ∈ R d .
As we shall see in Section 10, Assumptions (A1)-(A8) can be substantially relaxed in the expense of more technical estimates. We choose to present the main results with Assumptions (A1)-(A8) for the sake of readability.
An important object of study will be the operator L x,µ , paramaterized by x ∈ R d and µ ∈ P 2 (R d ) which acts on g ∈ C 2 (T d ) by Related to this operator we consider the measure π(·|x, µ) ∈ P(T d ), paramaterized by µ ∈ P 2 (R d ), whose densityπ(·|x, µ) satisfies the adjoint equation where both of these equations are given periodic boundary conditions. As we will see in Propositions C.1 and C.2, π and Φ are uniquely defined and π indeed admits a densityπ.
We wish to observe the behavior of the sequence of P(C([0, 1]; R d ) random variables as N → ∞. Specifically, letting ev : C([0, 1]; R d ) → R d be the evaluation map at time t, in Theorem 3.7 we see that, under assumptions (A1)-(A8), We seek to quantify the rate at which the convergence of the random measures given by Equation 5 to the law of the solution of Equation 6 occurs via deriving a large deviations principal for {µ N } N ∈N .
2.1. The Controlled Process. We start by constructing a controlled version of the system of mean-field SDEs 1 which will then allow us to us the weak convergence approach to large deviations of [21].
where E denotes the expectation with respect to P and | · | the Euclidean norm.
Given u N ∈ U N , we consider the controlled system of SDEs whereμ N (t) andμ N are the empirical measures ofX i,N (t) andX i,N respectively, For notational convenience, we now introduce some spaces of interest. Let X : Note that we construct Y and Z this way to allow for extension of the results of this paper to bounded time intervals other than [0, 1]. Also note that by Section 6.3 in [53] that Z is a Polish space and that by Theorem A.3.3. in [21] that Y is a Polish space. Note that if u ∈ U N for any N ∈ N, then u induces a Z-valued random variable r via Since for r ∈ Z, t → r(B × [0, t]) for B ∈ B(R d ) is absolutely continuous, there exists r t : [0, 1] → P(R m ) such that r(dzdt) = r t (dz)dt. Similarly, for m ∈ Y, there exists m t : [0, 1] → P(T d ) such that m(dydt) = m t (dy)dt.
Consider the McKean-Vlasov SDE parameterized by ν ∈ C([0, 1]; P(R d )) given by: forX ν ∈ X , m ∈ Y, and ρ ∈ Z. Q ∈ P(W) corresponds to a weak solution of (10) if there exists a filtered probability space (Ω,F,P), {F t } and an m-dimensionalF t -Brownian motion W such that (X ν , m, ρ) is añ F t -adapted Wvalued random variable that has distribution Q underP. Note thatX ν , m, and ρ are each random processes. A useful remark making the point that the measures ρ(ω), m(ω) and the processX ν (ω) all dependent on each other follows.

We are interested in particular in weak solutions
(For a description of B(P(R d )) see [21] Lemma A.5.1).
Since ν Q (t) = L(X νQ t ), we are thus interested in weak solutions to the limiting controlled McKean-Vlasov SDE Note that in the case R m zρ t (ω)(dz) = 0 and m t (dy) = π(dy|X t , L(X t )) for almost every ω ∈Ω and t ∈ [0, 1], this agrees with Equation 6.
The process triple (X, m, ρ) can be given explicitly as the coordinate process on the probability space (W, B(W), Q) endowed with the canonical filtration G t := σ (X s , m(s), ρ(s)), 0 ≤ s ≤ t . Thus, for ω = (φ, n, r) ∈ W, Thus, for g : Throughout this paper we will only integrate m(t, ω) against time intervals of the form [s, t], so we will We end this section with a discussion on well posedness of the limiting controlled McKean-Vlasov SDE (12).
Definition 2.2. We will say weak-sense uniqueness holds for Equation 12 if under the assumptions: ( Under the assumptions (A1)-(A8), weak-sense uniqueness as defined in Definition 2.2 holds for Equation (12).
Proof. Since the product of bounded Lipshitz functions is Lipshitz (Φ is Lipshitz in x, y because it has bounded first order derivatives and in µ by Corollary C.3), the averaged coefficients are Lipshitz continuous by Section 6.2 of [54]. Then the requirements for Theorem 2.1 in [62] are satisfied by the limiting system. By a slight modification of the proof to include the control term (in particular the Yamada-Wantanabe Theorem used in Lemma 2.3 does not directly apply, but a modification of its proof is simple given we consider relaxed controls ρ such that 1 0 R m |z| 2 ρ t (dz)dt < ∞) we get that for any fixed ρ there is a unique strong solution to Equation 10 under assumptions (A1)-(A6). Then, noting that by Remark 6.7 that Θ| B(X ×Y) is completely determined by Θ X , we get immediately that weak-sense uniqueness holds by the proof of Proposition C.2 in [30].

Statement of the Main Results
In order to state the main results of this paper, we need the following two definitions: Definition 3.1. We will say Θ ∈ P(W) is in V if (V1) Θ corresponds to a weak solutionX of (12).
Where here we are using the notation for the coordinate process given in Equation 13.
Essentially, condition (V4) says that the coordinate process m s (dy) is π(dy|X s , ν Θ (s)), the invariant measure for the fast dynamics, for Θ-a.e. ω ∈ W. A detailed explanation on why this is true is in Section 6.2 and in particular in Remark 6.7.
Definition 3.2. A function I : P(X ) → [0, ∞] is called a (good) rate function if for each M < ∞, the set {θ ∈ P(X ) : I(θ) ≤ M } is compact. We say that a Laplace principal holds for the family {µ N } N ∈N with rate function I if for any bounded, continuous F : P(X ) → R, It is well known that in our setting the Laplace principal holds if and only if {µ N } N ∈N satisfies a LDP with rate function I. See [21] Theorem 1.2.3.
In order to prove the Laplace Principal for {µ N } N ∈N , we make use of the following proposition: Proposition 3.3. The pre-limit expression in (14) can be written as for any F ∈ C b (P(X )) whereμ N is given by (8) with u N = (u N 1 , ..., u N N ) ∈ U N the control in Equation 7. Proof. By Proposition A.1 and [63] there is Borel measurable ψ i,N such that and by the characterization of B(P(X )) given in Lemma A.5.1 of [21] p : and is thus a Borel-measurable function of the driving Wiener processes for each N . Then Theorem 3.6 in [10] applies, giving us the desired result.
Proof. As is standard, in order to prove that {µ N } N ∈N satisfies the Laplace Principal with rate function I, we prove the Laplace Principal lower bound (42) in Section 7 and the Laplace Principal upper bound (44) in Section 8. The main tool in these proofs is the Variational Representation Theorem for Functionals of Brownian Motion, given in Proposition 3.3. Once we identify the law of large numbers result for the controlled process in Section 6, the Laplace Principal lower bound, i.e. lim inf follows immediately from Fatou's lemma, as seen in Section 7. Then to prove the Laplace Principal upper bound, in Section 8, given θ ∈ P(X ), we construct a probability space and a sequence of iid controls whose law corresponds to Θ Z for Θ ∈ V, Θ X = θ, that nearly reaches the infimum in the definition of I(θ). Once we show these two bounds, we get inf θ∈P(X ) so that Equation 14 is satisfied. In Section 9 we prove that the level sets of I are compact, so indeed I is a good rate function. Remark 3.6. We can see that the Law of the control in the definition of I depends on θ and the invariant measure π directly as follows: decomposing Θ ∈ V into stochastic kernels as Θ(dφdndr) = γ(dr|n, φ)λ(dn|φ)Θ X (dφ) = γ(dr|n, φ)λ(dn|φ)θ(dφ), we have by Remark 6.7 that λ(dn|φ) = δ π(·|φ(s),θ(s))⊗ds (dn), which we will abbreviate as δ π (dn). Here by θ(s) we mean θ • ev −1 (s). Then, using the fact that the control only appears linearly in the dynamics of Equation 12, In order to solve the variational problem which defines the rate function I, one must find a minimizing stochastic kernel γ.
In proving Theorem 3.4, we will also have proved the following convergence result: Theorem 3.7. Let ev : X → R d be the evaluation map at time t and {µ N } be as defined by Equation Proof. This follows immediately from the proofs in Section 6 by taking u N ≡ 0 for all N ∈ N.
Remark 3.8. It is worth noting that via an integration-by-parts argument, the diffusion term in Equation 6 can also be written as See [4] Chapter 3 Section 6.2.

Formal Connection to Rate Functions in the Existing Literature
The goal of this section is to connect, at least at a formal level, the rate function obtained in Theorem 3.4 to the ones obtained in the literature for δ = 1, which is the result of [16], and for N = 1, which is the result of [20]. As noted in Section 7.1 of [11], in the absence of multiple scales there is a heuristic connection between the variational form of the rate function of weakly interacting particles and that from the classical paper [16]. In order to connect this to our case, we also must compare the LDP for small noise diffusions from [27] to the rate function for combined small noise and averaging from [20].
Assume d = m, f ≡ 0, σ = I, and the initial conditions of all the particles is 0. Then π is Lebesgue measure and Φ = 0. The limiting system given by Equation 12 is given by Since in [20], the joint small noise and scale separation limit is being taken, we expect that we should be able to formally connect our rate function, given by Equation 15, to I DG , the rate function from [16], and I DS , the rate function from [20], by assuming that the analogous rate function to I DS which acts on P(X ) instead of X would be given through the same relationship as the small noise rate function from [27] to the mean-field rate function from [16].
The rate function in [16] for the empirical measure for strong solutions to if θ : [0, 1] → P(R d ) is absolutely continuous in the distribution sense (see Definition 4.1 in [16]) and I DG (θ) = +∞ otherwise. Hereθ(t) denotes the distributional derivative of θ at time t, D is the Schwartz space of compactly supported smooth test functions g : R d → R, and L 2 (θ(t)) acts on g ∈ D according to Under the current assumptions, the rate function for {X ǫ } ǫ>0 , the strong solutions to of [27] acts on φ ∈ X by: The rate function for {X δ } δ>0 , the strong solutions to is given in [20] acts on φ ∈ X by for φ absolutely continuous and +∞ otherwise. Here ǫ(δ) ↓ 0 as δ ↓ 0 such that ǫ δ → ∞ as δ ↓ 0 and r(x) := T d b(x, y)dy.
Comparing I DE to I DG and I DS , we see our rate function can be expected to be the same as I DG but with b(x, θ(t)) replaced by r(x, θ(t)) := T d b(x, y, θ(t))dy. in other words, in the notation of [16], we expect the rate function to have the representation if θ : [0, 1] → P(R d ) is absolutely continuous in the distribution sense and S(θ) = +∞ otherwise. Here L 3 (θ(t)) acts on g ∈ D according to Formally we can see the connection between our rate function given by Equation 16 and S by denoting V 1 the class of Θ ∈ V such that Θ-almost surely, ρ t (dz) = δ ∇v(t,Xt) (dz),for some v : [0, 1] × R d → R such that v(t, ·) ∈ D, ∀t ∈ [0, 1]. Applying Itô's formula, we get for g ∈ D and Θ ∈ V 1 with Θ X = θ that Taking expectations, dividing by h, and sending h → 0, we get g,θ(t) − L 3 (θ(t)) * θ(t) = ∇v(t, ·) · ∇g, θ(t) , ∀t ∈ [0, 1].
Since this holds for every Θ ∈ V 1 , it then stands that it holds for Θ ∈ V 1 along which the infimum in Thus we see if indeed the infimum over Θ ∈ V in Equation 16 can be restricted to V 1 , then our rate function is equivalent to S.

An example with a bistable confining potential
Even though we have chosen for presentation purposes to proove Theorem 3.4 under strong boundedness assumptions on the coefficients, the extended results of Section 10 show that Theorem 3.4 holds true under greater generality. In particular, if one can appropriately control the behavior of the solution to (4) then the results can be extended. As we explain in detail in Section 10, in the absence of such a general theory, one in principle would need to confirm this in a case by case situation. In this example, we do so for the popular interacting particle system considered in the classical work [15] where the drift is a bistable, confining potential.
Consider 1-D the system of weakly interacting particles from Dawson's classical paper [15], where the interacting potential is modified from 2π cos(2πx/δ) for some ǫ > 0. Then the system of controlled SDE's is given by 1], and W i t are independent 1-D Brownian motions.
In order to confirm that the result of Theorem 3.4 holds in this case, we need to verify that Assumptions (A1')-(A12') of Section 10 hold ( see Theorem 10.2). After doing so, we discuss how the typical events and large deviations principle look here, and provide an alternative variational form for the rate function.

5.1.
Verification that the necessary assumptions hold. Let us start by assuming that (1) (A1') in Section 10 holds.
. To see that the uncontrolled equation has a unique strong solution, one can write the system as a 2N dimensional SDE in the same way as in the proof of Proposition A.1 and use a standard truncation argument for SDEs with one-sided Lipshitz drift and the fact that the solution is nonexplosive (see [15] p.37 and [47]). In fact for this particular case, Theorem 1 in [61] directly applies for each N ∈ N. Thus (A2') holds.
To see (A3') holds, we can apply standard PDE theory as in the bounded case. Further, since we are in the case d = m = 1, we can solve for the densityπ of π explicitly. We get thatπ(y) is the solution to Solving, we getπ The cell problem takes the form −ǫ sin(2πy)Φ ′ (y) + 1 2 σ 2 Φ ′′ (y) = ǫ sin(2πy), Φ 1 -periodic, 1 0 Φ(y)π(y)dy = 0. and its unique solution is So, keeping in mind that Φ is 1-periodic and smooth, we have for some K > 0. Thus, (A3') and (A4') hold, and we attain some apriori bounds that will help us in proving (A9'). Since Φ is independent of µ, (A5') and (A11') hold trivially, and by our explicit representation (A6') holds. Since the only coefficient which depends on a measure is through a first order interaction through µ → ·, µ , and convergence in W 2 implies convergence of first moments, we also have that (A10') holds. The proof that (A7') and (A8') hold is given by Proposition 5.1, where we show that in this particular example we can take p = 4 and p 2 = 6.
For presentation purposes we offer the proof of Proposition 5.1 in Appendix D. We next confirm that (A9') is satisfied. We identify that ) and all the other terms are bounded uniformly in x and µ. This gives via the triangle and Cauchy-Schwarz inequalities that all the conditions in (A9') are satisfied.
Lastly, we check that weak-sense uniqueness as given in Definition 2.2 holds, so that (A12') is satisfied. For this, we first write down the limiting controlled system: where here we used Remark 3.8 to simplify the diffusion term. Since for all m ∈ Y and all t ∈ [0, 1], m t (dy) are bounded and c 1 (t) > 0, we quickly see that due to the assumed L 2 bound on the control in the conditions of Definition 2.2 that the proof goes through in the same way as in Appendix A of [15].

5.2.
Form of the limiting theorems and equivalent formulations. Now that we have confirmed that the necessary assumptions indeed hold for the system 18, let us discuss what the law of large numbers and large deviations principle look like.
Unlike the system offered in Section 4, the nontrivial interaction of this system with π makes it so we cannot make a direct formal connection to the rate funcion of [16] by the same means. However, we are able to prove an alternative variational form of the rate function, as stated in the following Theorem 5.2.
In preparation for stating this result, let us setŴ = X ×Ŷ where X = C([0, 1]; R d ) and We shall write thatΘ ∈ P(Ŵ) is inV if (V 1)Θ corresponds to a weak solutionX of (21), with (X,ρ) the coordinate process onŴ defined as in Equation 13.
The proof of Theorem 5.2 is deferred to Appendix E. Note that the proof of this theorem extends to the more general case where σ(x, y, µ) = σ(y, µ), f (x, y, µ) = f (y, µ), m = d, [∇ y Φ(y, µ) + I]σ(y, µ) is uniformly bounded in y and µ, and T d [∇ y Φ(y, µ) + I]σ(y, µ)π(dy|µ) is invertible with uniformly bounded inverse for all µ, where as with Theorem 10.2, the exponent on |X t | in (V 2) must be adapted to p in (A7'). For the sake of brivity, we leave the proof of this extension to the interested reader.
Lastly, even though we do not show this here, we do mention that along the same lines one is able to check that the required assumptions to satisfy Theorem 10.2 also hold for slightly more complicated cases where the empirical measure appears linearly in the coefficient which blows up in the drift. For example, one can consider a particle system of the form We leave the details of this example to the interested reader.

Limiting Behavior of the Controlled Empirical Measure
Our object of study in this section is the family of occupation measures {Q N } N ∈N ∈ P(W) defined by: for A × B × C ∈ B(W), ω ∈ Ω, ρ i,N the ordinary relaxed control corresponding to u i,N via Equation 9,X i,N as in (7), and m i,N (ω) ∈ Y given by (24) for I ∈ B([0, 1]) and D ∈ B(T d ). We use the convention that for s > 1, u N i (s) = 0, ∀i, N ∈ N. Remark 6.1. For a discussion of another possible choice of occupation measures, see Remark E.1 after the proof of Theorem 5.2 in Appendix E.
Assume that there exists B > 0 such that P almost-surely, We will prove the following two propositions: Proof. See Subsection 6.1. Z . This will follow analogously to p.88-89 of [11]. We recall here the arguments for the readers convenience. Observe that is a tightness function on R 1 . Namely, it is bounded from below and has relatively compact level sets. Indeed, boundedness from below is obvious and in order to confirm the second property, for c ∈ (0, ∞) let us set R c := {r ∈ R 1 : g(r) ≤ c}. Chebyshev's inequality for M > 0 gives that Therefore, R c is tight and thus relatively compact as a subset of R. Let {r n } n∈N ⊂ R c be such that {r n } n∈N converges weakly to r * ∈ R. We need to show that r * has finite first moment and that first moments of {r n } n∈N converge to the first moment of r * . By Jensen's inequality and Fatou's lemma (Thm A.3.12 in [21] Now letting M > 0, by Equation 26 and Hölder's inequality, we have for all r ∈ R c , So by reverse Fatou's Lemma we get Given that M may be taken to be arbitrarily large, we have Thus we have g is a tightness function on R 1 . Now define G : Then G is a tightness function on P(Z) (see Theorem A.3.17 in [21]). Thus in order to prove tightness of But this follows immediately from assumption (25), since by definition of G and Q N , denotes the set of subprobability measures on E, is compact by Corollary A.3.16 in [21] (this also works for M α (E), positive Borel measures µ on E with µ(E) ≤ α, for any α > 0). Then by the proof of Lemma 3.3.1 in [21], is closed in the topology of weak convergence (if a weakly converging sequence of measures on T d ×[0, 1] has the property that for each member of the sequence, its second marginal is Lebesgue measure, then this will also be true of the limiting measure), and hence Y = R 1 (T d ) is compact. Then P(Y) is compact, and hence P(P(Y)) is compact. Since {L(Q N )} N ∈N ⊂ P(P(Y)), and on a metrizable space compactness implies sequential compactness, we immediately get Q N Y is tight as a P(Y)-random variable.
is the space of maps from [0, 1] to E which are right continuous and have left-hand limits. Noting thatμ N (ω) ∈ P(X ) for each ω ∈ Ω, N ∈ N, we can treat {μ N } N ∈N as a sequence of C([0, 1]; P(R d ))-valued random variables, so that random variables will imply tightness as C([0, 1]; P(R d ))-valued random variables by Problem 25 on p.153 of [22].
We will use Theorem 3.8.6 in [22] together with Theorem 3.1 in [39] to show the tightness of Note that by Proposition 3.4.4. and Theorem 3.4.5. in [22], the class of functions Here by A ⊂⊂ B we mean A is a compact subset of B. Appealing to Theorem 8.6 and Remark 8.7 in [22], it will thus be sufficient to show: Lemma 6.5. For q(x, y) := 1 ∧ |x − y| and 0 ≤ t ≤ t + τ ≤ 1, 0 ≤ τ ≤ γ, there exists a family {ξ N (γ)} γ∈(0,1),N ∈N of non negative random variables and r > 0 such that Note in particular that the following compact containment condition required by Theorem 3.8.6 in [22] as provided in the following corollary follows directly from Lemma 6.4 and its proof.
We proceed with the proof of Lemma 6.4: By Prokhorov's Theorem, K * L ⊂⊂ P(R d ) for each L. Now we see that This bound approaches 0 as L → ∞, so the lemma is proved.
Now we prove Lemma 6.5.
Proof of Lemma 6.5. We take r = 1. Now we observe that for all 0 ≤ t ≤ t + τ ≤ 1, 0 ≤ τ ≤ γ and g ∈ G, Applying Itô's formula to h, we get (ignoring the arguments for notational convenience) In order to control the term that blows up as δ → 0, we define ψ l (x, y, µ) Equation 4. Then ψ l solves . Now applying Itô's formula (using Equations 51 and 52 in Proposition B.3 and the regularity of Φ from Proposition C.2) to ψ l , we get where in all coefficients where the argument is suppressed, the argument is (X i,N s ,X i,N s /δ,μ N s ). Solving for 1 δ t 0 f l h x l ds and plugging into our representation for h, we get Thus we have Firstly, we observe that Next, we observe that by Hölder's inequality, monotonicity, Assumption (A1), Proposition C.2, and Jensen's inequality. Similarly, we have In addition, Lastly, we observe that by Assumption (A1) and Proposition C.2. So, letting for C large enough to dominate all the constants in the previous bounds, we see that for each g ∈ G, This follows immediately since by Jensen's inequality 6.2. Identification of the Limit. Extract a convergent subsequence from {Q N } N ∈N and relabel with new indexes so that {Q N } N ∈N converges to some Q weakly as a P(W)-valued random variable. Let (Ω,F ,P) be the probability space on which Q lies. We wish to identify the limit Q as a member of V forP almost everyω ∈Ω. Our main tool here is the associated martingale problem to weak solutions of (10). An important element to the proof which is special to the joint limit as N → ∞, δ ↓ 0 is that we must first show that (V4) holds before identifying the SDE associated to Q to prove (V1). This is because, as proven in (41), in the prelimit there exists a term which is O(1) in N , but is in fact 0 in the limit due to the centering condition (A8). As the centering condition is a statement involving the invariant measure π (see Equation 3), it is necessary to the proof that we have already identified the Y component of the limiting Coordinate Process 13 as being concentrated on π.
We want to show that for almost everyω ∈Ω and ∀t ∈ [0, 1], Let g l : T d → R, l ∈ N be smooth and bounded with bounded derivatives and dense in C 2 b (T d ). This set exists by using Stone-Weierstrass and taking rational coefficients. LetȲ i,N =X i,N /δ. Considering the operator which acts on g ∈ C 2 b (T d ) by Note that by (A1), for t ∈ [0, 1] and fixed N ∈ N, is an F t -martingale. By definition, for t ∈ [0, 1], Consider now the operator which acts on g ∈ C 2 b (T d ) by B x,z,µ [g](y) := b(x, y, µ) + σ(x, y, µ)z · ∇g(y).
We will show the right hand side of Equation 28 converges in distribution to E Q T d ×[0,t] L 1X s ,νQ(s) f (y)m(dyds) and the left hand side converges in distribution to 0, so by a density argument the result holds.
The proof that the right hand side of Equation follows from the observation that We invoke Skorohod's representation theorem (Theorem 3.1.8 in [22]) to assume the convergence of Q N → Q holds with probability 1. Without making a distinction between the original probability space in the new one, we will prove so that by Chebyshev's inequality the convergence holds in probability and hence in distribution.
First we prove that We have By Assumption (A1) the bounded convergence theorem applies. So we can pass the limit through the integrals, and the results follows. Now we will show which together with triangle inequality and Equation 30 proves Equation 29.
To prove the left hand side of (28) converges to zero in distribution, we will show that as N → ∞, so the result will follow by Chebyshev's inequality. We first note that ≤ δ ∇g l ∞ C by Assumption (A1) and Itô Isometry. Also, Lastly, by Hölder's inequality, Assumption (25), and Assumption (A1).
Now we have that for each g l and t ∈ [0, 1], there exists a set N g l ,t such thatP (N g l ,t ) = 0 and Taking a countable dense set D ⊂ [0, 1] and letting N = ∪ l∈N ∪ t∈D N g l ,t , we haveP(N ) = 0 and So (V4) holds for almost everyω ∈Ω. In Remark 6.7 we discuss implications of (V4).
Given g ∈ C 2 b (R d ) and Θ ∈ P(W), define a real-valued process {M Θ g (t)} t∈[0,1] on (W, B(W), Θ) given by We will say Θ ∈ P(W) solves the martingale problem associated toX νΘ with initial distribution ν 0 if for all 0 ≤ s ≤ t ≤ 1 and g ∈ C 2 b (R d ), for {G t } t∈[0,1] the canonical filtration on the coordinate process as defined in Equation (13). To identify the limit Q as a weak solutionX νQ to 10, by the density argument offered at the end of this subsection, it suffices to show that for fixed h ∈ C ∞ c (R d ), 0 ≤ s ≤ t ≤ 1, and G s -measurable Ψ ∈ C b (W) that Theorem 6.8.
We note that it is enough to prove Theorem 6.8 for h ∈ C ∞ c (R d ), due to the fact that C ∞ c (R d ) is separating in the sense of Chapter 3 Section 4 in [22] (see [14] Definition 3.1 and Chapter 4 Section 8 in [22]). In order prove Theorem 6.8 we will prove Lemma 6.9 and Lemma 6.10. Lemma 6.9.
Then the conclusion follows. We proceed with the proof of Lemma 6.9.
Proof of Lemma 6.9. Unpacking the notation in Equation 35, we see what we are trying to show is that in distribution. We invoke Skorokhod's representation theorem to assume the convergence of Q N → Q occurs with probability 1, without making a distinction in the notation between the new probability space and the original one. Then we see Lemma 6.9 essentially follows from the definition of convergence of measures in the space P(W). The only caveats are that the integrand is also a function of the converging measures through its dependence on ν Q N , and that the integrand is not apriori in C b (W), since it grows linearly in the control. Thus, we will show and Once these limits are established, by triangle inequality and Chebyshev's inequality Lemma 6.9 will be proved.

Noting that h(φ(t)) − h(φ(s)) is bounded and writing A[h] as in Equation 39
, we see that only R m T d A 2 (φ(τ ), y, ν Q (s))zn s (dy)r s (dz) exhibits growth in the control r. Since the desired convergence occurs immediately by boundedness of the integrand and almost-sure convergence of Q N → Q, we only show work to show the convergence of this term. Let Then for all r ∈ Z, M > 0, t ∈ [0, 1], ψ 1 M (r, t) + ψ 2 M (r, t) = R m zr t (dz), by Assumption (A1), Proposotion C.2, and monotonicity by Assumption (25).
Taking N → ∞ then M → ∞ the result follows.
Now we prove Lemma 6.10.
Proof of Lemma 6.10. Again, we invoke Skorohod's representation theorem to assume the convergence of Q N → Q occurs with probability 1.
→ 0 as N → ∞, and so the conclusion will follow via Chebyshev's inequality. From Equation 27, we get that m i,N are defined as in Equation 24 and the arguments which are omitted are taken to be (X i,N τ ,X i,N τ /δ,μ N τ ). Thus By the same proofs as in tightness of Q N X in Subsection 6.1.3, using Assumption (A1) and Proposition C.2, we get for large enough N that where C depends only on the sup norms of Ψ and h and its first 3 derivatives. This vanishes as N → ∞, so once we prove the following (41), Lemma 6.10 will be proved.
Unlike the average of D i,N k , k = 1, 2, 3, 4, the term we wish to vanish in (41) is O(1) in N . However, as we will see, the fact that Q almost surely satisfies (V4) along with Remark 6.7 and the centering condition from Assumption (A8) will result in this term vanishing when we pass to the limit. We first observe that We can rewrite D N in terms of the occupation measures defined in Equation 23 as: And thus First we show the first of these terms vanishes as N → ∞. We have We note now that by Propositions A.3 and C.2 along with Assumption (A1) and Bounded Convergence Theorem, for fixed φ, ψ, n, m and τ So by Assumption (A1) and Proposition C.2, we have by Bounded Convergence Theorem that this term vanishes. Now we have By the same argument as above, we have that we can pass the limit through the expectation to get Since Q N → Q in P(W) almost surely and the integrand is bounded, we have via Proposition 4.6 on p.115 of [22] that

Now by Hölder's inequality and Tonelli's Theorem,
The first term in the product inside the expectation is bounded by Proposition C.2. For the second, we have that by writing Q| B(X ×Y (dψdn) = λ(dn|ψ)Q X (dψ). Since Q almost surely satisfies (V4), we have by Remark 6.7 that P-almost surely, for every τ ∈ [0, 1] Thus lim N →∞ E[D N ] is bounded by the expectation of a term which its almost surely bounded times a term which is almost-surely 0, and thus (41) holds and the proof of Lemma 6.10 is complete.
We have then that for each (s, t, Since there is a a countable collection of h ∈ C ∞ c (R d ) which is dense in C ∞ c (R d ), a countable collection (s, t) ∈ [0, 1] 2 which is dense in [0, 1] 2 , and countably many Φ ∈ C b (W) generating each of the countably many sigma algebras G s l , letting Z be the union over all these countable collections of Z (s,t,Ψ,h) , we have Z ∈F ,P(Z) = 0, and So Theorem 6.8 is proved.

6.2.3.
Proof of (V2). By Skorohod's representation theorem, we can invoke another probability space on which the convergence of Q N → Q occurs with probability 1. Without making a distinction in the notation between that probability space and our original one, we note that by Fatou's lemma where in the first equality we use that through Theorem 6.8 we identified Q to P-a.s. be the unique (by Proposition 2.3) deterministic measure in P(W) with Z-marginal Q Z solving the Martingale Problem 34.

Proof of (V3). This follows immediately from weak convergence, since by Proposition
where again in the last step we use that through Theorem 6.8 we identified Q toP-a.s. be the unique (by Proposition 2.3) deterministic measure in P(W) with Z-marginal Q Z solving the Martingale Problem 34.

The Laplace Principal Lower Bound
We now proceed with proving the Laplace Principal Lower Bound: It suffices to prove this bound along any subsequence such that the left hand side converges. Such a sequence exists since Note also that for this choice of controls, we have for all N ∈ N, Thus, since (25) is satisfied, by the proof of Theorem 4.4 in [10], we also get that it is enough to assume that P almost-surely, so the results of Section 6 apply with {v N } N ∈N as our choice of controls, and for {Q N } N ∈N as in Equation Since η is arbitrary the lower bound (42) is proved.

The Laplace Principal Upper Bound
In order to close the proof of Theorem 3.4, we now need to show for F ∈ C b (P(X )) that Given η > 0, take θ ∈ P(X ) such that Since the bound given in Equation 44 is trivial if the right hand side is +∞, we may assume it is finite. By the definition of I, there exists Θ ∈ V such that Θ X = θ. By merit of (V2), we get that letting (X, m, ρ) be the canonical process on W as defined by Equation 13 that there exists a m-dimensional G Θ t+ -Brownian motion W such that ((W, B(W), Θ), {G Θ t+ }, (X, m, ρ, W )) is a weak solution of (10). Note that we take the Θ-augmentation and right limit of G t so that we have a filtration that satisfies the usual conditions. By (V1) we know that the martingale problem (34) is satisfied by the coordinate process, and by Exercise 5.4.13 in [41] it is also satisfied with G Θ t+ in the place of G t . We note that in the dynamics of Equation 10 the control only appears linearly as Then we see that definingρ(ω)(I × D) = I δ u(t,ω) (D)dt for D ∈ B(R m ), I ∈ B([0, 1]), ω ∈ W, that ((W, B(W).Θ), {G Θ t+ }, (X, m,ρ, W )) is still a weak solution to (10) and the cost ofρ does not exceed that of ρ.
Let {W i } i∈N be independent m-dimensional GΘ t+ -Brownian motions and {X i,N } i∈{1,...,N } be the unique solution to the system of SDEs on (Ω ∞ , F ∞ , P ∞ ) Equation  24 withX i,N in the place ofX i,N . Since Equation 45 holds, sup N ∈N almostsurely, and the proof of tightness and the fact thatQ N →Q ∈ V weakly as a P(W)-valued random variable along some subsequence holds by the proofs in Section 6.
We wish now to conclude thatQ X =Θ X . By Proposition 2.3, weak-sense uniqueness as defined in Definition 2.2 holds, so it suffices to prove thatQ Z =Θ Z . By the mapping theorem (Theorem 2.7 in [5]) and continuity of the projection operator from W to Z, we can simply showQ N Z →Θ Z weakly as a P(Z)valued random variable. Sinceρ i,∞ are independent and identically-distributed under P ∞ with common distribution the same as that ofρ underΘ, forQ a limit point of {Q N } N ∈N defined on a probability space (Ω,F ,P), we have by Varadarajan's theorem ( [18] p.399) thatP-almost surely, ThereforeQ X =Θ X , and we have, where the infimum in the first line is taken to be over all stochastic bases, Since η is arbitrary, Equation 44 is proved.

Compactness of Level Sets
Consider I as defined in Equation 16. We want to prove that for each s ∈ [0, ∞), the set is a compact subset of P(X ). This will imply that indeed I is a good rate function.
Since in this section we are dealing with sequences of measures all of which coincide with weak solutions of Equation 12, but with possibly different controls, we introduce a new notation for the coordinate process which allows us to keep track of which measure the X -component of the coordinate process corresponds to. For this we use the paramaterized version of the limiting Equation 10.
For Q corresponding to a weak solution of Equation 12, Q also corresponds to a solution of Equation 10 with ν Q as defined in Equation 11 in the place of ν. Thus we consider the process triple (X νQ , m, ρ), which can be given explicitly as the coordinate process on the probability space (W, B(W), Q) endowed with the canonical filtration G t := σ (X νQ s , m(s), ρ(s)), 0 ≤ s ≤ t . Thus, for ω = (φ, n, r) ∈ W, Lemma 9.1. Fix K < ∞ and consider a sequence {Q N } N ∈N ⊂ P(W) such that for every N ∈ N, Q N is viable and Proof. As in Subsection 6.1, it suffices to show tightness of each of the marginals. It is worth noting that where before we were proving tightness of L(Q N ) in P(P(W)), here we have that Q N are deterministic measures and we are proving tightness of the measures themselves in P(Q N ).
Tightness of the Y-marginals follows in essentially the same way as in Subsection 6.1.
Tightness of the Z-marginals is also very similar to Subsection 6.1.1.
is a tightness function on R 1 , so since For the tightness of the X -marginals, we use that each Q N satisfies (V1); that is, Q N X = L(X ν Q N ). Via Theorem 2.4.10 in [41], it suffices to show that for every η > 0, where here we are using the notation from Equation 47. We have that by Chebyshev's inequality, we get via Hölder's inequality, Itô isometry, Assumption (A1) and Proposition C.2 that Then we have by Young's inequality that So by dominated convergence theorem, we have Lemma 9.2. Fix K < ∞ and consider a convergent sequence {Q N } N ∈N ⊂ P(W) such that for every N ∈ N, Q N is viable and Then for Q such that Q N → Q, Q is viable.
Proof. The fact that Q satisfies (V2) follows immediately from Fatou's lemma. Since by Proposition A.3 ν 0 = lim N →∞ ν Q N (0) = ν Q (0), (V3) is satisfied. We now prove Q satisfies (V1). As before, our tool here is the martingale problem associated to Equation 10. It suffices to show that for fixed h ∈ C ∞ c (R d ), 0 ≤ s ≤ t ≤ 1, and G s -measureable Ψ ∈ C b (W) that where M Q h is given in Equation 33. It suffices to show that since by (V1) Unlike in the previous proof of (V1), here the convergence is as a sequence of real numbers and not in distribution, since Q N are deterministic.
So that we can keep track of which measure m and ρ correspond to in the Coordinate Process 47 on W under Q N , we relabel it (X ν Q N , m N , ρ N ). Under Q, we keep the notation (X νQ , m, ρ). Invoking Skorohod's representation theorem to find another probability space on which the convergence of the random variables (X ν Q N , m N , ρ N ) → (X νQ , m, ρ) occurs for almost every ω ∈ Ω, we have By continuity and boundedness of g and convergence ofX ν Q N →X νQ along with bounded convergence theorem, By Assumption (A1) and Proposition C.2 along with bounded convergence thoerem, By continuity of the coefficients in x and µ from Assumption (A1) and Proposition C.2, along with the assumed uniform L 2 bound on the control and with the fact that the growth in the control is linear, if we can show that ν Q N (t) → ν Q (t) in P(R d ) for each t ∈ [0, 1] then this term will vanish by the same triangleinequality argument given in the proof of Lemma 6.9. But this follows immediately by the assumption that Q N → Q almost surely and Proposition A.3, and by Chebyshev's inequality and the same density argument as at the end of Subsection 6.2.2 we have that Q satisfies (V1).
Finally we prove that Q satisfies (V4). Again invoking Skorohod's representation theorem to find another probability space on which the convergence of the random variables (X ν Q N , m N , ρ N ) → (X νQ , m, ρ) occurs for almost every ω ∈ Ω, by Assumption (A1) and Dominated Convergence Theorem = 0 by continuity of L 1 x,µ in µ via Assumption (A1) and Proposition A.3. So Q satisfies (V4). Lemma 9.1 establishes precompactness of I s defined in (46). Now we will use both Lemmas 9.1 and 9.2 to prove the level sets I s are closed via showing lower-semicontinuity of I. Lemma 9.3. The functional I given in Equation 16 is lower semi-continuous.
Proof. Consider a sequence {θ N } ⊂ P(X ) with limit θ. We wish to show It suffices to consider the case there the left hand side is finite, so there is M ∈ [0, ∞) such that lim inf N →∞ I(θ N ) ≤ M . Then, recalling that by taking a subsequence of {θ N } if necessary, we can find measures Θ N such that Θ N X = θ N , and so lower semi-continuity of I is proved.

A More General Regime
For presentation purposes we proved Theorem 3.4 under quite strong boundedness assumptions for the coefficients. The proofs however make it clear that one can relax such assumptions at the expense of requiring more precise information on the behavior of the solution to Equation 4 and its derivatives.
In particular, let us make the following set of assumptions: The system of SDEs (1) admits a unique strong solution. (A3') There exists a unique invariant measure π(dy|x, µ) satisfying Equation 3 and the centering condition: holds. (A4') There exists a unique function Φ : is fully C 2 in the sense of Definition B.2. (A6') The partial derivatives in x and y of Φ, ∂ µ Φ, and ∇ x Φ, ∇ y Φ exist.
(A7') There exists p ≥ 2 such that forX i,N controlled by any u N ∈ U N satisfying sup N ∈N E 1 This assures that for all N , there exists a modification ofμ N ∈ C([0, 1]; P 2 (R d )) so thatμ N t is in the domain of the coefficients and Φ for all time.
Admittedly these assumptions seem somewhat contrived to meet the requirements of our proofs, but the reason for this presentation is two-fold. The first reason is that these assumptions encapsulate two major classes of SDEs for the particles: ones where the coefficients are bounded (in which case we can drop assumption (A8')), and ones where b is a confining potential and the growth of f and σ in x and µ is not too fast (as in Section 5). The second is that, with the notable exception of [54], regularity of the Cell Problem 4 with respect to the paramater µ ∈ P 2 (R d ) is not yet well-studied in the existing literature. Thus this regularity must be studied on a case-to-case basis in applications, though sufficient regularity can be proved either through explicitly solving for Φ or through the representation offered by Equation 54 in the proof of Proposition C.2.
The proofs go through in the same manner as under assumptions (A1)-(A8), with a few notable changes. The first is that the necessary results of Appendix A, C, and Proposition 2.3 are assumed rather than proven. The second is that the bounds in assumption (A9') are needed for the proof of tightness in Subsection 6.1. Those bounds yield the moment dependence on |X i,N t | p which then due to Assumption (A7')-(A8') allow us to conclude tightness and then convergence. (A11') is needed for technical reasons in the proof of Lemma 6.10.
Lastly, the class of viable measures in the definition of the rate function (16) must be modified: Definition 10.1. We will say Θ ∈ P(W) is in V ′ if (1) Θ satisfies (V1), (V3), and (V4) from Definition 3.1.
(2) Θ satisfies (V2)': Due to this modification, the proofs in Section 9 have a few minor differences: Lemmas 9.1 and 9.2 should instead be proved under the assumption that there exists K > 0 such that where here we are using the modified coordinate process notation of Equation 47. The proofs follow in essentially the same manner using the uniform integrability granted by Assumption (A8'). Then, when proving Lemma 9.3, we see that in fact the bound offered in Equation 48 imply that Equation 49 holds for some K by using the same construction of a prelimit system for each Q N given in Section 8, Fatou's Lemma, and Assumption (A7').

Conclusions and Future Work
We have derived a large deviations principal and law of large numbers for the empirical measure of a system of weakly interacting particles in a two-scale environment in the joint many-particle and averaging limit. We use weak convergence methods, and obtain a variational form of the rate function. We saw that for the system (1), the two limiting procedures commute.
The results of this paper bring to light many interesting problems to be explored in future work. Of significant interest is making the discussion of Section 4 rigorous; in other words, proving the equivalence of rate functions in variational form, such as from Theorem 3.4 or [11], to rate functions of the form found in [16]. A related problem is finding methods for solving constrained optimization problems over a space of measures where the constraint is determined by a class of joint laws for a controlled McKean-Vlasov SDE. This would allow for finding an explicit form of the rate function I in Theorem 3.4. Finally, an interesting extension of this work would be to proving a large deviations principal for systems whose coefficients depend on the "fast empirical measure" µ N,δ the empirical projection of the last coordinate of g, i.e. the function satisfyinĝ g(x, y, β 1 , ..., β N ) = g(x, y, One can then verify that (A5) then implies that for each N ∈ N, ∃C(N ) such that for allx 1 , so that by standard existence and uniqueness results for SDE's with globally Lipshitz coefficients, the proposition holds. See, for example, Theorem 5.2.1 in [49].
Proposition A.2. ForX i,N as in Equation 7 controlled by any u N ∈ U N satisfying almost surely for some This ensures that for all N , there exists a modification ofμ N ∈ C([0, 1]; P 2 (R d )) so thatμ N t is in the domain of the coefficients and Φ for all time.
Proof. It clearly suffices to prove the result assuming q ≥ 1. We use the same computations as in the proof of tightness of Q N X in Subsection 6.1.3, but taking h(x) = |x| 2 . Then, by Equation 27 in Subsection 6.1, we get Using that for any p, |∇h(x)| = p|x| p−1 , |∇∇h(x)| = p(p 2 − 2p + d) 1/2 |x| p−2 and when p = 2, we have ∇∇h x l = 0, l ∈ {1, ..., d}, we apply Cauchy Schwarz to all the inner products with ∇h, ∇∇h, and ∇∇h x l in B i,N k (t), k = 1, 2, 4, 5, 6, 8 and, using Assumption (A1) and Proposition C.2, Since C(d) is a constant depending only on the assumed bounds on the coefficients and the dimension d, 2γ with γ small enough that C(d)γ/2 < 1 and subtract that term to the left hand side. Using also our bound on the initial values assumed in (A6) and C r -inequality, we get E sup by Burkholder-Davis-Gundy inequality on the martingale terms, Assumption (A1), and Proposition C.2 by Hölder's, Jensen's, and C r inequalities by Jensen's inequality, monotonicity of the integrands, and the fact that N ≥ 1 |X i,N s | 2 dt q 1/2 1 + B q/2 by the assumed bound on the control by Young's inequality, Tonelli's Theorem, and Jensen's inequality. By Grönwall's inequality, we can conclude the proof of the proposition.
We end this section with a proposition regarding the mapping defined in Equation 11.

Appendix B. On Lions Differentiation
We will need the following two definitions from [12]: Definition B.1. Given a function u : P 2 (R d ) → R, we may define a lifting of u toũ : L 2 (Ω,F,P; R d ) → R viaũ(X) = u(L(X)) for X ∈ L 2 (Ω,F ,P; R d ).
Here we assumeΩ is a Polish space,F its Borel σ-field, andP is an atomless probability measure (sinceΩ is Polish, this is equivalent to every singleton having zero measure).
The Fréchet derivative ofũ can be viewed as an element of L 2 (Ω,F ,P; R d ) by identifying L 2 (Ω,F ,P; R d ) and its dual. From this, one can find that if u is L-differentiable at µ 0 ∈ P 2 (R d ), there is a deterministic measurable function ξ : R d → R d such that Dũ(X 0 ) = ξ(X 0 ), and that ξ is uniquely defined µ 0 -almost everywhere on R d . We denote this equivalence class of ξ ∈ L 2 (R d , µ 0 ; R d ) by ∂ µ u(µ 0 ) and call ∂ µ u(µ 0 )(·) : R d → R d the Lions derivative of u at µ 0 . Note that this definition is independent of the choice of X 0 and (Ω,F ,P). See [12] Section 5.2.
To avoid confusion when u depends on more variables than just (1) u is C 1 in the sense of L-differentiation, and its first derivative has a jointly continuous version For each fixed µ ∈ P 2 (R d ), the version of R d ∋ v → ∂ µ u(µ)(v) ∈ R d from the first condition is differentiable on R d in the classical sense and its derivative is given by a jointly continuous function ∈ R d×d such that for any µ ∈ P 2 (R d ) and X ∈ L 2 (Ω,F ,P; R d ) with L(X) = µ, ∂ 2 u(µ)(v)(X) gives the Fréchet derivative at X of L 2 (Ω,F,P; is also assumed to be continuous in the product topology. We recall now a useful connection between the Lion derivative as defined in B.1 and the empirical measure. Proposition B.3. For g : P 2 (R d ) → R d which is Fully C 2 in the sense of definition B.2, we can define the empirical projection of g, as g N : (R d ) N → R d given by Then g N is twice differentiable on (R d ) N , and for each β 1 , .., β N ∈ R d , (i, j) ∈ {1, ..., N } 2 , l ∈ {1, ..., d} and In particular, this holds for Φ(x, y, ·) for fixed x ∈ R d and y ∈ T d .
Appendix C. On the Operator L 1 and Related PDEs Proposition C.1. Under assumptions (A1)-(A8), the invariant measure π defined by Equation 3 is uniquely determined for each x ∈ R d and µ ∈ P 2 (R d ) and has a densityπ.
Proof. This follows immediately from Theorem 6.16 in [52].
Proof. Existence and uniqueness follows directly from Theorems 6.16 and 7.9 in [52].
Consider the frozen process on T d for fixed x ∈ R d , µ ∈ P 2 (R d ), y ∈ T d given by whereW t is a d-dimensional,F t -adapted Brownian motion on some probability space (Ω,F ,P) satisfying the usual conditions.
As per Proposition 4.1 in [54] and Section 11.6 in [52], Φ is given by Then the fact that Φ is fully C 2 and smooth in x and y and boundedness of Φ, along with regularity of Φ of the same type given in (A3) and (A4) follows from the unique representation of the cell problem given by Equation 54 and the regularity assumptions on the coefficients given by (A3), (A4), and (A7) (see, for example [50], [51] for general results on Euclidean space with no measure dependence, [4] Chapter 3 Section 6 for the case where the fast component is on the torus with no measure dependence, as well as [54] for when Φ depends on a measure). Differentiability in x an y can also be seen directly from standard interior regularity theory for elliptic PDEs (see [36], [28], and Proposition 5.1 in [19]). Though second differentiability in µ isn't addressed in [54], it follows by applying the same direct differentiation to the transition density of the semigroup associated to Equation 53 twice. See [14] and [13] for an example of where estimates on the second derivative of the transition density in µ are derived under different, weaker assumptions. In addition, in [54] the fast component of the process is on Euclidean space, so they make use of a certain confining assuption on the coefficients to prove Lemma 3.6. In our case this assumption is not needed, as the analagous ergodicity result to Lemma 3.6 is automatically given on a compact space via Theorem 6.16 in [52]. For a proof see Chapter 3 Section 3 in [4].
Proof. From Proposition C.2, we have ∂ µ g(x, y, µ)(·) is bounded in L 2 (R d , µ) for all x ∈ R d , y ∈ T d , µ ∈ P 2 (R d ). This implies that g(x, y, ·) is Lipshitz continuous with respect to W 2 for each x ∈ R d and y ∈ T d by Remark 5.27 in [12].
Appendix D. Proof of Proposition 5.1 Here we present the proof of Proposition 5.1.

Now we have
Also, by performing the exact same proof without the supremum over time in the expectation, (using Jensen's inequality and Itô Isometry instead of Burkholder-Davis-Gundy inequality for the martingale terms), we get from Equation 55 that for each t ∈ [0, 1], < ∞.
The fact that Θ ∈ V already fixes Θ X ×Y . Then we set Θ Z (C) =ΘŶ {r ∈Ŷ : ∃r ∈ C where r t (dz) = ZẐ * δ T×R [Φ ′ (y)+1]ẑrt(dydẑ) (dz), for a.e. t ∈ [0, 1]} for all C ∈ Z. Notice that in our definition of Z we do not require r t to be a probability measure (the proof of tightness in Subsection 6.1.3 still holds in the larger space of positive Borel measures on R m × [0, 1] with finite first moment rather than just probability measures with finite first moment since {Q N Z } N ∈N is uniformly bounded in the total variation norm, see for example Theorem 8.6.2 in [8]), and that since Φ ′ (y) is bounded, Indeed, with such a choice the proofs in Sections 6, 7, and 9 go through in essentially the same way. It is a well known result that the rate function for the Laplace Principal of a sequence of random variables on a given space is unique (see, for example, Theorem 1.3.1 in [21]). The proof of Theorem 5.2 shows that, at least in a certain subclass of our system, the rate function attained from the use of {Q N } N ∈N and {Q N } N ∈N agree. It is thus possibly a viable option to use {Q N } N ∈N to attain a rate function of the form offered in Theorem 5.2 for the general system.
The reason why we instead make the choice of occupation measures described by Equation 23 is because it makes the proof of the upper bound in Section 8 much easier. We would not be able to exploit weak-sense uniqueness (see Definition 2.2) as we did if we had used {Q N } N ∈N .