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# Moderate deviations for fully coupled multiscale weakly interacting particle
systems

Z.W. Bezemek, and K. Spiliopoulos  Boston University, Department of
Mathematics and Statistics
111 Cummington Mall, Boston, MA 02215, USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We consider a collection of fully coupled weakly interacting diffusion
processes moving in a two-scale environment. We study the moderate 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, which provide a convenient representation for the
moderate deviations rate function in a variational form in terms of an
effective mean field control problem. We rigorously obtain equivalent
representation for the moderate deviations rate function in an appropriate
“negative Sobolev” form, proving their equivalence, which is reminiscent of
the large deviations rate function form for the empirical measure of weakly
interacting diffusions obtained in the 1987 seminal paper by Dawson-Gärtner.
In the course of the proof we obtain related ergodic theorems and we consider
the regularity of Poisson type of equations associated to McKean-Vlasov
problems, both of which are topics of independent interest. A novel “doubled
corrector problem” is introduced in order to control derivatives in the
measure arguments of the solutions to the related Poisson equations used to
control behavior of fluctuation terms.

###### Key words and phrases:

interacting particle systems, multiscale processes, empirical measure,
moderate deviations

###### 2010 Mathematics Subject Classification:

60F10, 60F05

This work has been partially supported by the National Science Foundation (DMS
2107856) and Simons Foundation Award 672441. The authors of the paper would
like to thank both reviewers for a very careful and constructive review of
this article.

##  1\. Introduction

The purpose of this paper is to study the moderate deviations principle (MDP)
for slow-fast interacting particle systems. In particular, we consider the
system

(1) $\displaystyle dX^{i,\epsilon,N}_{t}$
$\displaystyle=\biggl{[}\frac{1}{\epsilon}b(X^{i,\epsilon,N}_{t},Y^{i,\epsilon,N}_{t},\mu^{\epsilon,N}_{t})+c(X^{i,\epsilon,N}_{t},Y^{i,\epsilon,N}_{t},\mu^{\epsilon,N}_{t})\biggr{]}dt+\sigma(X^{i,\epsilon,N}_{t},Y^{i,\epsilon,N}_{t},\mu^{\epsilon,N}_{t})dW^{i}_{t}$
$\displaystyle dY^{i,\epsilon,N}_{t}$
$\displaystyle=\frac{1}{\epsilon}\biggl{[}\frac{1}{\epsilon}f(X^{i,\epsilon,N}_{t},Y^{i,\epsilon,N}_{t},\mu^{\epsilon,N}_{t})+g(X^{i,\epsilon,N}_{t},Y^{i,\epsilon,N}_{t},\mu^{\epsilon,N}_{t})\biggr{]}dt$
$\displaystyle+\frac{1}{\epsilon}\biggl{[}\tau_{1}(X^{i,\epsilon,N}_{t},Y^{i,\epsilon,N}_{t},\mu^{\epsilon,N}_{t})dW^{i}_{t}+\tau_{2}(X^{i,\epsilon,N}_{t},Y^{i,\epsilon,N}_{t},\mu^{\epsilon,N}_{t})dB^{i}_{t}\biggr{]}$
$\displaystyle(X^{i,\epsilon,N}_{0},Y^{i,\epsilon,N}_{0})$
$\displaystyle=(\eta^{x},\eta^{y}){}$

on a filtered probability space
$(\Omega,\mathcal{F},\mathbb{P},\\{\mathcal{F}_{t}\\})$ with
$\\{\mathcal{F}_{t}\\}$ satisfying the usual conditions, where
$b,c,\sigma,f,g,\tau_{1},\tau_{2}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R})\rightarrow\mathbb{R}$,
$B^{i}_{t},W^{i}_{t}$ are independent standard 1-D $\mathcal{F}_{t}$-Brownian
motions for $i=1,...,N$, and $(\eta^{x},\eta^{y})\in\mathbb{R}^{2}$. Here and
throughout $\mathcal{P}_{2}(\mathbb{R})$ denotes the space of probability
measures on $\mathbb{R}$ with finite second moment, equipped with the
2-Wasserstein metric (see Appendix D). $\mu^{\epsilon,N}$ is defined by

(2)
$\displaystyle\mu^{\epsilon,N}_{t}=\frac{1}{N}\sum_{i=1}^{N}\delta_{X^{i,\epsilon,N}_{t}},t\in[0,T].$

In (1), $X^{i,\epsilon,N}$ and $Y^{i,\epsilon,N}$ represent the slow and fast
motion respectively of the $i^{\text{th}}$ component. Note that classical
models of interacting particles in a two-scale potential, see [BS, Dawson,
delgadino2020, GP], can be thought of as special cases of (1) with
$Y^{i,\epsilon,N}=X^{i,\epsilon,N}/\epsilon$.

Assume that $\epsilon(N)\rightarrow 0$ as $N\rightarrow\infty$. In our case,
moderate deviations amounts to studying the behavior of the empirical measure
of the particles, i.e., of $\mu^{\epsilon,N}$ in the regime between
fluctuations and large deviations behavior. In particular, if we denote by
$\mathcal{L}(X)$ the process at which $\mu^{\epsilon,N}$ converges to (the law
of the averaged McKean-Vlasov Equation (25)) and consider the moderate
deviation scaling sequence $\\{a(N)\\}_{N\in\mathbb{N}}$ such that
$a(N)>0,\forall N\in\mathbb{N}$ with $a(N)\rightarrow 0$ and
$a(N)\sqrt{N}\rightarrow\infty$ as $N\rightarrow\infty$, the moderate
deviations process is defined to be

(3) $\displaystyle Z^{N}_{t}\coloneqq
a(N)\sqrt{N}(\mu^{\epsilon,N}_{t}-\mathcal{L}(X_{t})),t\in[0,T].$

The goal of this paper is to derive the large deviations principle with speed
$a^{-2}(N)$ for the process $Z^{N}_{t}$, which is the moderate deviations
principle for the measure-valued process $\mu^{\epsilon,N}_{t}$. Notice that
if $a(N)=1$ then we get the standard fluctuations process whose limiting
behavior amounts to fluctuations around the law of large numbers,
$\mathcal{L}(X_{t})$, whereas if $a(N)=1/\sqrt{N}$ then we would be in the
large deviations regime.

We remark here that due to the effect of multiple scales, it turns out that a
relation between $\epsilon$ and $N$ is needed. So beyond requiring
$a(N)\rightarrow 0$ and $a(N)\sqrt{N}\rightarrow\infty$, we also require that
there exists $\rho\in(0,1)$ and $\lambda\in(0,\infty]$ such that
$a(N)\sqrt{N}\epsilon(N)^{\rho}\rightarrow\lambda$ as $N\rightarrow\infty$.
Note that this should be viewed as a restriction on the scaling sequence
$a(N)$, not on the relationship between $\epsilon$ and $N$, and in some
regimes we expect this assumption can be weakened. See Remark 6.2.

The presence of multiple scales is a common feature in a range of models used
in various disciplines ranging from climate modeling to chemical physics to
finance, see for example [BryngelsonOnuchicWolynes, feng2012small,
jean2000derivatives, HyeonThirumalai, majda2008applied, Zwanzig] for a
representative, but by no means complete list of references. Interacting
diffusions have also been the central topic of study in science and
engineering, see for example [BinneyTremaine, Garnier1, Garnier2, IssacsonMS,
Lucon2016, MotschTadmor2014] to name a few. In the absence of multiple scales,
i.e., when $\epsilon=1$, law of large numbers, fluctuations and large
deviations behavior as $N\rightarrow\infty$ has been studied in the
literature, see [Dawson, DG, BDF]. Analogously, in the case of $N=1$, the
behavior as $\epsilon\downarrow 0$, have been extensively studied in the
literature, see for example [Baldi, DS, FS, GaitsgoryNguyen, JS, MS,
MSImportanceSampling, Lipster, PV1, PV2, Spiliopoulos2013a,
Spiliopoulos2014Fluctuations, Spiliopoulos2013QuenchedLDP, Veretennikov,
VeretennikovSPA2000].

Homogenization of McKean-Vlasov equations (equations that are the limit of
$N\rightarrow\infty$ with $\epsilon$ fixed) has also been recently studied in
the literature, see e.g., [BezemekSpiliopoulosAveraging2022, HLL, KSS,
RocknerMcKeanVlasov]. These results can be thought of as looking at the limit
of the system (1) when first $N\rightarrow\infty$ and then
$\epsilon\rightarrow 0$. Large deviations for a special case of (1) has been
recently established in [BS] and in the absence of multiple scales in [BDF].
In [Orrieri] the author studies large deviations for interacting particle
systems in the absence of multiple scales but in the joint mean-field and
small-noise limit. In the absence of multiple scales, i.e., when $\epsilon=1$,
moderate deviations for interacting particle systems have been studied in
[BW].

The contributions of this work are fourfold. Firstly, we investigate the
combined limit $N\rightarrow\infty$ and $\epsilon\rightarrow 0$ for the fully
coupled interacting particle system of McKean-Vlasov type (1) through the lens
of moderate deviations. In order to do so, we use the weak convergence
methodology developed in [DE] which leads to the study of (appropriately
linearized) optimal stochastic control problems of McKean-Vlasov type, see for
example [CD, Lacker, Fischer]. The first main result of this paper is Theorem
3.2 which provides a variational representation of the moderate deviations
rate function.

Secondly, we rigorously re-express the obtained variational form of the rate
function in the “negative Sobolev” form given in Theorem 5.1 of the seminal
paper by Dawson-Gärtner [DG] in the absence of multiple scales. Hence, we
rigorously establish the equivalence of the two formulations in the moderate
deviations setting, see Proposition 4.3. A connection of this form was
recently established rigorously for the first time in the large deviations
setting in [BS].

Thirdly, in the process of establishing the MDP, we derive related ergodic
theorems for multiscale interacting particle systems that are of independent
interest. Due to the nature of moderate deviations, we need to consider
certain solutions of Poisson equations whose properties are considered for the
first time in this paper. In particular, we must control a term involving a
derivative in the measure argument of the solution to the Poisson Equation
(22) (known as the Cell-Problem in the periodic setting). Such terms are
unique to slow-fast interacting particle systems and slow-fast McKean-Vlasov
SDEs, and thus do not appear whatsoever in proofs of averaging in the one-
particle setting. Thus, the “doubled corrector problem” construction, (63),
and the method of proof of Proposition 6.3 are novel ideas here, see also
[BezemekSpiliopoulosAveraging2022].

Fourthly, in contrast to [BW], in this paper the coefficients of the model
need not depend on the measure parameter in an affine way. We allow the
coefficients of the interacting particle system (1) to have any dependence on
the measure $\mu$, so long that it is sufficiently smooth- see Corollary 3.4
and Remark 3.5. This is thanks to Lemma D.7, which is inspired by Lemma 5.10
in [DLR], and allows us to see that with sufficient regularity of a functional
on $\mathcal{P}_{2}(\mathbb{R})$, the $L^{2}$ error of that functional
evaluated at the empirical measure of $N$ IID random variables and the Law of
those random variables is $\mathcal{O}(1/N)$ as $N\rightarrow\infty$.

In Section 4.2, we make our general results concrete for a popular model of
interacting particles in a two-scale potential, see also [GP] for a motivating
example in this direction. In addition, we present in Section C.3 a number of
concrete examples where the conditions of this paper hold.

The identification of the optimal change of measure in the moderate deviations
lower bound through feedback controls together with the equivalence proof
between the variational formulation and the “negative Sobolev” form of the
rate function, open the door to a rigorous study of provably-efficient
accelerated Monte-Carlo schemes for rare events computation, analogous to what
has been accomplished in the one particle case, see e.g., [DSW,
MSImportanceSampling]. Exploring this is beyond the scope of this work and
will be addressed elsewhere.

In addition, [Dawson] remarks that phase transitions can occur at the level of
fluctuations for interacting particle systems. Since the moderate deviations
principle is essentially a large deviations statement around the fluctuations,
the results obtained in this paper can potentially be related to phase
transitions and allow to characterize them further. This dynamical systems
direction is left for future work as it is also outside the scope of this
paper.

In contrast to large deviations, the main difficulty with moderate deviations
lies in the tightness proof, where we use an appropriate coupling argument, as
well as in the fact that the space of signed measures is not completely
metrizable in the topology of weak convergence (see [DBDG] Remark 1.2, as well
as [RV] Remarks 2.2 and 2.3 for further discussion on related issues). Thus,
as we will see, we will have to study $Z^{N}$ as a distribution-valued process
on a suitable weighted Sobolev space. In addition, the presence of the
multiple scales complicates the required estimates because the ergodic
behavior needs to be accounted for as well. The coupling argument used in the
proof of tightness is non-standard in that the IID particle system used as an
intermediary process between the empirical measure $\mu^{\epsilon,N}$ from
Equation (2) and its homogenized McKean-Vlasov limit $\mathcal{L}(X)$ from
Equation (25) is not equal in distribution to $X$. Instead, it is an IID
system of slow-fast McKean Vlasov SDEs - see Equation (57). Thus our proof of
tightness is in a sense relying on the fact that the limits
$N\rightarrow\infty$ and $\epsilon\downarrow 0$ for the empirical measure (2)
commute at the level of the law of large numbers. For a further discussion of
this, see Remark 5.1 and the discussion at the beginning of Section 7.

The rest of the paper is organized as follows. In Section 2, we introduce the
appropriate topology for $Z^{N}$ and lay out our main assumptions. We also
introduce a quite useful multi-index notation that will allow us to circumvent
notational difficulties with various combinations of mixed derivatives that
appear throughout the paper. The derivation of the moderate deviations
principle is based on the weak convergence approach of [DE] which converts the
large deviations problem to weak convergence of an appropriate stochastic
control problem. The main result is presented in Section 3, Theorem 3.2. In
Section 4 we prove an alternative form of the rate function. This form
provides a rigorous connection in moderate deviations between the “variational
form” of the rate function for the empirical measure of weakly interacting
particle systems proved in Theorem 3.2 to the “negative Sobolev” form given in
Theorem 5.1 of the seminal paper by Dawson-Gärtner [DG]. Corollaries 3.4 and
4.6 specialize the discussion to the setting without multiscale structure and
thus generalizes the results of [BW]. Specific examples are presented in
Subsection 4.2. Section 5 formulates the appropriate stochastic control
problem.

Sections 6-10 are devoted to the proof of Theorem 3.2. Due to the presence of
the multiple scales, ergodic theorems are needed to characterize the behavior
as $\epsilon\downarrow 0$ of certain functionals of interest for the
controlled multiscale interacting particle system; this is the content of
Section 6. Tightness of the controlled system is proven in Section 7. In
Section 8 we establish the limiting behavior of the controlled system. The
Laplace principle lower bound is proven in Section 9. Section 10 contains the
proof of the Laplace principle upper bound as well as compactness of level
sets. Conclusions and directions for future work are in Section 11. Appendix A
provides a list of technical notation used throughout the manuscript for
convenience. A number of key technical estimates are presented in the
remainder of the appendix. In particular, Appendix B contains moments bounds
for the controlled system. Appendix C presents regularity results for the
Poisson equation needed to study the fluctuations. Even though related results
exist in the literature, the fully coupled McKean-Vlasov case is not covered
by the existing results, and therefore Appendix C contains the appropriate
discussion of the necessary extensions. Lastly, Appendix D contains necessary
results on differentiation of functions on spaces of measures.

##  2\. Notation, Topologies, and Assumptions

In order to construct an appropriate topology for the process $Z^{N}$ from
Equation (3), we follow the method of [BW, HM, KX]. Denote by $\mathcal{S}$
the space of functions $\phi\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\rightarrow\mathbb{R}$ which are infinitely differentiable
and satisfy $|x|^{m}\phi^{(k)}(x)\rightarrow 0$ as $|x|\rightarrow\infty$ for
all $m,k\in\mathbb{N}$. On $\mathcal{S}$, consider the sequence of inner
products $(\cdot,\cdot)_{n}$ and
$\mathinner{\\!\left\lVert\cdot\right\rVert}_{n}$ defined by

(4) $\displaystyle(\phi,\psi)_{n}$
$\displaystyle\coloneqq\sum_{k=0}^{n}\int_{\mathbb{R}}(1+x^{2})^{2n}\phi^{(k)}(x)\psi^{(k)}(x)dx,\qquad\mathinner{\\!\left\lVert\phi\right\rVert}_{n}\coloneqq\sqrt{(\phi,\phi)_{n}}$

for each $n\in\mathbb{N}$. As per [GV] p.82 (this specific example on p.84),
this sequence of seminorms induces a nuclear Fréchet topology on
$\mathcal{S}$. Let $\mathcal{S}_{n}$ be the completion of $\mathcal{S}$ with
respect to $\mathinner{\\!\left\lVert\cdot\right\rVert}_{n}$ and
$\mathcal{S}_{-n}=\mathcal{S}_{n}^{\prime}$ the dual space of
$\mathcal{S}_{n}$. We equip $\mathcal{S}_{-n}$ with dual norm
$\mathinner{\\!\left\lVert\cdot\right\rVert}_{-n}$ and corresponding inner
product $(\cdot,\cdot)_{-n}$. Then $\\{\mathcal{S}_{n}\\}_{n\in\mathbb{Z}}$
defines a sequence of nested Hilbert spaces with
$\mathcal{S}_{m}\subset\mathcal{S}_{n}$ for $m\geq n$. In addition we have for
each $n\in\mathbb{N}$, there exists $m>n$ such that the canonical embedding
$\mathcal{S}_{-n}\rightarrow\mathcal{S}_{-m}$ is Hilbert-Schmidt. In
particular, this holds for $m$ sufficiently large that
$\sum_{j=1}^{N}\mathinner{\\!\left\lVert\phi^{m}_{j}\right\rVert}_{n}<\infty$,
where $\\{\phi^{m}_{j}\\}_{j\in\mathbb{N}}$ is a complete orthonormal system
of $\mathcal{S}_{m}$. This allows us to use the results of [Mitoma] to see
that $\\{Z^{N}\\}_{N\in\mathbb{N}}$ is tight as a sequence of
$C([0,T];S_{-m})$-valued random variables for sufficiently large $m$. In
particular, we will require $m>7$ to be sufficiently large so that the
canonical embedding

(5) $\displaystyle\mathcal{S}_{-7}\rightarrow\mathcal{S}_{-m}\text{ is
Hilbert-Schmidt.}$

In the proof of the Laplace Principle, we will also make use of $w>9$ such
that

(6) $\displaystyle\mathcal{S}_{-m-2}\rightarrow\mathcal{S}_{-w}\text{ is
Hilbert-Schmidt.}$

When proving compactness of level sets of the rate function, we will in
addition make use of $r>11$ sufficiently large that the canonical embedding

(7) $\displaystyle\mathcal{S}_{-w-2}\rightarrow\mathcal{S}_{-r}\text{ is
Hilbert-Schmidt.}$

It will be useful to consider another system of seminorms on $\mathcal{S}$
given, for each $n\in\mathbb{N}$, by

(8)
$\displaystyle|\phi|_{n}\coloneqq\sum_{k=0}^{n}\sup_{x\in\mathbb{R}}|\phi^{(k)}(x)|{}$

Via a standard Sobolev embedding argument, one can show that for each
$n\in\mathbb{N}$, there exists $C(n)$ such that:

(9) $\displaystyle|\phi|_{n}\leq
C(n)\mathinner{\\!\left\lVert\phi\right\rVert}_{n+1},\forall\phi\in\mathcal{S}.$

Let $\bm{X}$ and $\bm{Y}$ be Polish spaces, and
$(\tilde{\Omega},\tilde{\mathcal{F}},\mu)$ be a measure space. We will denote
by $\mathcal{P}(\bm{X})$ the space of probability measures on $\bm{X}$ with
the topology of weak convergence,
$\mathcal{P}_{2}(\bm{X})\subset\mathcal{P}(\bm{X})$ the space of square
integrable probability measures on $\bm{X}$ with the 2-Wasserstein metric (see
Definition D.1), $\mathcal{B}(\bm{X})$ the Borel $\sigma$-field of $\bm{X}$,
$C(\bm{X};\bm{Y})$ the space of continuous functions from $\bm{X}$ to
$\bm{Y}$, $C_{b}(\bm{X})$ the space of bounded, continuous functions from
$\bm{X}$ to $\mathbb{R}$ with norm
$\mathinner{\\!\left\lVert\cdot\right\rVert}_{\infty}$, and
$L^{p}(\tilde{\Omega},\tilde{\mathcal{F}},\mu)$ the space of $p$-integrable
functions on $(\tilde{\Omega},\tilde{\mathcal{F}},\mu)$ (where if
$\tilde{\Omega}=\bm{X}$ and no $\sigma$-algebra is provided we assume it is
$\mathcal{B}(\bm{X})$). For
$\mu\in\mathcal{P}(\bm{X}),\nu\in\mathcal{P}(\bm{Y})$, we will denote the
product measure induced by $\mu$ and $\nu$ on $\bm{X}\times\bm{Y}$ by
$\mu\otimes\nu$. We will at times denote
$L^{2}(\bm{X}\times\bm{X},\mu\otimes\mu)$ by $L^{2}(\bm{X},\mu)\otimes
L^{2}(\bm{X},\mu)$. We will denote by $L^{1}_{\text{loc}}(\bm{X},\mu)$ the
space of locally integrable functions on $\bm{X}$. For
$U\subseteq\mathbb{R}^{d}$ open, we will denote by $C^{\infty}_{c}(U)$ the
space of smooth, compactly supported functions on $U$. $C^{k}_{b}(\mathbb{R})$
for $k\in\mathbb{N}$ will note the space of functions with $k$ continuous and
bounded derivatives on $\mathbb{R}$, with norm $|\cdot|_{k}$ as in Equation
(8), and $C^{1,k}_{b}([0,T]\times\mathbb{R})$ will denote continuous functions
$\psi$ on $[0,T]\times\mathbb{R}$ with a continuous, bounded time derivative
on $(0,T)$, denoted $\dot{\psi}$, such that
$\mathinner{\\!\left\lVert\psi\right\rVert}_{C^{1,k}_{b}([0,T]\times\mathbb{R})}\coloneqq\sup_{t\in[0,T],x\in\mathbb{R}}|\dot{\psi}(t,x)|+\sup_{t\in[0,T]}\mathinner{\\!\left\lVert\psi(t,\cdot)\right\rVert}_{C^{k}_{b}(\mathbb{R})}<\infty$.
$C^{k}_{b,L}(\mathbb{R})\subset C^{k}_{b}(\mathbb{R})$ is the space of
functions in $C^{k}_{b}(\mathbb{R})$ such that all $k$ derivatives are
Lipschitz continuous. For $\phi\in
L^{1}(\bm{X},\mu),\mu\in\mathcal{P}(\mathbb{R})$ we define
$\langle\mu,\phi\rangle\coloneqq\int_{\bm{X}}\phi(x)\mu(dx)$. Similarly, for
$Z\in\mathcal{S}_{-p},\phi\in\mathcal{S}_{p}$, we will denote the action of
$Z$ on $\phi$ by $\langle Z,\phi\rangle$. For $a,b\in\mathbb{R}$, we will
denote $a\vee b=\max\\{a,b\\}$ and $a\wedge b=\min\\{a,b\\}$. $C$ will be used
for a constant which may change from line to line throughout, and when there
are parameters $a_{1},...,a_{n}$ which $C$ depends on in an important manner,
will denote this dependence by $C(a_{1},...,a_{n})$. For all function spaces,
the codomain is assumed to be $\mathbb{R}$ unless otherwise denoted.

In the construction of the controlled system, we will also make use of the
space of measures on $\mathbb{R}^{d}\times[0,T]$ such that
$Q(\mathbb{R}^{d}\times[0,t])=t,\forall t\in[0,T]$. We will denote this space
$M_{T}(\mathbb{R}^{d})$. We equip $M_{T}(\mathbb{R}^{d})$ with the topology of
weak convergence of measures (thus making $M_{T}(\mathbb{R}^{d})$ a Polish
space by [DE] Theorem A.3.3). See also the proof of Lemma 3.3.1 in [DE] for
the fact that $M_{T}(\mathbb{R}^{d})$ is a closed subset of finite positive
Borel measures on $\mathbb{R}^{d}\times[0,T]$). For when dealing with the
occupation measures as defined in Equation (56), we will in particular take
$d=4$ and will interpret $Q(dx,dy,dz,dt)$ as $x$ denoting variable
representing the first coordinate in $\mathbb{R}^{4}$, $y$ the second, and $z$
the third and fourth.

For a mapping $\vartheta\mathrel{\mathop{\mathchar
58\relax}}[0,T]\rightarrow\mathcal{P}(\mathbb{R}^{d})$, it will be useful to
define an element of $M_{T}(\mathbb{R}^{d})$ induced by $\vartheta$ by

(10)
$\displaystyle\nu_{\vartheta}(A\times[0,t])\coloneqq\int_{0}^{t}\vartheta(s)[A]ds,\forall
t\in[0,T],A\in\mathcal{B}(\mathbb{R}^{d}).$

Due to the nature of the space we consider the sequence
$\\{Z^{N}\\}_{N\in\mathbb{N}}$ to live on, it is natural that we will have to
restrict the growth of the coefficients which appear in Equations (1) and (25)
in $x$. We will also need to ensure that the derivatives of the Poisson
Equation which appear in the definition the limiting coefficients in Equation
(23) exist and that the homogenized drift and diffusion coefficients in
Equation (24), which determine the limiting McKean-Vlasov Equation $X_{t}$
from Equation (25), are well-defined. In doing so, will be controlling many
mixed derivatives of functions in the Lions sense [NotesMFG] and in the
standard sense, it will be useful for us to borrow the multi-index notation
proposed in [CM] and employed in [CST]. For the reader’s convenience, we have
included in Appendix D a brief review on differentiation of functions on
spaces of measures. For a more comprehensive exposition on this, we refer the
interested reader to [CD] Chapter 5.

Furthermore, since we prove the moderate deviations principle via use of the
controlled particle system (55), we will only have up to second moments of the
controlled fast system (see Appendix B). It will be important to make sure
that terms which the controlled fast process enters in the intermediate proofs
of tightness, so naturally we will need some assumptions on the rate of
polynomial growth in $y$ of the coefficients which appear in Equations (1) and
(25) (See Remark 2.7). We thus extend the multi-index notation from the
aforementioned papers to track specific collections of mixed partial
derivatives, and to give us a clean way of tracking the rate of polynomial
growth in $y$ for those mixed partials in the coming definitions.

######  Definition 2.1.

Let $n,l$ be non-negative integers and $\bm{\beta}=(\beta_{1},...,\beta_{n})$
be an $n-$dimensional vector of non-negative integers. We call any ordered
tuple of the form $(n,l,\bm{\beta})$ a multi-index. For a function
$G\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R})\rightarrow\mathbb{R}$,
we will denote for a multi-index $(n,l,\bm{\beta})$, if this derivative is
well defined,

$\displaystyle
D^{(n,l,\bm{\beta})}G(x,\mu)[z_{1},...,z_{n}]=\partial_{z_{1}}^{\beta_{1}}...\partial_{z_{n}}^{\beta_{n}}\partial_{x}^{l}\partial_{\mu}^{n}G(x,\mu)[z_{1},...,z_{n}],$

As noted in the Remark D.3, for such a derivative to be well defined we
require for it to be jointly continuous in $x,\mu,z_{1},...,z_{n}$ where the
topology used in the measure component is that of
$\mathcal{P}_{2}(\mathbb{R})$.

We also define $\bm{\delta}^{(n,l,\bm{\beta})}G(x,\mu)[z_{1},...,z_{n}]$ in
the exact same way, with the Lions derivatives $\partial_{\mu}$ replaced by
linear functional derivatives $\frac{\delta}{\delta m}$; see Appendix D for
differentiation of functions on spaces of measures.

######  Definition 2.2.

For $\bm{\zeta}$ a collection of multi-indices of the form
$(n,l,\bm{\beta})\in\mathbb{N}\times\mathbb{N}\times\mathbb{N}^{n}$, we will
call $\bm{\zeta}$ a complete collection of multi-indices if for any
$(n,l,\bm{\beta})\in\bm{\zeta}$,
$\\{(k,j,\bm{\alpha}(k))\in\mathbb{N}\times\mathbb{N}\times\mathbb{N}^{k}\mathrel{\mathop{\mathchar
58\relax}}j\leq l,k\leq
n,\bm{\alpha}(k)=(\alpha_{1},...,\alpha_{k}),\exists\bm{\beta}(k)=(\beta(k)_{1},...,\beta(k)_{k})\in\binom{\bm{\beta}}{k}\text{
such that }\alpha_{p}\leq\beta(k)_{p},\forall p=1,...,k\\}\subset\bm{\zeta}$.
Here for a vector of positive integers $\beta=(\beta_{1},...,\beta_{n})$ and
$k\in\mathbb{N},k\leq n$, we are using the notation $\binom{\bm{\beta}}{k}$ to
represent the set of size $\binom{n}{k}$ containing all the $k$-dimensional
vectors of positive integers which can be obtained from removing $n-k$ entries
from $\bm{\beta}$.

######  Remark 2.3.

Definition 2.2 is enforcing that if collection of multi-indices contains a
multi-index representing some mixed derivative in $(x,\mu,z)$ as per
Definition 2.1, then it also contains all lower-order mixed derivatives of the
same type. For example, if $\bm{\zeta}$ is a collection of multi-indices
containing $(2,0,(1,1))$ (corresponding to
$\partial_{z_{1}}\partial_{z_{2}}\partial^{2}_{\mu}G(x,\mu)[z_{1},z_{2}]$)
then, in order to be complete, it must also contain the terms
$(2,0,(1,0)),(2,0,(0,1)),(2,0,0),(1,0,1),(1,0,0),$ and $(0,0,0)$
(corresponding to the terms
$\partial_{z_{1}}\partial^{2}_{\mu}G(x,\mu)[z_{1},z_{2}]$,
$\partial_{z_{2}}\partial^{2}_{\mu}G(x,\mu)[z_{1},z_{2}],\partial^{2}_{\mu}G(x,\mu)[z_{1},z_{2}],\partial_{z}\partial_{\mu}G(x,\mu)[z]$,
$\partial_{\mu}G(x,\mu)[z]$, and $G(x,\mu)$ respectively). This is a technical
requirement used in order to state the results in Appendix C.1 in a way that
allows the inductive arguments used therein to go through.

Using this multi-index notation, it will be useful to define some spaces
regarding regularity of functions in regards to these mixed derivatives. We
thus make the following modifications to Definition 2.13 in [CST]:

######  Definition 2.4.

For $\bm{\zeta}$ a collection of multi-indices of the form
$(n,l,\bm{\beta})\in\mathbb{N}\times\mathbb{N}\times\mathbb{N}^{n}$, we define
$\mathcal{M}_{b}^{\bm{\zeta}}(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$ to
be the class of functions $G\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R})\rightarrow\mathbb{R}$
such that $D^{(n,l,\bm{\beta})}G(x,\mu)[z_{1},...,z_{n}]$ exists and satisfies

(11) $\displaystyle\mathinner{\\!\left\lVert
G\right\rVert}_{\mathcal{M}_{b}^{\bm{\zeta}}(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))}\coloneqq\sup_{(n,l,\bm{\beta})\in\bm{\zeta}}\sup_{x,z_{1},...,z_{n}\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}|D^{(n,l,\bm{\beta})}G(x,\mu)[z_{1},...,z_{n}]|$
$\displaystyle\leq C.$

We denote the class of functions
$G\in\mathcal{M}_{b}^{\bm{\zeta}}(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$
such that:

(12)
$\displaystyle|D^{(n,l,\bm{\beta})}G(x,\mu)[z_{1},...,z_{n}]-D^{(n,l,\bm{\beta})}G(x^{\prime},\mu^{\prime})[z_{1}^{\prime},...,z_{n}^{\prime}]|$
$\displaystyle\leq
C_{L}\biggl{(}|x-x^{\prime}|+\sum_{i=1}^{N}|z_{i}-z^{\prime}_{i}|+\mathbb{W}_{2}(\mu,\mu^{\prime})\biggr{)}$

for all $(n,l,\bm{\beta})\in\bm{\zeta}$ and
$x,x^{\prime},z_{1},...,z_{n},z_{1}^{\prime},...,z_{n}^{\prime}\in\mathbb{\mathbb{R}},\mu,\mu^{\prime}\in\mathcal{P}_{2}(\mathbb{R})$
by
$\mathcal{M}_{b,L}^{\bm{\zeta}}(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$.
We define $\mathcal{M}_{b}^{\bm{\zeta}}(\mathcal{P}_{2}(\mathbb{R}))$ and
$\mathcal{M}_{b,L}^{\bm{\zeta}}(\mathcal{P}_{2}(\mathbb{R}))$ analogously,
where instead here $\bm{\zeta}$ is a collection of multi-indices of the form
$(n,\bm{\beta})\in\mathbb{N}\times\mathbb{N}^{n}$, and we take the $l=0$ in
the above multi-index notation for the derivatives.

We will also make use of the class of functions
$\mathcal{M}_{p}^{\bm{\zeta}}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$
which contains $G\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R})\rightarrow\mathbb{R}$
such that
$G(\cdot,y,\cdot)\in\mathcal{M}_{b}^{\bm{\zeta}}(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$
for all $y\in\mathbb{R}$, with all derivatives appearing in the definition of
$\mathcal{M}_{b}^{\bm{\zeta}}(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$
jointly continuous in $(x,y,\mathbb{W}_{2})$, and for each multi-index
$(n,l,\bm{\beta})\in\bm{\zeta}$,

(13)
$\displaystyle\sup_{x,z_{1},...,z_{n}\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}|D^{(n,l,\bm{\beta})}G(x,y,\mu)[z_{1},...,z_{n}]|$
$\displaystyle\leq C(1+|y|)^{q_{G}(n,l,\bm{\beta})},$

where $q_{G}(n,l,\bm{\beta})\in\mathbb{R}$. Similarly,
$\mathcal{M}_{p,L}^{\bm{\zeta}}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$
is defined as
$G\in\mathcal{M}_{p}^{\bm{\zeta}}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$
such that Equation (12) holds for $G(\cdot,y,\cdot)$ for each
$y\in\mathbb{R}$, where $C_{L}(y)$ grows at most polynomially in $y$.

We also define
$\mathcal{M}_{b}^{\bm{\zeta}}([0,T]\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$
to be the class of functions $G\mathrel{\mathop{\mathchar
58\relax}}[0,T]\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R})\rightarrow\mathbb{R}$
such that $G(\cdot,x,\mu)$ is continuously differentiable on $(0,T)$ for all
$x\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})$ with time derivative
denoted by $\dot{G}(t,x,\mu)$,
$G(t,\cdot,\cdot)\in\mathcal{M}_{b}^{\bm{\zeta}}(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$
for all $t\in[0,T]$, with (11) holding uniformly in $t$, and $G,\dot{G},$ and
all derivatives involved in the definition of
$\mathcal{M}_{b}^{\bm{\zeta}}(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$
are jointly continuous in time, measure, and space. We define for
$G\in\mathcal{M}_{b}^{\bm{\zeta}}([0,T]\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$

$\displaystyle\mathinner{\\!\left\lVert
G\right\rVert}_{\mathcal{M}_{b}^{\bm{\zeta}}([0,T]\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))}\coloneqq\sup_{t\in[0,T]}\mathinner{\\!\left\lVert
G(t,\cdot)\right\rVert}_{\mathcal{M}_{b}^{\bm{\zeta}}(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))}+\sup_{t\in[0,T],x\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}|\dot{G}(t,x,\mu)|.$

We denote the class of functions
$G\in\mathcal{M}_{b}^{\bm{\zeta}}([0,T]\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$
such that (12) holds uniformly in $t$ by
$\mathcal{M}_{b,L}^{\bm{\zeta}}([0,T]\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$.
Again, we define
$\mathcal{M}_{b}^{\bm{\zeta}}([0,T]\times\mathcal{P}_{2}(\mathbb{R}))$,
$\mathcal{M}_{b,L}^{\bm{\zeta}}([0,T]\times\mathcal{P}_{2}(\mathbb{R}))$, and
$\mathcal{M}_{p}^{\bm{\zeta}}([0,T]\times\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$
analogously.

At times we will want to consider Lions Derivatives bounded in
$L^{2}(\mathbb{R},\mu)$ rather than uniformly in $z$. Thus we define
$\tilde{\mathcal{M}}_{b}^{\bm{\zeta}}(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$
to be the class of functions $G\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R})\rightarrow\mathbb{R}$
such that $D^{(n,l,\bm{\beta})}G(x,\mu)[z_{1},...,z_{n}]$ exists and satisfies

(14) $\displaystyle\mathinner{\\!\left\lVert
G\right\rVert}_{\tilde{\mathcal{M}}_{b}^{\bm{\zeta}}(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))}$
$\displaystyle\coloneqq\sup_{(n,l,\bm{\beta})\in\bm{\zeta}}\sup_{x\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}\mathinner{\\!\left\lVert
D^{(n,l,\bm{\beta})}G(x,\mu)[\cdot]\right\rVert}_{L^{2}(\mu,\mathbb{R})^{\otimes
n}}$
$\displaystyle=\sup_{(n,l,\bm{\beta})\in\bm{\zeta}}\sup_{x\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}\biggl{(}\int_{\mathbb{R}}...\int_{\mathbb{R}}|D^{(n,l,\bm{\beta})}G(x,\mu)[z_{1},...,z_{n}]|^{2}\mu(dz_{1})...\mu(dz_{n})\biggr{)}^{1/2}\leq
C.$

We also define
$\tilde{\mathcal{M}}_{b}^{\bm{\zeta}}([0,T]\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$
analogously to
$\mathcal{M}_{b}^{\bm{\zeta}}([0,T]\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$,
$\tilde{\mathcal{M}}_{p}^{\bm{\zeta}}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$
analogously to
$\mathcal{M}_{p}^{\bm{\zeta}}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$,
and
$\tilde{\mathcal{M}}_{p}^{\bm{\zeta}}([0,T]\times\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$
analogously to
$\mathcal{M}_{p}^{\bm{\zeta}}([0,T]\times\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$.
We will denote the polynomial growth rate for
$G\in\tilde{\mathcal{M}}_{p}^{\bm{\zeta}}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$
and $(n,l,\bm{\beta})\in\bm{\zeta}$ as in Equation (13) but with the
$L^{2}(\mathbb{R},\mu)^{\otimes n}$-norm by
$\tilde{q}(n,l,\bm{\beta})\in\mathbb{R}$ to avoid confusion with polynomial
growth of the derivatives in the uniform norm. That is:

(15)
$\displaystyle\sup_{x\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}\biggl{(}\int_{\mathbb{R}}...\int_{\mathbb{R}}|D^{(n,l,\bm{\beta})}G(x,\mu)[z_{1},...,z_{n}]|^{2}\mu(dz_{1})...\mu(dz_{n})\biggr{)}^{1/2}\leq
C(1+|y|)^{\tilde{q}_{G}(n,l,\bm{\beta})}.$

Lastly, we define
$\mathcal{M}_{\bm{\delta},b}^{\bm{\zeta}}(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$
and
$\mathcal{M}_{\bm{\delta},p}^{\bm{\zeta}}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$
in the same way as
$\mathcal{M}_{b}^{\bm{\zeta}}(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$
and
$\mathcal{M}_{p}^{\bm{\zeta}}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$
respectively, but with with $D^{(n,l,\bm{\beta})}$ replaced by
$\bm{\delta}^{(n,l,\bm{\beta})}$. We also extend this in the natural way when
the spatial components are in higher dimensions (i.e. taking gradients and
using norms in $\mathbb{R}^{d}$).

Let us now introduce the main assumptions that are needed for the work of this
paper to go through.

  1. A1)

$0<\lambda_{-}\leq\tau_{1}^{2}(x,y,\mu)+\tau_{2}^{2}(x,y,\mu)\leq\lambda_{+}<\infty$,
$\forall x,y\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})$, and
$\tau_{1},\tau_{2}$ have two uniformly bounded derivatives in $y$ and which
are jointly continuous in $(x,y,\mathbb{W}_{2}).$

  2. A2)

There exists $\beta>0$ and $\kappa>0$ such that:

(16) $\displaystyle f(x,y,\mu)=-\kappa y+\eta(x,y,\mu)$

where $\eta$ is uniformly bounded in $x$ and $\mu$, and Lipschitz in the sense
of A9) in $x$, $\mu$, and $y$ with

$\displaystyle|\eta(x,y_{1},\mu)-\eta(x,y_{2},\mu)|\leq
L_{\eta}|y_{1}-y_{2}|,\forall
x\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})$

for $L_{\eta}$ such that $L_{\eta}-\kappa<0$, and

(17) $\displaystyle
2(f(x,y_{1},\mu)-f(x,y_{2},\mu))(y_{1}-y_{2})+3|\tau_{1}(x,y_{1},\mu)-\tau_{1}(x,y_{2},\mu)|^{2}$
$\displaystyle+3|\tau_{2}(x,y_{1},\mu)-\tau_{2}(x,y_{2},\mu)|^{2}\leq-\beta|y_{1}-y_{2}|^{2},$
$\displaystyle\forall
x,y_{1},y_{2}\in\mathbb{R},\mu\in\mathcal{P}(\mathbb{R}).$

Let $a(x,y,\mu)=\frac{1}{2}[\tau_{1}^{2}(x,y,\mu)+\tau_{2}^{2}(x,y,\mu)]$. For
$x\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})$, we define the differential
operator $L_{x,\mu}$ acting on $\phi\in C_{b}^{2}(\mathbb{R})$ by

(18) $\displaystyle
L_{x,\mu}\phi(y)=f(x,y,\mu)\phi^{\prime}(y)+a(x,y,\mu)\phi^{\prime\prime}(y).$

Note that under assumptions A1) and A2), there is a constant $C$ independent
of $x,y,\mu$ such that:

(19) $\displaystyle
2f(x,y,\mu)y+3|\tau_{1}(x,y,\mu)|^{2}+3|\tau_{2}(x,y,\mu)|^{2}\leq-\frac{\beta}{2}|y|^{2}+C,\forall
x,y\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R}).$

Thus by [PV1] Proposition 1 (see also [Veretennikov1987]), there exists a
$\pi(\cdot;x,\mu)$ which is the unique measure solving

(20) $\displaystyle L_{x,\mu}^{*}\pi=0.$

Moreover, for all $k>0$, there is $C_{k}\geq 0$ such that
$\sup_{x\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}\int_{\mathbb{R}}|y|^{k}\pi(dy;x,\mu)\leq
C_{k}$.

  1. A3)

For $\pi$ as in Equation (20),

(21) $\displaystyle\int_{\mathbb{R}}b(x,y,\mu)\pi(dy;x,\mu)=0,\forall
x\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R}),$

$b$ is jointly continuous in $(x,y,\mathbb{W}_{2})$, grows at most
polynomially in $y$ uniformly in
$x\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})$.

Having introduced the notation above, we can now present the law of large
numbers for the empirical measure $\mu^{\epsilon,N}$ from Equation 2 in the
joint limit as $\epsilon\downarrow 0,N\rightarrow\infty$. Under assumptions
A1)-A3), by Lemma C.1 we consider $\Phi$ the unique classical solution to:

(22) $\displaystyle L_{x,\mu}\Phi(x,y,\mu)$
$\displaystyle=-b(x,y,\mu),\qquad\int_{\mathbb{R}}\Phi(x,y,\mu)\pi(dy;x,\mu)=0.{}$

Let us define the functions

(23) $\displaystyle\gamma(x,y,\mu)$
$\displaystyle\coloneqq\gamma_{1}(x,y,\mu)+c(x,y,\mu)$
$\displaystyle\gamma_{1}(x,y,\mu)$ $\displaystyle\coloneqq
b(x,y,\mu)\Phi_{x}(x,y,\mu)+g(x,y,\mu)\Phi_{y}(x,y,\mu)+\sigma(x,y,\mu)\tau_{1}(x,y,\mu)\Phi_{xy}(x,y,\mu)$
$\displaystyle D(x,y,\mu)$ $\displaystyle\coloneqq
D_{1}(x,y,\mu)+\frac{1}{2}\sigma^{2}(x,y,\mu)$ $\displaystyle D_{1}(x,y,\mu)$
$\displaystyle=b(x,y,\mu)\Phi(x,y,\mu)+\sigma(x,y,\mu)\tau_{1}(x,y,\mu)\Phi_{y}(x,y,\mu).$

and

(24) $\displaystyle\bar{\gamma}(x,\mu)$
$\displaystyle\coloneqq\biggl{[}\int_{\mathbb{R}}\gamma(x,y,\mu)\pi(dy;x,\mu)\biggr{]},\qquad\bar{D}(x,\mu)\coloneqq\biggl{[}\int_{\mathbb{R}}D(x,y,\mu)\pi(dy;x,\mu)\biggr{]}.$

Then, by essentially the same arguments as in [BS], under the conditions
outlined below, $\mu^{\epsilon,N}$ converges in distribution to the
deterministic limit $\mathcal{L}(X)$ where $X$ satisfies the averaged McKean-
Vlasov SDE

(25) $\displaystyle dX_{t}$
$\displaystyle=\bar{\gamma}(X_{t},\mathcal{L}(X_{t}))dt+\sqrt{2\bar{D}(X_{t},\mathcal{L}(X_{t}))}dW^{2}_{t}\quad
X_{0}=\eta^{x}.$

Here $W^{2}_{t}$ is a Brownian motion on another filtered probability space
satisfying the usual conditions. In fact, we see here in Lemma 8.2 that in
fact this convergence occurs in $\mathcal{P}_{2}(\mathbb{R})$ for each
$t\in[0,T].$

######  Remark 2.5.

Using an integration-by-parts argument, one can find that the diffusion
coefficient $\bar{D}$ can be written in the alternative form

(26) $\displaystyle\bar{D}(x,\mu)$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}}\left([\tau_{2}(x,y,\mu)\Phi_{y}(x,y,\mu)]^{2}+[\sigma(x,y,\mu)+\tau_{1}(x,y,\mu)\Phi_{y}(x,y,\mu)]^{2}\right)\pi(dy;x,\mu),$

and hence is non-negative. See [Bensoussan] Chapter 3 Section 6.2 for a
similar computation.

We now introduce the remaining assumptions. Since we are dealing with
fluctuations, we will need to be able to obtain rates of averaging, and thus
there are several auxiliary Poisson equations involved in the proof of
tightness. When there is more specific structure to the system of equations
(1), these assumptions may be able to be verified on a case-by-case basis. In
Subsection C.3 we provide concrete examples for which all of the conditions
imposed in the paper hold. Remark 2.6 and mainly Remark 2.7 discuss the
meaning of these assumptions more thoroughly. In doing so, it will be useful
to define the following complete collections of multi-indices in the sense of
Definitions 2.1 and 2.2:

(27) $\displaystyle\hat{\bm{\zeta}}$
$\displaystyle\ni\\{(0,j_{1},0),(1,j_{2},j_{3}),(2,j_{4},(j_{5},j_{6})),(3,0,(j_{7},0,0))\mathrel{\mathop{\mathchar
58\relax}}j_{1}\in\\{0,1,...4\\},j_{2}+j_{3}\leq 4,j_{4}+j_{5}+j_{6}\leq
2,j_{7}=0,1\\}$ $\displaystyle\tilde{\bm{\zeta}}$
$\displaystyle\ni\\{(0,j_{1},0),(1,j_{2},j_{3}),(2,0,0)\mathrel{\mathop{\mathchar
58\relax}}j_{1}=0,1,2,j_{2}+j_{3}\leq 1\\}$
$\displaystyle\tilde{\bm{\zeta}}_{1}$
$\displaystyle\ni\\{(j_{1},j_{2},0)\mathrel{\mathop{\mathchar
58\relax}}j_{1}+j_{2}\leq 1\\}$ $\displaystyle\tilde{\bm{\zeta}}_{2}$
$\displaystyle\ni\\{(j,0,0)\mathrel{\mathop{\mathchar 58\relax}}j=0,1\\}$
$\displaystyle\tilde{\bm{\zeta}}_{3}$
$\displaystyle\ni\\{(0,j_{1},0),(1,0,j_{2})\mathrel{\mathop{\mathchar
58\relax}}j_{1}=0,1,2,j_{2}=0,1\\}$ $\displaystyle\bm{\zeta}_{x,l}$
$\displaystyle\ni\\{(0,j,0)\mathrel{\mathop{\mathchar
58\relax}}j=0,1,..,l\\},l\in\mathbb{N}$ $\displaystyle\bar{\bm{\zeta}}$
$\displaystyle\ni\\{(j,0,0)\mathrel{\mathop{\mathchar 58\relax}}j=0,1,2\\}$
$\displaystyle\bar{\bm{\zeta}}_{l}$
$\displaystyle\ni\\{(0,0,0),(1,0,j)\mathrel{\mathop{\mathchar
58\relax}}j=0,1,...,l\\},l\in\mathbb{N}.$

In the following set of assumptions, recall that for
$G\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R})\rightarrow\mathbb{R}$
and a multi-index $(n,l,\beta)$, $\tilde{q}_{G}(n,l,\beta)$ denotes the rate
of polynomial growth in $y$ of the mixed derivative of $G$ corresponding to
$(n,l,\beta)$ as per Equation (15) in Definition 2.4. Recall also the spaces
of functions of measures from Definition 2.4.

  1. A4)

Strong existence and uniqueness holds for the system of SDEs (1) for all
$N\in\mathbb{N}$, for the Slow-Fast McKean-Vlasov SDEs (57), and for the
limiting McKean-Vlasov SDE (25).

  2. A5)

$g$ and $\sigma$ are uniformly bounded, and $c,b$ grow at most linearly in $y$
uniformly in $x\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})$. All
coefficients are jointly continuous in $(x,y,\mathbb{W}_{2}).$

  3. A6)

There exists a unique strong solution
$\Phi\in\tilde{\mathcal{M}}_{p}^{\tilde{\bm{\zeta}}}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$
to Equation (22) with $\tilde{q}_{\Phi}(n,l,\bm{\beta})\leq
1,\forall(n,l,\bm{\beta})\in\tilde{\bm{\zeta}}$, and
$\Phi_{y}\in\tilde{\mathcal{M}}_{p}^{\tilde{\bm{\zeta}}_{2}}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$,
with $\tilde{q}_{\Phi_{y}}(n,l,\bm{\beta})\leq
1,\forall(n,l,\bm{\beta})\in\tilde{\bm{\zeta}_{2}}$. In addition, this can be
strengthened to $\tilde{q}_{\Phi}(0,k,0)\leq 0,k=0,1$ and
$\tilde{q}_{\Phi_{y}}(0,0,0)\leq 0$. (For Proposition 6.1 and Theorem 7.2).

  4. A7)

There exists a unique strong solution
$\chi\in\tilde{\mathcal{M}}_{p}^{\tilde{\bm{\zeta}}}(\mathbb{R}^{2}\times\mathbb{R}^{2}\times\mathcal{P}_{2}(\mathbb{R}))$
to Equation (63) with $\tilde{q}_{\chi}(n,l,\bm{\beta})\leq
1,\forall(n,l,\bm{\beta})\in\tilde{\bm{\zeta}}$, and
$\chi_{y}\in\tilde{\mathcal{M}}_{p}^{\tilde{\bm{\zeta}}_{1}}(\mathbb{R}^{2}\times\mathbb{R}^{2}\times\mathcal{P}_{2}(\mathbb{R}))$,
$\chi_{yy}\in\tilde{\mathcal{M}}_{p}^{(0,0,0)}(\mathbb{R}^{2}\times\mathbb{R}^{2}\times\mathcal{P}_{2}(\mathbb{R}))$
with $\tilde{q}_{\chi_{y}}(n,l,\bm{\beta})\leq
1,\forall(n,l,\bm{\beta})\in\tilde{\bm{\zeta}_{1}}$,
$\tilde{q}_{\chi_{yy}}(0,0,0)\leq 1$. In addition, this can be strengthened to
$\tilde{q}_{\chi}(0,k,0)\leq 0,k=0,1$ and $\tilde{q}_{\chi_{y}}(0,0,0)\leq 0$.
(For Proposition 6.3 and Theorem 7.2).

  5. A8)

For $F=\gamma,D,$ or
$\sigma\psi_{1}+[\tau_{1}\psi_{1}+\tau_{2}\psi_{2}]\Phi_{y}$ for any
$\psi_{1},\psi_{2}\in C^{\infty}_{c}([0,T]\times\mathbb{R}\times\mathbb{R})$,
there exists a unique strong solution
$\Xi\in\tilde{\mathcal{M}}_{p}^{\tilde{\bm{\zeta}}}([0,T]\times\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$
to Equation (64) with each of these choices of $F$,
$\tilde{q}_{\Xi}(n,l,\bm{\beta})\leq
2,\forall(n,l,\bm{\beta})\in\tilde{\bm{\zeta}}$, and
$\Xi_{y}\in\tilde{\mathcal{M}}_{p}^{\tilde{\bm{\zeta}}_{1}}([0,T]\times\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$
with $\tilde{q}_{\Xi_{y}}(n,l,\bm{\beta})\leq
2,\forall(n,l,\bm{\beta})\in\tilde{\bm{\zeta}}_{1}$. Moreover, we assume for
all choices of $F$, this can be strengthened to
$\tilde{q}_{\Xi}(n,l,\bm{\beta})\leq
1,\forall(n,l,\bm{\beta})\in\tilde{\bm{\zeta}}_{1}$ and
$\tilde{q}_{\Xi_{y}}(0,0,0)\leq 1$. (For Propositions 6.4/ 10.1 and Theorem
7.2).

  6. A9)

For $F=\gamma,\sigma+\tau_{1}\Phi_{y},\tau_{2}\Phi_{y},\tau_{1},\tau_{2}$:

$\displaystyle|F(x_{1},y_{1},\mu_{1})-F(x_{2},y_{2},\mu_{2})|\leq
C(|x_{1}-x_{2}|+|y_{1}-y_{2}|+\mathbb{W}_{2}(\mu_{1},\mu_{2})),\forall
x_{1},x_{2},y\in\mathbb{R},\mu_{1},\mu_{2}\in\mathcal{P}_{2}(\mathbb{R}).$

(Lemmas 7.4 and 7.5).

  7. A10)

$\bar{\gamma},\bar{D}^{1/2}\in\mathcal{M}_{b,L}^{\hat{\bm{\zeta}}}(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$.
(For Theorem 7.2).

  8. A11)

Consider the Poisson equation

(28) $\displaystyle
L^{2}_{x,\bar{x},\mu}\tilde{\chi}(x,\bar{x},y,\bar{y},\mu)$
$\displaystyle=-b(x,y,\mu)\Phi(\bar{x},\bar{y},\mu),\quad\int_{\mathbb{R}}\int_{\mathbb{R}}\tilde{\chi}(x,\bar{x},y,\bar{y},\mu)\pi(dy;x,\mu)\pi(d\bar{y},\bar{x},\mu)=0.$

where $L^{2}_{x,\bar{x},\mu}$ is as in Equation (60). Assume there exists a
unique strong solution
$\tilde{\chi}\in\tilde{\mathcal{M}}_{p}^{\tilde{\bm{\zeta}}_{3}}(\mathbb{R}^{2}\times\mathbb{R}^{2}\times\mathcal{P}_{2}(\mathbb{R}))$
and
$\tilde{\chi_{y}}\in\tilde{\mathcal{M}}_{p}^{\bm{\zeta}_{x,1}}(\mathbb{R}^{2}\times\mathbb{R}^{2}\times\mathcal{P}_{2}(\mathbb{R}))$
to Equation (28). (For Theorem 7.2).

  9. A12)

$\tau_{1},\tau_{2},f,\gamma,\sigma+\tau_{1}\Phi_{y},\tau_{2}\Phi_{y}\in\mathcal{M}^{\bar{\bm{\zeta}}}_{\bm{\delta},p}(\mathbb{R}\times\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R})).$
(For Lemmas 7.4 and 7.5).

  10. A13)

For $w$ as in Equation (6) and $\bar{\gamma}$,$\bar{D}$ as in Equation (24),
$\bar{\gamma},\bar{D}\in\mathcal{M}_{b}^{\bm{\zeta}_{x,w+2}}(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))\cap\mathcal{M}_{\bm{\delta},b}^{\bar{\bm{\zeta}}_{w+2}}(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$,
and

$\displaystyle\sup_{x\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}\mathinner{\\!\left\lVert\frac{\delta}{\delta
m}\bar{\gamma}(x,\mu)[\cdot]\right\rVert}_{w+2}+\sup_{x\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}\mathinner{\\!\left\lVert\frac{\delta}{\delta
m}\bar{D}(x,\mu)[\cdot]\right\rVert}_{w+2}<\infty.$

(For Lemmas 7.7, 8.6, 8.7 and Proposition 8.3).

######  Remark 2.6.

There is a current gap in the literature regarding rates of polynomial growth
for derivatives of solutions to Poisson Equations of the form (22), as
outlined in [GS] Remark A.1. Though in Proposition A.2 they state a result
partially amending this issue, the bounds provided likely are not tight. In
particular, under the assumption A2) which we require for moment bounds of the
fast process (and hence slow) process in Section B, their result cannot
provide boundedness of derivatives in $y$ of $\Phi$ from (22), or any of the
other auxiliary Poisson equations which we consider. This in turn also makes
it difficult to gain good rates of polynomial growth for derivatives in the
parameters $x$ and $\mu$. We need strict control of these rates of growth, for
the reasons outlined in Remark 2.7. Stronger bounds are derived in the 1-D
case in Proposition A.4 of [GS], so this makes gaining the necessary control
much easier in the current setting (see the results contained in Subsection
C.1 in the Appendix). Note also the much stricter assumptions imposed when
handling the multi-dimensional cell problem in Lemma C.6 (which is required to
establish sufficient conditions for A7)).

######  Remark 2.7.

Assumptions A1) and A2) are used in tandem for the existence and uniqueness of
the invariant measure $\pi$ from Equation (20). Such an invariant measure
exists under weaker recurrence conditions on $f$ (see, e.g. [PV1] Proposition
1 ), but we use the near-Ornstein–Uhlenbeck structure assumed in (16) and the
form of the retraction to the mean (17) in order to prove certain moment
bounds on the controlled fast process in the Appendix B, and (17) is also used
in order to gain sufficient conditions for the required regularity of the
Poisson Equations in Assumptions A6)\- A13) in Appendix C. In particular, (16)
is inspired by Assumption 4.1 (iii) in [JS] and is needed for Lemma B.2, and
(17) is a standard assumption for control of moments of SDEs over infinite
time horizons and for controlling solutions of related Cauchy problems (see
e.g. [RocknerMcKeanVlasov] Assumption A.1 Equation (2.3)).

The centering condition A3) is standard in the theory of stochastic
homogenization. Assumption A4) is required in order to apply the weak-
convergence approach to large deviations. In particular, it ensures that the
prelimit control representation (51) holds. This is known to hold, for
example, under global Lipschitz assumptions on all the coefficients (see, e.g.
[Wang] Theorem 2.1 and Section 6.1 in [RocknerMcKeanVlasov]), though can also
be proved under much weaker assumptions. These two assumptions, along with
existence and uniqueness of the invariant measure $\pi$ from Equation (20) and
the Poisson Equation $\Phi$ from Equation (22), can be seen as the crucial
hypothesis of this paper. The rest of the assumptions are technical and
essentially used to have sufficient conditions for tightness of the controlled
fluctuations processes $\tilde{Z}^{N}$ from Equation ((54)) (and, in the case
of Assumption A13), to have uniqueness of solutions to its limit (32)).

The boundedness and linear growth of the coefficients from Assumption A5) are
used to restrict the growth of the coefficients so that second moments of the
controlled fast process $\tilde{Y}^{i,\epsilon,N}$ from Equation (55) can be
proved in Appendix B, and to ensure that only knowing these second moment
bounds are sufficient for boundedness of the remainder terms in e.g. the
ergodic-type theorems of Section 6. The joint continuity assumption is used to
ensure that integrating the coefficients is a continuous function on the space
of measures.

The Assumptions A6)\- A13) are listed in terms of the Poisson Equations and
averaged coefficients (and hence implicitly in terms of $\Phi$ from Assumption
A6)) because these assumptions can be verified on a case-by-case basis when
the differential operator (18) or the inhomogeneities considered have some
special structure. See the Examples provided in Appendix C.3.

The growth required by the specific derivatives listed in Assumptions A6) \-
A8) are imposed in order to ensure that the remainder terms resulting form
Itô’s formula in the Ergodic-Type Theorems in Section 6 are bounded. In
particular, in Section 6, we are dealing with the controlled slow-fast system
(55), which due to the controls a priori being at best $L^{2}$ integrable (see
the bound (53)), we are only able to show that we have 2 bounded moments of
the fast component (see Appendix B). This is limiting, since the terms which
show up in the Ergodic-Type Theorems are products of derivatives of the
Poisson equation with the coefficients of the system (1), of which $c$ and $b$
may grow linearly as per assumption A5), and with the $L^{2}$ controls.

Using Assumption A6) as an example and unpacking the multi-index notation, we
are requiring $\Phi,\Phi_{x},\Phi_{y}$ are bounded, and
$\Phi_{xx},\partial_{\mu}\Phi,\partial_{\mu}\Phi_{x},\partial_{\mu}\Phi_{y},\partial_{z}\partial_{\mu}\Phi,\partial^{2}_{\mu}\Phi$
grow at most linearly in $y$ in their appropriate norms. Looking at the proof
of Proposition 6.1, since we are taking the $L^{2}$ norm of the remainder
terms $\tilde{B}_{1}-\tilde{B}_{8}$, we are essentially ensuring all the
products showing up in these terms are $L^{2}$ bounded. In particular, in
$\tilde{B}_{7}$, the controls are multiplied by $\Phi$ and $\Phi_{x}$, which
is why we end up needing those derivatives to be bounded. $\Phi_{y}$ being
bounded is needed elsewhere for essentially the same reason - see, e.g. the
proof of Proposition 10.2, where we use that $B^{N}_{t}$ is bounded in
$L^{2}$. The reasoning behind the Assumptions A7) and A8) are the exact same,
with additional regularity of $\chi_{y}$ and $\Xi_{y}$ (replacing
$\tilde{\bm{\zeta}}_{2}$ by $\tilde{\bm{\zeta}}_{1}$ means we are requiring
$\chi_{y}$ and $\Xi_{y}$ have an $x$ derivative which grows at most linearly
in addition to a $\mu$ derivative) and $\chi_{yy}$ required due to those
additional terms showing up in $\bar{B}_{2}$ in Proposition 6.3, $C_{2}$ in
Proposition 6.4, and $\bar{B}_{13}$ in Proposition 6.3 respectively.

The Lipschitz continuity imposed in Assumption A9) and the existence of two
linear functional derivatives which grow at most polynomially in $y$ uniformly
in $x,\mu$ imposed in Assumption A12) are used to couple the controlled
particles (55) to the auxiliary IID particles (57) in Subsection 7.2. In
particular, the terms required to be Lipschitz are those which show up in the
drift and diffusion of the processes which result from applying Proposition
6.1 to the controlled system and IID system respectively. The use of a
Lipschitz property in such a coupling argument is standard - see, e.g. Lemma 1
in [HM]. Since we don’t assume that the coefficients have linear interaction
with the measure, Assumption A12) is being used to apply Lemma D.7 to the
listed functions. The result of that Lemma is essentially the Assumption (S3)
made in [KX], which we are using in essentially the same manner that they are
in their coupling argument in Theorem 2.4.

Assumption A10) is tailored to ensure enough regularity of the coefficients of
the Cauchy Problem on Wasserstein Space for Theorem 7.2 to hold- see
[BezemekSpiliopoulosAveraging2022] (in particular Lemma 5.1 therein).
Assumption A11) is used to apply the same result, and requires the
introduction of the additional auxiliary Poisson equation A11) which is
defined similarly to $\chi$ from Assumption A7) but with a different
inhomogeneity due to an additional term which arises in
[BezemekSpiliopoulosAveraging2022] Proposition 4.4 due to the McKean-Vlasov
dynamics. The use of this specific control over the derivatives of
$\tilde{\chi}$ is discussed after the statement of Theorem 7.2.

Finally, Assumption A13) is needed for well-definedness/uniqueness of the
limiting Equation (32). See the analogous Assumptions 2.2/2.3 in [BW].

##  3\. Main Results

We are now ready to state our main result, which takes the form of Theorem 3.2
below. These results will be applied to a concrete class of examples of
interacting particle systems of the form (1) in Subsection 4.2.

We prove the large deviations principle for fluctuations process $\\{Z^{N}\\}$
from Equation (3) via means of the Laplace Principle. In other words, in
Theorem 3.2, we identify the rate function $I\mathrel{\mathop{\mathchar
58\relax}}C([0,T];\mathcal{S}_{-r})\rightarrow[0,+\infty]$ such that for $w$
as in Equation (6):

(29)
$\displaystyle\lim_{N\rightarrow\infty}-a^{2}(N)\log\mathbb{E}\exp\biggl{(}-\frac{1}{a^{2}(N)}F(Z^{N})\biggr{)}=\inf_{Z\in
C([0,T];S_{-w})}\\{I(Z)+F(Z)\\}$

for all $F\in C_{b}(C([0,T];\mathcal{S}_{-\tau}))$, for any $\tau\geq w$. In
particular, this holds for all $F\in C_{b}(C([0,T];\mathcal{S}_{-r}))$ for
$r>w+2$ as in Equation (7), and for such $F$ the right hand side is equal to
$\inf_{Z\in C([0,T];S_{-r})}\\{I(Z)+F(Z)\\}$ by construction of $I$ (see
Theorem 3.2). The equality (29) along with compactness of level sets of $I$
implies that $\\{Z^{N}\\}$ satisfies the large deviations principle with speed
$a^{-2}(N)$ and rate function $I$ via e.g. Theorem 1.2.3 in [DE].

In order to show (29), we will show in Section 9 that the Laplace principle
Lower Bound:

(30)
$\displaystyle\liminf_{N\rightarrow\infty}-a^{2}(N)\log\mathbb{E}\exp\biggl{(}-\frac{1}{a^{2}(N)}F(Z^{N})\biggr{)}\geq\inf_{Z\in
C([0,T];S_{-w})}\\{I(Z)+F(Z)\\},\forall F\in
C_{b}(C([0,T];\mathcal{S}_{-\tau}))$

for any $\tau\geq w$, with $w$ as in Equation (6), holds.

Then, in Section 10 we will prove the Laplace principle Upper Bound:

(31)
$\displaystyle\limsup_{N\rightarrow\infty}-a^{2}(N)\log\mathbb{E}\exp\biggl{(}-\frac{1}{a^{2}(N)}F(Z^{N})\biggr{)}\leq\inf_{Z\in
C([0,T];S_{-w})}\\{I(Z)+F(Z)\\},\forall F\in
C_{b}(C([0,T];\mathcal{S}_{-\tau}))$

for any $\tau\geq w$ holds and compactness of level sets of $I$ in
$C([0,T];\mathcal{S}_{-r})$, at which point the moderate deviations principle
of Theorem 3.2 will be established.

We now formulate the rate function. Consider the controlled limiting equation:

(32) $\displaystyle\langle Z_{t},\phi\rangle$
$\displaystyle=\int_{0}^{t}\langle
Z_{s},\bar{L}_{\mathcal{L}(X_{s})}\phi(\cdot)\rangle
ds+\int_{\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{2}\times[0,t]}\sigma(x,y,\mathcal{L}(X_{s}))z_{1}\phi^{\prime}(x)Q(dx,dy,dz,ds)$
$\displaystyle+\int_{\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{2}\times[0,t]}[\tau_{1}(x,y,\mathcal{L}(X_{s}))z_{1}+\tau_{2}(x,y,\mathcal{L}(X_{s}))z_{2}]\Phi_{y}(x,y,\mathcal{L}(X_{s}))\phi^{\prime}(x)Q(dx,dy,dz,ds)$
$\displaystyle\bar{L}_{\nu}\phi(x)$
$\displaystyle\coloneqq\bar{\gamma}(x,\nu)\phi^{\prime}(x)+\bar{D}(x,\nu)\phi^{\prime\prime}(x)+\int_{\mathbb{R}}\left(\frac{\delta}{\delta
m}\bar{\gamma}(z,\nu)[x]\phi^{\prime}(z)+\frac{\delta}{\delta
m}\bar{D}(z,\nu)[x]\phi^{\prime\prime}(z)\right)\nu(dz),\nu\in\mathcal{P}(\mathbb{R}).$

for all $\phi\in C^{\infty}_{c}(\mathbb{R})$. Here we recall the limiting
coefficients $\bar{\gamma},\bar{D}$ from Equation (24), the limiting McKean-
Vlasov Equation $X_{t}$ from Equation (25), and the linear functional
derivative $\frac{\delta}{\delta m}$ from Definition D.4.

######  Theorem 3.1.

Let assumptions A1) \- A13) hold. Then $\\{Z^{N}\\}_{N\in\mathbb{N}}$
satisfies the Laplace principle (29) with rate function $I$ given by

(33) $\displaystyle I(Z)=\inf_{Q\in
P^{*}(Z)}\biggl{\\{}\frac{1}{2}\int_{\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{2}\times[0,T]}\left(z_{1}^{2}+z_{2}^{2}\right)Q(dx,dy,dz,ds)\biggr{\\}}{}$

where $Q\in M_{T}(\mathbb{R}^{4})$ (recall this space from above Equation
(10)) is in $P^{*}(Z)$ if:

  1. ($P^{enumi}$1)

$(Z,Q)$ satisfies Equation (32)

  2. ($P^{enumi}$2)

$\int_{\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{2}\times[0,T]}\left(z_{1}^{2}+z_{2}^{2}\right)Q(dx,dy,dz,ds)<\infty$

  3. ($P^{enumi}$3)

Disintegrating
$Q(dx,dy,dz,ds)=\kappa(dz;x,y,s)\lambda(dy;x,s)Q_{(1,4)}(dx,ds)$,
$\lambda(dy;x,s)=\pi(dy;x,\mathcal{L}(X_{s}))$
$\nu_{\mathcal{L}(X_{\cdot})}$-almost surely, where $\pi$ is as in Equation
(20) and $\nu_{\mathcal{L}(X_{\cdot})}$ is as in Equation (10).

  4. ($P^{enumi}$4)

$Q_{(1,4)}=\nu_{\mathcal{L}(X_{\cdot})}$.

Here we use the convention that $\inf\\{\emptyset\\}=+\infty$.

Replacing assumption A13) by the following:

  1. A’13)

For $r$ as in Equation (7) and $\bar{\gamma}$,$\bar{D}$ as in Equation (24),
$\bar{\gamma},\bar{D}\in\mathcal{M}_{b}^{\bm{\zeta}_{x,r+2}}(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))\cap\mathcal{M}_{\bm{\delta},b}^{\bar{\bm{\zeta}}_{r+2}}(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$
(recalling these spaces from Definition 2.4 and these collections of multi-
indices from Equation (27)), and

$\displaystyle\sup_{x\in\mathbb{R},\mu\in\mathcal{P}(R)}\mathinner{\\!\left\lVert\frac{\delta}{\delta
m}\bar{\gamma}(x,\mu)[\cdot]\right\rVert}_{r+2}+\sup_{x\in\mathbb{R},\mu\in\mathcal{P}(R)}\mathinner{\\!\left\lVert\frac{\delta}{\delta
m}\bar{D}(x,\mu)[\cdot]\right\rVert}_{r+2}<\infty.$

we can in addition prove compactness of level sets of the rate function given
in (33) by extending it to a larger space. For a discussion of the necessity
of this extension, see the comments below Equation (2.10) and below Equation
(4.33) in [BW]. This yields the main result:

######  Theorem 3.2.

Let assumptions A1) \- A12) and A’13) hold. Then
$\\{Z^{N}\\}_{N\in\mathbb{N}}$ from Equation (3) satisfies the large deviation
principle on the space $C([0,T];\mathcal{S}_{-r})$, with $r$ as in Equation
(7), speed $a^{-2}(N)$ and good rate function $I$ given as in Equation (33).
Here we use the convention that $\inf\\{\emptyset\\}=+\infty$, and also impose
that $I(Z)=+\infty$ for $Z\in C([0,T];\mathcal{S}_{-r})\setminus
C([0,T];\mathcal{S}_{-w})$.

As is typically the case when using the weak convergence approach of [DE] to
prove a large deviations principle, the rate function (33) can also be
characterized by controls in feedback form:

######  Corollary 3.3.

In the setting of Theorem 3.1, we can alternatively characterize the rate
function as:

(34) $\displaystyle I^{o}(Z)=\inf_{h\in
P^{o}(Z)}\biggl{\\{}\frac{1}{2}\int_{0}^{T}\mathbb{E}\biggl{[}\int_{\mathbb{R}}|h(s,X_{s},y)|^{2}\pi(dy;X_{s},\mathcal{L}(X_{s}))\biggr{]}ds\biggr{\\}}$

where $h\mathrel{\mathop{\mathchar
58\relax}}[0,T]\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}^{2}$ is
in $P^{o}(Z)$ if:

  1. ($P^{o}$1)

$(Z,h)$ satisfies Equation (35) for all $t\in[0,T]$ and $\phi\in
C_{c}^{\infty}(\mathbb{R})$

  2. ($P^{o}$2)

$\int_{0}^{T}\mathbb{E}\biggl{[}\int_{\mathbb{R}}|h(s,X_{s},y)|^{2}\pi(dy;X_{s},\mathcal{L}(X_{s}))\biggr{]}ds<\infty.$

Here we define:

(35) $\displaystyle\langle Z_{t},\phi\rangle$
$\displaystyle=\int_{0}^{t}\langle
Z_{s},\bar{L}_{\mathcal{L}(X_{s})}\phi(\cdot)\rangle
ds+\int_{0}^{t}\mathbb{E}\biggl{[}\int_{\mathbb{R}}\sigma(X_{s},y,\mathcal{L}(X_{s}))h_{1}(s,X_{s},y)\phi^{\prime}(X_{s})\pi(dy;X_{s},\mathcal{L}(X_{s}))\biggr{]}ds$
$\displaystyle+\int_{0}^{t}\mathbb{E}\left[\int_{\mathbb{R}}[\tau_{1}(X_{s},y,\mathcal{L}(X_{s}))h_{1}(s,X_{s},y)+\tau_{2}(X_{s},y,\mathcal{L}(X_{s}))h_{2}(s,X_{s},y)]\times\right.$
$\displaystyle\hskip
199.16928pt\left.\times\Phi_{y}(X_{s},y,\mathcal{L}(X_{s}))\phi^{\prime}(X_{s})\pi(dy;X_{s},\mathcal{L}(X_{s}))\right.\biggr{]}ds$

Again, we use the convention that $\inf\\{\emptyset\\}=+\infty$. In the
setting of Theorem 3.2, we also impose that $I^{o}(Z)=+\infty$ for $Z\in
C([0,T];\mathcal{S}_{-r})\setminus C([0,T];\mathcal{S}_{-w})$.

###### Proof.

This follows from Jensen’s inequality and the affine dependence of the
coefficients on the controls. The details are omitted for brevity given that
the argument is standard, e.g., see Section 5 in [DS]. ∎

In addition, as a corollary to the proof of Theorem 3.2, we extend the results
from [BW] as follows:

######  Corollary 3.4.

(MDP without Multiscale Structure) Suppose that $b=f=g=\tau_{1}=\tau_{2}\equiv
0$ and $c(x,y,\mu)=c(x,\mu),\sigma(x,y,\mu)=\sigma(x,\mu)$. Let $v>4$ be
sufficiently large that the canonical embedding
$\mathcal{S}_{-4}\rightarrow\mathcal{S}_{-v}$ is Hilbert-Schmidt, $\rho>6$ be
sufficiently large that the canonical embedding
$\mathcal{S}_{-v-2}\rightarrow\mathcal{S}_{-\rho}$ is Hilbert-Schmidt, and
$\bar{\bm{\zeta}}$ as in (27). Assume also that
$\sigma,c\in\mathcal{M}_{\bm{\delta},b}^{\bm{\zeta}}(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$
and for $F(x,\mu)=c(x,\mu)$ or $\sigma(x,\mu)$:

  1. (1)

$\sup_{\mu\in\mathcal{P}_{2}(\mathbb{R})}|F(\cdot,\mu)|_{\rho+2}<\infty$

  2. (2)

$\sup_{x\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})}\mathinner{\\!\left\lVert\frac{\delta}{\delta
m}F(x,\mu)[\cdot]\right\rVert}_{\rho+2}<\infty$.

Here we recall the space $\mathcal{M}_{\bm{\delta},b}$ from Definition 2.4,
the collection of multi-indices $\bm{\zeta}$ from Equation (27), and the norms
on $\mathcal{S}$ defined in Equations (4) and (8). Then
$\\{Z^{N}\\}_{N\in\mathbb{N}}$ satisfies a large deviation principle on the
space $C([0,T];\mathcal{S}_{-\rho})$ with speed $a^{-2}(N)$ and good rate
function $\tilde{I}^{o}$ given by

(36)
$\displaystyle\tilde{I}^{o}(Z)=\inf_{h\in\tilde{P}^{o}(Z)}\biggl{\\{}\frac{1}{2}\int_{0}^{T}\mathbb{E}\biggl{[}|h(s,X_{s})|^{2}\biggr{]}ds\biggr{\\}}$

where $h\mathrel{\mathop{\mathchar
58\relax}}[0,T]\times\mathbb{R}\rightarrow\mathbb{R}$ is in $\tilde{P}^{o}(Z)$
if:

  1. ($P^{o}$1)

$(Z,h)$ satisfies Equation (37) for all $t\in[0,T]$ and $\phi\in
C_{c}^{\infty}(\mathbb{R})$

  2. ($P^{o}$2)

$\int_{0}^{T}\mathbb{E}\biggl{[}|h(s,X_{s})|^{2}\biggr{]}ds<\infty$

and $\inf\\{\emptyset\\}=+\infty$, $I(Z)=+\infty$ for $Z\in
C([0,T];\mathcal{S}_{-\rho})\setminus C([0,T];\mathcal{S}_{-v})$. Here we
define:

(37) $\displaystyle\langle Z_{t},\phi\rangle$
$\displaystyle=\int_{0}^{t}\langle
Z_{s},\tilde{L}_{\mathcal{L}(\tilde{X}_{s})}\phi(\cdot)\rangle
ds+\int_{0}^{t}\mathbb{E}\biggl{[}\sigma(\tilde{X}_{s},\mathcal{L}(\tilde{X}_{s}))h(s,\tilde{X}_{s})\phi^{\prime}(\tilde{X}_{s})\biggr{]}ds$
$\displaystyle\tilde{L}_{\nu}\phi(x)$
$\displaystyle=c(x,\nu)\phi^{\prime}(x)+\frac{\sigma^{2}(x,\nu)}{2}\phi^{\prime\prime}(x)+\int_{\mathbb{R}}\left(\frac{\delta}{\delta
m}c(z,\nu)[x]\phi^{\prime}(z)+\frac{1}{2}\frac{\delta}{\delta
m}[\sigma^{2}(z,\nu)[x]]\phi^{\prime\prime}(z)\right)\nu(dz)$
$\displaystyle\tilde{X}_{t}$
$\displaystyle=\eta^{x}+\int_{0}^{t}c(\tilde{X}_{s},\mathcal{L}(\tilde{X}_{s}))ds+\int_{0}^{t}\sigma(\tilde{X}_{s},\mathcal{L}(\tilde{X}_{s}))dW_{s}.$

###### Proof.

This follows from Theorem 3.2. The assumptions needed are vastly simplified
due to the absence of multiscale structure. In particular, we have no need for
the results from Section 6 and Subsection 7.1. The rate function can be posed
on a smaller space $C([0,T];\mathcal{S}_{-\rho})$ (agreeing with that of
Theorem 2.1 in [BW]) as opposed to the larger $C([0,T];\mathcal{S}_{-r})$ of
our Theorem 3.2 due to the IID system (57) not depending on $\epsilon$ in this
regime. In particular, this means
$\bar{X}^{\epsilon}_{t}\overset{d}{=}\tilde{X}_{t}$ in the proof of Lemma 7.6,
and hence the result is improved $C(T)|\phi|^{2}_{1}$ instead of
$C(T)|\phi|^{2}_{4}$. Similarly, in the result of Lemma 7.7, the bound on
$R^{N}_{t}(\phi)$ can be improved from $\bar{R}(N,T)|\phi|_{4}$ to
$\bar{R}(N,T)|\phi|_{3}$ using Lemma 7.5 and the proof method of Proposition
4.2 in [BW]. At this point tightness of $\\{\tilde{Z}^{N}\\}_{N\in\mathbb{N}}$
from Equation (54) can be proved in Proposition 7.8, but with the uniform
7-continuity of Equation (67) improved to uniform 4-continuity, and hence the
result holds with $w$ replaced by $v$. The remainder of the proofs in the
paper found in Subsection 7.4 and Sections 8, and 9 then go through verbatim
with $m$ and $w$ replaced by $v$ and $r$ replaced with $\rho$, but with the
simplifications assumed on the coefficients allowing us to set many terms
equal to $0$. In particular, in the controlled particle Equation (55), we can
set $\tilde{Y}^{i,\epsilon,N}\equiv 0$, and throughout the invariant measure
$\pi$ from Equation (20) can be set to $\delta_{0}$, which makes dealing with
the second marginals of the occupation measures $\\{Q^{N}\\}_{N\in\mathbb{N}}$
from Equation (56) trivial. Lastly, in Section 10, due to the lack of
multiscale structure, there is no need for an approximation argument in the
proof of Proposition 10.1, and hence existence of solutions to (37) can be
established in $C([0,T];\mathcal{S}_{-v})$ and compactness of level sets
established in $C([0,T];\mathcal{S}_{-\rho})$ exactly as in Subsections 4.4
and 4.5 of [BW]. ∎

######  Remark 3.5.

Note that, in contrast to [BW], which assumes a linear-in-measure form of the
coefficients of Equation (1) (without multiscale structure), i.e. that there
are $\beta,\alpha\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}^{2}\rightarrow\mathbb{R}$ such that
$c(x,\mu)=\int_{\mathbb{R}}\beta(x,z)\mu(dz),\sigma(x,\mu)=\int_{\mathbb{R}}\alpha(x,z)\mu(dz)$,
we do not suppose any particular form of $c(x,\mu)$, $\sigma(x,\mu)$ other
than that they have sufficient regularity for the proof of tightness and
existence/uniqueness of the limiting equation. We are able to do so via the
use of Lemma D.7 (which holds also in the case without dependence of the
function $p$ on $y$) and the assumption that
$\sigma,c\in\mathcal{M}_{\bm{\delta},b}^{\bm{\zeta}}(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$.
For the specific linear form of $c$ and $\sigma$ assumed by [BW],
$\frac{\delta}{\delta m}c(x,\mu)[z]=\beta(x,z)$ and $\frac{\delta}{\delta
m}\sigma(x,\mu)[z]=\alpha(x,z)$, so the condition (2) from Corollary 3.4 in
fact implies
$\sigma,c\in\mathcal{M}_{\bm{\delta},b}^{\bm{\zeta}}(\mathbb{R}\times\mathcal{P}_{2}(\mathbb{R}))$.
In addition, (1) and (2) are exactly the assumptions (a) and (b) from
Condition 2.3 of [BW] in this subcase, so indeed Corollary 3.4 provides a
strict generalization of their result. See also Corollary 4.6 where we further
extend this result to get an alternate form of the rate function analogous to
that of Dawson-Gärtner [DG].

It is also useful to characterize the way that the limiting equations
(32),(35), and (37) act on functions which depend both on time and space.
Hence we make the following remark:

######  Remark 3.6.

We can alternatively characterize the controlled limiting Equation (32) (and
analogously the ordinary controlled limiting Equations (35) and (37)) in terms
of how the $Z$ acts on $\psi\in C^{\infty}_{c}(U\times\mathbb{R})$, where $U$
is an open interval containing $[0,T]$. For Equation (35), this
characterization is:

(38) $\displaystyle\langle Z_{T},\psi(T,\cdot)\rangle=\int_{0}^{T}\langle
Z_{s},\dot{\psi}(s,\cdot)\rangle ds+\int_{0}^{T}\langle
Z_{s},\bar{L}_{\mathcal{L}(X_{s})}\psi(s,\cdot)\rangle ds$
$\displaystyle+\int_{0}^{T}\mathbb{E}\biggl{[}\int_{\mathbb{R}}\sigma(X_{s},y,\mathcal{L}(X_{s}))h_{1}(s,X_{s},y)\psi_{x}(s,X_{s})\pi(dy;X_{s},\mathcal{L}(X_{s}))\biggr{]}ds$
$\displaystyle+\int_{0}^{T}\mathbb{E}\biggl{[}\int_{\mathbb{R}}[\tau_{1}(X_{s},y,\mathcal{L}(X_{s}))h_{1}(s,X_{s},y)+\tau_{2}(X_{s},y,\mathcal{L}(X_{s}))h_{2}(s,X_{s},y)]\Phi_{y}(X_{s},y,\mathcal{L}(X_{s}))\psi_{x}(s,X_{s})\pi(dy;X_{s},\mathcal{L}(X_{s}))\biggr{]}ds$
$\displaystyle Z_{0}=0.$

This is analogous to the form of the limiting equation seen in [Orrieri]
(Remark 2.9) and [BW] (Remark 2.2).

##  4\. On the form of the rate function

###  4.1. Statement and Proof of Equivalent forms of the Rate Function

Here we prove an alternative form of the moderate deviations rate function
(33), which is analogous to the “negative Sobolev” form of the large
deviations rate function for the empirical measure of weakly interacting
diffusions found in Theorem 5.1 of the classical work of Dawson-Gärtner [DG].
This is the first time such a form of the rate function has been provided in
the moderate deviations setting, both with and without multiscale structure.
The result for the specialized case without multiscale structure can be found
as Corollary 4.6 below.

A direct connection between the variational form of the large deviations rate
function from [BDF] and the “negative Sobolev” form of [DG] was recently made
for the first time in [BS] Section 5.2. In contrast to the large deviations
setting, in the moderate deviations rate function (34), we already know the
controls $h$ are in feedback form, but rather than being feedback controls of
the limiting controlled processes $Z$ in Equation (35), they are feedback
controls of the law of large numbers $\mathcal{L}(X)$ from Equation (25).
Moreover, contrast to in the large deviations setting of [BS], here we handle
the dependence of the controls $h$ on the parameter $y$ do to the multiscale
structure and obtaining the “negative Sobolev” form of the rate function
uniformly.

In order to state the alternate form of the rate function we first need to
recall the following definition:

######  Definition 4.1.

(Definition 4.1 in [DG]) For a compact set $K\subset\mathbb{R}$, we will
denote the subspace of $C^{\infty}_{c}(\mathbb{R})$ which have compact support
contained in $K$ by $\mathcal{S}_{K}$. Let $I$ be an interval on the real
line. A map $Z\mathrel{\mathop{\mathchar
58\relax}}I\rightarrow\mathcal{S}^{\prime}$ is called absolutely continuous if
for each compact set $K\subset\mathbb{R}$, there exists a neighborhood of $0$
in $\mathcal{S}_{K}$ and an absolutely continuous function
$H_{K}\mathrel{\mathop{\mathchar 58\relax}}I\rightarrow\mathbb{R}$ such that

$\displaystyle|\langle Z(u),\phi\rangle-\langle
Z(v),\phi\rangle|\leq|H_{K}(u)-H_{K}(v)|$

for all $u,v\in I$ and $\phi\in U_{K}$.

It is also useful to recall the following result:

######  Lemma 4.2.

(Lemma 4.2 in [DG]) Assume the map $Z\mathrel{\mathop{\mathchar
58\relax}}I\rightarrow\mathcal{S}^{\prime}$ is absolutely continuous. Then the
real function $\langle Z,\phi\rangle$ is absolutely continuous for each
$\phi\in C^{\infty}_{c}(\mathbb{R})$ and the derivative in the distribution
sense

$\displaystyle\dot{Z}(t)\coloneqq\lim_{h\downarrow 0}h^{-1}[Z(t+h)-Z(t)]$

exists for Lebesgue almost-every $t\in I$.

Now we are ready to state the equivalent form of the rate function:

######  Proposition 4.3.

Let assumptions A1) \- A12) and A’13) hold. Assume also that
$\bar{D}(x,\mu)>0$ for all $x\in\mathbb{R},\mu\in\mathcal{P}_{2}$, where
$\bar{D}$ is as in Equation (24). Let $r$ be as in Equation (7). Consider
$I^{DG}\mathrel{\mathop{\mathchar
58\relax}}C([0,T];\mathcal{S}_{-r})\rightarrow[0,+\infty]$ given by:

(39) $\displaystyle I^{DG}(Z)$
$\displaystyle=\frac{1}{4}\int_{0}^{T}\sup_{\phi\in
C^{\infty}_{c}(\mathbb{R})\mathrel{\mathop{\mathchar
58\relax}}\mathbb{E}[\bar{D}(X_{t},\mathcal{L}(X_{t}))|\phi^{\prime}(X_{t})|^{2}]\neq
0}\frac{\biggl{|}\langle\dot{Z}_{t}-\bar{L}^{*}_{\mathcal{L}(X_{t})}Z_{t},\phi\rangle\biggr{|}^{2}}{\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))|\phi^{\prime}(X_{t})|^{2}\biggr{]}}dt$

if $Z(0)=0$, $Z$ is absolutely continuous in the sense if Definition 4.1, and
$Z\in C([0,T];\mathcal{S}_{-w})$, and $I^{DG}(Z)=+\infty$ otherwise. Here
$X_{t}$ is as in Equation (25), $\dot{Z}$ is the time derivative of $Z$ in the
distribution sense from Lemma 4.2 and
$\bar{L}^{*}_{\mathcal{L}(X_{s})}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{S}_{-w}\rightarrow\mathcal{S}_{-(w+2)}$ is the adjoint of
$\bar{L}_{\mathcal{L}(X_{s})}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{S}_{w+2}\rightarrow\mathcal{S}_{w}$ given in Equation (32)
(using here Lemma 8.6).

Then $\\{Z^{N}\\}_{N\in\mathbb{N}}$ from Equation (3) satisfies a large
deviation principle on the space $C([0,T];\mathcal{S}_{-r})$ with speed
$a^{-2}(N)$ and good rate function $I^{DG}$.

######  Remark 4.4.

Note that the assumption that $\bar{D}(x,\mu)>0$ for all $x\in\mathbb{R}$ and
$\mu\in\mathcal{P}_{2}(\mathbb{R})$ is not very restrictive. In particular,
via the representation for the density of the invariant measure $\pi$ given in
Equation (76), we know it is strictly positive for all $x,\mu$. Then via the
representation for $\bar{D}(x,\mu)$ given in Equation (26), we have that if
there is $x,\mu$ such that $\bar{D}(x,\mu)=0$, then for that $x,\mu$, we must
have

$\displaystyle[\tau_{2}(x,y,\mu)\Phi_{y}(x,y,\mu)]^{2}+[\sigma(x,y,\mu)+\tau_{1}(x,y,\mu)\Phi_{y}(x,y,\mu)]^{2}=0$

for Lebesgue-almost every $y\in\mathbb{R}$. This will only happen if $\sigma$
has a very specific relation to $f,\tau_{1},\tau_{2},b$ and hence $\Phi_{y}$.

In order to prove Proposition 4.3, we first prove the following Lemma, which
gives us a form of the rate function analogous to Equation (4.21) in [DG]:

######  Lemma 4.5.

Assume the same setup as Proposition 4.3. For $\psi\in
C^{\infty}_{c}(U\times\mathbb{R})$ and $Z\in S_{-w}$, define

(40) $\displaystyle F_{Z}(\psi)$ $\displaystyle=\langle
Z_{T},\psi(T,\cdot)\rangle-\int_{0}^{T}\langle
Z_{s},\dot{\psi}(s,\cdot)\rangle ds-\int_{0}^{T}\langle
Z_{s},\bar{L}_{\mathcal{L}(X_{s})}\psi(s,\cdot)\rangle ds$

and consider $J\mathrel{\mathop{\mathchar
58\relax}}C([0,T];\mathcal{S}_{-\rho})\rightarrow[0,+\infty]$ given by:

(41) $\displaystyle J(Z)$ $\displaystyle=\sup_{\psi\in
C^{\infty}_{c}(U\times\mathbb{R})}\biggl{\\{}F_{Z}(\psi)-\int_{0}^{T}\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))|\psi_{x}(t,X_{t})|^{2}\biggr{]}dt\biggr{\\}}$

if $Z_{0}=0$ and $Z\in C([0,T];\mathcal{S}_{-w})$, and $J(Z)=+\infty$
otherwise. Here $U$ is an open interval containing $[0,T]$. Then
$\\{Z^{N}\\}_{N\in\mathbb{N}}$ satisfies a large deviation principle on the
space $C([0,T];\mathcal{S}_{-r})$ with speed $a^{-2}(N)$ and good rate
function $J$.

###### Proof.

Since by Theorem 1.3.1 in [DE], the rate function for a sequence of random
variables is unique, it suffices to show that $I^{o}=J$, where $I^{o}$ is from
Corollary 3.3. We note that by Remark 3.6, we can replace ($P^{o}$1) in
definition of the multiscale ordinary rate function $I^{o}$ by $Z$ satisfying
Equation (38). We will also use the alternative form of $\bar{D}(x,\mu)$
provided by Equation (26) in Remark 2.5.

First we show $J\leq I^{o}$. Take $Z$ such $I^{o}(Z)<\infty$. Then $P^{o}(Z)$
is non-empty, and for any $h\in P^{o}(Z)$ and, by Equation (38), for any
$\psi\in C^{\infty}_{c}(U\times\mathbb{R})$:

$\displaystyle|F_{Z}(\psi)|=\biggl{|}\int_{0}^{T}\mathbb{E}\biggl{[}\int_{\mathbb{R}}\biggl{(}[\sigma(X_{s},y,\mathcal{L}(X_{s}))+\tau_{1}(X_{s},y,\mathcal{L}(X_{s}))\Phi_{y}(X_{s},y,\mathcal{L}(X_{s}))]h_{1}(s,X_{s},y)$
$\displaystyle+\tau_{2}(X_{s},y,\mathcal{L}(X_{s}))\Phi_{y}(X_{s},y,\mathcal{L}(X_{s}))h_{2}(s,X_{s},y)\biggr{)}\psi_{x}(s,X_{s})\pi(dy;X_{s},\mathcal{L}(X_{s}))\biggr{]}ds\biggr{|}$
$\displaystyle\leq\biggl{(}\int_{0}^{T}\mathbb{E}\biggl{[}\int_{\mathbb{R}}\biggl{(}[\sigma(X_{s},y,\mathcal{L}(X_{s}))+\tau_{1}(X_{s},y,\mathcal{L}(X_{s}))\Phi_{y}(X_{s},y,\mathcal{L}(X_{s}))]^{2}$
$\displaystyle\hskip
56.9055pt+[\tau_{2}(X_{s},y,\mathcal{L}(X_{s}))\Phi_{y}(X_{s},y,\mathcal{L}(X_{s}))]^{2}\biggr{)}\pi(dy;X_{s},\mathcal{L}(X_{s}))|\psi_{x}(s,X_{s})|^{2}\biggr{]}ds\biggr{)}^{1/2}$
$\displaystyle\times\biggl{(}\int_{0}^{T}\mathbb{E}\biggl{[}\int_{\mathbb{R}}|h_{1}(s,X_{s},y)|^{2}+|h_{2}(s,X_{s},y)|^{2}\pi(dy;X_{s},\mathcal{L}(X_{s}))\biggr{]}ds\biggr{)}^{1/2}$
$\displaystyle=\sqrt{2}\biggl{(}\int_{0}^{T}\mathbb{E}\biggl{[}\bar{D}(X_{s},\mathcal{L}(X_{s}))|\psi_{x}(s,X_{s})|^{2}\biggr{]}ds\biggr{)}^{1/2}\biggl{(}\int_{0}^{T}\mathbb{E}\biggl{[}\int_{\mathbb{R}}|h(s,X_{s},y)|^{2}\pi(dy;X_{s},\mathcal{L}(X_{s}))\biggr{]}ds\biggr{)}^{1/2}$

so in particular, if
$\int_{0}^{T}\mathbb{E}\biggl{[}\bar{D}(X_{s},\mathcal{L}(X_{s}))|\psi_{x}(s,X_{s})|^{2}\biggr{]}ds=0$,
then $F_{Z}(\psi)=0$. Then, observing that $\psi\in
C^{\infty}_{c}(U\times\mathbb{R})$ if and only if for any
$c\in\mathbb{R}\setminus\\{0\\}$, $c\psi\in C^{\infty}_{c}(U\times\mathbb{R})$
and that $F_{Z}$ is linear, we have:

$\displaystyle J(Z)$ $\displaystyle=\sup_{\psi\in
C^{\infty}_{c}(U\times\mathbb{R})\mathrel{\mathop{\mathchar
58\relax}}\int_{0}^{T}\mathbb{E}\biggl{[}\bar{D}(X_{s},\mathcal{L}(X_{s}))|\psi_{x}(s,X_{s})|^{2}\biggr{]}ds\neq
0}\biggl{\\{}F_{Z}(\psi)-\int_{0}^{T}\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))|\psi_{x}(t,X_{t})|^{2}\biggr{]}dt\biggr{\\}}\vee
0$ $\displaystyle=\sup_{\psi\in
C^{\infty}_{c}(U\times\mathbb{R})\mathrel{\mathop{\mathchar
58\relax}}\int_{0}^{T}\mathbb{E}\biggl{[}\bar{D}(X_{s},\mathcal{L}(X_{s}))|\psi_{x}(s,X_{s})|^{2}\biggr{]}ds\neq
0}\sup_{c\in\mathbb{R}}\biggl{\\{}cF_{Z}(\psi)-c^{2}\int_{0}^{T}\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))|\psi_{x}(t,X_{t})|^{2}\biggr{]}dt\biggr{\\}}\vee
0$ $\displaystyle=\sup_{\psi\in
C^{\infty}_{c}(U\times\mathbb{R})\mathrel{\mathop{\mathchar
58\relax}}\int_{0}^{T}\mathbb{E}\biggl{[}\bar{D}(X_{s},\mathcal{L}(X_{s}))|\psi_{x}(s,X_{s})|^{2}\biggr{]}ds\neq
0}\frac{|F_{Z}(\psi)|^{2}}{4\int_{0}^{T}\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))|\psi_{x}(t,X_{t})|^{2}\biggr{]}dt}.$

Returning to the above inequality and squaring both sides, we have

$\displaystyle\frac{|F_{Z}(\psi)|^{2}}{2\int_{0}^{T}\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))|\psi_{x}(t,X_{t})|^{2}\biggr{]}dt}\leq\int_{0}^{T}\mathbb{E}\biggl{[}\int_{\mathbb{R}}|h(s,X_{s},y)|^{2}\pi(dy;X_{s},\mathcal{L}(X_{s}))\biggr{]}ds,$

for all $\psi\in C^{\infty}_{c}(U\times\mathbb{R})$ such that
$\int_{0}^{T}\mathbb{E}\biggl{[}\bar{D}(X_{s},\mathcal{L}(X_{s}))|\psi_{x}(s,X_{s})|^{2}\biggr{]}ds\neq
0$ and all $h\in P^{o}(Z)$. So $J(Z)\leq I^{o}(Z)$.

Now we prove $J\geq I^{o}$. Assume without loss of generality that $J(Z)\leq
C<\infty$. Then, since

$\displaystyle J(Z)$ $\displaystyle=\sup_{\psi\in
C^{\infty}_{c}(U\times\mathbb{R})}\sup_{c\in\mathbb{R}}\biggl{\\{}cF_{Z}(\psi)-c^{2}\int_{0}^{T}\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))|\psi_{x}(t,X_{t})|^{2}\biggr{]}dt\biggr{\\}}=+\infty$

if there exists $\psi\in C^{\infty}_{c}(U\times\mathbb{R})$ such that
$F_{Z}(\psi)\neq 0$ and
$\int_{0}^{T}\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))|\psi_{x}(t,X_{t})|^{2}\biggr{]}dt=0$,
we have

$\displaystyle J(Z)=\sup_{\psi\in
C^{\infty}_{c}(U\times\mathbb{R})\mathrel{\mathop{\mathchar
58\relax}}\int_{0}^{T}\mathbb{E}\biggl{[}\bar{D}(X_{s},\mathcal{L}(X_{s}))|\psi_{x}(s,X_{s})|^{2}\biggr{]}ds\neq
0}\frac{|F_{Z}(\psi)|^{2}}{4\int_{0}^{T}\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))|\psi_{x}(t,X_{t})|^{2}\biggr{]}dt}.$

This shows that for all $\psi\in C^{\infty}_{c}(U\times\mathbb{R})$

(42) $\displaystyle\biggl{|}F_{Z}(\psi)\biggr{|}$ $\displaystyle\leq
2\sqrt{C}\biggl{(}\int_{0}^{T}\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))|\psi_{x}(t,X_{t})|^{2}\biggr{]}dt\biggr{)}^{1/2}.$

Now we borrow some notation from [DG] (see p. 270-271). We let for $t\in[0,T]$
$\nabla_{t},(\cdot,\cdot)_{t},$ and $|\cdot|_{t}$ be (formally) the Riemannian
gradient, inner product, and Riemannian norm in the tangent space of the
Riemannian structure on $\mathbb{R}$ induced by the diffusion matrix
$t\mapsto\bar{D}(\cdot,\mathcal{L}(X_{t}))$, i.e.:

$\displaystyle\nabla_{t}f$
$\displaystyle\coloneqq\bar{D}(\cdot,\mathcal{L}(X_{t}))\frac{df}{dx},\qquad(X,Y)_{t}\coloneqq\bar{D}(\cdot,\mathcal{L}(X_{t}))^{-1}XY,\qquad|X|_{t}\coloneqq(X,X)^{1/2}_{t}.$

In particular, note that

$\displaystyle|\nabla_{t}f|^{2}_{t}$
$\displaystyle=\bar{D}(\cdot,\mathcal{L}(X_{t}))|\frac{df}{dx}|^{2}.$

Now, as on p.279 in [DG], we define $L^{2}[0,T]$ to be the Hilbert space of
measurable maps $g\mathrel{\mathop{\mathchar
58\relax}}[0,T]\times\mathbb{R}\rightarrow\mathbb{R}$ with finite norm

$\displaystyle\mathinner{\\!\left\lVert
g\right\rVert}\coloneqq\biggl{(}\int_{0}^{T}\langle\mathcal{L}(X_{t}),|g(t,\cdot)|^{2}_{t}\rangle
dt\biggr{)}^{1/2}=\biggl{(}\int_{0}^{T}\mathbb{E}[\bar{D}(X_{t},\mathcal{L}(X_{t}))^{-1}|g(t,X_{t})|^{2}]dt\biggr{)}^{1/2}$

and inner product

$\displaystyle[g_{1},g_{2}]$
$\displaystyle\coloneqq\int_{0}^{T}\langle\mathcal{L}(X_{t}),(g_{1}(t,\cdot),g_{2}(t,\cdot))_{t}\rangle
dt=\int_{0}^{T}\mathbb{E}[\bar{D}(X_{t},\mathcal{L}(X_{t}))^{-1}g_{1}(t,X_{t})g_{2}(t,X_{t})]dt.$

Denote by $L^{2}_{\nabla}[0,T]$ the closure in $L^{2}[0,T]$ of the linear
subset $L$ consisting of all maps $(s,x)\mapsto\nabla_{s}\psi(s,x)$, $\psi\in
C^{\infty}_{c}(U\times\mathbb{R})$. Then $F_{Z}$ can be viewed as a linear
functional on $L$, and by the bound (42), is bounded. Then, by the Riesz
Representation Theorem, there exists $\bar{h}\in L^{2}_{\nabla}[0,T]$ such
that

(43) $\displaystyle
F_{Z}(\psi)=\int_{0}^{T}\langle\mathcal{L}(X_{s}),(\bar{h}(s,\cdot),\nabla_{s}\psi(s,\cdot))_{s}\rangle
ds=\int_{0}^{T}\mathbb{E}[h(s,X_{s})\psi_{x}(s,X_{s})]ds,\psi\in
C^{\infty}_{c}(U\times\mathbb{R}).$

Note that actually, $L^{2}_{\nabla}$ must be considered not as a class of
functions, but as a set of equivalence classes of functions agreeing
$\nu_{\mathcal{L}(X_{\cdot})}$-almost surely. This is of no consequence,
however, since the bound (42) ensures that $F_{Z}(\psi)=F_{Z}(\tilde{\psi})$
if $\psi_{x}$ and $\tilde{\psi}_{x}$ are in the same equivalence class (see
p.279 in [DG] and Appendix D.5 in [FK] for a more thorough treatment of the
space $L^{2}_{\nabla}[0,T]$ and its dual).

Consider $\tilde{h}(s,x,y)\mathrel{\mathop{\mathchar
58\relax}}[0,T]\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}^{2}$
given by

(44) $\displaystyle\tilde{h}_{1}(s,x,y)$
$\displaystyle=\frac{1}{2\bar{D}(x,\mathcal{L}(X_{s}))}[\sigma(x,y,\mathcal{L}(X_{s}))+\tau_{1}(x,y,\mathcal{L}(X_{s}))\Phi_{y}(x,y,\mathcal{L}(X_{s}))]\bar{h}(s,x)$
$\displaystyle\tilde{h}_{2}(s,x,y)$
$\displaystyle=\frac{1}{2\bar{D}(x,\mathcal{L}(X_{s}))}\tau_{2}(x,y,\mathcal{L}(X_{s}))\Phi_{y}(x,y,\mathcal{L}(X_{s}))\bar{h}(s,x).$

We have:

(45)
$\displaystyle\int_{0}^{T}\mathbb{E}\biggl{[}\int_{\mathbb{R}}|\tilde{h}(s,X_{s},y)|^{2}\pi(dy;X_{s},\mathcal{L}(X_{s}))\biggr{]}ds$
$\displaystyle=\int_{0}^{T}\mathbb{E}\biggl{[}\frac{|\bar{h}(s,X_{s})|^{2}}{4|\bar{D}(x,\mathcal{L}(X_{s}))|^{2}}\int_{\mathbb{R}}\biggl{(}[\sigma(X_{s},y,\mathcal{L}(X_{s}))+\tau_{1}(X_{s},y,\mathcal{L}(X_{s}))\Phi_{y}(X_{s},y,\mathcal{L}(X_{s}))]^{2}$
$\displaystyle\hskip
113.81102pt+[\tau_{2}(X_{s},y,\mathcal{L}(X_{s}))\Phi_{y}(X_{s},y,\mathcal{L}(X_{s}))]^{2}\biggr{)}\pi(dy;X_{s},\mathcal{L}(X_{s}))\biggr{]}ds$
$\displaystyle=\frac{1}{2}\int_{0}^{T}\mathbb{E}\biggl{[}\bar{D}(x,\mathcal{L}(X_{s}))^{-1}|\bar{h}(s,X_{s})|^{2}\biggr{]}ds<\infty.$

Moreover, for $\psi\in C^{\infty}_{c}(U\times\mathbb{R})$:

$\displaystyle\int_{0}^{T}\mathbb{E}\biggl{[}\int_{\mathbb{R}}\biggl{(}[\sigma(X_{s},y,\mathcal{L}(X_{s}))+\tau_{1}(X_{s},y,\mathcal{L}(X_{s}))\Phi_{y}(X_{s},y,\mathcal{L}(X_{s}))]\tilde{h}_{1}(s,X_{s},y)$
$\displaystyle+\tau_{2}(X_{s},y,\mathcal{L}(X_{s}))\Phi_{y}(X_{s},y,\mathcal{L}(X_{s}))\tilde{h}_{2}(s,X_{s},y)\biggr{)}\psi_{x}(s,X_{s})\pi(dy;X_{s},\mathcal{L}(X_{s}))\biggr{]}ds$
$\displaystyle=\int_{0}^{T}\mathbb{E}\biggl{[}\int_{\mathbb{R}}\biggl{(}[\sigma(X_{s},y,\mathcal{L}(X_{s}))+\tau_{1}(X_{s},y,\mathcal{L}(X_{s}))\Phi_{y}(X_{s},y,\mathcal{L}(X_{s}))]^{2}$
$\displaystyle+[\tau_{2}(X_{s},y,\mathcal{L}(X_{s}))\Phi_{y}(X_{s},y,\mathcal{L}(X_{s}))]^{2}\biggr{)}\pi(dy;X_{s},\mathcal{L}(X_{s}))\frac{\bar{h}(s,X_{s})}{2\bar{D}(X_{s},\mathcal{L}(X_{s})}\psi_{x}(s,X_{s})\biggr{]}ds$
$\displaystyle=\int_{0}^{T}\mathbb{E}\biggl{[}\bar{h}(s,X_{s})\psi_{x}(s,X_{s})\biggr{]}ds$
$\displaystyle=F_{Z}(\psi)\text{ by Equation }\eqref{eq:FZreiszrep}.$

Thus, $\tilde{h}\in P^{o}(Z)$ by definition. Take a sequence
$\\{\tilde{\psi}^{n}\\}\subset L$ such that
$\tilde{\psi}^{n}\rightarrow\bar{h}$ in $L^{2}[0,T]$. By virtue of
$\tilde{\psi}^{n}\in L$, we have for each $n$, there is $\psi^{n}\in
C^{\infty}_{c}(U\times\mathbb{R})$ such that
$\tilde{\psi}^{n}(s,x)=\nabla_{s}\psi^{n}(s,x)=\bar{D}(x,\mathcal{L}(X_{s}))\psi^{n}_{x}(s,x)$.

In particular, we have
$|\tilde{\psi}^{n}|_{t}^{2}\rightarrow|\bar{h}|_{t}^{2}$, so

$\displaystyle\int_{0}^{T}\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))|\psi^{n}_{x}(t,X_{t})|^{2}\biggr{]}dt\rightarrow\int_{0}^{T}\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))^{-1}|\bar{h}(t,X_{t})|^{2}\biggr{]}dt$

and $(\tilde{\psi}^{n},\bar{h})_{t}\rightarrow|\bar{h}|_{t}^{2}$, so

$\displaystyle\int_{0}^{T}\mathbb{E}\biggl{[}\psi^{n}_{x}(t,X_{t})h(t,X_{t})\biggr{]}dt\rightarrow\int_{0}^{T}\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))^{-1}|\bar{h}(t,X_{t})|^{2}\biggr{]}dt.$

Note that if
$\int_{0}^{T}\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))^{-1}|\bar{h}(t,X_{t})|^{2}\biggr{]}dt=0$,
we have by Equation (45), that the relation holds
$\int_{0}^{T}\mathbb{E}\biggl{[}\int_{\mathbb{R}}|\tilde{h}(s,X_{s},y)|^{2}\pi(dy)\biggr{]}ds=0$,
and hence $I^{o}(Z)=0$, so the desired bound is trivial.

Assuming then that
$\int_{0}^{T}\mathbb{E}\biggl{[}D(X_{t},\mathcal{L}(X_{t}))^{-1}|\bar{h}(t,X_{t})|^{2}\biggr{]}dt\neq
0$, we may choose a subsequence of $\\{\psi^{n}_{x}\\}$ such that
$\int_{0}^{T}\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))|\psi^{n}_{x}(t,X_{t})|^{2}\biggr{]}dt\neq
0,\forall n$. Then:

$\displaystyle J(Z)$
$\displaystyle\geq\frac{1}{4}\frac{\biggl{(}\int_{0}^{T}\mathbb{E}\biggl{[}\psi^{n}_{x}(s,X_{s})\bar{h}(s,X_{s})\biggr{]}ds\biggr{)}^{2}}{{\int_{0}^{T}\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))|\psi^{n}_{x}(t,X_{t})|^{2}\biggr{]}dt}}\text{
for all }n\in\mathbb{N}$
$\displaystyle\rightarrow\frac{1}{4}\int_{0}^{T}\mathbb{E}\biggl{[}D(X_{t},\mathcal{L}(X_{t}))^{-1}|\bar{h}(t,X_{t})|^{2}\biggr{]}dt\text{
as }n\rightarrow\infty.$

By Equation (45),

$\displaystyle\frac{1}{4}\int_{0}^{T}\mathbb{E}\biggl{[}D(X_{t},\mathcal{L}(X_{t}))^{-1}|\bar{h}(t,X_{t})|^{2}\biggr{]}dt=\frac{1}{2}\int_{0}^{T}\mathbb{E}\biggl{[}\int_{\mathbb{R}}|\tilde{h}(s,X_{s},y)|^{2}\pi(dy;X_{s},\mathcal{L}(X_{s}))\biggl{]}ds,$

so since $\tilde{h}\in P^{o}(Z)$:

$\displaystyle J(Z)$
$\displaystyle\geq\frac{1}{2}\int_{0}^{T}\mathbb{E}\biggl{[}\int_{\mathbb{R}}|\tilde{h}(s,X_{s},y)|^{2}\pi(dy;X_{s},\mathcal{L}(X_{s}))\biggl{]}ds\geq
I^{o}(Z).$

∎

Now we are ready to prove Proposition 4.3.

###### Proof of Proposition 4.3.

As noted, the form of the rate function proved in Lemma 4.5 is analogous to
that of Equation (4.21) in [DG]. We follow the proof of Lemma 4.8 in [DG],
making changes to account for the multiscale structure and the entry of
$\mathcal{L}(X_{s})$ rather than $Z_{s}$ in the subtracted term in Equation
(4.24), which comes the fact that we are looking at moderate deviations rather
than large deviations. We also use the specific information about the optimal
control from the proof of Lemma 4.5.

Once again, it is sufficient to show $I^{o}=I^{DG}$, or equivalently,
$J=I^{DG}$. First we show that $I^{o}=J\leq I^{DG}$. Let $Z\in
C([0,T];\mathcal{S}_{-w})$ be such that $I^{DG}(Z)<\infty$. Note that

$\displaystyle\sup_{\phi\in
C^{\infty}_{c}(\mathbb{R})\mathrel{\mathop{\mathchar
58\relax}}\mathbb{E}[\bar{D}(X_{t},\mathcal{L}(X_{t}))|\phi^{\prime}(X_{t})|^{2}]\neq
0}\biggl{\\{}\langle\dot{Z}_{t}-\bar{L}^{*}_{\mathcal{L}(X_{t})}Z_{t},\phi\rangle-\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))|\phi^{\prime}(X_{t})|^{2}\biggr{]}\biggr{\\}}$
$\displaystyle=\sup_{\phi\in
C^{\infty}_{c}(\mathbb{R})\mathrel{\mathop{\mathchar
58\relax}}\mathbb{E}[\bar{D}(X_{t},\mathcal{L}(X_{t}))|\phi^{\prime}(X_{t})|^{2}]\neq
0}\sup_{c\in\mathbb{R}}\biggl{\\{}c\langle\dot{Z}_{t}-\bar{L}^{*}_{\mathcal{L}(X_{t})}Z_{t},\phi\rangle-c^{2}\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))|\phi^{\prime}(X_{t})|^{2}\biggr{]}\biggr{\\}}$
$\displaystyle=\frac{1}{4}\sup_{\phi\in
C^{\infty}_{c}(\mathbb{R})\mathrel{\mathop{\mathchar
58\relax}}\mathbb{E}[\bar{D}(X_{t},\mathcal{L}(X_{t}))|\phi^{\prime}(X_{t})|^{2}]\neq
0}\frac{\biggl{|}\langle\dot{Z}_{t}-\bar{L}^{*}_{\mathcal{L}(X_{t})}Z_{t},\phi\rangle\biggr{|}^{2}}{\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))|\phi^{\prime}(X_{t})|^{2}\biggr{]}}$

for all $t\in[0,T]$. So for any $\psi\in C^{\infty}_{c}(U\times\mathbb{R})$:

$\displaystyle I^{DG}(Z)$ $\displaystyle=\frac{1}{4}\int_{0}^{T}\sup_{\phi\in
C^{\infty}_{c}(\mathbb{R})\mathrel{\mathop{\mathchar
58\relax}}\mathbb{E}[\bar{D}(X_{t},\mathcal{L}(X_{t}))|\phi^{\prime}(X_{t})|^{2}]\neq
0}\frac{\biggl{|}\langle\dot{Z}_{t}-\bar{L}^{*}_{\mathcal{L}(X_{t})}Z_{t},\phi\rangle\biggr{|}^{2}}{\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))|\phi^{\prime}(X_{t})|^{2}\biggr{]}}dt$
$\displaystyle=\int_{0}^{T}\sup_{\phi\in
C^{\infty}_{c}(\mathbb{R})\mathrel{\mathop{\mathchar
58\relax}}\mathbb{E}[\bar{D}(X_{t},\mathcal{L}(X_{t}))|\phi^{\prime}(X_{t})|^{2}]\neq
0}\biggl{\\{}\langle\dot{Z}_{t}-\bar{L}^{*}_{\mathcal{L}(X_{t})}Z_{t},\phi\rangle-\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))|\phi^{\prime}(X_{t})|^{2}\biggr{]}\biggr{\\}}dt$
$\displaystyle\geq\int_{0}^{T}\langle\dot{Z}_{t}-\bar{L}^{*}_{\mathcal{L}(X_{t})}Z_{t},\psi(t,\cdot)\rangle-\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))|\psi_{x}(t,X_{t})|^{2}\biggr{]}dt$
$\displaystyle=\langle Z_{T},\psi(T,\cdot)\rangle-\int_{0}^{T}\langle
Z_{t},\dot{\psi}(t,\cdot)\rangle dt-\int_{0}^{T}\langle
Z_{t},\bar{L}_{\mathcal{L}(X_{t})}\psi(t,\cdot)\rangle
dt-\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))|\psi_{x}(t,X_{t})|^{2}\biggr{]}dt,$

where in the last step we used Lemma 4.3 in [DG]. Then taking the supremum
over all $\psi\in C^{\infty}_{c}(U\times\mathbb{R})$, we get $I^{DG}(Z)\geq
J(Z)$, as desired.

Now we show that $I^{DG}\leq J=I^{o}$. Consider $Z\in
C([0,T];\mathcal{S}_{-w})$ such that $J(Z)<\infty$.

In Lemma 4.5, we proved for $\tilde{h}$ as in Equation (44), $\tilde{h}\in
P^{o}(Z)$. We also showed:

$\displaystyle\frac{1}{2}\int_{0}^{T}\mathbb{E}\biggl{[}\int_{\mathbb{R}}|\tilde{h}(s,X_{s},y)|^{2}\pi(dy)\biggr{]}ds\leq
J(Z)=I^{o}(Z)\leq\frac{1}{2}\int_{0}^{T}\mathbb{E}\biggl{[}\int_{\mathbb{R}}|h(s,X_{s},y)|^{2}\pi(dy)\biggr{]}ds,\forall
h\in P^{o}(Z),$

so that in fact

(46) $\displaystyle
J(Z)=I^{o}(Z)=\frac{1}{2}\int_{0}^{T}\mathbb{E}\biggl{[}\int_{\mathbb{R}}|\tilde{h}(s,X_{s},y)|^{2}\pi(dy)\biggr{]}ds=\frac{1}{4}\int_{0}^{T}\mathbb{E}\biggl{[}\bar{D}(X_{s},\mathcal{L}(X_{s}))^{-1}|\bar{h}(s,X_{s})|^{2}\biggr{]}ds,$

where in the last inequality we used Equation (45).

Now, by the fact that $\tilde{h}\in P^{o}(Z)$, we have by Equation (35) that
for all $0\leq s\leq t\leq T$ and $\phi\in C^{\infty}_{c}(\mathbb{R})$:

$\displaystyle\langle Z_{t},\phi\rangle-\langle Z_{s},\phi\rangle$
$\displaystyle=\int_{s}^{t}\langle
Z_{u},\bar{L}_{\mathcal{L}(X_{u})}\phi(\cdot)\rangle
du+\int_{s}^{t}\mathbb{E}\biggl{[}\int_{\mathbb{R}}\biggl{(}[\sigma(X_{u},y,\mathcal{L}(X_{u}))+\tau_{1}(X_{u},y,\mathcal{L}(X_{u}))\Phi_{y}(X_{u},y,\mathcal{L}(X_{u}))]\tilde{h}_{1}(u,X_{u},y)$
$\displaystyle+\tau_{2}(X_{u},y,\mathcal{L}(X_{u}))\Phi_{y}(X_{u},y,\mathcal{L}(X_{u}))\tilde{h}_{2}(u,X_{u},y)\biggr{)}\pi(dy;X_{u},\mathcal{L}(X_{u}))\phi^{\prime}(X_{u})\biggr{]}du$
$\displaystyle=\int_{s}^{t}\langle
Z_{u},\bar{L}_{\mathcal{L}(X_{u})}\phi(\cdot)\rangle
du+\int_{s}^{t}\mathbb{E}\biggl{[}\bar{h}(u,X_{u})\phi^{\prime}(X_{u})\biggr{]}du$

where $\bar{h}$ is as in Equation (43), so by Definition 4.1 and Lemma 8.6,
$Z$ is an absolutely continuous map from $[0,T]$ to $\mathcal{S}^{\prime}$.
Then, using Lemma 4.2, we have for each $\phi\in C^{\infty}_{c}(\mathbb{R})$:

(47) $\displaystyle\langle\dot{Z}_{t},\phi\rangle$
$\displaystyle=\mathbb{E}\biggl{[}\bar{h}(t,X_{t})\phi^{\prime}(X_{t})\biggr{]}+\langle
Z_{t},\bar{L}_{\mathcal{L}(X_{t})}\phi(\cdot)\rangle.$

Using a density argument, we can make sure this holds simultaneously for all
$\phi\in C^{\infty}_{c}(\mathbb{R})$ and Lebesgue almost every $t\in[0,T]$
(see p.280 of [DG]). This gives:

$\displaystyle I^{DG}(Z)$ $\displaystyle=\frac{1}{4}\int_{0}^{T}\sup_{\phi\in
C^{\infty}_{c}(\mathbb{R})\mathrel{\mathop{\mathchar
58\relax}}\mathbb{E}[\bar{D}(X_{t},\mathcal{L}(X_{t}))|\phi^{\prime}(X_{t})|^{2}]\neq
0}\frac{\biggl{(}\mathbb{E}\biggl{[}\bar{h}(t,X_{t})\phi^{\prime}(X_{t})\biggr{]}\biggr{)}^{2}}{\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))|\phi^{\prime}(X_{t})|^{2}\biggr{]}}dt$

For any $\phi\in C^{\infty}_{c}(\mathbb{R})$ and $t\in[0,T]$ such that
$\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))|\phi^{\prime}(X_{t})|^{2}\biggr{]}\neq
0$, we have

$\displaystyle\frac{\biggl{(}\mathbb{E}\biggl{[}\bar{h}(t,X_{t})\phi^{\prime}(X_{t})\biggr{]}\biggr{)}^{2}}{\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))|\phi^{\prime}(X_{t})|^{2}\biggr{]}}$
$\displaystyle=\frac{\biggl{(}\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))^{-1/2}\bar{h}(t,X_{t})\bar{D}(X_{t},\mathcal{L}(X_{t}))^{1/2}\phi^{\prime}(X_{t})\biggr{]}\biggr{)}^{2}}{\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))|\phi^{\prime}(X_{t})|^{2}\biggr{]}}$
$\displaystyle\leq\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))^{-1}|\bar{h}(t,X_{t})|^{2}\biggr{]}$

so

$\displaystyle I^{DG}(Z)$
$\displaystyle\leq\frac{1}{4}\int_{0}^{T}\mathbb{E}\biggl{[}\bar{D}(X_{t},\mathcal{L}(X_{t}))^{-1}|\bar{h}(t,X_{t})|^{2}\biggr{]}dt,$

and by Equation (46) we are done. ∎

As a corollary to the above result, we also get an alternative form of the
rate function in the setting without multiscale structure. This provides us
with rate functions with which it is more feasible to compare the likelihood
of rare events for the fluctuation process (3) as $N\rightarrow\infty$ in the
multiscale and non-multiscale setting as opposed to the variational form given
in Theorem 3.2 and Corollary 3.4. This analysis is outside the scope of this
paper, but is an interesting avenue for future research.

######  Corollary 4.6.

In the setting of Corollary 3.4, assume in addition $\sigma^{2}(x,\mu)>0$, for
all $x\in\mathbb{R}$ and $\mu\in\mathcal{P}_{2}(\mathbb{R})$. Consider
$\tilde{I}^{DG}\mathrel{\mathop{\mathchar
58\relax}}C([0,T];\mathcal{S}_{-\rho})\rightarrow[0,+\infty]$ given by :

(48)
$\displaystyle\tilde{I}^{DG}(Z)\coloneqq\frac{1}{2}\int_{0}^{T}\sup_{\phi\in
C^{\infty}_{c}(\mathbb{R})\mathrel{\mathop{\mathchar
58\relax}}\mathbb{E}[\sigma^{2}(X_{t},\mathcal{L}(X_{t}))|\phi^{\prime}(\tilde{X}_{t})|^{2}]\neq
0}\frac{|\langle\dot{Z}_{t}-\tilde{L}^{*}_{\mathcal{L}(\tilde{X}_{t})}Z_{t},\phi\rangle|^{2}}{\mathbb{E}[\sigma^{2}(X_{t},\mathcal{L}(X_{t}))|\phi^{\prime}(\tilde{X}_{t})|^{2}]}dt,$

if $Z(0)=0$, $Z$ is absolutely continuous in the sense if Definition 4.1, and
$Z\in C([0,T];\mathcal{S}_{-v})$, and $I^{DG}(Z)=+\infty$ otherwise. Here
$\tilde{X}_{t}$ is as in Corollary 3.4, $\dot{Z}$ is the time derivative of
$Z$ in the distribution sense from Lemma 4.2 and
$\tilde{L}^{*}_{\mathcal{L}(\tilde{X}_{s})}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{S}_{-v}\rightarrow\mathcal{S}_{-(v+2)}$ is the adjoint of
$\tilde{L}_{\mathcal{L}(\tilde{X}_{s})}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{S}_{v+2}\rightarrow\mathcal{S}_{v}$ given in Corollary 3.4
(using here Lemma 8.6).

Then $\\{Z^{N}\\}_{N\in\mathbb{N}}$ satisfies a large deviation principle on
the space $C([0,T];\mathcal{S}_{-\rho})$ with speed $a^{-2}(N)$ and good rate
function $\tilde{I}^{DG}$.

###### Proof.

This follows by the same proof as Proposition 4.3, removing the dependence of
the control on $y$ and setting $\Phi\equiv 0$.

∎

###  4.2. Examples: A Class of Aggregation-Diffusion Equations

A common form for interacting particle systems which are widely used in many
settings such as in biology, ecology, social sciences, economics, molecular
dynamics, and in study of spatially homogeneous granular media (see e.g., [MT,
Garnier1, BCCP, KCBFL]) is:

(49) $\displaystyle dX^{i,N}_{t}$
$\displaystyle=-V^{\prime}(X^{i,N}_{t})dt-\frac{1}{N}\sum_{j=1}^{N}W^{\prime}(X^{i,N}_{t}-X^{j,N}_{t})dt+\sigma
dW^{i}_{t},\quad X^{i,N}_{0}=\eta^{x}$

where $V\mathrel{\mathop{\mathchar 58\relax}}\mathbb{R}\rightarrow\mathbb{R}$
is a sufficiently smooth confining potential and $W\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\rightarrow\mathbb{R}$ is a sufficiently smooth
interaction potential. The class of systems (49) contains the system in the
seminal paper [Dawson], where many mathematical aspects of a model for
cooperative behavior in a bi-stable confining potential with attraction to the
mean are explored. This leads us to our first example:

######  Example 4.7.

Consider the system (49). Let $v,\rho$ be as in Corollary 3.4. Suppose
$V^{\prime},W^{\prime}\in C_{b}^{\rho+2}$,
$W^{\prime}\in\mathcal{S}_{\rho+2}$, and $\sigma>0$. Then
$\\{Z^{N}\\}_{N\in\mathbb{N}}=\\{a(N)\sqrt{N}[\frac{1}{N}\sum_{i=1}^{N}\delta_{X^{i,N}_{\cdot}}-\mathcal{L}(\tilde{X}_{\cdot})]\\}_{N\in\mathbb{N}}$
satisfies a large deviation principle on the space
$C([0,T];\mathcal{S}_{-\rho})$ with speed $a^{-2}(N)$ and good rate function
$\tilde{I}^{DG}$ given by:

$\displaystyle\tilde{I}^{DG}(Z)\coloneqq\frac{1}{2\sigma^{2}}\int_{0}^{T}\sup_{\phi\in
C^{\infty}_{c}(\mathbb{R})\mathrel{\mathop{\mathchar
58\relax}}\mathbb{E}[|\phi^{\prime}(\tilde{X}_{t})|^{2}]\neq
0}\frac{|\langle\dot{Z}_{t}-\tilde{L}^{*}_{\mathcal{L}(\tilde{X}_{t})}Z_{t},\phi\rangle|^{2}}{\mathbb{E}[|\phi^{\prime}(\tilde{X}_{t})|^{2}]}dt,$

if $Z(0)=0$, $Z$ is absolutely continuous in the sense if Definition 4.1, and
$Z\in C([0,T];\mathcal{S}_{-v})$, and $I^{DG}(Z)=+\infty$ otherwise.

Here $\tilde{X}_{t}$ satisfies:

$\displaystyle d\tilde{X}_{t}$
$\displaystyle=-V^{\prime}(\tilde{X}_{t})dt-\bar{\mathbb{E}}[W^{\prime}(x-\bar{X}_{t})]|_{x=\tilde{X}_{t}}dt+\sigma
dW_{t},\quad\tilde{X}_{0}=\eta^{x}$

and $\tilde{L}_{\mathcal{L}(\tilde{X}_{s})}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{S}_{v+2}\rightarrow\mathcal{S}_{v}$ acts on $\phi\in
C^{\infty}_{c}(\mathbb{R})$ by:

$\displaystyle\tilde{L}_{\mathcal{L}(\tilde{X}_{s})}\phi(x)$
$\displaystyle=-[V^{\prime}(x)+\mathbb{E}[W^{\prime}(x-\tilde{X}_{s})]]\phi^{\prime}(x)+\frac{\sigma^{2}}{2}\phi^{\prime\prime}(x)-\mathbb{E}[W^{\prime}(\tilde{X}_{s}-x)\phi^{\prime}(\tilde{X}_{s})].$

We are denoting by $\bar{X}_{t}$ an independent copy of $\tilde{X}_{t}$ on
another probability space $(\bar{\Omega},\bar{\mathcal{F}},\bar{\mathbb{P}})$,
and by $\bar{\mathbb{E}}$ the expectation on that space.

###### Proof.

Noting that $\sigma$ is constant and $\frac{\delta}{\delta
m}c(x,\mu)[z]=-W^{\prime}(x-z)$, the assumptions put forward in Corollary 3.4
can be directly verified. This example then immediately falls into the regime
of Corollary 4.6. ∎

In [GP], the authors make, among other modifications, a modification to $V$ in
Equation (49) so that it is a so-called rough-potential (see also [Zwanzig]
and [BS] Section 5), by letting $V^{\epsilon}(x)=V_{1}(x)+V_{2}(x/\epsilon)$,
where $V_{2}$ is sufficiently smooth and periodic. The system becomes:

$\displaystyle
dX^{i,\epsilon,N}_{t}=-[V_{1}^{\prime}(X^{i,\epsilon,N}_{t})+\frac{1}{\epsilon}V_{2}^{\prime}(X^{i,\epsilon,N}_{t}/\epsilon)]dt-\frac{1}{N}\sum_{j=1}^{N}W^{\prime}(X^{i,\epsilon,N}_{t}-X^{j,\epsilon,N}_{t})dt+\sigma
dW^{i}_{t}.$

Letting $Y^{i,\epsilon,N}_{t}=X^{i,\epsilon,N}_{t}/\epsilon$, we see this is a
subclass of systems of the form (1) with

$\displaystyle f(x,y,\mu)$ $\displaystyle=b(x,y,\mu)=-V_{2}^{\prime}(y),\quad
g(x,y,\mu)=c(x,y,\mu)=-V_{1}^{\prime}(x)-\langle\mu,W^{\prime}(x-\cdot)\rangle$
$\displaystyle\sigma(x,y,\mu)$
$\displaystyle=\tau_{1}(x,y,\mu)\equiv\sigma,\quad\tau_{2}\equiv 0,$

Keeping within our setting of a slow-fast system on $\mathbb{R}$, we consider
a version of this system where the fast and slow dynamics are allowed to be
different, and the fast system is not confined to the torus:

(50) $\displaystyle dX^{i,\epsilon,N}_{t}$
$\displaystyle=-[V_{1}^{\prime}(X^{i,\epsilon,N}_{t})+\frac{1}{\epsilon}V_{2}^{\prime}(Y^{i,\epsilon,N}_{t})]dt-\frac{1}{N}\sum_{j=1}^{N}W_{1}^{\prime}(X^{i,\epsilon,N}_{t}-X^{j,\epsilon,N}_{t})dt+\sigma
dW^{i}_{t}$ $\displaystyle dY^{i,\epsilon,N}_{t}$
$\displaystyle=-\frac{1}{\epsilon}[V_{3}^{\prime}(X^{i,\epsilon,N}_{t})+\frac{1}{\epsilon}V_{4}^{\prime}(Y^{i,\epsilon,N}_{t})]dt-\frac{1}{\epsilon}\frac{1}{N}\sum_{j=1}^{N}W_{2}^{\prime}(X^{i,\epsilon,N}_{t}-X^{j,\epsilon,N}_{t})dt+\frac{1}{\epsilon}\tau_{1}dW^{i}_{t}+\frac{1}{\epsilon}\tau_{2}dB^{i}_{t}$
$\displaystyle(X^{i,\epsilon,N}_{0},Y^{i,\epsilon,N}_{0})$
$\displaystyle=(\eta^{x},\eta^{y}).$

This falls into the class of systems (1) with

$\displaystyle b(x,y,\mu)$ $\displaystyle=-V_{2}^{\prime}(y),\quad
c(x,y,\mu)=-V_{1}^{\prime}(x)-\langle\mu,W_{1}^{\prime}(x-\cdot)\rangle,\quad\sigma(x,y,\mu)\equiv\sigma$
$\displaystyle f(x,y,\mu)$ $\displaystyle=-V_{4}^{\prime}(y),\quad
g(x,y,\mu)=-V_{3}^{\prime}(x)-\langle\mu,W_{2}^{\prime}(x-\cdot)\rangle,\quad\tau_{1}(x,y,\mu)\equiv\tau_{1},\quad\tau_{2}(x,y,\mu)\equiv\tau_{2}.$

######  Example 4.8.

Consider the system (50).

Suppose $V_{4}(y)=\frac{\kappa}{2}y^{2}+\tilde{\eta}(y)$ where $\kappa>0$ and
$\tilde{\eta}\in C^{2}_{b}(\mathbb{R})$ is even with
$\mathinner{\\!\left\lVert\tilde{\eta}^{\prime\prime}\right\rVert}_{\infty}<\kappa$,
$V_{1}^{\prime},V_{3}^{\prime},W_{1}^{\prime},W_{2}^{\prime}\in
C_{b}^{r+2}(\mathbb{R})$, $W_{1}^{\prime},W_{2}^{\prime}\in\mathcal{S}_{r+2}$
where $r$ is as in Equation (7), $\sigma,\tau_{2}\neq 0$, $V_{2}$ is even, and
$V_{2}^{\prime}$ is Lipschitz continuous and $O(|y|^{1/2})$ as
$|y|\rightarrow\infty$.

Then
$\\{Z^{N}\\}_{N\in\mathbb{N}}=\\{a(N)\sqrt{N}[\frac{1}{N}\sum_{i=1}^{N}\delta_{X^{i,\epsilon,N}_{\cdot}}-\mathcal{L}(X_{\cdot})]\\}_{N\in\mathbb{N}}$
satisfies a large deviation principle on the space $C([0,T];\mathcal{S}_{-r})$
with speed $a^{-2}(N)$ and good rate function $I^{DG}$ given by:

$\displaystyle I^{DG}(Z)$ $\displaystyle=\frac{1}{2[\sigma^{2}+2\alpha
a+2\sigma\tau_{1}\tilde{\alpha}]}\int_{0}^{T}\sup_{\phi\in
C^{\infty}_{c}(\mathbb{R})\mathrel{\mathop{\mathchar
58\relax}}\mathbb{E}[|\phi^{\prime}(X_{t})|^{2}]\neq
0}\frac{\biggl{|}\langle\dot{Z}_{t}-\bar{L}^{*}_{\mathcal{L}(X_{t})}Z_{t},\phi\rangle\biggr{|}^{2}}{\mathbb{E}\biggl{[}|\phi^{\prime}(X_{t})|^{2}\biggr{]}}dt$

if $Z(0)=0$, $Z$ is absolutely continuous in the sense if Definition 4.1, and
$Z\in C([0,T];\mathcal{S}_{-w})$, and $I^{DG}(Z)=+\infty$ otherwise. Here
$X_{t}$ satisfies:

$\displaystyle dX_{t}$
$\displaystyle=-[\tilde{\alpha}V_{3}^{\prime}(X_{t})+V_{1}^{\prime}(X_{t})]dt-\bar{\mathbb{E}}[\tilde{\alpha}W_{2}^{\prime}(x-\bar{X}_{t})+W_{1}^{\prime}(x-\bar{X}_{t})]|_{x=X_{t}}dt+[\sigma^{2}+2\alpha
a+2\sigma\tau_{1}\tilde{\alpha}]^{1/2}dW_{t}$ $\displaystyle X_{0}$
$\displaystyle=\eta^{x}$ $\displaystyle\tilde{\alpha}$
$\displaystyle=\int_{\mathbb{R}}\Phi^{\prime}(y)\pi(dy),\quad\alpha=\int_{\mathbb{R}}[\Phi^{\prime}(y)]^{2}\pi(dy),\quad
a=\frac{1}{2}[\tau_{1}^{2}+\tau_{2}^{2}]$

and $\bar{L}_{\mathcal{L}(X_{s})}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{S}_{w+2}\rightarrow\mathcal{S}_{w}$ acts on $\phi\in
C^{\infty}_{c}(\mathbb{R})$ by:

$\displaystyle\bar{L}_{\mathcal{L}(X_{s})}\phi(x)$
$\displaystyle\coloneqq-[\tilde{\alpha}V_{3}^{\prime}(x)+V_{1}^{\prime}(x)+\mathbb{E}[\tilde{\alpha}W_{2}^{\prime}(x-X_{s})+W_{1}^{\prime}(x-X_{s})]]\phi^{\prime}(x)+\frac{1}{2}[\sigma^{2}+2\alpha
a+2\sigma\tau_{1}\tilde{\alpha}]\phi^{\prime\prime}(x)$
$\displaystyle\qquad-\mathbb{E}[[\tilde{\alpha}W_{2}^{\prime}(X_{s}-x)+W_{1}^{\prime}(X_{s}-x)]\phi^{\prime}(X_{s})].$

Again, we are denoting by $\bar{X}_{t}$ an independent copy of $X_{t}$ on
another probability space $(\bar{\Omega},\bar{\mathcal{F}},\bar{\mathbb{P}})$
and by $\bar{\mathbb{E}}$ the expectation on that space.

###### Proof.

Once we show $\int_{\mathbb{R}}V_{2}^{\prime}(y)\pi(dy)=0$ for $\pi$ as in
Equation (20), it follows that assumptions A1) \- A12) and A’13) hold via
Example C.10 in the appendix. Via Remark 4.4, we also have $\bar{D}>0,\forall
x\in\mathbb{R},\mu\in\mathcal{P}_{2}(\mathbb{R})$. Then this example is an
immediate corollary of Proposition 4.3.

We know in this setting that $\pi$ admits a density of the form
$\pi(y)=C\exp\left(\frac{-V_{4}(y)}{a}\right)$, where $C$ is a normalizing
constant (see Equation (76) in the appendix). Then since $V_{2},V_{4}$ are
assumed even and hence $V_{2}^{\prime}\pi$ is odd, the result holds. ∎

##  5\. Overview of the approach and formulation of the Controlled System

We use the weak convergence approach of [DE] in order to prove the large
deviations principle for $Z^{N}$. As discussed in Section 3, we prove the
large deviations principle via proving $Z^{N}$ satisfies the Laplace principle
with speed $a^{-2}(N)$ and good rate function $I$ given by Equation (33) (see,
e.g. [DE] Section 1.2).

The method for this is to use the variational representation from [BD] to get
that for each $N\in\mathbb{N}$ and $F\in C_{b}(C([0,T];S_{-\tau}))$, $\tau\geq
w$, where $w$ is as in Equation (6),

(51)
$\displaystyle-a^{2}(N)\log\mathbb{E}\exp\biggl{(}-\frac{1}{a^{2}(N)}F(Z^{N})\biggr{)}$
$\displaystyle=\inf_{\tilde{u}^{N}}\mathbb{E}\biggl{[}\frac{1}{2}\frac{1}{N}\sum_{i=1}^{N}\int_{0}^{T}\left(|\tilde{u}^{N,1}_{i}(s)|^{2}+|\tilde{u}^{N,2}_{i}(s)|^{2}\right)ds+F(\tilde{Z}^{N})\biggr{]}$

where $\\{\tilde{u}^{N,k}_{i}\\}_{i\in\mathbb{N},k=1,2}$ are
$\\{\mathcal{F}_{t}\\}$-progressively-measurable processes such that

(52)
$\displaystyle\sup_{N\in\mathbb{N}}\frac{1}{N}\mathbb{E}\biggl{[}\sum_{i=1}^{N}\int_{0}^{T}\left(|\tilde{u}^{N,1}_{i}(s)|^{2}+|\tilde{u}^{N,2}_{i}(s)|^{2}\right)ds\biggr{]}<\infty.$

One can see that in fact the results of [BD] indeed imply the equality (51) by
following an argument along the same lines as Proposition 3.3. in [BS].

This bound on the controls can be improved when proving the Laplace principle
Lower Bound (30) to:

(53)
$\displaystyle\sup_{N\in\mathbb{N}}\frac{1}{N}\sum_{i=1}^{N}\int_{0}^{T}\left(|\tilde{u}^{N,1}_{i}(s)|^{2}+|\tilde{u}^{N,2}_{i}(s)|^{2}\right)ds<\infty,\mathbb{\mathbb{P}}-\text{
almost surely.}$

by the argument found in Theorem 4.4 of [BD]. Here $\tilde{Z}^{N}$ is given
by, for $\phi\in C^{\infty}_{c}(\mathbb{R})\mathrel{\mathop{\mathchar
58\relax}}$

(54) $\displaystyle\langle\tilde{Z}^{N}_{t},\phi\rangle$
$\displaystyle=a(N)\sqrt{N}(\langle\tilde{\mu}^{\epsilon,N}_{t},\phi\rangle-\langle\mathcal{L}(X_{t}),\phi\rangle),\quad\text{with}\quad\tilde{\mu}^{\epsilon,N}_{t}=\frac{1}{N}\sum_{i=1}^{N}\delta_{\tilde{X}^{i,\epsilon,N}_{t}},\quad
t\in[0,T].$

$\tilde{X}^{i,\epsilon,N}_{t}$ are solutions to:

(55) $\displaystyle
d\tilde{X}^{i,\epsilon,N}_{t}=\biggl{[}\frac{1}{\epsilon}b(\tilde{X}^{i,\epsilon,N}_{t},\tilde{Y}^{i,\epsilon,N}_{t},\tilde{\mu}^{\epsilon,N}_{t})+c(\tilde{X}^{i,\epsilon,N}_{t},\tilde{Y}^{i,\epsilon,N}_{t},\tilde{\mu}^{\epsilon,N}_{t})+\sigma(\tilde{X}^{i,\epsilon,N}_{t},\tilde{Y}^{i,\epsilon,N}_{t},\tilde{\mu}^{\epsilon,N}_{t})\frac{\tilde{u}^{N,1}_{i}(t)}{a(N)\sqrt{N}}\biggr{]}dt$
$\displaystyle+\sigma(\tilde{X}^{i,\epsilon,N}_{t},\tilde{Y}^{i,\epsilon,N}_{t},\tilde{\mu}^{\epsilon,N}_{t})dW^{i}_{t}$
$\displaystyle
d\tilde{Y}^{i,\epsilon,N}_{t}=\frac{1}{\epsilon}\biggl{[}\frac{1}{\epsilon}f(\tilde{X}^{i,\epsilon,N}_{t},\tilde{Y}^{i,\epsilon,N}_{t},\tilde{\mu}^{\epsilon,N}_{t})+g(\tilde{X}^{i,\epsilon,N}_{t},\tilde{Y}^{i,\epsilon,N}_{t},\tilde{\mu}^{\epsilon,N}_{t})+\tau_{1}(\tilde{X}^{i,\epsilon,N}_{t},\tilde{Y}^{i,\epsilon,N}_{t},\tilde{\mu}^{\epsilon,N}_{t})\frac{\tilde{u}^{N,1}_{i}(t)}{a(N)\sqrt{N}}$
$\displaystyle+\tau_{2}(\tilde{X}^{i,\epsilon,N}_{t},\tilde{Y}^{i,\epsilon,N}_{t},\tilde{\mu}^{\epsilon,N}_{t})\frac{\tilde{u}^{N,2}_{i}(t)}{a(N)\sqrt{N}}\biggr{]}dt+\frac{1}{\epsilon}\biggl{[}\tau_{1}(\tilde{X}^{i,\epsilon,N}_{t},\tilde{Y}^{i,\epsilon,N}_{t},\tilde{\mu}^{\epsilon,N}_{t})dW^{i}_{t}+\tau_{2}(\tilde{X}^{i,\epsilon,N}_{t},\tilde{Y}^{i,\epsilon,N}_{t},\tilde{\mu}^{\epsilon,N}_{t})dB^{i}_{t}\biggr{]}$
$\displaystyle(\tilde{X}^{i,\epsilon,N}_{0},\tilde{Y}^{i,\epsilon,N}_{0})=(\eta^{x},\eta^{y}).$

We couple the controls to the joint empirical measures of the fast and slow
process by defining occupation measures $\\{Q^{N}\\}_{N\in\mathbb{N}}\subset
M_{T}(\mathbb{R}^{4})$ in the following way: for
$A,B\in\mathcal{B}(\mathbb{R})$ and $C\in\mathcal{B}(\mathbb{R}^{2})$:

(56) $\displaystyle Q^{N}(A\times B\times C\times[0,t])$
$\displaystyle=\frac{1}{N}\sum_{i=1}^{N}\int_{0}^{t}\delta_{\tilde{X}^{i,\epsilon,N}_{s}}(A)\delta_{\tilde{Y}^{i,\epsilon,N}_{s}}(B)\delta_{(\tilde{u}^{N,1}_{i}(s),\tilde{u}^{N,2}_{i}(s))}(C)ds.$

The proof of the Inequalities (30) and (31) are attained by identifying limit
in distribution of $(\tilde{Z}^{N},Q^{N})$ as satisfying the limiting
controlled Equation (32). This identification of the limit is the subject of
Section 8. In order to identify this limit, we first need to establish
tightness of the sequence of random variables
$\\{(\tilde{Z}^{N},Q^{N})\\}_{N\in\mathbb{N}}$, as done in Section 7. The
proof of tightness relies on a combination of Ergodic-Type Theorems for the
system of controlled interacting particles (55) as proved in Section 6 and on
establishing rates of averaging for fully coupled McKean-Vlasov Equations, as
done in Subsection 7.1. These rates of averaging are needed do to a novel
coupling argument made in the proof of tightness (see Lemma 7.6) to the
following system of IID slow-fast McKean-Vlasov Equations:

(57) $\displaystyle d\bar{X}^{i,\epsilon}_{t}$
$\displaystyle=\biggl{[}\frac{1}{\epsilon}b(\bar{X}^{i,\epsilon}_{t},\bar{Y}^{i,\epsilon}_{t},\mathcal{L}(\bar{X}^{\epsilon}_{t}))+c(\bar{X}^{i,\epsilon}_{t},\bar{Y}^{i,\epsilon}_{t},\mathcal{L}(\bar{X}^{\epsilon}_{t}))\biggr{]}dt+\sigma(\bar{X}^{i,\epsilon}_{t},\bar{Y}^{i,\epsilon}_{t},\mathcal{L}(\bar{X}^{\epsilon}_{t}))dW^{i}_{t}$
$\displaystyle d\bar{Y}^{i,\epsilon}_{t}$
$\displaystyle=\frac{1}{\epsilon}\biggl{[}\frac{1}{\epsilon}f(\bar{X}^{i,\epsilon}_{t},\bar{Y}^{i,\epsilon}_{t},\mathcal{L}(\bar{X}^{\epsilon}_{t}))+g(\bar{X}^{i,\epsilon}_{t},\bar{Y}^{i,\epsilon}_{t},\mathcal{L}(\bar{X}^{\epsilon}_{t}))\biggr{]}dt$
$\displaystyle+\frac{1}{\epsilon}\biggl{[}\tau_{1}(\bar{X}^{i,\epsilon}_{t},\bar{Y}^{i,\epsilon}_{t},\mathcal{L}(\bar{X}^{\epsilon}_{t}))dW^{i}_{t}+\tau_{2}(\bar{X}^{i,\epsilon}_{t},\bar{Y}^{i,\epsilon}_{t},\mathcal{L}(\bar{X}^{\epsilon}_{t}))dB^{i}_{t}\biggr{]}$
$\displaystyle(\bar{X}^{i,\epsilon}_{0},\bar{X}^{i,\epsilon}_{0})$
$\displaystyle=(\eta^{x},\eta^{y}),$

where $\bar{X}^{\epsilon}$ is any particle that has common law with the
$\bar{X}^{i,\epsilon}$’s and $W^{i},B^{i}$ are the same driving Brownian
motions as in Equations (1) and (55).

We will also make use of the empirical measure on $N$ of the IID slow
particles from Equation (57):

(58)
$\displaystyle\bar{\mu}^{\epsilon,N}\coloneqq\frac{1}{N}\sum_{i=1}^{N}\delta_{\bar{X}^{i,\epsilon}_{t}}.$

######  Remark 5.1.

Note that these IID particles are what we get from replacing
$\mu^{\epsilon,N}$ by $\mathcal{L}(\bar{X}^{\epsilon})$ in Equation (1). Using
such an auxiliary process is a traditional proof method for tightness of
fluctuation processes related to empirical measures; See [HM] Theorem 1/Lemma
1, [LossFromDefault] Section 8, [DLR] Section 5.1, [KX] Theorem 2.4/3.1, [FM]
Lemma 3.2/Proposition 3.5/Section 4 for examples of this general approach..
However, a key difference here form those proofs is that the IID particles are
not copies of the limiting process (25), but instead are copies of the process
we would obtain from keeping $\epsilon>0$ fixed and sending
$N\rightarrow\infty$. As seen in [BS], the limit in distribution as
$N\rightarrow\infty$,$\epsilon\downarrow 0$ of the empirical measure
$\mu^{\epsilon,N}$ does not depend on the relative rates at which $\epsilon$
and $N$ go to their respective limits. Hence, we are able to treat each of the
problems separately, and obtain a rate of convergence of
$\tilde{\mu}^{\epsilon,N}$ from Equation (54) to $\bar{\mu}^{\epsilon,N}$ from
Equation (58) as $N\rightarrow\infty$ in $L^{2}$ (see Lemma 7.5), and a rate
of convergence of $\mathcal{L}(\bar{X}^{1,\epsilon}_{t})$ from Equation (57)
to $\mathcal{L}(X_{t})$ uniformly as an element of $S_{-m}$, where $X_{t}$ is
as in Equation (25) and $m$ is as in Equation (5). The latter is a problem of
independent interest in itself, and extends the current known results on
averaging for SDEs and McKean-Vlasov SDEs, which can be found in, e.g.
[RocknerFullyCoupled] and [RocknerMcKeanVlasov] respectively. The result is
contained in Subsection 7.1 as Theorem 7.2, and its proof is the subject of
the complimentary paper [BezemekSpiliopoulosAveraging2022].

##  6\. Ergodic-Type Theorems for the Controlled System (55)

In this section, we use the method of auxiliary Poisson equations to derive
rates of averaging in the form of Ergodic-Type Theorems for the controlled
particles (55). These results are used in the proof of tightness of the
controlled fluctuation process, as they allow us to couple the controlled
particles (55) to the IID slow-fast McKean-Vlasov Equations (57) - see Lemma
7.5. They also allow us to identify a prelimit representation for the
controlled fluctuations processes $\tilde{Z^{N}}$ from Equation (54) (see
Lemma 7.7), which informs the controlled limit proved in Section 8. In
particular, Proposition 6.1 is necessary to handle the terms
$\frac{1}{\epsilon}b$ appearing in the drift of the slow particles
$X^{i,N,\epsilon}$, $\tilde{X}^{i,N,\epsilon}$ in Equations (1),(55). This is
where the terms involving the solution $\Phi$ to the Poisson Equation (22) in
the limiting coefficients (23) come from. The same analysis is performed in
averaging fully-coupled standard diffusions - see e.g. [PV2] Theorem 4 and
[RocknerFullyCoupled] Lemma 4.4 - but here we must also account for the
dependence of the coefficients on the empirical measure, and hence derivatives
of $\Phi$ in its measure component appear in the remainder terms. One term
involving the derivative in the measure component of $\Phi$ a priori seems to
be $\mathcal{O}(1)$ in the limit, but is seen to vanish as
$N\rightarrow\infty$ in Proposition 6.3. Naturally such a term does not appear
in the setting without measure dependence of the coefficients, and is unique
to slow-fast interacting particle systems and slow-fast McKean-Vlasov SDEs.
Thus the “doubled Poisson equation” construction (see Equation (63)) and the
proof of Proposition 6.3 are novel to this paper and the related paper
[BezemekSpiliopoulosAveraging2022]. Proposition 6.4 is used to see that drift
and diffusion coefficients which depend on the fast particles
$\tilde{Y}^{i,\epsilon,N}$ from Equation (55) can be exchanged for those where
dependence on $\tilde{Y}^{i,\epsilon,N}$ is replaced with integration against
the invariant measure $\pi$ from Equation (20) at a cost of
$\mathcal{O}(\epsilon)$. This method is employed when establishing rates of
stochastic homogenization in the standard (one-particle) setting in e.g.
[Spiliopoulos2014Fluctuations] Lemma 4.1,[MS] Lemma B.5, and
[RocknerFullyCoupled] Lemma 4.2. There again, our setting is different than
the standard case in that we must compensate for the dependence of the
empirical measure of the coefficients, which yields terms involving the
derivative in the measure component of the auxiliary Poisson Equation (64).

######  Proposition 6.1.

Consider $\psi\in C^{1,2}_{b}([0,T]\times\mathbb{R})$. Under assumptions A1)
\- A6), we have for any $t\in[0,T]$:

$\displaystyle\frac{a(N)}{\sqrt{N}}\sum_{i=1}^{N}\mathbb{E}\biggl{[}\sup_{t\in[0,T]}\biggl{|}\int_{0}^{t}\frac{1}{\epsilon}b(i)\psi(s,\tilde{X}^{i,\epsilon,N}_{s})ds-\int_{0}^{t}\biggl{(}\gamma_{1}(i)\psi(s,\tilde{X}^{i,\epsilon,N}_{s})+D_{1}(i)\psi_{x}(s,\tilde{X}^{i,\epsilon,N}_{s})$
$\displaystyle+[\frac{\tau_{1}(i)}{a(N)\sqrt{N}}\tilde{u}^{N,1}_{i}(s)+\frac{\tau_{2}(i)}{a(N)\sqrt{N}}\tilde{u}^{N,2}_{i}(s)]\Phi_{y}(i)\psi(s,\tilde{X}^{i,\epsilon,N}_{s})\biggr{)}ds-\int_{0}^{t}\tau_{1}(i)\Phi_{y}(i)\psi(s,\tilde{X}^{i,\epsilon,N}_{s})dW_{s}^{i}$
$\displaystyle-\int_{0}^{t}\tau_{2}(i)\Phi_{y}(i)\psi(s,\tilde{X}^{i,\epsilon,N}_{s})dB_{s}^{i}-\int_{0}^{t}\frac{1}{N}\sum_{j=1}^{N}b(j)\partial_{\mu}\Phi(i)[j]\psi(s,\tilde{X}^{i,\epsilon,N}_{s})ds\biggr{|}^{2}\biggr{]}$
$\displaystyle\leq
C[\epsilon^{2}a(N)\sqrt{N}(1+T+T^{2})+\frac{a(N)}{\sqrt{N}}T^{2}]\mathinner{\\!\left\lVert\psi\right\rVert}^{2}_{C_{b}^{1,2}}$

where here $(i)$ denotes the argument
$(\tilde{X}^{i,\epsilon,N}_{s},\tilde{Y}^{i,\epsilon,N}_{s},\tilde{\mu}^{\epsilon,N}_{s})$
and similarly for $(j)$, $[j]$ denotes the argument
$\tilde{X}^{j,\epsilon,N}_{s}$, and $\Phi$ is as in (22). Here we recall the
definitions of $\gamma_{1},D_{1}$ from Equation (23).

###### Proof.

Using Lemma C.1 to gain appropriate differentiablity of $\Phi$, letting
$\Phi^{N}\mathrel{\mathop{\mathchar
58\relax}}\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{N}\rightarrow\mathbb{R}$
be the empirical projection of $\Phi$ and applying standard Itô’s formula and
Proposition D.6 to the composition
$\Phi^{N}(\tilde{X}^{i,\epsilon,N}_{s},\tilde{Y}^{i,\epsilon,N}_{s},(\tilde{X}^{1,\epsilon,N}_{s},...,\tilde{X}^{N,\epsilon,N}_{s}))$,
we get:

$\displaystyle\int_{0}^{t}\frac{1}{\epsilon}b(i)\psi(s,\tilde{X}^{i,\epsilon,N}_{s})ds-\int_{0}^{t}\biggl{(}\gamma_{1}(i)\psi(s,\tilde{X}^{i,\epsilon,N}_{s})+D_{1}(i)\psi_{x}(s,\tilde{X}^{i,\epsilon,N}_{s})$
$\displaystyle+[\frac{\tau_{1}(i)}{a(N)\sqrt{N}}\tilde{u}^{N,1}_{i}(s)+\frac{\tau_{2}(i)}{a(N)\sqrt{N}}\tilde{u}^{N,2}_{i}(s)]\Phi_{y}(i)\psi(s,\tilde{X}^{i,\epsilon,N}_{s})\biggr{)}ds-\int_{0}^{t}\tau_{1}(i)\Phi_{y}(i)\psi(s,\tilde{X}^{i,\epsilon,N}_{s})dW_{s}^{i}$
$\displaystyle-\int_{0}^{t}\tau_{2}(i)\Phi_{y}(i)\psi(s,\tilde{X}^{i,\epsilon,N}_{s})dB_{s}^{i}-\int_{0}^{t}\frac{1}{N}\sum_{j=1}^{N}b(j)\partial_{\mu}\Phi(i)[j]\psi(s,\tilde{X}^{i,\epsilon,N}_{s})ds=\sum_{k=1}^{8}\tilde{B}^{i,\epsilon,N}_{k}$

where:

$\displaystyle\tilde{B}^{i,\epsilon,N}_{1}(t)$
$\displaystyle=\epsilon[\Phi(\tilde{X}^{i,\epsilon,N}_{0},\tilde{Y}^{i,\epsilon,N}_{0},\tilde{\mu}^{\epsilon,N}_{0})\psi(0,\tilde{X}^{i,\epsilon,N}_{0})-\Phi(\tilde{X}^{i,\epsilon,N}_{t},\tilde{Y}^{i,\epsilon,N}_{t},\tilde{\mu}^{\epsilon,N}_{t})\psi(t,\tilde{X}^{i,\epsilon,N}_{t})]$
$\displaystyle\tilde{B}^{i,\epsilon,N}_{2}(t)$
$\displaystyle=\frac{1}{N}\int_{0}^{t}\sigma(i)\tau_{1}(i)\partial_{\mu}\Phi_{y}(i)[i]\psi(s,\tilde{X}^{i,\epsilon,N}_{s})ds$
$\displaystyle\tilde{B}^{i,\epsilon,N}_{3}(t)$
$\displaystyle=\epsilon\int_{0}^{t}\biggl{(}\Phi(i)\dot{\psi}(s,\tilde{X}^{i,\epsilon,N}_{s})+c(i)[\Phi_{x}(i)\psi(s,\tilde{X}^{i,\epsilon,N}_{s})+\Phi(i)\psi_{x}(s,\tilde{X}^{i,\epsilon,N}_{s})]+\frac{\sigma^{2}(i)}{2}[\Phi_{xx}(i)\psi(s,\tilde{X}^{i,\epsilon,N}_{s})$
$\displaystyle\qquad+2\Phi_{x}(i)\psi_{x}(s,\tilde{X}^{i,\epsilon,N}_{s})+\Phi(i)\psi_{xx}(s,\tilde{X}^{i,\epsilon,N}_{s})]\biggr{)}ds$
$\displaystyle\tilde{B}^{i,\epsilon,N}_{4}(t)$
$\displaystyle=\epsilon\int_{0}^{t}\frac{\sigma^{2}(i)}{2}[\frac{2}{N}\partial_{\mu}\Phi(i)[i]\psi_{x}(s,\tilde{X}^{i,\epsilon,N}_{s})+\frac{2}{N}\partial_{\mu}\Phi_{x}(i)[i]\psi(s,\tilde{X}^{i,\epsilon,N}_{s})]ds$
$\displaystyle\tilde{B}^{i,\epsilon,N}_{5}(t)$
$\displaystyle=\epsilon\int_{0}^{t}\frac{1}{N}\sum_{j=1}^{N}\biggl{\\{}c(j)\partial_{\mu}\Phi(i)[j]\psi(s,\tilde{X}^{i,\epsilon,N}_{s})+\frac{1}{2}\sigma^{2}(j)[\frac{1}{N}\partial^{2}_{\mu}\Phi(i)[j,j]+\partial_{z}\partial_{\mu}\Phi(i)[j]]\psi(s,\tilde{X}^{i,\epsilon,N}_{s})\biggr{\\}}ds$
$\displaystyle\tilde{B}^{i,\epsilon,N}_{6}(t)$
$\displaystyle=\epsilon\biggl{[}\int_{0}^{t}\sigma(i)[\Phi_{x}(i)\psi(s,\tilde{X}^{i,\epsilon,N}_{s})+\Phi(i)\psi_{x}(s,\tilde{X}^{i,\epsilon,N}_{s})]dW^{i}_{t}+\frac{1}{N}\sum_{j=1}^{N}\biggl{\\{}\int_{0}^{t}\sigma(j)\partial_{\mu}\Phi(i)[j]\psi(s,\tilde{X}^{i,\epsilon,N}_{s})dW^{j}_{s}\biggr{\\}}\biggr{]}$
$\displaystyle\tilde{B}^{i,\epsilon,N}_{7}(t)$
$\displaystyle=\epsilon\int_{0}^{t}\frac{\sigma(i)}{a(N)\sqrt{N}}\tilde{u}^{N,1}_{i}(s)[\Phi_{x}(i)\psi(s,\tilde{X}^{i,\epsilon,N}_{s})+\Phi(i)\psi_{x}(s,\tilde{X}^{i,\epsilon,N}_{s})]ds$
$\displaystyle\tilde{B}^{i,\epsilon,N}_{8}(t)$
$\displaystyle=\epsilon\int_{0}^{t}\frac{1}{N}\biggl{\\{}\sum_{j=1}^{N}\frac{\sigma(j)}{a(N)\sqrt{N}}\tilde{u}^{N,1}_{j}(s)\partial_{\mu}\Phi(i)[j]\psi(s,\tilde{X}^{i,\epsilon,N}_{s})\biggr{\\}}ds.$

Via Lemma B.1, the assumed linear growth of $b$ and $c$ in $y$ and boundedness
of $\sigma$, and the assumed bound (53) on the controls, one can check that
indeed $\tilde{\mu}^{N}_{t}\in\mathcal{P}_{2}(\mathbb{R})$ for each
$t\in[0,T]$ and $N\in\mathbb{N}$, and so there is no issue with the domain of
$\Phi$ and its derivatives being $\mathcal{P}_{2}(\mathbb{R})$.

Then, by multiple applications of Hölder’s inequality, and using the assumed
uniform in $x,\mu$ polynomial growth in $y$ of $\Phi$ and its derivatives from
Assumption A6):

$\displaystyle\frac{a(N)}{\sqrt{N}}\sum_{i=1}^{N}\mathbb{E}\biggl{[}\sup_{t\in[0,T]}|\tilde{B}^{i,\epsilon,N}_{1}(t)|^{2}\biggr{]}\leq\epsilon^{2}a(N)\sqrt{N}\mathinner{\\!\left\lVert\psi\right\rVert}^{2}_{\infty}$
$\displaystyle\frac{a(N)}{\sqrt{N}}\sum_{i=1}^{N}\mathbb{E}\biggl{[}\sup_{t\in[0,T]}|\tilde{B}^{i,\epsilon,N}_{2}(t)|^{2}\biggr{]}\leq
C\frac{a(N)}{\sqrt{N}}\frac{1}{N^{2}}\sum_{i=1}^{N}T\mathbb{E}\biggl{[}\int_{0}^{T}|\partial_{\mu}\Phi_{y}(i)[i]|^{2}ds\biggr{]}\mathinner{\\!\left\lVert\psi\right\rVert}^{2}_{\infty}$
$\displaystyle\qquad\leq
C\frac{a(N)}{\sqrt{N}}\frac{1}{N}\sum_{i=1}^{N}T\mathbb{E}\biggl{[}\int_{0}^{T}\mathinner{\\!\left\lVert\partial_{\mu}\Phi_{y}(i)[\cdot]\right\rVert}_{L^{2}(\mathbb{R},\tilde{\mu}^{N,\epsilon}_{s})}^{2}ds\biggr{]}\mathinner{\\!\left\lVert\psi\right\rVert}^{2}_{\infty}$
$\displaystyle\qquad\leq
C\frac{a(N)}{\sqrt{N}}T^{2}\biggl{(}1+\frac{1}{N}\sum_{i=1}^{N}\sup_{t\in[0,T]}\mathbb{E}\biggl{[}|\tilde{Y}^{i,\epsilon,N}_{t}|^{2\tilde{q}_{\Phi_{y}}(1,0,0)}\biggr{]}\biggr{)}\mathinner{\\!\left\lVert\psi\right\rVert}^{2}_{\infty}$
$\displaystyle\frac{a(N)}{\sqrt{N}}\sum_{i=1}^{N}\mathbb{E}\biggl{[}\sup_{t\in[0,T]}|\tilde{B}^{i,\epsilon,N}_{3}(t)|^{2}\biggr{]}\leq\epsilon^{2}a(N)\sqrt{N}T^{2}\biggl{(}1+\frac{1}{N}\sum_{i=1}^{N}\sup_{t\in[0,T]}\mathbb{E}\biggl{[}|\tilde{Y}^{i,\epsilon,N}_{t}|^{2}+|\tilde{Y}^{i,\epsilon,N}_{t}|^{2q_{\Phi}(0,2,0)}\biggr{]}\biggr{)}\mathinner{\\!\left\lVert\psi\right\rVert}^{2}_{C^{1,2}_{b}}$
$\displaystyle\frac{a(N)}{\sqrt{N}}\sum_{i=1}^{N}\mathbb{E}\biggl{[}\sup_{t\in[0,T]}|\tilde{B}^{i,\epsilon,N}_{4}(t)|^{2}\biggr{]}\leq
C\frac{a(N)}{\sqrt{N}}\frac{\epsilon^{2}}{N^{2}}\sum_{i=1}^{N}T\mathbb{E}\biggl{[}\int_{0}^{T}\biggl{|}\biggl{(}|\partial_{\mu}\Phi(i)[i]|+|\partial_{\mu}\Phi_{x}(i)[i]|\biggr{)}\biggr{|}^{2}ds\biggr{]}(\mathinner{\\!\left\lVert\psi\right\rVert}^{2}_{\infty}+\mathinner{\\!\left\lVert\psi_{x}\right\rVert}^{2}_{\infty})$
$\displaystyle\qquad\leq
C\frac{a(N)}{\sqrt{N}}\frac{\epsilon^{2}}{N}\sum_{i=1}^{N}T\mathbb{E}\biggl{[}\int_{0}^{T}\biggl{|}\biggl{(}\mathinner{\\!\left\lVert\partial_{\mu}\Phi(i)[\cdot]\right\rVert}_{L^{2}(\mathbb{R},\tilde{\mu}^{N,\epsilon}_{s})}+\mathinner{\\!\left\lVert\partial_{\mu}\Phi_{x}(i)[\cdot]\right\rVert}_{L^{2}(\mathbb{R},\tilde{\mu}^{N,\epsilon}_{s})}\biggr{)}\biggr{|}^{2}ds\biggr{]}(\mathinner{\\!\left\lVert\psi\right\rVert}^{2}_{\infty}+\mathinner{\\!\left\lVert\psi_{x}\right\rVert}^{2}_{\infty})$
$\displaystyle\qquad\leq
C\frac{a(N)}{\sqrt{N}}\epsilon^{2}T^{2}\biggl{(}1+\frac{1}{N}\sum_{i=1}^{N}\sup_{t\in[0,T]}\mathbb{E}\biggl{[}|\tilde{Y}^{i,\epsilon,N}_{t}|^{2(\tilde{q}_{\Phi}(1,0,0)\vee\tilde{q}_{\Phi}(1,1,0))}\biggr{]}\biggr{)}(\mathinner{\\!\left\lVert\psi\right\rVert}^{2}_{\infty}+\mathinner{\\!\left\lVert\psi_{x}\right\rVert}^{2}_{\infty})$

Here for $\tilde{B}_{1}$, we used the assumed boundedness of $\Phi$ from A6).
For $\tilde{B}_{2}$ we used the assumed polynomial growth in $y$ of
$\partial_{\mu}\Phi$ from A6) and the boundedness of $\sigma$ and $\tau_{1}$
from A5) and A1). For $\tilde{B}_{3}$ we used the assumed polynomial growth in
$y$ of $\Phi_{xx}$ and boundedness of $\Phi,\Phi_{x}$ from A6) and the
boundedness of $\sigma$ and the linear growth in $y$ of $c$ from A5). In
$\tilde{B}_{4}$ we used the assumed polynomial growth in $y$ of
$\partial_{\mu}\Phi$ and $\partial_{\mu}\Phi_{x}$ from A6) and the assumed
boundedness of $\sigma$ from A5).

For $\tilde{B}^{i,\epsilon,N}_{5}(t)$, we bound the two terms separately. For
the first, we use the assumed linear growth in $y$ of $c$ and polynomial
growth of $\partial_{\mu}\Phi$ in $y$ to get:

$\displaystyle\frac{a(N)}{\sqrt{N}}\sum_{i=1}^{N}\frac{\epsilon^{2}}{N^{2}}\mathbb{E}\biggl{[}\sup_{t\in[0,T]}\biggl{|}\int_{0}^{t}\sum_{j=1}^{N}c(j)\partial_{\mu}\Phi(i)[j]\psi(s,\tilde{X}^{i,\epsilon,N}_{s})ds\biggr{|}^{2}\biggr{]}$