# An oscillator driven by algebraically decorrelating noise

Chistophe Gomez  Aix Marseille Univ, CNRS, I2M, Marseille, France
<EMAIL_ADDRESS>,  Gautam Iyer  Department of Mathematical
Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA
<EMAIL_ADDRESS>,  Hai Le  Department of Mathematics, Pennsylvania State
University, State College PA 16802<EMAIL_ADDRESS>and  Alexei
Novikov  Department of Mathematics, Pennsylvania State University, State
College PA 16802<EMAIL_ADDRESS>
###### Abstract.

We consider a stochastically forced nonlinear oscillator driven by a
stationary Gaussian noise that has an algebraically decaying covariance
function. It is well known that such noise processes can be renormalized to
converge to _fractional_ Brownian motion, a process that has memory. In
contrast, we show that the renormalized limit of the nonlinear oscillator
driven by this noise converges to diffusion driven by standard (not
fractional) Brownian motion, and thus retains no memory in the scaling limit.
The proof is based on the study of a fast-slow system using the perturbed test
function method.

###### Key words and phrases:

Hamiltonian systems, Random perturbations, Algebraic decaying correlations,
Fractional processes, Diffusion-approximation

###### 2010 Mathematics Subject Classification:

Primary 37H05, 60H10; Secondary 60F05, 60G15.

This work has been partially supported by the National Science Foundation
under grants NSF DMS-1813943, DMS-1515187 to AN and HL, NSF DMS-1814147 to GI,
and the Center for Nonlinear Analysis.

##  1\. Introduction

Consider a stochastically forced nonlinear oscillator with 1 degree of
freedom:

(1.1) $\ddot{x}_{t}+f(x_{t})=\varepsilon v(t),\quad
x_{0}\in\mathbb{R},\quad\dot{x}_{0}=y_{0}\in\mathbb{R}\,.$

Here $f\colon\mathbb{R}\to\mathbb{R}$ is a given smooth function, and $v$ is a
stochastic process representing the noise. Our interest is to study the
asymptotic long-time behavior of $x$ when the noise $v$ has an algebraically
decaying covariance function. In this case, the rescaled noise converges to
fractional Brownian motion (fBm), a process that has a memory. We aim to study
how the nonlinear dynamics affects the limiting behavior, and study whether or
not the rescaled oscillator also converges to a process with memory.

A similar question was studied by Komorowski et al. [KNR14] in the case of a
passive tracer particle advected by a periodic shear flow. In this case, it
turns out that there is a certain parameter regime where the time rescaled
dynamics is Markovian and the memory effect of the noise is forgotten.
However, there is also a regime (namely the “very long-time” behavior when the
Hurst index of the driving noise is larger than $1/2$) where the memory effect
persists. In contrast, for the oscillator (1.1), we will show that the memory
effect _never_ persists, and the effective long-time behavior is always
Markovian. The main reason for this is that the oscillatory nature of the
deterministic dynamics cancels slowly decaying correlations, and destroys the
memory effect. To study (1.1) we cannot use the limit theorem for additive
functionals fBm’s used by [KNR14]. Instead, we recast (1.1) as a fast-slow
system and use the perturbed test function method. Even though this method was
introduced in the ’76 [PSV76, Kus84], it has applications to many problems
arising today [FGPSl07, DRV21, CV21], and is the only method we presently know
that can be used to prove our main result.

To describe our setup, we first introduce the Hamiltonian

$H(x,y)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\frac{1}{2}y^{2}+\int_{0}^{x}f(s)\,ds\,,$

and recast (1.1) as a stochastically perturbed Hamiltonian system

(1.2a) $\displaystyle\dot{x}_{t}=y_{t}=\partial_{y}H(x_{t},y_{t})\,,$ (1.2b)
$\displaystyle\dot{y}_{t}=\ddot{x}_{t}=-f(x_{t})+\varepsilon
v(t)=-\partial_{x}H(x_{t},y_{t})+\varepsilon v(t)\,.$

Setting
$X_{t}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}(x_{t},y_{t})^{T}$,
we rewrite (1.2a)–(1.2b) compactly as

(1.3) $\dot{X}_{t}=\nabla^{\bot}H(X_{t})+\varepsilon v(t)e_{2},\quad
X_{0}=(x_{0},y_{0})\in\mathbb{R}^{2}\,.$

where
$\nabla^{\bot}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}(\partial_{y},-\partial_{x})^{T}$,
and $e_{2}=(0,1)^{T}$.

To study the long-time behavior, we consider the time rescaled process
$X^{\varepsilon}(t)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}X(t/\varepsilon^{2})$,
and observe

(1.4)
$\dot{X}_{t}^{\varepsilon}=\frac{1}{\varepsilon^{2}}\nabla^{\bot}H(X_{t}^{\varepsilon})+\frac{1}{\varepsilon}v\left(\frac{t}{\varepsilon^{2}}\right)e_{2}\,,\quad
X_{0}^{\varepsilon}=X_{0}=(x_{0},y_{0})\in\mathbb{R}^{2}.$

When the noise $v$ is white, the behavior of $X^{\varepsilon}$ as
$\varepsilon\to 0$ is completely described by the averaging principle of
Freidlin and Wentzell [FW93, FW94, FW98]. To briefly explain this, we note
that in the absence of noise (i.e. when $v\equiv 0$), the process
$X^{\varepsilon}$ travels very fast along level sets of the Hamiltonian. In
the presence of noise, the process $X^{\varepsilon}$ will also diffuse slowly
across these level sets. To capture the limiting behavior we factor out the
fast motion by projecting to the _Reeb graph_ of the Hamiltonian, which has
the effect of identifying all closed trajectories of the Hamiltonian system to
points. Now, as $\varepsilon\to 0$, this projection converges to a diffusion
driven by a standard Brownian motion and we refer the reader to [FW93, FW94,
FW98] for details. (We also remark that when the noise $v$ is white, one may
also study the behavior of $X$ on time scales shorter than
$1/\varepsilon^{2}$. In some scenarios, a robust and stable limiting behavior
is also observed on these time scales and has been studied by several authors
[You88, YJ91, Bak11, HV14a, HV14b, HKPG16, TV16, HIK+18].)

In this paper, our interest is to study the case when the driving noise in
(1.4) is colored in time and has algebraically decaying correlations. It is
well known that this noise can be renormalized to converge to a fractional
Brownian motion (fBm), with Hurst parameter that is determined from the decay
rate of the correlation function (see for instance [Taq75, Taq77, Mar05]).
When the Hurst parameter is not half (which happens when the correlation
function decays like $1/t^{\gamma}$ with $\gamma\neq 1$), the renormalized
limit of the noise has memory and is a non-Markovian process. We aim to study
whether the memory effect is also present in the $\varepsilon\to 0$ limit of
the nonlinear oscillator (1.4), where the noise is combined with the
Hamiltonian dynamics.

At present, we are only able to analyze the scenario where the Hamiltonian is
smooth with exactly one non-degenerate critical point. In this case, it is
convenient to think about the dynamics of $X^{\varepsilon}$ in terms of
action-angle coordinates. The angular coordinate of $X^{\varepsilon}$ changes
very fast, and has no meaningful limit as $\varepsilon\to 0$. The action
coordinate of $X^{\varepsilon}$, on the other hand, changes slowly as a result
of the interaction between the noise and the averaged angular coordinate and
has a non-trivial limit $\varepsilon\to 0$. To study this we use the
Hamiltonian itself as a proxy for the action coordinate and state our results
in terms of convergence of the process $H(X^{\varepsilon})$. (To relate this
to the Freidlin–Wentzell framework, we note that when $H$ has exactly one non-
degenerate critical point, the Reeb graph has exactly one vertex and one edge,
and $H(X^{\varepsilon})$ is precisely the projection of $X^{\varepsilon}$ onto
the Reeb graph.)

In this paper, we prove two results. They show that even though the driving
noise has memory, the time correlations are destroyed by the oscillatory
dynamics of (1.4), and $H(X^{\varepsilon})$ converges to a diffusion driven by
standard Brownian motion (a process without memory). Roughly speaking our main
results are as follows:

  1. (1)

If the Hamiltonian $H$ is quadratic, and the noise $v$ is any stationary
Gaussian process with an algebraically decaying covariance function, then
$H(X^{\varepsilon})$ converges to a diffusion process driven by a Brownian
motion. In this particular case, the limiting diffusion is a rescaling of the
square of the 2-dimensional Bessel process.

  2. (2)

If the Hamiltonian is not quadratic (but still smooth with exactly one non-
degenerate critical point), then we can show $H(X^{\varepsilon})$ converges to
a diffusion driven by Brownian motions, provided the noise $v$ is chosen
suitably. (The noise $v$ is still a stationary Gaussian process with an
algebraically decaying covariance function. However, it must be expressible as
a superposition of Ornstein–Uhlenbeck processes as we make use of certain
exponential mixing estimates in the proof.)

###  1.1. Main results for a quadratic Hamiltonian.

We now state our results precisely when $H$ is quadratic. Hereafter we assume
that the noise $v$ is a stationary Gaussian process with covariance function

(1.5)
$R(t)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\bm{E}v(t)v(0)=\bm{E}v(t+s)v(s)\,.$

Our main result when the Hamiltonian is quadratic is as follows:

######  Proposition 1.1.

Let $H$ be the quadratic Hamiltonian

(1.6)
$H(x)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\frac{\lvert
x\rvert^{2}}{2}\,,\qquad x=(x_{1},x_{2})\in\mathbb{R}^{2}\,,$

and suppose the noise $v$ is a stationary Gaussian process whose covariance
function, $R$, is of the form

(1.7) $R(t)=\frac{L(t)}{t^{\gamma}}$

for some $\gamma\in(0,2)$ and a function $L$ that is slowly varying at
infinity. 111Recall a function $L$ is said to be slowly varying at infinity if
for every $s>0$ we have $L(st)/t\to 1$ as $t\to\infty$. If $\gamma\in(0,1]$,
we further assume that $L$ has the slow increase property

(1.8) $\lim_{t\to\infty}\frac{L^{\prime}(t)}{L(t)/t}=0\,.$

Then the family of processes $H(X^{\varepsilon})_{\varepsilon>0}$ converges in
distribution to $H(W^{D})$, where $W^{D}$ is a 2D Brownian motion with
$W_{0}=X_{0}$ and covariance matrix $D$ given by

(1.9)
$D_{11}=D_{22}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\lim_{x\to\infty}\int_{0}^{x}R(z)\cos(z)\,dz\qquad\text{and}\qquad
D_{12}=D_{21}=0\,.$

######  Remark 1.2.

For $\gamma\in(0,1]$ we note that (1.8) implies that $R^{\prime}<0$ near
infinity. Since $\cos(z)$ oscillates periodically, the limit in (1.9) exists
and is finite. For $\gamma\in(1,2)$, we note $R\in L^{1}(\mathbb{R})$ and so
the limit in (1.9) also exists and is finite.

As mentioned earlier, it is well known [Taq75, Taq77, Mar05] that the
renormalized noise converges to a fractional Brownian motion. For the reader’s
convenience, we state a result to that effect next.

######  Proposition 1.3.

Let $v$ be a stationary Gaussian process with covariance function $R$ given by
(1.7) for some $\gamma\in(0,2)$ and a slowly varying function $L$. If
$\gamma\in(1,2)$, we additionally suppose

(1.10) $\int_{0}^{\infty}R(t)\,dt=0\,.$

Let

(1.11)
$\sigma(\varepsilon)=\begin{dcases}L(\varepsilon^{-2})^{1/2}\varepsilon^{\gamma}&\gamma\neq
1\,,\\\
L(\varepsilon^{-2})^{1/2}\varepsilon\lvert\ln\varepsilon\rvert^{1/2}&\gamma=1\,,\end{dcases}\quad\text{and}\quad
u^{\varepsilon}(t)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\frac{1}{\sigma(\varepsilon)}\int_{0}^{t}v\left(\frac{s}{\varepsilon^{2}}\right)\,ds\,.$

Then, as $\varepsilon\to 0$, the family of processes
$(u^{\varepsilon})_{\varepsilon>0}$ converges in distribution to
$\sigma_{\mathcal{H}}B^{{\mathcal{H}}}$, where $B^{\mathcal{H}}$ is a standard
fractional Brownian motion with Hurst index ${\mathcal{H}}=1-\gamma/2$, and

(1.12)
$\sigma_{\mathcal{H}}^{2}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\begin{dcases}\frac{1}{\mathcal{H}\lvert
2\mathcal{H}-1\rvert}&\mathcal{H}\neq\frac{1}{2}\,,\\\
1&\mathcal{H}=\frac{1}{2}\,.\end{dcases}$

To reiterate our main point, we note that Proposition 1.1 implies

(1.13)
$H\Big{(}X_{0}+\frac{1}{\varepsilon^{2}}\int_{0}^{t}\nabla^{\perp}H(X^{\varepsilon}_{s})\,ds+\frac{1}{\varepsilon}\int_{0}^{t}v\Big{(}\frac{s}{\varepsilon^{2}}\Big{)}e_{2}\,ds\Big{)}\xrightarrow{\varepsilon\to
0}H(W^{D}_{t})\,,$

for a Brownian motion $W^{D}$ with covariance matrix $D$. However,

$\frac{1}{\sigma(\varepsilon)}\int_{0}^{t}v\Big{(}\frac{s}{\varepsilon^{2}}\Big{)}\,ds\xrightarrow{\varepsilon\to
0}\sigma_{\mathcal{H}}B^{H}_{t}\,,$

for a fBm $B^{\mathcal{H}}$ with Hurst index $\mathcal{H}$. Of course,
$H(W^{D})$ is a Markov process with no memory, but $B^{\mathcal{H}}$ is a non-
Markovian process with memory.

Note further that when $\gamma\in(0,1)$, we see
$\varepsilon\ll\sigma(\varepsilon)$ and so the term
$\frac{1}{\varepsilon}\int_{0}^{t}v(s/\varepsilon^{2})\,ds$ appearing in
(1.13) diverges as $\varepsilon\to 0$. On the other hand when $\gamma\in(1,2)$
(and equation (1.10) holds), we see $\sigma(\varepsilon)\ll\varepsilon$, and
so this term vanishes. In both cases, the oscillatory Hamiltonian term
contributes non-trivially and as a result, the time correlations are forgotten
and $H(X^{\varepsilon})$ converges to a memoryless Markov process.

We now explain the reason for the assumption (1.10), which is required in
Proposition 1.3 when $\gamma\in(1,2)$, but not in Proposition 1.1. If
$\gamma\in(1,2)$ by the central limit theorem one can show that
$\frac{1}{\varepsilon}\int_{0}^{t}v(s/\varepsilon^{2})\,ds$ converges to a
Brownian motion. The covariance of this Brownian motion is proportional to
$\int_{0}^{\infty}R(t)\,dt$. If this is $0$ (as required by assumption
(1.10)), then one can divide $\int_{0}^{t}v(s/\varepsilon^{2})\,ds$ by the
smaller factor $\varepsilon^{\gamma}=\sigma(\varepsilon)$, and look for a non-
trivial limit. This is precisely what is given by Proposition 1.3. The reason
the assumption (1.10) is not needed for Proposition 1.1 is because in (1.13)
the noise is only scaled by a factor of $1/\varepsilon$ and not
$1/\sigma(\varepsilon)$.

Finally, we briefly summarize the main idea of the proofs of Propositions 1.1
and 1.3. First, Proposition 1.3 only requires convergence of Gaussian
processes and can be proved by directly computing covariances. Similar results
are well known [Taq75, Taq77, Mar05]. Since the exact result we need isn’t
readily available, we prove it in Appendix B.

The proof of Proposition 1.1 requires a little more work. The main
simplification obtained from the assumption that $H$ is quadratic is that the
evolution of the angle coordinate is independent of the action coordinate. As
a result one can perform a (time dependent) rotation in space and prove
convergence of $H(X^{\varepsilon})$ by studying integrals of the form

$\frac{1}{\varepsilon}\int_{0}^{t}v\Big{(}\frac{s}{\varepsilon^{2}}\Big{)}\cos\Big{(}\frac{s}{\varepsilon^{2}}\Big{)}\,ds\qquad\text{and}\qquad\frac{1}{\varepsilon}\int_{0}^{t}v\Big{(}\frac{s}{\varepsilon^{2}}\Big{)}\sin\Big{(}\frac{s}{\varepsilon^{2}}\Big{)}\,ds\,.$

These are now Gaussian and one can study convergence by computing covariances
directly. We note that the study of similar integrals arises in the study of
random media where several authors [Mar05, FGPSl07, Gar97] have used
oscillatory dynamics to decorrelate the medium.

###  1.2. Main results for Hamiltonians with only one critical point.

We now study the case where the Hamiltonian has exactly one non-degenerate
critical point but is not necessarily quadratic. In this case we will need to
work with a stationary Gaussian noise process $v$ that can be obtained by
super-imposing Ornstein–Uhlenbeck processes and written in the form

$v(t)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\int_{S}e^{-\mu\lvert
p\rvert^{2\beta}(t-u)}\mathcal{B}(du,dp)\,.$

Here $S\subseteq\mathbb{R}$ is a bounded symmetric open interval, $\mu$,
$\beta$ are constants, and $\mathcal{B}$ is a Gaussian random measure that is
white in time and colored in space and will be constructed explicitly in
Section 3 below to ensure that the covariance of $v$ decays algebraically. Our
main theorem can now be stated as follows.

######  Theorem 1.4.

Let $H\colon\mathbb{R}^{2}\to\mathbb{R}$ be a smooth Hamiltonian with exactly
one non-degenerate minimum at $(0,0)$, and $v$ be the stationary Gaussian
noise described above. Then

(1.14) $H(X^{\varepsilon})\xrightarrow[\varepsilon\to
0]{\mathcal{D}}\mathcal{X}\,,$

where $\mathcal{X}$ is a diffusion driven by standard Brownian motion, with
infinitesimal generator

(1.15)
$L=\frac{1}{2\Lambda(\mathcal{X})}\partial_{\mathcal{X}}(\Sigma(\mathcal{X})\partial_{\mathcal{X}})\,.$

Here the coefficients $\Lambda$ and $\Sigma$ are defined by

$\Lambda(\mathcal{X})=\oint_{\\{H=\mathcal{X}\\}}\,\frac{dl}{|\nabla H|}\,,$

and

$\Sigma(\mathcal{X})=2\int_{u=0}^{\infty}du\,R(u)\oint_{\\{H=\mathcal{X}\\}}\partial_{y}H(\check{X}^{x,y}_{u})\partial_{y}H(x,y)\frac{dl(x,y)}{|\nabla
H(x,y)|}\,,$

where $\check{X}^{x,y}$ is the solution to

$\partial_{t}\check{X}^{x,y}_{t}=\nabla^{\perp}H(\check{X}^{x,y}_{t}),\qquad\text{with}\qquad\check{X}^{x,y}_{t=0}=(x,y)\,.$

Theorem 1.4 follows immediately from the more general Theorem 4.3, below,
concerning fast-slow systems, and the proof of Theorem 1.4 is presented
shortly after the statement of Theorem 4.3. Moreover, Theorem 4.3 also
explicitly characterizes the limiting system and can be used to explicitly
characterize the diffusion $\mathcal{X}$ in Theorem 1.4.

Note that when $H$ is not quadratic the convergence of $H(X^{\varepsilon})$
can’t be reduced to Gaussian integrals as in Proposition 1.1, and so the proof
of Theorem 1.4 is more involved. We prove Theorem 1.4 by first switching to
action-angle coordinates and converting (1.4) to a fast-slow system, where the
slow variables are mean-zero in the fast variable. We then use the perturbed
test function (PTF) method [Kus84] to prove convergence of this system. This
approach provides more information than just the convergence of
$H(X^{\varepsilon})$, as stated in Theorem 1.4, and we refer the reader to
Theorem 4.3, below, for the precise statement. Of course, the convergence of
action coordinate is equivalent to convergence of the processes
$H(X^{\varepsilon})$. However, our proof also identifies an asymptotic
diffusive behavior of the angle coordinate on the slow scale.

We remark that our fast-slow system is different from that recently considered
by Hairer and Li [HL20]. Indeed, in [HL20], the authors proved an averaging
principle for a coupled fast-slow system driven by fBm, with Hurst index
$\mathcal{H}>\frac{1}{2}$, and they obtained convergence in probability to the
naively averaged system. However, they require the fast variable to be driven
by an independent _standard_ Brownian motion. In the scenario that arises in
the present paper, the fast variable is driven by noise that converges to a
_fractional_ Brownian motion when renormalized, and the same noise is also
used to drive the slow variable. Moreover the limiting system we obtain is not
the naively averaged one, but a diffusion with an averaged generator akin to
the limiting behavior of fast-slow systems driven by standard Brownian noise
(see for instance [PS08]).

Theorem 1.4 is stated for Hamiltonians which have only one (non-degenerate)
critical point. The Reeb graph of such Hamiltonians is simply the half line,
and the limiting process in Theorem 1.4 is a diffusion on the half line. For
general Hamiltonians, Reeb graph may have multiple vertices, and the limiting
process should be a diffusion on this Reeb graph. When the driving noise is a
standard Brownian motion (as in the Freidlin–Wentzell theory), this diffusion
is characterized through its generator on each edge of the Reeb graph, and
certain gluing conditions on the vertices. While the proofs in the
Freidlin–Wentzell theory rely heavily on the Markov property, and do not apply
to our situation, the form of the infinitesimal generator (1.15) is similar to
that in [FW98, Equation 1.8]. As a result, we expect similar gluing conditions
to hold in our context.

Explicitly, for a sufficiently general class of Hamiltonians, we expect that
the process $(H(X^{\varepsilon}))_{\varepsilon>0}$ converges in distribution
to a diffusion process on the Reeb graph. The generator of this diffusion
should be

$L_{j}h_{j}(\mathcal{X})=\frac{1}{2\Lambda_{j}(\mathcal{X})}\partial_{\mathcal{X}}(\Sigma_{j}(\mathcal{X})\partial_{\mathcal{X}}h_{j})$

on each edge $I_{j}$. The coefficients $\Lambda_{j}$ and $\Sigma_{j}$ are
obtained by taking line integrals as follows. Let
$\mathcal{C}_{j}(\mathcal{X})$ the connected component of the level set
$\\{H=\mathcal{X}\\}$ corresponding to a point $\mathcal{X}\in I_{j}$. Now
$\Sigma_{j}$ and $\Lambda_{j}$ are given by

$\displaystyle\Sigma_{j}(\mathcal{X})=2\int_{u=0}^{\infty}du\,R(u)\oint_{C_{j}(\mathcal{X})}\partial_{y}H(\check{X}^{x,y}_{u})\partial_{y}H(x,y)\,\frac{dl(x,y)}{|\nabla
H(x,y)|}\,,$
$\displaystyle\Lambda_{j}(\mathcal{X})=\oint_{C_{j}(\mathcal{X})}\frac{dl(x,y)}{\lvert\nabla
H(x,y)\rvert}\,.$

To describe the gluing condition, consider an interior vertex $O_{k}$
corresponding to the saddle point $(\mathbf{x}_{k},\mathbf{y}_{k})$ and the
level set $\\{H=\mathcal{X}_{k}\\}$, at which the collection of edges
$\\{I_{j}\\}$ meet. (The typical situation is a “figure eight” scenario, which
has three edges meeting at the vertex $O_{k}$.) Now define

$\beta_{kj}=\lim_{\mathcal{X}\to\mathcal{X}_{k}}C_{j}(\mathcal{X})\,,$

where the limit is taken along the edge $I_{j}$. The gluing condition at the
vertex $O_{k}$ can then be written as

(1.16) $\sum_{j:\,I_{j}\sim
O_{k}}\pm\beta_{kj}\,h^{\prime}_{j}(H(\mathbf{x}_{k},\mathbf{y}_{k}))=0\,,$

where sign before $\beta_{kj}$ is positive if $\mathcal{X}>\mathcal{X}_{k}$
along the edge $I_{j}$, and negative if $\mathcal{X}<\mathcal{X}_{k}$ along
the edge $I_{j}$.

The main difference between our main result, Theorem 1.4, and the standard
Freidlin–Wentzell theory [FW98] is in the coefficient $\Sigma$. The definition
of $\Sigma$ in Theorem 1.4 involves an average of
$\partial_{y}H(\check{X}^{x,y}_{u})\partial_{y}H(x,y)$, with a shift of one
factor to the point $\check{X}^{x,y}_{t=u}$. In [FW98] the coefficient
$\Sigma$ depends only on $(\partial_{y}H(x,y))^{2}$. The reason for this
modification lies in the fact that in our case the oscillations of the
Hamiltonian system and the random fluctuations in (1.4) are strongly coupled,
and are both of order $O(1/\varepsilon^{2})$. The oscillatory behavior of the
Hamiltonian system and random fluctuations fully interact with each other and
produce an effective diffusion coefficient involving the correlation of
$\partial_{y}H(\check{X}^{x,y}_{u})$ with $\partial_{y}H(x,y)$ for pairs of
points on the same orbit. If the oscillatory behavior of the Hamiltonian
system is slower than the random fluctuations, then their coupling is weak and
we would obtain the same generators as in [FW98]. This will arise if, for
example, we consider white noise in (1.4) as the driving random force. This
dichotomy of limiting behaviors between strongly and weakly coupled setups is
well-known in other approximation-diffusion limits of random differential
equations with periodic components, (for instance, compare and contrast
Theorems 6.4 and 6.5 in [FGPSl07]). In the present work, we need the
oscillations of the Hamiltonian system and the random fluctuations to be of
the same order; otherwise, the shift $\check{X}^{x,y}_{u}$, and hence
$\Sigma$, is not well-defined. We hope to present a proof of this phenomenon
in future work.

###  1.3. Plan of this paper.

In Section 2 we prove Proposition 1.1, modulo two computational lemmas. The
proofs of these lemmas are relegated to Appendix A. The proof of Proposition
1.3 is presented in Appendix B. In Section 3 we construct the noise process
$v$ that will be used in Theorem 1.4. We also establish basic properties of
the noise in this section and put two computationally involved proofs in
Appendices C and D. In Section 4 we state a precise, more general, version of
Theorem 1.4 (Theorems 4.3 and 4.4), and prove Theorem 1.4. We prove regularity
of the coefficients of the limiting equation in Theorem 4.3 (equation (4.13))
in Appendix E, and study the limiting equation itself in Section 5. Finally,
we prove Theorem 4.3 in Sections 6, 7 and 8.

### Acknowledgments

The authors thank Lenya Ryzhik for suggesting this problem to us, and for many
helpful discussions.

##  2\. Proof of convergence for a quadratic Hamiltonian.

This section is devoted to proving Proposition 1.1. The main idea behind the
proof is that when $H$ is quadratic, the deterministic dynamics rotates with
constant angular speed in all trajectories. Performing a spatial rotation will
now reduce the problem to studying convergence of Gaussian processes, which
can be resolved by computing covariances.

###### Proof of Proposition 1.1.

When $H$ is given by (1.6), we use Duhamel’s formula to write the solution of
(1.4) as

$X^{\varepsilon}_{t}=\begin{pmatrix}\phantom{-}\cos(\frac{t}{\varepsilon^{2}})&\sin(\frac{t}{\varepsilon^{2}})\\\
-\sin(\frac{t}{\varepsilon^{2}})&\cos(\frac{t}{\varepsilon^{2}})\end{pmatrix}X_{0}+\frac{1}{\varepsilon}\int_{0}^{t}v\Big{(}\frac{\tau}{\varepsilon^{2}}\Big{)}\begin{pmatrix}\sin(\frac{t-\tau)}{\varepsilon^{2}})\\\
\cos(\frac{t-\tau}{\varepsilon^{2}})\end{pmatrix}\,d\tau.$

Let $M(t)$ be the rotation matrix

$M(t)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\begin{pmatrix}\cos(t)&\sin(t)\\\
-\sin(t)&\cos(t)\end{pmatrix}\,,$

and define the rotated process $Y^{\varepsilon}$ by
$Y^{\varepsilon}_{t}=M(-t/\varepsilon^{2})X^{\varepsilon}_{t}$. Clearly
$H(X^{\varepsilon})=H(Y^{\varepsilon})$, and so convergence of the processes
$H(X^{\varepsilon})$ reduces to convergence of the processes
$H(Y^{\varepsilon})$.

We claim that the processes $Y^{\varepsilon}$ itself converge to a Brownian
motion as $\varepsilon\to 0$. To see this, observe

$Y^{\varepsilon}_{t}=X_{0}+\frac{1}{\varepsilon}\int_{0}^{t}v\Big{(}\frac{\tau}{\varepsilon^{2}}\Big{)}\begin{pmatrix}-\sin(\frac{\tau}{\varepsilon^{2}})\\\
\phantom{-}\cos(\frac{\tau}{\varepsilon^{2}})\end{pmatrix}\,d\tau\,.$

We will now show that the second term above converges to a Brownian motion.
For this, define

(2.1) $\displaystyle
w_{1}^{\varepsilon}(t)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\frac{1}{\varepsilon}\int_{0}^{t}v\left(\frac{\tau}{\varepsilon^{2}}\right)\sin\left(\frac{\tau}{\varepsilon^{2}}\right)d\tau\,,$
(2.2) $\displaystyle
w_{2}^{\varepsilon}(t)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\frac{1}{\varepsilon}\int_{0}^{t}v\left(\frac{\tau}{\varepsilon^{2}}\right)\cos\left(\frac{\tau}{\varepsilon^{2}}\right)d\tau\,.$

Note, the noise $v$ (when normalized similarly by $\sigma(\varepsilon)$
without the periodic terms) converges to fBm. However, the expressions above
use a normalization factor of $\varepsilon$ instead of $\sigma(\varepsilon)$,
and have an oscillatory factor. For this reason we claim
$(w_{1}^{\varepsilon},w_{2}^{\varepsilon})$ converges to a Brownian motion. To
prove convergence of $w_{1}^{\varepsilon}$, $w_{2}^{\varepsilon}$, we first
state two lemmas:

######  Lemma 2.1.

For any $T>0$, there exists a constant $C>0$ such that for every
$s,t\in[0,T]$, $i\in\\{1,2\\}$, we have

(2.3)
$\bm{E}(w^{\varepsilon}_{i}(t)-w^{\varepsilon}_{i}(s))^{2}\leqslant\begin{dcases}C|t-s|^{1-\gamma}&\gamma\in(0,1)\,,\\\
C\lvert t-s\rvert^{2-\gamma}&\gamma\in[1,2)\,.\end{dcases}$

######  Lemma 2.2.

For every $s,t\geqslant 0$, and $i\in\\{1,2\\}$ we have

(2.4) $\displaystyle\lim_{\varepsilon\to
0}\bm{E}(w^{\varepsilon}_{i}(t)-w^{\varepsilon}_{i}(s))^{2}=D_{11}|t-s|\,,$
(2.5) $\displaystyle\lim_{\varepsilon\to
0}\bm{E}(w^{\varepsilon}_{1}(t)-w^{\varepsilon}_{1}(s))(w^{\varepsilon}_{2}(t)-w^{\varepsilon}_{2}(s))=0\,.$

The proofs of both Lemmas 2.1 and 2.2 are lengthy, but direct computations.
Thus, for clarity of presentation, we postpone the proofs to Appendix A. Once
Lemmas 2.1 and 2.2 are established, the proof of Proposition 1.1 follows
quickly. Indeed, Lemma 2.1 implies that the family
$(Y^{\varepsilon})_{\varepsilon>0}$ is tight on
$\mathcal{C}([0,T],\mathbb{R}^{2})$

To see this, it suffices to show that the processes $w^{\varepsilon}_{i}$ are
tight. Without loss of generality suppose $T=1$, and suppose $\gamma\in(0,1)$
(the case when $\gamma\in[1,2)$ is similar). Choose an integer $M$ such that
$M_{\gamma}=M(1-\gamma)>1$. Since $w_{i}^{\varepsilon}$ is Gaussian process,
we know

$\bm{E}(w^{\varepsilon}_{i}(t)-w^{\varepsilon}_{i}(s))^{2M}=C_{M}[\bm{E}(w^{\varepsilon}_{i}(t)-w^{\varepsilon}_{i}(s))^{2}]^{M}$

for some constant $C_{M}$. We will allow $C_{M}$ to change from line to line
as long as it only depends on $M$, and remains independent of $\varepsilon$.
The above implies

(2.6) $\bm{E}(w^{\varepsilon}_{i}(t)-w^{\varepsilon}_{i}(s))^{2M}\leqslant
C_{M}|t-s|^{M_{\gamma}}\,.$

Since $M_{\gamma}>1$ by choice, Kolmogorov’s criterion (see for instance
[EK86] Proposition 10.3) implies that the family $w_{i}^{\varepsilon}$ is
tight. This implies that the processes $Y^{\varepsilon}$ converge in
distribution along a subsequence to a continuous process $Y$.

Since each process $Y^{\varepsilon}$ is a Gaussian process, the limiting
process $Y$ must also be a Gaussian process. Now Lemma 2.2 implies $Y$ is a
Brownian motion in $\mathbb{R}^{2}$ with $Y_{0}=X_{0}$ and covariance matrix
$D$. Since the law of the limiting process is uniquely determined, the family
$Y^{\varepsilon}$ itself must converge in distribution, without having to
select a subsequence. Finally, since
$H(X^{\varepsilon}_{t})=H(Y^{\varepsilon}_{t})$, we obtain convergence of
$(H(X^{\varepsilon}))_{\varepsilon>0}$ as claimed. ∎

##  3\. Construction of the Noise.

In this section, we will construct noise $v$ that will be used in Theorem 1.4,
and establish a few properties that will be used in the course of the proof of
Theorem 1.4. We require the covariance function $R$ to be of the form

(3.1)
$R(t)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\int_{S}r(p)e^{-\mu|p|^{2\beta}|t|}\,dp\,,$

where $S=(-r_{s},r_{s})$ is a symmetric, bounded, open interval, $\mu,\beta>0$
are constants, and $r\colon S-\\{0\\}\to[0,\infty)$ is defined by

$r(p)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\frac{\lambda(p)}{|p|^{2\alpha}}\,.$

Here $\lambda\colon S\to\mathbb{R}$ is a smooth bounded even function such
that $\lambda(0)\neq 0$ and

$\int_{S}r(p)\,dp>0\,,$

and

(3.2) $\alpha<\frac{1}{2}\,.$

We will now construct a stationary Gaussian process $v$ with covariance
function $R$. The chosen form of $R$ allows us to construct $v$ by
superimposing Ornstein–Uhlenbeck processes as follows. Let $\xi$ be 2D white
noise and define a Gaussian random measure $\mathcal{B}$ by

$\mathcal{B}(du,dp)=\sqrt{2\mu
r(p)}\,|p|^{\beta}\mathbf{1}_{S}(p)\,\xi(du,dp)\,,$

Clearly, the covariance of $\mathcal{B}$ is given by

$\bm{E}\mathcal{B}(du,dp)\,\mathcal{B}(du^{\prime},dp^{\prime})=2\mu\,r(p)\,|p|^{2\beta}\delta(u-u^{\prime})\delta(p-p^{\prime})\,du\,du^{\prime}\,dp\,dp^{\prime}\,.$

Now define the measure-valued Gaussian random process $V$ by

$V(t,dp)=\int_{-\infty}^{t}e^{-\mu|p|^{2\beta}(t-u)}\,\mathcal{B}(du,dp)\,,$

and finally define the noise $v$ by

$v(t)=\int_{S}V(t,dp)\,.$

We now establish a few properties of the process $v$ that will be used in the
proof of Theorem 1.4.

######  Lemma 3.1.

The process $v$ defined above is a stationary Gaussian process with covariance
function $R$.

###### Proof.

Clearly $v$ is a Gaussian process. To see that $v$ is stationary with
covariance function $R$ we choose any $s\leqslant t$ and compute

$\displaystyle\bm{E}v(s)v(t)$
$\displaystyle=\begin{multlined}\int_{-\infty}^{s}\int_{S}\exp\Big{(}-\mu\big{(}\lvert
p\rvert^{2\beta}(s-u)+\lvert p\rvert^{2\beta}(t-u)\big{)}\big{)}\cdot
2\mu\lvert
p\rvert^{2\beta}r(p)\,du\,dp\end{multlined}\int_{-\infty}^{s}\int_{S}\exp\Big{(}-\mu\big{(}\lvert
p\rvert^{2\beta}(s-u)+\lvert p\rvert^{2\beta}(t-u)\big{)}\big{)}\cdot
2\mu\lvert p\rvert^{2\beta}r(p)\,du\,dp$
$\displaystyle=\int_{S}r(p)e^{-\mu|p|^{2\beta}(t-s)}\,dp=R(t-s)\,.\qed$

######  Lemma 3.2.

If $R$ is given by (3.1), then

$\lim_{t\to\infty}t^{\gamma}R(t)=c_{0}$

where $c_{0}$ and $\gamma$ are defined by

$\gamma\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\frac{1-2\alpha}{2\beta}\qquad\text{and}\qquad
c_{0}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\lambda(0)\int_{p\in\mathbb{R}}\frac{e^{-\mu\lvert
p\rvert^{2\beta}}}{\lvert p\rvert^{2\alpha}}\,dp\,.$

###### Proof.

By a change of variable, note that

$\displaystyle
t^{\gamma}R(t)=t^{\gamma}\int_{S}\frac{\lambda(p)}{|p|^{2\alpha}}e^{-\mu|p|^{2\beta}t}\,dp$
$\displaystyle=\int_{(t^{1/(2\beta)}S)}\frac{1}{|p^{\prime}|^{2\alpha}}\lambda\Big{(}\frac{p^{\prime}}{t^{1/(2\beta)}}\Big{)}e^{-\mu|p^{\prime}|^{2\beta}}\,dp^{\prime}\,,$

which converges to $c_{0}$ as $t\to\infty$. ∎

######  Remark 3.3.

Note that (3.2) already implies $\gamma>0$. If additionally we assume

(3.3) $2\alpha+4\beta>1$

then we will also have $\gamma<2$. In this case the process $v$ can be
renormalized as in (1.11) to converge to a fBm. For Theorem 1.4, however, the
assumption (3.3) is unnecessary.

Let $\\{\mathcal{G}_{t}\\}$ be the augmented filtration generated by $V$. That
is,

(3.4)
$\mathcal{G}_{t}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\sigma\big{(}\mathcal{N}\cup\\{\sigma(V(s,\cdot))\nonscript\>|\nonscript\>\mathopen{}\allowbreak
0\leqslant s\leqslant t\\}\big{)}\,,$

where $\mathcal{N}$ is the set of all null sets known at time infinity. Our
next lemma computes conditional expectations of $V(\cdot,dp)$ with respect to
the filtration $\mathcal{G}$.

######  Lemma 3.4.

We have for any $t,h\geqslant 0$

(3.5)
$\bm{E}\big{[}V(t+h,dp)|\mathcal{G}_{t}\big{]}=e^{-\mu|p|^{2\beta}h}\,V(t,dp)$

and

(3.6)
$\begin{split}\bm{E}\Big{[}V(t+h,dp)V(t+h,dq)\Big{|}\mathcal{G}_{t}\Big{]}&-\bm{E}\big{[}V(t+h,dp)|\mathcal{G}_{t}\big{]}\bm{E}\big{[}V(t+h,dq)|\mathcal{G}_{t}\big{]}\\\
&=(1-e^{-2\mu|p|^{2\beta}h})r(p)\delta(p-q)\,dp\,dq\,.\end{split}$

Our last result concerns the boundedness of our noise process.

######  Lemma 3.5.

Let $T>0$, $M>0$,

$D_{k,M}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}[0,T]\times
L^{\infty}([0,T],W_{k,M})$

with

$W_{k,M}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\big{\\{}\varphi\in
W^{1,k}(S):\quad\|\varphi\|_{W^{1,k}}\leqslant M\big{\\}}$

and where $W^{1,k}(S)$ stands for the Sobolev space with $k\in(1,\infty]$. We
have

(3.7) $\bm{E}\Big{[}\sup_{(t,\varphi)\in
D_{k,M}}\Big{|}V\Big{(}\frac{t}{\varepsilon^{2}},\varphi(t,\cdot)\Big{)}\Big{|}\Big{]}\leqslant
C+\frac{C(\varepsilon)}{\varepsilon}\,,$

and for any $n\in\mathbb{N}^{*}$

(3.8) $\sup_{\varepsilon}\sup_{t\in[0,T]}\bm{E}\Big{[}\sup_{\varphi\in
W_{k,M}}\Big{|}V\Big{(}\frac{t}{\varepsilon^{2}},\varphi\Big{)}\Big{|}^{n}\Big{]}\leqslant
C_{n}\,,$

where $C$, $C_{n}$ and $C(\varepsilon)$ are three positive constants where the
latter satisfies

$\lim_{\varepsilon\to 0}C(\varepsilon)=0.$

Here the notation $V(t/\varepsilon^{2},\varphi)$ denotes integral of a
function $\varphi(p)$ with respect to the measure $V(t/\varepsilon^{2},dp)$.
That is,

$V\Big{(}\frac{t}{\varepsilon^{2}},\varphi\Big{)}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\int_{S}\varphi(p)\,V\Big{(}\frac{t}{\varepsilon^{2}},dp\Big{)}\,,$

and we will drop the $S$ in the above notation for simplicity unless specified
otherwise. The proofs of these two Lemmas are computationally involved, and we
relegate them to Appendices C and D respectively.

##  4\. The Main Theorem

One canonical way to analyze integrable Hamiltonian systems is to use a set of
_action-angle coordinates_. These coordinates separate the slow and fast
motion, and preserves the Hamiltonian structure.

###  4.1. Action-angle coordinates

The Liouville-Arnold theorem [Arn89] asserts that there exists a symplectic
canonical transformation $\varphi\colon
X=(x,y)\mapsto(I,\theta)\in\mathbb{R}\times\mathbb{T}$, where the action
variable $I$ and the angle variable $\theta$ satisfy

(4.1) $K(I)=H(x,y)\qquad\text{and}\qquad\\{I,\theta\\}=1\,,$

where $K$ is a given one variable smooth enough increasing function such that
$K(0)=0$, $\\{\cdot,\cdot\\}$ stands for the standard Poisson bracket and is
defined by

$\\{g,h\\}=\partial_{x}g\partial_{y}h-\partial_{y}g\partial_{x}h\,,$

for our Hamiltonian system with one degree of freedom. The relation on the
right-hand side of (4.1) can be also stated as

$\nabla I\cdot\nabla^{\perp}\theta=1\,.$

Note that in the action-angle coordinate the Hamiltonian is a function of the
action coordinate alone.

The existence of such transformation $\varphi$ is guaranteed by the Liouville-
Arnold theorem, and can be constructed through a generating function [Arn89,
Section 50 pp. 281], and implicitly defined by the relations

(4.2)
$y=\partial_{x}\mathcal{S}(I,x)\,,\qquad\theta=\partial_{I}\mathcal{S}(I,x)\,,\qquad\text{and}\qquad
H(x,\partial_{x}\mathcal{S}(I,x))=K(I)\,.$

Such a construction is not unique. For convenience in the forthcoming analysis
we will choose $\mathcal{S}(I,0)=0$ so that

$\displaystyle\mathcal{S}(I,x)=\int_{0}^{x}\partial_{x^{\prime}}\mathcal{S}(I,x^{\prime})\,dx^{\prime}\,,$
$\displaystyle\partial_{I}\mathcal{S}(I,x)=\int_{0}^{x}\partial_{Ix^{\prime}}\mathcal{S}(I,x^{\prime})\,dx^{\prime}\,,$

and hence

$\partial_{I}\mathcal{S}(I,0)=0\,.$

The meaning of the second relation is that the zero angle corresponds to
either the positive part of the $y$-axis or the negative part. Consequently,
we have for any $I>0$

(4.3)
$\varphi^{-1}_{1}(I,\theta=0)=0\qquad\text{and}\qquad\varphi^{-1}_{2}(I,\theta=0)\neq
0\,,$

and in particular,

(4.4) $\partial_{I}\varphi^{-1}_{1}(I,\theta=0)=0\,.$

The identity (4.3) follows from the fact that $K$ vanishes only for $I=0$. For
any $I>0$, the $x$ and $y$ components cannot simultaneously vanish.

Now let us write $\varphi=(\varphi_{1},\varphi_{2})$ and define

$I_{t}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}I(X_{t})=\varphi_{1}(X_{t})\,,\qquad\text{and}\qquad\theta_{t}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\theta(X_{t})=\varphi_{2}(X_{t})\,.$

In the absence of random fluctuations (i.e. when $v=0$), the Hamiltonian
system $\dot{X}_{t}=\nabla^{\perp}H(X_{t})$ becomes

$\dot{I}_{t}=0\,,\qquad\text{and}\qquad\dot{\theta_{t}}=\omega(I_{t})\,,\qquad\text{with}\qquad\omega(I)=K^{\prime}(I)\,.$

In the presence of random fluctuations, the Hamiltonian system (1.3) becomes

(4.5) $\dot{I}_{t}=\varepsilon
v(t)a(I_{t},\theta_{t})\,,\qquad\text{and}\qquad\dot{\theta}_{t}=\omega(I_{t})+\varepsilon
v(t)b(I_{t},\theta_{t})\,,$

where

$a=e_{2}\cdot\nabla\varphi_{1}\circ\varphi^{-1}\,,\quad\text{and}\quad
b=e_{2}\cdot\nabla\varphi_{2}\circ\varphi^{-1}\,.$

######  Example 4.1.

For illustration purposes, we now explicitly compute the action-angle
variables when the Hamiltonian is quadratic. From (4.2), we have

$x=\sqrt{\frac{I}{\pi}}\cos\Big{(}2\pi\theta-\frac{\pi}{2}\Big{)}\qquad\text{and}\qquad
y=\sqrt{\frac{I}{\pi}}\sin\Big{(}2\pi\theta-\frac{\pi}{2}\Big{)}$

where $\theta$ gives the angle of $X=(x,y)^{T}$ from the negative part of the
vertical axis. One can then see that the action variable is a multiple of the
Hamiltonian,

$K(I)=\frac{I}{2\pi}\qquad\text{and}\qquad\omega(I)=\frac{1}{2\pi},$

while the angle variable corresponds to the angle on a trajectory, which is a
circle in the case of $H(X)=|X|^{2}/2$, with period $1$. In this case, the
functions $a$ and $b$ are as follows

(4.6) $a(I,\theta)=2\sqrt{\pi
I}\sin\Big{(}2\pi\theta-\frac{\pi}{2}\Big{)}\quad\text{and}\quad
b(I,\theta)=\frac{1}{2\sqrt{\pi
I}}\cos\Big{(}2\pi\theta-\frac{\pi}{2}\Big{)}\,.$

###  4.2. The main theorem

To study the long-time behavior of (4.5), we let

$I^{\varepsilon}_{t}=I_{t/\varepsilon^{2}}\qquad\text{and}\qquad\theta^{\varepsilon}_{t}=\theta_{t/\varepsilon^{2}}\,.$

Now (4.5) becomes

(4.7)
$\dot{I}_{t}^{\varepsilon}=\frac{1}{\varepsilon}v\Big{(}\frac{t}{\varepsilon^{2}}\Big{)}a(I_{t}^{\varepsilon},\theta_{t}^{\varepsilon})\,,\qquad\dot{\theta}_{t}^{\varepsilon}=\frac{\omega(I_{t}^{\varepsilon})}{\varepsilon^{2}}+\frac{1}{\varepsilon}v\Big{(}\frac{t}{\varepsilon^{2}}\Big{)}b(I_{t}^{\varepsilon},\theta_{t}^{\varepsilon})\,,$

with $I^{\varepsilon}_{0}=I_{0}$ and $\theta^{\varepsilon}_{0}=\theta_{0}$.
Using the approach in [Gar97] to separate fast and slow motions, we study this
system by splitting the angle variable $\theta_{t}^{\varepsilon}$ into two
parts

$\theta_{t}^{\varepsilon}=\psi_{t}^{\varepsilon}+\tau_{t}^{\varepsilon}\,.$

The evolution of $\tau_{t}^{\varepsilon}$ (fast motion) is obtained by
averaging (4.7) over the angular coordinate, and $\psi_{t}^{\varepsilon}$
(slow motion) is the remainder. That is we require

(4.8)
$\dot{\tau}_{t}^{\varepsilon}=\frac{\omega(I_{t}^{\varepsilon})}{\varepsilon^{2}}+\frac{1}{\varepsilon}v\Big{(}\frac{t}{\varepsilon^{2}}\Big{)}\langle
b(I_{t}^{\varepsilon},\cdot)\rangle$

with initial condition $\tau^{\varepsilon}_{0}=0$, and where we have the
notation

$\langle
g\rangle\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\int_{\theta=0}^{1}g(\theta)\,d\theta\,.$

Considering only the slow motion variables
$(I_{t}^{\varepsilon},\psi_{t}^{\varepsilon})$, the system (4.7) becomes

(4.9)
$\begin{split}\dot{I}_{t}^{\varepsilon}&=\frac{1}{\varepsilon}v\Big{(}\frac{t}{\varepsilon^{2}}\Big{)}A(I_{t}^{\varepsilon},\psi_{t}^{\varepsilon},\tau_{t}^{\varepsilon})\\\
\dot{\psi}_{t}^{\varepsilon}&=\frac{1}{\varepsilon}v\Big{(}\frac{t}{\varepsilon^{2}}\Big{)}B(I_{t}^{\varepsilon},\psi_{t}^{\varepsilon},\tau_{t}^{\varepsilon})\end{split}$

with

$A(I,\psi,\tau)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}a(I,\psi+\tau)\qquad\text{and}\qquad
B(I,\psi,\tau)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}b(I,\psi+\tau)-\langle
b(I,\cdot)\rangle\,.$

The above equations are coupled with the initial conditions
$I^{\varepsilon}_{0}=I_{0}$, and $\psi^{\varepsilon}_{0}=\theta_{0}$. Note
that $A$ and $B$ are both 1-periodic and mean-zero with respect to $\tau$.
This latter property is mandatory to deal with the long-range correlation case
as illustrated in Proposition 1.1. In fact, if the system possesses a
component with zero frequency, a noise with long-range correlations will
charge this component and cause the system to blow up as $\varepsilon\to 0$.

It is straightforward to see this mean-zero property in $\tau$ for $B$, but
for $A$ we use that

$\textrm{Jac}\,\varphi^{-1}(I,\theta)=[\textrm{Jac}\,\varphi(\varphi^{-1}(I,\theta))]^{-1}$

and that $\det\textrm{Jac}\,\varphi(X)=1$ since $\varphi$ is a symplectic
transformation, to obtain the following relations

(4.10)
$\begin{split}\partial_{I}\varphi^{-1}_{1}(I,\theta)&=\partial_{y}\varphi_{2}(X),\quad\partial_{I}\varphi^{-1}_{2}(I,\theta)=-\partial_{x}\varphi_{2}(X)\,,\\\
\partial_{\theta}\varphi^{-1}_{1}(I,\theta)&=-\partial_{y}\varphi_{1}(X),\quad\partial_{\theta}\varphi^{-1}_{2}(I,\theta)=\partial_{x}\varphi_{1}(X)\,.\end{split}$

Therefore, we have

$a(I,\theta)=-\partial_{\theta}\varphi^{-1}_{1}(I,\theta)$

which is clearly mean-zero with respect to $\theta$ since we have the
derivative of a periodic function. Note also that from these relations the
mean-zero property in $\theta$ for $b$ is not clear. This is the reason why we
introduce the compensation $\langle b\rangle$ in the definition of $B$.

Our main result obtains the limiting behavior of (4.9) as $\varepsilon\to 0$
under the following assumptions.

  * •

The function $K$ is smooth, and

(4.11) $\inf_{I\geqslant 0}K^{\prime}(I)=\inf_{I\geqslant
0}\omega(I)>\omega_{0}>0$

for some strictly positive number $\omega_{0}$.

  * •

There exist $r>0$, and positive constants $c_{1,r}$, $c_{2,r}>0$ such that for
any $I\in(0,r)$

(4.12) $c_{1,r}I\leqslant K(I)\leqslant
c_{2,r}I\qquad\text{and}\qquad|\omega^{\prime}(I)|\leqslant\frac{c_{2,r}}{I}\,.$

Note that these conditions imply that $K(0)=0$ and that $K$ is an increasing
function in $I$. These assumptions are not too restrictive since any
Hamiltonian satisfies (4.11) and (4.12) near non-degenerate critical points.
The following proposition is a consequence of the above assumptions, which
concerns bounds on the function $a$. The proof is given in Appendix E.

######  Proposition 4.2.

Under (4.11) and (4.12) there exist $r>0$ and a constant $C_{r}>0$ such that
for any $I\in(0,r)$ we have

$\sup_{\theta\in\mathbb{T}}\Big{(}|a(I,\theta)|+|\partial_{\theta}a(I,\theta)|\Big{)}\leqslant
C_{r}\sqrt{I}\,,$

and

$\sup_{\theta\in\mathbb{T}}|\partial_{I}a(I,\theta)|\leqslant\frac{C_{r}}{\sqrt{I}}\,.$

The proposition mainly describes the behavior of $a,\partial_{I}a$ and
$\partial_{\theta}a$ around $I=0$. These bounds will allow us to prove the
main result of the paper that characterizes the limiting process of the
sequence of processes $(I^{\varepsilon},\psi^{\varepsilon})_{\varepsilon>0}$.

######  Theorem 4.3.

Assume (4.11) and (4.12) hold, the family
$(I^{\varepsilon},\psi^{\varepsilon})_{\varepsilon>0}$ (defined in
(4.8)–(4.9)) converges in distribution in
$\mathcal{C}([0,\infty),\mathbb{R}^{2})$ to a process
$(I_{t},\psi_{t})_{t\geqslant 0}$, where $(I_{t})_{t\geqslant 0}$ is the
unique weak solution of the SDE

(4.13)
$\begin{split}dI_{t}&=\int_{\tau=0}^{1}\,a(I_{t},\tau)\,dW_{t}(I_{t},\tau)\,d\tau\\\
&\quad+\Big{[}\int_{u=0}^{\infty}R(u)\int_{\tau=0}^{1}\partial_{I}\big{(}a(I,\tau+\omega(I)u)\big{)}_{|I=I_{t}}a(I_{t},\tau)\\\
&\qquad\qquad+a(I_{t},\tau+\omega(I_{t})u)\partial_{I}a(I_{t},\tau)\,d\tau\,du\Big{]}\,dt\,,\end{split}$

with initial condition $I_{t=0}=I_{0}$. Here, $W$ is a real valued Brownian
field with covariance function

(4.14) $\begin{split}\bm{E}[W_{t}(y,\phi_{1})W_{s}(y,\phi_{2})]&=t\wedge
s\int_{u=0}^{\infty}R(u)\\\
&\quad\times\int_{\tau=0}^{1}\phi_{1}(\tau+\omega(y)u)\phi_{2}(\tau)+\phi_{1}(\tau)\phi_{2}(\tau+\omega(y)u)\,d\tau\,du\,,\end{split}$

for any $\phi_{1},\phi_{2}\in L^{2}_{0}(\mathbb{T})$ where

$L^{2}_{0}(\mathbb{T})\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\big{\\{}\phi\in
L^{2}(\mathbb{T}):\langle\phi\rangle=0\big{\\}}\,.$

Explicitly, we can write

$W_{t}(y,\tau)=\frac{1}{\sqrt{2}}\sum_{n\in\mathbb{Z}^{*}}e^{2i\pi
n\tau}R^{1/2}_{n}(y)(B^{1}_{t,n}-iB^{2}_{t,n})$

where $B^{1}_{t,n}$ and $B^{2}_{t,n}$ are two independent real valued
cylindrical Brownian motions satisfying $B^{1}_{t,-n}=B^{1}_{t,n}$,
$B^{2}_{t,-n}=-B^{2}_{t,n}$, and $R_{n}=2\int_{0}^{\infty}R(u)\cos(2\pi
n\omega(I)u)\,du$.

Also, we have

(4.15)
$\begin{split}d\psi_{t}&=\int_{\tau=0}^{1}\,b(I_{t},\tau)\,dW_{t}(I_{t},\tau)\,d\tau\\\
&\qquad+\Big{[}\int_{u=0}^{\infty}R(u)\int_{\tau=0}^{1}\partial_{I}\big{(}b(I,\tau+\omega(I)u)\big{)}_{|I=I_{t}}a(I_{t},\tau)\\\
&\qquad\qquad+b(I_{t},\tau+\omega(I_{t})u)\partial_{I}a(I_{t},\tau)\,d\tau\,du\,\Big{]}dt\,,\end{split}$

with $\psi_{t=0}=\theta_{0}$.

Note that the action variable $I$ does not depend on the slow angular motion
$\psi$. However, the distribution of the slow angular motion $\psi$ is
completely determined by the motion of the action variable $I$ and the
Brownian field.

We also note that the functions $\partial_{I}a$ and $\partial_{I}b$ appearing
in (4.13) and (4.15) may be singular at $I=0$. Nevertheless, since the minimum
of $H$ is non-degenerate, the function $a\partial_{I}a$ has no singularity at
$I=0$ and there is no singularity on the right-hand side of equation (4.13).
The term $a\partial_{I}b$ appearing in (4.15) may still have an $O(I^{-1/2})$
singularity at $I=0$. However, since the point $I=0$ is inaccessible
(Proposition 5.2), the right-hand side of (4.15) is well defined.

We now use Theorem 4.3 to prove Theorem 1.4.

###### Proof of Theorem 1.4.

Using action angle coordinates, we convert the system (1.4) to (4.9). Since
the Hamiltonian $H$ has exactly one non-degenerate critical point, the
assumptions (4.11)–(4.12) are satisfied. Now by Theorem 4.3 we see that the
family of processes $(I^{\varepsilon},\psi^{\varepsilon})_{\varepsilon>0}$
converges in distribution to the pair $(I,\psi)$ which solves (4.13), (4.15).
Since $H(X^{\varepsilon})=K(I^{\varepsilon})$, we now obtain Theorem 1.4 by
considering $\mathcal{X}=K(I)$. considering $\mathcal{X}=K(I)$. The
infinitesimal generator for $I$ can be written in the form

$\frac{1}{2}(\partial_{I}\tilde{\Sigma}(I)\partial_{I})$

where

$\tilde{\Sigma}(I)=2\int_{u=0}^{\infty}R(u)\int_{\tau=0}^{1}a(I,\tau+\omega(I)u)a(I,\tau)\,du\,d\tau\,.$

Now, using (E.1), (E.5) and that
$\varphi^{-1}(I,\tau+\omega(I)u)=\check{X}^{x,y}_{u}$ for
$\varphi^{-1}(I,\tau)=(x,y)$, together with
$\omega(I)=1/\Lambda(\mathcal{X})$, we have

$\tilde{\Sigma}(I)=\frac{2}{\omega(I)}\int_{u=0}^{\infty}du\,R(u)\oint_{\\{H=\mathcal{X}\\}}\partial_{y}H(\check{X}^{x,y}_{u})\partial_{y}H(x,y)|\nabla
H(x,y)|^{-1}dl(x,y)\,.$

Finally, using $\partial_{I}=\omega(I)\partial_{\mathcal{X}}$ provides the
desired result. ∎

For the quadratic Hamiltonian we can explicitly derive the stochastic
differential equations for $(I_{t})_{t\geqslant 0}$ and
$(\psi_{t})_{t\geqslant 0}$. Using equation (4.6) we obtain

(4.16) $dI_{t}=2\sqrt{mI_{t}}dB^{1}_{t}+2m\,dt\qquad\text{and}\qquad
d\psi_{t}=\frac{m}{2\pi\sqrt{I_{t}}}\,dB^{2}_{t}$

with

$m=\pi\int_{0}^{\infty}R(u)\cos(u)\,du\,,$

and where $B^{1}$ and $B^{2}$ are two independent standard Brownian motions.
One can easily remark that $(I_{t}/m)_{t\geqslant 0}$ is a 2-dimensional
squared Bessel process, so that $(I_{t}/2\pi)_{t\geqslant 0}$ has the same law
as $(|W_{t}|^{2}/2)_{t\geqslant 0}$, for $W$ being a 2-dimensional Brownian
motion with covariance matrix given by (1.9). Then, we recover the result of
Proposition 1.1.

The proof of Theorem 4.3 will be given in Section 6.

###  4.3. Generalization for Hamiltonian systems with one degree of freedom
and 2-dimensional noises

Theorem 4.3 can be readily extended to general Hamiltonian systems with one
degree of freedom and 2-dimensional noises. In fact, our proof does not depend
on the particular shapes of Hamiltonians for a one-degree of freedom
oscillator problems. Also, considering 2-dimensional noises brings no
additional difficulties except more tedious notations in the proof of Theorem
4.3. Here we state this extension of Theorem 4.3 for the sake of completeness,
but the detailed proof will be omitted.

We consider the following one-degree of freedom Hamiltonian system

$X^{\varepsilon}_{t}=\frac{1}{\varepsilon}\nabla^{\perp}H(X^{\varepsilon}_{t})+\frac{1}{\varepsilon}\mathbf{v}\Big{(}\frac{t}{\varepsilon^{2}}\Big{)}\,,\qquad
X^{\varepsilon}_{0}=X_{0}\,,$

where the Hamiltonian $H$ is as in Theorem 4.3, and $\mathbf{v}$ is a
2-dimensional noise with covariance function

$\mathbf{R}_{jl}(t-s)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\bm{E}[\mathbf{v}_{j}(t)\mathbf{v}_{l}(s)]=\int_{S}\frac{\lambda_{jl}(p)}{|p|^{2\alpha}}e^{-\mu|p|^{2\beta}|t-s|}\,dp\qquad
j,l\in\\{1,2\\}\,,$

where $(\lambda_{jl})_{j,l\in\\{1,2\\}}$ is a $2\times 2$ symmetric matrix
valued function with assumptions similar to the ones considered in Section 3.

Introducing the same action-angle coordinates as in Section 4.1, they now
satisfy

$\dot{I}_{t}^{\varepsilon}=\frac{1}{\varepsilon}\mathbf{v}\Big{(}\frac{t}{\varepsilon^{2}}\Big{)}\cdot\mathbf{A}(I_{t}^{\varepsilon},\psi_{t}^{\varepsilon},\tau_{t}^{\varepsilon})\qquad\text{and}\qquad\dot{\psi}_{t}^{\varepsilon}=\frac{1}{\varepsilon}\mathbf{v}\Big{(}\frac{t}{\varepsilon^{2}}\Big{)}\cdot\mathbf{B}(I_{t}^{\varepsilon},\psi_{t}^{\varepsilon},\tau_{t}^{\varepsilon}),$

with

$\dot{\tau}_{t}^{\varepsilon}=\frac{\omega(I_{t}^{\varepsilon})}{\varepsilon^{2}}+\frac{1}{\varepsilon}\mathbf{v}\Big{(}\frac{t}{\varepsilon^{2}}\Big{)}\cdot\langle
b(I_{t}^{\varepsilon},\cdot)\rangle\,,$

and where

$\mathbf{A}(I,\psi,\tau)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\begin{pmatrix}a_{1}(I,\psi+\tau)\\\
a_{2}(I,\psi+\tau)\end{pmatrix}\qquad\text{and}\qquad\mathbf{B}(I,\psi,\tau)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\begin{pmatrix}b_{1}(I,\psi+\tau)-\langle
b_{1}(I,\cdot)\rangle\\\ b_{2}(I,\psi+\tau)-\langle
b_{2}(I,\cdot)\rangle\end{pmatrix},$

with

$a_{j}(I,\theta)=e_{j}\cdot\nabla\varphi_{1}(\varphi^{-1}(I,\theta))\qquad\text{and}\qquad
b_{j}(I,\theta)=e_{j}\cdot\nabla\varphi_{2}(\varphi^{-1}(I,\theta))\,,$

for $j\in\\{1,2\\}$.

The extension of Theorem 4.3 is as follows.

######  Theorem 4.4.

The family $(I^{\varepsilon},\psi^{\varepsilon})_{\varepsilon>0}$ converges in
distribution in $\mathcal{C}([0,\infty),\mathbb{R}^{2})$ to a process
$(I_{t},\psi_{t})_{t\geqslant 0}$, where $(I_{t})_{t\geqslant 0}$ is the
unique weak solution of the SDE

(4.17) $\begin{split}dI_{t}&=\int_{\tau=0}^{1}\,a(I_{t},\tau)\cdot
dW_{t}(I_{t},\tau)\,d\tau\\\
&\quad+\int_{u=0}^{\infty}\int_{\tau=0}^{1}\partial_{I}\big{(}a^{T}(I,\tau+\omega(I)u)\big{)}_{|I=I_{t}}\mathbf{R}(u)a(I_{t},\tau)\\\
&\qquad\qquad+a^{T}(I_{t},\tau+\omega(I_{t})u)\mathbf{R}(u)\partial_{I}a(I_{t},\tau)\,d\tau\,du\,dt,\end{split}$

with initial condition $I_{t=0}=I_{0}$. Here, $W$ is a 2-dimensional real
valued Brownian field with covariance function

$\displaystyle\bm{E}[W^{j}_{t}(y,\phi_{1})W^{l}_{s}(y,\phi_{2})]$
$\displaystyle=(t\wedge s)\int_{u=0}^{\infty}R_{jl}(u)$
$\displaystyle\quad\times\int_{\tau=0}^{1}\phi_{1}(\tau+\omega(y)u)\phi_{2}(\tau)+\phi_{1}(\tau)\phi_{2}(\tau+\omega(y)u)\,d\tau\,du\,,$

for any $j,l\in\\{1,2\\}$ and $\phi_{1},\phi_{2}\in L^{2}_{0}(\mathbb{T})$
where

$L^{2}_{0}(\mathbb{T})\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\big{\\{}\phi\in
L^{2}(\mathbb{T}):\langle\phi\rangle=0\big{\\}}\,.$

Also, we have

(4.18) $\begin{split}d\psi_{t}&=\int_{\tau=0}^{1}\,b(I_{t},\tau)\cdot
dW_{t}(I_{t},\tau)\,d\tau\\\
&\quad+\int_{u=0}^{\infty}\int_{\tau=0}^{1}\partial_{I}\big{(}b^{T}(I,\tau+\omega(I)u)\big{)}_{|I=I_{t}}\mathbf{R}(u)a(I_{t},\tau)\\\
&\qquad\qquad+b^{T}(I_{t},\tau+\omega(I_{t})u)\mathbf{R}(u)\partial_{I}a(I_{t},\tau)\,d\tau\,du\,dt\,,\end{split}$

with $\psi_{t=0}=\theta_{0}$.

##  5\. Properties of the Limiting Equation for the Action Variable

In this section, we study properties of the equation (4.13). The main issue
concerns the behavior of $a$ at $I=0$, but we will see how to define a unique
global solution for this equation that does not reach $0$ with probability
one. In the case of a quadratic Hamiltonian, we have seen in (4.16) that
$(I_{t})_{t\geqslant 0}$ is related to a 2-dimensional squared Bessel process
(see (4.16)) which is known to reach $0$ with probability zero [KS91,
Proposition 3.22 p. 161].

###  5.1. Existence of a solution

First, we remark that the function $a$ is not Lipschitz in $I$ as it typically
has a square-root singularity near $0$. So it is not immediately apparent that
equation (4.13) has solutions. To construct solutions to equation
$\eqref{eq:I}$, define

(5.1) $\displaystyle
a_{n}(I)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\int_{\tau=0}^{1}a(I,\tau)e^{-2i\pi
n\tau}\,d\tau$ (5.2) $\displaystyle
R_{n}(I)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}2\int_{u=0}^{\infty}R(u)\cos(2\pi
n\omega(I)u)\,du\,.$

and let

$\displaystyle\mathbf{a}(I)$
$\displaystyle\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\sum_{n\in\mathbb{Z^{\ast}}}|a_{n}(I)|^{2}R_{n}(I)$
(5.3)
$\displaystyle=2\int_{u=0}^{\infty}R(u)\int_{\tau=0}^{1}a(I,\tau)a(I,\tau+\omega(I)u)\,d\tau\,du\,,$
$\displaystyle\mathbf{b}(I)$
$\displaystyle\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\int_{u=0}^{\infty}R(u)\int_{\tau=0}^{1}\Big{[}\partial_{I}(a(I,\tau+\omega(I)u))a(I,\tau)$
(5.4)
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\partial_{I}a(I,\tau)a(I,\tau+\omega(I)u)\Big{]}\,d\tau\,du\,.$

Note that equality in (5.1) is justified by using the Fourier series expansion
for $a(I,\tau)$, and using the fact that $a(I,\tau)$ is mean-zero in $\tau$.
That is,

$a(I,\tau)=\sum\limits_{n\in\mathbb{Z}^{*}}e^{2i\pi n\tau}a_{n}(I)\,.$

######  Definition 5.1.

A weak solution in the interval $(0,\infty)$ to equation (4.13) is a filtered
probability space $(\Omega,\mathcal{F},\mathcal{F}_{t},\bm{P})$ satisfying the
usual conditions and a continuous pair of processes $(I_{t},W_{t})$ such that
the followings hold.

  1. (1)

The process $I$ takes values in $[0,\infty]$ with $I_{0}\in(0,\infty)$,

  2. (2)

The process $W$ is a standard $\mathcal{F}_{t}$-adapted Brownian motion on the
Hilbert space $L^{2}_{0}(\mathbb{T})$ with covariance function given by
(4.14).

  3. (3)

For any $t,M>0$ we have

(5.5)
$\displaystyle\int_{s=0}^{t\wedge\zeta_{M}}(\mathbf{a}(I_{s})+|\mathbf{b}(I_{s})|)\,ds<\infty\,,$
(5.6) $\displaystyle
I_{t\wedge\zeta_{M}}=I_{0}+\int_{s=0}^{t\wedge\zeta_{M}}\int_{\tau=0}a(I_{s},\tau)\,dW_{s}(I_{s},\tau)+\int_{s=0}^{t\wedge\zeta_{M}}\mathbf{b}(I_{s})\,ds\quad\forall
t\geqslant 0\,,$

almost surely. Here
$\zeta_{M}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\zeta_{1,M}\wedge\zeta_{2,M}$
where $\zeta_{1,M}$ and $\zeta_{2,M}$ are defined by

$\displaystyle\zeta_{1,M}$
$\displaystyle\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\inf\\{t\geqslant
0\,|\,|I_{t}|\geqslant M\\}$ $\displaystyle\zeta_{2,M}$
$\displaystyle\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\inf\\{t\geqslant
0\,|\,|I_{t}|\leqslant 1/M\\}\,.$

For notational convenience we denote

(5.7)
$\zeta_{\infty}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\lim_{M\to\infty}\zeta_{M}=\inf\\{t\geqslant
0\,|\,I_{t}\not\in(0,\infty)\\}$

to be the first exit time of $I$ from $(0,\infty)$.

We will now work with the canonical probability space
$\mathcal{C}([0,\infty),\mathbb{R})$ and canonical filtration

$\mathcal{M}_{t}=\sigma(h_{s},\,0\leqslant s\leqslant t)\,.$

To construct a solution to (4.13), we note first that when restricted to the
interval $[1/M,M]$, the function $a$ is Lipschitz. Thus we know (4.13) has a
strong solution when truncated to this interval (see Proposition 6.1, below).
If we denote this truncated solution by $(I^{M},\zeta_{M})$ we obtain an
increasing sequence of stopping times $\zeta_{M}$, an increasing family of
$\sigma$-algebras $\mathcal{M}_{\zeta_{M}}$, and a sequence of probability
measures $\bm{P}^{M}_{I}$ such that

$\bm{P}^{M+1}_{I}=\bm{P}^{M}_{I}\qquad\text{on}\quad\mathcal{M}_{\zeta_{M}}\qquad\forall
M>0\,.$

As a result, we can define $\bm{P}^{0}$ on
$\bigcup_{M}\mathcal{M}_{\zeta_{M}}$ by

$\bm{P}^{0}_{I}(O)=\bm{P}^{M}_{I}(O)\qquad\text{for}\quad
O\in\mathcal{M}_{\zeta_{M}}\,,$

and extend $\bm{P}^{0}_{I}$ to a probability measure on the $\sigma$-algebra
$\mathcal{M}_{\zeta_{\infty}}$ such that for any $M\in\mathbb{N}$ we have

(5.8)
$\bm{P}^{0}_{I}=\bm{P}^{M}_{I}\qquad\text{on}\quad\mathcal{M}_{\zeta_{M}}\,.$

Note that

(5.9)
$\mathcal{M}_{\zeta_{\infty}}=\sigma\Big{(}\bigcup_{M}\mathcal{M}_{\zeta_{M}}\Big{)}\,,$

which can be easily verified from the fact that

$\mathcal{M}_{\zeta_{\infty}}=\sigma(h_{t\wedge\zeta_{\infty}},t\geqslant
0)\,,$

(see for instance [SV06, Lemma 1.3.3 p. 33]). As a result $\bm{P}^{0}_{I}$
provides a solution to (4.13), that we can denote by $I=(I_{t})_{t\geqslant
0}$, in the sense of Definition 5.1. Note that the process
$(I_{t})_{t\geqslant 0}$ is nonnegative with probability one thanks to the
Portmanteau theorem [Bil99, Theorem 2.1 p. 16].

In the next two sections, we will first show (using the bounds in Proposition
4.2) that $\bm{P}^{0}_{I}(\zeta_{\infty}=\infty)=1$. That is, the process $I$
does not reach $0$ or blow up in finite time. This will show that
$\bm{P}^{0}_{I}$ is the unique extension on
$\mathcal{M}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\sigma(\bigcup_{t\geqslant
0}\mathcal{M}_{t})$ satisfying (5.8). Additionally, we will show that
$I=(I_{t})_{t\geqslant 0}$ is the unique global solution to (4.13).

###  5.2. Non-explosion and inaccessibility to $0$

Our aim in this section is to show that the process $I$ can not explode in
finite time nor reach $0$.

######  Proposition 5.2.

Consider a weak solution to (4.13) (as in definition 5.1), and let
$\zeta_{\infty}$ be as in (5.7). Then

$\bm{P}^{0}_{I}(\zeta_{\infty}=\infty)=1\,.$

This result together with (5.8) implies that for any $t\geqslant 0$

$\lim_{M\to\infty}\bm{P}^{M}_{I}(\zeta_{M}\leqslant
t)=\lim_{M\to\infty}\bm{P}^{0}_{I}(\zeta_{M}\leqslant t)=0\,.$

Therefore, [SV06, Theorem 1.3.5 p. 34] guarantees that $\bm{P}^{0}_{I}$ is the
unique extension satisfying (5.8).

Before proving Proposition 5.2 we need to introduce the _scale function_

$p(x)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\int_{1}^{x}\exp\left(-2\int_{1}^{\xi}\frac{\mathbf{b}(\nu)}{\mathbf{a}(\nu)}\,d\nu\right)d\xi,\quad
x\in(0,\infty)$

where $\mathbf{a}$ and $\mathbf{b}$ are defined by (5.1) and (5.1)
respectively. Observe

(5.10)
$p^{\prime\prime}(x)=-2\frac{\mathbf{b}(x)}{\mathbf{a}(x)}p^{\prime}(x).$

Standard results express the exit probability of $I$ in terms of the scale
function, and we prove them in our context for completeness.

######  Lemma 5.3.

For any $0<x_{-}<x_{+}<\infty$ we have

$\bm{P}(I_{\zeta_{x_{-},x_{+}}}=x_{-})=\frac{p(x_{+})-p(I_{0})}{p(x_{+})-p(x_{-})}\quad\text{and}\quad\bm{P}(I_{\zeta_{x_{-},x_{+}}}=x_{+})=\frac{p(I_{0})-p(x_{-})}{p(x_{+})-p(x_{-})}\,,$

where $\zeta_{x_{-},x_{+}}=\zeta_{1,x_{+}}\wedge\zeta_{2,x_{-}}$.

###### Proof of Lemma 5.3..

The proof consists of applying the Itô formula for $p(I_{t})$ with
$t\leqslant\zeta_{x_{-},x_{+}}\leqslant\zeta_{\infty}$. To do this we rewrite
the stochastic integral of $\eqref{eq:I}$ in the Fourier domain, that is

$\int_{s=0}^{t}\int_{\tau=0}^{1}a(I_{s},\tau)\,dW_{s}(I_{s},\tau)\,d\tau=\int_{s=0}^{t}\sum_{n\in\mathbb{Z}^{\ast}}a_{n}(I_{s})R^{1/2}_{n}(I_{s})\frac{1}{\sqrt{2}}(dB^{1}_{s,n}-idB^{2}_{s,n})\,,$

providing an explicit semi-martingale representation for $(I_{t})_{t\geqslant
0}$. Therefore, applying the Itô formula to $p(I_{t})$ we have

$\begin{split}&p(I_{t\wedge\zeta_{x_{-},x_{+}}})=p(I_{0})+\int_{0}^{t\wedge\zeta_{x_{-},x_{+}}}p^{\prime}(I_{s})\,dI_{s}+\frac{1}{2}\int_{0}^{t\wedge\zeta_{x_{-},x_{+}}}p^{\prime\prime}(I_{s})d\langle
I\rangle_{s}\\\
&=p(I_{0})+\int_{0}^{t\wedge\zeta_{x_{-},x_{+}}}p^{\prime}(I_{s})\sum_{n\in\mathbb{Z}^{\ast}}a_{n}(I_{s})R^{1/2}_{n}(I_{s})\frac{1}{\sqrt{2}}(dB^{1}_{s,n}-idB^{2}_{s,n})\\\
&\qquad+\int_{0}^{t\wedge\zeta_{x_{-},x_{+}}}p^{\prime}(I_{s})\mathbf{b}(I_{s})\,ds\\\
&\qquad+\frac{1}{2}\int_{0}^{t\wedge\zeta_{x_{-},x_{+}}}-2\frac{\mathbf{b}(I_{s})}{\mathbf{a}(I_{s})}p^{\prime}(I_{s})\sum_{n\in\mathbb{Z}^{\ast}}|a_{n}(I_{s})|^{2}R_{n}(I_{s})\,ds\\\
&=p(I_{0})+\int_{0}^{t\wedge\zeta_{x_{-},x_{+}}}p^{\prime}(I_{s})\sum_{n\in\mathbb{Z}^{\ast}}a_{n}(I_{s})R^{1/2}_{n}(I_{s})\frac{1}{\sqrt{2}}(dB^{1}_{s,n}-idB^{2}_{s,n})\end{split}$

thanks to (5.10) and (5.1). Note that the stochastic integral on the right-
hand side of the last line is a martingale starting from $0$, so by taking the
expectation, and passing to the limit $t\to\infty$ we obtain

$p(I_{0})=\bm{E}[p(I_{\zeta_{x_{-},x_{+}}})]=p(x_{-})\bm{P}(I_{\zeta_{x_{-},x_{+}}}=x_{-})+p(x_{+})\bm{P}(I_{\zeta_{x_{-},x_{+}}}=x_{+})\,.$

Solving for $\bm{P}(I_{\zeta_{x_{-},x_{+}}}=x_{-})$ and
$\bm{P}(I_{\zeta_{x_{-},x_{+}}}=x_{+})$ we obtain the desired result. ∎

With this lemma, we can now turn to the proof of Proposition 5.2.

###### Proof of Proposition 5.2..

Let us start with the following remark. The term $\mathbf{b}$ can be written
as

$\mathbf{b}(I)=\frac{1}{2}\partial_{I}\mathbf{a}(I)\,,$

so that

$\begin{split}p(0_{+})&=\lim_{x\to
0_{+}}\int_{1}^{x}\exp\left(-2\int_{1}^{\xi}\frac{\mathbf{b}(\nu)}{\mathbf{a}(\nu)}d\nu\right)d\xi\\\
&=\lim_{x\to
0_{+}}\mathbf{a}(1)\int_{1}^{x}\exp\left(-\ln(\mathbf{a}(\xi))\right)\,d\xi\\\
&=\lim_{x\to
0_{+}}\mathbf{a}(1)\int_{1}^{x}\frac{d\xi}{\mathbf{a}(\xi)}\,.\end{split}$

Now, according to (5.1) together with Proposition 4.2, we have

$0\leqslant\mathbf{a}(\xi)\leqslant C\,\xi$

since for any $n\in\mathbb{Z}^{\ast}$

$\displaystyle 0\leqslant R_{n}(\xi)$
$\displaystyle=2\int_{0}^{\infty}R(u)\cos(2\pi n\omega(\xi)u)\,du$
$\displaystyle=2\int_{0}^{\infty}\Big{(}\int_{S}r(p)e^{-\mu|p|^{2\beta}u}\,dp\Big{)}\cos(2\pi
n\omega(\xi)u)\,du$
$\displaystyle=2\int_{S}\,dp\,r(p)\Big{(}\int_{0}^{\infty}\cos(2\pi
n\omega(\xi)u)e^{-\mu|p|^{2\beta}u}\,du\Big{)}$ (5.11)
$\displaystyle=\int_{S}dp\,r(p)\frac{2\mu|p|^{2\beta}}{\mu^{2}|p|^{4\beta}+4\pi^{2}n^{2}\omega^{2}(\xi)}\leqslant\frac{C_{\omega_{0}}}{n^{2}}\int_{S}|p|^{2\beta}r(p)\,dp\,.$

As a result, we have

$p(0_{+})\leqslant\lim_{x\to
0_{+}}\mathbf{a}(1)\int_{1}^{x}\frac{d\xi}{C\xi}=-\infty\,,$

which implies $p(0_{+})=-\infty$. To conclude the proof we consider two cases,
that is $p(\infty)=\infty$ and $p(\infty)<\infty$. Following the same lines as
[KS91, Proposition 5.22 p. 345], if $p(\infty)=\infty$ we directly have the
desired result, and if $p(\infty)<\infty$ we have

$\displaystyle\lim\limits_{x_{-}\to
0_{+}}\lim\limits_{x_{+}\to\infty}\bm{P}(I_{\zeta_{x_{-},x_{+}}}=x_{+})=\lim\limits_{x_{-}\to
0_{+}}\lim\limits_{x_{+}\to\infty}\frac{p(I_{0})-p(x_{-})}{p(x_{+})-p(x_{-})}=1\,.$

This implies $\bm{P}(\zeta_{\infty}=\zeta_{1,\infty})=1$ meaning that if
$I_{t}$ exits the interval $(0,\infty)$, it almost surely does so at $\infty$.
Here

$\zeta_{1,\infty}=\lim_{M\to\infty}\zeta_{1,M}\,.$

To prove that $\bm{P}(\zeta_{1,\infty}=\infty)=1$, and then conclude the proof
of the Proposition 5.2, we need the following result.

######  Lemma 5.4.

For any $T>0$ we have

$\lim_{M\to\infty}\bm{P}\Big{(}\sup_{t\in[0,T]}I_{t\wedge\zeta_{M}}\geqslant
M\Big{)}=0\,.$

Indeed, writing

$\begin{split}\bm{P}(\zeta_{1,\infty}<\infty)&=\lim_{T\to\infty}\bm{P}(\zeta_{1,\infty}\leqslant
T)\\\
&=\lim_{T\to\infty}\lim_{M_{0}\to\infty}\bm{P}\Big{(}\zeta_{1,\infty}\leqslant
T,\inf_{t\in[0,\zeta_{\infty})}I_{t}>1/M_{0}\Big{)},\\\
&=\lim_{T\to\infty}\lim_{M_{0}\to\infty}\lim_{M\to\infty}\bm{P}\Big{(}\zeta_{1,M}\leqslant
T,\inf_{t\in[0,\zeta_{M})}I_{t}>1/M_{0}\Big{)}\,.\end{split}$

To justify the second equality, we note that if $\zeta_{1,\infty}$ is finite,
then $\inf_{t\in[0,\zeta_{\infty})}I_{t}$ must be positive since
$\zeta_{1,\infty}=\zeta_{\infty}$ almost surely. We have for any $M>M_{0}$

$\begin{split}\bm{P}\Bigl{(}\zeta_{1,M}\leqslant
T,\inf_{t\in[0,\zeta_{M}]}I_{t}>\frac{1}{M_{0}}\Bigr{)}&\leqslant\bm{P}(\zeta_{1,M}\leqslant
T,\,\zeta_{M}=\zeta_{1,M})\\\
&\leqslant\bm{P}\Big{(}\sup_{t\in[0,T]}I_{t\wedge\zeta_{M}}\geqslant
M\Big{)}.\end{split}$

Therefore, Lemma 5.4 will imply $\bm{P}(\zeta_{1,\infty}<\infty)=0$.

###### Proof of Lemma 5.4..

First we remark that using Chebychev’s inequality

$\displaystyle\bm{P}\Big{(}\sup_{t\in[0,T]}I_{t\wedge\zeta_{M}}\geqslant
M\Big{)}$
$\displaystyle=\bm{P}\Big{(}\sup_{t\in[0,T]}I_{t\wedge\zeta_{M}}^{2}\geqslant
M^{2}\Big{)}$ (5.12)
$\displaystyle\leqslant\frac{1}{M^{2}}\bm{E}\sup_{t\in[0,T]}I_{t\wedge\zeta_{1,M}}^{2}\,.$

Let $T^{\prime}\leqslant T$, we also have

$\displaystyle\bm{E}\sup_{t\in[0,T^{\prime}]}I_{t\wedge\zeta_{M}}^{2}$
$\displaystyle\leqslant
3I_{0}+3\bm{E}\sup_{t\in[0,T^{\prime}]}N^{2}_{t}+3\bm{E}\sup_{t\in[0,T^{\prime}]}\Bigl{(}\int_{0}^{t\wedge\zeta_{M}}\mathbf{b}(I_{s})ds\Bigr{)}^{2}$
$\displaystyle=3I_{0}+J_{T^{\prime}}+K_{T^{\prime}}\,,$

and where, using (5.6)

$N_{t\wedge\zeta_{M}}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}I_{t\wedge\zeta_{M}}-I_{0}-\int_{0}^{t\wedge\zeta_{M}}\mathbf{b}(I_{s})\,ds=\int_{s=0}^{t\wedge\zeta_{M}}\int_{\tau=0}^{1}a(I_{s},\tau)\,dW_{s}(I_{s},\tau)\,d\tau$

is a martingale with quadratic variation

$\langle N\rangle_{t}=\int_{0}^{t\wedge\zeta_{M}}\mathbf{a}(I_{s})\,ds\,.$

For the term $J_{T^{\prime}}$ we apply Burkholder-Davis-Gundy inequality
[KS91, Theorem 3.28 p. 166] together with (5.2) from which we obtain

$J_{T^{\prime}}\leqslant
C\bm{E}\int_{0}^{T^{\prime}\wedge\zeta_{M}}\mathbf{a}(I_{s})\,ds\leqslant
C\int_{0}^{T^{\prime}}\bm{E}[\mathbf{a}(I_{s\wedge\zeta_{M}})]\,ds\\\
\leqslant
C_{\omega_{0}}\int_{S}|p|^{2\beta}|r(p)|\,dp\int_{s=0}^{T^{\prime}}\int_{\tau=0}^{1}\bm{E}[|a(I_{s\wedge\zeta_{M}},\tau)|^{2}]\,d\tau\,ds\,.$

Moreover, using Proposition 4.2, we have

$\begin{split}\int_{0}^{1}\bm{E}[|a(I_{s\wedge\zeta_{M}},\tau)|^{2}]\,d\tau&=\int_{0}^{1}\Bigl{(}\bm{E}[|a(I_{s\wedge\zeta_{M}},\tau)|^{2}\mathbf{1}_{(I_{s\wedge\zeta_{M}}\leqslant
r)}]\\\
&\qquad+\bm{E}[|a(I_{s\wedge\zeta_{M}},\tau)|^{2}\mathbf{1}_{(I_{s\wedge\zeta_{M}}>r)}]\Bigr{)}\,d\tau\\\
&\leqslant r\,C+C_{r}\bm{E}I_{s\wedge\zeta_{M}}^{2}\\\ &\leqslant
r\,C+C_{r}\bm{E}\sup_{t\in[0,s]}I_{t\wedge\zeta_{M}}^{2}\,,\end{split}$

and hence

$J_{T^{\prime}}\leqslant
C_{1}\,T+C_{2}\int_{0}^{T^{\prime}}\bm{E}\Big{(}\sup_{t\in[0,s]}I_{t\wedge\zeta_{M}}^{2}\Big{)}ds\,.$

Now, for $K_{T^{\prime}}$, we use Cauchy-Schwarz inequality

$\begin{split}\bm{E}\sup_{t\in[0,T^{\prime}]}\Bigl{(}\int_{0}^{t\wedge\zeta_{M}}\mathbf{b}(I_{s})\,ds\Bigr{)}^{2}&\leqslant
T\bm{E}\sup_{t\in[0,T^{\prime}]}\int_{0}^{t\wedge\zeta_{M}}\mathbf{b}^{2}(I_{s})\,ds\\\
&\leqslant
T\bm{E}\int_{0}^{T^{\prime}}\mathbf{b}^{2}(I_{s\wedge\zeta_{M}})\,ds\,.\end{split}$

From Proposition 4.2, for $I\leqslant r$, we again use the Cauchy-Schwarz
inequality

$\begin{split}|\mathbf{b}(I)|&\leqslant
C\,|\omega^{\prime}(I)|\,\sqrt{\int_{0}^{1}|a(I,\tau)|^{2}\,d\tau\int_{0}^{1}|\partial_{\tau}a(I,\tau)|^{2}\,d\tau}\\\
&+C_{\omega_{0}}\sqrt{\int_{0}^{1}|a(I,\tau)|^{2}\,d\tau\int_{0}^{1}|\partial_{I}a(I,\tau)|^{2}\,d\tau}\\\
&\leqslant C.\end{split}$

As a result,

$\begin{split}\bm{E}\int_{0}^{T^{\prime}}\mathbf{b}^{2}(I_{s\wedge\zeta_{M}})\,ds&=\bm{E}\int_{0}^{T^{\prime}}\mathbf{b}^{2}(I_{s\wedge\zeta_{M}})\mathbf{1}_{(I_{s\wedge\zeta_{M}}\leqslant
r)}\,ds\\\
&\qquad+\bm{E}\int_{0}^{T^{\prime}}\mathbf{b}^{2}(I_{s\wedge\zeta_{M}})\mathbf{1}_{(I_{s\wedge\zeta_{M}}>r)}\,ds\\\
&\leqslant
C_{1}\,T+C_{2,r}\bm{E}\int_{0}^{T^{\prime}}I_{s\wedge\zeta_{M}}^{2}\mathbf{1}_{(I_{s\wedge\zeta_{M}}>r)}\,ds\\\
&\leqslant
C_{1}\,T+C_{2,r}\int_{0}^{T^{\prime}}\bm{E}\Bigl{(}\sup_{t\in[0,s]}I_{t\wedge\zeta_{M}}^{2}\Bigr{)}\,ds\,,\end{split}$

and

$K_{T^{\prime}}\leqslant
C_{1}\,T^{2}+C_{2}\,T\int_{0}^{T^{\prime}}\bm{E}\Bigl{(}\sup_{t\in[0,s]}I_{t\wedge\zeta_{M}}^{2}\Bigr{)}\,ds\,.$

Finally, we have

$\bm{E}\sup_{t\in[0,T^{\prime}]}I_{t\wedge\zeta_{M}}^{2}\leqslant
C_{1}(1+T+T^{2})+C_{2}(1+T)\int_{0}^{T^{\prime}}\bm{E}\Bigl{(}\sup_{t\in[0,s]}I_{t\wedge\zeta_{M}}^{2}\Bigr{)}\,ds$

and then by Gronwall’s inequality with $T^{\prime}=T$

$\bm{E}\sup_{t\in[0,T]}I_{t\wedge\zeta_{M}}^{2}\leqslant
C_{1}(1+T+T^{2})e^{C_{2}(1+T)T}\,.$

This concludes the proof of the Lemma by letting $M\to\infty$ in (5.2).
Consequently, the proof of Proposition 5.2 is also complete. ∎

###  5.3. Uniqueness of solutions

The weak uniqueness property for equation (4.13) comes from the fact that this
equation has strong uniqueness on $\mathcal{M}_{\zeta_{2,M}}$ for any $M$, and
then weak uniqueness on $\mathcal{M}_{\zeta_{2,M}}$ for any $M$ [KS91,
Proposition 3.20 p. 309], since we avoid the lack of regularity at $0$ for the
action variable $I$. However, according to Proposition 5.2, we have
$\lim_{M\to\infty}\bm{P}^{0}_{I}(\zeta_{2,M}\leqslant T)=0$ for any $T>0$
which guarantees the weak uniqueness thanks to [SV06, Theorem 1.3.5 p. 34].

##  6\. Proof of the main theorem (Theorem 4.3)

###  6.1. Truncated Process

We first describe the strategy used to prove Theorem 4.3. In addition to the
difficulties concerning the behavior of $a$ around $I=0$ that we mentioned in
Theorem 4.3, we have no _apriori_ estimates for the process
$(I^{\varepsilon}_{t},\psi^{\varepsilon}_{t})_{t\geqslant 0}$ that are uniform
in $\varepsilon$, in probability. This is a problem when trying to apply
[Kus84, Theorem 4 p. 48] directly to prove the tightness property, and when
trying to identify the law of the subsequential limits. To bypass this we
follow the strategy developed in [SV06, Chapter 11] by introducing a
_truncated process_ that does not suffer from the above problems. To relate
the truncated process with the original one we introduce a family of stopping
times. As we will see, these times go to $\infty$ when we remove the
truncation and then finally obtain the weak convergence of the original
processes.

Let $M>0$ and consider the process
$(I^{\varepsilon,M}_{t},\psi^{\varepsilon,M}_{t})_{t\geqslant 0}$ which is the
unique solution to

(6.1)
$\begin{split}\dot{I}^{\varepsilon,M}_{t}&=\frac{1}{\varepsilon}v\Big{(}\frac{t}{\varepsilon^{2}}\Big{)}\phi_{M}(I^{\varepsilon,M}_{t},\psi^{\varepsilon,M}_{t})a(I^{\varepsilon,M}_{t},\psi^{\varepsilon,M}_{t}+\tau^{\varepsilon,M}_{t})\\\
\dot{\psi}^{\varepsilon,M}_{t}&=\frac{1}{\varepsilon}v\Big{(}\frac{t}{\varepsilon^{2}}\Big{)}\phi_{M}(I^{\varepsilon,M}_{t},\psi^{\varepsilon,M}_{t})(b(I^{\varepsilon,M}_{t},\psi^{\varepsilon,M}_{t}+\tau^{\varepsilon,M}_{t})-\langle
b(I^{\varepsilon,M}_{t},\cdot)\rangle)\end{split}$

coupled with the initial conditions $I^{\varepsilon,M}_{0}=I_{0}$, and
$\psi^{\varepsilon,M}_{0}=\theta_{0}$, with

$\dot{\tau}^{\varepsilon,M}_{t}=\frac{\omega(I^{\varepsilon,M}_{t})}{\varepsilon^{2}}+\frac{1}{\varepsilon}v\Big{(}\frac{t}{\varepsilon^{2}}\Big{)}\phi_{M}(I_{t}^{\varepsilon,M},\psi_{t}^{\varepsilon,M})\langle
b(I^{\varepsilon,M}_{t},\cdot)\rangle\,.$

Here $\phi_{M}$ is a smooth function on $\mathbb{R}^{2}$ such that

$0\leqslant\phi_{M}\leqslant 1,\quad\phi_{M}(I,\psi)=1\quad\text{if}\quad
1/M\leqslant I\leqslant M\quad\text{and}\quad|\psi|\leqslant M\,,$

and

$\phi_{M}(I,\psi)=0\quad\text{if}\quad I>2M\quad\text{or}\quad
I<1/(2M)\quad\text{or}\quad|\psi|>2M\,.$

Thanks to the truncation provided by the cutoff function $\phi_{M}$, for both
action and angle variables, the process
$(I^{\varepsilon,M}_{t},\psi^{\varepsilon,M}_{t})_{t\geqslant 0}$ does not
suffer from the regularity problem for the action variable around $0$ as
mentioned above. Also, its convergence in distribution in
$\mathcal{C}([0,\infty),\mathbb{R}^{2})$ can be proved using the perturbed-
test function method and martingale properties [Kus84, PSV76] thanks to the
boundedness of the process. All these points are developed more precisely in
Section 7.

To simplify the notation, we now omit the superscript $M$ and denote

$Y_{t}^{\varepsilon}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\begin{pmatrix}I_{t}^{\varepsilon,M}\\\
\psi_{t}^{\varepsilon,M}\end{pmatrix}\,.$

Now the system (6.1) can be rewritten as

(6.2)
$\dot{Y}^{\varepsilon}_{t}=\frac{1}{\varepsilon}v\Big{(}\dfrac{t}{\varepsilon^{2}}\Big{)}F(Y^{\varepsilon}_{t},\tau^{\varepsilon}_{t})$

with

(6.3)
$\dot{\tau}^{\varepsilon}_{t}=\frac{\omega(Y^{\varepsilon}_{t})}{\varepsilon^{2}}+\frac{1}{\varepsilon}v\Big{(}\dfrac{t}{\varepsilon^{2}}\Big{)}G(Y^{\varepsilon}_{t})\,.$

Here

(6.4)
$\begin{split}F(Y,\tau)&\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\begin{pmatrix}\phi_{M}(I,\psi)a(I,\psi+\tau)\\\
\phi_{M}(I,\psi)(b(I,\psi+\tau)-\langle b(I,\cdot)\rangle)\end{pmatrix},\\\
G(Y)&\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\phi_{M}(I,\psi)\langle
b(I,\cdot)\rangle\,,\end{split}$

and $a,b$ are defined in equation (4.5). Note that $F$ is a smooth bounded
function (with bounded derivatives) 1-periodic and mean-zero with respect to
$\tau$ (the truncation only affects $\psi$, not $\tau$), and
$\omega(Y_{t}^{\varepsilon})=\omega(I_{t}^{\varepsilon,M})$.

To relate the truncated and original processes we introduce first some
notations. In the remaining of the paper, the space of all possible outcomes
is $\mathcal{C}([0,\infty),\mathbb{R}^{2})$, and we denote the corresponding
canonical filtration by $\mathcal{M}_{t}=\sigma(y_{s},0\leqslant s\leqslant
t)$. The laws of the truncated and original processes on
$\mathcal{C}([0,\infty),\mathbb{R}^{2})$ will be denoted respectively by
$\bm{P}^{\varepsilon,M}$ and $\bm{P}^{\varepsilon}$. Now, we consider the
stopping times

$\eta_{1,M}(y)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\inf\\{t\geqslant
0:\|y_{t}\|\geqslant
M\\}\quad\text{and}\quad\eta_{2,M}(y)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\inf\\{t\geqslant
0:|y^{1}_{t}|\leqslant 1/M\\}\,,$

for any $y=(y^{1},y^{2})\in\mathcal{C}([0,\infty),\mathbb{R}^{2})$ and

(6.5)
$\eta_{M}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\eta_{1,M}\wedge\eta_{2,M}\,.$

It is not hard to see that $\eta_{1,M}$ and $\eta_{2,M}$ are lower semi-
continuous so that $\eta_{M}$ is also lower semi-continuous, that is their
lower level sets are closed subsets of
$\mathcal{C}([0,\infty),\mathbb{R}^{2})$. The latter property will be used to
obtain the convergence of the original process from the truncated one.

From the definition of the above stopping times and the cutoff function
$\phi_{M}$ it is clear that

(6.6)
$\bm{P}^{\varepsilon,M}=\bm{P}^{\varepsilon}\qquad\text{on}\quad\mathcal{M}_{\eta_{M}}\,.$

This is the relation that links the truncated and original processes and that
will be also used below in Section 6.2 to show the convergence of the original
process.

For now, we have the following convergence results for the truncated process.
We return the index $M$ for a moment to emphasize the truncation.

######  Proposition 6.1.

The family
$(Y^{\varepsilon})_{\varepsilon>0}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}(I^{\varepsilon,M},\psi^{\varepsilon,M})_{\varepsilon>0}$
converges in distribution in $\mathcal{C}([0,\infty),\mathbb{R}^{2})$ to the
unique solution $(Y_{t}^{M})_{t\geqslant
0}=(I_{t}^{M},\psi_{t}^{M})_{t\geqslant 0}$ to the martingale problem with
generator defined by

(6.7)
$\mathcal{L}h(y)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\int_{\tau=0}^{1}\int_{u=0}^{\infty}R(u)\tilde{F}(0,y,\tau)\cdot\nabla(\tilde{F}(u,y,\tau)\cdot\nabla
h(y))\,du\,d\tau\,,$

where

$\tilde{F}(u,y,\tau)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}F(y,\tau+\omega(y)u)\,,$

with $F$ defined in (6.4) and starting point $(I_{0},\psi_{0})$.

From this characterization, we can deduce the SDEs satisfied by the limiting
process $Y_{t}^{M}=(I_{t}^{M},\psi_{t}^{M})$ up to the stopping time
$\eta_{M}$.

######  Proposition 6.2.

Denoting
$\tilde{I}^{M}_{t}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}I^{M}_{t\wedge\eta_{M}}$,
we have

(6.8)
$\begin{split}d\tilde{I}^{M}_{t}&=\int_{\tau=0}^{1}\,a(\tilde{I}^{M}_{t},\tau)\,dW_{t\wedge\eta_{M}}(\tilde{I}^{M}_{t},\tau)\,d\tau\\\
&\quad+\int_{u=0}^{\infty}R(u)\Big{(}\int_{\tau=0}^{1}\partial_{I}\big{(}a(I,\tau+\omega(I)u)\big{)}_{|I=\tilde{I}^{M}_{t}}a(\tilde{I}^{M}_{t},\tau)\\\
&\qquad\qquad+a(\tilde{I}^{M}_{t},\tau+\omega(\tilde{I}^{M}_{t})u)\partial_{I}a(\tilde{I}^{M}_{t},\tau)\Big{)}\,d\tau\,du\,dt,\end{split}$

with $\tilde{I}^{M}_{t=0}=I_{0}$, and where $W$ is defined in (4.14). Also, we
have

(6.9)
$\begin{split}d\psi^{M}_{t\wedge\eta_{M}}&=\int_{\tau=0}^{1}\,b(\tilde{I}^{M}_{t},\tau)\,dW_{t\wedge\eta_{M}}(\tilde{I}^{M}_{t},\tau)\,d\tau\\\
&\quad+\int_{u=0}^{\infty}R(u)\Big{(}\int_{\tau=0}^{1}\partial_{I}\big{(}b(I,\tau+\omega(I)u)\big{)}_{|I=\tilde{I}^{M}_{t}}a(\tilde{I}^{M}_{t},\tau)\\\
&\qquad\qquad+b(\tilde{I}^{M}_{t},\tau+\omega(\tilde{I}^{M}_{t})u)\partial_{I}a(\tilde{I}^{M}_{t},\tau)\Big{)}\,d\tau\,du\,dt,\end{split}$

with $\psi^{M}_{t=0}=\theta_{0}$.

As we can see, this result is similar to the one of Theorem 4.3, with
coefficients $a$ and $b$ themselves, but involving the cutoff $M$ through the
stopping time $\eta_{M}$. The proofs of Proposition 6.1 and 6.2 are presented
in Sections 7 and 8, respectively. In the next section, we will show how these
two propositions can be used to prove Theorem 4.3.

###  6.2. Convergence of
$(I^{\varepsilon},\psi^{\varepsilon})_{\varepsilon>0}$

In order to prove Theorem 4.3, we need to remove the cutoff $M$ to show the
convergence in distribution of
$(I^{\varepsilon},\psi^{\varepsilon})_{\varepsilon>0}$ to
$(I_{t},\psi_{t})_{t\geqslant 0}$, where $I$, $\psi$ are defined by
(4.13)-(4.15). This is similar to Lemma 11.1.1 (p. 262) in [SV06], and we
adapt the proof to our situation.

In what follows, we denote by $\bm{P}^{M}$ and $\bm{P}^{0}$ the distribution
of $Y^{M}=(I^{M}_{t},\psi^{M}_{t})_{t\geqslant 0}$ (defined in Proposition
6.1) and $Y=(I_{t},\psi_{t})_{t\geqslant 0}$ (defined in Proposition 4.3),
respectively. Note that according to (6.9) and (4.15) the distributions
$\bm{P}^{M}$ and $\bm{P}^{0}$ are completely determined by their first
marginal $\bm{P}^{M}_{I}$ and $\bm{P}^{0}_{I}$. Also, in view of (6.8) and
(6.9), it is straightforward to see that (with $\eta_{M}$ defined in (6.5))

(6.10) $\bm{P}^{0}=\bm{P}^{M}\qquad\text{on}\quad\mathcal{M}_{\eta_{M}}\,.$

To prove this convergence, let $\Phi$ be a continuous bounded function on
$\mathcal{C}([0,\infty),\mathbb{R}^{2})$, and write

$\displaystyle\bm{E}^{\bm{P}^{\varepsilon}}[\Phi(y_{t},t\in[0,T])]-\bm{E}^{\bm{P}}[\Phi(y_{t},t\in[0,T])]$
$\displaystyle=\bm{E}^{\bm{P}^{\varepsilon}}[\Phi(y_{t},t\in[0,T])]-\bm{E}^{\bm{P}^{\varepsilon,M}}[\Phi(y_{t},t\in[0,T])]$
$\displaystyle\qquad+\bm{E}^{\bm{P}^{\varepsilon,M}}[\Phi(y_{t},t\in[0,T])]-\bm{E}^{\bm{P}^{M}}[\Phi(y_{t},t\in[0,T])]$
$\displaystyle\qquad+\bm{E}^{\bm{P}^{M}}[\Phi(y_{t},t\in[0,T])]-\bm{E}^{\bm{P}^{0}}[\Phi(y_{t},t\in[0,T])]\,.$

Considering the following decomposition

$\begin{split}\bm{E}^{\bm{P}^{\varepsilon}}[\Phi(y_{t},t\in[0,T])]&=\bm{E}^{\bm{P}^{\varepsilon}}[\Phi(y_{t},t\in[0,T])\mathbf{1}_{\\{\eta_{M}\leqslant
T\\}}]\\\
&\qquad+\bm{E}^{\bm{P}^{\varepsilon}}[\Phi(y_{t},t\in[0,T])\mathbf{1}_{\\{\eta_{M}>T\\}}]\end{split}$

with

$\begin{split}&\bm{E}^{\bm{P}^{\varepsilon}}[\Phi(y_{t},t\in[0,T])\mathbf{1}_{\\{\eta_{M}>T\\}}]=\bm{E}^{\bm{P}^{\varepsilon}}[\Phi(y_{t\wedge\eta_{M}},t\in[0,T])\mathbf{1}_{\\{\eta_{M}>T\\}}]\\\
&=\bm{E}^{\bm{P}^{\varepsilon,M}}[\Phi(y_{t\wedge\eta_{M}},t\in[0,T])\mathbf{1}_{\\{\eta_{M}>T\\}}]\\\
&=\bm{E}^{\bm{P}^{\varepsilon,M}}[\Phi(y_{t},t\in[0,T])\mathbf{1}_{\\{\eta_{M}>T\\}}]\\\
&=\bm{E}^{\bm{P}^{\varepsilon,M}}[\Phi(y_{t},t\in[0,T])]-\bm{E}^{\bm{P}^{\varepsilon,M}}[\Phi(y_{t},t\in[0,T])\mathbf{1}_{\\{\eta_{M}\leqslant
T\\}}]\,.\end{split}$

Here we used (6.6) in the second line. Using (6.6) again we note

$\begin{split}\bm{E}^{\bm{P}^{\varepsilon}}[\Phi(y_{t},t\in[0,T])]&-\bm{E}^{\bm{P}^{\varepsilon,M}}[\Phi(y_{t},t\in[0,T])]\\\
&\leqslant\|\Phi\|_{\infty}(\bm{P}^{\varepsilon}(\eta_{M}\leqslant
T)+\bm{P}^{\varepsilon,M}(\eta_{M}\leqslant T))\\\ &\leqslant
2\|\Phi\|_{\infty}\bm{P}^{\varepsilon,M}(\eta_{M}\leqslant T)\,.\end{split}$

The same results hold for $\bm{P}^{0}$ and $\bm{P}^{M}$ instead of
$\bm{P}^{\varepsilon}$ and $\bm{P}^{\varepsilon,M}$ respectively, but using
(6.10) instead of (6.6). As a result, we obtain

$\begin{split}\limsup_{\varepsilon\to
0}|\bm{E}^{\bm{P}^{\varepsilon}}[\Phi(y_{t},t\in[0,T])]&-\bm{E}^{\bm{P}^{0}}[\Phi(y_{t},t\in[0,T])]|\\\
&\leqslant 2\|\Phi\|_{\infty}\limsup_{\varepsilon\to
0}\bm{P}^{\varepsilon,M}(\eta_{M}\leqslant T)\\\
&+2\|\Phi\|_{\infty}\bm{P}^{0}(\eta_{M}\leqslant T)\,.\end{split}$

Notice that the stopping time $\eta_{M}$ is lower semi-continuous on
$\mathcal{C}([0,\infty),\mathbb{R}^{2})$, so that $(\eta_{M}\leqslant
M)\in\mathcal{M}_{\eta_{M}}$ is a closed subset of
$\mathcal{C}([0,\infty),\mathbb{R}^{2})$. According to the Portmanteau theorem
[Bil99, Theorem 2.1 p. 16] we have

$\limsup_{\varepsilon\to
0}|\bm{E}^{\bm{P}^{\varepsilon}}[\Phi(y_{t},t\in[0,T])]-\bm{E}^{\bm{P}^{0}}[\Phi(y_{t},t\in[0,T])]|\leqslant
4\|\Phi\|_{\infty}\bm{P}^{0}(\eta_{M}\leqslant T)\,,$

for any $M$. Moreover, we have

$\bm{P}^{0}(\eta_{M}\leqslant T)\leqslant\bm{P}^{0}(\zeta_{M}\leqslant
T)+\bm{P}^{0}(\zeta^{\prime}_{M}\leqslant T)\,.$

where for any any $y=(y^{1},y^{2})\in\mathcal{C}([0,\infty),\mathbb{R}^{2})$
we define

$\zeta^{\prime}_{M}(y)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\inf\\{t\geqslant
0:|y^{2}_{t}|\geqslant M\\}\,.$

By Proposition 5.2 we know

$\bm{P}^{0}(\zeta_{M}\leqslant T)=\bm{P}^{0}_{I}(\zeta_{M}\leqslant
T)\underset{M\to\infty}{\longrightarrow}0\,.$

For the other term, let $0<\eta<1$ and write

$\displaystyle\bm{P}^{0}(\zeta^{\prime}_{M}\leqslant T)\leqslant\bm{P}\Big{(}$
$\displaystyle\sup_{t\in[0,T]}|\psi_{t}|\geqslant
M,\inf_{t\in[0,T]}I_{t}>\eta,\sup_{t\in[0,T]}I_{t}<\frac{1}{\eta}\Big{)}$
$\displaystyle+\bm{P}\Big{(}\sup_{t\in[0,T]}I_{t}\geqslant\frac{1}{\eta}\Big{)}+\bm{P}\Big{(}\inf_{t\in[0,T]}I_{t}\leqslant\eta\Big{)}\,,$

then using Markov inequality and (4.15), we obtain the following for some
$C_{\eta}>0$

$\begin{split}\bm{P}\Big{(}&\sup_{t\in[0,T]}|\psi_{t}|\geqslant
M,\inf_{t\in[0,T]}I_{t}>\eta,\sup_{t\in[0,T]}I_{t}<\frac{1}{\eta}\Big{)}\\\
&\leqslant\frac{1}{M^{2}}\bm{E}\Big{[}\sup_{t\in[0,T]}|\psi_{t}|^{2}\mathbf{1}_{\\{\inf_{t\in[0,T]}I_{t}>\eta,\,\sup_{t\in[0,T]}I_{t}<\frac{1}{\eta}\\}}\Big{]}\\\
&\leqslant\frac{2}{M^{2}}\bm{E}\Big{[}\sup_{t\in[0,T]}\Big{|}\int_{\tau=0}^{1}\int_{s=0}^{t}b(I_{s},\tau)dW_{s}(I_{s},\tau)d\tau\Big{|}^{2}\mathbf{1}_{\\{\inf_{t\in[0,T]}I_{t}>\eta,\,\sup_{t\in[0,T]}I_{t}<\frac{1}{\eta}\\}}\Big{]}\\\
&\quad+\frac{C_{\eta}}{M^{2}}\,.\end{split}$

Indeed, since we are on the event
$\\{\inf_{t\in[0,T]}I_{t}>\eta,\,\sup_{t\in[0,T]}I_{t}<\frac{1}{\eta}\\}$ we
avoid the singularity of the function $b$ at $I=0$ and keep $I$ bounded.
Therefore, we can directly bound the drift term in (4.15) by bounding
continuous functions $a,b,\partial_{I}a,\partial_{I}b$ by some constant
$C_{\eta}$. We will subsequently let $C_{\eta}$ be a constant that only
depends on $\eta$ whose value may change from line to line. The details are as
follows.

Write the Fourier expansions of $a$ and $b$ as

$a(I,\tau)=\sum\limits_{n\in\mathbb{Z}^{*}}e^{2i\pi n\tau}a_{n}(I),\qquad
b(I,\tau)=\sum\limits_{n\in\mathbb{Z}}e^{2i\pi n\tau}b_{n}(I)\,.$

Considering only the first integrand in the drift term in (4.15) and
substituting the above expansions, we obtain

$\displaystyle\int_{u=0}^{\infty}R(u)\int_{\tau=0}^{1}\partial_{I}\big{(}b(I,\tau+\omega(I)u)\big{)}_{|I=I_{t}}a(I_{t},\tau)\,d\tau\,du$
$\displaystyle=\int_{u=0}^{\infty}R(u)\int_{\tau=0}^{1}\Big{[}\sum\limits_{n\in\mathbb{Z}}\Big{(}(\partial_{I}b)_{n}(I_{t})+2i\pi
n\omega^{\prime}(I_{t})ub_{n}(I_{t})\Big{)}e^{2i\pi n(\tau+\omega(I_{t})u)}$
$\displaystyle\hskip
120.00018pt\cdot\Big{(}\sum\limits_{n^{\prime}\in\mathbb{Z}^{*}}a_{n^{\prime}}(I_{t})e^{2i\pi
n^{\prime}\tau}\Big{)}\Big{]}\,d\tau\,du$ (6.11)
$\displaystyle=\int_{u=0}^{\infty}R(u)\sum\limits_{n\in\mathbb{Z}^{*}}\Big{(}(\partial_{I}b)_{n}(I_{t})+2i\pi
n\omega^{\prime}(I_{t})ub_{n}(I_{t})\Big{)}a_{-n}(I_{t})e^{2i\pi
n\omega(I_{t})u}\,du$ (6.12) $\displaystyle=R_{1}+R_{2}\,,$

where

$\displaystyle R_{1}$
$\displaystyle\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\int_{u=0}^{\infty}R(u)\sum\limits_{n\in\mathbb{Z}^{*}}(\partial_{I}b)_{n}(I_{t})a_{-n}(I_{t})\cos(2\pi
n\omega(I_{t})u)\,du$ $\displaystyle R_{2}$
$\displaystyle\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}-\int_{u=0}^{\infty}R(u)\sum\limits_{n\in\mathbb{Z}^{*}}(\partial_{I}b)_{n}(I_{t})a_{-n}(I_{t})\cdot
2\pi n\omega^{\prime}(I_{t})u\sin(2\pi n\omega(I_{t})u)\,du\,.$

Here (6.11) followed by expanding the double sum in $n$ and $n^{\prime}$,
integrating in $\tau$, and observing that the only terms that do not vanish
are those with $n+n^{\prime}=0$. The last equality (6.12) followed since the
expression is real-valued and so must equal its real part.

Using (5.2) to bound the integral in $u$ and bounding
$(\partial_{I}b)_{n}(I_{t})$, $a_{-n}(I_{t})$ by some constant $C_{\eta}$, we
have that

$|R_{1}|\leqslant
C_{\eta}\sum\limits_{n\in\mathbb{Z}^{*}}\frac{1}{n^{2}}\leqslant C_{\eta}\,.$

Similarly, we can estimate that

$\displaystyle\int_{u=0}^{\infty}$ $\displaystyle R(u)\cdot 2\pi nu\sin(2\pi
n\omega(I_{t})u)\,du$
$\displaystyle=\int_{u=0}^{\infty}\Big{(}\int_{S}r(p)e^{-\mu|p|^{2\beta}u}\,dp\Big{)}\cdot
2\pi nu\sin(2\pi n\omega(I_{t})u)\,du$
$\displaystyle=\int_{S}\,dp\,r(p)\Big{(}\int_{u=0}^{\infty}2\pi nu\sin(2\pi
n\omega(I_{t})u)e^{-\mu|p|^{2\beta}u}\,du\Big{)}$
$\displaystyle=2\int_{S}\,dp\,r(p)\frac{(2\pi
n)^{2}\omega(I_{t})\mu|p|^{2\beta}}{(\mu^{2}|p|^{4\beta}+4\pi^{2}n^{2}\omega^{2}(I_{t}))^{2}}\leq\frac{C_{\omega_{0}}}{n^{2}}\int_{S}|p|^{2\beta}r(p)\,dp\,.$

Therefore, we also have

$|R_{2}|\leq C_{\eta}\sum\limits_{n\in\mathbb{Z}^{*}}\frac{1}{n^{2}}\leq
C_{\eta}\,.$

Combining these estimates we have

$\displaystyle\Bigl{|}\int_{u=0}^{\infty}R(u)\int_{\tau=0}^{1}\partial_{I}\big{(}b(I,\tau+\omega(I)u)\big{)}_{|I=I_{t}}a(I_{t},\tau)$
$\displaystyle\qquad\qquad+b(I_{t},\tau+\omega(I_{t})u)\partial_{I}a(I_{t},\tau)\,d\tau\,du\,\Bigr{|}\leqslant
C_{\eta}\sum\limits_{n\in\mathbb{Z}^{*}}\frac{1}{n^{2}}\leqslant C_{\eta}\,.$

For the stochastic integral term, we introduce a smooth function $b^{\eta}$
such that for any $\tau\in(0,1)$ we have

$b^{\eta}(I,\tau)=b(I,\tau)\quad\text{if}\quad I>\eta\qquad\text{and}\qquad
b^{\eta}(I,\tau)=0\quad\text{if}\quad I<\frac{\eta}{2}\,.$

Then

$\begin{split}\bm{E}\Big{[}&\sup_{t\in[0,T]}\Big{|}\int_{\tau=0}^{1}\int_{s=0}^{t}b(I_{s},\tau)dW_{s}(I_{s},\tau)d\tau\Big{|}^{2}\mathbf{1}_{\\{\inf_{t\in[0,T]}I_{t}>\eta,\,\sup_{t\in[0,T]}I_{t}<\frac{1}{\eta}\\}}\Big{]}\\\
&=\bm{E}\Big{[}\sup_{t\in[0,T]}\Big{|}\int_{\tau=0}^{1}\int_{s=0}^{t}b^{\eta}(I_{s},\tau)dW_{s}(I_{s},\tau)d\tau\Big{|}^{2}\mathbf{1}_{\\{\inf_{t\in[0,T]}I_{t}>\eta,\,\sup_{t\in[0,T]}I_{t}<\frac{1}{\eta}\\}}\Big{]}\\\
&\leqslant C_{\eta}\,,\end{split}$

where the last inequality follows from the Burkholder-Davis-Gundy inequality
to bound the supremum of a continuous local martingale. Finally, for any
$0<\eta<1$ we have

$\limsup_{M\to\infty}\bm{P}^{0}(\zeta^{\prime}_{M}\leqslant
T)\leqslant\bm{P}\Big{(}\sup_{t\in[0,T]}I_{t}\geqslant\frac{1}{\eta}\Big{)}+\bm{P}\Big{(}\inf_{t\in[0,T]}I_{t}\leqslant\eta\Big{)},$

and

$\displaystyle\lim_{\eta\to
0}\bm{P}\Big{(}\inf_{t\in[0,T]}I_{t}\leqslant\eta\Big{)}$
$\displaystyle=\bm{P}\Big{(}\inf_{t\in[0,T]}I_{t}=0\Big{)}=0\,,$
$\displaystyle\lim_{\eta\to
0}\bm{P}\Big{(}\sup_{t\in[0,T]}I_{t}\geqslant\frac{1}{\eta}\Big{)}$
$\displaystyle=0\quad\quad(\text{by Lemma~{}\ref{bound_I}})\,.$

As a result, we obtain

$\lim_{\varepsilon\to
0}\bm{E}^{\bm{P}^{\varepsilon}}[\Phi(y_{t},t\in[0,T])]=\bm{E}^{\bm{P}^{0}}[\Phi(y_{t},t\in[0,T])]\,,$

which concludes the proof of Theorem 4.3.

It remains to prove Proposition 6.1, and we do this in the next section.

##  7\. Proof of Proposition 6.1

We will prove convergence of the family $(Y^{\varepsilon})_{\varepsilon>0}$ by
first showing tightness, and then showing all subsequential limits are
solutions of a well-posed martingale problem with generator (6.7). This
follows the classical approach to prove weak convergence of a sequence of
diffusions using martingale problem that is well developed in [SV06]. However,
because the processes involved are not diffusions (and often not even
Markovian), we use pseudo-generators and the perturbed test function method
(see for instance [Kus84]).

###  7.1. Pseudo-generator and Perturbed test function method

In this section, we follow [Kus84, Chapter 3]. Even though the pair of
processes $(Y_{t}^{\varepsilon},\tau_{t}^{\varepsilon})_{t\geqslant 0}$ is
Markov for each $\varepsilon>0$, the process
$(Y_{t}^{\varepsilon})_{t\geqslant 0}$ by itself is _not_ Markov. Since
analyzing the pair $(Y_{t}^{\varepsilon},\tau_{t}^{\varepsilon})_{t\geqslant
0}$ is difficult, we study the process $(Y_{t}^{\varepsilon})_{t\geqslant 0}$
on its own by using the pseudo-generator. This allows us to apply martingale
techniques to the non-Markovian process $(Y^{\varepsilon}_{t})_{t\geqslant
0}$.

####  7.1.1. Pseudo-generator

Let

$\mathcal{G}^{\varepsilon}_{t}\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\mathcal{G}_{t/\varepsilon^{2}}\,,$

where $\mathcal{G}_{t}$ is defined by (3.4), and $\mathcal{S}^{\varepsilon}$
be the set of all measurable functions $h(t)$, adapted to the filtration
$(\mathcal{G}^{\varepsilon}_{t})$, for which $\sup_{t\leqslant T}\bm{E}[\lvert
h(t)\rvert]<+\infty$, and where $T>0$ is fixed. The $\operatorname*{p-lim}$
and the pseudo-generator are defined as follows. Let $h$ and $h^{\delta}$ in
$\mathcal{S}^{\varepsilon}$ for all $\delta>0$. We say that
$h=\operatorname*{p-lim}_{\delta}h^{\delta}$ if

$\sup_{t,\delta}\bm{E}\lvert
h^{\delta}(t)\rvert<+\infty\quad\text{and}\quad\lim_{\delta\rightarrow
0}\bm{E}\lvert h^{\delta}(t)-h(t)\rvert=0\quad\forall t\geqslant 0\,.$

We say that $h\in\mathcal{D}(\mathcal{A}^{\varepsilon})$ the domain of
$\mathcal{A}^{\varepsilon}$ and $\mathcal{A}^{\varepsilon}h=g$ if $h$ and $g$
are in $\mathcal{S}^{\varepsilon}$ and

$\operatorname*{p-lim}_{\delta\to
0}\Bigl{(}\frac{\bm{E}^{\varepsilon}_{t}h(t+\delta)-h(t)}{\delta}-g(t)\Bigr{)}=0\,,$

where $\bm{E}^{\varepsilon}_{t}$ is the conditional expectation given
$\mathcal{G}^{\varepsilon}_{t}$. The difference with respect to the usual
infinitesimal generator is that we have average limit (via
$\operatorname*{p-lim}$) rather than pointwise limit. This greatly helps us
average out the noise in the limit $\varepsilon\to 0$. A useful result [Kus84,
Theorem 1, p. 39] about the pseudo-generator $\mathcal{A}^{\varepsilon}$ that
will be central in our proof is the following.

######  Proposition 7.1 (Theorem 1, p. 30 in [Kus84]).

Let $h\in\mathcal{D}(\mathcal{A}^{\varepsilon})$. Then

$M_{h}^{\varepsilon}(t)=h(t)-h(0)-\int_{0}^{t}\mathcal{A}^{\varepsilon}h(u)\,du$

is a $(\mathcal{G}^{\varepsilon}_{t})$-martingale.

In other words, we introduce an operator $\mathcal{A}^{\varepsilon}$ on
functions of $Y^{\varepsilon}$ having similar properties to those of an
infinitesimal operator of a Markov process; more specifically, it satisfies a
martingale problem.

####  7.1.2. Perturbed test function method

The main idea behind the perturbed test function method is the following. To
characterize the limiting process through a martingale problem with generator
$\mathcal{A}$, we can then try to compare $\mathcal{A}^{\varepsilon}h$ with
$\mathcal{A}h$ for some test function $h$. Unfortunately,
$\mathcal{A}^{\varepsilon}h$ may have singular terms in $\varepsilon$, and we
cannot proceed that way directly. Instead, try and extract the effective
statistical behavior of the system under consideration from
$\mathcal{A}^{\varepsilon}$ by introducing $h^{\varepsilon}$ an appropriate
perturbation of $h$. In other words, if we can show $\lim_{\varepsilon\to
0}\bm{E}|h^{\varepsilon}(t)-h(t)|=0$ and $\lim_{\varepsilon\to
0}\bm{E}|\mathcal{A}^{\varepsilon}h^{\varepsilon}(t)-\mathcal{A}h(Y_{t}^{\varepsilon})|=0$
for each $t$, then we can use Proposition 7.1 to conclude that the limiting
process $(Y_{t})_{t\geqslant 0}$ satisfies the martingale problem with
generator $\mathcal{A}$.

To depict how the perturbation $h^{\varepsilon}$ is built, let us briefly
remind its construction in the simpler situation

$\dot{Y}^{\varepsilon}_{t}=\frac{1}{\varepsilon}\mathbf{F}\Big{(}Y^{\varepsilon}_{t},\nu\Big{(}\frac{t}{\varepsilon^{2}}\Big{)},\frac{t}{\varepsilon^{2}}\Big{)}\qquad\text{and}\qquad\mathbf{F}(y,v,\tau)=v\,F(y,\tau)\,,$

where $\nu$ is a centered ergodic Markov process with invariant measure $\mu$,
and generator $\mathcal{L}_{\nu}$ satisfying the Fredholm alternative. In this
situation, to apply the perturbed test function method, we consider the
extended Markov process
$Z^{\varepsilon}_{t}=(Y^{\varepsilon}_{t},\nu(t/\varepsilon^{2}),\tau_{t/\varepsilon^{2}})$
whose infinitesimal generator is given by

$\mathcal{L}^{\varepsilon}=\frac{1}{\varepsilon^{2}}Q+\frac{1}{\varepsilon}\,\mathbf{F}(y,v,\tau)\cdot\nabla_{y}\qquad\text{with}\qquad
Q=\Big{(}\mathcal{L}_{\nu}+\frac{\partial}{\partial\tau}\Big{)}\,.$

Looking for a perturbation $h^{\varepsilon}$ of $h$ of the form

$h^{\varepsilon}(y,v,\tau)=h(y)+\varepsilon
h_{1}(y,v,\tau)+\varepsilon^{2}h_{2}(y,v,\tau)\,,$

we have

(7.1) $\displaystyle\mathcal{L}^{\varepsilon}h^{\varepsilon}(y,v,\tau)$
$\displaystyle=\frac{1}{\varepsilon}\big{(}Qh_{1}(y,v,\tau)+\mathbf{F}(y,v,\tau)\cdot\nabla_{y}h(y)\big{)}$
$\displaystyle+Qh_{2}(y,v,\tau)+\mathbf{F}(y,v,\tau)\cdot\nabla_{y}h_{1}(y,v,\tau)$
$\displaystyle+\mathcal{O}(\varepsilon)\,.$

To cancel the singular term we have to solve a Poisson problem of the form

$Qp=-q(y,v,\tau)\,.$

Assuming the centering condition

$\int_{0}^{1}\bm{E}_{\mu}[q(y,\nu(0),\tau)]\,d\tau=0\,,$

where the expectation is taken with respect to the invariant measure $\mu$ of
the process $\nu$, the Poisson problem admits

$p=\int_{0}^{\infty}e^{uQ}q\,du$

as a solution, so that

$p(Z^{\varepsilon}_{t})=\int_{0}^{\infty}\bm{E}\Big{[}q\Big{(}Y^{\varepsilon}_{t},\nu\Big{(}u+\frac{t}{\varepsilon^{2}}\Big{)},u+\frac{t}{\varepsilon^{2}}\Big{)}\,\Big{|}\,\nu(s/\varepsilon^{2}),\,s\leqslant
t\Big{]}\,du\,,$

from the Markov property. Remembering that $\nu$ is centered and $F$ is mean
zero with respect to $\tau$, we then have

(7.2)
$h_{1}(Z^{\varepsilon}_{t})=\int_{0}^{\infty}\bm{E}\Big{[}\mathbf{F}\Big{(}Y^{\varepsilon}_{t},\nu\Big{(}u+\frac{t}{\varepsilon^{2}}\Big{)},u+\frac{t}{\varepsilon^{2}}\Big{)}\cdot\nabla_{y}h(Y^{\varepsilon}_{t})\,\Big{|}\,\nu(s/\varepsilon^{2}),\,s\leqslant
t\Big{]}\,du\,.$

To treat the term of order 1 in (7.1), which still has fast oscillations
through $h_{1}$, we solve again the Poisson problem but this time with

(7.3)
$q(y,v,\tau)=\mathbf{F}(y,v,\tau)\cdot\nabla_{y}h_{1}(y,v,\tau)-\int_{0}^{1}\bm{E}_{\mu}[\mathbf{F}(y,\nu(0),\tau^{\prime})\cdot\nabla_{y}h_{1}(y,\nu(0),\tau^{\prime})]d\tau^{\prime}\,,$

which satisfies the above centering condition. As a result, we can chose

(7.4) $\displaystyle h_{2}(Z^{\varepsilon}_{t})$
$\displaystyle=\int_{0}^{\infty}\bm{E}\Big{[}(\mathbf{F}\cdot\nabla_{y}h_{1})\Big{(}Y^{\varepsilon}_{t},\nu\Big{(}u+\frac{t}{\varepsilon^{2}}\Big{)},u+\frac{t}{\varepsilon^{2}}\Big{)}\,\Big{|}\,\nu(s/\varepsilon^{2}),\,s\leqslant
t\Big{]}du$
$\displaystyle-\int_{0}^{1}\bm{E}_{\mu}[\mathbf{F}(Y^{\varepsilon}_{t},\nu(0),\tau^{\prime})\cdot\nabla_{y}h_{1}(Y^{\varepsilon}_{t},\nu(0),\tau^{\prime})]\,d\tau^{\prime}\,.$

Finally, due to the centering in (7.3), the expansion (7.1) becomes

$\displaystyle\mathcal{L}^{\varepsilon}h^{\varepsilon}(Z^{\varepsilon}_{t})$
$\displaystyle=\int_{0}^{1}\int_{0}^{\infty}\bm{E}_{\mu}[\mathbf{F}(Y^{\varepsilon}_{t},\nu(0),\tau)\cdot\nabla_{y}(\mathbf{F}(Y^{\varepsilon}_{t},\nu(u),\tau+u)\cdot\nabla_{y}h(Y^{\varepsilon}_{t}))]\,du\,d\tau$
$\displaystyle+\mathcal{O}(\varepsilon)\,,$

and we obtain the same expression as in (6.7).

However, considering non-Markovian fluctuations, as well as long-range
correlations, and a more complex periodic structure, make the construction of
$h^{\varepsilon}$ more delicate. The main concern is to handle the non-
integrability at infinity of the correlation function of $V$. To overcome this
difficulty we must take advantage of the periodic structure as in the proof of
Proposition 1.1. The details are in the next section where we will construct
$h^{\varepsilon}$ to show tightness of $(Y^{\varepsilon})_{\varepsilon>0}$,
which implies imply weak subsequential convergence of stochastic processes.

Before going into the proofs, we remind the reader that
$(Y^{\varepsilon})_{\varepsilon>0}$ is defined via a truncated process in
equation (6.2) and there is an omitted index $M$ in the functions $F$ and $G$
to indicate the truncation. This allows us to bound several expressions in the
proofs in terms of $M$.

###  7.2. Tightness

In this section, we prove the tightness of the family
$(Y^{\varepsilon})_{\varepsilon>0}$, seen as a family of continuous-time
processes. Then, according to [Bil99, Theorem 13.4] it suffices to prove the
tightness of $(Y^{\varepsilon})_{\varepsilon>0}$ in
$\mathcal{D}([0,T],\mathbb{R}^{2})$ (which is the set of càdlàg functions with
values in $\mathbb{R}^{2}$ equipped with the Skorokhod topology).

######  Proposition 7.2.

The family $(Y^{\varepsilon})_{\varepsilon>0}$ is tight in
$\mathcal{D}([0,T],\mathbb{R}^{2})$.

The proof of Proposition 7.2 consists of applying [Kus84, Theorem 4, p. 48].
In what follows, let $h$ be a bounded smooth function on $\mathbb{R}^{2}$ with
bounded derivatives, and set $h_{0}^{\varepsilon}(t)=h(Y^{\varepsilon}_{t})$.
Our goal is to construct perturbations to $h_{0}^{\varepsilon}$ as described
above. The pseudo-generator at $h_{0}^{\varepsilon}$ is then given by

(7.5)
$\mathcal{A}^{\varepsilon}h_{0}^{\varepsilon}(t)=\frac{1}{\varepsilon}\int
V\Big{(}\frac{t}{\varepsilon^{2}},dp\Big{)}F\big{(}Y^{\varepsilon}_{t},\tau^{\varepsilon}_{t}\big{)}\cdot\nabla
h(Y^{\varepsilon}_{t})$

which is simply differentiating $h_{0}^{\varepsilon}(t)$ with respect to $t$
(the derivative $\dot{Y}^{\varepsilon}_{t}$ is stated in equation (6.2)). The
goal now is to modify the test function $h^{\varepsilon}_{0}$ using a small
perturbation $h^{\varepsilon}_{1}$ so that the pseudo-generator
$\mathcal{A}^{\varepsilon}(h^{\varepsilon}_{0}+h^{\varepsilon}_{1})$ does not
blow up in $\varepsilon$ anymore. The first perturbation of
$h^{\varepsilon}_{0}$ is defined as follows.

$\displaystyle
h^{\varepsilon}_{1}(t)=\frac{1}{\varepsilon}\int_{t}^{\infty}du\int\bm{E}^{\varepsilon}_{t}\Big{[}V\Big{(}\frac{u}{\varepsilon^{2}},dp\Big{)}F\Big{(}Y^{\varepsilon}_{t},\tau^{\varepsilon}_{t}+\frac{u-t}{\varepsilon^{2}}\,\omega(Y^{\varepsilon}_{t})\Big{)}\Big{]}\cdot\nabla
h(Y^{\varepsilon}_{t})$
$\displaystyle=\varepsilon\int_{0}^{\infty}du\int\bm{E}^{\varepsilon}_{t}\Big{[}V\Big{(}u+\frac{t}{\varepsilon^{2}},dp\Big{)}F\Big{(}Y^{\varepsilon}_{t},\tau^{\varepsilon}_{t}+u\,\omega(Y^{\varepsilon}_{t})\Big{)}\Big{]}\cdot\nabla
h(Y^{\varepsilon}_{t})$ (7.6)
$\displaystyle=\varepsilon\int_{0}^{\infty}du\int\sum_{n\in\mathbb{Z}^{*}}\bm{E}^{\varepsilon}_{t}\Big{[}V\Big{(}u+\frac{t}{\varepsilon^{2}},dp\Big{)}\Big{]}e^{2i\pi
n(\tau^{\varepsilon}_{t}+u\,\omega(Y^{\varepsilon}_{t}))}F_{n}(Y^{\varepsilon}_{t})\cdot\nabla
h(Y^{\varepsilon}_{t}),$

where

$F_{n}(y)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\int_{0}^{1}F(y,\tau)e^{-2i\pi
n\tau}\,d\tau\,.$

According to Lemma 3.4, we have

$\bm{E}^{\varepsilon}_{t}V\Big{(}u+\frac{t}{\varepsilon^{2}},dp\Big{)}=e^{-\mu|p|^{2\beta}u}V\Big{(}\frac{t}{\varepsilon^{2}},dp\Big{)},$

so that

$\begin{split}h^{\varepsilon}_{1}(t)&=\varepsilon\sum_{n\in\mathbb{Z}^{*}}e^{2i\pi
n\tau^{\varepsilon}_{t}}\int
V\Big{(}\frac{t}{\varepsilon^{2}},dp\Big{)}F_{n}(Y^{\varepsilon}_{t})\cdot\nabla
h(Y^{\varepsilon}_{t})\int_{0}^{\infty}e^{-\mu|p|^{2\beta}u}e^{2i\pi
n\omega(Y^{\varepsilon}_{t})u}\,du\\\
&=\varepsilon\sum_{n\in\mathbb{Z}^{*}}e^{2i\pi
n\tau^{\varepsilon}_{t}}\int\frac{V(t/\varepsilon^{2},dp)}{\mu|p|^{2\beta}-2i\pi
n\,\omega(Y^{\varepsilon}_{t})}F_{n}(Y^{\varepsilon}_{t})\cdot\nabla
h(Y^{\varepsilon}_{t})\end{split}$

To apply [Kus84, Theorem 4 p. 48] and then prove Proposition 7.2, we only have
to prove the two following lemmas.

######  Lemma 7.3.

For any $T>0$, and $\eta>0$

$\lim_{\varepsilon\to 0}\bm{P}\Big{(}\sup_{t\in[0,T]}\lvert
h^{\varepsilon}_{1}(t)\rvert>\eta\Big{)}=0\qquad\text{and}\qquad\lim_{\varepsilon\to
0}\sup_{t\in[0,T]}\bm{E}\lvert h^{\varepsilon}_{1}(t)\rvert=0\,.$

######  Lemma 7.4.

For any $T>0$,
$\big{\\{}\mathcal{A}^{\varepsilon}\big{(}h^{\varepsilon}_{0}+h^{\varepsilon}_{1}\big{)}(t),\varepsilon\in(0,1),0\leqslant
t\leqslant T\big{\\}}$ is uniformly integrable.

Before going through the proofs of these two lemmas, we make one remark that
the test function $h^{\varepsilon}_{1}$, above, is well defined since we sum
over $n\neq 0$. This is because $F$ has mean zero respect to the $\tau$, and
explains why this mean-zero condition is crucial to the analysis. For $n=0$,
the above function $h^{\varepsilon}_{1}$ would not be defined in the case of
long-range correlations.

###### Proof of Lemma 7.3..

We rewrite $h_{1}^{\varepsilon}$ as

$h_{1}^{\varepsilon}(t)=\varepsilon
V\Big{(}\frac{t}{\varepsilon^{2}},\varphi_{t,\varepsilon}\Big{)},$

with

(7.7)
$\varphi_{t,\varepsilon}(p)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\sum_{n\in\mathbb{Z}^{*}}e^{2i\pi
n\tau^{\varepsilon}_{t}}\frac{1}{\mu|p|^{2\beta}-2i\pi
n\,\omega(Y^{\varepsilon}_{t})}F_{n}(Y^{\varepsilon}_{t})\cdot\nabla
h(Y^{\varepsilon}_{t})\,.$

By the Cauchy-Schwarz inequality

$\begin{split}|\varphi_{t,\varepsilon}(p)|&\leqslant
C_{h,\omega_{0}}\sum_{n\in\mathbb{Z}^{*}}\frac{1}{|n|}\|F_{n}(Y^{\varepsilon}_{t})\|\\\
&\leqslant
C_{h,\omega_{0}}\Big{(}\sum_{n\in\mathbb{Z}^{*}}\frac{1}{|n|^{2}}\Big{)}^{1/2}\Big{(}\sum_{n\in\mathbb{Z}^{*}}\|F_{n}(Y_{t}^{\varepsilon})\|^{2}\Big{)}^{1/2}\\\
&\leqslant
C_{h,\omega_{0}}\Big{(}\sum_{n\in\mathbb{Z}^{*}}\frac{1}{|n|^{2}}\Big{)}^{1/2}\Big{(}\int_{0}^{1}\|F(Y^{\varepsilon}_{t},\tau)\|^{2}\,d\tau\Big{)}^{1/2}\\\
&\leqslant C_{h,\omega_{0}}\sup_{\begin{subarray}{c}|y|\leqslant
2M\end{subarray}}\Big{(}\int_{0}^{1}\|F(y,\tau)\|^{2}\,d\tau\Big{)}^{1/2}.\end{split}$

If $\beta\geqslant 1/2$, we have the Lipschitz bound

$\begin{split}|\varphi_{t,\varepsilon}(p)-\varphi_{t,\varepsilon}(q)|&\leqslant(|p|^{2\beta}-|q|^{2\beta})C_{h,\omega_{0},F}\sum_{n\in\mathbb{Z}^{*}}\frac{1}{|n|^{2}}\\\
&\leqslant||p|-|q||\sup_{r\in
S}|r|^{2\beta-1}C_{h,\omega_{0},F}\sum_{n\in\mathbb{Z}^{*}}\frac{1}{|n|^{2}}\leqslant
L|p-q|\end{split}$

with

$L=\sup\limits_{r\in
S}|r|^{2\beta-1}C_{h,\omega_{0},F}\sum_{n\in\mathbb{Z}^{*}}\frac{1}{|n|^{2}}<\infty\,.$

Using inequality (3.7) of Lemma 3.5 with $k=\infty$, along with the
Chebychev’s inequality, we have

$\bm{P}\Big{(}\sup_{t\in[0,T]}\lvert
h^{\varepsilon}_{1}(t)\rvert>\eta\Big{)}\leqslant\frac{1}{\eta}\bm{E}\sup_{t\in[0,T]}\lvert
h^{\varepsilon}_{1}(t)\rvert\leqslant
C^{\prime}_{h,\omega_{0},F}\,C(\varepsilon)\,,$

with $\lim_{\varepsilon}C(\varepsilon)=0$.

If $\beta<1/2$, the function $\varphi_{t,\varepsilon}$ belongs to $W^{1,k}(S)$
with $k\in(1,1/(1-2\beta))$ since

$\int_{S}|\partial_{p}\varphi_{t,\varepsilon}(p)|^{k}\,dp\leqslant
C^{\prime\prime}_{h,\omega_{0},F}\int_{S}|p|^{k(2\beta-1)}\,dr<\infty\,,$

so that using inequality (3.7) of Lemma 3.5 but with $k\in(1,1/(1-2\beta))$
leads again to

$\bm{P}\Big{(}\sup_{t\in[0,T]}\lvert
h^{\varepsilon}_{1}(t)\rvert>\eta\Big{)}\leqslant
C^{\prime}_{h,\omega_{0},F}\,C(\varepsilon)\,.$

To prove the second limit in the lemma, one can just remark that

$\bm{E}\lvert
h^{\varepsilon}_{1}(t)\rvert\leqslant\bm{E}\sup_{t\in[0,T]}\lvert
h^{\varepsilon}_{1}(t)\rvert$

for any $t\in[0,T]$, and then concludes the proof of Lemma 7.3. ∎

###### Proof of Lemma 7.4..

For proving uniform integrability, it is sufficient to show that

$\sup_{\varepsilon,t}\bm{E}|\mathcal{A}^{\varepsilon}(h^{\varepsilon}_{0}+h^{\varepsilon}_{1})(t)|^{2}\leqslant
C\,.$

Following the definition of pseudo-generator in Section 7.1.1, we have that

$\displaystyle\mathcal{A}^{\varepsilon}$
$\displaystyle(h_{1}^{\varepsilon})(t)=-\mathcal{A}^{\varepsilon}(h_{0}^{\varepsilon})(t)$
$\displaystyle+\frac{1}{\varepsilon}\int_{t}^{\infty}du\int\bm{E}^{\varepsilon}_{t}\Big{[}V\Big{(}\frac{u}{\varepsilon^{2}},dp\Big{)}\Big{]}\frac{d}{ds}\Big{[}F\Big{(}Y^{\varepsilon}_{s},\tau^{\varepsilon}_{s}+\frac{u-s}{\varepsilon^{2}}\,\omega(Y^{\varepsilon}_{s})\Big{)}\cdot\nabla
h(Y^{\varepsilon}_{s})\Big{]}\Big{|}_{s=t}$

which is essentially differentiating $h^{\varepsilon}_{1}(t)$ with respect to
$t$ (we however remark that we do not differentiate the $t$ in
$\bm{E}_{t}^{\varepsilon})$. After some lengthy but straightforward
computations (similarly to those used to obtain (7.2)), we obtain

$\displaystyle\mathcal{A}^{\varepsilon}(h^{\varepsilon}_{1})(t)=-\mathcal{A}^{\varepsilon}(h^{\varepsilon}_{0})(t)+\sum_{n,m\in\mathbb{Z}^{*}}e^{2i\pi(n+m)\tau^{\varepsilon}_{t}}\iint\frac{V(t/\varepsilon^{2},dp)V(t/\varepsilon^{2},dp^{\prime})}{\mu|p|^{2\beta}-2i\pi
n\omega(Y^{\varepsilon}_{t})}$
$\displaystyle\qquad\qquad\cdot\Big{(}F_{n}(Y^{\varepsilon}_{t})^{T}\nabla^{2}h(Y^{\varepsilon}_{t})F_{m}(Y^{\varepsilon}_{t})+\nabla
h(Y^{\varepsilon}_{t})^{T}\textrm{Jac}F_{n}(Y^{\varepsilon}_{t})F_{m}(Y^{\varepsilon}_{t})$
$\displaystyle\qquad\qquad\qquad-\frac{2i\pi n}{\mu|p|^{2\beta}-2i\pi
n\omega(Y^{\varepsilon}_{t})}F_{n}(Y^{\varepsilon}_{t})\cdot\nabla
h(Y^{\varepsilon}_{t})F_{m}(Y^{\varepsilon}_{t})\cdot\nabla\omega(Y^{\varepsilon}_{t})\Big{)}$
$\displaystyle\qquad+\sum_{n\in\mathbb{Z}^{*}}2i\pi n\,e^{2i\pi
n\tau^{\varepsilon}_{t}}\iint\frac{V(t/\varepsilon^{2},dp)V(t/\varepsilon^{2},dp^{\prime})}{\mu|p|^{2\beta}-2i\pi
n\omega(Y^{\varepsilon}_{t})}$ $\displaystyle\qquad\qquad\cdot
F_{n}(Y^{\varepsilon}_{t})\cdot\nabla
h(Y^{\varepsilon}_{t})G(Y^{\varepsilon}_{t})$
$\displaystyle=-\mathcal{A}^{\varepsilon}(h^{\varepsilon}_{0})(t)+V\Big{(}\frac{t}{\varepsilon^{2}},\varphi^{1}_{t,\varepsilon}\Big{)}V\Big{(}\frac{t}{\varepsilon^{2}},\mathbf{1}_{S}\Big{)}+V\Big{(}\frac{t}{\varepsilon^{2}},\varphi^{2}_{t,\varepsilon}\Big{)}V\Big{(}\frac{t}{\varepsilon^{2}},\mathbf{1}_{S}\Big{)}$
(7.8)
$\displaystyle\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}-\mathcal{A}^{\varepsilon}(h^{\varepsilon}_{0})(t)+\mathcal{P}_{1}^{\varepsilon}(t)+\mathcal{P}_{2}^{\varepsilon}(t)$

where $\varphi_{t,\varepsilon}^{j}$, $j=1,2$ are defined similarly as in
(7.7). As in the proof of Lemma 7.3, it is not hard to see that the
$\varphi^{j}_{t,\varepsilon}$, $j=1,2$ are Lispschitz in $p$ on $S$, and that

$\begin{split}|\varphi^{1}_{t,\varepsilon}(p)|&\leqslant
C_{h,\omega_{0},\nabla\omega}\sum_{m\in\mathbb{Z}^{*}}|F_{m}(Y^{\varepsilon}_{t})|\sum_{n\in\mathbb{Z}^{*}}\frac{1}{|n|}\Big{(}(1+|n|)|F_{n}(Y^{\varepsilon}_{t})|+\|\textrm{Jac}F_{n}(Y^{\varepsilon}_{t})\|\Big{)}\\\
&\leqslant\tilde{C}_{h,\omega_{0},\nabla\omega}\sup_{\begin{subarray}{c}|y|\leqslant
2M\end{subarray}}\int_{0}^{1}\Big{(}|F(y,\tau)|^{2}+|\partial_{\tau}F(y,\tau)|^{2}+\|\textrm{Jac}F(y,\tau)\|^{2}\Big{)}\,d\tau\,.\end{split}$

The same lines give also

$|\varphi^{2}_{t,\varepsilon}(p)|\leqslant
C_{h,\omega_{0}}\sup_{\begin{subarray}{c}|y|\leqslant
2M\end{subarray}}\big{|}G(y)\big{|}\Big{(}\int_{0}^{1}|\partial_{\tau}F(y,\tau)|^{2}\,d\tau\Big{)}^{1/2}.$

Then, we can use inequality (3.8) of Lemma 3.5, and obtain

$\sup_{\varepsilon,t}\bm{E}[|\mathcal{P}_{j}^{\varepsilon}(t)|^{2}]\leqslant
C,\quad j=1,2\ ,.$

which concludes the proof of Lemma 7.4, and then Proposition 7.2 as well. ∎

###  7.3. Identification of the limit

In this section, we identify all the limit points of
$(Y^{\varepsilon})_{\varepsilon>0}$ via a well-posed martingale problem as
stated in the following proposition. By abuse of notations, we still denote by
$(Y^{\varepsilon})_{\varepsilon>0}$ a converging subsequence and by
$(Y_{t})_{t\geqslant 0}$ a limit point.

######  Proposition 7.5.

All the limit points $(Y_{t})_{t\geqslant 0}$ of
$(Y^{\varepsilon})_{\varepsilon>0}$ are solutions of a well-posed martingale
problem with generator

(7.9)
$\mathcal{L}h(y)=\int_{0}^{1}d\tau\int_{0}^{\infty}du\,R(u)\tilde{F}(0,y,\tau)\cdot\nabla_{y}(\tilde{F}(u,y,\tau)\cdot\nabla_{y}h(y))$

with

$\tilde{F}(u,y,\tau)=F(y,\tau+\omega(y)u)\,.$

To prove this proposition we use the notion of pseudo-generator introduced in
Section 7.1.1 and the perturbed-test-function technique that we have already
used in Section 7.2 for the proof of tightness. Thanks to the pseudo-generator
we can characterize the subsequential limits of
$(Y^{\varepsilon})_{\varepsilon>0}$ as solutions of a well-posed martingale
problem.

The outline of the proof is as follows. Recall in Section 7.2 we saw that
$\mathcal{A}^{\varepsilon}(h^{\varepsilon}_{0})$ has a singular
$1/\varepsilon$ term, and modified the test function $h^{\varepsilon}_{0}$
with a small perturbation $h^{\varepsilon}_{1}$ to remove this singular term.
As a result,
$\mathcal{A}^{\varepsilon}(h^{\varepsilon}_{0}+h^{\varepsilon}_{1})$ is not
singular any more. However, it is not yet the generator of a martingale
problem as
$\mathcal{A}^{\varepsilon}(h^{\varepsilon}_{0}+h^{\varepsilon}_{1})$ still has
oscillatory terms in the form of $e^{2i\pi n\tau_{t}^{\varepsilon}}$. We will
show that these terms essentially vanish as $\varepsilon\to 0$ by introducing
a small perturbation $h_{2}^{\varepsilon}(t)$ to cancel these oscillations,
and then construct another perturbation $h_{3}^{\varepsilon}(t)$ to cancel
oscillations that result from computing
$\mathcal{A}^{\varepsilon}(h_{2}^{\varepsilon})(t)$. After we have constructed
the perturbed test function

$h^{\varepsilon}(t)=h_{0}^{\varepsilon}(t)+h_{1}^{\varepsilon}(t)+h_{2}^{\varepsilon}(t)+h^{\varepsilon}_{3}(t)\,,$

we show that the pseudo-generator

$\mathcal{A}^{\varepsilon}(h_{0}^{\varepsilon}+h_{1}^{\varepsilon}+h_{2}^{\varepsilon}+h^{\varepsilon}_{3})(t)$

has the desired form of a generator related to a martingale problem, plus a
negligible term. Combining with tightness, we can show that the limiting
process $(Y_{t})_{t\geqslant 0}$ satisfies a martingale problem.

We now carry out this approach. For convenience, the proofs of needed
auxiliary lemmas will be deferred to Sections 7.4-7.8.

###### Proof of Proposition 7.5.

We split $\mathcal{A}^{\varepsilon}(h^{\varepsilon}_{0}+h^{\varepsilon}_{1})$
into three parts

$\mathcal{A}^{\varepsilon}(h^{\varepsilon}_{0}+h^{\varepsilon}_{1})(t)=\mathcal{A}^{\varepsilon}_{1}(t)+\mathcal{A}^{\varepsilon}_{2}(t)+\mathcal{P}_{2}^{\varepsilon}(t)\,,$

where $\mathcal{P}_{2}^{\varepsilon}$ is defined in (7.2), and
$\mathcal{A}^{\varepsilon}_{1}(t)$, $\mathcal{A}^{\varepsilon}_{2}(t)$ are
given by

$\displaystyle\mathcal{A}^{\varepsilon}_{1}(t)=\sum_{\begin{subarray}{c}n,m\in\mathbb{Z}^{*}\\\
n+m\neq 0\end{subarray}}e^{2i\pi(n+m)\tau^{\varepsilon}_{t}}\iint
V\Big{(}\frac{t}{\varepsilon^{2}},dp\Big{)}V\Big{(}\frac{t}{\varepsilon^{2}},dp^{\prime}\Big{)}W_{n,m}(p,Y^{\varepsilon}_{t})\,,$
(7.10)
$\displaystyle\mathcal{A}^{\varepsilon}_{2}(t)=\sum_{n\in\mathbb{Z}^{*}}\iint
V\Big{(}\frac{t}{\varepsilon^{2}},dp\Big{)}V\Big{(}\frac{t}{\varepsilon^{2}},dp^{\prime}\Big{)}W_{n}(p,Y^{\varepsilon}_{t})\,.$

Here

$\displaystyle W_{n,m}(p,Y^{\varepsilon}_{t})$
$\displaystyle\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\frac{1}{\mu|p|^{2\beta}-2i\pi
n\omega(Y^{\varepsilon}_{t})}\Big{(}F_{n}(Y^{\varepsilon}_{t})^{T}\nabla^{2}h(Y^{\varepsilon}_{t})F_{m}(Y^{\varepsilon}_{t})$
$\displaystyle\qquad\qquad+\nabla
h(Y^{\varepsilon}_{t})^{T}\textrm{Jac}F_{n}(Y^{\varepsilon}_{t})F_{m}(Y^{\varepsilon}_{t})\Big{)}$
(7.11) $\displaystyle\qquad-\frac{2i\pi n}{(\mu|p|^{2\beta}-2i\pi
n\omega(Y^{\varepsilon}_{t}))^{2}}F_{n}(Y^{\varepsilon}_{t})\cdot\nabla
h(Y^{\varepsilon}_{t})F_{m}(Y^{\varepsilon}_{t})\cdot\nabla\omega(Y^{\varepsilon}_{t})\,,$

and

$W_{n}(p,Y^{\varepsilon}_{t})=W_{n,-n}(p,Y^{\varepsilon}_{t})\,.$

The term $\mathcal{A}^{\varepsilon}_{2}$ no longer has oscillatory components
as $\mathcal{A}_{2}^{\varepsilon}$ only picks up the terms with $n+m=0$. We
now show that this term is related to the generator of a martingale problem.

######  Lemma 7.6.

For all $t\geqslant 0$, we have

$\lim_{\varepsilon\to
0}\bm{E}\Bigl{|}\int_{0}^{t}(\mathcal{A}^{\varepsilon}_{2}(u)-\mathcal{L}h(Y^{\varepsilon}_{u}))\,du\Bigr{|}=0\,.$

with

$\mathcal{L}h(y)=\sum_{n\in\mathbb{Z}^{*}}\int dp\,r(p)W_{n}(p,y)\,,$

where $W_{n}$ is defined through (7.3).

We present the proof of Lemma 7.6 in Section 7.7. To show the well-posedness
of the limiting martingale problem, we need to rewrite $\mathcal{L}$ as a
second-order differential operator, which eventually gives us the form in
(7.9). We do this in Section 7.8, below.

For the oscillatory term $\mathcal{A}^{\varepsilon}_{1}$, we introduce two
more small perturbations $h^{\varepsilon}_{2}$ and $h^{\varepsilon}_{3}$ to
prove that the oscillations will vanish as $\varepsilon\to 0$. Note that
$\mathcal{P}_{2}^{\varepsilon}$ also has oscillatory terms. These can be
treated in a manner similar to the oscillatory terms in
$\mathcal{A}^{\varepsilon}_{1}$, so that these terms vanish in the limit
$\varepsilon\to 0$. Since the treatment of this will be similar to the
treatment of $\mathcal{A}^{\varepsilon}_{1}$, we omit the details.

Similarly to the construction of $h_{1}^{\varepsilon}$ in equation (7.2), we
introduce the second perturbation

$\begin{split}h^{\varepsilon}_{2}&(t)=\int_{t}^{\infty}du\sum_{\begin{subarray}{c}n,m\in\mathbb{Z}^{*}\\\
n+m\neq
0\end{subarray}}e^{2i\pi(n+m)(\tau^{\varepsilon}_{t}+(u-t)\omega(Y^{\varepsilon}_{t})/\varepsilon^{2})}\\\
&\cdot\iint\Big{(}\bm{E}^{\varepsilon}_{t}\Big{[}V\Big{(}\frac{u}{\varepsilon^{2}},dp\Big{)}V\Big{(}\frac{u}{\varepsilon^{2}},dp^{\prime}\Big{)}\Big{]}-\bm{E}[V(0,dp)V(0,dp^{\prime})]\Big{)}W_{n,m}(p,Y^{\varepsilon}_{t})\,.\end{split}$

Making the change of variable $u\to t+\varepsilon^{2}u$, we have

$\begin{split}h^{\varepsilon}_{2}&(t)=\varepsilon^{2}\int_{0}^{\infty}du\sum_{\begin{subarray}{c}n,m\in\mathbb{Z}^{*}\\\
n+m\neq
0\end{subarray}}e^{2i\pi(n+m)(\tau^{\varepsilon}_{t}+u\omega(Y^{\varepsilon}_{t}))}W_{n,m}(p,Y^{\varepsilon}_{t})\\\
&\cdot\iint\Big{(}\bm{E}^{\varepsilon}_{t}\Big{[}V\Big{(}u+\frac{t}{\varepsilon^{2}},dp\Big{)}V\Big{(}u+\frac{t}{\varepsilon^{2}},dp^{\prime}\Big{)}\Big{]}-\bm{E}[V(0,dp)V(0,dp^{\prime})]\Big{)}\,.\end{split}$

According to formula (3.6), one has

$\begin{split}\bm{E}^{\varepsilon}_{t}\Big{[}V&\Big{(}u+\frac{t}{\varepsilon^{2}},dp\Big{)}V\Big{(}u+\frac{t}{\varepsilon^{2}},dp^{\prime}\Big{)}\Big{]}-\bm{E}[V(0,dp)V(0,dp^{\prime})]\\\
&=e^{-\mu(|p|^{2\beta}+|p^{\prime}|^{2\beta})u}\Big{(}V\Big{(}\frac{t}{\varepsilon^{2}},dp\Big{)}V\Big{(}\frac{t}{\varepsilon^{2}},dp^{\prime}\Big{)}-r(p)\delta(p-p^{\prime})\,dp\,dp^{\prime}\Big{)}\,,\end{split}$

so that

$\begin{split}h^{\varepsilon}_{2}(t)&=\varepsilon^{2}\sum_{\begin{subarray}{c}n,m\in\mathbb{Z}^{*}\\\
n+m\neq 0\end{subarray}}e^{2i\pi(n+m)\tau^{\varepsilon}_{t}}\\\
&\times\iint\frac{V(t/\varepsilon^{2},dp)V(t/\varepsilon^{2},dp^{\prime})-r(p)\delta(p-p^{\prime})\,dp\,dp^{\prime}}{\mu(|p|^{2\beta}+|p^{\prime}|^{2\beta})-2i\pi(n+m)\omega(Y^{\varepsilon}_{t})}W_{n,m}(p,Y^{\varepsilon}_{t})\,.\end{split}$

We claim that $h^{\varepsilon}_{2}$ is uniformly small.

######  Lemma 7.7.

We have

$\lim_{\varepsilon\to 0}\sup_{t\in[0,T]}\bm{E}|h^{\varepsilon}_{2}(t)|=0\,.$

For clarity of presentation we postpone the proof lf Lemma 7.7 to Section 7.4.
As before, following the definition of pseudo-generator in Section 7.1.1, we
have

$\mathcal{A}^{\varepsilon}(h_{2}^{\varepsilon})(t)=-\mathcal{A}^{\varepsilon}_{1}(t)+\mathcal{D}_{\varepsilon}(t)+\mathcal{R}^{1}_{\varepsilon}(t)\,,$

with

$\begin{split}\mathcal{D}_{\varepsilon}(t)&\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\sum_{\begin{subarray}{c}n,m\in\mathbb{Z}^{*}\\\
n+m\neq 0\end{subarray}}e^{2i\pi(n+m)\tau_{t}^{\varepsilon}}\int
dp\,r(p)W_{n,m}(p,Y^{\varepsilon}_{t})\,,\end{split}$

and $\mathcal{R}_{\varepsilon}^{1}(t)$ to be the remainder (an explicit
formula for $\mathcal{R}^{1}_{\varepsilon}(t)$ is presented in to Section 7.5,
below). Even though oscillatory terms are present in
$\mathcal{R}^{1}_{\varepsilon}$, we make the following claim.

######  Lemma 7.8.

We have

$\lim\limits_{\varepsilon\to
0}\sup_{t\in[0,T]}\bm{E}|\mathcal{R}^{1}_{\varepsilon}(t)|=0\,.$

The proof of this lemma is presented in Section 7.5, below. To treat the
oscillations in $\mathcal{D}_{\varepsilon}(t)$ we introduce the third
perturbation

$\displaystyle h^{\varepsilon}_{3}(t)$
$\displaystyle=\int_{t}^{\infty}du\sum_{\begin{subarray}{c}n,m\in\mathbb{Z}^{*}\\\
n+m\neq
0\end{subarray}}e^{2i\pi(n+m)(\tau_{t}^{\varepsilon}+(u-t)\omega(Y_{t}^{\varepsilon})/\varepsilon^{2})}e^{-\varepsilon(u-t)}\int
dp\,r(p)W_{n,m}(p,Y^{\varepsilon}_{t})$
$\displaystyle=\varepsilon^{2}\sum_{\begin{subarray}{c}n,m\in\mathbb{Z}^{*}\\\
n+m\neq
0\end{subarray}}\frac{e^{2i\pi(n+m)\tau_{t}^{\varepsilon}}}{\varepsilon^{3}-2i\pi(n+m)\omega(Y_{t}^{\varepsilon})}\int
dp\,r(p)W_{n,m}(p,Y^{\varepsilon}_{t})$

We will see that

$\mathcal{A}^{\varepsilon}(h^{\varepsilon}_{3})(t)=-\mathcal{D}_{\varepsilon}(t)+\mathcal{R}^{2}_{\varepsilon}(t)$

with explicit formula for $\mathcal{R}^{2}_{\varepsilon}(t)$ presented in
Section 7.6, below. We claim $h^{\varepsilon}_{3}$ and $R^{2}_{\varepsilon}$
both vanish as $\varepsilon\to 0$.

######  Lemma 7.9.

We have

$\lim_{\varepsilon\to
0}\sup_{t\in[0,T]}\bm{E}|h^{\varepsilon}_{3}(t)|=0\qquad\text{and}\qquad\lim\limits_{\varepsilon\to
0}\sup_{t\in[0,T]}\bm{E}|\mathcal{R}^{2}_{\varepsilon}(t)|=0\,.$

We will prove Lemma 7.9 in Section 7.6, below. As a result, we now have a
perturbed test function

$h(t)=h_{0}^{\varepsilon}(t)+h_{1}^{\varepsilon}(t)+h_{2}^{\varepsilon}(t)+h^{\varepsilon}_{3}(t)\,,$

with $h^{\varepsilon}_{0}(t)=h(Y^{\varepsilon}_{t})$, for which

(7.12)
$\mathcal{A}^{\varepsilon}(h_{0}^{\varepsilon}+h_{1}^{\varepsilon}+h_{2}^{\varepsilon}+h^{\varepsilon}_{3})(t)=\mathcal{A}^{\varepsilon}_{2}(t)+o(\varepsilon)\,.$

Using Theorem 7.1 we see that for all $N\geqslant 1$, any bounded continuous
function $\Psi$, and every sequence $0<s_{1}<\cdots<s_{N}\leqslant s<t$ we
must have

(7.13)
$\bm{E}\Big{[}\Psi\big{(}Y^{\varepsilon}_{s_{1}},\dots,Y^{\varepsilon}_{s_{N}}\big{)}\Big{(}M^{\varepsilon}_{h}(t)-M^{\varepsilon}_{h}(s)\Big{)}\Big{]}=0\,.$

Together with Lemmas 7.3, 7.6, 7.7, 7.8, 7.9, and estimate (7.12), we have

(7.14) $\lim_{\varepsilon\to
0}\bm{E}\Big{[}\Psi\big{(}Y^{\varepsilon}_{s_{1}},\dots,Y^{\varepsilon}_{s_{N}}\big{)}\Big{(}h(Y^{\varepsilon}_{t})-h(Y^{\varepsilon}_{s})-\int_{s}^{t}\mathcal{L}h(u)\,du\Big{)}\Big{]}=0\,,$

where $\mathcal{L}$ is defined in (7.9). Combining with
$Y^{\varepsilon}_{t=0}=Y_{0}$ for every $\varepsilon>0$, the equation (7.14)
implies that any subsequential limit $(Y_{t})_{t\geqslant 0}$ of
$(Y_{\varepsilon})_{\varepsilon>0}$ is a solution of the martingale problem
for the generator $\mathcal{L}$ with initial condition $Y_{t=0}=Y_{0}$. We
claim that the solution to this martingale problem is unique (we prove this in
Section 7.8, below). Given this, along with the tightness proven in
Proposition 7.2, Prokhorov’s theorem implies the weak convergence of the
sequence of processes $(Y^{\varepsilon})_{\varepsilon>0}$ to the process
$(Y_{t})_{t\geqslant 0}$ as desired. This completes the proof of Proposition
6.1. ∎

###  7.4. Proof of Lemma 7.7

We now prove Lemma 7.7. By using Lemma 3.5, the proof is similar that of Lemma
7.4.

###### Proof of Lemma 7.7.

We split the integrand in $h^{\varepsilon}_{2}$ into two parts as follows

$\begin{split}h^{\varepsilon}_{2}(t)&=\varepsilon^{2}\sum_{\begin{subarray}{c}n,m\in\mathbb{Z}^{*}\\\
n+m\neq 0\end{subarray}}e^{2i\pi(n+m)\tau^{\varepsilon}_{t}}\\\
&\qquad\cdot\iint\frac{V(t/\varepsilon^{2},dp)V(t/\varepsilon^{2},dp^{\prime})-r(p)\delta(p-p^{\prime})\,dp\,dp^{\prime}}{\mu(|p|^{2\beta}+|p^{\prime}|^{2\beta})-2i\pi(n+m)\omega(Y^{\varepsilon}_{t})}W_{n,m}(p,Y^{\varepsilon}_{t})\\\
&\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}h^{\varepsilon}_{21}(t)-h^{\varepsilon}_{22}(t)\end{split}$

where $h_{21}^{\varepsilon}$ contains the term
$V(t/\varepsilon^{2},dp)V(t/\varepsilon^{2},dp^{\prime})$ and
$h_{22}^{\varepsilon}$ contains the term $r(p)\delta(p-p^{\prime})$.

For $h_{22}^{\varepsilon}$, it is straightforward to see that

$\displaystyle|h^{\varepsilon}_{22}(t)|$
$\displaystyle\leqslant\varepsilon^{2}C_{h,\omega_{0},\nabla\omega}\int
dp\,r(p)$
$\displaystyle\qquad\cdot\sum_{\begin{subarray}{c}n,m\in\mathbb{Z}^{*}\\\
n+m\neq
0\end{subarray}}\frac{1}{|n(n+m)|}|F_{m}(Y^{\varepsilon}_{t})|\Big{(}\big{(}1+|n|)|F_{n}(Y^{\varepsilon}_{t})|+\|\textrm{Jac}F_{n}(Y^{\varepsilon}_{t})\|\Big{)}$
$\displaystyle\leqslant\varepsilon^{2}C_{h,\omega_{0},\nabla\omega}\int
dp\,r(p)$ (7.15)
$\displaystyle\qquad\cdot\sum_{n\in\mathbb{Z}^{*}}\frac{1}{|n|^{2}}\sup_{\begin{subarray}{c}|y|\leqslant
2M\end{subarray}}\int_{0}^{1}\Big{(}|F(y,\tau)|^{2}+|\partial_{\tau}F(y,\tau)|^{2}+\|\textrm{Jac}F(y,\tau)\|^{2}\Big{)}d\tau\,.$

We rewrite

$h^{\varepsilon}_{21}(t)=\varepsilon^{2}V\Big{(}\frac{t}{\varepsilon^{2}},\varphi_{1,t,\varepsilon}\Big{)},$

with

$\varphi_{1,t,\varepsilon}(p)=V\Big{(}\frac{t}{\varepsilon^{2}},\varphi_{2,t,p,\varepsilon}\Big{)}$

and

$\begin{split}\varphi_{2,t,p,\varepsilon}(p^{\prime})&=\sum_{\begin{subarray}{c}n,m\in\mathbb{Z}^{*}\\\
n+m\neq
0\end{subarray}}e^{2i\pi(n+m)\tau^{\varepsilon}_{t}}W_{n,m}(p,Y^{\varepsilon}_{t})\\\
&\qquad\cdot\frac{1}{\mu(|p|^{2\beta}+|p^{\prime}|^{2\beta})-2i\pi(n+m)\omega(Y^{\varepsilon}_{t})}\,.\end{split}$

For readers’ convenience, we pause to recall the meaning of the above
notation. The function $\varphi_{1,t,\varepsilon}$ of the variable $p$ is
obtained by applying the measure $V(t/\varepsilon^{2},dp^{\prime})$ to the
function $\varphi_{2,t,p,\varepsilon}(p^{\prime})$, where
$\varphi_{2,t,p,\varepsilon}$ (a function of the variable $p^{\prime}$), is
defined as above. Finally, $h_{21}^{\varepsilon}(t)$ is obtained by applying
the measure $V(t/\varepsilon^{2},dp)$ to the function
$\varphi_{1,t,\varepsilon}(p)$.

For $h^{\varepsilon}_{22}$, one has

$|\varphi_{2,t,p,\varepsilon}(p^{\prime})|\leqslant C\,,$

uniformly in $p$ and $p^{\prime}$. Following the same lines as for the first
perturbation of the test function $h^{\varepsilon}_{1}$ but using inequality
(3.8) of Lemma 3.5 (with $k=\infty$ if $\beta\geqslant 1/2$ and
$k\in(1,1/(1-2\beta))$ if $\beta<1/2$), we have for some $C^{\prime}>0$

$\sup_{p\in S}|\varphi_{1,t,\varepsilon}(p)|\leqslant\sup_{\varphi\in
W_{k,C^{\prime}}}\Big{|}V\Big{(}\frac{t}{\varepsilon^{2}},\varphi\Big{)}\Big{|}\,.$

In fact, the derivative of $\varphi_{2,t,p,\varepsilon}$ in $p^{\prime}$ can
be uniformly bounded in $p$ in the corresponding $L^{k}(S)$ space.

Now, let us look at the derivative of $\varphi_{1,t,\varepsilon}(p)$ in $p$.
One can write

$\partial_{p}\varphi_{1,t,\varepsilon}(p)=|p|^{2\beta-1}V\Big{(}\frac{t}{\varepsilon^{2}},\varphi^{\prime}_{2,t,p,\varepsilon}\Big{)}+V\Big{(}\frac{t}{\varepsilon^{2}},\varphi^{\prime\prime}_{2,t,p,\varepsilon}\Big{)},$

where $\varphi^{\prime}_{2,t,p,\varepsilon}$ and
$\varphi^{\prime\prime}_{2,t,p,\varepsilon}$ belong to some
$W_{k,C^{\prime\prime}}$, and where $C^{\prime\prime}$ does not depend on $p$.
Then,

$|\partial_{p}\varphi_{1,t,\varepsilon}(p)|\leqslant(|p|^{2\beta-1}+1)\sup_{\varphi\in
W_{k,C^{\prime\prime}}}\Big{|}V\Big{(}\frac{t}{\varepsilon^{2}},\varphi\Big{)}\Big{|}\,.$

As a result, considering $\tilde{C}=\max(C^{\prime},C^{\prime\prime})$, and

$\tilde{h}^{\varepsilon}_{21}(t)\stackrel{{\scriptstyle\scriptscriptstyle\textup{def}}}{{=}}\frac{h^{\varepsilon}_{21}(t)}{\sup_{\varphi\in
W_{k,\tilde{C}}}|V(t/\varepsilon^{2},\varphi)|}=\varepsilon^{2}V\Big{(}\frac{t}{\varepsilon^{2}},\tilde{\varphi}_{1,t,\varepsilon}\Big{)}\,,$

with

$\tilde{\varphi}_{1,t,\varepsilon}=\frac{\varphi_{1,t,\varepsilon}}{\sup_{\varphi\in
W_{k,\tilde{C}}}|V(t/\varepsilon^{2},\varphi)|},$

we have

$\|\tilde{\varphi}_{1,t,\varepsilon}\|_{W^{1,k}(S)}\leqslant 1.$

Therefore,

$|\tilde{h}^{\varepsilon}_{21}(t)|\leqslant\varepsilon^{2}\sup_{\varphi\in
W_{k,\tilde{C}}}\Big{|}V\Big{(}\frac{t}{\varepsilon^{2}},\varphi\Big{)}\Big{|},$

and hence

$\bm{E}|h^{\varepsilon}_{21}(t)|\leqslant\varepsilon^{2}\bm{E}\sup_{\varphi\in
W_{k,\max(1,\tilde{C})}}\Big{|}V\Big{(}\frac{t}{\varepsilon^{2}},\varphi\Big{)}\Big{|}^{2}=\varepsilon^{2}\bm{E}\sup_{\varphi\in
W_{k,\max(1,\tilde{C})}}|V(0,\varphi)|^{2}\,.$

Using Lemma 3.5, this concludes the proof. ∎

###  7.5. Proof of Lemma 7.8

We devote this section to proving Lemma 7.8. By following the computations in
(7.2), we obtain the following explicit formula for
$\mathcal{R}_{1}^{\varepsilon}(t)$:

$\displaystyle\mathcal{R}_{1}^{\varepsilon}(t)=\varepsilon
v\Big{(}\frac{t}{\varepsilon^{2}}\Big{)}\sum_{\begin{subarray}{c}n,m\in\mathbb{Z}^{*}\\\
n+m\neq
0\end{subarray}}\iint\frac{2i\pi(n+m)e^{2i\pi(n+m)\tau^{\varepsilon}_{t}}}{\mu(|p|^{2\beta}+|p^{\prime}|^{2\beta})-2i\pi(n+m)\omega(Y^{\varepsilon}_{t})}$