# Slow-fast dynamics in stochastic Lotka-Volterra systems

Julien Barré1, Bastien Fernandez2 and Grégoire Panel1

###### Abstract

We investigate the large population dynamics of a family of stochastic
particle systems with three-state cyclic individual behaviour and parameter-
dependent transition rates. On short time scales, the dynamics turns out to be
approximated by an integrable Hamiltonian system whose phase space is foliated
by periodic trajectories. This feature suggests to consider the effective
dynamics of the long-term process that results from averaging over the rapid
oscillations. We establish the convergence of this process in the large
population limit to the solutions of an explicit stochastic differential
equation. Remarkably, this averaging phenomenon is complemented by the
convergence of stationary measures. The proof of averaging follows the
Stroock-Varadhan approach to martingale problems and relies on a fine analysis
of the system’s dynamical features.

1 Institut Denis Poisson, Université d’Orléans, Université de Tours and CNRS,
Orléans, France.

2 Laboratoire de Probabilités, Statistique et Modélisation

Univ. Paris Cité - CNRS - Sorbonne Univ.

F-75013 Paris, France

##  1 Introduction

Stochastic interacting particle systems with cyclic structure, sometimes
called stochastic Lotka-Volterra (LV) systems or rock-paper-scissors games,
play an important role in modelling in a large variety of different fields:
ecology and population dynamics [4, 5, 7, 30], evolutionary game theory [18,
33], dynamics of Bose-Einstein condensates [27], chemical reaction networks
[34], etc. To describe the long term dynamics of these models is therefore an
important challenge in the theoretical and mathematical sciences.

In the idealised large population limit, these stochastic systems are usually
well-approximated by systems of ordinary differential equations (of LV type),
see for instance [2, 18, 19, 27]; thus considerations about deterministic
dynamics could suffice in principle. Yet, in the more realistic case of finite
populations, the deterministic approximations are valid only on short time
scales. Stochasticity must be integrated into the analysis in order to
apprehend the long-term behaviors. In fact, dramatic finite-size effects can
generate various phenomena, such as extinction events and other averaging
phenomena, which are not captured by the deterministic limit, see e.g. [5, 8,
11, 22, 32] for examples in the physics literature. From a rigorous
mathematical viewpoint, extinction events have been described in examples of
particle systems, in particular in population dynamics [13, 14]. However, to
the best of our knowledge, no mathematical characterisation of an emerging
averaging phenomena in stochastic LV systems has been given in the literature.
Of note, averaging is ubiquitous in particle systems without cyclic behaviour,
when the slow-fast time scale separation naturally materializes in the
original variables. Various mathematical results have been obtained in this
context, see e.g. multiscale chemical reaction and gene networks [3, 10, 24,
25] and structured population dynamics [9, 29].

In order to mathematically address averaging in stochastic LV systems as it
emerges from oscillatory behaviours that involve all the original variables,
we consider in this paper a simple example of particle system with cyclic
state space. In few words, the system is a Markov process that can be defined
as follows (see section 2.1 for details). The state space is
$\mathbb{Z}_{3}=\mathbb{Z}/3\mathbb{Z}$ and a particle in state
$i\in\mathbb{Z}_{3}$ can only jump to state $i+1$. The jumps are independent
and for each particle in state $i$, occur with rate $a+N_{i+1}$ where
$a\in\mathbb{R}^{+}$ is an intrinsic rate and $N_{i+1}\in\mathbb{N}$ is the
number of particles in state $i+1$. The population size
$N=\sum_{i\in\mathbb{Z}_{3}}N_{i}$ is constant so that the phase space is the
two-dimensional simplicial grid.

Figure 1: Trajectories of the particle system in the two-dimensional
simplicial grid (Left $a=0.2$, Right $a=1.3$, $N=2000$ in the main pictures,
$N=200$ in the right insets). The initial condition are located at the center
of the grid. The colors stand for the time in $[0,1]$, from blue ($t=0$) to
red ($t=1$). Left inset: Time series of the slow variable
$z(t)=\prod_{i\in\mathbb{Z}_{3}}x_{i}(t)$ for $N=2000$, where
$x_{i}=\tfrac{N_{i}(t)}{N}$.

As intended, such extensive transition rates promote rapid oscillations in
phase space when $N$ is large (see illustrations on Fig. 1, in particular
compare the main pictures $N=2000$ from the corresponding right insets
$N=200$). In fact, the deterministic flow in the large population limit, which
turns out to approximate the short time scale dynamics when $N$ is large (see
Proposition 2.2 below), consists of an integrable Hamiltonian system whose
two-dimensional simplex phase space is foliated by periodic trajectories (on
which the slow variable $z=\prod_{i\in\mathbb{Z}_{3}}x_{i}$ remains constant).
This suggests to consider the one-dimensional transverse dynamics of the
variable $z$ that results from averaging the fast motions on the periodic
loops. The main result of this paper (Theorem 3.1) states that for large $N$,
the slow-time scale transverse dynamics is indeed approximated by a diffusion
process with $a$-dependent drift. In short terms, a slow-fast dynamics emerges
in this system in the large population limit.

Technically speaking, the stochastic process that governs the dynamics of the
particle system. can be regarded as a random perturbation of a dynamical
system with a conservation law. Yet, the oscillation period diverges at the
phase space boundary (independently of the population size) and this prevents
us to apply the standard techniques in this setting [17, 31]. Instead, our
proof follows the Stroock-Varadhan approach to martingale problems [35] and
relies on the compactness-uniqueness argument in this context. The core
argument (section 3.3) is a proof of the $L^{1}$-convergence of martingales
which is tailored to the specific nature of the process and in particular, to
its behaviour close to the boundary.

Remarkably, for this particle system, the averaging phenomenon is further
complemented by the large $N$ convergence of stationary measures. Indeed, for
every $N$, the (unique) stationary measure of the process on the simplicial
grid is a product measure which converges to a Dirichlet distribution in the
large population limit (Proposition 2.1). Moreover, the push-forward measure
on the transverse variable $z$ induced by this distribution turns out to be
stationary for the semi-group associated with the diffusion process
(Proposition 2.8). Together with the specification of the nature of the
boundary points of this process (Lemma 2.7), these properties indicate that
the visits of the particles’ system to the boundaries of the simplex are
frequent for $a<1$ and become sparse when $a\geq 1$, as illustrated in Fig. 1.

Our results are limited here to a simple model with three states playing
symmetric roles, but the ideas and techniques can be useful in a broader
context. For instance, the analysis can be extended to state-dependent
transition rates, which may provide clues to answer the important question:
when extinction is possible, which species survives? The answer is counter
intuitive as shown in the physics literature [5]. The extension to more than
three states is more challenging, as the deterministic dynamics may not be
periodic any more; however, when it is periodic, the techniques of this
article may allow to prove the convergence of the slow dynamics to a multi
dimensional diffusion process.

##  2 Definitions and preliminary considerations

###  2.1 The stochastic particle system

We consider the two-dimensional simplex $S$ defined by

$S=\left\\{\mathrm{x}:=(x_{1},x_{2})\in(\mathbb{R}^{+})^{2}\ \text{such that}\
x_{1}+x_{2}\leq 1\right\\},$

and given $N\in\mathbb{N}$ (where $\mathbb{N}=\\{0,1,2,\cdots\\}$), let
$S_{N}=S\cap\tfrac{1}{N}\mathbb{N}^{2}$ be the two-dimensional simplicial
grid, whose vertices $\mathrm{v}_{i}$ are the points with coordinates
$(\mathrm{v}_{i})_{j}=\delta_{ij},\,j=1,2$ (where $\delta_{ij}$ is the
Kronecker symbol) (see Fig. 2).

Figure 2: Illustration of the simplicial grid $S_{N}$ of step size
$\tfrac{1}{N}$.

The time evolution of the particle system in $S_{N}$ is governed by the (jump)
Markov process $\\{P_{N}^{\mathrm{x}}\\}_{\mathrm{x}\in S_{N}}$ induced by the
generator $L_{N}$ defined by111Throughout the paper, the notations $i+1$ and
$i-1$ mean respectively $i+1\ \text{mod}\ 3$ and $i-1\ \text{mod}\ 3$.

$L_{N}f(\mathrm{x})=N\sum_{i\in\mathbb{Z}_{3}}x_{i}(a+Nx_{i+1})\left(f(\mathrm{x}+\frac{\mathrm{u}_{i}}{N})-f(\mathrm{x})\right)\quad\forall\mathrm{x}\in
S_{N},\ f:S_{N}\to\mathbb{R}$

where $x_{3}:=1-x_{1}-x_{2}$, $a\in\mathbb{R}^{+}$ and
$\mathrm{u}_{i}=v_{i+1}-v_{i}$ for
$i\in\mathbb{Z}_{3}:=\mathbb{Z}/3\mathbb{Z}$.

As mentioned in the introduction, this process represents the stochastic time
evolution of a population of individuals with cyclic state space and extensive
transition rates and is inspired by the modelling in various fields [16, 32].
In particular, the definition above suggests various natural extensions of
this process, such as increasing the number of states, from three to an
arbitrary $d\in\mathbb{N}$, or allowing any particle of a site $i$ to jump on
the site $j$. Notice that most of the approach and considerations in this
paper can be adapted to these extensions without additional conceptual
difficulties.

A nice feature of this process is that it is ergodic for every
$N\in\mathbb{N}$ and $a>0$ and its invariant measure turns out to be the
following product measure [16]

$\mu_{N,a}(\mathrm{x})=C_{N,a}\prod_{i\in\mathbb{Z}_{3}}\frac{\Gamma(Nx_{i}+a)}{\Gamma(Nx_{i}+1)},\quad\forall\mathrm{x}\in
S_{N},$

where $\Gamma$ stands for the Gamma function and $C_{N,a}$ is the
normalisation constant. For $a=0$, the three vertices
$\\{\mathrm{v}_{i}\\}_{i\in\mathbb{Z}_{3}}$ are absorbing states.

The measure $\mu_{N,a}$ can be regarded as an atomic measure in $S$. Under
this viewpoint, this measure can be shown to weakly converge to the Dirichlet
measure, namely the absolutely continuous measure $\mu_{a}$ on $S$ with
density $\rho_{a}$ defined by

$\rho_{a}(\mathrm{x})=C_{a}z(\mathrm{x})^{a-1},\ \forall\mathrm{x}\in S\
\text{where}\ z(\mathrm{x})=\prod_{i\in\mathbb{Z}_{3}}x_{i},$

and again $C_{a}$ is the normalisation constant. The convergence is claimed in
the following statement, whose proof is given in Appendix A.

######  Proposition 2.1.

For every $a>0$, we have

$\lim_{N\to\infty}\mu_{N,a}=\mu_{a},$

in the weak sense.

###  2.2 The deterministic approximation on short time scales

When interested in the temporal process associated with $L_{N}$ for large $N$,
the Taylor theorem applied to the first order expansion of
$f(\mathrm{x}+\frac{\mathrm{u}_{i}}{N})$ at $\mathrm{x}$ suggests to consider
the operator ${\cal L}_{\text{fast}}$ defined by (NB: $f^{\prime}$ denotes the
Fréchet derivative of $f$.)

${\cal
L}_{\text{fast}}f(\mathrm{x})=\sum_{i\in\mathbb{Z}_{3}}x_{i}x_{i+1}f^{\prime}(\mathrm{x})\mathrm{u}_{i}=\sum_{j=1,2}x_{j}(x_{j-1}-x_{j+1})\partial_{x_{j}}f(\mathrm{x}),\
f\in C^{1}(S),$

so that we have $\lim_{N\to\infty}\tfrac{1}{N}L_{N}f(\mathrm{x})={\cal
L}_{\text{fast}}f(\mathrm{x})$. Let $F$ be the vector field on $S$ defined by

$(F(\mathrm{x}))_{j}=x_{j}(x_{j-1}-x_{j+1}),\ j=1,2.$

This vector field defines a semi-flow on $S$, under which this simplex is
invariant. Let $t\mapsto X^{\rm fast}_{\mathrm{x}_{0}}(t)$ be the solution of
$\dot{\mathrm{x}}=F(\mathrm{x})$ with initial condition $\mathrm{x}_{0}\in S$.
Then for any $f\in C^{1}(S)$, we have

$\tfrac{d}{dt}f(X^{\rm fast}_{\mathrm{x}_{0}}(t))={\cal
L}_{\text{fast}}f(X^{\rm fast}_{\mathrm{x}_{0}}(t)),\ t>0.$

The convergence $\tfrac{1}{N}L_{N}f(\mathrm{x})\to{\cal
L}_{\text{fast}}f(\mathrm{x})$ suggests that the process associated with the
particle system can be approximated on the time scales of the order
$\frac{1}{N}$ by the deterministic semi-flow. In order to formalize this
approximation, given $T>0$, let $\mathbb{D}([0,T],S_{N})$ (resp.
$\mathbb{D}([0,T],S)$) be the set of càdlàg functions from $[0,T]$ into
$S_{N}$ (resp. $S$) and let ${\cal F}_{T,N}$ (resp. ${\cal F}_{T}$) be the
natural filtration associated with $\mathbb{D}([0,T],S_{N})$ (resp.
$\mathbb{D}([0,T],S)$). The set $\mathbb{D}([0,T],S)$ is endowed with the
Skorokhod metric. We denote by $X_{N}(t)$ where $t\in[0,T]$ and
$X_{N}\in\mathbb{D}([0,T],S_{N})$ the stochastic process on
$(\mathbb{D}([0,T],S_{N}),{\cal F}_{T,N})$ associated with the Markov process
and the initial measure $\delta_{X_{N}(0)}$. Clearly, $X_{N}(t)$ can be seen
as a process taking values in $S$

######  Proposition 2.2.

Assume that the sequence of initial conditions
$\\{X_{N}(0)\\}_{N\in\mathbb{N}}$ converges in law to some $\mathrm{x}_{0}\in
S$. Then, for every $T>0$, the sequence of time-scaled processes
$\\{X_{N}(\frac{t}{N}):t\in[0,T]\\}_{N\in\mathbb{N}}$ converges in law in
$\mathbb{D}([0,T],S)$ to the trajectory arc $\\{X^{\rm
fast}_{\mathrm{x}_{0}}(t):t\in[0,T]\\}$ of the solution of
$\dot{\mathrm{x}}=F(\mathrm{x})$ with initial condition $\mathrm{x}_{0}$.

This statement can be proved using a compactness-uniqueness argument just as
in the proof of Theorem 3.1, Chap. 3 in [4]. See also the proof of Theorem 3.1
below for the details of a similar argument.

###  2.3 Analysis of the deterministic dynamics

According to the expression of $F$, the semi-flow associated with
$\dot{\mathrm{x}}=F(\mathrm{x})$ is an instance of a Lotka-Volterra system.
Actually, this dynamics can be regarded as a Hamiltonian system with
Hamiltonian function $(x_{1},x_{2})\mapsto x_{1}x_{2}(1-x_{1}-x_{2})$.

The dynamics can be analysed in full details and its essential features have
already been identified [2, 32]. In particular, there are four stationary
points, namely the centre $(\tfrac{1}{3},\tfrac{1}{3},\tfrac{1}{3})$ and the
vertices $\mathrm{v}_{i}$ of $S$. Each boundary edge of $S$ is invariant under
the semi-flow and the dynamics on each edge consists of heteroclinic
trajectories between the two corresponding vertices.

In addition, in
$\text{Int}(S)\setminus(\tfrac{1}{3},\tfrac{1}{3},\tfrac{1}{3})$, the interior
of the simplex except the centre, the level sets of the functional $z$ \-
which takes values in $I:=(0,\tfrac{1}{27})$ \- constitute a foliation by
invariant loops on which the trajectories $t\mapsto X^{\rm
fast}_{\mathrm{x}_{0}}(t)$ are periodic, with period say
$T(z(\mathrm{x}_{0}))$, and counterclockwise motion (see Fig. 3). The periodic
trajectories and their period can be semi-explicitly computed, see Appendix
B.1 for the corresponding computations. In particular, the period diverges
when approaching the boundary edges of $S$.

Figure 3: Color plot and level sets of the function $\mathrm{x}\mapsto
z(\mathrm{x})$ in the simplex $S$.

In the sequel, we shall need the following additional properties of the period
function. Of note, we use the symbol $z$ for the variable in $I$ and also
$T(z)$ as an abbreviation of the notation of the period. Moreover, given two
real functions $u$ and $v\neq 0$ and $x_{0}\in\mathbb{R}$, we write $u(x)\sim
v(x)$ as $x\to x_{0}^{\pm}$ if $\lim_{x\to x_{0}^{\pm}}\frac{u(x)}{v(x)}=1$.

######  Lemma 2.3.

(i) The function $z\mapsto T(z)$ is $C^{\infty}$ on $I$ and
$T(\frac{1}{27}^{-})=2\pi\sqrt{3}$.

(ii) We have $T(z)\sim-3\ln z$ as $z\to 0^{+}$.

The proof is given in Appendix B.2. In addition, we shall also need some
properties of the (signed) area enclosed in the loop $z(\mathrm{x})=z$ and
defined by222Lemma 2.5 below shows that in fact
$A(z)=\int_{0}^{T(z)}x_{i+1}\dot{x}_{i}dt$ for every $i\in\mathbb{Z}_{3}$.

$A(z)=\int_{0}^{T(z)}x_{2}\dot{x}_{1}dt,\quad z\in I,$

which can be regarded as the action of the Hamiltonian system. The desired
properties of this function are listed in the next statement, whose proof is
given in Appendix B.3.

######  Lemma 2.4.

(i) The function $z\mapsto A(z)$ is $C^{\infty}$ and negative on $I$.
Moreover, we have $A^{\prime}(z)=T(z)$ for all $z\in I$.

(ii) $A(0^{+})=-\frac{1}{2}$ and $A(z)\sim-2\pi\sqrt{3}(\frac{1}{27}-z)$ as
$z\to\frac{1}{27}^{-}$.

###  2.4 The averaged generator: definition and explicit expression

Following the approach to Anosov averaging [31], the Proposition 2.2 and the
periodic motions of the system $\dot{\mathrm{x}}=F(\mathrm{x})$ suggest to
consider averaging the dynamics associated with the next-order approximation
of $L_{N}$. To that goal, consider first the operator ${\cal L}_{\text{slow}}$
that collects the $N$-independent terms in the (second-order) expansion of
$L_{N}$, and defined for $f\in C^{2}(S)$ as follows

$\displaystyle{\cal L}_{\text{slow}}f(\mathrm{x})$
$\displaystyle=a\sum_{i\in\mathbb{Z}_{3}}x_{i}f^{\prime}(\mathrm{x})\mathrm{u}_{i}+\tfrac{1}{2}\sum_{i\in\mathbb{Z}_{3}}x_{i}x_{i+1}f^{\prime\prime}(\mathrm{x})(\mathrm{u}_{i},\mathrm{u}_{i})$
$\displaystyle=a\sum_{j=1,2}(x_{j+1}-x_{j})\partial_{x_{j}}f(\mathrm{x})+\tfrac{1}{2}\sum_{j=1,2}x_{j}(x_{j-1}+x_{j+1})\partial_{x_{j}}^{2}f(\mathrm{x})-\sum_{j=1,2}x_{j}x_{j+1}\partial_{x_{j},x_{j+1}}^{2}f(\mathrm{x})$

Given $z\in I$ and a fonction $f$ defined on the loop of period $T(z)$, let
the time average $\langle f\rangle_{z}$ be defined by

$\langle f\rangle_{z}=\frac{1}{T(z)}\int_{0}^{T(z)}f(X^{\rm
fast}_{\mathrm{x}_{0}}(t))dt,$

(where $\mathrm{x}_{0}$ is any point on the loop). Then, using the notation
$f_{I}$ for the functions defined on $I$ (or $\bar{I}$), the averaged operator
${\cal L}_{\text{avg}}$ is defined by

${\cal L}_{\text{avg}}f_{I}(z)=\langle{\cal L}_{\text{slow}}(f_{I}\circ
z)\rangle_{z},\ z\in I,\ f_{I}\in C^{2}(I).$

An explicit expression of this operator can be obtained based on the analysis
of the dynamics generated by $F$. The results are summarized in the following
statement.

######  Lemma 2.5.

The average $\langle x_{i+1}\dot{x}_{i}\rangle_{z}$ does not depend on
$i\in\mathbb{Z}_{3}$. Letting $m(z):=-\langle x_{i+1}\dot{x}_{i}\rangle_{z}$,
the averaged operator can be expressed as the following second-order
differential operator

${\cal L}_{\text{\rm
avg}}f_{I}(z)=3(am(z)-z)f_{I}^{\prime}(z)+3zm(z)f_{I}^{\prime\prime}(z),\quad
z\in I,\ f_{I}\in C^{2}(I).$

Given the definition of $A(z)$ above, we have $m(z)=-\frac{A(z)}{T(z)}>0$ for
all $z\in I$ and Lemma 2.4 implies that $m\in C^{\infty}(I)$, which yields
that ${\cal L}_{\text{avg}}f_{I}\in C^{0}(I)$. Moreover, Lemma 2.4 and Lemma
2.5 both imply that for $f_{I}\in C^{2}(\overline{I})$; hence ${\cal
L}_{\text{avg}}f$ can be extended by continuity to the boundary of $I$ as
follows

${\cal L}_{\text{avg}}f_{I}(0)=0\quad\text{and}\quad{\cal
L}_{\text{avg}}f_{I}(\tfrac{1}{27})=-\tfrac{1}{9}f_{I}^{\prime}(\tfrac{1}{27}),$

which in particular yields the following match

${\cal L}_{\text{avg}}f_{I}(\tfrac{1}{27})={\cal L}_{\text{slow}}(f_{I}\circ
z)(\tfrac{1}{3},\tfrac{1}{3},\tfrac{1}{3}).$ (1)

Proof of the Lemma. For $f_{I}\in C^{1}(I)$ and $\mathrm{x}\in\text{Int}(S)$,
direct computations yield the following expression

$\displaystyle{\cal L}_{\text{slow}}(f_{I}\circ z)$
$\displaystyle=\Big{(}a\sum_{i\in\mathbb{Z}_{3}}\big{(}x^{2}_{i-1}x_{i+1}-z\big{)}-3z\Big{)}f_{I}^{\prime}\circ
z+\tfrac{1}{2}\sum_{i\in\mathbb{Z}_{3}}z\Big{(}x_{i}x^{2}_{i-1}+x_{i}x^{2}_{i+1}-2z\Big{)}f_{I}^{\prime\prime}\circ
z$

Together with the relations
$z(\mathrm{x})\tfrac{x_{i-1}}{x_{i}}=x_{i-1}^{2}x_{i+1}$ and
$z(\mathrm{x})\tfrac{x_{i+1}}{x_{i}}=x_{i-1}x_{i+1}^{2}$, this suggests
consider the averages $\langle x_{i}x_{i+1}^{2}\rangle_{z}$ and $\langle
x_{i}x_{i-1}^{2}\rangle_{z}$ in order to compute of the expression of ${\cal
L}_{\text{avg}}$. We have the following statement.

######  Claim 2.6.

For every $z\in I$, we have $\langle x_{i}x_{i+1}^{2}\rangle_{z}=\langle
x_{i}x_{i-1}^{2}\rangle_{z}$ for all $i\in\mathbb{Z}_{3}$ and these quantities
do not depend on $i$.

Proof: By periodicity of the trajectories, we have for every $f\in C^{1}(S)$
and $z\in I$

$0=\langle\tfrac{d}{dt}f\rangle_{z}=\langle{\cal
L}_{\text{fast}}f\rangle_{z}=\langle\sum_{j=1,2}x_{j}(x_{j-1}-x_{j+1})\partial_{x_{j}}f\rangle_{z}$

In particular, for $f(\mathrm{x})=x_{i}x_{i+1}$ for some $i\in\mathbb{Z}_{3}$,
we get $\langle x_{i-1}x_{i+1}^{2}-x_{i+1}x_{i-1}^{2}\rangle_{z}=0$, which
immediately yields the desired equality. In order to prove that the quantities
do not depend on $i$, apply the equality above with $f(\mathrm{x})=x_{i+1}$,
which combined with the previous one yields $\langle
x_{i}x_{i+1}^{2}\rangle_{z}=\langle x_{i+1}x_{i-1}^{2}\rangle_{z}$. $\Box$

The expression of ${\cal L}_{\text{avg}}$ then immediately follows from the
relation

$\langle x_{i}x_{i+1}^{2}\rangle_{z}-z=\langle
x_{i}x_{i+1}(x_{i+1}-x_{i-1})\rangle_{z}=-\langle
x_{i+1}\dot{x}_{i}\rangle_{z}$

Lemma 2.5 is proved. $\Box$

###  2.5 The stochastic differential equation and its solutions

Consider the (one-dimensional) stochastic differential equation associated
with ${\cal L}_{\text{avg}}$, namely

$dZ(t)=b(Z(t))dt+\sigma(Z(t))dW(t),\quad Z(0)=z_{0}\in I$ (2)

where $b(z)=3(am(z)-z)$, $\sigma(z)=\sqrt{6zm(z)}$ and $W(t)$ is some Brownian
motion. Lemma 2.4 and the fact that $m(z)=-\frac{A(z)}{T(z)}$ for all $z\in I$
imply that both functions $b$ and $\sigma$ are smooth on $I$ and
$\sigma^{2}>0$. These conditions ensure the existence and uniqueness of a
solution $Z_{z_{0}}(t)$ for $t$ up to the so-called explosion time
$T_{\text{ex}}(z_{0})$, namely the time it takes for the solution to reach the
boundary of $I$, see for instance [21, 26]. Of course, we have
$T_{\text{ex}}>0$ a.s.

The explosion time depends on the nature of the boundary points, which can be
evaluated using Feller’s test [21, 26]. This nature depends on the parameter
$a$ and is given in the following statement, which uses the Feller’s
classification in chapter 8.1 of [15].

######  Lemma 2.7.

In the SDE (2), the boundary point $z=\frac{1}{27}$ is entrance for all
$a\in\mathbb{R}^{+}$. Moreover the boundary point $z=0$ is entrance for $a\geq
1$, regular for $a\in(0,1)$ and exit for $a=0$.

Proof: We follow the arguments in chapter 8.1 of [15]. Using
$A^{\prime}(z)=T(z)$ in Lemma 2.4, the averaged generator can be recast as

$\mathcal{L}_{\rm avg}=\tfrac{d}{ds(z)}\left(\tfrac{d}{dp(z)}\right),$ (3)

where the scale function $p$ and the speed function $s$ are the positive
functions whose differential are respectively given by the following
equalities

$dp(z)=-\frac{1}{z^{a}A(z)}dz\quad\text{and}\quad
ds(z)=\frac{z^{a-1}T(z)}{3}dz,\quad z\in I.$

The approximations in Lemmas 2.3 and 2.4 imply the following ones

$\tfrac{dp(z)}{dz}\sim
2z^{-a}\quad\text{and}\quad\tfrac{ds(z)}{dz}\sim-z^{a-1}\ln z\quad\text{as}\
z\to 0^{+},$

and

$\tfrac{dp(z)}{dz}\sim\tfrac{27^{a}}{2\pi\sqrt{3}}\left(\tfrac{1}{27}-z\right)^{-1}\quad\text{and}\quad\tfrac{ds(z)}{dz}\sim\tfrac{18\pi\sqrt{3}}{27^{a}}\quad\text{as}\
z\to\tfrac{1}{27}^{-}.$

Explicit computations then yield the following estimates for every $r\in I$
(recall that $a\in\mathbb{R}^{+}$)

$\left|\int_{r}^{0}s(z)dp(z)\right|<+\infty\ \text{iff}\
a\in[0,1)\quad\text{and}\quad\left|\int_{r}^{0}p(z)ds(z)\right|<+\infty\
\text{iff}\ a>0,$

and

$\int_{r}^{\frac{1}{27}}s(z)dp(z)=+\infty\quad\text{and}\quad\int_{r}^{\frac{1}{27}}p(z)ds(z)<+\infty,$

from where the characterization of the boundary points $z=0$ and
$z=\frac{1}{27}$ immediately follow. $\Box$

As a consequence of the Lemma, for $a\geq 1$, we have $T_{\text{\rm
ex}}=+\infty$ a.s., and hence existence and uniqueness of solutions of the SDE
(2) for all $t>0$, a.s. For $a\in[0,1)$, the trajectory hits the boundary
point $z=0$ a.s. Yet, for $a=0$, existence and uniqueness of solutions for all
$t>0$ a.s. follows by letting $Z(t)=0$ for $t>T_{\rm ex}$.

For $a\in(0,1)$, the existence of solutions of the SDE (2) extends for $t$
beyond $T_{\text{ex}}$ but uniqueness is not granted in general and requires
to specify the behavior at the boundary point $z=0$ [20]. These features, as
well as those for $a>1$ can be expressed through the semi-group associated
with ${\cal L}_{\text{avg}}$. Following the definitions in [15], given $a>0$
consider the domain $\mathcal{D}_{\text{avg}}$ defined by

$\mathcal{D}_{\text{avg}}=\left\\{\begin{array}[]{ccl}\left\\{f_{I}\in
C^{2}(I)\cap C^{0}(\bar{I})~{}:~{}\mathcal{L}_{\rm avg}f_{I}\in
C^{0}(\bar{I})\right\\}&\text{if}&a\geq 1,\\\ \left\\{f_{I}\in C^{2}(I)\cap
C^{0}(\bar{I})~{}:~{}\mathcal{L}_{\rm avg}f_{I}\in
C^{0}(\bar{I}),~{}\lim_{z\to
0^{+}}z^{a}A(z)f_{I}^{\prime}(z)=0\right\\}&\text{if}&a\in(0,1).\end{array}\right.$

In particular, the choice of $\mathcal{D}_{\text{avg}}$ for $a\in(0,1)$
corresponds to an instantaneous reflexion at $z=0$. Theorem 1.1, Chap. 8 in
[15] states that, with these definitions of $\mathcal{D}_{\text{avg}}$, the
operator ${\cal L}_{\text{avg}}$ generates a Feller semi-group on
$C^{0}(\overline{I})$, for every $a>0$ (NB: In particular, in the case
$a\in(0,1)$ for which $z=0$ is regular, the set $\mathcal{D}_{0}$ in [15] is
defined with $q_{0}=0$.)

Our next result states that the measure on $\overline{I}$ with density
$z\mapsto z^{a-1}T(z)$ \- which, thanks to the equality $A^{\prime}(z)=T(z)$,
results to be the push-forward under $\mathrm{x}\mapsto z(\mathrm{x})$ of the
limit measure $\mu_{a}$ in Proposition 2.1 \- is a stationary measure of the
Feller semi-group (NB: Recall that for $a=0$, the point $z=0$ is exit).

######  Proposition 2.8.

For every $a>0$, we have

$\int_{0}^{\frac{1}{27}}z^{a-1}T(z)\mathcal{L}_{\rm avg}f_{I}(z)dz=0,\ \forall
f_{I}\in\mathcal{D}_{\rm avg}.$

Proof: For $z_{0}<z_{1}\in I$, we obtain after direct integration

$\int_{z_{0}}^{z_{1}}z^{a-1}T(z)\mathcal{L}_{\rm
avg}f_{I}(z)dz=\left[\tfrac{df_{I}(z)}{dp(z)}\right]_{z_{0}}^{z_{1}}.$

Therefore, all we have to show is $\lim_{z\to
0^{+}}\tfrac{df_{I}(z)}{dp(z)}=\lim_{z\to\frac{1}{27}^{-}}\tfrac{df_{I}(z)}{dp(z)}=0$.
We focus on the first limit; the second one follows from similar argument.
When $z=0$ is a regular boundary ($a\in(0,1)$), this is a consequence of the
choice of $\mathcal{D}_{\text{avg}}$. When $z=0$ is entrance ($a\geq 1$), the
Taylor formula implies that we have

$\tfrac{df_{I}(z)}{dp(z)}=\frac{1}{p(z)-p(r)}\left(f_{I}(z)-f_{I}(r)+\int_{r}^{z}\,\mathcal{L}_{\rm
avg}f_{I}(z)\,p(z)\,\mathrm{d}s(z)\right),\ r\in(z,\tfrac{1}{27}).$

The term inside the brackets converges when $z\to 0^{+}$ because we have
$f_{I},\mathcal{L}_{\rm avg}f_{I}\in C^{0}(\bar{I})$ and
$|\int_{r}^{0}p(z)ds(z)|<+\infty$. The result then follows from the limit
$\lim_{z\to 0^{+}}|p(z)|=+\infty$. $\Box$

##  3 Main result: Averaging

Given $T>0$, let $\mathbb{D}([0,T],\overline{I})$ be the set of càdlàg
functions from $[0,T]$ into $\overline{I}=[0,\frac{1}{27}]$ and let ${\cal
G}_{T}$ be the natural filtration associated with
$\mathbb{D}([0,T],\overline{I})$. The process $X_{N}(t)$ induces a stochastic
process $Z_{N}(t):=z(X_{N}(t))$ on $(\mathbb{D}([0,T],\overline{I}),{\cal
G}_{T})$. We are now in position to formulate the main result of the paper.

######  Theorem 3.1.

Assume that the sequence of initial conditions
$\\{Z_{N}(0)\\}_{N\in\mathbb{N}}$ converges in law to some $z_{0}\in I$.
If $a\geq 1$ or $a=0$, then for any $T>0$, the sequence of processes
$\\{Z_{N}(t):t\in[0,T]\\}_{N\in\mathbb{N}}$ converges in law to the weak
solution of the SDE (2) with initial condition $z_{0}$.
If $0<a<1$, then for any $T>0$, the sequence
$\\{Z_{N}(t):t\in[0,T]\\}_{N\in\mathbb{N}}$ is relatively compact.
Furthermore, any limit point is a weak solution of the SDE (2), with initial
condition $z_{0}$.

The proof of this statement follows the Stroock-Varadhan approach to
martingale problems [35]. The first step is a standard compactness argument
(see the presentation in [23], especially Corollary 2.3.3 therein) for the
semi-martingale structure associated with a stochastic process, the process
$Z_{N}$ in our case. Then we prove that every limit point of a subsequence
must satisfy the martingale problem associated with the SDE. That part of the
proof is specific to the particle system under consideration as it relies in
particular on various features of the short times deterministic dynamics. The
convergence follows suit when the SDE has a unique solution, ie. for $a\geq 1$
and $a=0$. In the other case ($a\in(0,1)$), while the SDE solutions are not
unique beyond $T_{\text{ex}}$, Proposition 2.1 and Proposition 2.8 suggest
that $\\{Z_{N}\\}$ should converge to the solution of the SDE with
instantaneous reflection at the boundary $z=0$, provided it is defined and
unique. This property remains to be proved.

The rest of this section is devoted to the proof of Theorem 3.1 which, for the
sake of clarity, is decomposed into three subsections.

###  3.1 Proof of compactness

Recall that $\\{P_{N}^{\mathrm{x}}\\}_{\mathrm{x}\in S_{N}}$ denotes the
Markov process generated by $L_{N}$. For every $\mathrm{x}\in S_{N}$, the
probability measure $P_{N}^{\mathrm{x}}$ solves the martingale problem
associated with $L_{N}$ and the initial condition $\mathrm{x}$. In particular
the stochastic process $M_{N}$ on $(\mathbb{D}([0,T],S_{N}),{\cal F}_{T,N})$
with values in $\mathbb{R}$, defined by

$t\mapsto M_{N}(t)=Z_{N}(t)-A_{N}(t)\,,\quad\text{where}\quad
A_{N}(t):=\int_{0}^{t}(L_{N}z)(X_{N}(s))ds\,,$

is a martingale relative to $P_{N}^{X_{N}(0)}$. Moreover, that $L_{N}z$ is
bounded in $S_{N}$ implies that $A_{N}$ is a process of finite variation.
Consequently $M_{N}$ is bounded and then is locally square integrable.

According to Corollary 2.3.3 in [23], in order to prove that the laws
associated with $\\{Z_{N}\\}$ form a tight family, it suffices to show that

  * •

the sequences $\\{A_{N}\\}$ and $\\{\langle M_{N}\rangle\\}$ satisfy the
Aldous condition and

  * •

the sequence of the laws of $\sup_{t\in[0,T]}|A_{N}(t)|$ (resp.
$\sup_{t\in[0,T]}|\langle M_{N}(t)\rangle|$) is tight in $\mathbb{R}$.

Below we focus on proving the Aldous condition. The proof of the other
condition is similar and left to the reader.

In order to prove the Aldous condition for $A_{N}$, we observe that the Markov
inequality implies that for every $0\leq t<t^{\prime}$, we have

$P_{N}^{\mathrm{x}}(|A_{N}(t^{\prime})-A_{N}(t)|\geq\eta)\leq\frac{1}{\eta}\mathbb{E}_{\mathrm{x}}\left(\int_{t}^{t^{\prime}}|L_{N}z(X_{N}(s))|ds\right),$

Explicit calculations using that ${\cal L}_{\rm fast}z=0$ imply that the
integrand $|L_{N}z(X_{N}(s))|$ is uniformly bounded in $N$, and so the
probability $P_{N}^{\mathrm{x}}(|A_{N}(t^{\prime})-A_{N}(t)|\geq\eta)$ can be
made arbitrarily small, uniformly in $N$, by taking $t^{\prime}-t$
sufficiently small.

Moreover, the increasing process $\langle M_{N}\rangle$ is given by

$\langle M_{N}(t)\rangle=\int_{0}^{t}q_{N}z(X_{N}(s))ds$

where $q_{N}$ is the quadratic operator defined for any function
$f:S_{N}\to\mathbb{R}$ by

$q_{N}f(\mathrm{x})=\sum_{i\in\mathbb{Z}_{3}}x_{i}(\frac{a}{N}+x_{i+1})N^{2}\left(f(\mathrm{x}+\frac{\mathrm{u}_{i}}{N})-f(\mathrm{x})\right)^{2}.$
(4)

A similar argument as above applies to prove the Aldous condition for $\langle
M_{N}\rangle$, using the mean value theorem for $z$.

###  3.2 Extending the time scale of the convergence to the deterministic
approximation

Let $\|\cdot\|_{1}$ denote the $\ell^{1}$-norm in $S$.

######  Lemma 3.2.

Let $\\{T_{N}\\}_{N\in\mathbb{N}}$ be a sequence in $\mathbb{R}^{+}$ such that
$T_{N}\leq C\frac{\log\log N}{N}$ for all $N\in\mathbb{N}$, for some $C>0$.
Then we have

$\lim_{N\to\infty}\sup_{t\in[0,T_{N}]}\sup_{X_{N}(0)\in
S_{N}}\mathbb{E}_{P_{N}^{X_{N}(0)}}\left\|X_{N}(t)-X^{\rm
fast}_{X_{N}(0)}(Nt)\right\|_{1}=0.$

Proof. Given $f\in C^{2}(S)$, consider the martingale $M_{N}^{f}$ relative to
$P_{N}^{X_{N}(0)}$ and defined by

$t\mapsto M_{N}^{f}(t):=f(X_{N}(t))-\int_{0}^{t}L_{N}f(X_{N}(s))ds$ (5)

Using also the relation $f(X^{\rm
fast}_{X_{N}(0)}(Nt))=f(X_{N}(0))+\int_{0}^{t}N{\cal L}_{\text{fast}}f(X^{\rm
fast}_{X_{N}(0)}(s))ds$, we then get the estimate

$\displaystyle\mathbb{E}_{P_{N}^{X_{N}(0)}}\left|f(X_{N}(t))-f(X^{\rm
fast}_{X_{N}(0)}(Nt))\right|\leq$
$\displaystyle\mathbb{E}_{P_{N}^{X_{N}(0)}}\left|M_{N}^{f}(t)-f(X_{N}(0))\right|$
$\displaystyle+\int_{0}^{t}\mathbb{E}_{P_{N}^{X_{N}(0)}}\left|L_{N}f(X_{N}(s))-N{\cal
L}_{\text{fast}}f(X_{N}(s))\right|ds$
$\displaystyle+N\int_{0}^{t}\mathbb{E}_{P_{N}^{X_{N}(0)}}\left|{\cal
L}_{\text{fast}}f(X_{N}(s))-{\cal L}_{\text{fast}}f(X^{\rm
fast}_{X_{N}(0)}(s))\right|ds.$

We estimate each term in the RHS separately. By the Hölder inequality, we have

$\mathbb{E}_{P_{N}^{X_{N}(0)}}\left|M_{N}^{f}(t)-f(X_{N}(0))\right|\leq\left(\mathbb{E}_{P_{N}^{X_{N}(0)}}\left|M_{N}^{f}(t)-f(X_{N}(0))\right|^{2}\right)^{\tfrac{1}{2}}=\left(\mathbb{E}_{P_{N}^{X_{N}(0)}}\langle
M_{N}^{f}(t)\rangle\right)^{\tfrac{1}{2}},$

where $\langle M_{N}^{f}(t)\rangle=\int_{0}^{t}q_{N}f(X_{N}(s))ds$ and the
quadratic operator $q_{N}$ is defined in (4). This definition implies that for
every $f\in C^{1}(S)$, there exists $K_{f}>0$ such that

$\sup_{\mathrm{x}\in S,N\in\mathbb{N}}|q_{N}f(\mathrm{x})|\leq K_{f}$

from where we get the upper bound
$\mathbb{E}_{P_{N}^{X_{N}(0)}}\left|M_{N}^{f}(t)-f(X_{N}(0))\right|\leq
K_{f}\sqrt{t}$.

Similarly, given $f\in C^{2}(S)$, the Taylor theorem applied to the first
order expansion of $f(\mathrm{x}+\frac{\mathrm{u}_{i}}{N})$ at $\mathrm{x}$
implies the inequality

$\int_{0}^{t}\mathbb{E}_{P_{N}^{X_{N}(0)}}\left|L_{N}f(X_{N}(s))-N{\cal
L}_{\text{fast}}f(X_{N}(Ns))\right|ds\leq K_{f}t,$

provided that $K_{f}$ is chosen sufficiently large. By choosing $K_{f}$ even
larger if necessary so that the following inequality holds

$|{\cal L}_{\text{fast}}f(\mathrm{x})-{\cal L}_{\text{fast}}f(\mathrm{y})|\leq
K_{f}\|\mathrm{x}-\mathrm{y}\|_{1},\quad\forall\mathrm{x},\mathrm{y}\in S,$

and by collecting all the estimates, we finally obtain

$\mathbb{E}_{P_{N}^{X_{N}(0)}}\left|f(X_{N}(t))-f(X^{\rm
fast}_{X_{N}(0)}(Nt))\right|\leq
K_{f}\left(\sqrt{t}+t+N\int_{0}^{t}E_{P_{N}^{X_{N}(0)}}\left\|X_{N}(s)-X^{\rm
fast}_{X_{N}(0)}(Ns)\right\|_{1}ds\right).$

Applying this inequality to the functions $f(\mathrm{x})=x_{i}$
($i\in\mathbb{Z}_{3}$), and using the Gronwall inequality, we finally get

$\sup_{X_{N}(0)\in S_{N}}\mathbb{E}_{P_{N}^{X_{N}(0)}}\left\|X_{N}(t)-X^{\rm
fast}_{X_{N}(0)}(Nt)\right\|_{1}\leq K\left(\sqrt{t}+t\right)e^{NKt},$

for some $K>0$. The Lemma then immediately follows from the fact that
$T_{N}\leq C\frac{\log\log N}{N}$ implies that
$\lim_{N\to\infty}(\sqrt{T_{N}}+T_{N})e^{NKT_{N}}=0$. $\Box$

###  3.3 Identification of the limits, end of the proof of Theorem 3.1

The tightness of the sequence $\\{Z_{N}\\}$ (section 3.1) implies that, up to
passing to a subsequence, this sequence converges in law to a process $Z$ with
values in $\overline{I}$. To complete the proof of Theorem 3.1, it remains to
prove that $Z$ must be a solution of the SDE (2). The core argument is to
establish that $Z$ must solve the martingale problem associated with ${\cal
L}_{\text{avg}}$ for a sufficiently large set of functions. To that goal, we
first invoke a slight extension of the Skorokhod’s representation theorem, see
Theorem C.1 in Appendix C, according to which there exist a common probability
space $(\mathrm{E},\Omega,\mathrm{P})$ in which the random variables
$z(X_{N})$ pointwise converge to $Z$. Then, we consider the martingales
$M_{N}^{f_{I}\circ z}$ defined by

$t\mapsto M_{N}^{f_{I}\circ z}(t)=f_{I}(Z_{N}(t))-\int_{0}^{t}L_{N}(f_{I}\circ
z)(X_{N}(s))ds.$

The key argument of the proof of Theorem 3.1 is the following
$L^{1}$-convergence of the martingales.

######  Proposition 3.3.

For every $f_{I}\in C^{3}(\overline{I})$ and $T>0$, we have

$\lim_{N\to\infty}\mathbb{E}_{\mathrm{P}}\left|M_{N}^{f_{I}\circ
z}(t)-f_{I}(Z(t))-\int_{0}^{t}{\cal L}_{\text{\rm
avg}}f_{I}(Z(s))ds\right|=0,\quad\forall t\in[0,T].$

The end of the proof of the Theorem 3.1 uses again standard arguments. The
Proposition implies in particular that the process $t\mapsto
f_{I}(Z(t))+\int_{0}^{t}{\cal L}_{\text{\rm avg}}f_{I}(Z(s))ds$ equipped with
the probability $\mathrm{P}$ is a martingale for every $f_{I}\in
C^{3}(\overline{I})$, see e.g. Lemma 3.6, Chap. 2 in [12], and hence for
$f_{I}(Z)=Z$ and $f_{I}(Z)=Z^{2}$. Moreover, the process is continuous (proved
in the proof of the proposition below). Hence by (an adaptation to
$\overline{I}$ of the) Proposition 4.6, Chap. 5 in [26], one defines a
Brownian motion $W$ such that the process is a solution of the SDE (2).

Proof of the Proposition. We are going to expand the difference
$M_{N}^{f_{I}\circ z}(t)-f_{I}(Z(t))+\int_{0}^{t}{\cal
L}_{\text{avg}}f_{I}(Z(s))ds$ into a telescopic sum for which the elements can
be controlled using characteristic features of the various processes involved
in the approximation.

To that goal, let $k_{N}=\lfloor\tfrac{tN}{\log\log N}\rfloor$ and
$T_{N}=\tfrac{t}{k_{N}}$. Using the definition of $M_{N}^{f_{I}\circ z}$
above, we write

$\mathbb{E}_{\mathrm{P}}\left|M_{N}^{f_{I}\circ
z}(t)-f_{I}(Z(t))+\int_{0}^{t}{\cal
L}_{\text{avg}}f_{I}(Z(s))ds\right|\leq\mathbb{E}_{\mathrm{P}}\left|f_{I}(Z_{N}(t))-f_{I}(Z(t))\right|+\sum_{\ell=1}^{5}\mathbb{E}_{\mathrm{P}}\left|Q_{N,\ell}^{f_{I}}(t)-Q_{N,\ell+1}^{f_{I}}(t)\right|,$
(6)

where the boundary terms

$Q_{N,1}^{f_{I}}(t)=\displaystyle\int_{0}^{t}L_{N}(f_{I}\circ
z)(X_{N}(s))ds\quad\text{and}\quad Q_{N,6}^{f_{I}}(t)=\int_{0}^{t}{\cal
L}_{\text{avg}}f_{I}(Z(s))ds$

in the telescopic sum are already known (NB: While $Q_{N,6}^{f_{I}}(t)$
actually does not depend on $N$, using this notation simplifies the expression
(6)) and the interior terms $Q_{N,\ell}^{f_{I}}(t)$ are defined by

$Q_{N,\ell}^{f_{I}}(t)=\begin{cases}{\displaystyle\int_{0}^{t}{\cal
L}_{\text{slow}}(f_{I}\circ z)(X_{N}(s))ds}\quad\text{if}\quad\ell=2\\\
{\displaystyle\sum_{k=0}^{k_{N}-1}\int_{kT_{N}}^{(k+1)T_{N}}{\cal
L}_{\text{slow}}(f_{I}\circ z)(X^{\rm
fast}_{X_{N}(kT_{N})}(Ns))ds}\quad\text{if}\quad\ell=3\\\ {\displaystyle
T_{N}\sum_{k=0}^{k_{N}-1}{\cal
L}_{\text{avg}}f_{I}(Z_{N}(kT_{N}))}\quad\text{if}\quad\ell=4\\\
{\displaystyle T_{N}\sum_{k=0}^{k_{N}-1}{\cal
L}_{\text{avg}}f_{I}(Z(kT_{N}))}\quad\text{if}\quad\ell=5\\\ \end{cases}$

We now prove that each term in the RHS of (6) vanishes in the limit of large
$N$.

Proof of convergence of
$\mathbb{E}_{\mathrm{P}}\left|f_{I}(Z_{N}(t))-f_{I}(Z(t))\right|$ and
$\mathbb{E}_{\mathrm{P}}\left|Q_{N,4}^{f_{I}}(t)-Q_{N,5}^{f_{I}}(t)\right|$.
We first write

$\mathbb{E}_{\mathrm{P}}\left|f_{I}(Z_{N}(t))-f_{I}(Z(t))\right|\leq\mathbb{E}_{\mathrm{P}}\left\|f_{I}(Z_{N})-f_{I}(Z)\right\|_{\infty},\
t\in[0,T],$

where $\|\cdot\|_{\infty}$ denotes the uniform norm on $[0,T]$. Similarly, we
have

$\mathbb{E}_{\mathrm{P}}\left|Q_{N,4}^{f_{I}}(t)-Q_{N,5}^{f_{I}}(t)\right|\leq
t\mathbb{E}_{\mathrm{P}}\left\|{\cal L}_{\text{avg}}f_{I}(Z_{N})-{\cal
L}_{\text{avg}}f_{I}(Z)\right\|_{\infty},\ t\in[0,T].$

The amplitude of the jumps of $Z_{N}$ is at most $\tfrac{1}{N}$ and the
mapping $(Z(t))_{t\in\mathbb{R}^{+}}\mapsto\sup_{t\in[0,T]}|Z(t)-Z(t^{-})|$ is
continuous in the Skorokhod topology of $\mathbb{D}([0,T],\overline{I})$;
hence the limit $Z$ must be continuous a.s. and the convergence $Z_{N}\to Z$
a.s. must occur in the sense of $\|\cdot\|_{\infty}$. The desired convergences
then follow from the dominated convergence theorem using that $f_{I}$ and
${\cal L}_{\text{avg}}f_{I}$ are uniformly continuous over $\overline{I}$.

Proof of convergence of
$\mathbb{E}_{\mathrm{P}}\left|Q_{N,1}^{f_{I}}(t)-Q_{N,2}^{f_{I}}(t)\right|$.
Given $f\in C^{3}(S)$, the Taylor theorem applied to the second order
expansion of $f(\mathrm{x}+\frac{\mathrm{u}_{i}}{N})$ at $\mathrm{x}$ implies
the existence of $K_{f}>0$ such that we have

$\left|L_{N}f(\mathrm{x})-N{\cal L}_{\text{fast}}f(\mathrm{x})-{\cal
L}_{\text{slow}}f(\mathrm{x})\right|\leq\frac{K_{f}}{N}.$

Applying this inequality to $f_{I}\circ z$ with $f_{I}\in
C^{3}(\overline{I})$, and using the property ${\cal
L}_{\text{fast}}(f_{I}\circ z)=0$ (which follows from the fact that the
function $\mathrm{z}$ is invariant under the flow generated by $F$), we
immediately obtain the desired convergence

$\lim_{N\to\infty}\mathbb{E}_{\mathrm{P}}\left|Q_{N,1}^{f_{I}}(t)-Q_{N,2}^{f_{I}}(t)\right|=0,\
\forall t\in[0,T].$

Proof of convergence of
$\mathbb{E}_{\mathrm{P}}\left|Q_{N,2}^{f_{I}}(t)-Q_{N,3}^{f_{I}}(t)\right|$.
Given $f_{I}\in C^{2}(\overline{I})$, let $K_{s}$ be the Lipschitz constant of
${\cal L}_{\text{slow}}(f_{I}\circ z)$ with respect to the
$\|\cdot\|_{1}$-norm. We have

$\displaystyle\mathbb{E}_{\mathrm{P}}\left|Q_{N,2}^{f_{I}}(t)-Q_{N,3}^{f_{I}}(t)\right|$
$\displaystyle\leq
K_{s}\sum_{k=0}^{k_{N}-1}\int_{kT_{N}}^{(k+1)T_{N}}\mathbb{E}_{\mathrm{P}}\|X_{N}(s)-X^{\rm
fast}_{X_{N}(kT_{N})}(Ns)\|_{1}ds$ $\displaystyle\leq
K_{s}t\sup_{t\in[0,T_{N}]}\sup_{X_{N}(0)\in
S_{N}}\mathbb{E}_{P_{N}^{X_{N}(0)}}\left\|X_{N}(t)-X^{\rm
fast}_{X_{N}(0)}(Nt)\right\|_{1},$

and then Lemma 3.2 immediately imply

$\lim_{N\to\infty}\mathbb{E}_{\mathrm{P}}\left|Q_{N,2}^{f_{I}}(t)-Q_{N,3}^{f_{I}}(t)\right|=0,\
\forall t\in[0,T].$

Proof of convergence of
$\mathbb{E}_{\mathrm{P}}\left|Q_{N,3}^{f_{I}}(t)-Q_{N,4}^{f_{I}}(t)\right|$.
The proof of convergence for this term follows from considerations about
localisation in $S$ and related dynamical estimates. Writing

$\mathbb{E}_{\mathrm{P}}\left|Q_{N,3}^{f_{I}}(t)-Q_{N,4}^{f_{I}}(t)\right|\leq
T_{N}\sum_{k=0}^{k_{N}-1}\mathbb{E}_{\mathrm{P}}\left|\frac{1}{T_{N}}\int_{kT_{N}}^{(k+1)T_{N}}{\cal
L}_{\text{slow}}(f_{I}\circ z)(X^{\rm fast}_{X_{N}(kT_{N})}(Ns))ds-{\cal
L}_{\text{avg}}f_{I}(Z_{N}(kT_{N}))\right|$ (7)

and given $r\in I$ (to be specified later on), for each term in the sum of the
RHS, we consider separately the cases $Z_{N}(kT_{N})\in[0,r)$ and
$Z_{N}(kT_{N})\in[r,\tfrac{1}{27}]$.

In the second case, we use that given $t>0$ and $\mathrm{x}\in S$ such that
$z(\mathrm{x})\geq r$,
$\mathrm{x}\neq(\tfrac{1}{3},\tfrac{1}{3},\tfrac{1}{3})$, the definition of
the averaged generator ${\cal L}_{\text{avg}}$ and the periodicity $X^{\rm
fast}_{\mathrm{x}}(s+T(z(\mathrm{x})))=X^{\rm fast}_{\mathrm{x}}(s)$ imply

$\displaystyle\left|\tfrac{1}{NT_{N}}\int_{0}^{NT_{N}}{\cal
L}_{\text{slow}}(f_{I}\circ z)(X^{\rm fast}_{\mathrm{x}}(s))ds-{\cal
L}_{\text{avg}}f_{I}(Z(\mathrm{x}))\right|$
$\displaystyle\leq\tfrac{1}{NT_{N}}\left|\int_{0}^{NT_{N}}{\cal
L}_{\text{slow}}(f_{I}\circ z)(X^{\rm
fast}_{\mathrm{x}}(s))ds-\lfloor\frac{NT_{N}}{T(z(\mathrm{x}))}\rfloor\int_{0}^{T(z(\mathrm{x}))}{\cal
L}_{\text{slow}}(f_{I}\circ z)(X^{\rm fast}_{\mathrm{x}}(s))ds\right|$
$\displaystyle+\left|\left(\frac{T(z(\mathrm{x}))}{NT_{N}}\lfloor\frac{NT_{N}}{T(z(\mathrm{x}))}\rfloor-1\right){\cal
L}_{\text{avg}}f_{I}(Z(\mathrm{x}))\right|$
$\displaystyle\leq\tfrac{1}{NT_{N}}\left|\int_{T(z(\mathrm{x}))\lfloor\frac{NT_{N}}{T(z(\mathrm{x}))}\rfloor}^{NT_{N}}{\cal
L}_{\text{slow}}(f_{I}\circ z)(X^{\rm
fast}_{\mathrm{x}}(s))ds\right|+\frac{T(z(\mathrm{x}))}{NT_{N}}\left|{\cal
L}_{\text{avg}}f_{I}(Z(\mathrm{x}))\right|.$

Accordingly, and using also that the period $T(z(\mathrm{x}))$ is bounded over
those points $\mathrm{x}\in S$ for which $z(\mathrm{x})\geq r$ (see Lemma 2.3
(i)), we conclude that the RHS vanishes in the limit $N\to\infty$, since
$\lim_{N\to\infty}NT_{N}=+\infty$. For
$\mathrm{x}=(\tfrac{1}{3},\tfrac{1}{3},\tfrac{1}{3})$, the result immediately
follows from the equality (1).

The first case $Z_{N}(kT_{N})\in[0,r)$ corresponds to the neighborhood of the
simplex boundaries, where the control of averaging is more elusive. Recall
from the comments after Lemma 2.5 that ${\cal L}_{\text{avg}}f_{I}(0^{+})=0$.
Hence by taking $r$ sufficiently small, for each (putative) term ${\cal
L}_{\text{avg}}f_{I}(Z_{N}(kT_{N}))$ in (7), we can make its contribution
arbitrarily small (and the same comment applies to the maximal total
contribution $T_{N}\sum_{k=0}^{k_{N}-1}\left|{\cal
L}_{\text{avg}}f_{I}(Z_{N}(kT_{N}))\right|$).

In order to address the remaining integral term in (7), we observe that when
$Z_{N}(t)\in[0,r)$, we have $z(X^{\rm fast}_{X_{N}(t)}(Ns))\in[0,r)$ for all
$s\in\mathbb{R}^{+}$. Therefore, at any $s$, the point $X^{\rm
fast}_{X_{N}(t)}(Ns)$ may be close to one of the vertices $\mathrm{v}_{i}$ of
$S$. An explicit calculation shows that

$\lim_{\mathrm{x}\to\mathrm{v}_{i}}{\cal L}_{\text{slow}}(f_{I}\circ
z)(\mathrm{x})=0,\ i\in\mathbb{Z}_{3}.$

As before, to choose some neighbourhoods $V_{i,r}$ of the $\mathrm{v}_{i}$
with sufficiently small radius $r$ implies that the contribution of the
integral terms in (7), for those $X^{\rm
fast}_{X_{N}(kT_{N})}(Ns)\in\bigcup_{i\in\mathbb{Z}_{3}}V_{i,r}$, can be made
arbitrarily small, uniformly in $N$.

W.l.o.g. we may assume that $\bigcup_{i\in\mathbb{Z}_{3}}V_{i,r}$ is invariant
under the cyclic permutation of coordinates $(x_{i})\mapsto(x_{i+1})$. Then,
if a trajectory $X^{\rm fast}_{X_{N}(t)}(Ns)$ leaves a set $V_{i,r}$, then it
must travel to $V_{i+1,r}$. In the intermediate region between $V_{i,r}$ and
$V_{i+1,r}$, the norm $\|F(\mathrm{x})\|$ of the vector field is bounded below
(Indeed, one easily checks that this is the case when restricted to the
segment $\mathrm{v}_{i}\mathrm{v}_{i+1}$, part of an edge in the boundary
$\partial S$. Then apply a continuity argument). Therefore, the transit time
of $X^{\rm fast}_{X_{N}(t)}(Ns)$ must be bounded from below by, say
$\tfrac{t_{r}}{N}$ for some $t_{r}>0$. It follows that, in the interval
$[kT_{N},(k+1)T_{N}]$ the total time the trajectory spends outside
$\bigcup_{i\in\mathbb{Z}_{3}}V_{i,r}$, cannot exceed $3T_{N}\tfrac{t_{r}}{N}$.
The corresponding total contribution of the integral terms in (7), for those
$X^{\rm fast}_{X_{N}(kT_{N})}(Ns)$ in the transit regions between the
$V_{i,r}$, then cannot exceed $3\tfrac{t_{r}k_{N}}{N}$, which vanishes when
$NT_{N}\to\infty$. This completes the proof that

$\lim_{N\to\infty}\mathbb{E}_{\mathrm{P}}\left|Q_{N,3}^{f_{I}}(t)-Q_{N,4}^{f_{I}}(t)\right|=0,\
\forall t\in[0,T].$

Proof of convergence of
$\mathbb{E}_{\mathrm{P}}\left|Q_{N,5}^{f_{I}}(t)-Q_{N,6}^{f_{I}}(t)\right|$.
The quantity ${\displaystyle T_{N}\sum_{k=0}^{k_{N}-1}{\cal
L}_{\text{avg}}f_{I}(Z(kT_{N}))}$ can be regarded as a Riemann sum for the
integral $\int_{0}^{t}{\cal L}_{\text{avg}}f_{I}(Z(s))ds$; hence the desired
convergence follows since ${\cal L}_{\text{avg}}f_{I}\circ Z$ in a.s
continuous over $[0,T]$.

The proof of the Proposition is complete. $\Box$

Acknowledgements

This work has been supported by the ANR-19-CE40-0023 (PERISTOCH). We are
grateful to Nils Berglund, Nicolas Fournier and Luc Hillairet for fruitful
discussions and relevant comments.

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##  Appendix A Proof of Proposition 2.1

One ingredient is the following convergence of the uniform atomic measure
$\nu_{N}=\tfrac{1}{|S_{N}|}\sum_{\mathrm{x}\in S_{N}}\delta_{\mathrm{x}}$ on
$S$ to the Lebesgue measure.

######  Claim A.1.

We have $\lim_{N\to\infty}\nu_{N}=\text{Leb}_{S}$.

Sketch of proof of the Claim: This property is a consequence of the fact that
every continuous function on $S$ is Riemann integrable on this set. Consider
the restriction to $S$ of the Voronoï triangulation of the points in $S_{N}$.
Since the distribution of these points is uniform and isotropic, the cells
associated with points in the interior of $S$ are all copies of the same
polyhedron $P_{N}$, whose diameter vanishes as $N\to\infty$. Therefore, given
$f\in C^{0}(S)$, the sum

$\text{Vol}(P_{N})\sum_{\mathrm{x}\in\text{Int}(S_{N})}f(\mathrm{x})+\sum_{\mathrm{x}\in
S_{N}\setminus\text{Int}(S_{N})}\text{Vol}(P_{N}(\mathrm{x})\cap
S)f(\mathrm{x})$

(where $P_{N}(\mathrm{x})$ is the cell at $\mathrm{x}\in
S_{N}\setminus\text{Int}(S_{N})$) can be viewed as a Darboux sum for the
integral $\int_{S}f(\mathrm{x})d\mathrm{x}$. The claimed weak convergence then
readily follows from the following estimates

$\frac{1}{|S_{N}|}<\frac{\text{Vol}(P_{N})}{\text{Vol}(S)}<\frac{1}{|\text{Int}(S_{N})|}\quad\text{and}\quad\lim_{N\to\infty}\frac{|S_{N}\setminus\text{Int}(S_{N})|}{|S_{N}|}=0.$

$\Box$

Independently, the Wendel limit (see e.g. [1], page 257, 6.1.46) implies

$\lim_{N\to\infty}\frac{1}{N^{a-1}}\frac{\Gamma(Nx+a)}{\Gamma(Nx+1)}=x^{a-1},\
\forall x\geq 0\ \text{if}\ a\geq 1\ (\forall x>0\ \text{if}\ a<1)$

which suggests to consider the convergence of densities

$\lim_{N\to\infty}\rho_{N,a}(\mathrm{x})=\rho_{a}(\mathrm{x})\quad\text{where}\quad\rho_{N,a}(\mathrm{x})=\frac{1}{N^{3(a-1)}}\prod_{i\in\mathbb{Z}_{3}}\frac{\Gamma(Nx_{i}+a)}{\Gamma(Nx_{i}+1)}.$

An analysis of the sign of the derivative of the function
$y\mapsto\ln\left(\tfrac{\Gamma(yx+a)}{y^{a-1}\Gamma(yx+1)}\right)$ for $x\geq
0$ yields the following conclusion

  * If $a<1$, the sequence $\left\\{\tfrac{1}{N^{a-1}}\frac{\Gamma(Nx+a)}{\Gamma(Nx+1)}\right\\}_{N\in\mathbb{N}}$ is increasing for all $x>0$.

  * If $a\geq 1$, the sequence $\left\\{\tfrac{1}{N^{a-1}}\frac{\Gamma(Nx+a)}{\Gamma(Nx+1)}\right\\}_{N\in\mathbb{N}}$ is non-increasing for all $x\geq 0$.

Dini’s Theorem then implies that the convergence to the Wendel limit is
uniform in every compact set included in $(0,1]$ when $a<1$ (resp. in $[0,1]$
when $a\geq 1$).

In order to prove that $\lim_{N\to\infty}\mu_{N,a}=\mu_{a}$, we separate the
case $a\geq 1$ and $a\in(0,1)$.

If $a\geq 1$, the proof is immediate. Indeed, given $f\in C^{0}(S)$, the
uniform convergence above and Claim A.1 respectively imply

$\lim_{N\to\infty}\int_{S}f(\mathrm{x})\left(\rho_{N,a}(\mathrm{x})-\rho_{a}(\mathrm{x})\right)d\nu_{N}(\mathrm{x})=0\
\text{and}\
\lim_{N\to\infty}\int_{S}f(\mathrm{x})\rho_{a}(\mathrm{x})d\nu_{N}(\mathrm{x})=\int_{S}f(\mathrm{x})\rho_{a}(\mathrm{x})d\mathrm{x}$

which immediately yields

$\lim_{N\to\infty}\int_{S}f(\mathrm{x})\rho_{N,a}(\mathrm{x})d\nu_{N}(\mathrm{x})=\int_{S}f(\mathrm{x})\rho_{a}(\mathrm{x})d\mathrm{x},$

and then the desired convergence, considering that for $f(\mathrm{x})=1$, this
previous relation gives
$\lim_{N\to\infty}\left(C_{N,a}|S_{N}|N^{3(a-1)}\right)^{-1}=C_{a}^{-1}$.

For $a\in(0,1)$, the proof is more involved because $\rho_{a}$ is not defined
on the boundary of $S$ while this set has positive measure for $\mu_{N,a}$
(hence the limitations on the domains where uniform convergence holds). Let
again $f\in C^{0}(S)$ and given $\epsilon>0$ arbitrary, let $\delta>0$ be
sufficiently small so that

$\int_{S\setminus
S(\delta)}\left|f(\mathrm{x})\right|\rho_{a}(\mathrm{x})d\mathrm{x}<\tfrac{\epsilon}{5}\quad\text{where}\quad
S(\delta)=\left\\{\mathrm{x}\in S\ :\ x_{i}\geq\delta,\ \forall i\right\\}.$

Besides, a similar decomposition as for $a\geq 1$ and uniform convergence on
$S(\delta)$ imply that we have

$\left|\int_{S(\delta)}f(\mathrm{x})\rho_{N,a}(\mathrm{x})d\nu_{N}(\mathrm{x})-\int_{S(\delta)}f(\mathrm{x})\rho_{a}(\mathrm{x})d\mathrm{x}\right|<\tfrac{\epsilon}{5}$

provided that $N$ is sufficiently large. It remains to consider the term
$\int_{S\setminus
S(\delta)}\left|f(\mathrm{x})\right|\rho_{N,a}(\mathrm{x})d\nu_{N}(\mathrm{x})$,
which we separate into two integrals, using the decomposition

$S\setminus S(\delta)=(S\setminus\text{Int}(S))\cup(\text{Int}(S)\setminus
S(\delta)).$

On one hand, the inequality $\rho_{N,a}\leq\rho_{a}$ and Claim A.1 imply

$\int_{\text{Int}(S)\setminus
S(\delta)}\left|f(\mathrm{x})\right|\rho_{N,a}(\mathrm{x})d\nu_{N}(\mathrm{x})\leq\int_{\text{Int}(S)\setminus
S(\delta)}\left|f(\mathrm{x})\right|\rho_{a}(\mathrm{x})d\nu_{N}(\mathrm{x})\leq\int_{S\setminus
S(\delta)}\left|f(\mathrm{x})\right|\rho_{a}(\mathrm{x})d\mathrm{x}+\tfrac{\epsilon}{5}\leq\tfrac{2\epsilon}{5}$

provided that $N$ is sufficiently large. On the other hand, control of the
integral

$\int_{S\setminus\text{Int}(S)}\left|f(\mathrm{x})\right|\rho_{N,a}(\mathrm{x})d\nu_{N}(\mathrm{x})$

is provided by the following property.

######  Claim A.2.

We have
$\lim_{N\to\infty}\int_{S\setminus\text{Int}(S)}\left|f(\mathrm{x})\right|\rho_{N,a}(\mathrm{x})d\nu_{N}(\mathrm{x})=0$.

Proof of the Claim: Since $f$ is bounded, it suffices to prove the result for
$f(\mathrm{x})=1$. To that goal, the objects above are considered in arbitrary
dimension (and the explicit dependence on $a$ is removed). Namely, let
$D\in\mathbb{N}$, $\mathbb{Z}_{D}=\mathbb{Z}/D\mathbb{Z}$ and

$S_{D}=\left\\{(x_{i})_{i\in\mathbb{Z}_{D}}\in(\mathbb{R}^{+})^{\mathbb{Z}_{D}}\
\text{such that}\ \sum_{i\in Z_{D}}x_{i}=1\right\\},\quad
S_{N,D}=S_{D}\cap\tfrac{1}{N}\mathbb{N}^{D},$

and

$\rho_{D}(\mathrm{x})=C_{D}\left(\prod_{i\in\mathbb{Z}_{D}}x_{i}\right)^{a-1},\quad\rho_{N,D}(\mathrm{x})=\frac{1}{N^{D(a-1)}}\prod_{i\in\mathbb{Z}_{D}}\frac{\Gamma(Nx_{i}+a)}{\Gamma(Nx_{i}+1)}$

and $\nu_{N,D}=\tfrac{1}{|S_{N,D}|}\sum_{\mathrm{x}\in
S_{N,D}}\delta_{\mathrm{x}}$. Now, consider the integral $I_{N,D}$ defined by
$I_{N,D}=\int_{S_{D}\setminus\text{Int}(S_{D})}\rho_{N,D}(\mathrm{x})d\nu_{N,D}(\mathrm{x})$.
The permutation symmetry implies that we have

$\displaystyle I_{N,D}$ $\displaystyle\leq
D\int_{S_{D}\cap\\{\mathrm{x}:x_{D}=0\\}}\rho_{N,D}(\mathrm{x})d\nu_{N,D}(\mathrm{x})$
$\displaystyle=D\frac{|S_{N,D-1}|}{|S_{N,D}|}\frac{\Gamma(a)}{N^{a-1}}\int_{S_{D-1}}\rho_{N,D-1}(\mathrm{x})d\nu_{N,D-1}(\mathrm{x})$
$\displaystyle=D\frac{D(D-1)N}{N+D-1}\frac{\Gamma(a)}{N^{a}}\left(I_{N,D-1}+\int_{\text{Int}(S_{D-1})}\rho_{N,D-1}(\mathrm{x})d\nu_{N,D-1}(\mathrm{x})\right)$
$\displaystyle\leq\frac{D(D-1)N}{N+D-1}\frac{\Gamma(a)}{N^{a}}\left(I_{N,D-1}+\frac{2}{C_{D-1}}\right),$

provided that $N$ is sufficiently large, where we used
$|S_{N,D}|={N+D-1\choose D-1}$, the inequality $\rho_{N,D-1}\leq\rho_{D-1}$,
Claim A.1 and
$\int_{S_{D-1}}\rho_{D-1}(\mathrm{x})d\mathrm{x}=\tfrac{1}{C_{D-1}}$. The
limit $\lim_{N\to\infty}I_{N,D}=0$ then easily follows by iterating this
inequality over the decreasing values of $D$ and using also that $a>0$. $\Box$

##  Appendix B Analysis of the dynamics $\dot{\mathrm{x}}=F(\mathrm{x})$

###  B.1 Expressions of the solutions

The equation $\dot{\mathrm{x}}=F(\mathrm{x})$ for $\mathrm{x}\in S$ gives the
following system of two coupled ODEs

$\left\\{\begin{array}[]{l}\dot{x_{1}}=x_{1}(1-x_{1}-2x_{2})\\\
\dot{x_{2}}=x_{2}(2x_{1}+x_{2}-1)\end{array}\right.$

This system indicates that $x_{1}$ must increase when $x_{2}<x_{3}$ and must
decrease when $x_{3}<x_{2}$. Similar considerations apply to $x_{2}$. Solving
the equation $x_{1}x(1-x-x_{1})=z$ for $x$ (assuming $x_{1}\neq 0$) yields
that we must have

$x_{2}(x_{1})=\tfrac{1-x_{1}-\sqrt{(1-x_{1})^{2}-\frac{4z}{x_{1}}}}{2}\quad\text{and}\quad
x_{3}(x_{1})=\tfrac{1-x_{1}+\sqrt{(1-x_{1})^{2}-\frac{4z}{x_{1}}}}{2}$

when $x_{1}$ increases and the RHS are exchanged when $x_{1}$ decreases. In
particular, when $x_{1}$ increases, it must satisfy the following ODE

$\dot{x_{1}}=x_{1}\sqrt{(1-x_{1})^{2}-\frac{4z}{x_{1}}}.$

Moreover, by continuity, those values of $x\in(0,1)$ for which
$x_{2}(x)=x_{3}(x)$ must be the roots of the polynomial $x(1-x)^{2}-4z=0$. It
is direct to show that such roots exist and there are two of them, say
$x^{\text{min}}(z)<x^{\text{max}}(z)$, iff $z\in I$. Alternatively
$x^{\text{min}}(z)=\min_{t\in\mathbb{R}^{+}}x_{1}(t)$ and
$x^{\text{max}}(z)=\max_{t\in\mathbb{R}^{+}}x_{1}(t)$. An explicit computation
of the roots of the cubic polynomial yields the following expressions

$\displaystyle
x^{\text{min}}(z)=\tfrac{2}{3}\left(1-\sin\left(\tfrac{\theta(27z)}{3}+\tfrac{\pi}{6}\right)\right)$
$\displaystyle
x^{\text{max}}(z)=\tfrac{2}{3}\left(1+\sin\left(\tfrac{\theta(27z)}{3}-\tfrac{\pi}{6}\right)\right)$

(and we have $x^{\text{add}}(z)=\frac{2}{3}(1+\cos\tfrac{\theta(27z)}{3})$ for
the third root) where the angle $\theta(z)$ is defined by the relations

$\left\\{\begin{array}[]{l}\cos\theta(z)=2z-1\\\
\sin\theta(z)=2\sqrt{z(1-z)}\end{array}\right.$

In particular, the function $z\mapsto\theta(z)$ is decreasing over $(0,1)$
with range $(0,\pi)$ (and $\theta(1-z)=\pi-\theta(z)$), which implies the
following limits

$x^{\text{min}}(0^{+})=0,\ x^{\text{max}}(0^{+})=1\quad\text{and}\quad
x^{\text{min}}(\tfrac{1}{27}^{-})=x^{\text{max}}(\tfrac{1}{27}^{-})=\tfrac{1}{3},$

as expected. Also, the function $z\mapsto\theta(z)$ is $C^{\infty}$ over
$(0,1)$ (and its derivative is equal to $-\frac{1}{\sqrt{z(1-z)}}$).

The dynamics $\dot{\mathrm{x}}=F(\mathrm{x})$ commutes with cyclic
permutations of coordinates $x_{i}\mapsto x_{i+1}$. The permutation do not
affect the product $x_{1}x_{2}x_{3}$; hence invariant loops are also invariant
under these permutations. This implies the existence of $T>0$ such that
$x_{i}(\cdot+T)=x_{i+1}(\cdot)$ for all $i$. Therefore, all trajectories must
all be periodic.

Moreover, the autonomous equation for $x_{1}$ in the previous subsection
implies that the half-period $\frac{T(z)}{2}$ can be defined as the time it
takes for $x_{1}$ to transit from $x^{\text{min}}(z)$ to $x^{\text{max}}(z)$,
namely we have

$T(z)=2\int_{x^{\text{min}}(z)}^{x^{\text{max}}(z)}\tfrac{dx}{\sqrt{x^{2}(1-x)^{2}-4zx}}.$

###  B.2 Proof of Lemma 2.3

Proof that the function $z\mapsto T(z)$ is $C^{\infty}$ on $I$. Given an
initial point $(x_{1}^{\text{in}},x_{2}^{\text{in}})$, let $t\mapsto
x_{2}(t,x_{1}^{\text{in}},x_{2}^{\text{in}})$ be the value of the second
coordinate of the solution of the above system of coupled ODE’s. Using that
$x_{2}=\tfrac{1-x^{\text{min}}(z)}{2}$ (resp.
$x_{2}=\tfrac{1-x^{\text{max}}(z)}{2}$) when $x_{1}$ reaches
$x^{\text{min}}(z)$ (resp. $x^{\text{max}}(z)$), the period $T(z)$ can be
(also) specified using the relation

$x_{2}(\tfrac{T(z)}{2},x^{\text{min}}(z),\tfrac{1-x^{\text{min}}(z)}{2})=\tfrac{1-x^{\text{max}}(z)}{2}.$

This relation can be regarded as an equation for $T(z)$ given $z$. In fact,
the function $x_{2}(\frac{t}{2},x_{1}^{\text{in}},x_{2}^{\text{in}})$ is
$C^{\infty}$ jointly in $t$ and $(x_{1}^{\text{in}},x_{2}^{\text{in}})$,
because the vector field itself is $C^{\infty}$. The functions $z\mapsto
x^{\text{min}}(z)$ and $z\mapsto x^{\text{max}}(z)$ are $C^{\infty}$ on $I$
(inherited from the same property for $\theta(z)$). Also, for every $z\in I$,
the derivative

$\frac{d}{dt}x_{2}(\tfrac{T(z)}{2},x^{\text{min}}(z),\tfrac{1-x^{\text{min}}(z)}{2})=\dot{x}_{2}(x^{\text{max}}(z),\tfrac{1-x^{\text{max}}(z)}{2})=\tfrac{1-x^{\text{max}}(z)}{2}\left(x^{\text{max}}(z)-\tfrac{1-x^{\text{max}}(z)}{2}\right)\neq
0,$

does not vanish. That $z\mapsto T(z)$ is $C^{\infty}$ then follows from the
implicit function theorem.

Proof that $T(\frac{1}{27}^{-})$ exists and is finite. We observe that,
together with the explicit expression of $T(z)$ above, the equality
$x(1-x)^{2}-4z=(x-x^{\text{min}}(z))(x^{\text{max}}(z)-x)(x^{\text{add}}(z)-x)$
readily implies

$\tfrac{2}{\sqrt{\overline{\chi}(z)}}\int_{0}^{1}\tfrac{dx}{\sqrt{x(1-x)}}\leq
T(z)\leq\tfrac{2}{\sqrt{\underline{\chi}(z)}}\int_{0}^{1}\tfrac{dx}{\sqrt{x(1-x)}},$

where

$\underline{\chi}(z)=\min_{x\in(x^{\text{min}}(z),x^{\text{max}}(z))}(x^{\text{add}}(z)-x)x\quad\text{and}\quad\overline{\chi}(z)=\max_{x\in(x^{\text{min}}(z),x^{\text{max}}(z))}(x^{\text{add}}(z)-x)x.$

We have the limit $x^{\text{add}}(\frac{1}{27}^{-})=\frac{4}{3}$ which implies
the following ones
$\underline{\chi}(\frac{1}{27}^{-})=\overline{\chi}(\frac{1}{27}^{-})=\frac{1}{3}$.
Together with $\int_{0}^{1}\frac{dx}{\sqrt{x(1-x)}}=\pi$, these behaviours
yield the limit $T(\frac{1}{27}^{-})=2\pi\sqrt{3}$.

Figure 4:  An example for $z=0.01$ of periodic trajectory of the dynamics
$\dot{\mathrm{x}}=F(\mathrm{x})$ (thin (blue) curve). The trajectory can be
decomposed into 6 arcs, all of the same duration and which can be deduced one
another using cyclic permutations of coordinates and/or time reversal; from
$A$ (defined as the point where $x_{1}=x^{\rm max}(z)$) to $B$ (defined as
$x_{1}=x_{2}=\tfrac{1-x^{\text{min}}(z)}{2}$), then to $C$ ($x_{2}=x^{\rm
max}(z)$), etc. The (purple) dashed lines materialize the neighborhood $V_{r}$
of the vertex $\mathrm{v}_{1}$, such that ${\rm
max}(x_{2},x_{3}=1-x_{1}-x_{2})\leq r$. The point $A_{r}$ is where the
trajectory leaves $V_{r}$ (hence the coordinate $x_{2}$ of $A_{r}$ is $r$) and
the segment $A^{\prime}_{r}B^{\prime}$ on the edge
$\mathrm{v}_{1}\mathrm{v}_{2}$ is the segment that the arc $A_{r}B$ approaches
as $z\to 0^{+}$.

Proof that $T(z)\sim-3\ln z$ when $z\to 0^{+}$. We decompose any periodic
trajectory into 6 arcs, all of the same duration and which can be deduced one
another using cyclic permutations of coordinates and/or time reversal, see
Fig. 4. Accordingly, let $\tau(z)=\frac{T(z)}{6}$ be the transit time in each
arc. In particular, $\tau(z)$ is the transit time from $A$ (where
$x_{1}=x^{\rm max}(z)$) to $B$ ($x_{1}=x_{2}=\tfrac{1-x^{\text{min}}(z)}{2}$).
We aim at finding an equivalent for $\tau(z)$ as $z$ tends to $0^{+}$.

To that goal, let $r\in(0,\frac{1}{3})$ and consider the neighborhood of the
vertex $\mathrm{v}_{1}=(1,0)$, defined as $V_{r}=\\{(x_{1},x_{2}):{\rm
max}(x_{2},1-x_{1}-x_{2})\leq r\\}$. Let $z\in I$ be sufficiently small so
that $A\in V_{r}$. Let then $A_{r}$ be the intersection point of the arc from
$A$ to $B$ and the boundary of $V_{r}$. Let $\tau_{A\to A_{r}}$ (resp.
$\tau_{A_{r}\to B}$) denote the transit time from $A$ to $A_{r}$ (resp. from
$A_{r}$ to $B$). We clearly have

$\tau(z)=\tau_{A\to A_{r}}+\tau_{A_{r}\to B}.$

We now estimate the two durations separately. For the latter, we have

$\lim_{z\to 0^{+}}A_{r}=A^{\prime}_{r}\quad\text{and}\quad\lim_{z\to
0^{+}}B=B^{\prime},$

and the continuity of the vector field $F$ implies that $\lim_{z\to
0^{+}}\tau_{A_{r}\to B}=\tau_{A^{\prime}_{r}\to B}$ which is finite for every
$r>0$ (and diverges as $r\to 0^{+}$).

For the former, we observe that inside $V_{r}$, and hence along the arc from
$A$ to $A_{r}$, the coordinate $x_{1}$ is not smaller than $1-2r$. Using also
that $x_{2}\geq 0$, we obtain from the equation of the dynamics, the following
inequalities for the time derivative of the coordinate $x_{2}$ in the arc
between $A$ and $A_{r}$

$1-3r\leq\tfrac{\dot{x}_{2}}{x_{2}}\leq 1,$

from where direct integration yields the estimates

$(1-3r)\tau_{A\to A_{r}}\leq\ln x_{2}(A_{r})-\ln\tfrac{1-x^{\text{\rm
max}}(z)}{2}\leq\tau_{A\to A_{r}},$

and then

$1+\frac{\ln x_{2}(A_{r})+\tau_{A_{r}\to B}}{-\ln\tfrac{1-x^{\text{\rm
max}}(z)}{2}}\leq\frac{\tau(z)}{-\ln\tfrac{1-x^{\text{\rm
max}}(z)}{2}}\leq\tfrac{1}{1-3r}+\frac{\frac{\ln
x_{2}(A_{r})}{1-3r}+\tau_{A_{r}\to B}}{-\ln\tfrac{1-x^{\text{\rm
max}}(z)}{2}}.$

Now use that $x^{\text{\rm max}}(z)\sim 1-2\sqrt{z}$ as $z\to 0^{+}$,
$\lim_{z\to 0^{+}}x_{2}(A_{r})=x_{2}(A^{\prime}_{r})=r\sin\frac{\pi}{3}$ and
the limit $\lim_{z\to 0^{+}}\tau_{A_{r}\to B}$ above to obtain

$1\leq\liminf_{z\to 0^{+}}\frac{\tau(z)}{-\frac{1}{2}\ln z}\leq\limsup_{z\to
0^{+}}\frac{\tau(z)}{-\frac{1}{2}\ln z}\leq\tfrac{1}{1-3r}.$

The desired conclusion finally follows by taking the limit $r\to 0$.

###  B.3 Proof of Lemma 2.4

When evaluated for the trajectory such that $x_{1}(0)=x^{\text{min}}(z)$,
using the change of variable $t\mapsto x_{1}(t)$, the quantity $A(z)$ can be
written as follows

$\displaystyle A(z)$
$\displaystyle=\int_{x^{\text{min}}(z)}^{x^{\text{max}}(z)}\tfrac{1-x_{1}-\sqrt{(1-x_{1})^{2}-\frac{4z}{x_{1}}}}{2}dx_{1}+\int_{x^{\text{max}}(z)}^{x^{\text{min}}(z)}\tfrac{1-x_{1}+\sqrt{(1-x_{1})^{2}-\frac{4z}{x_{1}}}}{2}dx_{1}$
$\displaystyle=-\int_{x^{\text{min}}(z)}^{x^{\text{max}}(z)}\sqrt{(1-x_{1})^{2}-\frac{4z}{x_{1}}}dx_{1}.$

The smoothness of the integration bounds and of the integrand imply, via the
Leibniz integral rule, that the function $z\mapsto A(z)$ is $C^{\infty}$ and
$I^{\prime}(z)=T(z)$ on $I$.

From the expression above, $A(z)$ can be interpreted as the area enclosed in
the loop $z(\mathrm{x})=z$. Accordingly, we clearly have
$A(0^{+})=\frac{1}{2}$ and $A(\frac{1}{27}^{-})=0$. From the relation
$A^{\prime}(z)=T(z)$ on $I$ and $T(\frac{1}{27}^{-})=2\pi\sqrt{3}$, we
conclude that $A(z)\sim-2\pi\sqrt{3}(\frac{1}{27}-z)$ when
$z\to\frac{1}{27}^{-}$.

##  Appendix C A slight extension of the Skorokhod’s representation theorem

######  Theorem C.1.

Let $(E,{\cal B},P)$ and $(E_{N},{\cal B}_{N},P_{N})$ for $N\in\mathbb{N}$ be
probability spaces, where $E$ and $E_{N}$ are metric spaces, ${\cal B}$ and
${\cal B}_{N}$ their respective Borel $\sigma$-algebras and where the support
of $P$ is separable. Assume the existence of measurable functions
$f_{N}:E_{N}\to E$ such that $P_{N}\circ f_{N}^{-1}\Rightarrow P$ as
$N\to\infty$ (weak convergence). Then there exist random variables $Y$ and
$Y_{N}$ defined on a common probability space $(\mathrm{E},\Omega,\mathrm{P})$
such that the laws of $Y$ and $Y_{N}$ are respectively $P$ and $P_{N}$ and we
have

$\lim_{N\to\infty}Y_{N}(\omega)=Y(\omega)\ \text{for every}\
\omega\in\mathrm{E}.$

The proof is a direct adaptation mutatis mutandis of the proof of Theorem 6.7,
Chap. 1 in [6].