techniques and a detailed analysis of the SSDEs. This analysis will be done in
several steps. In Sections 7–9 we first deal with a simplified system, the
mean-field model, for which we derive the McKean-Vlasov limit and the mean-
field finite-systems scheme. In Sections 10–12 we extend our analysis to
finitely many hierarchical levels. In particular, we go through the following
list of systems of increasing complexity, each being a simplified version of
the system defined by (2.19) and each capturing a key feature:

  1. 1.

Two-colour mean-field finite-systems scheme (Section 10.1).

  2. 2.

Two-level hierarchical mean-field system (Section 11.1).

  3. 3.

Finite-level mean-field system (Section 12.1).

In Section 12 we put the pieces together to prove the multi-scaling for the
infinite-level system given in Theorems 4.2 and 4.4.

###  5 Main results $N\to\infty$: Orbit and cluster formation

In the hierarchical mean-field limit we say that the system clusters when the
colonies gradually form larger and larger mono-type blocks. In Section 5.1 we
determine whether, in the hierarchical mean-field limit, the system clusters
along successive space-time scales. How this happens is captured by the
interaction chain. We introduce a sequence of scaling factors
$(A_{k})_{k\in\mathbb{N}_{0}}$, where $A_{k}$ is defined in terms of the rates
$(c_{k})_{k\in\mathbb{N}_{0}}$, $(e_{k})_{k\in\mathbb{N}_{0}}$,
$(K_{k})_{k\in\mathbb{N}_{0}}$ and the factors $(E_{k})_{k\in\mathbb{N}_{0}}$.
Using these scaling factors, we analyse the orbit of the renormalisation
transformation and establish _universality_ : $A_{k}\mathcal{F}^{(k)}g$
converges as $k\to\infty$ to the Fisher-Wright diffusion function,
irrespective of the choice of $g$. In Section 5.2 we show how the scaling
factors $A_{k}$ are connected to the _growth of mono-type clusters_. In
Section 5.3 we identify the asymptotics of $A_{k}$ as $k\to\infty$ in terms of
the model parameters.

####  5.1 Orbit of renormalisation transformations

$n-1$$n$$n+1$01$Q^{(n)}\left(z_{n+1},dz_{0}\right)$$Q^{[n]}\left(z_{n+1},dz_{n}\right)$$Q^{[n-1]}\left(z_{n},dz_{n-1}\right)$$Q^{[0]}\left(z_{1},dz_{0}\right)$$k-1$$k$$k+1$$D_{k-1}$$D_{k}$$D_{k+1}$010110$\mathcal{F}^{(k-1)}g$$\mathcal{F}^{(k)}g$$\mathcal{F}^{(k+1)}g$$\mathcal{F}g$$g$
Figure 10: _Left:_ The interaction chain that connects successive hierarchical
levels downwards. The arrows on the right correspond to (4.20), the arrow on
the left corresponds to (5.1). _Right:_ The renormalisation transformation
that connects successive hierarchical levels upwards. The vertical arrows
correspond to (4.17). The horizontal arrows represent the interaction with the
effective seed-bank. The arrows on the left represent the resampling driven by
the renormalised diffusion function.

To determine whether clustering occurs, we start from larger and larger time
scales and use the interaction chain to see whether mono-type clusters are
formed in the single colonies. Recall the kernels introduced in (4.20) that
describe the connection between subsequent hierarchical levels in the
interaction chain. Define the following composition of kernels (see Fig. 10):

(5.1) $=Q^{[n]}\circ\cdots\circ Q^{[0]},\qquad n\in\mathbb{N}.$

In words, $Q^{(n)}(z_{n},{\rm d}z_{0})$ is the probability density to see the
population of a single colony in state $z_{0}$ given that the $(n+1)$-block
average equals $z_{n}$.

In Section 3 we identified the clustering regime for fixed $N<\infty$. In this
section we identify the clustering regime in the hierarchical mean-field
limit. In the clustering regime, in the hierarchical mean-field limit, an
interesting question is to determine how $\mathcal{F}^{(n)}g$ (recall (4.17))
scales with $n$. We identify the scaling and show that it does _not_ depend on
$g$ (see Fig. 11).

To state the clustering result, abbreviate

(5.2)
$=(\vartheta_{n},\overbrace{\vartheta_{n},\cdots,\vartheta_{n}}^{n+1\text{
times }},\theta_{y_{n+1}},\theta_{y_{n+2}},\cdots).$

######  Theorem 5.1.

[Renormalised scaling] Let $c_{k}$ be as in (2.5), $e_{k}$ and $K_{k}$ as in
(LABEL:defKem) and $E_{k}$ as in (4.5). Define

(5.3)
$=\frac{1}{2}\sum_{k=0}^{n-1}\frac{E_{k}}{c_{k}}\frac{(E_{k}c_{k}+e_{k})}{(E_{k}c_{k}+e_{k})+E_{k}K_{k}e_{k}},\qquad
n\in\mathbb{N}.$

Then

(5.4)
$\lim_{n\to\infty}Q^{(n)}\bigl{(}\bar{\vartheta}^{(n)},\,\cdot\,\bigr{)}=(1-\theta)\,\delta_{(0,0^{\mathbb{N}_{0}})}+\theta\,\delta_{(1,1^{\mathbb{N}_{0}})}$

if and only if

(5.5) $\lim_{n\to\infty}A_{n}=\infty.$

Moreover, if (5.5) holds, then for all $g\in\mathcal{G}$,

(5.6)
$\lim_{n\to\infty}A_{n}\mathcal{F}^{(n)}g=g_{\mathrm{FW}}\quad\text{pointwise},$

with $g_{\mathrm{FW}}(x)=x(1-x)$, $x\in[0,1]$.

The proof of Theorem 5.1 is given in Section 13. The scaling factors $A_{n}$
can be interpreted as clustering coefficients: in Section 5.2 we will show
that the faster the $A_{n}$ grow to infinity, the faster we expect to see
clusters grow. The property in (5.4) corresponds to the _clustering regime_.
According to (5.6), even though $A_{n}$ depends on the choice of the sequences
$\underline{K},\underline{e},\underline{c}$ in (2.5) and (LABEL:defKem), the
limit $A_{n}\mathcal{F}^{(n)}g$ as $n\to\infty$ is _universal_ : irrespective
of the choice of $g\in\mathcal{G}$, the limit is the standard Fisher-Wright
diffusion function $g_{\text{FW}}$. Thus, $g_{\text{FW}}$ is the _global
attractor of the renormalisation transformation_ (see Fig. 11).

$g$$A_{1}\mathcal{F}^{(1)}g$$A_{2}\mathcal{F}^{(2)}g$$A_{3}\mathcal{F}^{(3)}g$$A_{4}\mathcal{F}^{(4)}g$$g_{\text{FW}}$
Figure 11: Flow of the iterates $\mathcal{F}^{(n)}g$, $n\in\mathbb{N}_{0}$, of
the renormalisation transformation acting on the class $\mathcal{G}$. After
multiplication by $A_{n}$, the flow is globally attracted by $g_{\text{FW}}$.

####  5.2 Growth of mono-type clusters

In the _clustering regime_ we are interested in how fast mono-type clusters
grow in space over time. For the system on $\mathbb{Z}^{d}$ and $\Omega_{N}$
without seed-bank the growth rate has been studied in detail. Different growth
rates were found for _strongly recurrent_ and _critically recurrent_
migration. Typical examples on $\Omega_{N}$ are migrations with coefficients
$c_{k}=c^{k}$ with $c\in(0,1)$, respectively, $c_{k}=C$ with $C\in(0,\infty)$.
Typical examples on $\mathbb{Z}$, respectively, $\mathbb{Z}^{2}$ are
migrations with zero mean and finite variance. For these models the following
behaviour occurs.

  * •

In the strongly recurrent case, mono-type clusters grow fast and cover a
volume that increases at time $t$ at a rate that is given by the Green
function up to time $t$ of the underlying random walk, times a certain random
constant that can be determined explicitly and that is independent of the
diffusion function $g\in\mathcal{G}$ [EF96], [Kle96]. The cluster growth is
monitored by considering families of balls growing in time at such a speed
that, starting from a translation invariant and ergodic initial state, the
mean of the configuration in the ball is still close to the starting mean but
begins to move. _Fast clustering_ means that the cluster covers multiples of a
scale that eventually lies in every finite family of balls with the above
property.

  * •

In the critically recurrent case, the volume grows only moderately fast, like
$N^{(1-U)t}$ as $t\to\infty$, with $U\in[0,1]$ a random variable. In other
words, the cluster sizes have random orders of magnitude, an effect known as
_diffusive clustering_. For $c_{k}=C\in(0,\infty)$, $k\in\mathbb{N}_{0}$, the
distribution of $U$ can be identified by studying the fraction of active
individuals of type $1$ in a ball of size $N^{(1-u)t}$, which can be shown to
converge to $V(\log\tfrac{1}{1-u})$ as $t\to\infty$ with $(V(s))_{s\geq 0}$
the standard Fisher-Wright diffusion, irrespective of the choice of
$g\in\mathcal{G}$ [FG94], [FG96].

  * •

For more general migration it is possible that mono-type clusters grow slower
than any positive power of $t$ as $t\to\infty$. This occurs for recurrent
migration in which the Green function up to time $t$ grows like $o(\log t)$.
For this regime only few results are available [DGV95].

From the perspective of explaining _universality_ in $g\in\mathcal{G}$ in the
hierarchical mean-field limit $N\to\infty$, the above type of behaviour has
been studied in detail in [DGV95] and [GdHKK14] for the Fleming-Viot model,
respectively, the Cannings model without seed-bank. The renormalisation
analysis for the model with seed-bank allows us to study how the seed-bank
affects the cluster growth. In what follows we give a sketch of _three regimes
of cluster growth_.

###### Types of clustering.

If a ball in $\Omega_{N}$ lies in a mono-type cluster, then the block average
of the active and the dormant components in this ball are all close to either
$0$ or $1$. We can therefore analyse the growth rate of mono-type clusters by
analysing at which hierarchical level block averages hit $0$ or $1$ in the
limit as $N\to\infty$. To that end, we look at the interaction chain
$M^{k}_{-l(k)}$ for $k\to\infty$, where the _level scaling function_
$l\colon\,\mathbb{N}_{0}\to\mathbb{N}_{0}$ is non-decreasing with
$\lim_{k\to\infty}l(k)=\infty$ and is suitably chosen such that we obtain a
_non-trivial clustering limiting law_ , i.e.,

(5.7)
$\lim_{k\to\infty}\mathcal{L}\bigl{[}M^{k}_{-l(k)}\bigr{]}=\mathcal{L}\bigl{[}\hat{\theta}\,\bigr{]},$

where the limiting sequence of random frequencies $\hat{\theta}$ satisfies

(5.8)
$0<\mathbb{P}\bigl{(}\hat{\theta}\in\\{0^{\mathbb{N}_{0}},1^{\mathbb{N}_{0}}\\}\bigr{)}<1.$

In line with [DGV95] and [DG96], in order to analyse the growth of mono-type
clusters on multiple space-time scales in the hierarchical mean-field limit,
it is natural to consider a family of non-decreasing functions
$l_{\chi}\colon\,\mathbb{N}_{0}\to\mathbb{N}_{0}$, $\chi\in
I\subseteq[0,\infty)$, called the _cluster scales_ , satisfying (5.7)–(5.8):

  * (1)

Fast clustering: $\lim_{k\to\infty}l_{\chi}(k)/k=1$ for all $\chi\in I$.

  * (2)

Diffusive clustering: $\lim_{k\to\infty}l_{\chi}(k)/k=\kappa(\chi)$ for all
$\chi\in[0,1]$, where $\chi\mapsto\kappa(\chi)$ is continuous and non-
increasing with $\kappa(0)=1$ and $\kappa(1)=0$.

  * (3)

Slow clustering: $\lim_{k\to\infty}l_{\chi}(k)/k=0$ for all $\chi\in I$. (This
regime borders with the regime of coexistence.)

We write $(M^{\infty}_{\chi})_{\chi\in I}$ with
$M^{\infty}_{\chi}=\lim_{k\to\infty}M^{k}_{-l_{\chi}(k)}$ to denote the
_cluster process_.

######  Remark 5.2.

Examples are:

  * (1)

$I=\mathbb{N}_{0}$, $l_{\chi}(k)=k-\chi$.

  * (2)

$I=[0,1]$, $l_{\chi}(k)=\lfloor(1-\chi)k\rfloor$.

  * (3)

$I=[0,1]$, $l_{\chi}(k)=\lfloor L(k^{1-\chi})\rfloor$ with $L(0)=0$, $L$ non-
decreasing and sublinear.

In words, the clusters cover blocks of level: (1) $k-\chi$; (2)
$\lfloor(1-\chi)k\rfloor$; (3) $L(k^{1-\chi})$. For the model without seed-
bank and with migration coefficients $c_{k}=c^{k}$ with $c\in(0,1)$, case (1)
is realised with a Markov chain $(M^{\infty}_{l})_{l\in\mathbb{N}_{0}}$ as
scaling limit, while for $c_{k}=C$, case (2) is realised with a time-
transformed Fisher-Wright diffusion in $\chi$ as scaling limit. (For finite
$N$, this corresponds to the first and the second example given in the first
paragraph of this section.) For the model without seed-bank, these scales have
been shown to satisfy the required conditions. Case (3) also appears for the
model without seed-bank, but detailed information on scales and scaling limits
is lacking. As we will see below, seed-banks can slow down cluster growth, so
case (3) is worthwhile to be studied in more detail.  $\blacksquare$

Recall (5.3). Fast clustering corresponds to $A_{k}\gg k$, diffusive
clustering to $A_{k}\asymp k$, and slow clustering to $A_{k}\ll k$ for large
$k$. Theorem 5.3 below shows that, subject to (3.10) and (3.11), all three
regimes are possible for the model with seed-bank. The regimes are the same as
for the model without seed-bank when $\rho<\infty$, but different when
$\rho=\infty$.

For systems without seed-bank, examples of the three types of clustering can
be found in the literature: diffusive clustering in [Arr79], [CG86] (voter
model on $\mathbb{Z}$, respectively, $\mathbb{Z}^{2}$) and in [DG93a], [FG94],
[DGV95], [Kle96] (interacting Fleming-Viot processes on $\Omega_{N}$ with
$N<\infty$, respectively, $N\to\infty$), all types of clustering in [DG96],
[Kle97] , [Win02] (interacting Feller diffusions on $\Omega_{N}$ with
$N\to\infty$) and in [GdHKK14], [GdHK18] (interacting Cannings processes on
$\Omega_{N}$ in non-random and random environment with $N\to\infty$).

For the model with seed-bank we have to use the asymptotics of
$(A_{k})_{k\in\mathbb{N}}$ to identify the set $I$ and the family
$(l_{\chi}(\cdot))_{\chi\in I}$, and show that
$(M^{k}_{-l_{\chi}(k)})_{\chi\in I}$ converges as $k\to\infty$ to a Markov
process, which we want to identify.

###### Computations.

In the following we demonstrate how we can carry out the above task. The key
idea is to study first and second moments of the interaction chain, as well as
sums of variances in order to get a handle on the quadratic variation process.
To that end we calculate

(5.9) $V^{k}_{-l}=\text{ the conditional variance of the active part of
}M^{k}_{-l}\text{ given }M^{k}_{-(l+1)}$

and consider the sum of random variables
$A_{k,n}=\sum_{-(k+1)\leq-l\leq-n}V^{k}_{-l}$, $n\in\mathbb{N}_{0}$. In order
for the system to cluster, we must have $\lim_{k\to\infty}A_{k,n}=0$ for every
$n\in\mathbb{N}_{0}$. The _volatility profile_ is given by

(5.10) $(p_{\chi}(k))_{\chi\in[0,1]},\qquad
p_{\chi}(k)=A_{k,l_{\chi}(k)}/A_{k,0}.$

This profile is a _random variable_ that depends on the interaction chain up
to $M^{k}_{-l_{\chi}(k)}$. Since
$A_{k,l_{\chi}(k)}=A_{k,0}-A_{l_{\chi}(k)-1,0}$, we have
$p_{\chi}(k)=(A_{k,0}-A_{l_{\chi}(k)-1,0})/A_{k,0}$. For diffusive clustering,
for instance, we want to show that

(5.11) $\lim_{k\to\infty}p_{\chi}(k)=1-\kappa(\chi),\qquad\chi\in[0,1],$

while for fast clustering the limit is $0$ and for slow clustering the limit
is $1$. From (5.6) we know that the scaled renormalised diffusion function
$A_{n}\mathcal{F}^{(n)}(g)$ tends to the standard Fisher-Wright diffusion
function as $n\to\infty$. Since the latter hits the boundary $\\{0,1\\}$ after
some finite time, the coefficients $A_{n}$ describe the speed at which the
interaction chain hits this boundary. We next make this idea precise and show
how it can be used to obtain information about the growth of mono-type
clusters.

The kernels defined in Section 5.1 allow us to compute the first and second
moments of all the block averages, which will be done in Section 13.1
(Propositions 13.1–13.4). In particular, using the interaction chain starting
between at $-n$ and running until $-m$ with $-n<-m\leq 0$, and considering the
$m$-block averages on time scale $N^{m}t$ in the limit $N\to\infty$, we find
that the variance of the active component $x^{n}_{m}$ of $M^{n}_{-m}$ equals

(5.12)
$\mathbb{V}\mathrm{ar}(x^{n}_{m})=\mathbb{E}\left[(x^{n}_{m}-\vartheta_{n})^{2}\right]=A_{m}^{n}(\mathcal{F}^{(n+1)}g)(\vartheta_{n}),$

where

(5.13)
$=\frac{1}{2}\sum_{k=m}^{n}\frac{E_{k}}{c_{k}}\frac{(E_{k}c_{k}+e_{k})}{(E_{k}c_{k}+e_{k})+E_{k}K_{k}e_{k}}.$

(Note that $A_{n}=A^{n-1}_{0}$.) On the other hand, since $x^{n}_{m}\in(0,1)$
we have $\mathbb{V}\mathrm{ar}(x^{n}_{m})\in(0,1)$ and
$A^{n}_{m}(\mathcal{F}^{(n+1)})(\vartheta_{n})\in(0,1)$. Taking $m=0$, we get
$(\mathcal{F}^{(n+1)}g)(\vartheta_{n})\in(0,\frac{1}{A^{n}_{0}})$. This
implies that

(5.14)
$(\mathcal{F}^{(n+1)}g)(\vartheta_{n})=\int_{[0,1]^{2}}(\mathcal{F}^{m}g)(x_{m})\,Q^{(n)}_{m}((\vartheta_{n},\theta_{y,n}),{\rm
d}z_{m})\in\left(0,\tfrac{1}{A^{n}_{0}}\right).$

Since $\lim_{n\to\infty}A_{n}(\mathcal{F}^{(n+1)}g)=g_{\text{FW}}$, for large
enough $m,n$ we can approximate $\mathcal{F}^{(m)}g\approx
g_{\text{FW}}/A^{m}_{0}$. Therefore

(5.15)
$\int\frac{g_{\text{FW}}}{A^{m}_{0}}(x_{m})\,Q^{(n)}_{m}((\vartheta_{n},\theta_{y,n}),{\rm
d}z_{m})\in\left(0,\tfrac{1}{A^{n}_{0}}\right),$

or, equivalently,

(5.16)
$\int_{[0,1]^{2}}x_{m}(1-x_{m})\,Q^{(n)}_{m}((\vartheta_{n},\theta_{y,n}),{\rm
d}z_{m})\in\left(0,\tfrac{A^{m}_{0}}{A^{n}_{0}}\right).$

Hence, if $A^{m}_{0}/A^{n}_{0}<\epsilon$ with $\epsilon>0$ small, then we know
that with high probability the system on time scale $n$ has clusters with a
radius of size $m$. (Note that for the interaction chain this means that the
variance is almost entirely centred between $n$ and $m$.) Therefore the speed
at which $A^{m}_{0}/A^{n}_{0}$ converges to zero as $m,n\to\infty$ says
something about the speed at which monotype clusters form.

To capture the cluster growth, we must decide how we let $m,n\to\infty$. For
this we look for clusters of radius $l_{\chi}(n)$ with $\chi\in I$. Put

(5.17) $f^{n}(l_{\chi}(n))=\frac{A^{l_{\chi}(n)}_{0}}{A^{n}_{0}},$

and define, for $\epsilon>0$,

(5.18) $\mathcal{X}^{n}_{\epsilon}=\inf\\{\chi\in
I\colon\,f^{n}(l_{\chi}(n))<\epsilon\\}.$

Then the three types of clustering correspond to:

  * (1)

Fast clustering: $\lim_{n\to\infty}l_{\mathcal{X}^{n}_{\epsilon}}(n)/n=1$.

  * (2)

Diffusive clustering: $\lim_{n\to\infty}l_{\mathcal{X}^{n}_{\epsilon}}(n)/n=R$
for some random variable $R$ taking values in $(0,1)$.

  * (3)

Slow clustering: $\lim_{n\to\infty}l_{\mathcal{X}^{n}_{\epsilon}}(n)/n=0$.

In terms of the interaction chain starting from $-k$ with $k\to\infty$, in
view of (5.12) this corresponds to the variance in the interaction chain being
concentrated near the beginning, being spread out or being concentrated near
the end.

####  5.3 Rates of scaling for renormalised diffusion function

For the system without seed-bank, we have $K_{k}=e_{k}=0$ and $E_{k}=1$ for
all $k\in\mathbb{N}_{0}$. Hence

(5.19) $A_{n}=\frac{1}{2}\sum_{k=0}^{n-1}\frac{1}{c_{k}}$

and (5.4) holds if and only if

(5.20) $\sum_{k\in\mathbb{N}_{0}}\frac{1}{c_{k}}=\infty.$

Various subcases were analysed in [BCGdH95]. For the system with seed-bank
because $E_{0}=1$ and $E_{k}<1$ (see (4.5)), it follows from (5.3) that

(5.21) $A_{n}<\frac{1}{2}\sum_{k=0}^{n-1}\frac{1}{c_{k}}.$

Thus we see that the seed-bank _weakens clustering_ , i.e., enhances genetic
diversity, even in the hierarchical mean-field limit.

We identify the clustering regime in the setting where the coefficients are
_asymptotically polynomial_ , as in (3.10), or are _pure exponential_ , as in
(3.11). It turns out that there is a delicate interplay between the migration
and the seed-bank, resulting in 4 different scalings for asymptotically
polynomial coefficients and 8 different scalings for pure exponential
coefficients.

######  Theorem 5.3.

[Rates of scaling for diffusion function] Let $\rho$ be as defined in (2.14).

  * (I)

If $\rho<\infty$, then (5.5) holds if and only (5.20) hold, and

(5.22) $A_{n}\sim\frac{1}{2(1+\rho)}\sum_{m=0}^{k-1}\frac{1}{c_{k}}.$

  * (II)

If $\rho=\infty$, then (5.5) holds in the following cases:

    * $\bullet$

Subject to (3.10) if and only if $-\phi\leq\alpha\leq 1$, with

(5.23) $\displaystyle-\phi<\alpha<1\colon$ $\displaystyle A_{n}\sim
C_{1}\,n^{\alpha+\phi},$ $\displaystyle-\phi=\alpha<1\colon$ $\displaystyle
A_{n}\sim C_{2}\,\log n,$ $\displaystyle-\phi<\alpha=1\colon$ $\displaystyle
A_{n}\sim C_{3}\,\frac{n^{1+\phi}}{\log n},$
$\displaystyle-\phi=\alpha=1\colon$ $\displaystyle A_{n}\sim C_{4}\,\log\log
n,$

where

(5.24)
$\begin{array}[]{ll}&C_{1}=\frac{1}{2AF}\frac{1-\alpha}{\alpha+\phi},\quad
C_{2}=\frac{1}{2AF}(1-\alpha),\quad C_{3}=\frac{1}{2AF}\frac{1}{1+\phi},\quad
C_{4}=\frac{1}{2AF}.\\\ &\end{array}$

The values of $B,\beta$ play no role for the clustering, nor for the
asymptotics.

    * $\bullet$

Subject to (3.11) if and only if $Kc\leq 1\leq K$, with

(5.25) $\displaystyle c<Ke,\,Kc<1\colon$ $\displaystyle
A_{n}\sim\hat{C}_{1}\,(Kc)^{-(n-1)},$ $\displaystyle c<Ke,\,Kc=1\colon$
$\displaystyle A_{n}\sim\bar{C}_{1}\,n,$ $\displaystyle c=Ke,\,Kc<1\colon$
$\displaystyle A_{n}\sim\hat{C}_{2}\,(Kc)^{-(n-1)},$ $\displaystyle
c=Ke,\,Kc=1\colon$ $\displaystyle A_{n}\sim\bar{C}_{2}\,n,$ $\displaystyle
c>Ke,\,Kc<1\colon$ $\displaystyle A_{n}\sim\hat{C}_{3}\,(Kc)^{-(n-1)},$
$\displaystyle c>Ke,\,Kc=1\colon$ $\displaystyle A_{n}\sim\bar{C}_{3}\,n,$
$\displaystyle c<1=K\colon$ $\displaystyle
A_{n}\sim\tilde{C}_{1}\,n^{-1}\,c^{-(n-1)},$ $\displaystyle c=1=K\colon$
$\displaystyle A_{n}\sim\tilde{C}_{2}\log n,$

where

(5.26)
$\begin{array}[]{llll}&\hat{C}_{1}=\frac{K-1}{2K(1-Kc)},&\hat{C}_{2}=\frac{(K-1)^{2}}{2(2K-1)(1-Kc)},&\hat{C}_{3}=\frac{K-1}{2(1-Kc)},\\\\[8.5359pt]
&\bar{C}_{1}=\frac{K-1}{2K},&\bar{C}_{2}=\frac{(K-1)^{2}}{2(2K-1)},&\bar{C}_{3}=\frac{K-1}{2},\\\\[8.5359pt]
&\tilde{C}_{1}=\frac{1}{2(1-c)},&\tilde{C_{2}}=\frac{1}{2}.&\end{array}$

The value of $e$ plays no role for the clustering, but does for the
asymptotics.

The proof of Theorem 5.3 is given in Section 13. Part (I) shows that for
$\rho<\infty$ the clustering regime is the same as for the system without
seed-bank. The scaling of $A_{n}$ is controlled by the migration and is
reduced by a factor $1/(1+\rho)$ with respect to the seed-bank. Part (II)
shows that for $\rho=\infty$ the clustering regime is different from that for
the system without seed-bank. Clustering is harder to achieve: since
$\lim_{k\to\infty}E_{k}=0$ the growth rate of $A_{n}$ is _strictly smaller_
than without seed-bank.

Furthermore, subject to (3.10), if $-\phi<\alpha<1$, then the growth rate of
$A_{n}$ drops down from $\asymp n^{1+\phi}$ without seed-bank to $\asymp
n^{\alpha+\phi}$ with seed-bank, while if $-\phi=\alpha=1$, then it drops down
from $\asymp\log n$ to $\asymp\log\log n$. Similarly, subject to (3.11), if
$Kc<1<K$, then the growth rate of $A_{n}$ drops down from $\asymp c^{-n}$ to
$\asymp(Kc)^{-n}$, while if $c=K=1$, then it drops down from $\asymp n$ to
$\asymp\log n$.

Returning to the observations made in Section 5.2, we see that the three
clustering regimes also appear in the model with seed-bank, both for
$\rho<\infty$ and $\rho=\infty$, and in the latter case are accompanied by
different migration coefficients. The scaling results mentioned in Section 5.2
can in principle be deduced from the asymptotics of $A_{n}$ as $n\to\infty$ in
Theorem 5.3. It would be interesting to work out the details and to identify
the limiting processes that control the cluster growth.

##  Part II PREPARATIONS AND PROOFS

###  6 Proofs: $N<\infty$, identification of clustering regime

In this section we prove Theorems 3.2–3.3. The integral criterion for
$\rho=\infty$ in (3.8) is explained in Section 6.1. Theorem 3.2 is proved in
Section 6.2 and Theorem 3.3 in Section 6.3.

####  6.1 Explanation of clustering criterion for infinite seed-bank

Recall Fig. 6. Suppose that $g=dg_{\mathrm{FW}}$, so that we have a dual. We
will show that the integral criterion in (3.8) determines whether or not two
dual lineages coalesce with probability 1. Since two lineages in the dual can
only coalesce when they are active at the same site, we need to keep track of
the probabilities that the lineages are active at a given time. Because the
lineages can only migrate when they are active, we also need to keep track of
the total time they are active up to a given time.

Recall the renewal interpretation of the dual process (see Remark 2.9). We
argue heuristically as follows. If $\rho=\infty$, then the activity times
$\sigma_{k}$ are much smaller than the sleeping times $\tau_{k}$, and we may
assume that $\tau_{k}+\sigma_{k}\asymp\tau_{k}$, $k\to\infty$. Discretising
time, we can use the results from [AB16] for the intersection of two
independent renewal processes. Then the integral criterion in (3.8) can be
interpreted as follows:

  * •

If $\gamma\in(0,1)$, then the probability for each of the lineages to be
active at time $s$ decays like $\asymp\varphi(s)^{-1}s^{-(1-\gamma)}$ [AB16].
Hence the total time they are active up to time $s$ is
$\asymp\varphi(s)^{-1}s^{\gamma}$. Because the lineages only move when they
are active, the probability that the two lineages meet at time $s$ is $\asymp
a^{(N)}_{\varphi(s)^{-1}s^{\gamma}}(0,0)$. Hence the total hazard is
$\asymp\int_{1}^{\infty}{\rm
d}s\,[\varphi(s)^{-1}s^{-(1-\gamma)}]^{2}\,a^{(N)}_{\varphi(s)^{-1}s^{\gamma}}(0,0)$.
After the transformation $t=t(s)=\varphi(s)^{-1}s^{\gamma}$, the latter turns
into the integral in (3.8), modulo a constant. When carrying out this
transformation, we need that $s\varphi^{\prime}(s)/\varphi(s)\to 0$, which
follows from (3.7), and $\varphi(t(s))/\varphi(s)\asymp 1$, which follows from
the bound we imposed on $\psi$ in (3.7) together with the fact that
$\log\varphi(s)/\log s\to 0$. This computation is spelled out in Appendix A.

  * •

If $\gamma=1$, then the probability for each of the lineages to be active at
time $s$ decays like $\hat{\varphi}(s)^{-1}$ [AB16], and so the total time
they are active up to time $s$ is $\asymp s\hat{\varphi}(s)^{-1}$. Recall from
(3.6) that $\hat{\varphi}(t)=\mathbb{E}[\tau\wedge t]$ is also slowly
varying.) Hence the total hazard is $\asymp\int_{1}^{\infty}{\rm
d}s\,[\hat{\varphi}(s)^{-1}]^{2}\,a^{(N)}_{\hat{\varphi}(s)^{-1}s}(0,0)$.
After the transformation $t=t(s)=\hat{\varphi}(s)^{-1}s$ (for which we can use
the same type of computation as in Appendix A), the latter turns into the
integral in (3.8), modulo a constant.

####  6.2 Scaling of wake-up time and migration kernel for infinite seed-bank

We can prove Theorem 3.2 by direct computation via assumptions (3.10)–(3.11).
We start by computing $\gamma$. Afterwards we compute $\hat{\varphi}(t)$ and
$a_{t}^{\Omega_{N}}(0,0)$.

###### Computation of $\gamma$.

Recall (2.39), which reads

(6.1)
$\mathbb{P}(\tau>t)=\frac{1}{\chi}\sum_{m\in\mathbb{N}_{0}}K_{m}\frac{e_{m}}{N^{m}}\,{\rm
e}^{-(e_{m}/N^{m})t}.$

Since we are interested in the asymptotic behaviour of $\mathbb{P}(\tau>t)$ as
$t\to\infty$, we need to consider only large values of $t$. For large values
of $t$, only large values of $m$ (for which $\frac{e_{m}}{N^{m}}$ is small)
contribute to the sum in (6.1). Hence we can estimate the latter by an
integral and insert the assumptions made in (3.10)–(3.11). Subsequently, using
the change of variable $s=\frac{e_{m}}{N^{m}}$ and taking the logarithm to
express $m$ in terms of $s$, we obtain the following values of $\gamma$ after
extracting the $t$-dependence:

(6.2)
$\displaystyle\eqref{regvar}\quad\Longrightarrow\quad\gamma=1,\quad\varphi(t)\asymp(\log
t)^{-\alpha},$
$\displaystyle\eqref{pureexp}\quad\Longrightarrow\quad\gamma=\gamma_{N,K,e}=\frac{\log(N/Ke)}{\log(N/e)},\quad\varphi(t)\asymp
1.$

In order to guarantee that $\rho=\infty$, we must require that
$\alpha\in(-\infty,1]$, respectively, $K\in[1,\infty)$ (while $\beta$,
respectively, $e$ play no role). Subject to (3.10),

(6.3) $\hat{\varphi}(t)\asymp\left\\{\begin{array}[]{ll}(\log
t)^{1-\alpha},&\alpha\in(-\infty,1),\\\\[5.69046pt] \log\log
t,&\alpha=1,\end{array}\right.$

while subject to (3.11),

(6.4)
$\hat{\varphi}(t)\asymp\left\\{\begin{array}[]{ll}1,&K\in(1,\infty),\\\\[5.69046pt]
\log t,&K=1.\end{array}\right.$

###### Computation of $a^{\Omega_{N}}_{t}(0,0)$.

To compute $a^{\Omega_{N}}_{t}(0,0)$, we first rewrite the migration kernel
$a^{\Omega_{N}}(\cdot,\cdot)$ in (2.6) as

(6.5) $a^{\Omega_{N}}(0,\eta)=\frac{r_{\|\eta\|}}{N^{\|\eta\|-1}(N-1)}$

with

(6.6)
$r_{\|\eta\|}=\frac{1}{D(N)}\frac{N-1}{N}\sum_{l\geq\|\eta\|}\frac{c_{l-1}}{N^{l-1}}\frac{1}{N^{l-\|\eta\|}},$

where $D(N)$ is a renormalisation constant such that
$\sum_{j\in\mathbb{N}}r_{j}=1$. For transition kernels of the form (6.5), the
time-$t$ transition kernel $a^{\Omega_{N}}_{t}(\cdot,\cdot)$ was computed in
[FG94] with the help of Fourier analysis, see also [DGW05]. Namely,

(6.7) $a^{\Omega_{N}}_{t}(0,\eta)=\sum_{j\geq
k}K_{jk}(N)\,\frac{\exp[-h_{j}(N)t]}{N^{j}},\qquad t\geq
0,\quad\eta\in\Omega_{N}\colon\,d_{\Omega_{N}}(0,\eta)=k\in\mathbb{N}_{0},$

where

(6.8) $K_{jk}(N)=\left\\{\begin{array}[]{ll}0,&j=k=0,\\\ -1,&j=k>0,\\\
N-1,&\mbox{otherwise},\end{array}\right.\qquad j,k\in\mathbb{N}_{0},$

and

(6.9) $h_{j}(N)=\frac{N}{N-1}\,r_{j}(N)+\sum_{i>j}r_{i}(N),\qquad
j\in\mathbb{N}.$

The expressions in (6.6)–(6.9) simplify considerably in the limit as
$N\to\infty$, namely, the term with $i=j$ dominates and

(6.10) $h_{j}(N)\sim r_{j}(N)\sim\frac{c_{j-1}}{D(N)N^{j-1}},\quad
j\in\mathbb{N},\qquad D(N)\sim c_{0}.$

We show why this is true for $h_{j}(N)$ (the argument for $r_{j}(N)$ and
$D(N)$ is similar). Write

(6.11) $\displaystyle h_{j}(N)$
$\displaystyle=\frac{N}{N-1}\,r_{j}(N)+\sum_{i>j}r_{i}(N)$
$\displaystyle=\frac{1}{D(N)}\left(\sum_{l\geq
j}\frac{c_{l-1}}{N^{l-1}}\frac{1}{N^{l-j}}+\frac{N-1}{N}\sum_{l>j}\frac{c_{l-1}}{N^{l-1}}\sum_{i<j\leq
l}\frac{1}{N^{l-i}}\right)$
$\displaystyle=\frac{1}{D(N)}\frac{c_{j-1}}{N^{j-1}}\left(1+\left[1+O\left(\frac{1}{N}\right)\right]\,\left(\frac{c_{j-1}}{N^{j-1}}\right)^{-1}\sum_{l>j}\frac{c_{l-1}}{N^{l-1}}\right).$

Hence it suffices to show that

(6.12)
$\limsup_{N\to\infty}\left(\frac{c_{j-1}}{N^{j-1}}\right)^{-1}\sum_{l>j}\frac{c_{l-1}}{N^{l-1}}=0,\qquad
j\in\mathbb{N}.$

To do so, note that, since $\limsup_{k\to\infty}\frac{1}{k}\log c_{k}<\log N$
by (2.7), for $N$ large enough we have

(6.13) $\sup_{k\in\mathbb{N}_{0}}c_{k}^{1/k}<N.$

Let
$\bar{N}=\inf\\{N\in\mathbb{N}\colon\,\sup_{k\in\mathbb{N}_{0}}c_{k}^{1/k}<N\\}$.
Then

(6.14)
$\displaystyle\limsup_{N\to\infty}\left(\frac{c_{j-1}}{N^{j-1}}\right)^{-1}\sum_{l>j}\frac{c_{l-1}}{N^{l-1}}\leq\limsup_{N\to\infty}\frac{1}{c_{j-1}}\sum_{l>j}\frac{\bar{N}^{l-1}}{N^{l-1}}N^{j-1}$
$\displaystyle=\frac{\bar{N}^{j-1}}{c_{j-1}}\limsup_{N\to\infty}\frac{\frac{\bar{N}}{N}}{1-\frac{\bar{N}}{N}}=0,\qquad
j\in\mathbb{N},$

which settles (6.12).

To understand what (6.9) gives for finite $N$, note that for asymptotically
polynomial coefficients (recall (3.10))

(6.15)
$\displaystyle\left(\frac{c_{j-1}}{N^{j-1}}\right)^{-1}\sum_{l>j}\frac{c_{l-1}}{N^{l-1}}=[1+o(1)]\,\frac{N^{j-1}}{F(j-1)^{-\phi}}\sum_{l>j}\frac{{F(l-1)^{-\phi}}}{N^{l-1}}$
$\displaystyle=[1+o(1)]\,\sum_{l>j}\frac{{(l-1)^{-\phi}}}{(j-1)^{-\phi}}N^{-(l-j)}=[1+o(1)]\,\sum_{k\geq
1}\left(1+\frac{{k}}{j-1}\right)^{-\phi}N^{-k},\qquad j\in\mathbb{N}.$

For $\phi\geq 0$ the right-hand side is bounded from above by $\sum_{k\geq
1}N^{-k}=\frac{1}{N-1}$ and for $\phi<0$ by $N^{-1}\sum_{k\geq
1}(1+k)^{-\phi}N^{-(k-1)}\leq N^{-1}C_{\phi}$. On the other hand, for pure
exponential coefficients (recall (3.11)),

(6.16)
$\left(\frac{c_{j-1}}{N^{j-1}}\right)^{-1}\sum_{l>j}\frac{c_{l-1}}{N^{l-1}}=\sum_{k\geq
1}\left(\frac{c}{N}\right)^{-k}=\frac{c}{N-c}.$

Hence, for both choices of coefficients we have the following:

(6.17) For $N\to\infty$ the quantities $h_{j}(N),r_{j}(N)$ are bounded from
above
---
and below by positive finite constants times the right-hand side of
(6.10) uniformly in $j\in\mathbb{N}$.

Picking $\eta=0$ ($k=0$) in (6.7), we obtain

(6.18)
$a_{t}^{\Omega_{N}}(0,0)=\sum_{j\in\mathbb{N}}(N-1)\frac{\exp[-h_{j}(N)t]}{N^{j}}.$

Since we are interested in the asymptotic behaviour of
$a_{t}^{\Omega_{N}}(0,0)$, only large values of $j$ are relevant and we can
estimate the sum in (6.18) by an integral. To do so, we change variables by
putting $s=h_{j}(N)$ and exploit (6.17). Take the logarithm to express $j$ in
terms of $s$, compute ${\rm d}s/{\rm d}j$, and extract the $t$-dependence.
This gives

(6.19) $\displaystyle\eqref{regvar}\quad\Longrightarrow\quad
a_{t}^{\Omega_{N}}(0,0)\asymp t^{-1}\log^{\phi}t,$
$\displaystyle\eqref{pureexp}\quad\Longrightarrow\quad
a_{t}^{\Omega_{N}}(0,0)\asymp t^{-1-\delta_{N,c}},$

where

(6.20) $\delta_{N,c}=\frac{\log c}{\log(N/c)}.$

####  6.3 Hierarchical clustering

In this section we prove Theorem 3.3 by substituting the results of Theorem
3.2 into the clustering criterion in (3.8).

Combining (3.9), (6.2)–(6.4) and (6.19)–(6.20), we find the following
clustering criterion for _fixed_ $N$ and infinite seed-bank:

  * •

Subject to (3.10), clustering prevails if and only if

(6.21) $-\phi\leq\alpha\leq 1.$

  * •

Subject to (3.11), clustering prevails if and only

(6.22) $\delta_{N,c}\leq-\frac{1-\gamma_{N,K,e}}{\gamma_{N,K,e}}.$

In view of (6.2) and (3.14), the condition in (6.22) amounts to

(6.23) $\log N\times\log(Kc)\leq\log c\times\log(K^{2}e),$

where we use that $c<N$ and $Ke<N$ (recall (2.7) and (2.12)). The condition in
(6.23) holds for all $N$ when

(6.24) $Kc=1\text{ with
}\left\\{\begin{array}[]{ll}c=1,&K^{2}e\in(0,\infty),\\\ c>1,&K^{2}e\geq 1,\\\
c<1,&K^{2}e\leq 1.\end{array}\right.$

It also holds for $N$ large enough when $Kc<1$ and fails for $N$ large enough
when $Kc>1$. Thus, for infinite seed-bank, clustering prevails for $N$ large
enough if and only if

(6.25) $Kc\leq 1\leq K,$

which is the analogue of (6.21).

###  7 Preparation: $N\to\infty$, McKean-Vlasov process and mean-field system

To analyse the scaling of our hierarchical system in the hierarchichal mean-
field limit $N\to\infty$, we first need to understand simpler systems. In this
section we consider the mean-field system consisting of a _single hierarchy_ ,
and introduce the following:

  1. 1.

McKean-Vlasov process (Section 7.1).

  2. 2.

Mean-field system and McKean-Vlasov limit (Section 7.2).

For each we derive a key proposition that will play a crucial role in our
analysis of the truncated system with _finitely many hierarchies_ in Sections
10–12 and the full system with _infinitely many hierarchies_ in Section 12.
The proofs of the propositions stated in this section will be given in
Sections 7.3 and 7.4.

####  7.1 McKean-Vlasov process

In this section we introduce the McKean-Vlasov process, which will play an
important role in our analysis of the mean-field system to be introduced in
Sections 7.2–8.1. (In the full system the effective process introduced in
(4.9) will be seen to be an example of a McKean-Vlasov process.)

For $g\in\mathcal{G}$ and $c,K,e\in(0,\infty)$, consider the single-colony
process

(7.1) $z(t)=(x(t),y(t))_{t\geq 0},$

taking values in $[0,1]^{2}$, with initial law $\mathcal{L}[(x(0),y(0))]=\mu$
and with components evolving according to

(7.2) $\displaystyle{\rm d}x(t)=c\,[\mathbb{E}[x(t)]-x(t)]\,{\rm
d}t+\sqrt{g(x(t))}\,{\rm d}w(t)+Ke\,[y(t)-x(t)]\,{\rm d}t,$ $\displaystyle{\rm
d}y(t)=e\,[x(t)-y(t)]\,{\rm d}t,$

where $\mathbb{E}$ denotes expectation with respect to $\mu$. With the help of
Itô-calculus we can compute the expectation $\mathbb{E}[x(t)]$. Indeed, from
(7.2) we get

(7.3) $\displaystyle\frac{{\rm d}}{{\rm d}t}\mathbb{E}[x(t)]$
$\displaystyle=Ke\,\big{[}\mathbb{E}[y(t)]-\mathbb{E}[x(t)]\big{]},$
$\displaystyle\frac{{\rm d}}{{\rm d}t}\mathbb{E}[y(t)]$
$\displaystyle=e\,\big{[}\mathbb{E}[x(t)]-\mathbb{E}[y(t)]\big{]}.$

Define

(7.4)
$\theta_{x}=\mathbb{E}^{\mu}[x(0)],\qquad\theta_{y}=\mathbb{E}^{\mu}[y(0)],\qquad\theta=\mathbb{E}^{\mu}\left[\frac{x(0)+Ky(0)}{1+K}\right].$

Note that (7.3) implies that $\theta$ is a preserved quantity, i.e.,

(7.5)
$\mathbb{E}^{\mu}\left[\frac{x(0)+Ky(0)}{1+K}\right]=\mathbb{E}^{\mu}\left[\frac{x(t)+Ky(t)}{1+K}\right]=\theta,\qquad
t\geq 0.$

Solving (7.3), we find

(7.6) $\displaystyle\mathbb{E}[x(t)]$
$\displaystyle=\theta+\frac{K}{1+K}(\theta_{x}-\theta_{y})\,{\rm
e}^{-(K+1)et},$ $\displaystyle\mathbb{E}[y(t)]$
$\displaystyle=\theta-\frac{1}{1+K}(\theta_{x}-\theta_{y})\,{\rm
e}^{-(K+1)et}.$

In particular, from (4.51) we see that

(7.7) $\lim_{t\to\infty}(\mathbb{E}[x(t)],\mathbb{E}[y(t)])=(\theta,\theta).$

Hence, in equilibrium we can replace $\mathbb{E}[x(t)]$ in (7.2) by $\theta$.
After inserting (7.6) into (7.2), we can use [YW71, Theorem 1, Remark on
p.156] to show that for every deterministic initial state
$(x(0),y(0))\in[0,1]^{2}$ the SSDE in (7.2) has a unique strong solution. We
will refer to this solution as the _McKean-Vlasov process_.

######  Remark 7.1.

[Self-consistency] To prove uniqueness of the solution to (7.2) we can also
use [Gar88], where self-consistent mean-field dynamics are treated in detail.
The solution has the Feller property. $\blacksquare$

######  Proposition 7.2.

[Single-colony McKean-Vlasov process: equilibrium] For every initial law
$\mu\in\mathcal{P}([0,1]^{2})$ satisfying

(7.8)
$\mathbb{E}^{\mu}\left[\frac{x(0)+Ky(0)}{1+K}\right]=\theta,\qquad\theta\in[0,1],$

the process in (7.1) converges to a unique equilibrium,

(7.9) $\lim_{t\to\infty}\mathcal{L}[(x(t),y(t))]=\Gamma_{\theta},$

and

(7.10) $\Gamma_{\theta}\in\mathcal{P}([0,1]^{2}),$

satisfies

(7.11) $\theta=\int_{[0,1]^{2}}x\,\Gamma_{\theta}({\rm d}x,{\rm
d}y)=\int_{[0,1]^{2}}y\,\Gamma_{\theta}({\rm d}x,{\rm d}y).$

The proof of Proposition 7.2 is given in Section 7.3. Note that
$\Gamma_{\theta}=\Gamma_{\theta}^{g,c,K,e}$ depends on all the parameters
appearing in (7.2). In Section 8 we will see that $\Gamma_{\theta}$ is
continuous as a function of $\theta$.

######  Remark 7.3.

[Non-linear Markov process] Note that (7.1) is a _non-linear_ Markov process:
the evolution not only depends on the current state $z(t)$, but also on the
current law $\mathcal{L}[z(t)]$ via the expectation $\mathbb{E}[x(t)]$
appearing in the SSDE (7.2). This is different from the model without seed-
bank, where the non-linearity is replaced by a drift towards $\theta$ that is
constant in time. In equilibrium we can replace $\mathbb{E}[x(t)]$ by $\theta$
in (7.2), but before equilibrium is reached we cannot, because
$t\mapsto\mathbb{E}[x(t)]$ is not constant, as is clear from (4.51). Note that
$\mathbb{E}[x(t)]$ is a linear functional of $z(0)$. This fact will play an
important role in the renormalisation analysis in Section 13. $\blacksquare$

####  7.2 Mean-field system and McKean-Vlasov limit

In this section we consider a simplified version of the SSDE in (2.20),
namely, we restrict to the finite geographic space

(7.12) $=\\{0,1,\ldots,N-1\\},\qquad N\in\mathbb{N}.$

In this simplified version, the migration kernel $a^{\Omega_{N}}(\cdot,\cdot)$
is replaced by $a^{[N]}(\xi,\eta)=cN^{-1}$ for all $(\xi,\eta)\in[N]$, where
$c\in(0,\infty)$ is a constant. The seed-bank consists of only _one colour_
and the exchange rates between active and dormant are given by $Ke,e$. The
state space is

(7.13) $S=\mathfrak{s}^{[N]},\qquad\mathfrak{s}=[0,1]^{2},$

the system is denoted by

(7.14) $Z^{[N]}(t)=\big{(}X^{[N]}(t),Y^{[N]}(t)\big{)}_{t\geq
0},\qquad\big{(}X^{[N]}(t),Y^{[N]}(t)\big{)}=\big{(}x^{[N]}_{i}(t),y^{[N]}_{i}(t)\big{)}_{i\in[N]},$

and its components evolve according to the SSDE

(7.15) $\displaystyle{\rm
d}x^{[N]}_{i}(t)=\frac{c}{N}\sum_{j\in[N]}\big{[}x^{[N]}_{j}(t)-x^{[N]}_{i}(t)\big{]}\,{\rm
d}t+\sqrt{g\big{(}x^{[N]}_{i}(t)\big{)}}\,{\rm
d}w_{i}(t)+Ke\,\big{[}y^{[N]}_{i}(t)-x^{[N]}_{i}(t)\big{]}\,{\rm d}t,$
$\displaystyle{\rm
d}y^{[N]}_{i}(t)=e\,\big{[}x^{[N]}_{i}(t)-y^{[N]}_{i}(t)\big{]}\,{\rm
d}t,\qquad i\in[N],$

which is the special case of (2.20) obtained by setting
$a^{\Omega_{N}}(\eta,\xi)=0$ if $d(\eta,\xi)>1$ and $K_{m}=e_{m}=0$ for $m\geq
1$. It is natural to take an _exchangeable random initial state_ , because the
evolution preserves exchangeability. According to De Finetti’s theorem, there
is no loss of generality in taking an i.i.d. initial state, i.e.,

(7.16)
$\mathcal{L}\big{[}X^{[N]}(0),Y^{[N]}(0)\big{]}=\mu^{\otimes[N]},\qquad\qquad\mu\in\mathcal{P}\left([0,1]^{2}\right).$

By [SS80, Theorem 3.1], the SSDE in (7.15) is the unique weak solution of a
well-posed martingale problem. By [SS80, Theorem 3.2], for every deterministic
initial state $(X^{[N]}(0),Y^{[N]}(0))$, (7.15) has a unique strong solution.
We are interested in the limit $N\to\infty$. For the limiting process we
define

(7.17) $(Z(t))_{t\geq 0}=(X(t),Y(t))_{t\geq
0}=\big{(}(x_{i}(t),y_{i}(t))_{i\in\mathbb{N}_{0}}\big{)}_{t\geq 0}$

with components evolving according to (7.2), i.e.,

(7.18) $\displaystyle{\rm d}x_{i}(t)=c\,[\mathbb{E}[x_{i}(t)]-x_{i}(t)]\,{\rm
d}t+\sqrt{g(x_{i}(t))}\,{\rm d}w(t)+Ke\,[y_{i}(t)-x_{i}(t)]\,{\rm d}t,$
$\displaystyle{\rm d}y_{i}(t)=e\,[x_{i}(t)-y_{i}(t)]\,{\rm d}t,\qquad
i\in\mathbb{N}_{0},$

with $\mathcal{L}\left[(X(0),Y(0))\right]=\mu$ for some exchangeable
$\mu\in\mathcal{P}(([0,1]^{2})^{\otimes[\mathbb{N}_{0}]}$. Note that (7.18)
consists of i.i.d. copies of the single-colony McKean-Vlasov process in (7.1),
labelled by $i\in\mathbb{N}_{0}$.

######  Proposition 7.4.

[Infinite-system McKean-Vlasov limit: convergence]
Suppose that $\mathcal{L}[(X^{[N]}(0),Y^{[N]}(0))]=\mu^{[N]}$ is exchangeable
and

(7.19) $\theta=\mathbb{E}^{\mu^{[N]}}\left[\frac{x(0)+Ky(0)}{1+K}\right].$

Then

(7.20)
$\lim_{N\to\infty}\mathcal{L}\Big{[}\big{(}X^{[N]}(t),Y^{[N]}(t)\big{)}_{t\geq
0}\Big{]}=\mathcal{L}\big{[}(X(t),Y(t))_{t\geq 0}\big{]}$

with

(7.21) $\mathcal{L}\big{[}(X(0),Y(0))_{t\geq
0}\big{]}=\mu,\qquad\mu=\lim_{N\to\infty}\mu^{[N]},$

where the limit is the McKean-Vlasov process in (7.1)–(7.2).

The proof of Proposition 7.4 is given in Section 7.4. For the system without
seed-bank the McKean-Vlasov limit was proved in [Gar88]. The fact that the
components decouple is a property referred to as _propagation of chaos_.

####  7.3 Proof of equilibrium and ergodicity

In this section we prove Proposition 7.2.

###### Proof.

Note that, by (4.51), we can rewrite (7.2) as

(7.22) $\displaystyle{\rm
d}x(t)=c\,\left[\theta+\frac{K}{1+K}(\theta_{x}-\theta_{y})\,{\rm
e}^{-(K+1)et}-x(t)\right]\,{\rm d}t+\sqrt{g(x(t))}\,{\rm d}w(t)$
$\displaystyle\qquad\qquad\qquad+Ke\,[y(t)-x(t)]\,{\rm d}t,$
$\displaystyle{\rm d}y(t)=e\,[x(t)-y(t)]\,{\rm d}t.$

Existence and uniqueness of a strong solution is again standard (see e.g.
[YW71, Theorem 1] and recall Remark 7.1). We start by proving existence and
uniqueness of the equilibrium. Afterwards we show that the solution converges
to this equilibrium.

Consider two copies $(x_{1},y_{1})$ and $(x_{2},y_{2})$ of the system defined
in (LABEL:gh554), with $\mathcal{L}[(x_{1}(0),y_{1}(0))]=\mu_{1}$ and
$\mathcal{L}[(x_{2}(0),y_{2}(0))]=\mu_{2}$, where $\mu_{1}$ and $\mu_{2}$
satisfy

(7.23)
$\mathbb{E}^{\mu_{1}}\left[\frac{x_{1}(0)+Ky_{1}(0)}{1+K}\right]=\theta=\mathbb{E}^{\mu_{2}}\left[\frac{x_{2}(0)+Ky_{2}(0)}{1+K}\right]$

for some $\theta\in[0,1]$. Write

(7.24)
$\theta_{x_{1}}=\mathbb{E}^{\mu_{1}}[x_{1}(0)],\quad\theta_{y_{1}}=\mathbb{E}^{\mu_{1}}[y_{1}(0)],\quad\theta_{x_{2}}=\mathbb{E}^{\mu_{2}}[x_{2}(0)],\quad\theta_{y_{2}}=\mathbb{E}^{\mu_{2}}[y_{2}(0)].$

Couple the two systems by coupling their Brownian motions. Denote the coupled
process by

(7.25) $\displaystyle(\bar{z}(t))_{t\geq 0}=(z_{1}(t),z_{2}(t))_{t\geq
0},\quad z_{1}(t)=(x_{1}(t),y_{1}(t)),\quad z_{2}(t)=(x_{2}(t),y_{2}(t)),$
$\displaystyle\mathcal{L}(\bar{z}(0))=\mu_{1}\times\mu_{2},$

which has a unique strong solution. Put

(7.26) $\Delta(t)=x_{1}(t)-x_{2}(t),\qquad\delta(t)=y_{1}(t)-y_{2}(t).$

To show that the equilibrium is unique, it is enough to show that

(7.27) $\lim_{t\to\infty}\mathbb{E}\left[|\Delta(t)|+EK|\delta(t)|\right]=0.$

Using a generalised form of Itô’s formula, we find

(7.28) $\displaystyle{\rm d}|\Delta(t)|$ $\displaystyle=({\rm
sign}\,\Delta(t))\,{\rm d}\Delta(t)+{\rm d}L_{t}^{0}$ $\displaystyle=({\rm
sign}\,\Delta(t))\,c\,\left[\frac{K}{1+K}\big{(}(\theta_{x_{1}}-\theta_{x_{2}})-(\theta_{y_{1}}-\theta_{y_{2}})\big{)}\,{\rm
e}^{-(K+1)et}-\Delta(t)\right]\,{\rm d}t$ $\displaystyle\qquad+({\rm
sign}\,\Delta(t))\,\left(\sqrt{g(x_{1}(t))}-\sqrt{g(x_{2}(t))}\,\right){\rm
d}w(t)$ $\displaystyle\qquad+({\rm
sign}\,\Delta(t))\,Ke\,[\delta(t)-\Delta(t)]\,{\rm d}t,$

where we use that the local time $L_{t}^{0}$ (see [RW00, Section IV.43]) of
$\Delta(t)$ at $0$ equals $0$, since $g$ is Lipschitz (see [RW00, Proposition
V.39.3]). Again using Itô’s formula, we also find

(7.29) ${\rm d}|\delta(t)|=({\rm sign}\,\delta(t))\,{\rm d}\delta(t)=({\rm
sign}\,\delta(t))\,e\,[\Delta(t)-\delta(t)]\,{\rm d}t.$

Taking expectations in (7.28)–(7.29), we get

(7.30) $\displaystyle\frac{{\rm d}}{{\rm
d}t}\mathbb{E}[|\Delta(t)|+K|\delta(t)|]$
$\displaystyle=\mathbb{E}\left[c\,\left[({\rm
sign}\,\Delta(t))\frac{K}{1+K}\big{(}(\theta_{x_{1}}-\theta_{x_{2}})-(\theta_{y_{1}}-\theta_{y_{2}})\big{)}\,{\rm
e}^{-(K+1)et}-\left|\Delta(t)\right|\right]\right]$
$\displaystyle\quad+Ke\,\mathbb{E}\Big{[}\left({\rm sign}\,\Delta(t)-{\rm
sign}\,\delta(t)\right)(\delta(t)-\Delta(t))\Big{]}$
$\displaystyle=\mathbb{E}\left[c\,({\rm
sign}\,\Delta(t))\frac{K}{1+K}\big{(}(\theta_{x_{1}}-\theta_{x_{2}})-(\theta_{y_{1}}-\theta_{y_{2}})\big{)}\,{\rm
e}^{-(K+1)et}\right]$
$\displaystyle\quad-c\,\mathbb{E}[\left|\Delta(t)\right|]$
$\displaystyle\quad-2Ke\,\mathbb{E}\left[1_{\\{{\rm sign}\,\delta(t)\neq{\rm
sign}\Delta(t)\\}}\left(|\delta(t)|+|\Delta(t)|\right)\right].$

Define

(7.31)
$h(t)=c\,\mathbb{E}[\left|\Delta(t)\right|]+2Ke\,\mathbb{E}\left[1_{\\{{\rm
sign}\,\delta(t))\neq{\rm
sign}\,\Delta(t)\\}}\left(|\delta(t)|+|\Delta(t)|\right)\right].$

Then $h(t)$ satisfies

  1. 1.

$h(t)>0$.

  2. 2.

$0\leq\int_{0}^{\infty}{\rm d}t\,h(t)\leq
1+K+c\left|(\theta_{x_{1}}-\theta_{x_{2}})-(\theta_{y_{1}}-\theta_{y_{2}})\right|\frac{K}{K+1}\frac{1}{e(K+1)}\left[1-{\rm
e}^{-(K+1)et}\right]$.

  3. 3.

$h$ is differentiable with $h^{\prime}$ bounded (see [GHO20, Appendix D]).

Hence it follows that $\lim_{t\to\infty}h(t)=0$, which implies that

(7.32) $\lim_{t\to\infty}\mathbb{E}\left[|\Delta(t)|\right]=0.$

We are left to prove that $\lim_{t\to\infty}\mathbb{E}[|\delta(t)|]=0$. To do
so, we define

(7.33) $f(t)=\mathbb{E}[|\delta(t)|],\qquad G(t)=e\,\mathbb{E}[({\rm
sign}\,\delta(t))\Delta(t)].$

Note that $G$ is bounded and continuous. Taking expectations in (7.29), we
find

(7.34) $\frac{{\rm d}}{{\rm d}t}f(t)=-ef(t)+G(t),$

Solving (7.34) explicitly, we find that

(7.35) $f(t)=f(r)\,{\rm e}^{-e(t-r)}+\int_{r}^{t}{\rm d}s\,{\rm
e}^{-e(t-s)}G(s),\qquad r,t\in\mathbb{R},\,t>r\geq 0.$

By (7.32), for each $\epsilon>0$ we can find an $r\in\mathbb{R}$ such that
$\mathbb{E}[|\Delta(s)|]<\epsilon$ for all $s>r$, and hence
$\sup_{t>r}|G(t)|<\epsilon$. Therefore

(7.36) $f(t)\leq f(r)\,{\rm e}^{-e(t-r)}+\epsilon$

and, since $|f|<1$, we find, for each $\epsilon>0$,

(7.37) $\lim_{t\to\infty}f(t)<\epsilon.$

Therefore $\lim_{t\to\infty}\mathbb{E}[|\delta(t)|]=0$, which completes the
proof of uniqueness of the equilibrium for given $\theta$.

To prove existence of the equilibrium, let $(t_{n})_{n\in\mathbb{N}}$ be any
increasing sequenc eof times such that $\lim_{n\to\infty}t_{n}=\infty$. Let
$\mu=\mathcal{L}[(x(0),y(0))]$ be any initial measure of the system in
(LABEL:gh554) with $\mathbb{E}^{\mu}[\frac{x(0)+Ky(0)}{1+K}]=\theta$, and let
$\mu(t_{n})=\mathcal{L}[(x(t_{n}),y(t_{n}))]$. Since the state space is
compact, the sequence $(\mu(t_{n}))_{n\in\mathbb{N}}$ is tight, and by
Prohorov’s theorem we can find a converging subsequence
$(\mu(t_{n_{k}}))_{k\in\mathbb{N}}$. Put
$\nu=\lim_{k\to\infty}\mu(t_{n_{k}})$. We will show that $\nu$ is invariant.
To that end, recall from Section 7.1 that

(7.38) $\mathbb{E}^{\mu}\left[\frac{x(t)+Ky(t)}{1+K}\right]=\theta,\qquad
t\geq 0.$

Hence we can use the coupling in (7.25) to show that the system starting in
$\mu$ and the system starting $\mu(t)$ converge to the same law as
$t\to\infty$, from which it follows that
$\lim_{k\to\infty}\mu(t+t_{n_{k}})=\nu$. Let $(S_{t})_{t\geq 0}$ denote the
semigroup of the system in (LABEL:gh554). By the Feller property for
semigroups,

(7.39)
$S_{t}\nu=\lim_{k\to\infty}S_{t}\mu(t_{n_{k}})=\lim_{k\to\infty}S_{t_{n_{k}}}\left(S_{t}\mu\right)=\nu,$

where in the last equality we use the uniqueness of the equilibrium given
$\theta$. Thus, $\nu$ is an invariant measure. To exhibit its dependence on
$\theta$ we write $\nu_{\theta}$. Using the same coupling as in (7.25), and
starting from $\mu\times\nu_{\theta}$ with $\nu_{\theta}$ the invariant
measure just obtained, we see that for every $\theta$ the system in (7.2)
converges to a unique equilibrium measure $\nu_{\theta}$, and so (7.11) is
immediate from (7.38). ∎

####  7.4 Proof of McKean-Vlasov limit

In this section we give a sketch of the proof of Proposition 7.4. In Chapters
8-12 we encounter more difficult versions of Proposition 7.4. There we will
give the proofs in full detail.

###### Proof.

Since we start from a distribution $\mu(0)$ that is exchangeable, Aldous’s
ergodic theorem gives

(7.40)
$\lim_{N\to\infty}\frac{1}{N}\sum_{j\in[N]}x_{j}(0)=\mathbb{E}^{\mu(0)}[x_{0}]\quad\mathbb{P}\text{-a.s.}$

By Ioffe’s theorem [DGV95, Eqs. (1.1)–(1.2)], tightness of the associated
sequence of processes (uniformly on the state space) follows from boundedness
of the generator as an operator. To apply the generator criterion in [JM86] we
must show propagation of chaos and prove the weak law of large numbers

(7.41)
${}\lim_{N\to\infty}\frac{1}{N}\sum_{j\in[N]}x_{j}(t)=\mathbb{E}[x_{0}(t)].$

The propagation of chaos and the weak law of large numbers for $t>0$ therefore
follows from [Gar88, Section 4]. Since the martingale problem is well-posed
[Gar88, Section 2], the limiting process exists and is unique. ∎

###  8 Proofs: $N\to\infty$, mean-field finite-systems scheme

In Sections 8.1 we introduce the so called mean-field finite-systems scheme
for the mean-field system introduced in Section 7.2. In Section 8.2 we outline
the _abstract scheme_ behind the proof behind the mean-field finite-systems
scheme. The computations in the proof of the abstract scheme are long and
technical, and are deferred to Section 9.

####  8.1 Mean-field finite-systems scheme

In this section we describe the limiting dynamics of the finite system in
(7.14) from a _multiple space-time scale_ viewpoint. To do so, we need the
following limiting SSDE for the infinite system
$Z(t)=(z_{i}(t))_{i\in\mathbb{N}_{0}}=\left(x_{i}(t),y_{i}(t)\right)_{i\in\mathbb{N}_{0}}$,
with initial law $\mathcal{L}[Z(0)]=\mu^{\otimes\mathbb{N}_{0}}$, evolving
according to

(8.1) $\displaystyle{\rm d}x_{i}(t)=c\,[\theta-x_{i}(t)]\,{\rm
d}t+\sqrt{g(x_{i}(t))}\,{\rm d}w_{i}(t)+Ke\,[y_{i}(t)-x_{i}(t)]\,{\rm d}t,$
$\displaystyle{\rm d}y_{i}(t)=e\,[x_{i}(t)-y_{i}(t)]\,{\rm d}t,\
i\in\mathbb{N}_{0},$

where $\theta$ is defined in (7.11). Note that each component of (8.1) is an
autonomous copy of the McKean-Vlasov process in (7.2) in equilibrium.

For the multiscale analysis we will need the following ingredients:

  1. 1.

The _estimator_ for the finite system is defined by

(8.2)
$\bar{\Theta}^{[N]}(t)=\bar{\Theta}^{[N]}\big{(}Z^{[N]}(t)\big{)}=\frac{1}{N}\sum_{i\in[N]}\frac{x^{[N]}_{i}(t)+Ky_{i}^{[N]}(t)}{1+K}$

and its active and dormant counterparts

(8.3) $\displaystyle\bar{\Theta}_{x}^{[N]}(t)$
$\displaystyle=\frac{1}{N}\sum_{i\in[N]}x^{[N]}_{i}(t),$
$\displaystyle\bar{\Theta}_{y}^{[N]}(t)$
$\displaystyle=\frac{1}{N}\sum_{i\in[N]}y^{[N]}_{i}(t).$

  2. 2.

The _time scale_ $N$, on which
$\lim_{N\to\infty}\mathcal{L}[\bar{\Theta}^{[N]}(L(N))-\bar{\Theta}^{[N]}(0)]=\delta_{0}$
for all $L(N)$ such that $\lim_{N\to\infty}L(N)=\infty$ and
$\lim_{N\to\infty}L(N)/N=0$, but not for $L(N)=N$. In words, $N$ is the time
scale on which $\bar{\Theta}^{[N]}(\cdot)$ starts evolving, i.e.,
$\left(\bar{\Theta}^{[N]}(Ns)\right)_{s>0}$ is not a fixed process. When we
scale time by $N$, putting $t=Ns$, we view $s$ as the “fast time scale” and
$t$ as the “slow time scale”.

  3. 3.

The _invariant measure_ , i.e., the equilibrium measure of a single component
in (8.1) written

(8.4) $\Gamma_{\theta},$

and the _invariant measure_ of the infinite system in (8.1), written
$\nu_{\theta}=\Gamma_{\theta}^{\otimes\mathbb{N}_{0}}$, with $\theta\in[0,1]$
controlled by the initial measure (recall (7.4)–(4.51)).

  4. 4.

The _renormalisation transformation_
$\mathcal{F}\colon\,\mathcal{G}\to\mathcal{G}$,

(8.5) $(\mathcal{F}g)(\theta)=\int_{[0,1]^{2}}g(x)\,\nu_{\theta}({\rm d}x,{\rm
d}y_{0}),\quad\theta\in[0,1],$

where $\nu_{\theta}$ is the equilibrium measure of 8.1. Note that
$\mathcal{F}$ is the same transformation as defined in (4.16), but for the
truncated system. Note that we can also write

(8.6) $(\mathcal{F}g)(\theta)=\int_{[0,1]^{2}}g(x)\,\Gamma_{\theta}({\rm
d}x,{\rm d}y_{0}),\quad\theta\in[0,1],$

where $\Gamma_{\theta}$ is as defined in (8.4).

  5. 5.

The _macroscopic observable_ $\left(\bar{\Theta}(s)\right)_{s>0}$ satisfying
the SSDE

(8.7) ${\rm
d}\bar{\Theta}(s)=\frac{1}{1+K}\sqrt{\mathbb{E}^{\Gamma_{\bar{\Theta}(s)}}[g(u)]}\,{\rm
d}w(s)=\frac{1}{1+K}\sqrt{(\mathcal{F}g)(\bar{\Theta}(s))}\,{\rm d}w(s),$

To obtain the multi-scale limit dynamics for the system in (7.14), we speed up
time by a factor $N$ and define the process

(8.8)
$\left(x_{1}^{[N]}(s),y_{1}^{[N]}(s)\right)_{s>0}=\left({\Theta}^{[N]}_{x}(Ns),{\Theta}^{[N]}_{y}(Ns)\right)_{s>0},$

which is the analogue of the $1$-block average in (2.22). We use the lower
index $1$ to indicate that the average is taken over $[N]$ components. Using
(7.15), we see that the dynamics of (8.8) is given by the SSDE

(8.9) $\displaystyle{\rm d}x_{1}^{[N]}(s)$
$\displaystyle=\sqrt{\frac{1}{N}\sum_{i\in[N]}g\big{(}x_{i}(Ns)\big{)}}\,{\rm
d}w(s)+NKe\left[y_{1}^{[N]}(s)-x_{1}^{[N]}(s)\right]{\rm d}s,$
$\displaystyle{\rm d}y_{1}^{[N]}(s)$
$\displaystyle=Ne\left[x_{1}^{[N]}(s)-y_{1}^{[N]}(s)\right]{\rm d}s.$

$\dots$$0$$1$$N-2$$N-1$block average Figure 12: Given the value of the block
average, the $N\gg 1$ constituent components equilibrate on a time scale that
is fast with respect to the time scale on which the block average fluctuates.
Consequently, the volatility of the block average is the expectation of the
volatility of the constituent components under the conditional quasi-
equilibrium.

In (8.9), in the limit as $N\to\infty$ infinite rates appear in the exchange
between the active and the dormant population. However, looking at the process

(8.10)
$\left(\frac{x_{1}^{[N]}(s)+Ky_{1}^{[N]}(s)}{1+K}\right)_{s>0}=\left(\bar{\Theta}^{[N]}(Ns)\right)_{s>0}$

we see that the terms carrying a factor $N$ in front cancel out. Consequently,
for the process in (8.10) we can use ideas from [DG93a] to prove tightness as
$N\to\infty$ in the _classical topology_ of continuum path processes. We will
show in Section 9.3 that

(8.11)
$\displaystyle\lim_{N\to\infty}\mathcal{L}\left[\left(\left[x_{1}^{[N]}(s)-y_{1}^{[N]}(s)\right]\right)_{s\geq
0}\right]=\mathcal{L}\left[(0)_{s\geq 0}\right]$ $\displaystyle\mbox{ in the
\emph{Meyer-Zheng topology}}.$

Combining (8.10) and (8.11), we obtain the multiple space-time scaling
behaviour of the system in (7.14).

######  Proposition 8.1.

[Mean-field: finite-systems scheme] Suppose that the SSDE in (7.15) has
initial measure $\mathcal{L}[Z^{[N]}(0)]=\mu^{\otimes[N]}$ for some
$\mu\in\mathcal{P}\left([0,1]^{2}\right)$. Let

(8.12) $\theta=\mathbb{E}^{\mu}\left[\frac{x+Ky_{0}}{1+K}\right].$

  * (a)

For the averages in (8.8),

(8.13)
$\displaystyle\lim_{N\to\infty}\mathcal{L}\left[\left(x_{1}^{[N]}(s),y_{0,1}^{[N]}(s)\right)_{s>0}\right]=\mathcal{L}\left[\left(x_{1}^{\mathbb{N}_{0}}(s),y_{0,1}^{\mathbb{N}_{0}}(s)\right)_{s>0}\right]$
$\displaystyle\text{in the Meyer-Zheng topology},$

where the limit process is the unique solution of the SSDE

(8.14) $\displaystyle{\rm d}x_{1}^{\mathbb{N}_{0}}(s)$
$\displaystyle=\frac{1}{1+K}\sqrt{(\mathcal{F}g)\big{(}x_{1}^{\mathbb{N}_{0}}(s)\big{)}}\,{\rm
d}w(s),$ $\displaystyle y_{0,1}^{\mathbb{N}_{0}}(s)$
$\displaystyle=x_{1}^{\mathbb{N}_{0}}(s),$

with initial state

(8.15)
$\left(x_{1}^{\mathbb{N}_{0}}(0),y_{0,1}^{\mathbb{N}_{0}}(0)\right)=\left(\theta,\theta\right).$

  * (b)

For the weighted sum of the averages in (8.10),

(8.16)
$\lim_{N\to\infty}\mathcal{L}\left[\left(\bar{\Theta}^{[N]}(Ns)\right)_{s>0}\right]=\mathcal{L}\left[\left(\bar{\Theta}(s)\right)_{s>0}\right],$

where the limit is the macroscopic observable in (8.7) with initial state

(8.17) $\bar{\Theta}(0)=\theta.$

  * (c)

Define

(8.18) $\nu_{\theta}(s)=\int_{[0,1]}Q_{s}\bigl{(}\theta,{\rm
d}\theta^{\prime}\bigr{)}\,\nu_{\theta^{\prime}}\in\mathcal{P}([0,1]^{2}),$

where $Q_{s}(\theta,\cdot)$ is the time-$s$ marginal law of the process
$(\bar{\Theta}(s))_{s>0}$ starting from $\theta\in[0,1]$ (note that
$\nu_{\theta}(0)=\nu_{\theta}$). Then, for every $s\in(0,\infty)$,

(8.19)
$\displaystyle\lim_{N\to\infty}\mathcal{L}\left[\left(X^{[N]}(Ns+t),Y^{[N]}(Ns+t)\right)_{t>0}\right]=\mathcal{L}\left[(Z^{\nu_{\theta}(s)}(t))_{t>0}\right]$

where, conditional on $\bar{\Theta}(s)=\theta$,
$(z^{\nu_{\theta}(s)}(t))_{t\geq 0}$ is the random process in (8.1) and
$z^{\nu_{\theta}(s)}(0)$ is drawn according to $\nu_{\theta}(s)$ (which is a
mixture of random processes in equilibrium).

The proof of Proposition 8.1 is given in Section 8.2.

The result in Part (a) shows that the limit dynamics of the averages follows a
similar type of diffusion as a single colony, but with four important changes:

  * •

For the limit of the time-scaled average in (8.10) the diffusion function $g$
is replaced by a _renormalised diffusion function_ $\mathcal{F}g$, defined by
(8.5) (recall Fig. 12). In section 8.2 we will show that
$\mathcal{F}\mathcal{G}\subset\mathcal{G}$, i.e., $\mathcal{F}$ preserves the
class of diffusion functions defined in (2.15).

  * •

The average of the dormant population is the same as the average of the active
population, and hence the term that accounts for the exchange between the
active and the dormant population _vanishes_. This happens because when time
is speeded up by a factor $N$ also the rates of exchange between active and
dormant are speeded up by a factor $N$ (see (8.9)). Hence the exchange rates
become infinitely large, which implies that the active and the dormant
population equilibrate instantly in the Meyer-Zheng topology.

  * •

Since we take the average over all the components, the migration terms in
(7.15) cancel out against each other.

  * •

Comparing the system in (7.14) with the system of interacting Fisher-Wright
diffusions in the mean-field limit studied in [DG93b], we see from (8.14) that
the single-colour seed-bank _slows down the average_ by a factor $1/(1+K)$,
but does not change the system qualitatively. This is a direct consequence of
the fact that the averages of the active and the dormant population
equilibrate (due to the infinite rates), while only individuals in the active
part of the population resample.

The result in Part (b) shows that the limit dynamics of the averages in (8.10)
follows an autonomous SDE, with convergence in the classical topology, i.e.,
in $C_{b}([0,\infty),[0,1])$. The Brownian motion in (7.1) is taken to be
independent of the initial state. The result in Part (c) says that, on time
scale $1$ and starting from time $Ns$ with $N\to\infty$, the system has a
_McKean-Vlasov limit_ , i.e., exhibits propagation of chaos, with components
that are versions of a McKean-Vlasov process with a random initial state whose
law depends on $s$. So, in particular, the components become independent, and
we see _decoupling_ The proof of Part (c) will use Part (b). The proof of Part
(a) will follow from Part (b) after we use the Meyer-Zheng topology.

######  Remark 8.2.

[Basic multi-scale] Note that Proposition 8.1 already reveals several
phenomena that we encountered in Theorems 4.2 and 4.4, capturing the
hierarchical multiscale behaviour. Even for the one-layer mean-field system we
find decoupling of components, the occurrence of a renormalisation
transformation, equalisation of the seed-bank with the active population, and
the need for the Meyer-Zheng topology. Later we will see that the role of the
macroscopic observable $\bar{\Theta}$ is the same as that of the effective
process. $\blacksquare$

######  Remark 8.3.

[Interchange of limits] The notation
$x_{1}^{\mathbb{N}_{0}},y_{0,1}^{\mathbb{N}_{0}}$ indicates that the limit
arises from taking averages over $[N]$ and letting $N\to\infty$. Note that,
for i.i.d. initial states,

(8.20)
$x_{1}^{\mathbb{N}_{0}}(0)=\lim_{N\to\infty}x_{1}^{[N]}(0)=\lim_{N\to\infty}\frac{1}{N}\sum_{i\in[N]}x_{i}(0)=\theta_{\bar{\Theta}}\qquad\mathbb{P}\text{-a.s.}$

On the other hand, picking any sequence of times $L(N)$ such that
$\lim_{N\to\infty}L(N)=\infty$ and $\lim_{N\to\infty}L(N)/N=0$, we get

(8.21)
$x_{1}^{\mathbb{N}_{0}}(0+)=\lim_{N\to\infty}x_{1}^{[N]}\big{(}\tfrac{L(N)}{N}t\big{)}=\lim_{N\to\infty}\frac{1}{N}\sum_{i\in[N]}x_{i}(L(N)t)=\theta\qquad\mathbb{P}\text{-a.s.}$

The mismatch between (8.20) and (8.21) indicates that we must be careful with
interchanging the limits $N\to\infty$ and $s\downarrow 0$. This is why (8.13),
which lives on the fast time scale, is restricted to $s>0$.  $\blacksquare$

####  8.2 Abstract scheme behind finite-systems scheme

To prove Proposition 8.1, we follow the abstract scheme outlined in [DGV95, p.
2314–2315] and based on [DG93b], [DG93a]. Below we state the abstract scheme
for our model. The scheme consists of 4 steps, each of the steps consists of a
series of propositions and lemmas. The proofs of these are given in Section 9.

###### Step 1. Equilibrium of the single components.

This step fixes the one-dimensional distributions of the single components
when $t,N\to\infty$ in a combined way, and is the equivalent of [DG93b,
Proposition 1]. Recall that $\bar{\Theta}^{[N]}$ is defined in (8.2).

######  Proposition 8.4.

[Equilibrium for the infinite system] Let $(N_{k})_{k\in\mathbb{N}}$ be a
sequence in $\mathbb{N}$. Fix $s>0$. Let $L(N)$ be such that
$\lim_{N\to\infty}L(N)=\infty$ and $\lim_{N\to\infty}L(N)/N=0$, and suppose
that

(8.22)
$\displaystyle\lim_{k\to\infty}\mathcal{L}\left[\bar{\Theta}^{[N_{k}]}(N_{k}s)\right]=P_{s},$
$\displaystyle\lim_{k\to\infty}\mathcal{L}\left[\sup_{0\leq t\leq
L(N_{k})}\left|\bar{\Theta}^{[N_{k}]}(N_{k}s)-\bar{\Theta}^{[N_{k}]}(N_{k}s-t)\right|\right]=\delta_{0},$
$\displaystyle\lim_{k\to\infty}\mathcal{L}\bigl{(}X^{[N_{k}]}(N_{k}s),Y^{[N_{k}]}(N_{k}s)\bigr{)}=\nu(s).$

Then $\nu(s)$ is of the form

(8.23) $\nu(s)=\int_{[0,1]}P_{s}({\rm d}\theta)\,\nu_{\theta},$

where $\nu_{\theta}$ is the equilibrium measure of the process defined in
(8.1).

Proposition 8.4 follows from the following seven lemmas, which are the
analogues of the five lemmas used in [DG93b, p. 477–478] for the system
without seed-bank.

The first lemma establishes convergence of the infinite system in (8.1) to its
equilibrium.

######  Lemma 8.5.

[Convergence for the infinite system] Let $\mu$ be an exchangeable probability
measure on $([0,1]^{2})^{\mathbb{N}_{0}}$. Then for the system $(Z(t))_{t\geq
0}$ given by (8.1) with $\mathcal{L}(Z(0))=\mu$,

(8.24) $\lim_{t\to\infty}\mathcal{L}[Z(t)]=\nu_{\theta},$

where $\nu_{\theta}$ is of the form

(8.25) $\nu_{\theta}=\Gamma_{\theta}^{\otimes\mathbb{N}_{0}},$

with $\Gamma_{\theta}$ the equilibrium of the single-colony process defined in
(8.4). Moreover, $\nu_{\theta}$ is ergodic.

The second lemma establishes the continuity of the equilibrium with respect to
its center of drift $\theta$.

######  Lemma 8.6.

[Continuity of the equilibrium] Let $\mathcal{P}([0,1]^{\mathbb{N}_{0}})$
denote the space of probability measures on $[0,1]^{\mathbb{N}_{0}}$. The
mapping $[0,1]\to\mathcal{P}([0,1]^{\mathbb{N}_{0}})$ given by

(8.26) $\theta\mapsto\nu_{\theta}$

is continuous. Furthermore, if $h$ is a Lipschitz function on $[0,1]$, then
also $\mathcal{F}h$ defined by

(8.27)
$(\mathcal{F}h)(\theta)=\mathbb{E}^{\nu_{\theta}}[h(\cdot)]=\int_{([0,1]^{2})^{\mathbb{N}_{0}}}\nu_{\theta}({\rm
d}z)\,h(x_{0})$

is a Lipschitz function on $[0,1]$.

The third lemma characterises the speed at which the estimators
${{\Theta}}^{[N]}_{x}$ and ${{\Theta}}^{[N]}_{y}$ converge to each other.

######  Lemma 8.7.

[Comparison of empirical averages] Let $({{\Theta}}^{[N]}_{x}(t))_{t\geq 0}$
and $({\bar{\Theta}}^{[N]}_{y}(t))_{t\geq 0}$ be defined as in (8.3), and
define

(8.28) $={{\Theta}}^{[N]}_{x}(t)-{{\Theta}}^{[N]}_{y}(t).$

Then

(8.29)
$\mathbb{E}\left[\left|{\Delta}^{[N]}_{\bar{\Theta}}(t)\right|\right]\leq\sqrt{\mathbb{E}\left[\left({\Delta}^{[N]}_{\bar{\Theta}}(0)\right)^{2}\right]}\,{\rm
e}^{-(Ke+e)t}+\sqrt{\frac{\|g\|}{N(Ke+e)}}.$

######  Remark 8.8.

[Key estimate for Meyer-Zheng convergence] The estimate in (8.29) in Lemma 8.7
will be the key estimate to show convergence of the active and dormant
$1$-block in Meyer-Zheng topology. Note that if we look at times $Ns$ for
$s>0$, then (8.29) shows that
$\mathbb{E}[|{\Delta}^{[N]}_{\bar{\Theta}}(Ns)|]$ is
$\mathcal{O}(\sqrt{1/N})$.

The fourth lemma compares the finite system with an infinite system. To that
end we construct both the finite and the infinite system on the same state-
space by considering the finite system $(X^{[N]}(t),Y^{[N]}(t))$ as an element
of $([0,1]^{2})^{\mathbb{N}_{0}}$ via periodic continuation. Let $L(N)$ be
such that $\lim_{N\to\infty}L(N)=\infty$ and $\lim_{N\to\infty}L(N)/N=0$, and
define the distribution by continuing the configuration of
$(X^{[N]}(Ns-L(N)),Y^{[N]}(Ns-L(N)))$ periodically to
$([0,1]^{2})^{\mathbb{N}_{0}}$. Define

(8.30) $\bar{\Theta}^{[N]}=\bar{\Theta}^{[N]}(Ns-L(N)).$

Note that

(8.31) $\displaystyle\bar{\Theta}^{[N]}$
$\displaystyle=\frac{\frac{1}{N}\sum_{j\in[N]}x^{[N]}_{j}(Ns-L(N))+\frac{K}{N}\sum_{j\in[N]}y^{[N]}_{j}(Ns-L(N))}{1+K}$
$\displaystyle=\frac{1}{N}\sum_{j\in[N]}\frac{x_{j}^{\mu_{N}}(0)+Ky_{j}^{\mu_{N}}(0)}{1+K}.$

Thus, $\bar{\Theta}^{[N]}$ is a random variable whose law depends on
$\mathcal{L}\left[X^{[N]}(Ns-L(N)),Y^{[N]}(Ns-L(N))\right]$ $=\mu_{N}$. The
infinite system with initial law $\mu_{N}$ is denoted by

(8.32)
$=\big{(}x^{\mu_{N}}_{i}(t),y^{\mu_{N}}_{i}(t)\big{)}_{i\in\mathbb{N}_{0},\
t\geq 0}$

and evolves according to

(8.33) $\displaystyle{\rm
d}x^{\mu_{N}}_{i}(t)=c\big{[}\bar{\Theta}^{[N]}-x^{\mu_{N}}_{i}(t)\big{]}\,{\rm
d}t+\sqrt{g(x^{\mu_{N}}_{i}(t))}\,{\rm
d}w_{i}(t)+Ke\,[y^{\mu_{N}}_{i}(t)-x^{\mu_{N}}_{i}(t)]\,{\rm d}t,$
$\displaystyle{\rm
d}y^{\mu_{N}}_{i}(t)=e\,[x^{\mu_{N}}_{i}(t)-y^{\mu_{N}}_{i}(t)]\,{\rm
d}t,\qquad i\in\mathbb{N}_{0},$

where $\\{w_{i}\\}_{i\in\mathbb{N}_{0}}$ is a collection of independent
Brownian motions.

######  Lemma 8.9.

[Comparison of finite and infinite systems] Fix $s>0$ and assume that, for any
$L(N)$ satisfying $\lim_{N\to\infty}L(N)=\infty$ and
$\lim_{N\to\infty}L(N)/N=0$,

(8.34) $\displaystyle\lim_{N\to\infty}\sup_{0\leq t\leq
L(N)}\big{|}\bar{\Theta}^{[N]}(Ns)-\bar{\Theta}^{[N]}(Ns-t)\big{|}=0\ \text{
in probability}.$

Let

(8.35) $\bigl{(}X^{\mu_{N}}(t),Y^{\mu_{N}}(t)\bigr{)}_{t\geq 0}$

be the infinite system defined in (LABEL:gh45ainf) starting in the
distribution $\mu_{N}$, where $\mu_{N}$ is defined by continuing the
configuration of $(X^{[N]}(Ns-L(N)),Y^{[N]}(Ns-L(N)))$ periodically to
$([0,1]^{2})^{\mathbb{N}_{0}}$. Similarly, view $(X^{[N]}(t),Y^{[N]}(t))$ as
an element of $([0,1]^{2})^{\mathbb{N}_{0}}$ by periodic continuation. Then,
for all $t\geq 0$,

(8.36)
$\displaystyle\lim_{N\to\infty}\bigl{|}\mathbb{E}\bigl{[}f\bigl{(}X^{\mu_{N}}(t),Y^{\mu_{N}}(t)\bigr{)}-f\bigl{(}X^{[N]}(Ns-L(N)+t),Y^{[N]}(Ns-L(N)+t)\bigr{)}\bigr{]}\bigr{|}=0$
$\displaystyle\qquad\forall\,f\in\mathcal{C}\bigl{(}([0,1]^{2})^{\mathbb{N}_{0}},\mathbb{R}\bigr{)}.$

Before we can prove that the infinite system
$(X^{\mu_{N}}(t),Y^{\mu_{N}}(t))_{t\geq 0}$ converges to some limiting system
as $N\to\infty$, we need the following regularity property for the estimator
$\bar{\Theta}^{[N]}$. This is stated in our fifth lemma.

######  Lemma 8.10.

[Stability of the estimator for the conserved quantity] Define $\mu_{N}$ as in
Lemma 8.9. Let $(x_{i},y_{i})_{i\in[N]}$ be distributed according to the
exchangeable probability measure $\mu_{N}$ on $([0,1]^{2})^{\mathbb{N}_{0}}$
restricted to $([0,1]^{2})^{[N]}$. Suppose that $\lim_{N\to\infty}\mu_{N}=\mu$
for some exchangeable probability measure $\mu$ on
$([0,1]^{2})^{\mathbb{N}_{0}}$. Define a random variable $\phi$ on
$(\mu,([0,1]^{2})^{\mathbb{N}_{0}})$ by putting

(8.37)
$\phi=\lim_{n\to\infty}\frac{1}{n}\sum_{i\in[n]}\frac{x_{i}+Ky_{i}}{1+K},$

and a random variable $\phi_{N}$ on $(\mu_{N},([0,1]^{2})^{\mathbb{N}_{0}})$
by putting

(8.38) $\phi_{N}=\frac{1}{N}\sum_{i\in[N]}\frac{x_{i}+Ky_{i}}{1+K}.$

Then

(8.39) $\lim_{N\to\infty}\mathcal{L}[\phi_{N}]=\mathcal{L}[\phi].$

In the sixth lemma we state the convergence of the law
$\mathcal{L}[(X^{\mu_{N}}(t),Y^{\mu_{N}}(t))]$ to the law of a limiting system
as $N\to\infty$.

######  Lemma 8.11.

[Uniformity of the ergodic theorem for the infinite system] Let $\mu_{N}$ be
defined as in Lemma 8.9. Since $(\mu_{N})_{N\in\mathbb{N}}$ is tight, it has
convergent subsequences. Let $(N_{k})_{k\in\mathbb{N}}$ be a subsequence such
that $\mu=\lim_{k\to\infty}\mu_{N_{k}}$. Define

(8.40)
$\bar{\Theta}=\lim_{N\to\infty}\frac{1}{N}\sum_{i\in[N]}\frac{x_{i}^{\mu}+Ky_{i}^{\mu}}{1+K}\,\qquad\text{
in }L_{2}(\mu),$

and let $(X^{\mu}(t),Y^{\mu}(t))_{t\geq 0}$ be the infinite system evolving
according to

(8.41) $\displaystyle{\rm
d}x^{\mu}_{i}(t)=c\left[\bar{\Theta}-x^{\mu}_{i}(t)\right]{\rm
d}t+\sqrt{g(x^{\mu}_{i}(t))}\,{\rm
d}w_{i}(t)+Ke\,[y^{\mu}_{i}(t)-x^{\mu}_{i}(t)]\,{\rm d}t,$ $\displaystyle{\rm
d}y^{\mu}_{i}(t)=e\,[x^{\mu}_{i}(t)-y^{\mu}_{i}(t)]\,{\rm d}t,\qquad
i\in\mathbb{N}_{0}.$

Then

  1. 1.

For all $t\geq 0$,

(8.42)
$\displaystyle\lim_{k\to\infty}\bigl{|}\mathbb{E}\bigl{[}f\bigl{(}X^{\mu_{N_{k}}}(t),Y^{\mu_{N_{k}}}(t)\bigr{)}\bigr{]}-\mathbb{E}\bigl{[}f\bigl{(}X^{\mu}(t),Y^{\mu}(t)\bigr{)}\bigr{]}\bigr{|}=0,$
$\displaystyle\qquad\forall\,f\in\mathcal{C}\bigl{(}([0,1]^{2})^{\mathbb{N}_{0}},\mathbb{R}\bigr{)}.$

  2. 2.

There exists a sequence $(\bar{L}(N))_{N\in\mathbb{N}}$ satisfying
$\lim_{N\to\infty}\bar{L}(N)=\infty$ and $\lim_{N\to\infty}\bar{L}(N)/N=0$
such that

(8.43) $\displaystyle\lim_{k\to\infty}$
$\displaystyle\bigl{|}\mathbb{E}\bigl{[}f\bigl{(}X^{[N_{k}]}(N_{k}s-L(N_{k})+\bar{L}(N_{k})),Y^{[N_{k}]}(N_{k}s-L(N_{k})+\bar{L}(N_{k}))\bigr{)}$
$\displaystyle\qquad-f\bigl{(}X^{\mu_{N_{k}}}(\bar{L}(N_{k})),Y^{\mu_{N_{k}}}(\bar{L}(N_{k}))\bigr{)}\bigr{|}\bigr{]}$
$\displaystyle+\bigl{|}\mathbb{E}\bigl{[}f\bigl{(}X^{\mu_{N_{k}}}(\bar{L}(N_{k})),Y^{\mu_{N_{k}}}(\bar{L}(N_{k}))\bigr{)}\bigr{]}-\mathbb{E}\bigl{[}f\bigl{(}X^{\mu}(\bar{L}(N_{k})),Y^{\mu}(\bar{L}(N_{k}))\bigr{)}\bigr{]}\bigr{|}=0$
$\displaystyle\qquad\forall\,f\in\mathcal{C}\bigl{(}([0,1]^{2})^{\mathbb{N}_{0}},\mathbb{R}\bigr{)}.$

######  Remark 8.12.

[Existence of $\bar{\Theta}$] Note that the limit in (8.40) is well-defined by
the ergodic theorem in $L_{2}$, since $\mu$ is the limit of translation
invariant measures and hence is itself translation invariant.

In the seventh lemma we provide a coupling of two copies of the finite system
starting from different measures.

######  Lemma 8.13.

[Coupling of finite systems] Let $(X^{[N],1},Y^{[N],1})$ be a finite system
evolving according to (7.15) and starting from some exchangeable measure. Let
$\mu^{[N],1}$ be the measure obtain by periodic continuation of the
configuration of $(X^{[N],1}(0),Y^{[N],1}(0))$. Similarly, let
$(X^{[N],2},Y^{[N],2})$ be a finite system evolving according to (7.15) and
starting from some exchangeable measure. Let $\mu^{[N],2}$ be the measure
obtain by periodic continuation of the configuration of
$(X^{[N],2}(0),Y^{[N],2}(0))$. Let $\tilde{\mu}$ be any weak limit point of
the sequence of measures
$\\{\mu^{[N],1}\times\mu^{[N],2}\\}_{N\in\mathbb{N}}$. Define random variables
$\bar{\Theta}^{[N],1}$ on $(\mu^{[N],1},([0,1]^{2})^{\mathbb{N}_{0}})$,
$\bar{\Theta}^{[N],2}$ on $(\mu^{[N],1},([0,1]^{2})^{\mathbb{N}_{0}})$ and
$\bar{\Theta}_{1}$ and $\bar{\Theta}_{2}$ on
$(\mu,([0,1]^{2})^{\mathbb{N}_{0}})$ by

(8.44)
$\displaystyle\bar{\Theta}^{[N],1}=\frac{1}{N}\sum_{i\in[N]}\frac{x^{[N],1}_{i}+Ky^{[N],1}_{i}}{1+K},\qquad\bar{\Theta}^{[N],2}=\frac{1}{N}\sum_{i\in[N]}\frac{x^{[N],2}_{i}+Ky^{[N],2}_{i}}{1+K},$
$\displaystyle\bar{\Theta}_{1}=\lim_{n\to\infty}\frac{1}{n}\sum_{i\in[n]}\frac{x^{1}_{i}+Ky^{1}_{i}}{1+K},\qquad\bar{\Theta}_{2}=\lim_{n\to\infty}\frac{1}{n}\sum_{i\in[n]}\frac{x^{2}_{i}+Ky^{2}_{i}}{1+K},$

and let $(\bar{\Theta}^{[N],1}(t))_{t\geq 0}$ and
$(\bar{\Theta}^{[N],2}(t))_{t\geq 0}$ be defined according to (8.2) for
$(X^{[N]}_{1},Y^{[N]}_{1})$, respectively, $(X^{[N]}_{2},Y^{[N]}_{2})$. Assume
that

(8.45) $\displaystyle\lim_{N\to\infty}\sup_{0\leq t\leq
L(N)}\big{|}\bar{\Theta}^{[N],k}(0)-\bar{\Theta}^{[N],k}(t)\big{|}=0\ \text{
in probability},\quad k\in\\{1,2\\},$

and suppose that $\tilde{\mu}(\\{\bar{\Theta}_{1}=\bar{\Theta}_{2}\\})=1$.
Then, for any sequence $t(N)\to\infty$,

(8.46)
$\lim_{N\to\infty}\mathbb{E}\left[\big{|}x^{[N],1}_{i}(t(N))-x^{[N],2}_{i}(t(N))\big{|}+K\big{|}y^{[N],1}_{i}(t(N))-y^{[N],2}_{i}(t(N))\big{|}\right]=0.$

###### Step 2. Convergence of the estimator.

This step is the equivalent of [DG93b, Proposition 2]. We first prove the
tightness of the estimator $\bar{\Theta}^{[N]}$ in path space. After that we
settle convergence of the finite-dimensional distributions and identify the
limit.

######  Proposition 8.14.

[Convergence of average sum process]

(8.47)
$\lim_{N\to\infty}\mathcal{L}\left[\left(\bar{\Theta}^{[N]}(Ns)\right)_{s>0}\right]=\mathcal{L}[(\bar{\Theta}(s))_{s>0}],$

where $(\bar{\Theta}(s))_{s>0}$ evolves according to

(8.48) ${\rm
d}\bar{\Theta}(s)=\frac{1}{(1+K)}\sqrt{(\mathcal{F}g)(\bar{\Theta}(s))}\,{\rm
d}w(s).$

Proposition 8.14 follows from the following three lemmas, which are the
equivalent of the three lemmas used in [DG93b, p. 488–493] for the system
without seed-bank.

######  Lemma 8.15.

[Martingale property of average sum process]

  1. (1)

The process $(\bar{\Theta}^{[N]}(Ns))_{s>0}$ is a square-integrable martingale
with continuous paths and increasing process

(8.49)
$\left\langle\bar{\Theta}^{[N]}(Ns)\right\rangle_{s>0}=\frac{1}{(1+K)^{2}}\int_{0}^{s}{\rm
d}r\,\frac{1}{N}\sum_{i\in[N]}{g\big{(}x_{i}^{[N]}(Nr)\big{)}}.$

  2. (2)

Let $L(N)$ be such that $\lim_{N\to\infty}L(N)=\infty$ and
$\lim_{N\to\infty}L(N)/N=0$. Then

(8.50) $\lim_{N\to\infty}\sup_{0\leq t\leq
L(N)}\big{|}\bar{\Theta}^{[N]}(Ns)-\bar{\Theta}^{[N]}(Ns-t)\big{|}=0\text{ in
probability.}$

  3. (3)

$\left(\mathcal{L}[(\bar{\Theta}^{[N]}(Ns))_{s>0}]\right)_{N\in\mathbb{N}}$ is
tight as a sequence of probability measures on
$\mathcal{C}([0,\infty),[0,1])$.

######  Lemma 8.16.

[Martingale property of limit process] Let $(N_{k})_{k\in\mathbb{N}}$ be any
subsequence such that

(8.51)
$\lim_{k\to\infty}\mathcal{L}\left[\big{(}\bar{\Theta}^{[N_{k}]}(N_{k}s)\big{)}_{s>0}\right]=\mathcal{L}\bigl{[}(\bar{\Theta}(s))_{s>0}\bigr{]}.$

Then $(\bar{\Theta}(s))_{s>0}$ is a square-integrable martingale with
continuous paths, and

(8.52) $\left(\bar{\Theta}^{2}(s)-\int_{0}^{s}{\rm
d}r\,\frac{1}{(1+K)^{2}}\,\mathbb{E}^{\nu_{\bar{\Theta}(r)}}[g(x_{0})]\right)_{s>0}$

is a martingale.

######  Lemma 8.17.

[Uniqueness] The following martingale problem has a unique solution:

(8.53) $\displaystyle(\bar{\Theta}_{s})_{s>0}\text{ is a continuous martingale
with values in }[0,1],$
$\displaystyle\left(\bar{\Theta}^{2}(s)-\frac{1}{(1+K)^{2}}\int_{0}^{s}{\rm
d}r\,\mathbb{E}^{\nu_{\bar{\Theta}(r)}}[g(x_{0})]\right)_{s>0}\text{ is a
martingale.}$

The solution of (8.53) is given by the diffusion generated by
$\mathbb{E}^{\nu_{u}}[g(\cdot)]\frac{\partial^{2}}{\partial u^{2}}$.

###### Step 3. Convergence of the averages in the Meyer-Zheng topology.

Recall the definition of the Meyer-Zheng topology in Section 4.1. We have to
prove the following proposition.

######  Proposition 8.18.

[Convergence in Meyer-Zheng topology] If

(8.54)
$\lim_{N\to\infty}\mathcal{L}\left[\left(\bar{\Theta}(Ns)\right)_{s>0}\right]=\mathcal{L}\left[\left({\bar{\Theta}}(s)\right)_{s>0}\right],$

then

(8.55)
$\displaystyle\lim_{N\to\infty}\mathcal{L}\Bigl{[}\bigl{(}x_{1}^{[N]}(t),y_{1}^{[N]}(t)\bigr{)}_{t\geq
0}\Bigr{]}=\mathcal{L}\Bigl{[}\bigl{(}x_{1}^{\mathbb{N}_{0}}(t),y_{1}^{\mathbb{N}_{0}}(t)\bigr{)}_{t\geq
0}\Bigr{]}$ $\displaystyle\mbox{ in the Meyer-Zheng topology},$

where $(x_{1}^{\mathbb{N}_{0}}(t),y_{1}^{\mathbb{N}_{0}}(t))_{t\geq 0})$
evolves according to (8.14).

To prove Proposition 8.18 we will use Lemma 8.7 in combination with the
following three general lemmas about the Meyer-Zheng topology, which are
proven in Appendix B.3.

######  Lemma 8.19.

[Path convergence in probability in the Meyer-Zheng topology] Let
$((Z_{n}(t))_{t\geq 0})_{n\in\mathbb{N}}$ and $(Z(t))_{t\geq 0}$ be stochastic
processes on the Polish space $(E,d)$. If, for all $t\geq 0$,

(8.56) $\lim_{n\to\infty}\mathbb{E}\left[d(Z_{n}(t),Z(t))\right]=0,$

then,

(8.57) $\lim_{n\to\infty}(Z_{n}(t))_{t\geq 0}=(Z(t))_{t\geq 0}\text{ in
probability in the Meyer-Zheng topology}.$

######  Lemma 8.20.

[Convergence of the joint law of paths in the Meyer-Zheng topology] Let
$(X_{n})_{n\in\mathbb{N}},(Y_{n})_{n\in\mathbb{N}},X$ be stochastic processes
on a metric space $(E,d)$ and let $c\in E$ be a constant. If
$\lim_{n\to\infty}\mathcal{L}[X_{n}]=\mathcal{L}[X]$ in the Meyer-Zheng
topology and for all $t\geq 0$,
$\lim_{n\to\infty}\mathbb{E}[d(Y_{n}(t),c)]=0$, then
$\lim_{n\to\infty}\mathcal{L}[(X_{n},Y_{n})]=\mathcal{L}[(X,c)]$ in the Meyer-
Zheng topology.

######  Lemma 8.21.

[Continuous mapping theorem in the Meyer-Zheng topology] Let $f\colon\,E\to E$
be a continuous function and $x\in M_{E}[0,\infty)$.

  * (a)

The function

(8.58) $h\colon\,\Psi\to\Psi,\qquad\psi_{x}\to\psi_{f(x)},$

is continuous.

  * (b)

If the stochastic processes $(X_{n})_{n\in\mathbb{N}},X$ on state space
$(E,d)$ satisfy

(8.59) $\lim_{n\to\infty}\mathcal{L}[X_{n}]=\mathcal{L}[X]\text{ in the Meyer-
Zheng topology},$

then

(8.60) $\lim_{n\to\infty}\mathcal{L}[f(X_{n})]=\mathcal{L}[f(X)]\text{ in the
Meyer-Zheng topology.}$

Note that Lemma 8.21 allows us to use the continuous mapping theorem in the
Meyer-Zheng topology.

###### Step 4. Mean-field finite-systems scheme.

Use Steps 1– 4 to prove Proposition 8.1.

Having completed the abstract scheme of steps 1–4, we set out to prove the
constituent propositions and lemmas.

###  9 Proofs: $N\to\infty$, mean-field, proof of abstract scheme

In Sections 9.1–9.4 we prove the propositions and the lemmas stated in Steps
1–4 in Section 8.2.

####  9.1 Proof of step 1. Equilibrium of the single components

We start by proving Proposition 8.4 with the help of the seven lemmas stated
in Step 1 of Section 8.2. Afterwards we prove each of the lemmas.

######  $\bullet$ Proof of Proposition 8.4

###### Proof.

We use an argument similar to the one used in [DG93b, Section 2 (i)]. Let
$(L(N))_{N\in\mathbb{N}}$ be any sequence satisfying
$\lim_{N\to\infty}L(N)=\infty$ and $\lim_{N\to\infty}L(N)/N=0$. Let $\mu_{N}$
be the measure on $([0,1]^{2})^{\mathbb{N}_{0}}$ obtained by periodic
continuation of $\mathcal{L}[X^{[N]}(Ns-L(N)),Y^{[N]}(Ns-L(N))]$. Note that
$([0,1]^{2})^{\mathbb{N}_{0}}$ is compact. Hence, letting
$(N_{k})_{k\in\mathbb{N}}$ be the subsequence in Proposition 8.4, we can pass
to a further subsequence and obtain

(9.1) $\lim_{k\to\infty}\mu_{N_{k}}=\mu.$

Since we assumed that $\mathcal{L}[X^{[N]}(0),Y^{[N]}(0)]$ is exchangeable and
the dynamics preserves exchangeability, the measures $\mu_{N_{k}}$ are
exchangeable and also the limiting law $\mu$ is exchangeable. Define $\phi$ as
in (8.37) in Lemma 8.10. Then we can condition on $\phi$ and write

(9.2) $\mu=\int_{[0,1]}\mu_{\rho}\,{\rm d}\Lambda(\rho),$

where $\Lambda(\cdot)=\mathcal{L}[\phi]$. By assumption we know that

(9.3)
$\lim_{k\to\infty}\mathcal{L}\big{[}\bar{\Theta}^{[N_{k}]}(N_{k}s)\big{]}=P_{s}$

and

(9.4) $\lim_{k\to\infty}\mathcal{L}\left[\sup_{0\leq t\leq
L(N_{k})}\left|\bar{\Theta}^{[N_{k}]}(N_{k}s)-\bar{\Theta}^{[N_{k}]}(N_{k}s-t)\right|\right]=\delta_{0}.$

Hence

(9.5)
$\displaystyle\lim_{k\to\infty}\mathcal{L}\left[\bar{\Theta}^{[N_{k}]}(N_{k}s-L(N_{k}))\right]=P_{s}.$

Recall that

(9.6)
$\Lambda=\mathcal{L}[\phi]=\mathcal{L}\left[\lim_{n\to\infty}\frac{1}{n}\sum_{i\in[n]}\frac{x_{i}+Ky_{i}}{1+K}\right]\quad\text{
on }(\mu,([0,1]^{2})^{\mathbb{N}_{0}}).$

By Lemma 8.10, if
$\phi_{N_{k}}=\frac{1}{N_{k}}\sum_{i\in[N_{k}]}\frac{x_{i}+Ky_{i}}{1+K}$ on
$(\mu_{N_{k}},([0,1]^{2})^{\mathbb{N}_{0}})$, then
$\lim_{k\to\infty}\mathcal{L}[\phi_{N_{k}}]=\mathcal{L}[\phi]$. Taking the
subsequence $(\mu_{N_{k}})_{k\in\mathbb{N}}$, we get
$\Lambda(\cdot)=P_{s}(\cdot)$, and hence

(9.7) $\mu=\int_{[0,1]}\mu_{\rho}\,{\rm d}P_{s}(\rho).$

Let $\bar{L}(N)$ be the sequence constructed in Lemma 8.11[b]. We can require
that $\bar{L}(N)\leq L(N)$ for all $N\in\mathbb{N}$. Write

(9.8)
$\displaystyle\mathcal{L}\bigl{[}X^{[N_{k}]}(N_{k}s-L(N_{k})+\bar{L}(N_{k})),Y^{[N_{k}]}(N_{k}s-L(N_{k})+\bar{L}(N_{k}))\bigr{]}$
$\displaystyle=\mathcal{L}\bigl{[}X^{[N_{k}]}(N_{k}s-L(N_{k})+\bar{L}(N_{k})),Y^{N_{k}}(N_{k}s-L(N_{k})+\bar{L}(N_{k}))\bigr{]}$
$\displaystyle\quad-\mathcal{L}\bigl{[}X^{\mu_{N_{k}}}(\bar{L}(N_{k})),Y^{\mu_{N_{k}}}(\bar{L}(N_{k}))\bigr{]},$
$\displaystyle\quad+\mathcal{L}\bigl{[}X^{\mu_{N_{k}}}(\bar{L}(N_{k})),Y^{\mu_{N_{k}}}(\bar{L}(N_{k}))\bigr{]}-\mathcal{L}\bigl{[}X^{\mu}(\bar{L}(N_{k})),Y^{\mu}(\bar{L}(N_{k}))\bigr{]}$
$\displaystyle\quad+\mathcal{L}\bigl{[}X^{\mu}(\bar{L}(N_{k})),Y^{\mu}(\bar{L}(N_{k}))\bigr{]}.$

By Lemma 8.11, the first and the second term tend to zero as $k\to\infty$.
Hence

(9.9)
$\mathcal{L}\bigl{[}X^{[N_{k}]}(N_{k}s-L(N_{k})+\bar{L}(N_{k})),Y^{[N_{k}]}(N_{k}s-L(N_{k})+\bar{L}(N_{k}))\bigr{]}$

tends to $\mathcal{L}\bigl{[}X^{\mu}(L(N_{k})),Y^{\mu}(L(N_{k}))\bigr{]}$ as
$k\to\infty$. By (9.7),

(9.10)
$\mathcal{L}\bigl{[}X^{\mu}(\bar{L}(N_{k})),Y^{\mu}(\bar{L}(N_{k}))\bigr{]}=\int_{[0,1]}\mathcal{L}\bigl{[}X^{\mu_{\rho}}(\bar{L}(N_{k})),Y^{\mu_{\rho}}(\bar{L}(N_{k}))\bigr{]}\,{\rm
d}P_{s}(\rho).$

Since $\lim_{k\to\infty}\bar{L}(N_{k})=\infty$, by Lemma 8.5 we have

(9.11)
$\lim_{k\to\infty}\mathcal{L}\bigl{[}X^{\mu_{\rho}}(\bar{L}(N_{k})),Y^{\mu_{\rho}}(\bar{L}(N_{k}))\bigr{]}=\nu_{\rho}.$

Therefore, by (9.8) and Lemma 8.6,

(9.12)
$\displaystyle\lim_{k\to\infty}\mathcal{L}\bigl{[}X^{[N_{k}]}(N_{k}s-L(N_{k})+\bar{L}(N_{k})),Y^{[N_{k}]}(N_{k}s-L(N_{k})+\bar{L}(N_{k}))\bigr{]}$
$\displaystyle\qquad=\int_{[0,1]}\nu_{\rho}\,{\rm d}P_{s}(\rho).$

To show that

(9.13)
$\lim_{k\to\infty}\mathcal{L}\bigl{[}X^{[N_{k}]}(N_{k}s),Y^{[N_{k}]}(N_{k}s)\bigr{]}=\int_{[0,1]}\nu_{\rho}\,{\rm
d}P_{s}(\rho).$

we invoke Lemma 8.13. Let $(X^{[N],1},Y^{[N],1})$ be the finite system
starting from

(9.14) $\mathcal{L}\big{[}X^{[N]}(Ns-L(N)),Y^{[N]}(Ns-L(N))\big{]},$

and let $(\bar{L}(N))_{N\in\mathbb{N}}$ be the sequence such that (9.12)
holds. Let $(X^{[N],2}.Y^{[N],2})$ be the finite system starting from

(9.15)
$\mathcal{L}\big{[}X^{[N]}(Ns-\bar{L}(N)),Y^{[N]}(Ns-\bar{L}(N))\big{]}.$

Choose for the sequence $t(N)$ in Lemma 8.13 the sequence $\bar{L}(N)$. Let
$\mu^{[N],1}$ be defined by periodic continuation of
$(X^{[N]}(Ns-L(N)),Y^{[N]}(Ns-L(N)))$, and $\mu^{[N],2}$ by periodic
continuation of $(X^{[N]}(Ns-\bar{L}(N)),Y^{[N]}(Ns-\bar{L}(N)))$. Defining
$\bar{\Theta}_{1}$ and $\bar{\Theta}_{2}$ according to (8.44), where for
$\mu^{[N],2}$ we replace $L(N)$ by $\bar{L}(N)$, we get

(9.16)
$\lim_{k\to\infty}|\bar{\Theta}^{N_{k}}_{1}-\bar{\Theta}^{N_{k}}_{2}|=\lim_{k\to\infty}\big{|}\bar{\Theta}^{N_{k}}(N_{k}s-L(N_{k}))-\bar{\Theta}^{N_{k}}(N_{k}s-\bar{L}(N_{k}))\big{|}=0\text{
in probability}$

by the assumptions in (8.22). Hence, if $\mu$ is any weak limit point of the
sequence $(\mu^{[N_{k}],1}\times\mu^{[N_{k}],2})_{k\in\mathbb{N}}$, then

(9.17) $\mu(\bar{\Theta}_{1}=\bar{\Theta}_{2})=1.$

By passing to a further subsequence, we can now apply Lemma 8.13, to obtain

(9.18)
$\lim_{k\to\infty}\mathbb{E}\left[|x^{N_{k}}_{i,1}(\bar{L}(N_{k}))-x^{N_{k}}_{i,2}(\bar{L}(N_{k}))|+K|y^{N_{k}}_{i,1}(\bar{L}(N_{k}))-y^{N_{k}}_{i,2}(\bar{L}(N_{k}))|\right]=0.$

Note that

(9.19)
$\displaystyle\mathcal{L}\big{[}X_{1}(\bar{L}(N_{k}),Y_{1}(\bar{L}(N_{k})))\big{]}$
$\displaystyle=\mathcal{L}\big{[}X^{[N_{k}]}(N_{k}s-L(N_{k})+\bar{L}(N_{k})),Y^{[N_{k}]}(N_{k}s-L(N_{k})+\bar{L}(N_{k}))\big{]},$
$\displaystyle\mathcal{L}\big{[}X_{2}(\bar{L}(N_{k}),Y_{2}(\bar{L}(N_{k})))\big{]}=\mathcal{L}\big{[}X^{[N_{k}]}(N_{k}s),Y^{[N_{k}]}(N_{k}s)\big{]}.$

Moreover, we know from (9.12) that

(9.20)
$\lim_{k\to\infty}\mathcal{L}\left[X^{[N_{k}]}(N_{k}s-L(N_{k})+\bar{L}(N_{k})),Y^{[N_{k}]}(N_{k}s-L(N_{k})+\bar{L}(N_{k}))\right]=\int_{[0,1]}\nu_{\rho}P_{s}({\rm
d}\rho).$

Combining (9.18)–(9.20), we find that

(9.21)
$\mathcal{L}\big{[}X^{[N_{k}]}(N_{k}s),Y^{[N_{k}]}(N_{k}s)\big{]}=\int_{[0,1]}\nu_{\rho}P_{s}({\rm
d}\rho).$

∎

In the remainder of this section we prove Lemmas 8.5–8.11 and 8.13.

######  $\bullet$ Proof of Lemma 8.5

###### Proof.

Since the components of the infinite system in (8.1) evolve independently, it
is enough to show that each component converges to $\Gamma_{\theta}$. This
convergence follows from the proof of Proposition 7.2 (see Section 7.3). Hence
the infinite system defined by (7.18) converges to
$\nu_{\theta}=\Gamma_{\theta}^{\otimes\mathbb{N}_{0}}$. Ergodicity of
$\nu_{\theta}$ with respect to translations follows from Kolmogorov’s zero-one
law. ∎

######  $\bullet$ Proof of Lemma 8.7

###### Proof.

Using the definition of ${{\Theta}}^{[N]}_{x}(t)$, ${{\Theta}}^{[N]}_{y}(t)$
in (8.3) and the SSDE in (7.15), we find the following evolution for the
averages:

(9.22) $\displaystyle{\rm d}{{\Theta}}^{[N]}_{x}(t)$
$\displaystyle=\frac{1}{N}\sum_{i\in[N]}\sqrt{g(x_{i}(t))}\,{\rm
d}w_{i}(t)+Ke\,[{{\Theta}}^{[N]}_{y}(t)-{{\Theta}}^{[N]}_{x}(t)]\,{\rm d}t,$
$\displaystyle{\rm d}{{\Theta}}^{[N]}_{y}(t)$
$\displaystyle=e\,[{{\Theta}}^{[N]}_{x}(t)-{{\Theta}}^{[N]}_{y}(t)]\,{\rm
d}t.$

Consequently,

(9.23) $\displaystyle{\rm
d}\big{(}{\Delta}^{[N]}_{\bar{\Theta}}(t)\big{)}^{2}$
$\displaystyle=2{\Delta}^{[N]}_{\bar{\Theta}}(t)\,{\rm
d}{\Delta}^{[N]}_{\bar{\Theta}}(t)+2{\rm
d}\langle{\Delta}^{[N]}_{\bar{\Theta}}\rangle(t)$
$\displaystyle={\Delta}^{[N]}_{\bar{\Theta}}(t)\frac{1}{N}\sum_{i\in[N]}\sqrt{g(x_{i}(t))}\,{\rm
d}w_{i}(t)-(Ke+e)\,\big{(}{\Delta}^{[N]}_{\bar{\Theta}}(t)\big{)}^{2}\,{\rm
d}t$ $\displaystyle\qquad+2\frac{1}{N^{2}}\sum_{i\in[N]}g(x_{i}(t))\ {\rm
d}t,$

and hence

(9.24) $\frac{{\rm d}}{{\rm
d}t}\,\mathbb{E}\left[\big{(}{\Delta}^{[N]}_{\bar{\Theta}}(t)\big{)}^{2}\right]=-2(Ke+e)\,\mathbb{E}\left[\big{(}{\Delta}^{[N]}_{\bar{\Theta}}(t)\big{)}^{2}\right]+\frac{2}{N^{2}}\sum_{i\in[N]}g(x_{i}(t))$

and

(9.25)
$\mathbb{E}\left[\big{(}{\Delta}^{[N]}_{\bar{\Theta}}(t)\big{)}^{2}\right]=\mathbb{E}\left[\big{(}{\Delta}^{[N]}_{\bar{\Theta}}(0)\big{)}^{2}\right]{\rm
e}^{-2(Ke+e)t}+\int_{0}^{t}{\rm d}r\,{\rm
e}^{-2(Ke+e)(t-r)}\frac{2}{N^{2}}\sum_{i\in[N]}g(x_{i}(r)).$

Therefore we get the bound

(9.26)
$\mathbb{E}\left[\big{|}{\Delta}^{[N]}_{\bar{\Theta}}(t)\big{|}\right]\leq\sqrt{\mathbb{E}\left[\big{(}{\Delta}^{[N]}_{\bar{\Theta}}(0)\big{)}^{2}\right]}{\rm
e}^{-(Ke+e)t}+\sqrt{\frac{2\|g\|}{N(Ke+e)}}.$

∎

######  $\bullet$ Proof of Lemma 8.9

###### Proof.

To compare the systems in (7.15) and (LABEL:gh45ainf), we couple them via
their Brownian motions. Therefore for all $i\in[N]$ we assume that the
evolution in (7.15) and (LABEL:gh45ainf) is driven by the same Brownian
motion, $\tilde{w}_{i}=w_{i}$. If $i\notin[N]$, then we set $w_{i}=w_{j}$ for
$j=i\mod N$. We denote the coupled process by
$\tilde{z}(t)=(\tilde{z}_{i}(t))_{i\in\mathbb{N}_{0}}=(\tilde{z}^{[N]}_{i}(t),\tilde{z}^{\mu_{N}}_{i}(t))_{i\in\mathbb{N}_{0}}$,
where $\tilde{z}^{[N]}_{i}(t)=(\tilde{x}^{[N]}_{i}(t),\tilde{y}^{[N]}_{i}(t))$
and
$\tilde{z}^{\mu_{N}}_{i}(t)=(\tilde{x}^{\mu_{N}}_{i}(t),\tilde{y}^{\mu_{N}}_{i}(t))$.
The tilde indicates that we are considering the coupled process, and

(9.27) $\displaystyle\mathcal{L}[\tilde{z}(0)]$
$\displaystyle=\mathcal{L}\big{[}X^{[N]}(Ns-L(N)),Y^{[N]}(Ns-L(N))\big{]}\times\mu_{N}$
$\displaystyle=\mathcal{L}\big{[}X^{[N]}(Ns-L(N)),Y^{[N]}(Ns-L(N))\big{]}^{2}.$

Define

(9.28)
$\Delta^{N}_{i}(t)=\tilde{x}_{i}^{[N]}(t)-\tilde{x}_{i}^{\mu_{N}}(t),\qquad\delta^{N}_{i}(t)=\tilde{y}_{i}^{[N]}(t)-\tilde{y}_{i}^{\mu_{N}}(t).$

To prove that the coupling is successful, we show that, for all $t\geq 0$,

(9.29)
$\lim_{N\to\infty}\mathbb{E}\left[|\Delta_{i}^{N}(t)|+K|\delta_{i}^{N}(t)|\right]=0\quad\forall\,i\in\mathbb{N}_{0}.$

From now on we will only consider sites $i\in[0,N]$ for which both infinite
systems have the same Brownian motion.

From (7.15) and (LABEL:gh45ainf) it follows that

(9.30) $\displaystyle{\rm
d}\left[|\Delta^{N}_{i}(t)|+K|\delta^{N}_{i}(t)|\right]$
$\displaystyle\quad=({\rm sign}\,\Delta^{N}_{i}(t))\ {\rm
d}\Delta^{N}_{i}(t)+{\rm d}L_{t}^{0}+K\,{\rm sign}\,\delta^{N}_{i}(t)\ {\rm
d}\delta^{N}_{i}(t)$ $\displaystyle\quad=-c\ ({\rm sign}\,\Delta^{N}_{i}(t))\
\Delta^{N}_{i}(t)\,{\rm d}t+c\,({\rm sign}\,\Delta^{N}_{i}(t))\
\left[\bar{\Theta}^{[N]}(t)-\bar{\Theta}^{[N]}\right]\,{\rm d}t$
$\displaystyle\qquad+c\,({\rm sign}\,\Delta^{N}_{i}(t))\
\left[{\Theta}^{[N]}_{x}(t)-\bar{\Theta}^{[N]}(t)\right]\,{\rm d}t$
$\displaystyle\qquad+({\rm sign}\,\Delta^{N}_{i}(t))\
\left(\sqrt{g(x^{[N]}_{i}(t))}\,-\sqrt{g(x^{\mu_{N}}_{i}(t))}\,\right){\rm
d}w_{i}(t)$ $\displaystyle\qquad+({\rm
sign}\,\Delta^{N}_{i}(t))\,Ke\,[\delta^{N}_{i}(t)-\Delta^{N}_{i}(t)]\,{\rm
d}t$ $\displaystyle\qquad+({\rm
sign}\,\delta^{N}_{i}(t))\,Ke\,[\Delta^{N}_{i}(t)-\delta^{N}_{i}(t)]\,{\rm
d}t,$

where we use that the local time is zero, since $g$ is Lipschitz (see [RW00,
Proposition V.39.3]).

Taking expectations in (9.30), we find

(9.31) $\displaystyle\frac{{\rm d}}{{\rm
d}t}\mathbb{E}[|\Delta^{N}_{i}(t)|+K|\delta^{N}_{i}(t)|]$
$\displaystyle=-c\,\mathbb{E}\left[|\Delta^{N}_{i}(t)|\right]$
$\displaystyle\quad+c\ \mathbb{E}\left[({\rm sign}\,\Delta^{N}_{i}(t))\
\big{[}\bar{\Theta}^{[N]}(t)-\bar{\Theta}^{[N]}\big{]}\right]$
$\displaystyle\quad+c\,\mathbb{E}\left[({\rm sign}\,\Delta^{N}_{i}(t))\
\big{[}{\Theta}^{[N]}_{x}(t)-\bar{\Theta}^{[N]}(t)\big{]}\right]$
$\displaystyle\quad+Ke\,\mathbb{E}\left[\big{(}{\rm
sign}\,\Delta^{N}_{i}(t)-{\rm
sign}\,\delta^{N}_{i}(t)\big{)}\big{[}\delta^{N}_{i}(t)-\Delta^{N}_{i}(t)\big{]}\right].$

Note that we can rewrite (9.31) as

(9.32) $\displaystyle\frac{{\rm d}}{{\rm
d}t}\mathbb{E}[|\Delta^{N}_{i}(t)|+K|\delta^{N}_{i}(t)|]$
$\displaystyle=-c\,\mathbb{E}\left[|\Delta^{N}_{i}(t)|\right]$
$\displaystyle\quad-2Ke\,\mathbb{E}\left[1_{{\rm
sign}\,\Delta^{N}_{i}(t)\neq{\rm
sign}\,\delta^{N}_{i}(t)}\,\,[|\delta^{N}_{i}(t)|+|\Delta^{N}_{i}(t)|]\right]$
$\displaystyle\quad+c\ \mathbb{E}\left[({\rm sign}\,\Delta^{N}_{i}(t))\
\big{[}\bar{\Theta}^{[N]}(t)-\bar{\Theta}^{[N]}\big{]}\right]$
$\displaystyle\quad+c\,\mathbb{E}\left[({\rm sign}\,\Delta^{N}_{i}(t))\
\big{[}{\Theta}^{[N]}_{x}(t)-\bar{\Theta}^{[N]}(t)\big{]}\right].$

It therefore follows that

(9.33) $\displaystyle\mathbb{E}[|\Delta^{N}_{i}(t)|+K|\delta^{N}_{i}(t)|]$
$\displaystyle=\mathbb{E}[|\Delta^{N}_{i}(0)|+K|\delta^{N}_{i}(0)|]$
$\displaystyle\quad-c\int_{0}^{t}{\rm
d}r\,\mathbb{E}\left[|\Delta^{N}_{i}(r)|\right]$
$\displaystyle\quad-2Ke\int_{0}^{t}{\rm d}r\,\mathbb{E}\left[1_{{\rm
sign}\,\Delta^{N}_{i}(r)\neq{\rm
sign}\,\delta^{N}_{i}(r)}\,\,[|\delta^{N}_{i}(r)|+|\Delta^{N}_{i}(r)|]\right]$
$\displaystyle\quad+\int_{0}^{t}{\rm d}r\,c\,\mathbb{E}\left[({\rm
sign}\,\Delta^{N}_{i}(r))\
\big{[}\bar{\Theta}^{[N]}(r)-\bar{\Theta}^{[N]}\big{]}\right]$
$\displaystyle\quad+\int_{0}^{t}{\rm d}r\,c\,\mathbb{E}\left[({\rm
sign}\,\Delta^{N}_{i}(r))\
\big{[}{\Theta}^{[N]}_{x}(r)-\bar{\Theta}^{[N]}(r)\big{]}\right].$

Note that, by the choice of initial distribution for the coupling, we have

(9.34) $\mathbb{E}[|\Delta^{N}_{i}(0)|+K|\delta^{N}_{i}(0)|]=0.$

Therefore we get

(9.35) $\displaystyle 0$
$\displaystyle\leq\mathbb{E}[|\Delta^{N}_{i}(t)|+K|\delta^{N}_{i}(t)|]$
$\displaystyle\leq-c\int_{0}^{t}{\rm
d}r\,\mathbb{E}\left[|\Delta^{N}_{i}(r)|\right]$
$\displaystyle\quad-2Ke\int_{0}^{t}{\rm d}r\,\mathbb{E}\left[1_{{\rm
sign}\,\Delta^{N}_{i}(r)\neq{\rm
sign}\,\delta^{N}_{i}(r)}\,\,[|\delta^{N}_{i}(r)|+|\Delta^{N}_{i}(r)|]\right]$
$\displaystyle\quad+\int_{0}^{t}{\rm
d}r\,c\,\mathbb{E}\left[\big{|}\bar{\Theta}^{[N]}(r)-\bar{\Theta}^{[N]}\big{|}\right]$
$\displaystyle\quad+\int_{0}^{t}{\rm
d}r\,c\,\mathbb{E}\left[\big{|}{\Theta}^{[N]}_{x}(r)-\bar{\Theta}^{[N]}(r)\big{|}\right]$
$\displaystyle\leq t\left(\sup_{0\leq r\leq t}c\
\mathbb{E}\left[\big{|}\bar{\Theta}^{[N]}(r)-\bar{\Theta}^{[N]}\big{|}\right]+c\,\mathbb{E}\left[\big{|}{\Theta}^{[N]}_{x}(r)-\bar{\Theta}^{[N]}(r)\big{|}\right]\right).$

Hence, by the assumption in (8.34) and Lemma 8.7 (recall (9.27)) , we see
that, for all $t>0$,

(9.36)
$\lim_{N\to\infty}\mathbb{E}[|\Delta^{N}_{i}(t)|+K|\delta^{N}_{i}(t)|]=0.$

Therefore, for every Lipschitz function
$f\in\mathcal{C}\bigl{(}([0,1]),\mathbb{R}\bigr{)}$ of $x_{i}(t)$,

(9.37)
$\lim_{n\to\infty}\left|\mathbb{E}[f(x_{i}^{[N]}(t))-f(x_{i}^{\mu_{N}}(t))]\right|\leq\lim_{n\to\infty}\text{Lip}f\,\mathbb{E}[||\Delta^{N}_{i}(L(N))||]=0,$

and the same holds for Lipschitz functions of $y_{i}$. Using that the
Lipschitz functions are dense in
$\mathcal{C}\bigl{(}([0,1]),\mathbb{R}\bigr{)}$, we obtain that the result
actually holds for all
$f\in\mathcal{C}\bigl{(}([0,1]^{2})^{\mathbb{N}_{0}},\mathbb{R}\bigr{)}$
depending on finitely many components. This in turn implies that the result
holds for all
$f\in\mathcal{C}\bigl{(}([0,1]^{2})^{\mathbb{N}_{0}},\mathbb{R}\bigr{)}$. ∎

######  $\bullet$ Proof of Lemma 8.10

###### Proof.

Define

(9.38) $D^{N}(Z)=\frac{1}{N}\sum_{j\in[N]}\frac{x_{j}+Ky_{j}}{1+K},\qquad
D(Z)=\lim_{N\to\infty}D^{N}(Z)\text{ in }L_{2}(\mu).$

Since $\mu$ is translation invariant with
$\int_{[0,1]^{2}}\frac{x_{0}+Ky_{0}}{1+K}\,{\rm d}\mu<1$, the
$L_{2}(\mu)$-limit $D(Z)$ exists by the ergodic theorem. Since, by assumption,
$\mu_{N}\to\mu$ as $N\to\infty$ for all fixed $M\in\mathbb{N}_{0}$, we have

(9.39)
$\lim_{N\to\infty}\mathcal{L}_{\mu_{N}}[D^{M}(Z)]=\mathcal{L}_{\mu}[D^{M}(Z)].$

Therefore, in order to prove Lemma 8.10, we are left to show

(9.40) $\lim_{M\to\infty}\sup_{N\geq
M}\|D^{M}(Z)-D^{N}(Z)\|_{L_{2}(\mu_{N})}=0.$

This can be done by using Fourier transforms and spectral densities, and to do
so we follow the same strategy as in [DG93b, Lemma 2.5].

Define

(9.41)
$\bar{\theta}^{N}=\mathbb{E}^{\mu_{N}}\left[\frac{x_{0}+Ky_{0}}{1+K}\right].$

Since $\mu_{N}$ is translation invariant on $\mathbb{N}_{0}$, by Herglotz’s
theorem there exists a unique measure $\lambda_{N}$ such that, for all
$j,k\in\mathbb{N}_{0}$,

(9.42)
$\mathbb{E}^{\mu_{N}}\left[\left(\frac{x_{j}+Ky_{j}}{1+K}-\bar{\theta}^{N}\right)\left(\frac{x_{k}+Ky_{k}}{1+K}-\bar{\theta}^{N}\right)\right]=\int_{(-\pi,\pi]}\lambda_{N}({\rm
d}u)\,{\rm e}^{\mathrm{i}(j-k)u}.$

For $N\in\mathbb{N}_{0}$, define

(9.43) $D^{N}(u)=\frac{1}{N}\sum_{j\in[N]}{\rm e}^{\mathrm{i}ju}.$

By (9.42), it follows that

(9.44)
$\left\|D^{M}(Z)-D^{N}(Z)\right\|_{L_{2}(\mu_{N})}=\left\|D^{M}(u)-D^{N}(u)\right\|_{L_{2}(\lambda_{N})}.$

Polynomials of the type $D^{N}(u)$ are called trigonometric polynomials and
satisfy:

  1. 1.

$\lim_{N\to\infty}D^{N}(u)=1_{\\{0\\}}(u)$.

  2. 2.

For $\delta>0$ and $M<\infty$ there exists an $\epsilon(M,\delta)$ such that,
for all $N\geq M$,

(9.45) $|D^{N}(u)-1_{\\{0\\}}(u)|\leq
1_{(-\delta,\delta)\backslash\\{0\\}}+\epsilon(M,\delta)\text{ with
}\epsilon(M,\delta)\to 0\text{ as }M\to\infty.$

Hence it follows that

(9.46) $\left\|D^{M}(u)-D^{N}(u)\right\|^{2}_{L_{2}(\lambda_{N})}\leq
2\lambda_{N}((-\delta,\delta)\backslash\\{0\\})+2\epsilon(M,\delta).$

Now let $M\to\infty$, to obtain

(9.47) $\sup_{N\geq M}\|D^{M}(u)-D^{N}(u)\|_{L_{2}(\lambda_{N})}\leq
2\lambda_{N}((-\delta,\delta)\backslash\\{0\\}).$

Subsequently let $\delta\to 0$, so that
$(-\delta,\delta)\backslash\\{0\\}\to\emptyset$ and

(9.48) $\lim_{M\to\infty}\sup_{N\geq
M}\|D^{M}(u)-D^{N}(u)\|^{2}_{L_{2}(\lambda_{N})}=0.$

∎

######  $\bullet$ Proof of Lemma 8.11

###### Proof.

We first prove Lemma 8.11[1]. Afterwards we construct
$(\bar{L}(N))_{N\in\mathbb{N}}$ to prove Lemma 8.11[2].

Since $\lim_{k\to\infty}\mu_{N_{k}}=\mu$, Lemma 8.10 implies that
$\lim_{k\to\infty}\mathcal{L}[\bar{\Theta}^{[N_{k}]}]=\mathcal{L}[\bar{\Theta}]$.
For ease of notation we drop the subsequence notation in the remainder of this
proof. By Skohorod’s theorem we can construct the random variables
$(z_{i}^{\mu_{N}})_{N\in\mathbb{N}}$ and $z_{i}^{\mu}$ on one probability
space such that $\lim_{N\to\infty}z_{i}^{\mu_{N}}=z_{i}$ a.s. Then, as in the
proof of Lemma 8.10, we obtain

(9.49) $\lim_{N\to\infty}\mathbb{E}[|\bar{\Theta}^{[N]}-\bar{\Theta}|]=0.$

To prove the claim we couple the two infinite systems, namely,

(9.50) $\displaystyle{\rm
d}x^{\mu_{N}}_{i}(t)=c\left[\bar{\Theta}^{[N]}-x^{\mu_{N}}_{i}(t)\right]{\rm
d}t+\sqrt{g(x^{\mu_{N}}_{i}(t))}\,{\rm
d}w_{i}(t)+Ke\,[y^{\mu_{N}}_{i}(t)-x^{\mu_{N}}_{i}(t)]\,{\rm d}t,$
$\displaystyle{\rm
d}y^{\mu_{N}}_{i}(t)=e\,[x^{\mu_{N}}_{i}(t)-y^{\mu_{N}}_{i}(t)]\,{\rm
d}t,\qquad i\in\mathbb{N}_{0},$

and

(9.51) $\displaystyle{\rm
d}x^{\mu}_{i}(t)=c\left[\bar{\Theta}-x^{\mu}_{i}(t)\right]{\rm
d}t+\sqrt{g(x^{\mu}_{i}(t))}\,{\rm
d}w_{i}(t)+Ke\,[y^{\mu}_{i}(t)-x^{\mu}_{i}(t)]\,{\rm d}t,$ $\displaystyle{\rm
d}y^{\mu}_{i}(t)=e\,[x^{\mu}_{i}(t)-y^{\mu}_{i}(t)]\,{\rm d}t,\qquad
i\in\mathbb{N}_{0},$

are coupled by using the same Brownian motions in (LABEL:ainf) and
(LABEL:binf). Like before, define
$\Delta^{\mu_{N}}_{i}=x_{i}^{\mu_{N}}-x_{i}^{\mu}$ and
$\delta^{\mu_{N}}_{i}=y_{i}^{\mu_{N}}-y_{i}^{\mu}$. By the construction with
Skohorod’s theorem, we have that

(9.52)
$\lim_{N\to\infty}\mathbb{E}\big{[}|\Delta^{\mu_{N}}_{i}(0)|+K|\delta^{\mu_{N}}_{i}(0)|\big{]}=0.$

To prove that, for all $t\geq 0$,

(9.53)
$\lim_{N\to\infty}\mathbb{E}\big{[}|\Delta^{\mu_{N}}_{i}(t)|+K|\delta^{\mu_{N}}_{i}(t)|\big{]}=0,$

we proceed as in the proof of Lemma 8.9. By Itô-calculus, we find that

(9.54)
$\displaystyle\mathbb{E}\big{[}|\Delta^{\mu_{N}}_{i}(t)|+K|\delta^{\mu_{N}}_{i}(t)|\big{]}$
$\displaystyle=\mathbb{E}\big{[}|\Delta^{\mu_{N}}_{i}(0)|+K|\delta^{\mu_{N}}_{i}(0)|\big{]}$
$\displaystyle\quad-c\int_{0}^{t}{\rm
d}r\,\mathbb{E}\left[|\Delta^{\mu_{N}}_{i}(r)|\right]$
$\displaystyle\quad-2Ke\int_{0}^{t}{\rm d}r\,\mathbb{E}\left[1_{{\rm
sign}\,\Delta^{\mu_{N}}_{i}(r)\neq{\rm
sign}\,\delta^{\mu_{N}}_{i}(r)}\,\,[|\delta^{\mu_{N}}_{i}(r)|+|\Delta^{\mu_{N}}_{i}(r)|]\right]$
$\displaystyle\quad+\int_{0}^{t}{\rm d}r\,c\
\mathbb{E}\left[\big{|}\bar{\Theta}^{[N]}-\bar{\Theta}\big{|}\right]$
$\displaystyle\leq\mathbb{E}\big{[}|\Delta^{\mu_{N}}_{i}(0)|+K|\delta^{\mu_{N}}_{i}(0)|\big{]}+tc\,\mathbb{E}\big{[}\big{|}\bar{\Theta}^{[N]}-\bar{\Theta}\big{|}\big{]}.$

From (9.54) it the follows that, for every $t\geq 0$,

(9.55)
$\lim_{N\to\infty}\mathbb{E}\big{[}|\Delta^{\mu_{N}}_{i}(t)|+K|\delta^{\mu_{N}}_{i}(t)|\big{]}=0.$

We next construct the sequence $(\bar{L}(N))_{N\in\mathbb{N}}$. From (9.36)
and (9.55),we have

(9.56)
$\lim_{N\to\infty}\mathbb{E}\big{[}|\Delta^{N}_{i}(t)|+K|\delta^{N}_{i}(t)|\big{]}+\mathbb{E}\big{[}|\Delta^{\mu_{N}}_{i}(t)|+K|\delta^{\mu_{N}}_{i}(t)|\big{]}=0.$

Let $(t_{k})_{k\in\mathbb{N}}$ be an increasing sequence such that
$\lim_{k\to\infty}t_{k}=\infty$ and $\lim_{k\to\infty}t_{k}/k=0$. By (9.56),
we can for each $k$ find an $N_{k}\in\mathbb{N}$ such that, for all $N\geq
N_{k}$,

(9.57)
$\mathbb{E}\big{[}|\Delta^{N}_{i}(t_{k})|+K|\delta^{N}_{i}(t_{k})|\big{]}+\mathbb{E}\big{[}|\Delta^{\mu_{N}}_{i}(t_{k})|+K|\delta^{\mu_{N}}_{i}(t_{k})|\big{]}<\frac{1}{k}.$

Requiring that $N_{k+1}>N_{k}$, we obtain a strictly increasing sequence
$(N_{k})_{k\in\mathbb{N}}$ that partitions $\mathbb{N}$. Set

(9.58) $=\sum_{k\in\mathbb{N}}t_{k}1_{\\{N_{k},\cdots,N_{k+1}-1\\}}(N).$

We show that $\bar{L}(N)$ satisfies the required properties:

  * •

$\lim_{N\to\infty}\mathbb{E}[|\Delta^{N}_{i}(\bar{L}(N))|+K|\delta^{N}_{i}(\bar{L}(N))|]+\mathbb{E}[|\Delta^{\mu_{N}}_{i}(\bar{L}(N))|+K|\delta^{\mu_{N}}_{i}(\bar{L}(N))|]=0$.
To proof this, we fix $\epsilon>0$ and let $K$ be such that
$\frac{1}{K}<\epsilon$. Then, for all $N\geq N_{K}$,

(9.59)
$\displaystyle\mathbb{E}\big{[}|\Delta^{N}_{i}(\bar{L}(N))|+K|\delta^{N}_{i}(\bar{L}(N))|\big{]}+\mathbb{E}\big{[}|\Delta^{\mu_{N}}_{i}(\bar{L}(N))|+K|\delta^{\mu_{N}}_{i}(\bar{L}(N))|\big{]}$
$\displaystyle\quad=\sum_{k\in\mathbb{N}}\mathbb{E}\big{[}|\Delta^{N}_{i}(t_{k})|+K|\delta^{N}_{i}(t_{k})|\big{]}+\mathbb{E}\big{[}|\Delta^{\mu_{N}}_{i}(t_{k})|+K|\delta^{\mu_{N}}_{i}(t_{k})|\big{]}\,1_{\\{N_{k},\cdots,N_{k-1}\\}}(N)$
$\displaystyle\quad<\frac{1}{K}<\epsilon.$

We conclude that

(9.60)
$\lim_{N\to\infty}\mathbb{E}\big{[}|\Delta^{N}_{i}(\bar{L}(N))|+K|\delta^{N}_{i}(\bar{L}(N))|\big{]}+\mathbb{E}\big{[}|\Delta^{\mu_{N}}_{i}(\bar{L}(N))|v+K|\delta^{\mu_{N}}_{i}(\bar{L}(N))|\big{]}=0.$

  * •

$\lim_{N\to\infty}\bar{L}(N)=\infty$. By (9.58), for each $k\in\mathbb{N}$
there exists an $N_{k}\in\mathbb{N}$ such that, for all $N\geq N_{k}$,
$\bar{L}(N)\geq t_{k}$ and $t_{k}\to\infty$. We conclude that
$\lim_{N\to\infty}\bar{L}(N)=\infty$.

  * •

$\lim_{N\to\infty}\bar{L}(N)/N=0$. Recall that $\lim_{k\to\infty}t_{k}/k=0$
and $N_{k}\geq k$ by construction. Hence
$\lim_{N\to\infty}\bar{L}(N)/N\leq\lim_{N\to\infty}\sum_{k\in\mathbb{N}}(t_{k}/k)]\,1_{\\{N_{k},\cdots,N_{k-1}\\}}(N)=0$.

Choosing $(t_{k})_{k\in\mathbb{N}}=(L(N))_{N\in\mathbb{N}}$, we complete the
proof of Lemma 8.11. ∎

######  $\bullet$ Proof of Lemma 8.6

###### Proof.

The goal is to prove that $\nu_{\theta}$ is continuous in $\theta$. To do so,
let $(\theta_{n})_{n\in\mathbb{N}}$ be a sequence in $[0,1]$ (note that
$\theta_{n}$ is not a random variable) such that
$\lim_{n\to\infty}\theta_{n}=\theta$. Couple the two infinite systems

(9.61) $\displaystyle{\rm
d}x^{n}_{i}(t)=c\left[\theta_{n}-x^{n}_{i}(t)\right]{\rm
d}t+\sqrt{g(x^{n}_{i}(t))}\,{\rm
d}w_{i}(t)+Ke\,[y^{n}_{i}(t)-x^{n}_{i}(t)]\,{\rm d}t,$ $\displaystyle{\rm
d}y^{n}_{i}(t)=e\,[x^{n}_{i}(t)-y^{n}_{i}(t)]\,{\rm d}t,\qquad
i\in\mathbb{N}_{0},$

and

(9.62) $\displaystyle{\rm d}x_{i}(t)=c\left[\theta-x_{i}(t)\right]{\rm
d}t+\sqrt{g(x_{i}(t))}\,{\rm d}w_{i}(t)+Ke\,[y_{i}(t)-x_{i}(t)]\,{\rm d}t,$
$\displaystyle{\rm d}y_{i}(t)=e\,[x_{i}(t)-y_{i}(t)]\,{\rm d}t,\qquad
i\in\mathbb{N}_{0},$

via their Brownian motions, like in the proof of Lemma 8.11. Let
$\mathcal{L}[(x_{i}^{n}(0),y_{i}^{n}(0))]=\delta_{(\theta_{n},\theta_{n})}$
and $\mathcal{L}[(x_{i}(0),y_{i}(0))]=\delta_{(\theta,\theta)}$. As before,
define $\Delta^{n}_{i}=x_{i}^{n}-x_{i}$ and $\delta^{n}_{i}=y_{i}^{n}-y_{i}$.
Note that

(9.63)
$\lim_{n\to\infty}\mathbb{E}\big{[}|\Delta^{n}_{i}(0)|+K|\delta^{n}_{i}(0)|\big{]}=0.$

By a similar argument as in the proof of Lemma 8.11, we obtain that, for all
$t\geq 0$,

(9.64)
$\lim_{n\to\infty}\mathbb{E}\big{[}|\Delta^{n}_{i}(t)|+K|\delta^{n}_{i}(t)|\big{]}=0.$

Hence we can construct a sequence $(L(n))_{n\in\mathbb{N}}$ satisfying
$\lim_{n\to\infty}L(n)=\infty$ and $\lim_{n\to\infty}L(n)/n=0$ such that

(9.65)
$\lim_{n\to\infty}\mathbb{E}\big{[}|\Delta^{n}_{i}(L(n))|+K|\delta^{n}_{i}(L(n))|\big{]}=0.$

To prove the continuity of the equilibrium $\nu_{\theta}$ in $\theta$, we
reason as follows. First note that we can couple the system in (LABEL:dainf)
starting from $\delta_{(\theta_{n},\theta_{n})}$ with the system in
(LABEL:dainf) starting from $\nu_{\theta_{n}}$. By the uniqueness and
convergence to equilibrium (see Lemma 8.5), we see that this coupling is
successful. Similarly, we can couple the system in (LABEL:dbinf) starting from
$\delta_{(\theta,\theta)}$ with the system in (LABEL:dbinf) starting from
$\nu_{\theta_{n}}$, and see the coupling succesful. Finally, we use (9.65) to
obtain

(9.66)
$\displaystyle\lim_{n\to\infty}\mathbb{E}^{\nu_{\theta_{n}}\times\nu_{\theta}}\big{[}|\Delta^{n}_{i}(L(n))|+K|\delta^{n}_{i}(L(n))|\big{]}$
$\displaystyle\leq\lim_{n\to\infty}\mathbb{E}^{\nu_{\theta_{n}}\times\delta_{(\theta_{n},\theta_{n})}}\big{[}|\Delta^{n}_{i}(L(n))|+K|\delta^{n}_{i}(L(n))|\big{]}$
$\displaystyle\quad+\lim_{n\to\infty}\mathbb{E}^{\delta_{(\theta,\theta)}\times\delta_{(\theta_{n},\theta_{n})}}\big{[}|\Delta^{n}_{i}(L(n))|+K|\delta^{n}_{i}(L(n))|\big{]}$
$\displaystyle\quad+\lim_{n\to\infty}\mathbb{E}^{\nu_{\theta}\times\delta_{(\theta,\theta)}}\big{[}|\Delta^{n}_{i}(L(n))|+K|\delta^{n}_{i}(L(n))|\big{]}=0.$

Let $f$ be a Lipschitz function. Then, by the equilibrium property of
$\nu_{\theta_{n}}$ and $\nu_{\theta}$,

(9.67)
$\displaystyle\lim_{n\to\infty}\left|\mathbb{E}^{\nu_{\theta_{n}}}\big{[}f(x^{n}(0))\big{]}-\mathbb{E}^{\nu_{\theta}}\big{[}f(x(0))\big{]}\right|$
$\displaystyle\quad=\lim_{n\to\infty}\left|\mathbb{E}^{\nu_{\theta_{n}}}\big{[}f(x^{n}(L(n)))\big{]}-\mathbb{E}^{\nu_{\theta}}\big{[}f(x(L(n)))\big{]}\right|$
$\displaystyle\quad=\lim_{n\to\infty}\left|\mathbb{E}^{\nu_{\theta_{n}}\times\nu_{\theta}}\big{[}f(x^{n}(L(n)))-f(x(L(n)))\big{]}\right|$
$\displaystyle\quad\leq\lim_{n\to\infty}(\mathrm{Lip}f)\,\mathbb{E}^{\nu_{\theta_{n}}\times\nu_{\theta}}\big{[}|(x^{n}(L(n)))-(x(L(n)))|\big{]}=0.$

We can also show this if $f$ is a Lipschitz function of the $y$ component.
Hence $\nu_{\theta}$ is indeed continuous as a function of $\theta$. ∎

######  $\bullet$ Proof of Lemma 8.13

###### Proof.

Note that for all $N\in\mathbb{N}$ fixed, by Itô-calculus we find from (7.15)
that

(9.68) $\displaystyle\frac{{\rm d}}{{\rm
d}t}\mathbb{E}\left[|x^{[N],1}_{i}(t)-x^{[N],2}_{i}(t)|+K|y^{[N],1}_{i}(t)-y^{[N],2}_{i}(t)|\right]$
$\displaystyle=-\frac{2c}{N}\sum_{j\in[N]}\mathbb{E}\left[|x^{[N]}_{j,1}(t)-x^{[N]}_{j,2}(t)|\,1_{\big{\\{}{\rm
sign}\big{(}x^{[N]}_{j,1}(t)-x^{[N]}_{j,2}(t)\big{)}\neq{\rm
sign}\big{(}x^{[N],1}_{i}(t)-x^{[N],2}_{i}(t)\big{)}\big{\\}}}\right]$
$\displaystyle\qquad-2Ke\,\mathbb{E}\Big{[}|x^{[N],1}_{i}(t)-x^{[N],2}_{i}(t)|$
$\displaystyle\qquad\qquad+|y^{[N],1}_{i}(t)-y^{[N],2}_{i}(t)|\,1_{\big{\\{}{\rm
sign}\big{(}x^{[N],1}_{i}(t)-x^{[N],2}_{i}(t)\big{)}\neq{\rm
sign}\big{(}y^{[N],1}_{i}(t)-y^{[N],2}_{i}(t)\big{)}\big{\\}}}\Big{]}.$

Therefore, for each $N\in\mathbb{N}$,

(9.69)
$t\mapsto\mathbb{E}\left[|x^{[N],1}_{i}(t)-x^{[N],2}_{i}(t)|+K|y^{[N],1}_{i}(t)-y^{[N],2}_{i}(t)|\right]\text{
is decreasing}.$

Fix any $t(N)\to\infty$. By the assumption in Lemma 8.13, the proofs of Lemma
8.9 and Lemma 8.11 imply that (9.56) holds for both $(X^{[N],1},Y^{[N],1})$
and $(X^{[N],2},Y^{[N],2})$. Using the construction in the proof of Lemma
8.11, we can construct _one_ sequence $(l(N))_{N\in\mathbb{N}}$, satisfying
$l(N)\leq t(N)$, $\lim_{N\to\infty}l(N)=\infty$ and
$\lim_{N\to\infty}l(N)/N=0$, such that (9.60) with $\bar{L}(N)$ replaced by
$l(N)$ holds for both the systems arising from $(X^{[N],1},Y^{[N],1})$ and
$(X^{[N],2},Y^{[N],2})$.

Write

(9.70)
$\displaystyle\mathbb{E}\left[|x^{[N],1}_{i}(l(N))-x^{[N],2}_{i}(l(N))|+K|y^{[N],1}_{i}(l(N))-y^{[N],2}_{i}(l(N))|\right]$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\left[|x^{[N],1}_{i}(l(N))-x^{\mu,1}_{i}(l(N))|+K|y^{[N],1}_{i}(l(N))-y^{\mu,1}_{i}(l(N))|\right]$
$\displaystyle+\mathbb{E}\left[|x^{\mu,1}_{i}(l(N))-x^{\mu,2}_{i}(l(N))|+K|y^{\mu,1}_{i}(l(N))-y^{\mu,2}_{i}(l(N))|\right]$
$\displaystyle+\mathbb{E}\left[|x^{\mu,2}_{i}(l(N))-x^{[N],2}_{i}(l(N))|+K|y^{\mu,2}_{i}(l(N))-y^{[N],2}_{i}(l(N))|\right].$

Note that in the right-hand side of the inequality the first and the third
term tend to zero by (9.60). The second term tends to zero because
$\mu\\{\bar{\Theta}_{1}=\bar{\Theta}_{2}\\}=1$, and hence Lemma 8.5 can be
applied. Therefore

(9.71)
$\lim_{N\to\infty}\mathbb{E}\left[|x^{[N],1}_{i}(l(N))-x^{[N],2}_{i}(l(N))|+K|y^{[N],1}_{i}(l(N))-y^{[N],2}_{i}(l(N))|\right]=0.$

Using the monotonicity in (9.69), we get

(9.72)
$\displaystyle\lim_{N\to\infty}\mathbb{E}\left[|x^{[N],1}_{i}(t(N))-x^{[N],2}_{i}(t(N))|+K|y^{[N],1}_{i}(t(N))-y^{[N],2}_{i}(t(N))|\right]$
$\displaystyle\leq\lim_{N\to\infty}\mathbb{E}\left[|x^{[N],1}_{i}(l(N))-x^{[N],2}_{i}(l(N))|+K|y^{[N],1}_{i}(l(N))-y^{[N],2}_{i}(l(N))|\right]=0.$

∎

Combining the proofs of Proposition 8.4, Lemma 8.11 and Lemma 8.13, we obtain
the following corollary. This corollary turns out to be important in Section
9.2 in the proof of Lemma 8.16 to obtain the limiting evolution of the
$1$-blocks.

######  Corollary 9.1.

Fix $s>0$. Let $\mu_{N}$ be the measure obtained by periodic configuration of

(9.73) $\mathcal{L}\big{[}X^{[N]}(Ns-L(N)),Y^{[N]}(Ns-L(N))\big{]},$

and let $\mu$ be a weak limit point of the sequence
$(\mu_{N})_{N\in\mathbb{N}}$. Let

(9.74)
$\bar{\Theta}=\lim_{N\to\infty}\frac{1}{N}\sum_{i\in[N]}\frac{x_{i}^{\mu}+Ky_{i}^{\mu}}{1+K}\text{
in }L^{2}(\mu).$

and let $(X^{\nu_{\bar{\Theta}}},Y^{\nu_{\bar{\Theta}}})$ be the infinite
system evolving according to (LABEL:binfb) and starting from its equilibrium
measure. Consider the finite system $(X^{[N]},Y^{[N]})$ as a system on
$([0,1]\times[0,1])^{\mathbb{N}_{0}}$ obtained by periodic continuation.
Construct $(X^{[N]},Y^{[N]})$ and
$(X^{\nu_{\bar{\Theta}}},Y^{\nu_{\bar{\Theta}}})$ on one probability space.
Then, for all $t\geq 0$,

(9.75)
$\displaystyle\lim_{N\to\infty}\mathbb{E}\left[\big{|}x_{i}^{[N]}(Ns+t)-x_{i}^{\nu_{\bar{\Theta}}}(t)\big{|}\right]+K\,\mathbb{E}\left[\big{|}y_{i}^{[N]}(Ns+t)-y_{i}^{\nu_{\bar{\Theta}}}(t)\big{|}\right]=0,\qquad\forall\,i\in[N].$

###### Proof.

By Proposition 8.4 we have that
$\lim_{k\to\infty}\mathcal{L}[X^{[N_{k}]}(N_{k}s)+Y^{[N_{k}]}(N_{k}s)]=\nu(s)=\nu_{\bar{\Theta}}$.
Let $\nu_{N_{k}}$ be defined by periodic continuation of the configuration of
$(X^{[N_{k}]}(N_{k}s),Y^{[N_{k}]}(N_{k}s))$ and let
$\nu=\lim_{k\to\infty}\nu_{N_{k}}$, so $\nu=\nu_{\bar{\Theta}}$. Construct the
process $(X^{[N]}(t),Y^{[N]}(t))_{t\geq 0}$,
$(X^{\nu_{N_{k}}}(t),Y^{\nu_{N_{k}}}(t))_{t\geq 0}$ and
$(X^{\nu}(t),Y^{\nu}(t))_{t\geq 0}$ on one probability space and use for all
processes the same Brownian motions. Then the couplings in the proofs of Lemma
8.9 and Lemma 8.11 imply (9.75). ∎

####  9.2 Proof of step 2. Convergence of the estimator

In this section we prove the three lemmas stated in Step 2 of Section 8.2.
Afterwards we prove Proposition 8.14 with the help of these lemmas.

######  $\bullet$ Proof of Lemma 8.15

###### Proof.

Recall the definition of $\bar{\Theta}^{[N]}(t)$ in (8.2). It follows from the
SSDE in (7.15) that

(9.76) ${\rm
d}\bar{\Theta}^{[N]}(t)=\frac{1}{1+K}\frac{1}{N}\sum_{i\in[N]}\sqrt{g\big{(}x_{i}^{[N]}(t)\big{)}}\,{\rm
d}w_{i}(t).$

Hence we see that $t\mapsto\bar{\Theta}^{[N]}(t)$ is a continuous martingale.
By Itô’s formula we have

(9.77)
$\displaystyle\mathbb{E}\big{[}\big{(}\bar{\Theta}^{[N]}(t)\big{)}^{2}\big{]}$
$\displaystyle=\mathbb{E}\big{[}\big{(}\bar{\Theta}^{[N]}(0)\big{)}^{2}\big{]}+\frac{1}{(1+K)^{2}}\int_{0}^{t}{\rm
d}r\,\frac{1}{N^{2}}\sum_{i\in[N]}g\big{(}x_{i}^{[N]}(r)\big{)}$
$\displaystyle\leq 1+\frac{1}{N}\frac{\|g\|}{(1+K)^{2}}\,t.$

Since $g$ is a bounded function, we get that $t\mapsto\bar{\Theta}^{[N]}(t)$
is square integrable. It follows that,

(9.78)
$\left(\big{(}\bar{\Theta}^{[N]}(Ns+t)-\bar{\Theta}^{[N]}(Ns)\big{)}^{2}\right)_{t\geq
0}$

is a sub-martingale. Therefore, defining the stopping time

(9.79) $S^{N}_{\epsilon}=\inf\left\\{t\geq
0:\big{(}\bar{\Theta}^{[N]}(Ns+t)-\bar{\Theta}^{[N]}(Ns)\big{)}^{2}\geq\epsilon\right\\}\wedge
L(N),$

we find, by the continuity of $t\mapsto\bar{\Theta}^{[N]}(Ns+t)$ and the
optional sampling theorem, that

(9.80)
$\mathbb{P}\left(S^{N}_{\epsilon}\in[Ns,Ns+L(N))\right)\leq\frac{1}{\epsilon^{2}}\mathbb{E}\left[\big{(}\bar{\Theta}^{[N]}(Ns+L(N))-\bar{\Theta}^{[N]}(Ns)\big{)}^{2}\right].$

Combining (9.77) and (9.80), we find

(9.81) $\lim_{N\to\infty}\sup_{0\leq t\leq
L(N)}\big{|}\bar{\Theta}^{[N]}(Ns+t)-\bar{\Theta}^{[N]}(Ns)\big{|}=0\text{ in
probability.}$

To obtain the increasing process, note that by Itô-calculus it follows from
(9.76) that

(9.82) $\big{\langle}\bar{\Theta}^{[N]}(Ns)\big{\rangle}_{s\geq
0}=\frac{1}{(1+K)^{2}}\int_{0}^{s}{\rm
d}r\,\frac{1}{N}\sum_{i\in[N]}{g\big{(}x_{i}^{[N]}(Nr)\big{)}}.$

We are left to show that the sequence of processes
$((\bar{\Theta}^{[N]}(Ns))_{s\geq 0})_{N\in\mathbb{N}}$ is tight. Note that
(9.82) implies that, for all $N\in\mathbb{N}$ and $s\geq 0$,

(9.83)
$\big{\langle}\bar{\Theta}^{[N]}(Ns)\big{\rangle}\leq\frac{\|g\|}{(1+K)^{2}}\,s$

and

(9.84) $\frac{{\rm d}}{{\rm
d}s}\big{\langle}\bar{\Theta}^{[N]}(Ns)\big{\rangle}\leq\frac{\|g\|}{(1+K)^{2}}.$

Hence the sequence $(\langle\bar{\Theta}^{[N]}(Ns)\rangle_{s\geq
0})_{N\in\mathbb{N}}$ is equicontinuous. Therefore, by the Arzela-Ascoli
theorem (see e.g. [Bil99, Theorem 7.2]), for each $T\geq 0$ the set
$(\langle\bar{\Theta}^{[N]}(Ns)\rangle_{0\leq s\leq T})_{N\in\mathbb{N}}$ is
relatively compact in $C([0,T],\mathbb{R})$, the space of continuous functions
from $[0,T]\to\mathbb{R}$. Therefore the set of laws
$(\mathcal{L}\langle\bar{\Theta}^{[N]}(Ns)\rangle_{0\leq s\leq
T})_{N\in\mathbb{N}}$ is tight in $\mathcal{P}(C([0,T],\mathbb{R}))$. Hence it
follows that $(\mathcal{L}\langle\bar{\Theta}^{[N]}(Ns)\rangle_{s\geq
0})_{N\in\mathbb{N}}$ is tight in $\mathcal{P}(C([0,\infty),\mathbb{R}))$, the
set of probability laws on $C([0,\infty),\mathbb{R})$.

Since $(\bar{\Theta}^{[N]}(Ns)-\bar{\Theta}^{[N]}(0))_{s\geq 0}$ is a
stochastic integral, we can represent it as a time-transformed Brownian motion
(see e.g. [RY99, Chapter 5]):

(9.85) $\big{(}\bar{\Theta}^{[N]}(Ns)-\bar{\Theta}^{[N]}(0)\big{)}_{s\geq
0}=w\big{(}\langle\bar{\Theta}^{[N]}(Ns)\rangle\big{)}_{s\geq 0}.$

Let $\chi$ be a standard normal random variable. Then

(9.86)
$\displaystyle\mathbb{E}\left[\left(w\big{(}\langle\bar{\Theta}^{[N]}(Ns)\rangle\big{)}-w\big{(}\langle\bar{\Theta}^{[N]}(Nr)\rangle\big{)}\right)^{2}\right]$
$\displaystyle\leq\mathbb{E}\left[\left(\langle\bar{\Theta}^{[N]}(Ns)\rangle-\langle\bar{\Theta}^{[N]}(Nr)\rangle\right)^{2}\right]\mathbb{E}\left[\chi^{4}\right]$
$\displaystyle\leq\left(s-r\right)^{2}\frac{\|g\|^{2}}{(1+K)^{4}}\,\mathbb{E}\left[\chi^{4}\right].$

Hence it follows from Kolmogorov’s criterion for weak compactness (see e.g.
[RY99, Chapter XIII, Theorem 1.8]) that the sequence
$(\mathcal{L}[(\bar{\Theta}^{[N]}(Ns))_{s\geq 0}])_{N\in\mathbb{N}}$ is tight
in $\mathcal{P}(C([0,\infty),\mathbb{R}))$. ∎

######  $\bullet$ Proof of Lemma 8.16

###### Proof.

For ease of notation we will suppress the subsequence notation and assume that

(9.87)
$\lim_{N\to\infty}\mathcal{L}\left[\big{(}\bar{\Theta}^{[N]}(Ns)\big{)}_{s>0}\right]=\mathcal{L}\bigl{[}(\bar{\Theta}(s))_{s>0}\bigr{]}.$

The processes $(\bar{\Theta}^{[N]}(Ns))_{s\geq 0}$ are martingales, see
(9.76), measurable w.r.t. the canonical filtration $(\mathcal{F}_{s})_{s\geq
0}$ and so are there weak limit points. Therefore also the weak limit point
$(\bar{\Theta}(s))_{s>0}$ is a martingale, (see [DG93b, Section 3]). To obtain
(8.52), we use the following strategy. Recall from the proof of Lemma 8.15
that the sequence
$\\{\langle\bar{\Theta}^{[N]}(Ns)\rangle_{s>0}\\}_{N\in\mathbb{N}}$ is tight.
Hence, in order to prove that

(9.88)
$\lim_{N\to\infty}\mathcal{L}\left[\langle\bar{\Theta}^{[N]}(Ns)\rangle_{s>0}\right]=\mathcal{L}\left[\left(\int_{0}^{s}{\rm
d}r\frac{1}{(1+K)^{2}}\,\mathbb{E}^{\nu_{\bar{\Theta}(r)}}[g(x_{0})]\right)_{s>0}\right],$

it is enough to prove that the finite-dimensional distributions of
$(\langle\bar{\Theta}^{[N]}(Ns)\rangle_{s>0})_{N\in\mathbb{N}}$ converge to
the finite-dimensional distribution of $(\int_{0}^{s}{\rm
d}r\,\frac{1}{(1+K)^{2}}\mathbb{E}^{\nu_{\bar{\Theta}(r)}}[g(x_{0})])_{s>0}$.
We will prove a slightly stronger result, namely,

(9.89)
$\lim_{N\to\infty}\mathbb{E}\left[\left|\langle\bar{\Theta}^{[N]}(Ns)\rangle-\int_{0}^{s}{\rm
d}r\,\frac{1}{(1+K)^{2}}\,\mathbb{E}^{\nu_{\bar{\Theta}(r)}}[g(x_{0})]\right|\right]=0.$

Once we have obtained (9.89) and hence (9.88), by Skohorod’s theorem we can
construct the processes
$(\langle\bar{\Theta}^{[N]}(Ns)\rangle_{s>0})_{N\in\mathbb{N}}$ on a single
probability space, to obtain

(9.90) $\lim_{N\to\infty}\langle\bar{\Theta}^{[N]}(Ns)\rangle_{s\geq
0}=\left(\int_{0}^{s}{\rm
d}r\,\frac{1}{(1+K)^{2}}\mathbb{E}^{\nu_{\bar{\Theta}(r)}}[g(x_{0})]\right)_{s\geq
0}\quad a.s.$

Using the continuity of Brownian motion, we get that (recall (9.85))

(9.91)
$\displaystyle\lim_{N\to\infty}\big{(}\bar{\Theta}^{[N]}(Ns)\big{)}_{s>0}$
$\displaystyle=\lim_{N\to\infty}\left[w\big{(}\langle\bar{\Theta}^{[N]}(Ns)\rangle\big{)}_{s>0}+\bar{\Theta}^{[N]}(0)\right]$
$\displaystyle=w\left(\int_{0}^{s}{\rm
d}r\,\frac{1}{(1+K)^{2}}\,\mathbb{E}^{\nu_{\bar{\Theta}(r)}}[g(x_{0})]\right)_{s>0}+\vartheta_{0}\quad
a.s.$

Therefore we can choose a version of $(\bar{\Theta}(s))_{s>0}$ such that

(9.92)
$\lim_{N\to\infty}\big{(}\bar{\Theta}^{[N]}(Ns)\big{)}_{s>0}=\lim_{N\to\infty}\big{(}\bar{\Theta}(s)\big{)}_{s>0}\quad
a.s.$

and

(9.93)
$\lim_{N\to\infty}\big{(}\bar{\Theta}^{[N]}(Ns),\langle\bar{\Theta}^{[N]}(Ns)\rangle\big{)}_{s>0}=\left(\bar{\Theta}(s),\langle\bar{\Theta}(s)\rangle\right)_{s>0}\quad
a.s.$

By the continuous mapping theorem, (8.52) follows. The martingale property
follows from the fact that
$\big{(}\bar{\Theta}^{[N]}(Ns)^{2}-\langle\bar{\Theta}^{[N]}(Ns)\rangle\big{)}_{s>0}$
are martingales. Therefore, to finish the proof of Lemma 8.16 we are left to
prove (9.89).

To prove (9.89), define the empirical measures on $[0,1]$ by

(9.94) $U^{[N]}(Ns)=\frac{1}{N}\sum_{i\in[N]}\delta_{x_{i}(Ns)}.$

Note that we can write

(9.95)
$\displaystyle\mathbb{E}\left[\left|\langle\bar{\Theta}^{[N]}(Ns)\rangle-\int_{0}^{s}{\rm
d}r\,\frac{1}{(1+K)^{2}}\,\mathbb{E}^{\nu_{\bar{\Theta}(r)}}[g(x_{0})]\right|\right]$
$\displaystyle=\frac{1}{(1+K)^{2}}\,\mathbb{E}\left[\left|\int_{0}^{s}{\rm
d}r\,\,\mathbb{E}^{U^{[N]}(Nr)}[g(x_{0})]-\int_{0}^{s}{\rm
d}r\,\,\mathbb{E}^{\nu_{\bar{\Theta}(r)}}[g(x_{0})]\right|\right]$
$\displaystyle\leq\frac{1}{(1+K)^{2}}\int_{0}^{s}{\rm
d}r\,\mathbb{E}\left[\left|\mathbb{E}^{U^{[N]}(Nr)}[g(x_{0})]-\mathbb{E}^{\nu_{\bar{\Theta}(r)}}[g(x_{0})]\right|\right].$

Hence, to prove (9.89) it is enough to prove that, for all $r>0$,

(9.96)
$\lim_{N\to\infty}\mathbb{E}\left[\left|\mathbb{E}^{U^{[N]}(Nr)}[g(x_{0})]-\mathbb{E}^{\nu_{\bar{\Theta}(r)}}[g(x_{0})]\right|\right]=0$

and apply the dominated convergence theorem.

To prove (9.96), we will use the coupling arguments from Section 9.1, as well
as ergodicity and invariance under the evolution of $\nu_{\bar{\Theta}}$. As
before, let $z^{[N]}(t)$ denote the $[N]$-component system
$(x_{i}^{[N]}(t),y_{i}^{[N]}(t))_{t\geq 0}$ evolving according to (7.15),
viewed as a system on $\mathbb{N}_{0}$ obtained by periodic continuation and
with initial law
$\mathcal{L}[z^{[N]}(0)]=\mathcal{L}[x_{i}^{[N]}(Nr-L(N)),y_{i}^{[N]}(Nr-L(N))]$.
Let $(z^{\mu_{N}}(t))_{t\geq 0}$ denote the infinite system
$(x_{i}^{\mu_{N}}(t),y_{i}^{\mu_{N}}(t))_{t\geq 0}$ evolving according to
(LABEL:gh45ainf) with initial law $\mu_{N}$ obtained by periodic continuation
of the configuration of $(x_{i}^{[N]}(Nr-L(N)),y_{i}^{[N]}(Nr-L(N)))$, and let
$\mu$ be a weak limit point of the sequence $\mu_{N}$. Note that, for all
$r>0$, by Lemma 8.10 we have that
$\lim_{N\to\infty}\bar{\Theta}^{[N]}(Nr)=\bar{\Theta}(r)$ for
$\bar{\Theta}(r)=\lim_{n\to\infty}\frac{1}{n}\sum_{i\in[n]}\frac{x_{i}+Ky_{i}}{1+K}\text{
in }L_{2}(\mu)$. Let $\bar{L}(N)$ be the sequence constructed in Corollary
9.1. Then we can write

(9.97)
$\displaystyle\mathbb{E}\left[\left|\mathbb{E}^{U^{[N]}(Nr)}[g(x_{0})]-\mathbb{E}^{\nu_{\bar{\Theta}}(r)}[g(x_{0})]\right|\right]$
$\displaystyle\leq\mathbb{E}\left[\left|\frac{1}{N}\sum_{i\in[N]}g(x_{i}^{[N]}(Nr))-\frac{1}{N}\sum_{i\in[N]}g\big{(}x_{i}^{\nu_{\bar{\Theta}}(r)}(\bar{L}(N))\big{)}\right|\right]$
$\displaystyle\qquad+\mathbb{E}\left[\left|\frac{1}{N}\sum_{i\in[N]}g\big{(}x_{i}^{\nu_{\bar{\Theta}(r)}}(\bar{L}(N))\big{)}-\mathbb{E}^{\nu_{\bar{\Theta}(r)}}[g(x_{0})]\right|\right]$
$\displaystyle\leq(\text{Lip}\,g)\,\mathbb{E}\left[\left|x_{0}^{[N]}(Nr)-x_{0}^{\nu_{\bar{\Theta}}(r)}(\bar{L}(N))\right|\right]$
$\displaystyle\qquad+\mathbb{E}\left[\left|\frac{1}{N}\sum_{i\in[N]}g\big{(}x_{i}^{\nu_{\bar{\Theta}(r)}}(\bar{L}(N))\big{)}-\mathbb{E}^{\nu_{\bar{\Theta}(r)}}[g(x_{0})]\right|\right],$

where in the second inequality we use the Lipschitz property of $g$ and the
translation invariance of the system. Note that the first term tends to $0$ as
$N\to\infty$ by Corollary 9.1. Finally, note that by Lemma 8.5
$(x_{i})_{i\in\mathbb{N}_{0}}$ is a sequence of bounded i.i.d. random
variables under $\nu_{\bar{\Theta}(r)}$. Hence the last term tends to zero by
the law of large numbers. ∎

######  $\bullet$ Proof of Lemma 8.17

###### Proof.

Note that, since $g$ is Lipschitz, the function

(9.98) $\theta\mapsto\mathbb{E}^{\nu_{\theta}}[g],$

is Lipschitz by Lemma 8.6. Hence, by [YW71, Theorem 1], the SDE

(9.99) ${\rm
d}\Phi(s)=\frac{1}{(1+K)}\sqrt{\mathbb{E}^{\nu_{\Phi}(s)}[g]}\,{\rm d}w(s)$

has a pathwise unique solution (see i̧te[p315–317]KS91) and a unique solution
in law (see [YW71, Proposition 1]). This implies that the martingale problem
with generator

(9.100) $\frac{1}{(1+K)^{2}}\,\mathbb{E}^{\nu_{\Phi}}[g]\frac{{\rm d}}{{\rm
d}\Phi^{2}}$

has a unique solution. In particular, choosing $f(\Phi)=\Phi^{2}$, we see that
the martingale problem implies that

(9.101) $\left(\Phi^{2}(s)-\frac{1}{(1+K)^{2}}\int_{0}^{s}{\rm
d}u\,\mathbb{E}^{\nu_{\Phi(u)}}[g(x_{0})]\right)_{s>0}$

is a martingale.

Since $(\bar{\Theta}(s))_{s>0}$ is a continuous bounded martingale, it could
be written as a time transformed Brownian motion. The uniqueness of the
martingale problem in (8.53) now follows from the fact that the quadratic
variation of a martingale is unique. ∎

######  $\bullet$ Proof of Proposition 8.14

###### Proof.

Combining Lemma 8.16–8.17, all converging subsequences of
$(\mathcal{L}[(\bar{\Theta}^{[N]}(Ns))_{s\geq 0}])_{N\in\mathbb{N}}$ converge
to the same limit, which is the unique process satisfying the martingale
problem in (8.53). ∎

####  9.3 Proof of step 3. Convergence of the $1$-blocks in the Meyer-Zheng
topology

In this section we prove Proposition 8.18 stated in Step 3 of Section 8.2. The
Lemmas 8.19, 8.20 and 8.21 are proven in Appendix B.

######  $\bullet$ Proof of Proposition 8.18

###### Proof.

By Proposition 8.14 we have that

(9.102)
$\lim_{N\to\infty}\mathcal{L}[\bar{\Theta}^{[N]}(Ns)_{s>0}]=\mathcal{L}[(\bar{\Theta}(s))_{s>0}]$

in the normal topology and therefore (see B Lemma B.1)

(9.103)
$\lim_{N\to\infty}\mathcal{L}[\bar{\Theta}^{[N]}(Ns)_{s>0}]=\mathcal{L}[(\bar{\Theta}(s))_{s>0}]\,\text{
in Meyer-Zheng topology}.$

By Lemma 8.7, for $s>0$,

(9.104)
$\lim_{N\to\infty}\mathbb{E}\left[\left|\bar{\Theta}^{[N]}(Ns)-x_{1}^{[N]}(s)\right|\right]=0$

and therefore, by Lemma 8.19,

(9.105)
$\lim_{N\to\infty}d_{P}(\psi_{\bar{\Theta}^{[N]}},\psi_{x_{1}^{[N]}})=0\text{
in probability}.$

To apply the above results to our model, we recall the following basic result
(see [Bil99, Chapter 1]), which also holds for the Meyer-Zheng topology.

######  Lemma 9.2.

Let $X_{n}$, $Y_{n}$ be random variables. If

(9.106) $\lim_{n\to\infty}\mathcal{L}[X_{n}]=\mathcal{L}[X]$

and

(9.107) $\lim_{n\to\infty}d(X_{n},Y_{n})=0\text{ in probability},$

then

(9.108) $\lim_{n\to\infty}\mathcal{L}[Y_{n}]=\mathcal{L}(X).$

Applying Lemma (9.2) to our case, we obtain

(9.109)
$\lim_{N\to\infty}\mathcal{L}[(x_{1}^{[N]}(s))_{s>0}]=\mathcal{L}[(\bar{\Theta}(s))_{s>0}]\,\text{
in the Meyer-Zheng topology}.$

The argument for

(9.110)
$\lim_{N\to\infty}\mathcal{L}[(y_{1}^{[N]}(s))_{s>0}]=\mathcal{L}[(\bar{\Theta}(s))_{s>0}]\,\text{
in the Meyer-Zheng topology}$

follows in the same way. By Lemma 8.20, we obtain

(9.111)
$\lim_{N\to\infty}\mathcal{L}[(x_{1}^{[N]}(s),y_{1}^{[N]}(s)-x_{1}^{[N]}(s))_{s>0}]=\mathcal{L}[(\bar{\Theta}(s),0)_{s>0}]\,\text{
in the Meyer-Zheng topology}.$

Applying Lemma 8.21 with $f(x,y)=f(x,y+x)$ and the continuous mapping theorem,
we obtain

(9.112)
$\lim_{N\to\infty}\mathcal{L}\left[\left(x_{1}^{[N]}(s),y_{1}^{[N]}(s)\right)_{s>0}\right]=\mathcal{L}[(\bar{\Theta}(s),\bar{\Theta}(s))_{s>0}]\,\text{
in the Meyer-Zheng topology.}$

∎

####  9.4 Proof of step 4. Mean-field finite-systems scheme

######  $\bullet$ Proof of Proposition 8.1

###### Proof.

Proposition 8.1(b) follows directly from Propostion 8.14. The proof of
Proposition 8.1(a) follows from Proposition 8.1(b) and Proposition 8.18.

To prove Proposition 8.1(c) fix $t>0$. Consider the processes
$(X^{[N]}(sN+t),Y^{[N]}(sN+t))_{t\geq 0}$ as processes on
$([0,1]^{2})^{\mathbb{N}_{0}}$ by periodic continuation. Since
$([0,1]^{2})^{\mathbb{N}_{0}}$ is compact, the sequence
$(X^{[N]}(sN+t),Y^{[N]}(sN+t))_{N\in\mathbb{N}}$ is tight and hence has a
converging subsequence. Let $(\bar{\Theta}(s))_{s>0}$ be the limiting process
obtained in Proposition (8.14). This has continuous paths and is the unique
solution of a well-posed martingale problem, and hence is a Markov process.
Denote by $Q_{s}$ the time-$s$ semigroup corresponding to
$(\bar{\Theta}(s))_{s\geq 0}$. Combining Proposition 8.4, Proposition 8.14 and
Lemma 8.15, we get that, for each converging subsequence,

(9.113)
$\lim_{k\to\infty}\mathcal{L}\left[X^{[N_{k}]}(sN_{k}+t),Y^{[N_{k}]}(sN_{k}+t)\right]=\int
Q_{s}\left(\theta,{\rm d}\theta^{\prime}\right)\nu_{\theta^{\prime}}=\nu(s),$

and hence it follows that, for all $t\geq 0$,

(9.114)
$\lim_{N\to\infty}\mathcal{L}\left[X^{[N]}(sN+t),Y^{[N]}(sN+t)\right]=\int
Q_{s}\left(\theta,{\rm d}\theta^{\prime}\right)\nu_{\theta^{\prime}}=\nu(s).$

Let $(X^{\nu(s)}(t),Y^{\nu(s)}(t))_{t\geq 0}$ be the infinite system defined
in (7.18), starting from initial measure $\nu(s)$. Then it follows from
Corollary 9.1 that we can construct the processes
$(X^{[N]}(sN+t),Y^{[N]}(sN+t))_{t\geq 0}$ and
$(X^{\nu(s)}(t),Y^{\nu(s)}(t))_{t\geq 0}$ on one probability space such that,
for all $t\geq 0$,

(9.115)
$\lim_{N\to\infty}\mathbb{E}\left[\big{|}z^{\nu(s)}_{(i,R_{i})}(t)-z^{[N]}_{(i,R_{i})}(sN+t)\big{|}\right]=0\qquad\forall\,(i,R_{i})\in\mathbb{Z}\times\\{A,D\\}.$

Hence we see that the finite-dimensional distributions of the process
$(X^{[N]}(sN+t),Y^{[N]}(sN+t))_{t\geq 0}$ converge to the finite-dimensional
distributions of te process $(X^{\nu(s)}(t),Y^{\nu(s)}(t))_{t\geq 0}$.

Since we want that

(9.116)
$\lim_{N\to\infty}\mathcal{L}\left[\big{(}X^{[N]}(sN+t),Y^{[N]}(sN+t)\big{)}_{t\geq
0}\right]=\mathcal{L}\left[\big{(}X^{\nu(s)}(t),Y^{\nu(s)}(t)\big{)}_{t\geq
0}\right],$

we are left to show the tightness of
$\mathcal{L}[(X^{[N]}(sN+t),Y^{[N]}(sN+t))_{t\geq 0}]_{N\in\mathbb{N}}$ in the
path space
$\mathcal{C}([0,\infty),\left([0,1]\times[0,1]\right)^{\mathbb{N}_{0}})$.
Since $([0,1]^{2})^{\mathbb{N}_{0}}$ is endowed with the product topology, it
is enough to show for all sequence components $(x^{[N]}_{i}(t))_{t\geq 0}$ and
$(y^{[N]}_{i}(t))_{t\geq 0}$ that they are tight in path space (see [Bil99,
Theorem 7.3]).

To prove that the components are tight, we use a tightness criterion for semi-
martingales by Joffe and Metivier, [JM86, Proposition 3.2.3]. To use this
criterion, we have to show that for all $i\in[N]$ the components
$(x^{[N]}_{i}(t),y^{[N]}_{i}(t))$ are $\mathcal{D}$-semi-martingales as
defined in [JM86, Definition 3.1.1]. To do this, let
$\mathcal{C}^{*}\subset\mathcal{C}_{b}\left([0,1]^{2}\right)$ be the set of
polynomials on $[0,1]^{2}$, and define the operator
$\colon\,(\mathcal{C}^{*}\times[0,1]^{2}\times[0,\infty),\Omega)\to\mathbb{R}$
by

(9.117)
$G_{\dagger}^{[N]}(f,(x,y),t,\omega)=\left[\frac{c}{N}\sum_{j\in[N]}[x^{[N]}_{j}(t,\omega)-x]+K\,[y-x]\right]\frac{\partial
f}{\partial x}\\\ +\frac{1}{2}g(x)\frac{\partial^{2}f}{\partial
x^{2}}+e\,[x-y]\frac{\partial f}{\partial y}.$

We use the subscript $\dagger$ to emphasize that $G^{[N]}_{\dagger}$ is the
operator of a $\mathcal{D}$-semi-martingale and not a generator. Below we
check in 4 steps that the component processes
$(x_{i}^{[N]}(t),y_{i}^{[N]}(t))_{t\geq 0}$ are indeed $\mathcal{D}$-semi-
martingales.

  1. 1.

The functions

(9.118) $f_{1}(x_{i},y_{i})=x_{i},\quad f_{2}(x_{i},y_{i})=y_{i},$

are in $\mathcal{C}^{*}$, and so are $f_{1}^{2}$, $f_{1}f_{2}$, $f_{2}^{2}$.

  2. 2.

For every
$((x,y),t,\omega)\in\left([0,1]^{2}\times[0,\infty)\times\Omega\right)$, the