mapping $f\mapsto G_{\dagger}^{[N]}(f,(x,y),t,\omega)$ is linear on
$\mathcal{C}^{*}$ and $G_{\dagger}^{[N]}(f,\cdot,t,\omega)\in\mathcal{C}^{*}$.

  3. 3.

Let $(\mathcal{F}_{s})_{s\geq 0}$ be the filtration generated by the Brownian
motions $((w_{i}(s))_{s\geq 0})_{i\in[N]}$, and let $\mathcal{P}$ be the
$\sigma$-algebra generated by the predictable sets, i.e., sets of the form
$(s,t]\times F$ for $F\in\mathcal{F}_{s}$. Since the component processes
$(x_{j}^{[N]}(t))_{t\geq 0}$ are continuous, $((x,y),t,\omega)\mapsto
G^{[N]}_{\dagger}(f,(x,y),t,\omega)$ is
$\mathcal{B}([0,1]^{2})\otimes\mathcal{P}$ measurable for every
$f\in\mathcal{C}^{*}$, where $\mathcal{P}$ is the $\sigma$-algebra generated
by the sets of the form $(s,t]\times F$ for $F\in\mathcal{F}_{s}$.

  4. 4.

Applying Itô’s formula to the SSDE in (7.15), we obtain, for every
$f\in\mathcal{C}^{*}$,

(9.119) $\displaystyle f\big{(}x_{i}^{[N]}(t),y_{i}^{[N]}(t)\big{)}$
$\displaystyle=f\big{(}x_{i}^{[N]}(0),y_{i}^{[N]}(0)\big{)}$
$\displaystyle\qquad+\int_{0}^{t}{\rm
d}s\,\frac{c}{N}\sum_{j\in[N]}\big{[}x^{[N]}_{j}(s,\omega)-x^{[N]}_{i}(s)\big{]}\,\frac{\partial
f}{\partial x}\big{(}x_{i}^{[N]}(t),y_{i}^{[N]}(t)\big{)}$
$\displaystyle\qquad+\frac{1}{2}\int_{0}^{t}{\rm
d}w_{i}(s)\,\sqrt{g(x_{i}^{[N]}(s))}\,\,\frac{\partial f}{\partial
x}\big{(}x_{i}^{[N]}(t),y_{i}^{[N]}(t)\big{)}$
$\displaystyle\qquad+\int_{0}^{t}{\rm
d}s\,Ke\big{[}y_{i}^{[N]}(s)-x_{i}^{[N]}(s)\big{]}\,\frac{\partial f}{\partial
x}\big{(}x_{i}^{[N]}(t),y_{i}^{[N]}(t)\big{)}$
$\displaystyle\qquad+\int_{0}^{t}{\rm
d}s\,e\big{[}x_{i}^{[N]}(s)-y_{i}^{[N]}(s)\big{]}\,\frac{\partial f}{\partial
y}\big{(}x_{i}^{[N]}(t),y_{i}^{[N]}(t)\big{)}$
$\displaystyle\qquad+\int_{0}^{t}{\rm
d}s\,g(x_{i}^{[N]}(s))\,\,\frac{\partial^{2}f}{\partial
x^{2}}\big{(}x_{i}^{[N]}(t),y_{i}^{[N]}(t)\big{)}.$

Therefore

(9.120) $\displaystyle M^{[N],f}(t,\omega)$
$\displaystyle=f\big{(}x_{i}^{[N]}(t,\omega),y_{i}^{[N]}(t,\omega)\big{)}-f\big{(}x_{i}^{[N]}(0,\omega),y_{i}^{[N]}(0,\omega)\big{)}$
$\displaystyle\qquad-\int_{0}^{t}{\rm
d}s\,G_{\dagger}^{[N]}\big{(}f(x_{i}^{[N]}(s,\omega),y_{i}^{[N]}(s,\omega)),s,\omega\big{)}$

is a square-integrable martingale on $\left(\Omega,(\mathcal{F}_{s})_{s\geq
0},\mathbb{P}\right)$.

To check that the sequence of component processes
$((x_{i}^{[N]}(t),y_{i}^{[N]}(t)))_{N\in\mathbb{N}}$ is tight, we need the
local characteristics of the $\mathcal{D}$-semi-martingale, which are defined
in [JM86, Definition 3.1.2] as (recall (9.118))

(9.121) $\displaystyle b^{[N]}_{1}((x,y),t,\omega)$
$\displaystyle=G^{[N]}_{\dagger}(f_{1},(x,y),t,\omega),$ $\displaystyle
b^{[N]}_{2}((x,y),t,\omega)$
$\displaystyle=G^{[N]}_{\dagger}(f_{2},(x,y),t,\omega),$ $\displaystyle
a^{[N]}_{(1,1)}((x,y),t,\omega)$
$\displaystyle=G^{[N]}_{\dagger}(f_{1}f_{1},(x,y),t,\omega)-2x\,b_{1}((x,y),t,\omega),$
$\displaystyle a^{[N]}_{(2,1)}((x,y),t,\omega)$
$\displaystyle=G^{[N]}_{\dagger}(f_{1}f_{2},(x,y),t,\omega)-x\,b_{2}((x,y),t,\omega)-yb_{1}((x,y),t,\omega),$
$\displaystyle a^{[N]}_{(1,2)}((x,y),t,\omega)$
$\displaystyle=a_{(2,1)}((x,y),t,\omega),$ $\displaystyle
a^{[N]}_{(2,2)}((x,y),t,\omega)$
$\displaystyle=G^{[N]}_{\dagger}(f_{2}f_{2},(x,y),t,\omega)-2y\,b_{2}((x,y),t,\omega).$

Hence

(9.122) $\displaystyle b^{[N]}_{1}((x,y),t,\omega)$
$\displaystyle=\frac{c}{N}\sum_{j\in[N]}[x_{j}^{[N]}(t,\omega)-x]+Ke[y-x],$
$\displaystyle b^{[N]}_{2}((x,y),t,\omega)$ $\displaystyle=e[x-y],$
$\displaystyle a^{[N]}_{(1,1)}((x,y),t,\omega)$ $\displaystyle=2g(x),$
$\displaystyle a^{[N]}_{(1,2)}((x,y),t,\omega)$
$\displaystyle=a_{(2,1)}((x,y),t,\omega)=0,$ $\displaystyle
a^{[N]}_{(2,2)}((x,y),t,\omega)$ $\displaystyle=0.$

Here, $b^{[N]}_{i}$ and $a^{[N]}_{i,j}$, $i,j\in\\{1,2\\}$, are called the
local coefficients of first and second order. We check that the hypotheses
[JM86, H1, H2, H3 in Section 3.2.1] are satisfied.

  * $H_{1}$:

For all $N\in\mathbb{N}$,

(9.123)
$\sum_{i\in\\{1,2\\}}|b^{[N]}_{i}((x,y),t,\omega)|^{2}+\sum_{i,j\in\\{1,2\\}}|a^{[N]}_{(i,j)}((x,y),t,\omega)|^{2}\leq
4(c+Ke+e)^{2}+2||g||^{2}.$

Hence, choosing as positive adapted process the constant process $1$ and
letting the constant be equal to $4(c+Ke+e)^{2}+2||g||^{2}$, we see that
$H_{1}$ is satisfied.

  * $H_{2}$:

Since the component processes are bounded by $1$, also $H_{2}$ is satisfied.

  * $H_{3}$:

Since the increasing càdlàg function $(A^{[N]}(t))_{t\geq 0}$ in [JM86,
Definition 3.1.1] is in our case $A^{[N]}(t)=t$, also $H_{3}$ is satisfied.

Since $H_{1}$, $H_{2}$, $H_{3}$ are met, [JM86, Proposition 3.2.3] implies
that $((x_{i}^{[N]}(t),y_{i}^{[N]}(t))_{t>0})_{N\in\mathbb{N}}$ are tight in
the space of càdlàg paths $\mathcal{D}((0,\infty],[0,1]^{2})$. Hence (9.116)
indeed holds. ∎

###  10 Preparation: $N\to\infty$, two-colour mean-field

In this section we extend the results obtained in Section 8.1 to a mean-field
system where the seed-bank consists of two colours, one colour that interacts
on the slow time scale and one colour that interacts on the fast time scale.
To do so we follow the set-up used in Chapter 4. In particular, we highlight
the role of the second colour. Section 10.1 builds up the setting and states
the main scaling result: Proposition 10.2. Section 10.2 provides the proof of
this proposition based on a series of lemmas, which are stated and proved
first.

####  10.1 Two-colour mean-field finite-systems scheme

###### Setup.

In this section we consider a simplified version of our SSDE in (2.20) on the
finite geographic space

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

The migration kernel $a^{\Omega_{N}}(\cdot,\cdot)$ is replaced by the
migration kernel $a^{[N]}(i,j)=c_{0}N^{-1}$ for all $i,j\in[N]$, where
$c_{0}\in(0,\infty)$ is a constant. The seed-bank consists of _two colours_ ,
labeled $0$ and $1$. The exchange rates between the active and the colour-$0$
dormant population are given by $K_{0}e_{0},e_{0}$. The exchange rates between
active and the colour-$1$ dormant population are given by
$\frac{K_{1}e_{1}}{N},\frac{e_{1}}{N}$. The state space is

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

and the system, consisting of three components, is denoted by

(10.3) $\displaystyle
Z^{[N]}(t)=\big{(}X^{[N]}(t),(Y_{0}^{[N]}(t),Y_{1}^{[N]}(t))\big{)}_{t\geq
0},$
$\displaystyle\big{(}X^{[N]}(t),(Y_{0}^{[N]}(t),Y_{1}^{[N]}(t))\big{)}=\big{(}x_{i}(t),(y_{i,0}(t),y_{i,1}(t))\big{)}_{i\in[N]}.$

The components of $(Z^{[N]}(t))_{t\geq 0}$ evolve according to the SSDE

(10.4) $\displaystyle{\rm
d}x^{[N]}_{i}(t)=\frac{c_{0}}{N}\sum_{j\in[N]}[x^{[N]}_{j}(t)-x^{[N]}_{i}(t)]\,{\rm
d}t+\sqrt{g(x^{[N]}_{i}(t))}\,{\rm d}w_{i}(t)$
$\displaystyle\qquad\qquad+K_{0}e_{0}\,[y^{[N]}_{i,0}(t)-x^{[N]}_{i}(t)]\,{\rm
d}t+\frac{K_{1}e_{1}}{N}\,[y^{[N]}_{i,1}(t)-x^{[N]}_{i}(t)]\,{\rm d}t,$
$\displaystyle{\rm
d}y^{[N]}_{i,0}(t)=e_{0}\,[x^{[N]}_{i}(t)-y^{[N]}_{i,0}(t)]\,{\rm d}t,$
$\displaystyle{\rm
d}y^{[N]}_{i,1}(t)=\frac{e_{1}}{N}\,[x^{[N]}_{i}(t)-y^{[N]}_{i,1}(t)]\,{\rm
d}t,\qquad i\in[N],$

which is a special case of (2.20). The initial state is
$\mu(0)=\mu^{\otimes[N]}$ for some $\mu\in\mathcal{P}([0,1]^{3})$. The SSDE in
(10.4) has a unique weak solution coming from a well-posed martingale problem
[SS80, Theorem 3.1]. By [SS80, Theorem 3.2], (10.4) has a unique strong
solution for every deterministic initial state $Z^{[N]}(0)$. Therefore the
solutino of (10.4) is Feller and Markov for any initial law. The SSDE in
(10.4) can alternatively be written as

(10.5) $\displaystyle{\rm
d}x^{[N]}_{i}(t)=c_{0}\left[\frac{1}{N}\sum_{j\in[N]}x^{[N]}_{j}(t)-x^{[N]}_{i}(t)\right]\,{\rm
d}t+\sqrt{g\big{(}x^{[N]}_{i}(t))\big{)}}\,{\rm d}w_{i}(t)$
$\displaystyle\qquad\qquad\qquad+K_{0}e_{0}\,[y^{[N]}_{i,0}(t)-x^{[N]}_{i}(t)]\,{\rm
d}t+\frac{K_{1}e_{1}}{N}\,[y^{[N]}_{i,1}(t)-x^{[N]}_{i}(t)]\,{\rm d}t,$
$\displaystyle{\rm
d}y^{[N]}_{i,0}(t)=e_{0}\,[x^{[N]}_{i}(t)-y^{[N]}_{i,0}(t)]\,{\rm d}t,$
$\displaystyle{\rm
d}y^{[N]}_{i,1}(t)=\frac{e_{1}}{N}\,[x^{[N]}_{i}(t)-y^{[N]}_{i,1}(t)]\,{\rm
d}t,\qquad i\in[N].$

So the migration term for a single colony can be interpreted as a drift
towards the average of the active population. We are interested in
$\mathcal{L}[(Z^{[N]}(t)))_{t\geq 0}]$ in the limit as $N\to\infty$, on time
scales $t$ and $Ns$. Heuristically, analysing the SSDE in (10.5), we can
foresee the following results, which are made precise in Proposition 10.2.

$\bullet$ On time scale $1=N^{0}$ (space-time scale $0$), in the limit as
$N\to\infty$ the colour-$1$ dormant population $(Y^{[N]}_{1}(t))_{t\geq 0}$ in
(10.4) converges to a constant process, since the single components $y_{i,1}$
do not move on time scale $t$. The components of
$(X^{[N]}(t),Y_{0}^{[N]}(t))_{t\geq 0}$ converge to i.i.d. copies of the
single-colony McKean-Vlasov process in (7.1), where in the corresponding SSDE
the parameters $c,e,K$ are replaced by $c_{0},e_{0},K_{0}$ and $E=1$. So on
time scale $t$ we only see the colour-$0$ dormant population interacting with
the active population, and the colour-$1$ dormant population is not yet coming
into play. Therefore the colour-$0$ dormant population is the effective seed-
bank on time scale $1$, and the process

(10.6) $z_{0}^{\mathrm{eff},[N]}(t)=(x^{[N]}_{0}(t),y^{[N]}_{0,0}(t))_{t\geq
0}$

is called the effective process on level $0$. Note that the active population
has a drift towards $\frac{1}{N}\sum_{j\in[N]}x_{j}(t)$, which in the McKean-
Vlasov limit is replaced by $\mathbb{E}[x(t)]$ given by (4.51).

$\bullet$ On time scale $N$ (space-time scale $1$), we look at the averages

(10.7) $\displaystyle(z_{1}^{[N]}(s))_{s>0}$
$\displaystyle=\left(x_{1}^{[N]}(s),(y_{0,1}^{[N]}(s),y_{1,1}^{[N]}(s))\right)_{s>0}$
$\displaystyle=\left(\frac{1}{N}\sum_{i\in[N]}x^{[N]}_{i}(Ns),\left(\frac{1}{N}\sum_{i\in[N]}y^{[N]}_{i,0}(Ns),\frac{1}{N}\sum_{i\in[N]}y^{[N]}_{i,1}(Ns)\right)\right)_{s>0}.$

Again the lower index $1$ indicates that the average is the analogue of the
1-block average defined in (2.22). Using (10.4), we see that the dynamics of
the system in (10.7) is given by the SSDE

(10.8) $\displaystyle{\rm d}x_{1}^{[N]}(s)$
$\displaystyle=\sqrt{\frac{1}{N}\sum_{i\in[N]}g(x^{[N]}_{i}(Ns))}\,{\rm
d}w(s)+NK_{0}e_{0}\left[y_{0,1}^{[N]}(s)-x_{1}^{[N]}(s)\right]{\rm d}s$
$\displaystyle\qquad+K_{1}e_{1}\left[y_{1,1}^{[N]}(s)-x_{1}^{[N]}(s)\right]{\rm
d}s,$ $\displaystyle{\rm d}y_{0,1}^{[N]}(s)$
$\displaystyle=Ne_{0}\left[x_{1}^{[N]}(s)-y_{0,1}^{[N]}(s)\right]{\rm d}s,$
$\displaystyle{\rm d}y_{1,1}^{[N]}(s)$
$\displaystyle=e_{1}\left[x_{1}^{[N]}(s)-y_{1,1}^{[N]}(s)\right]{\rm d}s.$

Thus, as in the mean-field system with one-colour, on time scale $N$ infinite
rates appear in the interaction of the active population with the colour-$0$
dormant population. Therefore in the limit as $N\to\infty$ the path becomes
rougher and rougher at rarer and rarer times. Using the _Meyer-Zheng topology_
we can prove that
$\lim_{N\to\infty}y_{0,1}^{[N]}(s)=\lim_{N\to\infty}x_{1}^{[N]}(s)$ most of
the time. On the other hand, on time scale $N$, $x_{1}^{[N]}(s)$ has a non-
trivial interaction with $y_{1,1}^{[N]}(s)$, and therefore we say that on time
scale $N$ the colour-$1$ dormant population is the effective seed-bank. Note
that for the evolution of the average
$\frac{x_{1}^{[N]}(s)+K_{0}y_{0,1}^{[N]}(s)}{1+K_{0}}$ the rates with a factor
$N$ in front cancel out. We will use the quantity
$\frac{x_{1}^{[N]}(s)+K_{0}y_{0,1}^{[N]}(s)}{1+K_{0}}$ to obtain results in
the classical path-space topology. We call

(10.9)
$(z_{1}^{[N],\mathrm{eff}}(s))_{s>0}=\left(\frac{x_{1}^{[N]}(s)+K_{0}y_{0,1}^{[N]}(s)}{1+K_{0}},y_{1,1}(s)\right)_{s>0}$

the effective process on space-time scale 1. We will call space-time scale $1$
also level $1$.

###### Scaling limit.

To describe the limiting dynamics of the system in (10.4), we need the
infinite-dimensional process

(10.10) $(Z(t))_{t\geq 0}=\big{(}(z_{i}(t))_{t\geq
0}\big{)}_{i\in\mathbb{N}_{0}}=\big{(}(x_{i}(t),(y_{i,0}(t),y_{i,1}(t)))_{t\geq
0}\big{)}_{i\in\mathbb{N}_{0}}$

with state space $([0,1]^{3})^{\mathbb{N}_{0}}$ that evolves according to

(10.11) $\displaystyle{\rm d}x_{i}(t)$ $\displaystyle=c_{0}[\theta-
x_{i}(t)]\,{\rm d}t+\sqrt{g(x_{i}(t))}\,{\rm
d}w_{i}(t)+K_{0}e_{0}\,[y_{i,0}(t)-x_{i}(t)]\,{\rm d}t,$ $\displaystyle{\rm
d}y_{i,0}(t)$ $\displaystyle=e_{0}\,[x_{i}(t)-y_{i,0}(t)]\,{\rm d}t,$
$\displaystyle y_{i,1}(t)$ $\displaystyle=y_{i,1},\qquad i\in\mathbb{N}_{0}.$

Here, $\theta\in[0,1]$ and $y_{i,1}\in[0,1]$ for all $i\in\mathbb{N}_{0}$. We
will also need the limiting effective process

(10.12) $(Z^{\mathrm{eff}}(t))_{t\geq
0}=\big{(}(z^{\mathrm{eff}}_{i}(t))_{t\geq
0}\big{)}_{i\in\mathbb{N}_{0}}=\big{(}(x_{i}^{\mathrm{eff}}(t),y^{\mathrm{eff}}_{i,0}(t))_{t\geq
0}\big{)}_{i\in\mathbb{N}_{0}}$

with state space $([0,1]^{2})^{\mathbb{N}_{0}}$ that evolves according to

(10.13) $\displaystyle{\rm d}x^{\mathrm{eff}}_{i}(t)$
$\displaystyle=c_{0}[\theta-x^{\mathrm{eff}}_{i}(t)]\,{\rm
d}t+\sqrt{g\big{(}x^{\mathrm{eff}}_{i}(t)\big{)}}\,{\rm
d}w(t)+K_{0}e_{0}\,[y^{\mathrm{eff}}_{i,0}(t)-x^{\mathrm{eff}}_{i}(t)]\,{\rm
d}t,$ $\displaystyle{\rm d}y^{\mathrm{eff}}_{i,0}(t)$
$\displaystyle=e_{0}\,[x^{\mathrm{eff}}_{i}(t)-y^{\mathrm{eff}}_{i,0}(t)]\,{\rm
d}t,\qquad i\in\mathbb{N}_{0}.$

Like for the one-colour mean-field finite-systems scheme, we need the
following list of ingredients to formally state our multi-scaling properties:

  1. 1.

For positive times $t>0$, we define the so-called _estimators_ for the finite
system by:

(10.14) $\displaystyle\bar{\Theta}^{(1),[N]}(t)$
$\displaystyle=\frac{1}{N}\sum_{i\in[N]}\frac{x^{[N]}_{i}(t)+K_{0}y^{[N]}_{i,0}(t)}{1+K_{0}},$
$\displaystyle\Theta^{(1),[N]}_{x}(t)$
$\displaystyle=\frac{1}{N}\sum_{i\in[N]}x^{[N]}_{i}(t),$
$\displaystyle\Theta^{(1),[N]}_{y_{0}}(t)$
$\displaystyle=\frac{1}{N}\sum_{i\in[N]}y^{[N]}_{i,0}(t),$
$\displaystyle\Theta^{(1),[N]}_{y_{1}}(t)$
$\displaystyle=\frac{1}{N}\sum_{i\in[N]}y^{[N]}_{i,1}(t).$

We abbreviate

(10.15) $\displaystyle{\bf\Theta}^{(1),[N]}(t)$
$\displaystyle=\left({\Theta}_{x}^{(1),[N]}(t),\Theta^{(1),[N]}_{y_{0}}(t),\Theta^{(1),[N]}_{y_{1}}(t)\right),$
$\displaystyle{\bf\Theta}^{\mathrm{eff},(1),[N]}(t)$
$\displaystyle=\left(\bar{\Theta}^{(1),[N]}(t),\Theta^{(1),[N]}_{y_{1}}(t)\right).$

We refer to $({\bf\Theta}^{\mathrm{eff},(1),[N]}(t))_{t\geq 0}$ as the
_effective estimator process_ and to $({\bf\Theta}^{(1),[N]}(t))_{t\geq 0}$ as
the _estimator process_.

  2. 2.

The _time scale_ $Ns$ is such that
$\mathcal{L}[\bar{\Theta}^{[N]}(Ns-L(N))-\bar{\Theta}^{[N]}(Ns)]=\delta_{0}$
for all $L(N)$ satisfying $\lim_{N\to\infty}$ $L(N)=\infty$ and
$\lim_{N\to\infty}L(N)/N=0$, but not for $L(N)=N$. In words, $Ns$ is the time
scale on which $\bar{\Theta}^{[N]}(\cdot)$ starts evolving, i.e.,
$(\bar{\Theta}^{[N]}(Ns))_{s>0}$ is no longer a fixed process. When we scale
time by $Ns$, we will use $s$ as a time index, which indicates the “fast time
scale”. The “slow time scale” will be indicated by $t$. Thus, the time scales
for the two-colour mean-field system are the same as the time scales for the
one-colour mean-field system.

######  Remark 10.1.

[Notation] The upper index $1$ in $\bar{\Theta}^{(1)}$ and
$\Theta_{y_{1}}^{(1)}$ is used to indicate that we are working with a system
of level 1, so the system that lives on space-time scale 1. This can later be
easily generalized to levels $2$ and $k$. $\blacksquare$

  3. 3.

The _invariant measure_ (i.e., the equilibrium measure) for the evolution of a
single colony in (10.11), written

(10.16) $\Gamma_{\theta,\theta,y_{1}},$

and the _invariant measure_ of the infinite system in (10.11), written
$\nu_{\theta,\theta,{\bf
y_{1}}}=\Gamma_{\theta,\theta,y_{1}}^{\otimes\mathbb{N}_{0}}$ with
$\theta\in[0,1]$ and ${\bf y}_{1}\in[0,1]^{\mathbb{N}_{0}}$ a random variable.
The existence of the invariant measure $\nu_{\theta}$ and the convergence of
$\mathcal{L}[Z(t)_{t\geq 0}]$ towards $\nu_{\theta}$ will be shown in the
proof of Proposition 10.2.

  4. 4.

The invariant measure of the effective process in (10.13),

(10.17) $\Gamma^{\mathrm{eff}}_{\theta},$

and the invariant measure for the full process,
$\nu^{\mathrm{eff}}_{\theta}=(\Gamma^{\mathrm{eff}}_{\theta})^{\otimes\mathbb{N}_{0}}$.

  5. 5.

The renormalisation transformation
$\mathcal{F}\colon\,\mathcal{G}\to\mathcal{G}$,

(10.18)
$(\mathcal{F}g)(\theta)=\int_{([0,1]^{2})^{\mathbb{N}_{0}}}g(x_{0})\,\nu^{\mathrm{eff}}_{\theta}({\rm
d}x_{0},{\rm d}y_{0,0}),\quad\theta\in[0,1],$

where $\Gamma_{\theta}^{\mathrm{eff}}$ is the equilibrium measure of (10.16).
Note that this is the same transformation as defined in (4.16), but for the
truncated system. Since $\nu^{\mathrm{eff}}_{\theta}$ is a product measure, we
can write

(10.19)
$(\mathcal{F}g)(\theta)=\int_{[0,1]^{2}}g(x)\,\Gamma^{\mathrm{eff}}_{\theta}({\rm
d}x,{\rm d}y_{0}),\quad\theta\in[0,1],$

  6. 6.

The limiting 1-block process
$(z_{1}(s))_{s>0}=(x_{1}(s),(y_{0,1}(s),y_{1,1}(s)))_{s>0}$ evolving according
to

(10.20) $\displaystyle{\rm d}x_{1}(s)$
$\displaystyle=\frac{1}{1+K_{0}}\left[\sqrt{(\mathcal{F}g)\big{(}x_{1}(s)\big{)}}\,{\rm
d}w(s)+K_{1}e_{1}\left[y_{1,1}(s)-x_{1}(s)\right]{\rm d}s\right],$
$\displaystyle y_{0,1}(s)$ $\displaystyle=x_{1}(s),$ $\displaystyle{\rm
d}y_{1,1}(s)$ $\displaystyle=e_{1}\left[x_{1}(s)-y_{1,1}(s)\right]\,{\rm d}s,$

where $\mathcal{F}g$ is defined in (10.20). The effective process
$(z^{\mathrm{eff}}_{1}(s))_{s>0}=(x^{\mathrm{eff}}_{1}(s),y^{\mathrm{eff}}_{1,1}(s))_{s>0}$
on space-time scale $1$,

(10.21) $\displaystyle{\rm d}x^{\mathrm{eff}}_{1}(s)$
$\displaystyle=\frac{1}{1+K_{0}}\left[\sqrt{(\mathcal{F}g)(x^{\mathrm{eff}}_{1}(s))}\,{\rm
d}w(s)+K_{1}e_{1}\,[y^{\mathrm{eff}}_{1,1}(s)-x^{\mathrm{eff}}_{1}(s)]\,{\rm
d}s\right],$ $\displaystyle{\rm d}y^{\mathrm{eff}}_{1,1}(s)$
$\displaystyle=e_{1}\,[x^{\mathrm{eff}}_{1}(s)-y^{\mathrm{eff}}_{1,1}(s)]\,{\rm
d}s.$

We are now ready to state the scaling limit for the evolution of the averages
in (10.7), which we refer to as the _mean-field finite-systems scheme with two
colours_.

######  Proposition 10.2.

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

(10.22)
$\vartheta_{0}=\mathbb{E}^{\mu}\left[\frac{x+K_{0}y_{0}}{1+K_{0}}\right],\qquad\qquad\theta_{y_{1}}=\mathbb{E}^{\mu}\left[y_{1}\right].$

  * (a)

For the effective estimator process defined in (10.15),

(10.23)
$\lim_{N\to\infty}\mathcal{L}\left[\left({\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns)\right)_{s>0}\right]=\mathcal{L}\left[\left(z_{1}^{\mathrm{eff}}(s)\right)_{s>0}\right],$

where the limit is determined by the unique solution of the SSDE (10.21), with
initial state

(10.24) $\displaystyle
z_{1}^{\mathrm{eff}}(0)=\left(x_{1}^{\mathrm{eff}}(0),y^{\mathrm{eff}}_{1}(0)\right)=\left(\vartheta_{0},\theta_{y_{1}}\right).$

  * (b)

Assume for the $1$-dormant single components that

(10.25)
$\lim_{N\to\infty}\mathcal{L}\left[Y_{1}^{[N]}(Ns)\Big{|}{\bf\Theta}^{(1),[N]}(Ns)\right]=P^{z_{1}(s)}_{Y_{1}(s)}.$

Define

(10.26)
$\Gamma^{\mathrm{eff}}_{(\vartheta_{0},\theta_{y_{1}})}(s)=\int_{[0,1]^{2}}S_{s}\bigl{(}(\vartheta_{0},\theta_{y_{1}}),{\rm
d}(u_{x},u_{y})\bigr{)}\,\Gamma^{\mathrm{eff}}_{u_{x}}\in\mathcal{P}([0,1]^{2}),$

where $S_{s}((\vartheta_{0},\theta_{y_{1}}),\cdot)$ is the time-$s$ marginal
law of the process $(z_{1}^{\mathrm{eff}}(s))_{s>0}$ starting from
$(\theta_{0},\theta_{y_{1}})\in[0,1]^{2}$ and $\Gamma^{\mathrm{eff}}_{u_{x}}$
is the equilibrium distribution of the system in (10.13) with $\theta=u_{x}$
(note that
$\Gamma^{\mathrm{eff}}_{\vartheta_{0},\theta_{y_{1}}}(0)=\Gamma^{\mathrm{eff}}_{\vartheta_{0}}$).
Let $(z^{\mathrm{eff},\Gamma_{(\vartheta_{0},\theta_{y_{1}})}(s)}(t))_{t\geq
0}$ be the process with initial law
$z^{\mathrm{eff},\Gamma_{(\vartheta_{0},\theta_{y_{1}})}(s)}(0)$ drawn
according to $\Gamma^{\mathrm{eff}}_{(\vartheta_{0},\theta_{y_{1}})}(s)$
(which is a mixture of random processes in equilibrium) that, conditional on
$x_{1}^{\mathrm{eff}}(s)=\theta$, evolves according to (10.13). Then, for
every $s\in(0,\infty)$,

(10.27)
$\lim_{N\to\infty}\mathcal{L}\left[\left(z^{\mathrm{eff},[N]}_{0}(Ns+t)\right)_{t\geq
0}\right]=\mathcal{L}\left[(z^{\Gamma^{\mathrm{eff}}_{(\vartheta_{0},\theta_{y_{1}})}(s)}(t))_{t\geq
0}\right].$

  * (c)

For the averages in (10.7),

(10.28)
$\displaystyle\lim_{N\to\infty}\mathcal{L}\left[\left(z_{1}^{[N]}(s)\right)_{s>0}\right]=\mathcal{L}\left[\left(z_{1}(s)\right)_{s>0}\right]$
$\displaystyle\text{in the Meyer-Zheng topology},$

where the limit process is the unique solution of the SSDE in (10.20) with
initial state

(10.29)
$z_{1}(0)=\left(x_{1}(0),y_{0,1}(0),y_{1,1}(0)\right)=\left(\vartheta_{0},\vartheta_{0},\theta_{y_{1}}\right).$

  * (d)

Assume 10.25 and define

(10.30)
$\nu(s)=\int_{[0,1]^{3}}S_{s}\bigl{(}(\vartheta_{0},\vartheta_{0},\theta_{y_{1}}),{\rm
d}(u_{x},u_{x},u_{y_{1}})\bigr{)}\,\int_{[0,1]^{\mathbb{N}_{0}}}P^{(u_{x},u_{x},u_{y_{1}})}_{Y_{1}(s)}({\rm
d}{\bf y_{1}})\,\nu_{u_{x},{\bf y_{1}}},$

where $S_{s}((\vartheta_{0},\vartheta_{0},\theta_{y_{1}}),\cdot)$ is the
time-$s$ marginal law of the process $\left(z_{1}(s)\right)_{s>0}$ in (10.20),
starting from $(\vartheta_{0},\vartheta_{0},\theta_{y_{1}})\in[0,1]^{3}$, and
$\nu_{u_{x},{\bf y_{1}}}$ is the equilibrium distribution of the system in
(10.11) with $\theta=u_{x}$ and $(y_{i,1})_{i\in\mathbb{N}_{0}}=\bf{y_{1}}$,
(note that $\nu(0)=\nu_{\vartheta_{0},(y_{i,1}(0))_{i\in\mathbb{N}_{0}}}$).
Let $(z^{\nu(s)}(t))_{t\geq 0}$ be the process on
$([0,1]^{3})^{\mathbb{N}_{0}}$ with initial measure $z^{\nu(s)}(0)$ drawn
according to $\nu(s)$ (which is a mixture of random processes in equilibrium)
that conditional on $x_{1}(s)=\theta$ and $Y_{1}(s)={\bf y_{1}}$ evolves
according to (10.11) with $\theta=u_{x}$ and
$(y_{i,1})_{i\in\mathbb{N}_{0}}={\bf y_{1}}$. Then, for every
$s\in(0,\infty)$,

(10.31) $\lim_{N\to\infty}\mathcal{L}\left[\left(Z^{[N]}(Ns+t)\right)_{t\geq
0}\right]=\mathcal{L}\left[(z^{\nu(s)}(t))_{t\geq 0}\right].$

######  Remark 10.3 (Law of $1$-dormant single components).

Note that

(10.32)
$\left(\mathcal{L}\left[Y_{1}^{[N]}(Ns)\Big{|}\left(\bar{\Theta}^{[N]}(Ns),\Theta_{y_{1}}^{[N]}(Ns)\right)\right]\right)_{N\in\mathbb{N}_{0}}$

is a tight sequence of measures. Hence there exist weak limit points. In
Section 11 we will see that if there is a higher layer in the hierarchy, then
we can show that all weak limit points of (10.32) are the same and we can
identify the limit. For Theorems 4.2 and 4.4 we do not need this assumption,
since there will alwyas be multiple higher levels. $\blacksquare$

####  10.2 Proof of the two-colour mean-field finite-systems scheme

The proof of Proposition 10.2, the finite-systems scheme with one level and
two colours, follows the strategy used in Section 9 for the proof of
Proposition 8.1. Like for the one-colour finite-systems scheme, we denote the
slow time scale by $t$ and the fast time scale by $s$. The proof consists of
the following 6 steps:

  1. 1.

Tightness of the effective estimator processes defined in (10.15).

(10.33)
$\big{(}({\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns))_{s>0}\big{)}_{N\in\mathbb{N}}$

  2. 2.

Stability property of $({\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns+t))_{t>0}$,
i.e., for $L(N)$ satisfying $\lim_{N\to\infty}L(N)=\infty$ and
$\lim_{N\to\infty}L(N)/N=0$, and all $\epsilon>0$,

(10.34) $\lim_{N\to\infty}\mathbb{P}\left[\sup_{0\leq t\leq
L(N)}\left|\bar{\Theta}^{(1),[N]}(Ns)-\bar{\Theta}^{(1),[N]}(Ns-t)\right|>\epsilon\right]=0.$

and

(10.35) $\lim_{N\to\infty}\mathbb{P}\left[\sup_{0\leq t\leq
L(N)}\left|\Theta_{y_{1}}^{(1),[N]}(Ns)-\Theta_{y_{1}}^{(1),[N]}(Ns-t)\right|>\epsilon\right]=0.$

  3. 3.

Equilibrium of the infinite system and the one-dimensional distribution of the
effective single components $(Z(Ns+t))_{t>0}$, analogous to Proposition 8.4.

  4. 4.

Limiting evolution of the effective processes
$(({\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns))_{s>0})_{N\in\mathbb{N}}$.

  5. 5.

Evolution of the $1$-blocks in the Meyer-Zheng topology.

  6. 6.

Proof of Proposition 10.2.

###### Step 1: Tightness of the 1-block estimators.

######  Lemma 10.4.

[Tightness of the 1-block estimator] Let

(10.36) $({\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns))_{s>0}$

be defined as in (10.14). Then
$(\mathcal{L}[({\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns))_{s>0}])_{N\in\mathbb{N}}$
is a tight sequence of probability measures on
$\mathcal{C}((0,\infty),[0,1]^{2})$.

###### Proof.

To prove tightness of
$(({\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns))_{s>0})_{N\in\mathbb{N}}$, we will
prove for all $\epsilon>0$ that the set of measures
$(({\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns))_{s\geq\epsilon})_{N\in\mathbb{N}}$
is tight. To do so, fix $\epsilon>0$. We will again use [JM86, Proposition
3.2.3]. From (10.4) we find that the 1-block averages
$({\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns))_{s>0}$ evolve according to

(10.37) $\displaystyle{\rm d}\bar{\Theta}^{(1),[N]}(Ns)$
$\displaystyle=\frac{1}{1+K_{0}}\Bigg{[}\sqrt{\frac{1}{N}\sum_{i\in[N]}g\big{(}x^{[N]}_{i}(Ns)\big{)}}\,{\rm
d}w_{i}(s)$
$\displaystyle\qquad+K_{1}e_{1}\left[\Theta_{y_{1}}^{(1),[N]}-\frac{1}{N}\sum_{i\in[N]}x^{[N]}_{i}(Ns)\right]\,{\rm
d}s\Bigg{]},$ $\displaystyle{\rm d}\Theta^{(1),[N]}_{y_{1}}(Ns)$
$\displaystyle=e_{1}\left[\frac{1}{N}\sum_{i\in[N]}x^{[N]}_{i}(Ns)-\Theta^{(1),[N]}_{y_{1}}(Ns)\right]\,{\rm
d}s.$

To use [JM86, Proposition 3.2.3], we define $\mathcal{C}^{*}$ as the set of
polynomials on $([0,1]^{2})$. Note that
$({\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns))_{s\geq\epsilon}$ is a semi-
martingale. Applying Itô’s formula, we get

(10.38) $\displaystyle f\left({\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns)\right)$
$\displaystyle=f\left({\bf\Theta}^{\mathrm{eff},(1),[N]}(N\epsilon)\right)$
$\displaystyle\quad+\int_{\epsilon}^{s}{\rm
d}w_{i}(r)\,\frac{1}{1+K_{0}}\sqrt{\frac{1}{N}\sum_{i\in[N]}g\big{(}x^{[N]}_{i}(Nr)\big{)}}\,\frac{\partial
f}{\partial x}\left({\bf\Theta}^{\mathrm{eff},(1),[N]}(Nr)\right)$
$\displaystyle\quad+\int_{\epsilon}^{s}{\rm
d}r\,\frac{K_{1}e_{1}}{1+K_{0}}\left[\Theta_{y_{1}}^{(1),[N]}(Nr)-\frac{1}{N}\sum_{i\in[N]}x^{[N]}_{i}(Nr)\right]\frac{\partial
f}{\partial x}\left({\bf\Theta}^{\mathrm{eff},(1),[N]}(Nr)\right)$
$\displaystyle\quad+\int_{\epsilon}^{s}{\rm
d}r\,e_{1}\left[\frac{1}{N}\sum_{i\in[N]}x^{[N]}_{i}(Nr)-\Theta_{y_{1}}^{(1),[N]}(Nr)\right]\,\frac{\partial
f}{\partial y}\left({\bf\Theta}^{\mathrm{eff},(1),[N]}(Nr)\right)$
$\displaystyle\quad+\int_{\epsilon}^{s}{\rm
d}r\,\frac{1}{2(1+K_{0})^{2}}\frac{1}{N}\sum_{i\in[N]}g\big{(}x^{[N]}_{i}(Nr)\big{)}\,\frac{\partial^{2}f}{\partial
x^{2}}\left({\bf\Theta}^{\mathrm{eff},(1),[N]}(Nr)\right)$

for all $f\in\mathcal{C}^{*}$. Hence, if we define the operator

(10.39) $\displaystyle G^{(1),[N]}_{\dagger}$
$\displaystyle\colon\,(\mathcal{C}^{*},[0,1]^{2},[\epsilon,\infty),\Omega)\to\mathbb{R},$
$\displaystyle G^{(1),[N]}_{\dagger}(f,(x,y),s,\omega)$
$\displaystyle=\frac{K_{1}e_{1}}{1+K_{0}}\left[y-\frac{1}{N}\sum_{i\in[N]}x^{[N]}_{i}(Ns,\omega)\right]\frac{\partial
f}{\partial x}$
$\displaystyle\qquad+e_{1}\left[\frac{1}{N}\sum_{i\in[N]}x^{[N]}_{i}(Ns,\omega)-y\right]\frac{\partial
f}{\partial y}$
$\displaystyle\qquad+\frac{1}{2(1+K_{0})^{2}}\frac{1}{N}\sum_{i\in[N]}g(x^{[N]}_{i}(Ns,\omega))\,\frac{\partial^{2}f}{\partial
x^{2}},$

then we see that the process
$({\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns))_{s\geq\epsilon}$ is a
$\mathcal{D}$-semi-martingale for all $\epsilon>0$. For all $\epsilon>0$ the
conditions $H_{1},\ H_{2},\ H_{3}$ are satisfied as before. Therefore we
conclude from [JM86, Proposition 3.2.3] that the sequence
$(({\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns))_{s\geq\epsilon})_{N\in\mathbb{N}}$
is tight. Since this is true for all $\epsilon>0$, we conclude that
$(\mathcal{L}[({\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns))_{s>0}])_{N\in\mathbb{N}}$
is tight. ∎

###### Step 2: Stability of the $1$-block estimators.

######  Lemma 10.5.

[Stability property of the 1-block estimator] Let
${\bf\Theta}^{\mathrm{eff},(1),[N]}(t)$ be defined as in (10.14). For any
$L(N)$ satisfying $\lim_{N\to\infty}L(N)=\infty$ and
$\lim_{N\to\infty}L(N)/N=0$,

(10.40) $\lim_{N\to\infty}\sup_{0\leq t\leq
L(N)}\left|\bar{\Theta}^{(1),[N]}(Ns)-\bar{\Theta}^{(1),[N]}(Ns-t)\right|=0\text{
in probability}$

and

(10.41) $\lim_{N\to\infty}\sup_{0\leq t\leq
L(N)}\left|\Theta_{y_{1}}^{(1),[N]}(Ns)-\Theta_{y_{1}}^{(1),[N]}(Ns-t)\right|=0\text{
in probability. }$

###### Proof.

Fix $\epsilon>0$. From the SSDE (10.4) we obtain that, for $N$ large enough,

(10.42) $\displaystyle\mathbb{P}\Bigg{(}\sup_{0\leq t\leq
L(N)}\Bigg{|}\bar{\Theta}^{(1),[N]}(Ns)-\bar{\Theta}^{(1),[N]}(Ns-t)\Bigg{|}>\epsilon\Bigg{)}$
$\displaystyle=\mathbb{P}\Bigg{(}\sup_{0\leq t\leq
L(N)}\frac{1}{1+K_{0}}\Bigg{|}\int_{Ns-t}^{Ns}{\rm
d}r\,\frac{K_{1}e_{1}}{N}\left[\Theta^{(1),[N]}_{y_{1}}(r)-\frac{1}{N}\sum_{i\in[N]}x^{[N]}_{i}(r)\right]$
$\displaystyle\qquad\qquad\qquad+\int_{Ns-t}^{Ns}{\rm
d}w_{i}(r)\,\frac{1}{N}\sum_{i\in[N]}\sqrt{g\big{(}x^{[N]}_{i}(r)\big{)}}\,\Bigg{|}>\epsilon\Bigg{]}$
$\displaystyle\leq\mathbb{P}\Bigg{(}\Bigg{|}\frac{L(N)2K_{1}e_{1}}{N(1+K_{0})}\Bigg{|}+\sup_{0\leq
t\leq L(N)}\Bigg{|}\frac{1}{1+K_{0}}\int_{Ns-t}^{Ns}{\rm
d}w_{i}(r)\,\frac{1}{N}\sum_{i\in[N]}\sqrt{g(x^{[N]}_{i}(r))}\,\Bigg{|}>\epsilon\Bigg{)}$
$\displaystyle=\mathbb{P}\Bigg{(}\sup_{0\leq t\leq
L(N)}\,\Bigg{|}\frac{1}{1+K_{0}}\int_{Ns-t}^{Ns}{\rm
d}w_{i}(r)\,\frac{1}{N}\sum_{i\in[N]}\sqrt{g\big{(}x^{[N]}_{i}(r)\big{)}}\,\Bigg{|}>\epsilon-\frac{L(N)2K_{1}e_{1}}{N(1+K_{0})}\Bigg{)}$
$\displaystyle\leq\mathbb{P}\Bigg{(}\sup_{0\leq t\leq
L(N)}\Bigg{|}\frac{1}{1+K_{0}}\int_{Ns-t}^{Ns}{\rm
d}w_{i}(r)\,\frac{1}{N}\sum_{i\in[N]}\sqrt{g\big{(}x^{[N]}_{i}(r)\big{)}}\,\Bigg{|}>\frac{\epsilon}{2}\Bigg{)}.$

Applying the same optional stopping argument as used in the proof of Lemma
8.15, we find (10.40). For (10.41), note that

(10.43) $\displaystyle\mathbb{P}$ $\displaystyle\Bigg{(}\sup_{0\leq t\leq
L(N)}\Bigg{|}\Theta_{y_{1}}^{(1),[N]}(Ns)-\Theta_{y_{2}}^{(1),[N]}(Ns-t)\Bigg{|}>\epsilon\Bigg{)}$
$\displaystyle=\mathbb{P}\Bigg{(}\sup_{0\leq t\leq
L(N)}\frac{1}{1+K_{0}}\Bigg{|}\int_{Ns-t}^{Ns}{\rm
d}r\,\frac{e_{1}}{N}\left[\Theta^{(1),[N]}_{y_{1}}(r)-\frac{1}{N}\sum_{i\in[N]}x_{i}^{[N]}(r)\right]\Bigg{|}>\epsilon\Bigg{)}$
$\displaystyle\leq\mathbb{P}\Bigg{(}\frac{2e_{1}L(N)}{(1+K_{0})N}>\epsilon\Bigg{)}.$

Let $N\to\infty$ to obtain (10.41). ∎

###### Step 3: Equilibrium for the infinite system.

To derive the equilibrium of the single components in the infinite system, we
derive the following analoque of Proposition 8.4. Recall that the finite
system is denoted by $Z^{[N_{k}]}$ in (10.3), and recall the list of
ingredients in Section 10.1.

######  Proposition 10.6.

[Equilibrium for the infinite 2-colour system] Let $(N_{k})_{k\in\mathbb{N}}$
be a sequence in $\mathbb{N}$. Fix $s>0$. Let $L(N)$ satisfy
$\lim_{N\to\infty}L(N)=\infty$ and $\lim_{N\to\infty}L(N)/N=0$, and suppose
that

(10.44)
$\displaystyle\lim_{k\to\infty}\mathcal{L}\left[{\bf\Theta}^{\mathrm{eff},(1),[N_{k}]}(N_{k}s)\right]=P_{{\bf\Theta^{\mathrm{eff}}}(s)},$
$\displaystyle\lim_{k\to\infty}\mathcal{L}\left[Y_{1}^{[N_{k}]}(N_{k}s)\Big{|}{\bf\Theta}^{\mathrm{eff},(1),[N_{k}]}(N_{k}s)\right]=P^{{\bf\Theta}^{\mathrm{eff},(1)}(s)}_{Y_{1}(s)},$
$\displaystyle\lim_{k\to\infty}\mathcal{L}\left[\sup_{0\leq t\leq
L(N_{k})}\left|\bar{\Theta}^{[N_{k}]}(N_{k}s)-\bar{\Theta}^{[N_{k}]}(N_{k}s-t)\right|+\left|{\Theta}_{y_{1}}^{[N_{k}]}(N_{k}s)-{\Theta_{y_{1}}}^{[N_{k}]}(N_{k}s-t)\right|\right]$
$\displaystyle\qquad\qquad\qquad\qquad\,\,\,=\delta_{0},$
$\displaystyle\lim_{k\to\infty}\mathcal{L}\bigl{[}Z^{[N_{k}]}(N_{k}s)\bigr{]}=\nu(s).$

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

(10.45) $\nu(s)=\int_{[0,1]^{2}}P_{{\bf\Theta^{\mathrm{eff}}}(s)}({\rm
d}\theta,{\rm
d}\theta_{y})\,\int_{[0,1]^{\mathbb{N}_{0}}}P^{(\theta,\theta_{y})}_{Y_{1}(s)}({\rm
d}{\bf y_{1}})\,\nu_{\theta,{\bf y_{1}}},$

where ${\bf y_{1}}=(y_{i,1})_{i\in\mathbb{N}_{0}}$ is a sequence with elements
in $[0,1]$, and $\nu_{\theta,{\bf y_{1}}}$ is the equilibrium measure of the
process in (10.10) evolving according to (10.11) with
$(y_{i,1})_{i\in\mathbb{N}_{0}}$ given by the sequence ${\bf y_{1}}=$.

###### Preparation for the proof of Proposition 10.6.

The proof of Proposition 10.6 follows the same line of argument as used in the
proof of Proposition 8.4. We need lemmas that are similar to Lemmas 8.5-8.11,
but this time in the setting of the two-colour hierarchical mean-field finite-
systems scheme. Afterwards we prove Proposition 10.6.

######  Lemma 10.7.

[Convergence for the infinite system] Let $\mu$ be an exchangeable probability
measure on $([0,1]^{3})^{\mathbb{N}_{0}}$. Then for the system $(Z(t))_{t\geq
0}$ given by (10.10) with $\mathcal{L}[Z(0)]=\mu$,

(10.46) $\lim_{t\to\infty}\mathcal{L}[Z(t)]=\nu_{\theta,{\bf y_{1}}},$

where $\nu_{\theta,{\bf y_{1}}}$ is of the form

(10.47) $\nu_{\theta,{\bf
y_{1}}}=\prod_{i\in\mathbb{N}_{0}}\Gamma_{\theta,y_{i,1}}$

with $\Gamma_{\theta,y_{i,1}}$ the equilibrium of the $i$th single-component
process in (10.11).

###### Proof.

For each component of the infinite system in (10.10) the $1$-dormant single
component process $(y_{i,1}(t))_{t\geq 0}$ does not move on time scale $t$.
Hence, given the states of $1$-dormant single components, we can use a similar
argument as in the proof of Proposition 7.2 (see Section 7.3) to show that the
single components converge to an equilibrium measure
$\Gamma_{\theta,y_{i,1}}$. Since the single components do not interact, the
claim in Lemma 10.7 follows. ∎

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

######  Lemma 10.8.

[Continuity of the equilibrium] Let
$\mathcal{P}(([0,1]^{3})^{\mathbb{N}_{0}})$ denote the space of probability
measures on $([0,1]^{3})^{\mathbb{N}_{0}}$. The mapping
$[0,1]\times[0,1]^{\mathbb{N}_{0}}\to\mathcal{P}(([0,1]^{3})^{\mathbb{N}_{0}})$
given by

(10.48) $(\theta,{\bf y_{1}})\mapsto\nu_{\theta,{\bf y_{1}}}$

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

(10.49) $(\mathcal{F}h)(\theta)=\mathbb{E}^{\nu_{\theta,{\bf
y_{1}}}}[h(\cdot)]=\int_{([0,1]^{3})^{\mathbb{N}_{0}}}\nu_{\theta,{\bf
y_{1}}}({\rm d}z)\,h(x_{0})$

is a Lipschitz function on $[0,1]$, whose values are independent of ${\bf
y_{1}}$.

###### Proof.

Lemma 10.8 follows from the proof of Lemma 10.12. ∎

The third lemma characterises the speed at which the estimators
$({\Theta}^{[N]}_{x}(t))_{t\geq 0}$ and $({\Theta}^{[N]}_{y}(t))_{t\geq 0}$
converge to each other when $N\to\infty$ and $t\to\infty$.

######  Lemma 10.9.

[Comparison of empirical averages] Let $(\Theta^{(1),[N]}_{x}(t))_{t\geq 0}$
and $(\Theta^{(1),[N]}_{y_{0}}(t))_{t\geq 0}$ be defined as in (10.14). Then

(10.50)
$\displaystyle\mathbb{E}\left[\left|\Theta^{(1),[N]}_{x}(t)-\Theta^{(1),[N]}_{y_{0}}(t)\right|\right]$
$\displaystyle\leq\sqrt{\mathbb{E}\left[\left(\Theta^{(1),[N]}_{x}(0)-\Theta^{(1),[N]}_{y_{0}}(0)\right)^{2}\right]}{\rm
e}^{-(K_{0}e_{0}+e_{0})t}$
$\displaystyle\quad+\sqrt{\frac{2}{K_{0}e_{0}+e_{0}}\left[\frac{||g||}{N}+\frac{4K_{1}e_{1}}{N}\right]}.$

###### Proof.

From (10.4) it follows via Itô-calculus that

(10.51) $\displaystyle\frac{{\rm d}}{{\rm
d}t}\mathbb{E}\left[\left(\Theta^{(1),[N]}_{x}(t)-\Theta^{(1),[N]}_{y_{0}}(t)\right)^{2}\right]$
$\displaystyle=-2(K_{0}e_{0}+e_{0})\,\mathbb{E}\left[\left(\Theta^{(1),[N]}_{x}(t)-\Theta^{(1),[N]}_{y_{0}}(t)\right)^{2}\right]$
$\displaystyle\qquad+h^{[N]}(t),$

where

(10.52) $\displaystyle h^{[N]}(t)$
$\displaystyle=\mathbb{E}\left[\frac{2K_{1}e_{1}}{N}\left(\Theta^{(1),[N]}_{x}(t)-\Theta^{(1),[N]}_{y_{0}}(t)\right)\left[\Theta_{y_{1}}^{(1),[N]}(t)-\Theta^{(1),[N]}_{x}(t)\right]\right]$
$\displaystyle\quad+\frac{2}{N^{2}}\sum_{i\in[N]}\mathbb{E}\big{[}g\big{(}x^{[N]}_{i}(r)\big{)}\big{]}.$

Hence

(10.53)
$\displaystyle\mathbb{E}\left[\left(\Theta^{(1),[N]}_{x}(t)-\Theta^{(1),[N]}_{y_{0}}(t)\right)^{2}\right]$
$\displaystyle=\mathbb{E}\left[\left(\Theta^{(1),[N]}_{x}(0)-\Theta^{(1),[N]}_{y_{0}}(0)\right)^{2}\right]{\rm
e}^{-2(K_{0}e_{0}+e_{0})t}$ $\displaystyle\qquad+\int_{0}^{t}{\rm d}r\,{\rm
e}^{-2(K_{0}e_{0}+e_{0})(t-r)}h^{[N]}(r).$

Take the square root on both sides and use Jensen’s inequality to get (10.50).
∎

Like for the mean-field system with one colour, we need to compare the finite
system in (10.3) with an infinite system. To derive the analogue of Lemma 8.9,
let $L(N)$ satisfy $\lim_{N\to\infty}L(N)=\infty$ and
$\lim_{N\to\infty}L(N)/N=0$. Define the measure $\mu_{N}$ on
$([0,1]^{3})^{\mathbb{N}_{0}}$ by continuing the configuration of

(10.54)
$Z^{[N]}(Ns-L(N))=\left(X^{[N]}(Ns-L(N)),\left(Y_{0}^{[N]}(Ns-L(N)),Y_{1}^{[N]}(Ns-L(N))\right)\right)$

periodically to $([0,1]^{3})^{\mathbb{N}_{0}}$. Let

(10.55)
$\bar{\Theta}^{(1),[N]}=\frac{1}{N}\sum_{i\in[N]}\frac{x^{[N]}_{i}(Ns-L(N))+K_{0}y^{[N]}_{i,0}(Ns-L(N))}{1+K_{0}}.$

Let

(10.56) $(Z^{\mu_{N}}(t))_{t\geq
0}=\bigl{(}X^{\mu_{N}}(t),\left(Y_{0}^{\mu_{N}}(t),Y_{1}^{\mu_{N}}(t)\right)\bigr{)}_{t\geq
0}$

be the infinite system evolving according to

(10.57) $\displaystyle{\rm d}x^{\mu_{N}}_{i}(t)$
$\displaystyle=c_{0}\,[\bar{\Theta}^{(1),[N]}-x^{\mu_{N}}_{i}(t)]\,{\rm
d}t+\sqrt{g\big{(}x^{\mu_{N}}_{i}(t)\big{)}}\,{\rm
d}w_{i}(t)+K_{0}e_{0}\,[y^{\mu_{N}}_{i,0}(t)-x^{\mu_{N}}_{i}(t)]\,{\rm d}t,$
$\displaystyle{\rm d}y^{\mu_{N}}_{i,0}(t)$
$\displaystyle=e_{0}\,[x^{\mu_{N}}_{i}(t)-y^{\mu_{N}}_{i,0}(t)]\,{\rm d}t,$
$\displaystyle y^{\mu_{N}}_{i,1}(t)$
$\displaystyle=y^{\mu_{N}}_{i,1}(0),\qquad i\in\mathbb{N}_{0},$

starting from initial distribution $\mu_{N}$. Then the following lemma is the
equivalent of Lemma 8.9 for the two-colour mean-field system.

######  Lemma 10.10.

[Comparison of finite and infinite systems] Fix $s>0$, and let $L(N)$ satisfy
$\lim_{N\to\infty}L(N)=\infty$ and $\lim_{N\to\infty}L(N)/N$. Suppose that

(10.58) $\displaystyle\lim_{N\to\infty}\sup_{0\leq t\leq
L(N)}\left|\bar{\Theta}^{(1),[N]}(Ns)-\bar{\Theta}^{(1),[N]}(Ns-t)\right|=0\
\text{ in probability}.$

Then, for all $t\geq 0$,

(10.59)
$\lim_{k\to\infty}\left|\mathbb{E}\left[f\big{(}Z^{\mu_{N}}(t)\big{)}-f\big{(}Z^{[N]}(Ns-L(N)+t)\big{)}\right]\right|=0\qquad\forall\,f\in\mathcal{C}\bigl{(}([0,1]^{3})^{\mathbb{N}_{0}},\mathbb{R}\bigr{)}.$

###### Proof.

We proceed as in the proof of Lemma 8.9. We rewrite the SSDE in (10.4) as

(10.60) $\displaystyle{\rm
d}x^{[N]}_{i}(t)=c_{0}\,\big{[}\Theta^{(1),[N]}-x^{[N]}_{i}(t)\big{]}\,{\rm
d}t$
$\displaystyle\qquad\qquad\qquad+c_{0}\,\big{[}\bar{\Theta}^{(1),[N]}(t)-\Theta^{(1),[N]}\big{]}\,{\rm
d}t+c_{0}\,\big{[}\Theta^{(1),[N]}_{x}(t)-\bar{\Theta}^{(1),[N]}(t)\big{]}\,{\rm
d}t$
$\displaystyle\qquad\qquad\qquad+\sqrt{g\big{(}x^{[N]}_{i}(t)\big{)}}\,{\rm
d}w_{i}(t)$
$\displaystyle\qquad\qquad\qquad+K_{0}e_{0}\,\big{[}y^{[N]}_{i,0}(t)-x^{[N]}_{i}(t)\big{]}{\rm
d}t+\frac{K_{1}e_{1}}{N}\big{[}y^{[N]}_{i,1}(t)-x_{i}^{[N]}(t)\big{]}{\rm
d}t,$ $\displaystyle{\rm
d}y^{[N]}_{i,0}(t)=e_{0}\,\big{[}x^{[N]}_{i}(t)-y^{[N]}_{i,0}(t)\big{]}\,{\rm
d}t,$ $\displaystyle{\rm
d}y^{[N]}_{i,1}(t)=\frac{e_{1}}{N}\,\big{[}x^{[N]}_{i}(t)-y^{[N]}_{i,1}(t)\big{]}\,{\rm
d}t,\qquad i\in[N].$

As before, we consider the finite system in (LABEL:m24b) as a system on
$([0,1]^{3})^{\mathbb{N}_{0}}$ by periodic continuation, and we couple the
finite system in (LABEL:m24b) and the infinite system in (10.59) via there
Brownian motions. 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
$\tilde{z}^{[N]}_{i}(t)=(\tilde{x}^{[N]}_{i}(t),\tilde{y}^{[N]}_{i,0}(t),\tilde{y}^{[N]}_{i,1}(t))$
and
$\tilde{z}^{\mu_{N}}_{i}(t)=(\tilde{x}^{\mu_{N}}_{i}(t),\tilde{y}^{\mu_{N}}_{i,0}(t),\tilde{y}^{\mu_{N}}_{i,1}(t))$.
We define

(10.61) $\displaystyle\Delta_{i,0}^{[N]}(t)$
$\displaystyle=\tilde{x}_{i}^{[N]}(t)-\tilde{x}_{i}^{\mu_{N}}(t),$
$\displaystyle\delta_{i,0}^{[N]}(t)$
$\displaystyle=\tilde{y}_{i,0}^{[N]}(t)-\tilde{y}_{i,0}^{\mu_{N}}(t),$
$\displaystyle\delta_{i,1}^{[N]}(t)$
$\displaystyle=\tilde{y}_{i,1}^{[N]}(t)-\tilde{y}_{i,1}^{\mu_{N}}(t).$

As in the proof of Lemma 8.9, we have to show that, for all $t\geq 0$,

(10.62)
$\lim_{N\to\infty}\mathbb{E}[|\Delta^{[N]}_{i}(t)|]=0,\qquad\lim_{N\to\infty}\mathbb{E}[|\delta^{[N]}_{i,0}(t)|]=0,\qquad\lim_{N\to\infty}\mathbb{E}[|\delta^{[N]}_{i,1}(t)|]=0.$

To prove the third limit in (10.63), note that, by (10.57), (LABEL:m24b) and
the choice of the initial measure in the coupling,

(10.63) $y_{i,1}^{[N]}(t)=y_{i,1}^{[N]}(0)+\frac{e_{1}}{N}\int_{0}^{t}{\rm
d}r\,\big{[}x_{i}^{[N]}(r)-y_{i,1}^{[N]}(r)\big{]}=y_{i,1}^{\mu_{N}}(t)+\frac{e_{1}}{N}\int_{0}^{t}{\rm
d}r\,\big{[}x_{i}^{[N]}(r)-y_{i,1}^{[N]}(r)\big{]}.$

Hence

(10.64) $\lim_{N\to\infty}\mathbb{E}[|\delta^{[N]}_{i,1}(L(N))|]=0.$

To prove the first two limits in (10.63), we argue as in the proof of Lemma
8.9, but we need to add extra drift terms towards the first seed-bank. Using
Itô-calculus, we obtain

(10.65) $\displaystyle\frac{{\rm d}}{{\rm
d}t}\mathbb{E}[|\Delta_{i}^{[N]}(t)|+K|\delta^{[N]}_{i,0}(t)|]$
$\displaystyle=-c\,\mathbb{E}[\Delta_{i}^{[N]}(t)]$
$\displaystyle\quad-2K_{0}e_{0}\,\mathbb{E}\left[[|\Delta_{i}^{[N]}(t)|+|\delta^{[N]}_{i}(t)|]\,1_{\\{{\rm
sign}\Delta_{i}^{[N]}(t)\neq{\rm sign}\delta_{i,0}^{[N]}(t)\\}}\right]$
$\displaystyle\quad+c\,{\rm
sign}\Delta_{i}^{[N]}(t)\big{[}\bar{\Theta}^{(1),[N]}(t)-\bar{\Theta}^{(1),[N]}\big{]}$
$\displaystyle\quad+c\,{\rm
sign}\Delta_{i}^{[N]}(t)\big{[}\bar{\Theta}_{x}^{(1),[N]}(t)-\bar{\Theta}^{(1),[N]}(t)\big{]}$
$\displaystyle\quad+\frac{K_{1}e_{1}}{N}\,{\rm
sign}\Delta_{i}^{[N]}(t)\big{[}\delta_{i,1}^{[N]}(t)-\Delta_{i}^{[N]}(t)\big{]}.$

This can be rewritten as

(10.66) $\displaystyle
0\leq\mathbb{E}[|\Delta_{i}^{[N]}(t)|+K_{0}|\delta^{[N]}_{i,0}(t)|]$
$\displaystyle\leq\mathbb{E}[|\Delta_{i}^{[N]}(0)|+K|\delta^{[N]}_{i,0}(0)|]-c\int_{0}^{t}{\rm
d}r\,\mathbb{E}[\Delta_{i}^{[N]}(r)]$
$\displaystyle\quad-2K_{0}e_{0}\int_{0}^{t}{\rm
d}r\,\mathbb{E}\left[[|\Delta_{i}^{[N]}(r)|+|\delta^{[N]}_{i,0}(r)|]\,1_{\\{{\rm
sign}\Delta_{i}^{[N]}(t)\neq{\rm sign}\delta_{i,0}^{[N]}(t)\\}}\right]$
$\displaystyle\quad+c\,\int_{0}^{t}{\rm
d}r\,|\bar{\Theta}^{(1),[N]}(r)-{\Theta}^{(1),[N]}|$
$\displaystyle\quad+c\,\int_{0}^{t}{\rm
d}r\,|\bar{\Theta}_{x}^{(1),[N]}(r)-\bar{\Theta}^{(1),[N]}(r)|$
$\displaystyle\quad+\frac{K_{1}e_{1}}{N}\int_{0}^{t}{\rm
d}r\,\left|\delta_{i,1}^{[N]}(r)-\Delta_{i}^{[N]}(r)\right|.$

By the construction of the measure $\mu_{N}$, we have

(10.67)
$\lim_{N\to\infty}\mathbb{E}[|\Delta_{i}^{[N]}(0)|+K_{0}|\delta^{[N]}_{i}(0)|]=0.$

Therefore, for all $t\geq 0$,

(10.68)
$\lim_{N\to\infty}\mathbb{E}[|\Delta_{i}^{[N]}(t)|+K_{0}|\delta^{[N]}_{i,0}(t)|]=0.$

Combine this with (10.64) and use that Lipschitz functions are dense in the
set of bounded continuous functions. Then, as in the proof of Lemma 8.9, we
get the claim in (10.59). ∎

Before we can prove that the infinite system
$(X^{\mu_{N}}(t),Y_{0}^{\mu_{N}}(t),Y_{1}^{\mu_{N}}(t))_{t\geq 0}$ converges
to a limiting system as $N\to\infty$, we need the following regularity
property for the estimators $(\bar{\Theta}^{[N]},\Theta_{y_{1}}^{[N]})$.

######  Lemma 10.11.

[Stability of the estimator for the conserved quantity] Define $\mu_{N}$ as in
Lemma 10.10. Let $(x^{N}_{i},y^{N}_{i,0},y^{N}_{i,1})_{i\in[N]}$ be
distributed according to the exchangeable probability measure $\mu_{N}$ on
$([0,1]^{3})^{\mathbb{N}_{0}}$ restricted to $([0,1]^{3})^{[N]}$. Suppose that
$\lim_{N\to\infty}\mu_{N}=\mu$ for some exchangeable probability measure $\mu$
on $([0,1]^{3})^{\mathbb{N}_{0}}$. Define the random variable $\phi$ on
$(\mu,([0,1]^{3})^{\mathbb{N}_{0}})$ by putting

(10.69) $\displaystyle\phi$ $\displaystyle=(\phi_{1},\phi_{2}),$
$\displaystyle\phi_{1}$
$\displaystyle=\lim_{n\to\infty}\frac{1}{n}\sum_{i\in[n]}\frac{x_{i}+Ky_{i,0}}{1+K},\qquad\phi_{2}=\lim_{n\to\infty}\frac{1}{n}\sum_{i\in[n]}y_{i,1,}$

and the random variable $\phi^{[N]}$on
$(\mu_{N},([0,1]^{3})^{\mathbb{N}_{0}})$ by putting

(10.70) $\displaystyle\phi^{[N]}$
$\displaystyle=(\phi_{1}^{[N]},\phi_{2}^{[N]})$ $\displaystyle\phi^{[N]}_{1}$
$\displaystyle=\frac{1}{N}\sum_{i\in[N]}\frac{x^{N}_{i}+Ky^{N}_{i,0}}{1+K},\qquad\phi^{[N]}_{2}$
$\displaystyle=\frac{1}{N}\sum_{i\in[N]}y^{N}_{i,1}.$

Then

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

###### Proof.

We can use a similar argument as in the proof of Lemma 8.10. Define

(10.72)
$D^{[N]}(Z)=\left(\frac{1}{N}\sum_{j\in[N]}\frac{x_{j}+K_{0}y_{j,0}}{1+K_{0}},\frac{1}{N}\sum_{j\in[N]}y_{i,1}\right).$

Then we can proceed as in the proof of Lemma 8.10, using Fourier analysis for
both components of $D^{[N]}(Z)$ separately. ∎

In the fifth and final lemma we state the convergence of
$\mathcal{L}[(X^{\mu_{N}}(t),Y_{0}^{\mu_{N}}(t),Y_{1}^{\mu_{N}}(t))]$ to the
law of a limiting system as $N\to\infty$.

######  Lemma 10.12.

[Uniformity of the ergodic theorem for the infinite system] Let $\mu_{N}$ be
defined as in (10.56). 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

(10.73)
$\Theta=\lim_{N\to\infty}\frac{1}{N}\sum_{i\in[N]}\frac{x_{i}^{\mu}+Ky_{i,0}^{\mu}}{1+K}\,\qquad\text{
in }L_{2}(\mu).$

Let
$Z^{\mu}(t)=\bigl{(}X^{\mu}(t),Y_{0}^{\mu}(t),Y_{1}^{\mu}(t)\bigr{)}_{t\geq
0}$ be the infinite system evolving according to

(10.74) $\displaystyle{\rm
d}x^{\mu}_{i}(t)=c\left[\Theta-x^{\mu}_{i}(t)\right]{\rm
d}t+\sqrt{g(x^{\mu}_{i}(t))}\,{\rm
d}w_{i}(t)+Ke\,[y^{\mu}_{i,1}(t)-x^{\mu}_{i}(t)]\,{\rm d}t,$
$\displaystyle{\rm
d}y^{\mu}_{i,0}(t)=e\,[x^{\mu}_{i}(t)-y^{\mu}_{i,1}(t)]\,{\rm d}t,$
$\displaystyle{\rm d}y^{\mu}_{i,1}(t)=y^{\mu}_{i,1}(0),\qquad
i\in\mathbb{N}_{0}.$

and let
$Z^{\mu_{N_{k}}}(t)=(X^{\mu_{N_{k}}}(t),Y_{0}^{\mu_{N_{k}}}(t),Y_{1}^{\mu_{N_{k}}}(t))_{t\geq
0}$ be the infinite system defined in (10.56). Then

  1. 1.

For all $t\geq 0$,

(10.75)
$\displaystyle\lim_{k\to\infty}\bigl{|}\mathbb{E}\bigl{[}f\bigl{(}Z^{\mu_{N_{k}}}(t)\bigr{)}\bigr{]}-\mathbb{E}\bigl{[}f\bigl{(}Z^{\mu}(t)\bigr{)}\bigr{]}\bigr{|}=0\qquad\forall\,f\in\mathcal{C}\bigl{(}([0,1]^{2})^{\mathbb{N}_{0}},\mathbb{R}\bigr{)}.$

  2. 2.

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

(10.76)
$\displaystyle\lim_{k\to\infty}\bigl{|}\mathbb{E}\bigl{[}f\bigl{(}Z^{[N_{k}]}(N_{k}s-L(N_{k})+\bar{L}(N_{k}))\bigr{)}-f\bigl{(}Z^{\mu_{N_{k}}}(\bar{L}(N_{k}))\bigr{)}\bigr{|}\bigr{]}$
$\displaystyle\quad+\bigl{|}\mathbb{E}\bigl{[}f\bigl{(}Z^{\mu_{N_{k}}}(\bar{L}(N_{k}))\bigr{)}\bigr{]}-\mathbb{E}\bigl{[}f\bigl{(}Z^{\mu}(\bar{L}(N_{k}))\bigr{)}\bigr{]}\bigr{|}=0\qquad\forall\,f\in\mathcal{C}\bigl{(}([0,1]^{2})^{\mathbb{N}_{0}},\mathbb{R}\bigr{)}.$

###### Proof.

As in the proof of Lemma 8.11, we can construct
$(z_{i}^{\mu_{N}})_{i\in\mathbb{N}_{0}}$ and
$(z_{i}^{\mu})_{i\in\mathbb{N}_{0}}$ on one probability space. Then

(10.77) $\lim_{N\to\infty}y^{\mu_{N}}_{i,1}(0)=y^{\mu}_{i,1}(0)\text{ a.s. }$

and

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

Via a similar coupling as in Lemma (10.10), it follows via Itô-calculus that
(10.75) holds. Combining (10.64), (10.68), (10.77) and (10.78), we obtain, via
a similar construction as in the proof of Lemma 8.11, a sequence $\bar{L}(N)$
such that

(10.79)
$\displaystyle\lim_{N\to\infty}\mathbb{E}[|\Delta^{N}_{i}(\bar{L}(N))|+K_{0}|\delta^{N}_{i,0}(\bar{L}(N))|]+K_{1}|\delta^{N}_{i,1}(\bar{L}(N))|]$
$\displaystyle+\mathbb{E}[|\Delta^{\mu_{N}}_{i}(\bar{L}(N))|+K_{0}|\delta^{\mu_{N}}_{i,0}(\bar{L}(N))|]+K_{1}|\delta^{\mu_{N}}_{i,1}(\bar{L}(N))|]=0.$

As in the proof of Lemma 8.11, we can again use Lipschitz functions to
conclude (10.76). ∎

######  Lemma 10.13.

[Coupling of finite systems] Let

(10.80) $Z^{[N],1}=(X^{[N],1},Y_{0}^{[N],1},Y_{1}^{[N],1})$

be the finite system evolving according to (10.4) starting from an
exchangeable initial measure. Let $\mu^{[N],1}$ be the measure obtained by
periodic continuation of the configuration of $Z^{[N],1}(0)$. Similarly, let

(10.81) $Z^{[N],2}=(X^{[N],2},Y_{0}^{[N],2},Y_{1}^{[N],2})$

be the finite system evolving according to (10.4) starting from an
exchangeable initial measure. Let $\mu^{[N],2}$ be the measure obtained by
periodic continuation of the configuration of $Z^{[N],2}(0)$. 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 the random variables
$\bar{\Theta}^{[N],1}$ and $\bar{\Theta}^{[N],2}$ on
$(([0,1]^{3})^{\mathbb{N}_{0}}\times([0,1]^{3})^{\mathbb{N}_{0}},\mu^{[N],1}\times\mu^{[N],2})$
and $\bar{\Theta}_{1}$ and $\bar{\Theta}_{2}$ on
$(([0,1]^{3})^{\mathbb{N}_{0}}\times([0,1]^{3})^{\mathbb{N}_{0}},\bar{\mu})$
by

(10.82)
$\displaystyle\bar{\Theta}^{[N],1}=\frac{1}{N}\sum_{i\in[N]}\frac{x^{[N],1}_{i}+K_{0}y^{[N],1}_{i,0}}{1+K_{0}},\qquad\bar{\Theta}^{[N],2}=\frac{1}{N}\sum_{i\in[N]}\frac{x^{[N],2}_{i}+K_{0}y^{[N],2}_{i,0}}{1+K_{0}},$
$\displaystyle\bar{\Theta}^{1}=\lim_{n\to\infty}\frac{1}{n}\sum_{i\in[n]}\frac{x^{1}_{i}+K_{0}y^{1}_{i,0}}{1+K_{0}},\qquad\bar{\Theta}^{2}=\lim_{n\to\infty}\frac{1}{n}\sum_{i\in[n]}\frac{x^{2}_{i}+K_{0}y^{2}_{i,0}}{1+K_{0}},$

and let $(\bar{\Theta}^{(1),[N],1}(t))_{t\geq 0}$ and
$(\bar{\Theta}^{(1),[N],2}(t))_{t\geq 0}$ be defined as in (10.14) for
$Z^{[N],1}$, respectively, $Z^{[N],2}$. Suppose that

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

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

(10.84)
$\displaystyle\lim_{N\to\infty}\mathbb{E}\bigl{[}|x^{[N],1}_{i}(t(N))-x^{[N],2}_{i}(t(N))|+K_{0}|y^{[N],1}_{i,0}(t(N))-y^{[N],2}_{i,0}(t(N))|$
$\displaystyle\qquad+K_{1}|y^{[N],1}_{i,1}(t(N))-y^{[N],2}_{i,1}(t(N))|\bigr{]}=0.$

###### Proof.

Via standard Itô-calculus we obtain from (10.4) that

(10.85) $\displaystyle\frac{{\rm d}}{{\rm
d}t}\mathbb{E}\left[|x^{[N],1}_{i}(t)-x^{[N],2}_{i}(t)|+K_{0}|y^{[N],1}_{i,0}(t)-y^{[N],2}_{i,0}(t)|+K_{1}|y^{[N],1}_{i,1}(t)-y^{[N],2}_{i,1}(t)|\right]$
$\displaystyle=-\frac{2c}{N}\sum_{j\in[N]}\mathbb{E}\left[|x^{[N],1}_{j}(t)-x^{[N],2}_{j}(t)|1_{\\{{\rm
sign}(x^{[N],1}_{j}(t)-x^{[N],2}_{j}(t))\neq{\rm
sign}(x^{[N],1}_{i}(t)-x^{[N],2}_{i}(t))\\}}\right]$
$\displaystyle\qquad-2K_{0}e_{0}\,\mathbb{E}\bigg{[}|x^{[N],1}_{i}(t)-x^{[N],2}_{i}(t)|+K|y^{[N],1}_{i,0}(t)-y^{[N],2}_{i,0}(t)|$
$\displaystyle\qquad\qquad\times 1_{\\{{\rm
sign}(x^{[N],1}_{i}(t)-x^{[N],2}_{i}(t))\neq{\rm
sign}(y^{[N],1}_{i,0}(t)-y^{[N],2}_{i,0}(t))\\}}\bigg{]}$
$\displaystyle\qquad-2\frac{K_{1}e_{1}}{N}\,\mathbb{E}\bigg{[}|x^{[N],1}_{i}(t)-x^{[N],2}_{i}(t)|+K_{1}|y^{[N],1}_{i,0}(t)-y^{[N],2}_{i,0}(t)|$
$\displaystyle\qquad\qquad\times 1_{\\{{\rm
sign}(x^{[N],1}_{i}(t)-x^{[N],2}_{i}(t))\neq{\rm
sign}(y^{[N],1}_{i,1}(t)-y^{[N],2}_{i,1}(t))\\}}\bigg{]}.$

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

(10.86) $\displaystyle
t\mapsto\mathbb{E}\left[|x^{[N],1}_{i}(t)-x^{[N],2}_{i}(t)|+K_{0}|y^{[N],1}_{i,0}(t)-y^{[N],2}_{i,0}(t)|+K_{1}|y^{[N],1}_{i,1}(t)-y^{[N],2}_{i,1}(t)|\right]$

is a decreasing function. Hence we can use the same strategy as in the proof
of Lemma 8.13 to finish the proof. ∎

######  $\bullet$ Proof of Proposition 10.6

###### Proof.

We follow a similar argument as in the proof of Proposition 8.4. Let $L(N)$
satisfy $\lim_{N\to\infty}$ $L(N)=\infty$ and $\lim_{N\to\infty}L(N)/N=0$. Let
$\mu_{N}$ be the measure on $([0,1]^{3})^{\mathbb{N}_{0}}$ obtained by
periodic continuation of $\mathcal{L}[Z^{[N]}(Ns-L(N))]$. Note that
$([0,1]^{3})^{\mathbb{N}_{0}}$ is compact. Hence, letting
$(N_{k})_{k\in\mathbb{N}}$ be the subsequence in Proposition 10.6, we can pass
to a possibly further subsequence and obtain

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

Since we assumed that $\mathcal{L}[Z^{[N]}(0)]$ is exchangeable and the
dynamics preserve exchangeability, the measures $\mu_{N_{k}}$ are translation
invariant and also the limiting law $\mu$ is translation invariant.

Let $\phi=(\phi_{1},\phi_{2})$ be defined as in (10.69) in Lemma 10.11. Then
we can condition on $\phi=(\phi_{1},\phi_{2})$ and write

(10.88) $\mu=\int_{[0,1]^{2}}\mu_{\rho}\,{\rm d}\Lambda(\rho),$

where $\Lambda(\cdot)=\mathcal{L}[\phi]=\mathcal{L}[(\phi_{1},\phi_{2})]$ and
$\rho=(\rho_{1},\rho_{2})$. By assumption we know that

(10.89)
$\lim_{k\to\infty}\mathcal{L}\left[{\bf\Theta}^{\mathrm{eff},(1),[N_{k}]}(N_{k}s)\right]=P_{{\bf\Theta^{\mathrm{eff}}}(s)}(\cdot)$

and

(10.90) $\displaystyle\lim_{k\to\infty}\mathcal{L}\left[\sup_{0\leq t\leq
L(N_{k})}\left|\bar{\Theta}^{[N_{k}]}(N_{k}s)-\bar{\Theta}^{[N_{k}]}(N_{k}s-t)\right|+\left|{\Theta}_{y_{1}}^{[N_{k}]}(N_{k}s)-{\Theta_{y_{1}}}^{[N_{k}]}(N_{k}s-t)\right|\right]$
$\displaystyle\qquad\qquad\qquad=\delta_{0}.$

Hence

(10.91)
$\displaystyle\lim_{k\to\infty}\mathcal{L}\left[{\bf\Theta}^{\mathrm{eff},(1),[N_{k}]}(N_{k}s-L(N_{k}))\right]=P_{{\bf\Theta^{\mathrm{eff}}}(s)}(\cdot).$

Recall that

(10.92)
$\Lambda(\cdot)=\mathcal{L}\left[\lim_{n\to\infty}\left(\frac{1}{n}\sum_{i\in[n]}\frac{x_{i}+Ky_{i,0}}{1+K},\frac{1}{n}\sum_{i\in[n]}y_{i,1}\right)\right]\quad\text{
on }(\mu,([0,1]^{2})^{\mathbb{N}_{0}}).$

By Lemma 8.10, if

(10.93)
$\phi^{N_{k}}=(\phi_{1}^{N_{k}},\phi_{2}^{N_{k}})=\left(\frac{1}{N_{k}}\sum_{i\in[N_{k}]}\frac{x_{i}+Ky_{i,0}}{1+K},\frac{1}{N_{k}}\sum_{i\in[N_{k}]}y^{[N_{k}]}_{i,1}\right)\text{
on }(\mu_{N_{k}},([0,1]^{3})^{\mathbb{N}_{0}}),$

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
$\Lambda(\cdot)=P_{{\bf\Theta^{\mathrm{eff}}}(s)}(\cdot)$, and hence

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

Let $\bar{L}(N)$ be the sequence constructed in Lemma 10.12[b]. By
construction we can require that $\bar{L}(N)\leq L(N)$ for all
$N\in\mathbb{N}$. Write

(10.95)
$\displaystyle\mathcal{L}\bigl{[}Z^{[N_{k}]}(N_{k}s-L(N_{k})+\bar{L}(N_{k}))\bigr{]}$
$\displaystyle=\mathcal{L}\bigl{[}Z^{[N_{k}]}(N_{k}s-L(N_{k})+\bar{L}(N_{k}))\bigr{]}-\mathcal{L}\bigl{[}Z^{\mu_{N_{k}}}(\bar{L}(N_{k}))\bigr{]},$
$\displaystyle\quad+\mathcal{L}\bigl{[}Z^{\mu_{N_{k}}}(\bar{L}(N_{k}))\bigr{]}-\mathcal{L}\bigl{[}Z^{\mu}(\bar{L}(N_{k}))\bigr{]}$
$\displaystyle\quad+\mathcal{L}\bigl{[}Z^{\mu}(\bar{L}(N_{k}))\bigr{]}.$

By Lemma 10.12 the first and second differences tend to zero as $k\to\infty$.
Hence

(10.96)
$\lim_{k\to\infty}\mathcal{L}\bigl{[}Z^{[N_{k}]}(N_{k}s-L(N_{k})+\bar{L}(N_{k}))\bigr{]}=\mathcal{L}\bigl{[}Z^{\mu}(\bar{L}(N_{k}))\bigr{]}.$

By (10.88),

(10.97)
$\mathcal{L}\bigl{[}Z^{\mu}(\bar{L}(N_{k}))\bigr{]}=\int_{[0,1]^{2}}\mathcal{L}\bigl{[}Z^{\mu_{\rho}}(\bar{L}(N_{k}))\bigr{]}\,P_{{\bf\Theta^{\mathrm{eff}}}(s)}({\rm
d}\rho).$

For the infinite system $(Z^{\mu_{\rho}}(t))_{t\geq
0}=\left(X^{\mu_{\rho}}(t),Y_{0}^{\mu_{\rho}}(t),Y_{1}^{\mu_{\rho}}(t)\right)_{t\geq
0}$ we have

(10.98) $Y^{\mu}_{1}(t)=Y^{\mu}_{1}(0)\,a.s.$

and hence, since $\lim_{k\to\infty}\bar{L}(N_{k})/N_{k}=0$ by (10.44),

(10.99)
$\lim_{k\to\infty}\mathcal{L}[Y_{1}^{\mu_{\rho}}(\bar{L}(N_{k}))]=\mathcal{L}[Y_{1}^{\mu_{\rho}}(0)]\,\,\forall\,\rho\in[0,1].$

Therefore

(10.100)
$\lim_{k\to\infty}\mathcal{L}[Y_{1}^{\mu_{\rho}}(\bar{L}(N_{k}))]=P_{Y_{1}(s)}^{\rho}(\cdot)$

and

(10.101)
$\displaystyle\mathcal{L}\bigl{[}X^{\mu_{\rho}}(\bar{L}(N_{k})),Y_{0}^{\mu_{\rho}}(\bar{L}(N_{k})),Y_{1}^{\mu_{\rho}}(\bar{L}(N_{k}))\bigr{]}$
$\displaystyle\qquad=\int\mathcal{L}\bigl{[}X_{1}^{\mu_{\rho}}(\bar{L}(N_{k})),Y_{0}^{\mu_{\rho}}(\bar{L}(N_{k})),{\bf
y_{1}}\bigr{]}{\rm d}P_{Y_{1}(s)}^{\rho}({\rm d}{\bf y_{1}}).$

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

(10.102)
$\displaystyle\lim_{k\to\infty}\mathcal{L}\bigl{[}Z^{\mu_{\rho}}(\bar{L}(N_{k}))\bigr{]}$
$\displaystyle=\lim_{k\to\infty}\mathcal{L}\bigl{[}X_{1}^{\mu_{\rho}}(\bar{L}(N_{k})),Y_{0}^{\mu_{\rho}}(\bar{L}(N_{k})),Y_{1}^{\mu_{\rho}}(\bar{L}(N_{k}))\bigr{]}$
$\displaystyle=\int\nu_{\rho,{\bf y_{1}}}P_{Y_{1}(s)}^{\rho}({\rm d}{\bf
y_{1}}).$

Therefore, by (9.8), (10.97) and Lemma 8.6,

(10.103)
$\displaystyle\lim_{k\to\infty}\mathcal{L}\bigl{[}Z^{[N_{k}]}(N_{k}s-L(N_{k})+\bar{L}(N_{k}))\bigr{]}=\int_{[0,1]}P_{{\bf\Theta^{\mathrm{eff}}}(s)}({\rm
d}\rho)\int\nu_{\rho,{\bf y_{1}}}P_{Y_{1}(s)}^{\rho}({\rm d}{\bf y_{1}}).$

To finish the proof, we proceed as in the proof of Proposition 8.4 and invoke
Lemma 10.13. Let $Z^{[N],1}=(X^{[N],1},Y_{0}^{[N],1},Y_{1}^{[N],1})$ be the
finite system starting from

(10.104)
$\mathcal{L}\bigl{[}Z^{[N]}(Ns-L(N))\bigr{]}=\mathcal{L}\bigl{[}X^{[N]}(Ns-L(N)),Y_{0}^{[N]}(Ns-L(N)),Y_{1}^{[N]}(Ns-L(N))\bigr{]}.$

Let $(\bar{L}(N))_{N\in\mathbb{N}}$ be the sequence constructed in Lemma
10.12. Let $Z^{[N],2}=(X^{[N],2},Y_{0}^{[N],2},Y_{1}^{[N],2})$ be the finite
system starting from

(10.105)
$\mathcal{L}\bigl{[}X^{[N]}(Ns-\bar{L}(N))\bigr{]}=\mathcal{L}\left[X^{[N]}(Ns-\bar{L}(N)),Y_{0}^{[N]}(Ns-\bar{L}(N)),Y_{1}^{[N]}(Ns-\bar{L}(N))\right].$

Choose for $t(N)$ in Lemma 10.13 the sequence $\bar{L}(N)$. Let $\mu^{[N],1}$
be defined by the periodic continuation of the configuration of
$Z^{[N]}(Ns-L(N))$ and $\mu^{[N],2}$ be defined by periodic continuation of
the configuration of $Z^{[N]}(Ns-\bar{L}(N))$. Define $\Theta_{1}$ and
$\Theta_{2}$ according to (8.44), where under $\mu^{[N],2}$ we replace $L(N)$
by $\bar{L}(N)$. Then, by the assumptions in (10.44),

(10.106)
$\displaystyle\lim_{k\to\infty}|\Theta^{(1),[N_{k}],1}-\Theta^{(1),[N_{k}],2}|$
$\displaystyle=\lim_{k\to\infty}|\Theta^{N_{k}}(N_{k}s-L(N_{k}))-\Theta^{N_{k}}(N_{k}s-\bar{L}(N_{k}))|$
$\displaystyle=0\text{ in probability}.$

Using 10.63 we see that also, for all $i\in[N]$,

(10.107)
$\displaystyle\lim_{k\to\infty}|y^{[N_{k}],1}_{i,1}(0)-y^{[N_{k}],2}_{i,1}(0)|$
$\displaystyle=\lim_{k\to\infty}|y_{i,1}^{[N_{k}]}(N_{k}s-L(N_{k}))-y_{i,1}^{[N_{k}]}(N_{k}s-\bar{L}(N_{k}))|$
$\displaystyle=0\text{ in probability}.$

Therefore, if $\mu$ is any weak limit point of the sequence
$\left(\mu^{[N_{k}],1}\times\mu^{[N_{k}],2}\right)_{k\in\mathbb{N}}$, then

(10.108) $\mu(\\{\Theta_{1}=\Theta_{2},\,Y^{1}_{1}=Y^{2}_{1}\\})=1.$

Hence, by possibly passing to a further subsequence, we can now apply Lemma
10.13 to obtain, for all $i$,

(10.109) $\displaystyle\lim_{k\to\infty}\mathbb{E}\Bigl{[}$
$\displaystyle|x^{[N_{k}],1}_{i}(\bar{L}(N_{k}))-x^{[N_{k}],2}_{i}(\bar{L}(N_{k}))|$
$\displaystyle\qquad+K_{0}\,|y_{i,0}^{[N_{k}],1}(\bar{L}(N_{k}))-y_{i,0}^{[N_{k}],2}(\bar{L}(N_{k}))|$
$\displaystyle\qquad+K_{1}\,|y_{i,1}^{[N_{k}],1}(\bar{L}(N_{k}))-y_{i,1}^{[N_{k}],2}(\bar{L}(N_{k}))|\Bigr{]}=0.$

Hence

(10.110)
$\lim_{N\to\infty}\Big{(}\mathcal{L}[Z^{[N],1}(\bar{L}(N_{k}))]-\mathcal{L}[Z^{[N],2}(\bar{L}(N_{k}))]\Big{)}=\delta_{0}$

and therefore

(10.111)
$\lim_{k\to\infty}\mathcal{L}\bigl{(}Z^{[N_{k}]}(N_{k}s)\bigr{)}=\int_{[0,1]}P_{{\bf\Theta^{\mathrm{eff}}}(s)}({\rm
d}\rho)\int\nu_{\rho,{\bf y_{1}}}P_{Y_{1}(s)}^{\rho}({\rm d}{\bf y_{1}}).$

This concludes the proof of Proposition 10.6. ∎

Like for the one-colour mean-field system, Proposition 10.6 and Lemmas
10.7–10.13 give rise to the following corollary, which will be important to
derive the evolution of the $1$-blocks on time scale $Ns$.

######  Corollary 10.14.

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

(10.112)
$Z^{[N]}(Ns-L(N))=(X^{[N]}(Ns-L(N)),Y_{0}^{[N]}(Ns-L(N)),Y_{1}^{[N]}(Ns-L(N))),$

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

(10.113)
$\Theta=\lim_{N\to\infty}\frac{1}{N}\sum_{i\in[N]}\frac{x_{i}^{\mu}+Ky_{i}^{\mu}}{1+K}\,\qquad\text{
in }L_{2}(\mu),$

and let
$(Z^{\nu_{\Theta}}(t))_{t>0}=(X^{\nu_{\Theta}}(t),Y_{0}^{\nu_{\Theta}}(t),Y_{1}^{\nu_{\Theta}}(t))_{t>0}$
be the infinite system evolving according to (LABEL:binfb2) starting from its
equilibrium measure. Consider the finite system $Z^{[N]}$ as a system on
$([0,1]^{3})^{\mathbb{N}_{0}}$ by periodic continuation. Construct
$(Z^{[N]}(t))_{t>0}$ and $(Z^{\nu_{\Theta}}(t))_{t>0}$ on one probability
space. Then, for all $t\geq 0$,

(10.114) $\displaystyle\lim_{N\to\infty}$
$\displaystyle\mathbb{E}\left[\left|x_{i}^{[N]}(Ns+t)-x_{i}^{\nu_{\Theta}}(t)\right|\right]+K_{0}\,\mathbb{E}\left[\left|y_{i,0}^{[N]}(Ns+t)-y_{i,0}^{\nu_{\Theta}}(t)\right|\right]$
$\displaystyle\qquad+K_{1}\,\mathbb{E}\left[\left|y_{i,1}^{[N]}(Ns+t)-y_{i,1}^{\nu_{\Theta}}(t)\right|\right]=0\qquad\forall\,i\in[N].$

###### Proof.

Proceed as in the proof of Corollary 9.1, but use the setup of the two-colour
mean-field system and therefore replace Proposition 8.4, Lemma 8.11 and Lemma
8.13 by, respectively Proposition 10.6, Lemma 10.12 and Lemma 10.13. ∎

###### Step 4: Limiting evolution of the $1$-blocks.

######  Lemma 10.15.

[Limiting evolution of the 1-blocks] Let $(z_{1}^{\mathrm{eff}}(s))_{s>0}$ be
the process defined in (10.21) with initial state

(10.115) $z_{1}^{\mathrm{eff}}(0)=(\vartheta_{0},\theta_{y_{1}}).$

Then

(10.116)
$\lim_{N\to\infty}\mathcal{L}\left[\big{(}{\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns)\big{)}_{s>0}\right]=\mathcal{L}\left[(z_{1}^{\mathrm{eff}}(s))_{s>0}\right].$

###### Proof.

By [YW71], the SSDE in (10.21) has a unique strong solution. Therefore the
process $(z^{\mathrm{eff}}_{1}(s))_{s>0}$ is Markov. Its generator $G$ is
given by

(10.117) $G=\frac{K_{1}e_{1}}{1+K_{0}}\,(y-x)\,\frac{\partial}{\partial
x}+e_{1}(x-y)\,\frac{\partial}{\partial
y}+\frac{1}{(1+K_{0})^{2}}\,(\mathcal{F}g)(x)\,\frac{\partial^{2}}{\partial
x^{2}},$

and hence $(z_{1}^{\mathrm{eff}}(s))_{s\geq 0}$ solves the martingale problem
for $G$. We will use [JM86, Theorem 3.3.1], to prove that (10.116) holds.

Define

(10.118)
$(\vartheta_{0}^{N},\vartheta_{y_{1}}^{N})=\left(\bar{\Theta}^{(1),[N]}(0),\Theta^{(1),[N]}_{y_{1}}(0)\right).$

Since we start from an i.i.d. law, by the law of large numbers we have that

(10.119)
$\lim_{N\to\infty}{\bf\Theta}^{\mathrm{eff},(1),[N]}(0)=\lim_{N\to\infty}(\vartheta_{0}^{N},\vartheta_{y_{1}}^{N})=\left(\vartheta_{0},\theta_{y_{1}}\right)\qquad
a.s.$

By the SSDE in (10.37) and an optional sampling argument, we have, for all
$N\in\mathbb{N}$,

(10.120) $\lim_{s\downarrow
0}\left(\bar{\Theta}^{(1),[N]}(Ns),\Theta^{(1),[N]}(Ns)\right)=(\vartheta_{0}^{N},\vartheta_{y_{1}}^{N})\quad\text{
a.s.}$

Therefore we can continuously extend the process
$({\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns))_{s>0}$ to $0$ and, in particular,

(10.121)
$\lim_{N\to\infty}\mathcal{L}\left[{\bf\Theta}^{\mathrm{eff},(1),[N]}(0)\right]=\mathcal{L}\left[z_{1}^{\mathrm{eff}}(0)\right].$

Since we already showed that the processes

(10.122) $\big{(}{\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns)\big{)}_{s>0}$

are $\mathcal{D}$-semimartingales, and are trivially bounded, we are left to
show that

(10.123) $\displaystyle\lim_{N\to\infty}\int_{0}^{s}{\rm
d}r\,\mathbb{E}\Big{[}\Big{|}G^{(1),[N]}_{\dagger}\big{(}f,{\bf\Theta}^{\mathrm{eff},(1),[N]}(Nr),r,\cdot\big{)}-(Gf)\big{(}{\bf\Theta}^{\mathrm{eff},(1),[N]}(Nr)\big{)}\Big{|}\Big{]}=0.$

Here, $G^{(1),[N]}_{\dagger}$ is the operator defined in (10.39). Since we are
working on the space $\mathcal{C}^{*}$ of polynomials on $[0,1]^{2}$, all
derivatives of $f\in\mathcal{C}^{*}$ are bounded. Hence, by dominated
convergence, it is enough to prove that, for all $s>0$,

(10.124)
$\displaystyle\lim_{N\to\infty}\mathbb{E}^{[N]}\Big{[}\Big{|}G^{(1),[N]}_{\dagger}\big{(}f,{\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns),s,\cdot\big{)}-(Gf)\big{(}{\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns)\big{)}\Big{|}\Big{]}=0.$

Note that

(10.125)
$\displaystyle\mathbb{E}\Big{[}\Big{|}G^{(1),[N]}_{\dagger}\big{(}f,{\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns),s,\cdot\big{)}-(Gf)\big{(}{\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns)\big{)}\Big{|}\Big{]}$
$\displaystyle=\mathbb{E}\Bigg{[}\Bigg{|}\frac{K_{1}e_{1}}{1+K_{0}}\,\Big{[}\Theta^{(1),[N]}_{y_{1}}(Ns)-\frac{1}{N}\sum_{i\in[N]}x_{i}(Ns,\omega)\Big{]}\,\frac{\partial
f}{\partial x}\big{(}{\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns)\big{)}$
$\displaystyle\qquad+e_{1}\Big{[}\frac{1}{N}\sum_{i\in[N]}x_{i}(Ns,\omega)-\Theta^{(1),[N]}_{y_{1}}(Ns)\Big{]}\frac{\partial
f}{\partial y}\big{(}{\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns)\big{)}$
$\displaystyle\qquad+\frac{1}{(1+K_{0})^{2}}\frac{1}{N}\sum_{i\in[N]}g(x_{i}(Ns,\omega))\,\frac{\partial^{2}f}{\partial
x^{2}}\big{(}{\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns)\big{)}$
$\displaystyle\qquad-\frac{K_{1}e_{1}}{1+K_{0}}\big{[}\Theta^{(1),[N]}_{y_{1}}(Ns)-\bar{\Theta}^{(1),[N]}(Ns)\big{]}\,\frac{\partial
f}{\partial x}\big{(}{\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns)\big{)}$
$\displaystyle\qquad-
e_{1}\big{[}\bar{\Theta}^{(1),[N]}(Ns)-\Theta^{(1),[N]}_{y_{1}}(Ns)\big{]}\,\frac{\partial
f}{\partial y}\big{(}{\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns)\big{)}$
$\displaystyle\qquad-\frac{1}{(1+K_{0})^{2}}(\mathcal{F}g)(\bar{\Theta}^{(1),[N]}(Ns))\,\frac{\partial^{2}f}{\partial
x^{2}}\big{(}{\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns)\big{)}\Bigg{|}\Bigg{]}.$

Hence

(10.126)
$\displaystyle\lim_{N\to\infty}\mathbb{E}\left[\Big{|}G^{(1),[N]}_{\dagger}\big{(}f,{\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns),s,\cdot\big{)}-(Gf)\big{(}{\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns)\big{)}\Big{|}\right]$
$\displaystyle\leq\lim_{N\to\infty}\mathbb{E}\left[\frac{K_{1}e_{1}}{1+K_{0}}\,\left|\bar{\Theta}^{(1),[N]}(Ns)-\frac{1}{N}\sum_{i\in[N]}x_{i}(Ns,\omega)\right|\,\left|\frac{\partial
f}{\partial
x}\big{(}{\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns)\big{)}\right|\right]$
$\displaystyle+\lim_{N\to\infty}\mathbb{E}\left[e_{1}\left|\frac{1}{N}\sum_{i\in[N]}x_{i}(Ns,\omega)-\bar{\Theta}^{(1),[N]}(Ns)\right|\,\left|\frac{\partial
f}{\partial
y}\big{(}{\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns)\big{)}\right|\right]$
$\displaystyle+\lim_{N\to\infty}\mathbb{E}\left[\frac{1}{(1+K_{0})^{2}}\left|\frac{1}{N}\sum_{i\in[N]}g(x_{i}(Ns,\omega))-(\mathcal{F}g)\big{(}\bar{\Theta}^{(1),[N]}(Ns)\big{)}\right|\,\left|\frac{\partial^{2}f}{\partial
x^{2}}\big{(}{\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns)\big{)}\right|\right].$

Note that each of the derivatives is bounded by aconstant because we work on
$\mathcal{C}^{*}$. The first and the second term tend to zero by Lemma 10.9.
For the third term we can use a similar argument as used in (9.97), since we
showed Lemmas 10.7–10.13 for the single components in the mean-field system
with two colours. ∎

###### Step 5: Evolution of the averages in the Meyer-Zheng topology.

In this section we prove the following proposition

######  Proposition 10.16.

[Convergence in the Meyer-Zheng topology] Suppose that the effective estimator
process defined in (10.15) satisfies

(10.127)
$\lim_{N\to\infty}\mathcal{L}\left[\left({\bf\Theta}^{\mathrm{eff},(1),[N]}(Ns)\right)_{s>0}\right]=\mathcal{L}\left[\left(z_{1}^{\mathrm{eff}}(s)\right)_{s>0}\right].$

Then for the averages in (10.7),

(10.128)
$\displaystyle\lim_{N\to\infty}\mathcal{L}\left[\left(z_{1}^{[N]}(s)\right)_{s>0}\right]=\mathcal{L}\left[\left(z_{1}(s)\right)_{s>0}\right]$
$\displaystyle\text{in the Meyer-Zheng topology},$

where the limiting process $\left(z_{1}(s)\right)_{s>0}$ is defined as in
(10.20).

To prove Proposition 10.16 we need the following characterisation of
continuous functions in the Meyer-Zheng topology

######  Lemma 10.17.

[Convergence of marginals in the Meyer-Zheng topology] Let $(E,d)$ be a Polish
space with metric $d$. Suppose that $(X_{n}(s),Y_{n}(s))_{s>0}$ is a
stochastic process with state space $E^{2}$. If

(10.129)
$\lim_{n\to\infty}\mathcal{L}\left[(X_{n}(s),Y_{n}(s))_{s>0}\right]=\mathcal{L}\left[(X(s),Y(s))_{s>0}\right]\text{
in the Meyer-Zheng topology},$

then the marginals also converge in the Meyer-Zheng topology, i.e.,

(10.130)
$\displaystyle\lim_{n\to\infty}\mathcal{L}\left[(X_{n}(s))_{s>0}\right]=\mathcal{L}\left[(X(s))_{s>0}\right]\text{
in the Meyer-Zheng topology},$
$\displaystyle\lim_{n\to\infty}\mathcal{L}\left[(Y_{n}(s))_{s>0}\right]=\mathcal{L}\left[(Y(s))_{s>0}\right]\text{
in the Meyer-Zheng topology}.$

The proof of Lemma 10.17 is given in Appendix B.3.

###### Proof of Proposition 10.16.

By Lemma 10.9, we have that, for all $s>0$,

(10.131)
$\lim_{n\to\infty}\mathbb{E}\left[\left|\bar{\Theta}^{[N]}(Ns)-x_{1}^{[N]}(s)\right|\right]=0$

and

(10.132)
$\lim_{n\to\infty}\mathbb{E}\left[\left|\bar{\Theta}^{[N]}(Ns)-y_{0,1}^{[N]}(s)\right|\right]=0.$

Applying Lemmas 8.19, 8.20 and 8.21, like in the proof of Proposition 8.18, we
obtain

(10.133)
$\displaystyle\lim_{N\to\infty}\mathcal{L}\left[\left(x_{1}^{[N]}(s),y_{0,1}^{[N]}(s),\bar{\Theta}^{[N]}(Ns),\Theta^{[N]}_{y_{1,1}}(Ns)\right)_{s>0}\right]$
$\displaystyle=\mathcal{L}\left[\left(x^{\mathrm{eff}}_{1}(s),x^{\mathrm{eff}}_{1}(s),x^{\mathrm{eff}}_{1}(s),y_{1}^{\mathrm{eff}}(s)\right)_{s>0}\right]\text{
in the Meyer-Zheng topology.}$

Applying Lemma 10.17, we get the claim. ∎

###### Step 6: Proof of the two-colour mean-field finite-systems scheme.

###### Proof.

The proof of Proposition 10.2(a) follows directly from Lemma 10.15. The proof
of Proposition 10.2(b) is a consequence of Proposition 10.2(d). The proof of
Prosition 10.2(c) follows from Proposition 10.2(a) by applying Proposition
10.16. The proof of Proposition (10.2)(d) follows by the same argument as used
in the proof Proposition 8.1(c) in Section 9.4. In this argument we have to
replace the two-component system
$Z^{[N]}(Ns+t)=(X^{[N]}(Ns+t),Y^{[N]}(Ns+t))_{t\geq 0}$ by the three-component
system
$Z^{[N]}(Ns+t)=(X^{[N]}(Ns+t),Y_{0}^{[N]}(Ns+t),Y_{1}^{[N]}(Ns+t))_{t\geq 0}$
and use the infinite system defined in 10.11 instead of the infinite system
defined in (8.1). We now use the two-dimensional transition kernel in (10.30),
which controls the transition probabilities of the two-dimensional process
$(\bar{\Theta}^{(1)}(s),\Theta_{y_{1}}^{(1)}(s))_{s>0}$, instead of the one-
dimensional transition kernel in (8.18). ∎

###  11 Preparation: $N\to\infty$, two-level three-colour

To get a proper understanding of how the migration comes into play on
different space-time scales, we next look at a two-level mean-field system
where the geographic space consists of two layers and the seed-bank consist of
three layers, corresponding three colours $0$, $1$, $2$. In Section 11.1 we
give the set-up of the two-level three-colour mean-field model. In Section
11.2 we give a scheme to prove the analysis of the two-level three-colour
mean-field model. Finally, in Section 11.3 we prove the steps of the scheme
given in Section 11.2.

####  11.1 Two-level three-colour mean-field finite-systems scheme

We consider a restricted version of the SSDE in (2.20) on the finite
geographic space

(11.1) $[N^{2}]=\\{0,1,\ldots,N^{2}-1\\},\qquad N\in\mathbb{N}.$

This space should be interpreted as grouping the $N$-blocks consisting of $N$
colonies together, i.e.

(11.2) $[N^{2}]=\bigcup_{l=0}^{N-1}\\{Nl,Nl+1,\cdots,Nl+N-1\\}.$

With this interpretation we can use the metric $d_{[N^{2}]}$ that is induced
by the metric $d_{\Omega_{N}}$ on the hierarchical group $\Omega_{N}$ (recall
(2.4)). The migration kernel $a^{\Omega_{N}}(\cdot,\cdot)$ is restricted to
$[N^{2}]$ by setting all migration rates outside the 2-block equal to $0$,
i.e., $c_{k}=0$ for all $k\geq 2$. Hence the migration kernel is given by

(11.3) $a^{[N^{2}]}(i,j)=1_{\\{d_{[N^{2}]}(i,j)\leq
1\\}}\frac{c_{0}}{N}+\frac{c_{1}}{N^{3}},$

where $c_{0},c_{1}\in(0,\infty)$ are constants. The seed-bank of the
restricted system consists of _three colours_ , labeled $0$ $1$ and $2$, with
exchange rates given by $K_{0}e_{0},e_{0}$,
$\frac{K_{1}e_{1}}{N},\frac{e_{1}}{N}$ and
$\frac{K_{2}e_{2}}{N^{2}},\frac{e_{2}}{N^{2}}$ respectively. The state space
of the restricted system is

(11.4) $S=\mathfrak{s}^{[N^{2}]},\qquad\mathfrak{s}=[0,1]\times[0,1]^{3},$

and the restricted system is denoted by

(11.5) $\displaystyle(Z^{[N^{2}]}(t))_{t\geq 0}$
$\displaystyle=\Big{(}X^{[N^{2}]}(t),\left(Y_{0}^{[N^{2}]}(t),Y_{1}^{[N^{2}]}(t),Y_{2}^{[N^{2}]}(t)\right)\Big{)}_{t\geq
0},$
$\displaystyle\Big{(}X^{[N^{2}]}(t),\left(Y_{0}^{[N^{2}]}(t),Y_{1}^{[N^{2}]}(t),Y_{2}^{[N^{2}]}(t)\right)\Big{)}$
$\displaystyle=\Big{(}x^{[N^{2}]}_{i}(t),\left(y^{[N^{2}]}_{i,0}(t),y^{[N^{2}]}_{i,1}(t),y^{[N^{2}]}_{i,2}(t)\right)\Big{)}_{i\in[N^{2}]}.$

The components of the restricted system $(Z^{[N^{2}]}(t))_{t\geq 0}$ evolve
according to the SSDE

(11.6) $\displaystyle{\rm
d}x^{[N^{2}]}_{i}(t)=\frac{c_{0}}{N}\sum_{j\in[N^{2}]}1_{\\{d_{[N^{2}]}(i,j)\leq
1\\}}[x^{[N^{2}]}_{j}(t)-x^{[N^{2}]}_{i}(t)]\,{\rm d}t$
$\displaystyle\qquad\qquad\qquad+\frac{c_{1}}{N^{3}}\sum_{j\in[N^{2}]}[x^{[N^{2}]}_{j}(t)-x^{[N^{2}]}_{i}(t)]\,{\rm
d}t+\sqrt{g(x^{[N^{2}]}_{i}(t))}\,{\rm d}w_{i}(t)$
$\displaystyle\qquad\qquad\qquad+K_{0}e_{0}\,[y^{[N^{2}]}_{i,0}(t)-x^{[N^{2}]}_{i}(t)]\,{\rm
d}t$
$\displaystyle\qquad\qquad\qquad+\frac{K_{1}e_{1}}{N}\,[y^{[N^{2}]}_{i,1}(t)-x^{[N^{2}]}_{i}(t)]\,{\rm
d}t$
$\displaystyle\qquad\qquad\qquad+\frac{K_{2}e_{2}}{N^{2}}\,[y^{[N^{2}]}_{i,2}(t)-x^{[N^{2}]}_{i}(t)]\,{\rm
d}t,$ $\displaystyle{\rm
d}y^{[N^{2}]}_{i,0}(t)=e_{0}\,[x^{[N^{2}]}_{i}(t)-y^{[N^{2}]}_{i,0}(t)]\,{\rm
d}t,$ $\displaystyle{\rm
d}y^{[N^{2}]}_{i,1}(t)=\frac{e_{1}}{N}\,[x^{[N^{2}]}_{i}(t)-y^{[N^{2}]}_{i,1}(t)]\,{\rm
d}t,$ $\displaystyle{\rm
d}y^{[N^{2}]}_{i,2}(t)=\frac{e_{2}}{N^{2}}\,[x^{[N^{2}]}_{i}(t)-y^{[N^{2}]}_{i,2}(t)]\,{\rm
d}t,\qquad i\in[N^{2}],$

which is a special case of (2.20). By [SS80, Theorem 3.1], the SSDE in (11.6)
is the unique solution. It is important to note that we can write the SSDE
also

(11.7) $\displaystyle{\rm
d}x^{[N^{2}]}_{i}(t)=c_{0}\left[\frac{1}{N}\sum_{j\in[N]_{i}}x^{[N^{2}]}_{j}(t)-x^{[N^{2}]}_{i}(t)\right]\,{\rm
d}t$
$\displaystyle\qquad\qquad\qquad+\frac{c_{1}}{N}\left[\frac{1}{N^{2}}\sum_{j\in[N^{2}]}x^{[N^{2}]}_{j}(t)-x^{[N^{2}]}_{i}(t)\right]\,{\rm
d}t+\sqrt{g(x^{[N^{2}]}_{i}(t))}\,{\rm d}w_{i}(t)$
$\displaystyle\qquad\qquad\qquad+K_{0}e_{0}\,[y^{[N^{2}]}_{i,0}(t)-x^{[N^{2}]}_{i}(t)]\,{\rm
d}t+\frac{K_{1}e_{1}}{N}\,[y^{[N^{2}]}_{i,1}(t)-x^{[N^{2}]}_{i}(t)]\,{\rm d}t$
$\displaystyle\qquad\qquad\qquad+\frac{K_{2}e_{2}}{N^{2}}\,[y^{[N^{2}]}_{i,2}(t)-x^{[N^{2}]}_{i}(t)]\,{\rm
d}t,$ $\displaystyle{\rm
d}y^{[N^{2}]}_{i,0}(t)=e_{0}\,[x^{[N^{2}]}_{i}(t)-y^{[N^{2}]}_{i,0}(t)]\,{\rm
d}t,$ $\displaystyle{\rm
d}y^{[N^{2}]}_{i,1}(t)=\frac{e_{1}}{N}\,[x^{[N^{2}]}_{i}(t)-y^{[N^{2}]}_{i,1}(t)]\,{\rm
d}t,$ $\displaystyle{\rm
d}y^{[N^{2}]}_{i,2}(t)=\frac{e_{2}}{N^{2}}\,[x^{[N^{2}]}_{i}(t)-y^{[N^{2}]}_{i,2}(t)]\,{\rm
d}t,\qquad i\in[N^{2}],$

where $[N]_{i}$ denotes the set of colonies in the 1-block around site
$i\in[N^{2}]$. Therefore the migration term for a single colony in the two-
level mean-field system can be interpreted as a drift towards the $1$-block
average of the active population at rate $c_{0}$ and a drift towards the
$2$-block average of the active population at rate $\frac{c_{1}}{N}$. We are
interested in (11.7) on time scales $N^{0}$, $N$ and $N^{2}$. On time scale
$N^{0}$ we will look at the single colonies, i.e., space-time scale $0$. On
time scale $N$ we will look at the $1$-block averages, i.e., space-time scale
$1$ and on time scale $N^{2}$ we will look at the $2$-block averages, i.e.,
space-time scale $2$. In the sequel we will focus on site $0$, the 1-block
around site $0$ and the 2-block around site $0$. We will suppress this site
from the notation, but instead use subscripts $0$, $1$, $2$ to indicate when
we look at a single colony, a 1-block average or a 2-block average. We will
use the convention that in the subscript of a dormant population the first
subscript denotes the colour and the second subscript denotes the level of the
block, so $y_{0,1}$ is the $1$-block average around site $0$ of the dormant
population with colour $0$, while $y_{1,0}$ is the $1$-dormant single colony
at site $0$. Heuristically, we can read off the following results from the
SSDE in (11.7).

$\bullet$ On time scale $1=N^{0}$ (i.e., space-time scale $0$) in the limit as
$N\to\infty$, the colour-$1$ dormant population and the colour-$2$ dormant
population do not yet move. Hence

(11.8)
$\big{(}y^{[N^{2}]}_{1,0}(t_{0}),y^{[N^{2}]}_{2,0}(t_{0})\big{)}_{t_{0}\geq
0},$

converges as $N\to\infty$ to the constant processes on time scale $t_{0}$.
Therefore the colour $1$-dormant population and the colour $2$-dormant
population are both slow seed-banks on space-time scale $0$. The components
$((x_{0}^{[N^{2}]}(t_{0}),y_{0,0}^{[N^{2}]}(t_{0})))_{t_{0}\geq 0}$ converge
to i.i.d. copies of the single-colony McKean-Vlasov process in (7.1), where in
the corresponding SSDE the parameters $e,K,c$ are replaced by
$c_{0},e_{0},K_{0}$ and $E=1$. So, on time scale $1$ we only see the colour
$0$-dormant population evolve. Therefore the colour-$0$ dormant population is
the _effective seed-bank_ on time scale $t_{0}$. The process

(11.9) $(z^{[N^{2}]}_{0}(t_{0}))_{t_{0}\geq
0}=(x^{[N^{2}]}_{0}(t_{0}),y^{[N^{2}]}_{0,0}(t_{0}))_{t_{0}\geq 0},$

will be called _the single colony effective process._

$\bullet$ On time scale $N^{1}$ (i.e., space-time scale $1$), we look at the
averages

(11.10)
$\displaystyle(z_{1}^{[N^{2}]}(t_{1}))_{t_{1}>0}=\left(x_{1}^{[N^{2}]}(t_{1}),\left(y_{0,1}^{[N^{2}]}(t_{1}),y_{1,1}^{[N^{2}]}(t_{1}),y_{2,1}^{[N^{2}]}(t_{1})\right)\right)_{t_{1}>0}$
$\displaystyle=\left(\frac{1}{N}\sum_{i\in[N]}x^{[N^{2}]}_{i}(Nt_{1}),\left(\frac{1}{N}\sum_{i\in[N]}y^{[N^{2}]}_{i,0}(Nt_{1}),\frac{1}{N}\sum_{i\in[N]}y^{[N^{2}]}_{i,1}(Nt_{1}),\frac{1}{N}\sum_{i\in[N]}y^{[N^{2}]}_{i,2}(Nt_{1})\right)\right)_{t_{1}>0}.$

(Recall Remark 2.4 to appreciate the notation.) We use the lower index $1$ to
indicate that the average is the analogue of the 1-block average defined in
(2.22). Using (11.6), we see that the dynamics of the system in (11.10) is
given by the SSDE

(11.11) $\displaystyle{\rm d}x_{1}^{[N^{2}]}(t_{1})$
$\displaystyle=c_{1}\left[\frac{1}{N^{2}}\sum_{j\in[N^{2}]}x_{j}(Nt_{1})-x_{1}(t_{1})\right]{\rm
d}t_{1}+\sqrt{\frac{1}{N}\sum_{i\in[N]}g(x_{i}(Nt_{1}))}\,{\rm d}w(t_{1})$
$\displaystyle\qquad\qquad+NK_{0}e_{0}\left[y_{0,1}^{[N^{2}]}(t_{1})-x_{1}^{[N^{2}]}(t_{1})\right]{\rm
d}t_{1}$
$\displaystyle\qquad\qquad+K_{1}e_{1}\left[y_{1,1}^{[N^{2}]}(t_{1})-x_{1}^{[N^{2}]}(t_{1})\right]{\rm
d}t_{1}$
$\displaystyle\qquad\qquad+\frac{K_{2}e_{2}}{N}\left[y_{2,1}^{[N^{2}]}(t_{1})-x_{1}^{[N^{2}]}(t_{1})\right]{\rm
d}t_{1},$ $\displaystyle{\rm d}y_{0,1}^{[N^{2}]}(t_{1})$
$\displaystyle=Ne_{0}\left[x_{1}^{[N^{2}]}(t_{1})-y_{0,1}^{[N^{2}]}(t_{1})\right]{\rm
d}t_{1},$ $\displaystyle{\rm d}y_{1,1}^{[N^{2}]}(t_{1})$
$\displaystyle=e_{1}\left[x_{1}^{[N^{2}]}(t_{1})-y_{1,1}^{[N^{2}]}(t_{1})\right]{\rm
d}t_{1},$ $\displaystyle{\rm d}y_{2,1}^{[N^{2}]}(t_{1})$
$\displaystyle=\frac{e_{2}}{N}\left[x_{1}^{[N^{2}]}(t_{1})-y_{1,1}^{[N^{2}]}(t_{1})\right]{\rm
d}t_{1}.$

In the limit $N\to\infty$ we expect that the colour $2$-dormant population
does not move, since it only interacts with the active population at rate
$\frac{e_{2}}{N}$. Therefore we expect $(y_{2,1}^{[N^{2}]}(t))_{t>0}$ to
converge to a constant process and hence we say that the colour $2$-dormant
population behaves like a _slow seed-bank_. The colour $1$-dormant population,
however, has a non-trivial interaction with the active population and
therefore is the _effective seed-bank_ on space-time scale $1$. The colour
$0$-dormant population has, in the limit as $N\to\infty$, an infinitely strong
interaction with the active population. Therefore we expect that, in the limit
as $N\to\infty$, its path becomes rougher and rougher at rarer and rarer
times. We will need to use the _Meyer-Zheng topology_ to prove that

(11.12)
$\lim_{N\to\infty}y_{0,1}^{[N^{2}]}(t_{1})=\lim_{N\to\infty}x_{1}^{[N^{2}]}(t_{1})\text{
for most }t_{1}.$

Therefore the colour $0$-dormant population equalizes with the active
population, due to its infinitely strong interaction with the active
population. Hence at space-time scale $1$, the colour $0$-dormant population
behaves like a _fast seed-bank._ If we look at the active population, then we
see that it feels a drift towards the $2$-block average of the active
population, and resamples at a rate that is the 1-block average of the
resampling rates in the single colonies. Furthermore, in the limit as
$N\to\infty$, it feels an infinitely fast drift towards the colour $0$-dormant
population, has a non-trivial interaction with the colour $1$-dormant
population, and its interaction with the colour $2$-dormant population cancels
out. As long as we focus on the combination

$\frac{x_{1}^{[N^{2}]}(t_{1})+K_{0}y_{0,1}^{[N^{2}]}(t_{1})}{1+K_{0}},$

we see that the colour-$0$ terms with the factor $N$ in front cancel out. This
will allow us to do most of the analysis in the path space topology, without
using the Meyer-Zheng topology. The process

$\left(\frac{x_{1}^{[N^{2}]}(t_{1})+K_{0}y_{0,1}^{[N^{2}]}(t_{1})}{1+K_{0}},y^{[N^{2}]}_{1,1}(t_{1})\right)_{t_{1}>0}$

will therefore be called the _effective process_.

$\bullet$ On time scale $N^{2}$ (i.e., space-time scale $2$) we look at the
equivalent of the $2$-block averages in (2.22),

(11.13)
$\displaystyle\left(x_{2}^{[N^{2}]}(t_{2}),(y_{0,2}^{[N^{2}]}(t_{2}),y_{1,2}^{[N^{2}]}(t_{2}),y_{2,2}^{[N^{2}]}(t_{2}))\right)_{t_{2}>0}$
$\displaystyle=\Bigg{(}\frac{1}{N^{2}}\sum_{i\in[N^{2}]}x^{[N^{2}]}_{i}(N^{2}t_{2}),$
$\displaystyle\qquad\qquad\Bigg{(}\frac{1}{N^{2}}\sum_{i\in[N^{2}]}y^{[N^{2}]}_{i,0}(N^{2}t_{2}),\frac{1}{N^{2}}\sum_{i\in[N^{2}]}y^{[N^{2}]}_{i,1}(N^{2}t_{2}),\frac{1}{N^{2}}\sum_{i\in[N^{2}]}y^{[N^{2}]}_{i,2}(N^{2}t_{2})\Bigg{)}\Bigg{)}_{t_{2}>0},$

which evolves according to the SSDE

(11.14) $\displaystyle{\rm d}x_{2}^{[N^{2}]}(t_{2})$
$\displaystyle=\sqrt{\frac{1}{N^{2}}\sum_{i\in[N^{2}]}g(x_{i}(N^{2}t_{2}))}\,{\rm
d}w(t_{2})$
$\displaystyle\qquad\qquad+N^{2}K_{0}e_{0}\left[y_{0,2}^{[N^{2}]}(t_{2})-x_{2}^{[N^{2}]}(t_{2})\right]{\rm
d}t_{2}$
$\displaystyle\qquad\qquad+NK_{1}e_{1}\left[y_{1,2}^{[N^{2}]}(t_{2})-x_{2}^{[N^{2}]}(t_{2})\right]{\rm
d}t_{2}$
$\displaystyle\qquad\qquad+K_{2}e_{2}\left[y_{2,2}^{[N^{2}]}(t_{2})-x_{2}^{[N^{2}]}(t_{2})\right]{\rm
d}t_{2},$ $\displaystyle{\rm d}y_{0,2}^{[N^{2}]}(t_{2})$
$\displaystyle=N^{2}e_{0}\left[x_{2}^{[N^{2}]}(t_{2})-y_{0,2}^{[N^{2}]}(t_{2})\right]{\rm
d}t_{2},$ $\displaystyle{\rm d}y_{1,2}^{[N^{2}]}(t_{2})$
$\displaystyle=Ne_{1}\left[x_{2}^{[N^{2}]}(t_{2})-y_{1,2}^{[N^{2}]}(t_{2})\right]{\rm
d}t_{2},$ $\displaystyle{\rm d}y_{2,2}^{[N^{2}]}(t_{2})$
$\displaystyle=e_{2}\left[x_{2}^{[N^{2}]}(t_{2})-y_{2,2}^{[N^{2}]}(t_{2})\right]{\rm
d}t_{2}.$

In this case we see that migration in the active component cancels out and the
resampling rate is given by the average over the complete population. In the
limit as $N\to\infty$, we see that the active population interacts at an
infinitely fast rate with the $0$-dormant population as well as with the
colour $1$-dormant population. Hence both the colour $0$ and the colour $1$
seed-banks are fast seed-banks and we expect equalisation of the active
population and the colour $0$-dormant population and the colour $1$-dormant
population in Meyer-Zheng topology. The active population, in the limit as
$N\to\infty$, has a non-trivial interaction with the colour $2$-dormant
population, and hence the colour $2$-dormant population is the _effective
seed-bank_ on time scale $N^{2}$. Looking at the quantity

(11.15)
$\frac{x_{2}^{[N^{2}]}(t_{2})+K_{0}y_{0,2}^{[N^{2}]}(t_{2})+K_{1}y_{1,2}^{[N^{2}]}(t_{2})}{1+K_{0}+K_{1}},$

for which we find

(11.16) $\displaystyle{\rm
d}\left[\frac{x_{2}^{[N^{2}]}(t_{2})+K_{0}y_{0,2}^{[N^{2}]}(t_{2})+K_{1}y_{1,2}^{[N^{2}]}(t_{2})}{1+K_{0}+K_{1}}\right]$
$\displaystyle=\frac{1}{1+K_{0}+K_{1}}\sqrt{\frac{1}{N^{2}}\sum_{i\in[N^{2}]}g(x_{i}(N^{2}t_{2}))}\,{\rm
d}w(t_{2})+K_{2}e_{2}\left[y_{2,2}^{[N^{2}]}(t_{2})-x_{2}^{[N^{2}]}(t_{2})\right]{\rm
d}t_{2},$

we see that the infinite rates cancel out. We will call

(11.17)
$\left(\frac{x_{2}^{[N^{2}]}(t_{2})+K_{0}y_{0,2}^{[N^{2}]}(t_{2})+K_{1}y_{1,2}^{[N^{2}]}(t_{2})}{1+K_{0}+K_{1}},y_{2,2}^{[N^{2}]}(t_{2})\right)_{t_{2}>0}$

_the effective process._ Using the effective process we can analyse our system
in path space.

######  $\blacktriangleright$ Scaling limit.

Let $(z_{0}(t))_{t\geq 0}=(x_{0}(t),(y_{0,0}(t),y_{1,0}(t),y_{2,0}(t)))_{t\geq
0}$ be the process evolving according to

(11.18) $\displaystyle{\rm d}x_{0}(t)=c_{0}\,[\theta-x_{0}(t)]\,{\rm
d}t+\sqrt{g(x_{0}(t))}\,{\rm d}w(t)$
$\displaystyle\qquad\qquad+K_{0}e_{0}\,[y_{0,0}(t)-x_{0}(t)]\,{\rm d}t,$
$\displaystyle{\rm d}y_{0,0}(t)=e_{0}\,[x_{0}(t)-y_{0,0}(t)]\,{\rm d}t,$
$\displaystyle y_{1,0}(t)=y_{1,0},$ $\displaystyle y_{2,0}(t)=y_{2,0},$

where $\theta\in[0,1]$, $y_{1,0}\in[0,1]$ and $y_{2,0}\in[0,1]$. The process
$(z_{0}(t))_{t\geq 0})$ will be the limiting process for the single colonies.
The corresponding single colony effective processes are given by

(11.19) $\displaystyle{\rm d}x^{\mathrm{eff}}_{0}(t)$
$\displaystyle=c_{0}[\theta-x^{\mathrm{eff}}_{0}(t)]\,{\rm
d}t+\sqrt{g(x^{\mathrm{eff}}_{0}(t))}\,{\rm
d}w(t)+K_{0}e_{0}\,[y^{\mathrm{eff}}_{0,0}(t)-x^{\mathrm{eff}}_{0}(t)]\,{\rm
d}t,$ $\displaystyle{\rm d}y^{\mathrm{eff}}_{0,0}(t)$
$\displaystyle=e_{0}\,[x^{\mathrm{eff}}_{0}(t)-y^{\mathrm{eff}}_{0,0}(t)]\,{\rm
d}t,\qquad i\in\mathbb{N}_{0},$

where $\theta\in[0,1]$. By [YW71], (11.18) and (11.19) have a unique strong
solution. Like for the one-colour mean-field finite-systems scheme, we need
the following list of ingredients to formally state the multi-scale analysis:

  1. 1.

For positive times $t>0$, we define the following _$1$ -block estimators_ for
the finite system:

(11.20) $\displaystyle\bar{\Theta}^{(1),[N^{2}]}(t)$
$\displaystyle=\frac{1}{N}\sum_{i\in[N]}\frac{x^{[N^{2}]}_{i}(t)+K_{0}y^{[N^{2}]}_{i,0}(t)}{1+K_{0}},$
$\displaystyle\Theta^{(1),[N^{2}]}_{x}(t)$
$\displaystyle=\frac{1}{N}\sum_{i\in[N]}x^{[N^{2}]}_{i}(t),$
$\displaystyle\Theta^{(1),[N^{2}]}_{y_{0}}(t)$
$\displaystyle=\frac{1}{N}\sum_{i\in[N]}y^{[N^{2}]}_{i,0}(t),$
$\displaystyle\Theta^{(1),[N^{2}]}_{y_{1}}(t)$
$\displaystyle=\frac{1}{N}\sum_{i\in[N]}y^{[N^{2}]}_{i,1}(t),$
$\displaystyle\Theta^{(1),[N^{2}]}_{y_{2}}(t)$
$\displaystyle=\frac{1}{N}\sum_{i\in[N]}y^{[N^{2}]}_{i,2}(t).$

We abbreviate

(11.21) $\displaystyle{\bf\Theta}^{(1),[N^{2}]}(t)$
$\displaystyle=\left({\Theta}_{x}^{(1),[N^{2}]}(t),\left(\Theta^{(1),[N^{2}]}_{y_{0}}(t),\Theta^{(1),[N^{2}]}_{y_{1}}(t),\Theta^{(1),[N^{2}]}_{y_{2}}(t)\right)\right),$
$\displaystyle{\bf\Theta}^{\mathrm{aux},(1),[N^{2}]}(t)$
$\displaystyle=\left(\bar{\Theta}^{(1),[N^{2}]}(t),\Theta^{(1),[N^{2}]}_{y_{1}}(t),\Theta^{(1),[N^{2}]}_{y_{2}}(t)\right),$
$\displaystyle{\bf\Theta}^{\mathrm{eff},(1),[N^{2}]}(t)$
$\displaystyle=\left(\bar{\Theta}^{(1),[N^{2}]}(t),\Theta^{(1),[N^{2}]}_{y_{1}}(t)\right).$

We call $({\bf\Theta}^{(1),[N^{2}]}(t))_{t>0}$ the _$1$ -block estimator
process_, $({\bf\Theta}^{\mathrm{aux},(1),[N^{2}]}(t))_{t>0}$ the _auxiliary
$1$-block estimator process_ and
$({\bf\Theta}^{\mathrm{eff},(1),[N^{2}]}(t))_{t>0}$ the _effective $1$-block
estimator process_. The auxiliary $1$-block estimator will be useful in the
proofs. For $t>0$, we define the following _$2$ -block estimators_ for the
finite system:

(11.22) $\displaystyle\bar{\Theta}^{(2),[N^{2}]}(t)$
$\displaystyle=\frac{1}{N^{2}}\sum_{i\in[N^{2}]}\frac{x^{[N^{2}]}_{i}(t)+K_{0}y^{[N^{2}]}_{i,0}(t)+K_{1}y^{[N^{2}]}_{i,1}(t)}{1+K_{0}+K_{1}},$
$\displaystyle\Theta^{(2),[N^{2}]}_{x}(t)$
$\displaystyle=\frac{1}{N^{2}}\sum_{i\in[N^{2}]}x^{[N^{2}]}_{i}(t),$
$\displaystyle\Theta^{(2),[N^{2}]}_{y_{0}}(t)$
$\displaystyle=\frac{1}{N^{2}}\sum_{i\in[N^{2}]}y^{[N^{2}]}_{i,0}(t),$
$\displaystyle\Theta^{(2),[N^{2}]}_{y_{1}}(t)$
$\displaystyle=\frac{1}{N^{2}}\sum_{i\in[N^{2}]}y^{[N^{2}]}_{i,1}(t),$
$\displaystyle\Theta^{(2),[N^{2}]}_{y_{2}}(t)$
$\displaystyle=\frac{1}{N^{2}}\sum_{i\in[N^{2}]}y^{[N^{2}]}_{i,2}(t).$

We abbreviate

(11.23) $\displaystyle{\bf\Theta}^{(2),[N^{2}]}(t)$
$\displaystyle=\left({\Theta}_{x}^{(2),[N^{2}]}(t),\left(\Theta^{(2),[N^{2}]}_{y_{0}}(t),\Theta^{(2),[N^{2}]}_{y_{1}}(t),\Theta^{(2),[N^{2}]}_{y_{2}}(t)\right)\right),$
$\displaystyle{\bf\Theta}^{\mathrm{eff},(2),[N^{2}]}(t)$
$\displaystyle=\left(\bar{\Theta}^{(2),[N^{2}]}(t),\Theta^{(2),[N^{2}]}_{y_{1}}(t)\right).$

We call $({\bf\Theta}^{\mathrm{eff},(2),[N^{2}]}(t))_{t>0}$ the _effective
$2$-block estimator process_ and $({\bf\Theta}^{(2),[N^{2}]}(t))_{t>0}$ as the
_$2$ -block estimator process_.

  2. 2.

The _time scale_ $N$ for which
$\mathcal{L}[\bar{\Theta}^{(1),[N^{2}]}(Nt_{1}-L(N))-\bar{\Theta}^{(1),[N^{2}]}(Nt_{1})]=\delta_{0}$
for all $L(N)$ such that $L(N)\to\infty$ and $\lim_{N\to\infty}L(N)/N=0$, but
not for $L(N)=N$. In words, $N$ is the time scale on which
$\bar{\Theta}^{(1),[N^{2}]}(\cdot)$ starts evolving, i.e.,
$(\bar{\Theta}^{(1),[N^{2}]}(Nt_{1}))_{t_{1}>0}$ is no longer a fixed process.
When we use time scale $N$, we will use $t_{1}$ as a time index, which
indicates the “faster time scale”. For the “slow time scale” we use $t_{0}$ as
time index.

The _time scale_ $N^{2}$ for which
$\mathcal{L}[\bar{\Theta}^{(2),[N^{2}]}(N^{2}t_{2}-L(N)N)-\bar{\Theta}^{(2),[N^{2}]}(N^{2}t_{2})]=\delta_{0}$
for all $L(N)$ such that $L(N)\to\infty$ and $\lim_{N\to\infty}L(N)/N=0$, but
not for $L(N)=N$. In words, $N^{2}$ is the time scale on which
$\bar{\Theta}^{(2),[N^{2}]}(\cdot)$ starts evolving, i.e.,
$(\bar{\Theta}^{(2),[N^{2}]}(N^{2}t_{2}))_{t_{2}>0},$ is no longer a fixed
process. When we use time scale $N^{2}$, we will use $t_{2}$ as a time index,
which indicates the “fastest time scale”.

  3. 3.

The _invariant measure_ for the evolution of a single colony in (11.18),
written

(11.24) $\Gamma_{\theta,{\bf y_{0}}}^{(0)},\qquad{\bf
y_{0}}=(\theta,y_{1,0},y_{2,0}),$

and the invariant measure of the level-$0$ effective process evolving
according to (11.19), written

(11.25) $\Gamma_{\theta}^{\mathrm{eff},(0)}.$

  4. 4.

The renormalisation transformation
$\mathcal{F}\colon\,\mathcal{G}\to\mathcal{G}$,

(11.26)
$(\mathcal{F}g)(\theta)=\int_{[0,1]^{2}}g(x)\,\Gamma^{\mathrm{eff},(0)}_{\theta}({\rm
d}x,{\rm d}y_{0}),\quad\theta\in[0,1],$

where $\Gamma_{\theta}^{\mathrm{eff},(0)}$ is the equilibrium measure in
(11.25). Note that this is the same transformation as defined in (4.16), but
defined for the truncated system. Later we will study iterates of the
renormalisation transformation. Therefore we will write
$\mathcal{F}^{(1)}g=\mathcal{F}g$, to indicate that we apply the
renormalisation transformation only once.

  5. 5.

The limiting $1$-block process is given by
$(z_{1}(t))_{t>0})=(x_{1}(t),(y_{0,1}(t),y_{1,1}(t),y_{2,1}(t)))_{t>0}$ and
evolves according to

(11.27) $\displaystyle{\rm d}x_{1}(t)$
$\displaystyle=\frac{1}{1+K_{0}}\Bigg{[}c_{1}[\theta-x_{1}(t)]\,{\rm
d}t+\sqrt{(\mathcal{F}^{(1)}g)(x_{1}(t))}\,{\rm d}w(t)$
$\displaystyle\qquad+K_{1}e_{1}\,[y_{1,1}(t)-x_{1}(t)]\,{\rm d}t\Bigg{]},$
$\displaystyle y_{0,1}(t)$ $\displaystyle=x_{1}(t),$ $\displaystyle{\rm
d}y_{1,1}(t)$ $\displaystyle=e_{1}\,[x_{1}(t)-y_{1,1}(t)]\,{\rm d}t,$
$\displaystyle y_{2,1}(t)$ $\displaystyle=y_{2,1},$

where $\theta\in[0,1]$, and $y_{2,1}\in[0,1]$, and $\mathcal{F}^{(1)}$ is the
renormalisation transformation defined in (11.26). The limiting $1$-block
process for the auxiliary estimator process is given by
$(z_{1}^{\mathrm{aux}}(t))_{t>0}=(x^{\mathrm{aux}}_{1}(t),y^{\mathrm{aux}}_{1,1}(t),y^{\mathrm{aux}}_{2,1}(t))_{t>0}$
and evolves according to

(11.28) $\displaystyle{\rm d}x^{\mathrm{aux}}_{1}(t)$
$\displaystyle=\frac{1}{1+K_{0}}\Bigg{[}c_{1}[\theta-x^{\mathrm{aux}}_{1}(t)]\,{\rm
d}t+\sqrt{(\mathcal{F}^{(1)}g)(x^{\mathrm{aux}}_{1}(t))}\,{\rm d}w(t)$
$\displaystyle\qquad+K_{1}e_{1}\,[y^{\mathrm{aux}}_{1,1}(t)-x^{\mathrm{aux}}_{1}(t)]\,{\rm
d}t\Bigg{]}\,,$ $\displaystyle{\rm d}y^{\mathrm{aux}}_{1,1}(t)$
$\displaystyle=e_{1}\,[x^{\mathrm{aux}}_{1}(t)-y^{\mathrm{aux}}_{1,1}(t)]\,{\rm
d}t,$ $\displaystyle y^{\mathrm{aux}}_{2,1}(t)$ $\displaystyle=y_{2,1},$

for $\theta\in[0,1]$. The auxiliary estimator process turns out to be
important in the next section. The effective limiting $1$-block process is
given by
$(z_{1}^{\mathrm{eff}}(t))_{t>0})=(x^{\mathrm{eff}}_{1}(t),y^{\mathrm{eff}}_{1,1}(t))_{t>0}$
and evolves according to

(11.29) $\displaystyle{\rm d}x^{\mathrm{eff}}_{1}(t)$
$\displaystyle=\frac{1}{1+K_{0}}\Bigg{[}c_{1}[\theta-x^{\mathrm{eff}}_{1}(t)]\,{\rm
d}t+\sqrt{(\mathcal{F}^{(1)}g)(x^{\mathrm{eff}}_{1}(t))}\,{\rm d}w(t)$
$\displaystyle\qquad+K_{1}e_{1}\,[y^{\mathrm{eff}}_{1,1}(t)-x^{\mathrm{eff}}_{1}(t)]\,{\rm
d}t\Bigg{]},$ $\displaystyle{\rm d}y^{\mathrm{eff}}_{1,1}(t)$
$\displaystyle=e_{1}\,[x^{\mathrm{eff}}_{1}(t)-y^{\mathrm{eff}}_{1,1}(t)]\,{\rm
d}t,$

for $\theta\in[0,1]$. By [YW71], (11.27), (11.28) and (11.29) have a unique
strong solution.

  6. 6.

The _invariant measure_ of the infinite system in (11.27), written

(11.30) $\Gamma_{\theta,y_{1}}^{(1)},\qquad y_{1}=(\theta,\theta,y_{2,1}),$

and the invariant measures of the level-$1$ limiting estimator process
evolving according to (11.28) and the level-$1$ effective process evolving
according to (11.29),

(11.31) $,\,\Gamma_{\theta}^{\mathrm{eff},(1)}.$

  7. 7.

The first iteration of the renormalisation transformation,

(11.32)
$(\mathcal{F}^{(2)}g)(\theta)=\int_{[0,1]^{2}}(\mathcal{F}g)(x)\,\Gamma^{\mathrm{eff},(1)}_{\theta}({\rm
d}x,{\rm d}y_{1}),\quad\theta\in[0,1].$

Hence

(11.33)
$(\mathcal{F}^{(2)}g)(\theta)=\int_{[0,1]^{2}}\Gamma^{\mathrm{eff},(1)}_{\theta}({\rm
d}u,{\rm d}v)\int_{[0,1]^{2}}g(x)\,\Gamma^{\mathrm{eff},(0)}_{u}({\rm d}x,{\rm
d}y).$

  8. 8.

The limiting $2$-block process
$(z_{2}(t))_{t>0}=(x_{2}(t),(y_{0,2}(t),y_{1,2}(t),y_{2,2}(t)))_{t>0}$ evolves
according to

(11.34) $\displaystyle{\rm d}x_{2}(t)$
$\displaystyle=\frac{1}{1+K_{0}+K_{1}}\left[\sqrt{(\mathcal{F}^{(2)}g)\big{(}x_{2}(t)\big{)}}\,{\rm
d}w(t)+K_{2}e_{2}\left[y_{2,2}(t)-x_{2}(t)\right]{\rm d}t\right],$
$\displaystyle y_{0,2}(t)$ $\displaystyle=x_{2}(t),$ $\displaystyle{\rm
d}y_{1,2}(t)$ $\displaystyle=x_{2}(t),$ $\displaystyle{\rm d}y_{2,2}(t)$
$\displaystyle=e_{2}\left[x_{2}(t)-y_{2,2}(t)\right]\,{\rm d}t,$

where $\mathcal{F}^{(2)}g$ is defined as in (11.32). The limiting effective
$2$-block process on space-time scale $2$ is
$(z^{\mathrm{eff}}_{2}(t))_{t>0})=(x^{\mathrm{eff}}_{2}(t),y^{\mathrm{eff}}_{2,2}(t))_{t>0}$
and evolves according to

(11.35) $\displaystyle{\rm d}x^{\mathrm{eff}}_{2}(t)$
$\displaystyle=\frac{1}{1+K_{0}+K_{1}}\left[\sqrt{(\mathcal{F}^{(2)}g)(x^{\mathrm{eff}}_{2}(t))}\,{\rm
d}w(t)+K_{2}e_{2}\,[y^{\mathrm{eff}}_{2,2}(t)-x^{\mathrm{eff}}_{2}(t)]\,{\rm
d}t\right],$ $\displaystyle{\rm d}y^{\mathrm{eff}}_{2,2}(t)$
$\displaystyle=e_{2}\,[x^{\mathrm{eff}}_{2}(t)-y^{\mathrm{eff}}_{2,2}(t)]\,{\rm
d}t.$

We are now ready to state the scaling limit for the evolution of the averages
in (10.7).

######  Proposition 11.1.

[Mean-field: two-level three-colour finite-systems scheme] Suppose that
$\mu(0)=\mu^{\otimes[N^{2}]}$ for some
$\mu\in\mathcal{P}\left([0,1]\times[0,1]^{2}\right)$. Let

(11.36)
$\displaystyle\vartheta_{0}=\mathbb{E}^{\mu}\left[\frac{x+K_{0}y_{0}}{1+K_{0}}\right],\quad\vartheta_{1}=\mathbb{E}^{\mu}\left[\frac{x+K_{0}y_{0}+K_{1}y_{1}}{1+K_{0}+K_{1}}\right],$
$\displaystyle\theta_{y_{1}}=\mathbb{E}^{\mu}\left[y_{1}\right],\quad\theta_{y_{2}}=\mathbb{E}^{\mu}\left[y_{2}\right].$

and recall the limiting process $(z_{2}(t))_{t>0}$ in (11.34) and the limiting
process $(z_{1}(t))_{t>0}$ in (11.27). Assume for the $2$-dormant $1$-blocks
that

(11.37)
$\lim_{N\to\infty}\mathcal{L}\left[Y_{2,1}^{[N^{2}]}(Nt_{2})\Big{|}{\bf\Theta}^{(2),[N^{2}]}(N^{2}t_{2})\right]=P^{z_{2}(t_{2})},$

and for the $2$-dormant $0$-blocks (= single colonies) that

(11.38)
$\lim_{N\to\infty}\mathcal{L}\left[Y_{2,0}^{[N^{2}]}(Nt_{2}+Nt_{1})\Big{|}{\bf\Theta}^{\mathrm{eff},(1),[N^{2}]}(N^{2}t_{2}+Nt_{1})\right]=P^{z_{1}(t_{1})}.$

Then the following hold:

  * (a)

For the effective $2$-block estimator process defined in (11.23),

(11.39)
$\lim_{N\to\infty}\mathcal{L}\left[\left({\bf\Theta}^{\mathrm{eff},(2),[N^{2}]}(N^{2}t_{2})\right)_{t_{2}>0}\right]=\mathcal{L}\left[\left(z_{2}^{\mathrm{eff}}(t_{2})\right)_{t_{2}>0}\right],$

where the limit is determined by the unique solution of the SSDE (11.35) with
initial state

(11.40) $\displaystyle
z_{2}^{\mathrm{eff}}(0)=\left(x_{2}^{\mathrm{eff}}(0),y^{\mathrm{eff}}_{2}(0)\right)=\left(\vartheta_{1},\theta_{y_{2}}\right).$

  * (b)

For the effective $1$-block estimator process defined in (11.21),

(11.41)
$\displaystyle\lim_{N\to\infty}\mathcal{L}\left[\left({\bf\Theta}^{\mathrm{eff},(1),[N^{2}]}(N^{2}t_{2}+Nt_{1})\right)_{t_{1}>0}\,\right]=\mathcal{L}\left[(z_{1}^{\mathrm{eff}}(t_{1}))_{t_{1}>0}\right],$

where, conditional on $x^{\mathrm{eff}}_{2}(t_{2})=u$, the limit process is
the unique solution of the SSDE in (11.29) with $\theta$ replaced by $u$ and
with initial measure $\Gamma_{u}^{\mathrm{eff},(1)}$.

  * (c)

For the single colony effective process defined in (11.9),

(11.42)
$\displaystyle\lim_{N\to\infty}\mathcal{L}\left[\left(z_{0}^{\mathrm{eff},[N^{2}]}(N^{2}t_{2}+Nt_{1}+t_{0})\right)_{t_{0}\geq
0}\,\right]=\mathcal{L}\left[(z_{0}^{\mathrm{eff}}(t_{0}))_{t_{0}\geq
0}\right],$

where, conditional on $x_{1}^{\mathrm{eff}}(t_{1})=v$, the limit process is
the unique solution of the SSDE in (11.19) with $\theta$ replaced by $v$ and
with initial measure $\Gamma_{v}^{\mathrm{eff},(0)}$.

  * (d)

For the $2$-block estimator process defined in (11.23),

(11.43)
$\displaystyle\lim_{N\to\infty}\mathcal{L}\left[\left({\bf\Theta}^{(2),[N^{2}]}(N^{2}t_{2})\right)_{t_{2}>0}\right]=\mathcal{L}\left[\left(z_{2}(t_{2})\right)_{t_{2}>0}\right]$
$\displaystyle\text{in the Meyer-Zheng topology},$

where the limit process is the unique solution of the SSDE in (11.34) with
initial state

(11.44)
$z_{2}(0)=\left(\vartheta_{1},(\vartheta_{1},\vartheta_{1},\theta_{y_{2}})\right).$

  * (e)

Fix $t_{2}>0$. Assume (11.37). Define

(11.45) $\displaystyle\Gamma^{(1)}(t_{2})=$
$\displaystyle\int_{[0,1]^{4}}S^{[2]}_{t_{2}}\bigl{(}(\vartheta_{1},(\vartheta_{1},\vartheta_{1},\theta_{y_{2}})),{\rm
d}(u_{x},u_{x},u_{x},u_{y_{2,2}})\bigr{)}$
$\displaystyle\int_{[0,1]}P^{(u_{x},u_{x},u_{x},u_{y_{2,2}})}({\rm
d}y_{2,1})\,\Gamma^{(1)}_{(u_{x},(u_{x},u_{x},y_{2,1}))}\in\mathcal{P}([0,1]^{4}),$

where $\Gamma^{(1)}_{(u_{x},(u_{x},u_{x},y_{2,1}))}$ is the equilibrium
measure in (11.30) and
$S^{[2]}_{t_{2}}((\vartheta_{1},(\vartheta_{1},\vartheta_{1},\theta_{y_{2}})),\cdot)$
is the time-$t_{2}$ law of the limiting process $(z_{2}(t_{2}))_{t_{2}>0}$ in
(11.43) starting from
$(\vartheta_{1},(\vartheta_{1},\vartheta_{1},\theta_{y_{2}}))\in[0,1]\times[0,1]^{3}$.

Let $(z^{\Gamma^{(1)}(t_{2})}(t_{1}))_{t_{1}\geq 0}$ be the random process
that conditioned on $z_{2}(t_{2})=(\theta,(\theta,\theta,y_{2,2}))$ moves
according to (11.27) with $\theta=\theta$ and $y_{2,1}(0)=y_{2,1}$ and with
$z^{\Gamma^{(1)}(t_{2})}(0)$ be drawn according to $\Gamma^{(1)}(t_{2})$
(which is a mixture of random processes in equilibrium). Then for the
$1$-block estimator process defined in (11.21),

(11.46)
$\displaystyle\lim_{N\to\infty}\mathcal{L}\left[\left({\bf\Theta}^{(1),[N^{2}]}(N^{2}t_{2}+Nt_{1})\right)_{t_{1}>0}\right]=\mathcal{L}\left[(z^{\Gamma^{(1)}(t_{2})}(t_{1}))_{t_{1}>0}\right]$
$\displaystyle\qquad\qquad\qquad\text{ in the Meyer-Zheng topology}.$

  * (f)

Let $z_{1}(t_{1})$ be the limiting process obtained in (e). Assume (11.38).
Define, for $t_{2}\in(0,\infty)$,

(11.47) $\displaystyle\Gamma^{(0)}(t_{2})$
$\displaystyle=\int_{[0,1]^{4}}\Gamma^{(1)}(t_{2})({\rm
d}z_{1})\,\int_{[0,1]}P^{z_{1}}({\rm
d}y_{2,0})\,\Gamma^{(0)}_{(x_{1},(x_{1},y_{1,1}.y_{2,0}))},$

where $\Gamma^{(1)}(t_{2})$ is as defined in (11.45). Let
$(z^{\Gamma^{(0)}(t_{2})}(t_{0}))_{t_{0}\geq 0}$ be the random process in
(11.18) with $z^{\Gamma^{(0)}(t_{2})}(0)$ drawn according to
$\Gamma^{(0)}(t_{2})$ which is a mixture of random processes in equilibrium.
Then

(11.48)
$\displaystyle\lim_{N\to\infty}\mathcal{L}\Big{[}\Big{(}z_{0}^{[N^{2}]}(N^{2}t_{2}+Nt_{1}+t_{0})\Big{)}_{t_{0}\geq
0}\Big{]}=\mathcal{L}\left[(z^{\Gamma^{(0)}(t_{2})}(t_{0}))_{t_{0}\geq
0}\right].$

######  Remark 11.2.

Note that Proposition 11.1(f) does not depend on the choice of $t_{1}$,
because $\Gamma^{(1)}(t_{2})$ is already a mixture of equilibrium measures of
the $1$-block process. $\blacksquare$

######  Remark 11.3.

Note that in Propostion 11.1(f)
$\Gamma^{(0)}_{(x_{1},(x_{1},y_{1,1},y_{2,0}))}$ is the equilibrium measure of
(11.18) (see also (11.24)), where $y_{1,0}=y_{1,1}$. This means that all
colour $1$-dormant single colonies equal the current state of the colour
$1$-dormant $1$-block. We say that given the state of the $1$-dormant
$1$-block, the $1$-dormant single colonies become deterministic. This effect
occurs once a slow seed-bank, in this case the colour $1$ seed-bank, is
already in equilbrium on the space-time scale where it is effective, in this
case space-time-scale $1$. Since we start at times $N^{2}t_{2}$, the
$1$-dormant $1$-blocks are already in equilibrium. This will turn out to be
the reason that the single colour $1$-dormant colonies are equal to the
current value of the $1$-dormant $1$-block averages. Note that at time
$N^{2}t_{2}$ the $2$-dormant $2$-blocks do not yet have reached equilibrium.
Hence the colour $2$-dormant $1$-blocks and the colour $2$-dormant single
colonies do not equal the instantaneous value of the $2$-dormant $2$-block
averages. In the Section 11.3.8 we will treat this effect in detail.
$\blacksquare$

####  11.2 Scheme for the two-level three-colour mean-field analysis.

In this section we give a scheme to prove Proposition 11.1. The proof of the
steps in the scheme will be written in Section 11.3. To analyse the two-level
hierarchical mean-field system we use the results obtained in Sections 8.2, 9
and 10.2.

The scheme for the two-level three-colour hierarchical mean-field system comes
in 11 steps. Recall the estimators defined in (11.20) and (11.22).

  1. 1.

Tightness of the effective $2$-block estimator processes

(11.49)
$\left(\left({\bf\Theta}^{\mathrm{eff},(2),[N^{2}]}(N^{2}t_{2})\right)_{t_{2}>0}\right)_{N\in\mathbb{N}}=\left(\left(\bar{\Theta}^{(2),[N^{2}]}(N^{2}t_{2}),\Theta_{y_{2}}^{(2),[N^{2}]}(N^{2}t_{2})\right)_{t_{2}>0}\right)_{N\in\mathbb{N}}.$

  2. 2.

Stability property of the 2-block estimators, i.e., for $L(N)$ such that
$\lim_{N\to\infty}L(N)=\infty$ and $\lim_{N\to\infty}L(N)/N=0$,

(11.50) $\lim_{N\to\infty}\sup_{0\leq t\leq
L(N)}\left|\bar{\Theta}^{(2),[N^{2}]}(N^{2}t_{2})-\bar{\Theta}^{(2),[N^{2}]}(N^{2}t_{2}-Nt)\right|=0\text{
in probability}$

and

(11.51) $\lim_{N\to\infty}\sup_{0\leq t\leq
L(N)}\left|\Theta_{y_{2}}^{(2),[N^{2}]}(N^{2}t_{2})-\Theta_{y_{2}}^{(2),[N^{2}]}(N^{2}t_{2}-Nt)\right|=0\text{
in probability. }$

  3. 3.

Tightness of the effective $1$-block estimator process (recall (11.21)),

(11.52)
$\left(\left({\bf\Theta}^{\mathrm{aux},(1),[N^{2}]}(N^{2}t_{2}+Nt_{1})\right)_{t_{1}>0}\right)_{N\in\mathbb{N}}.$

  4. 4.

Stability property of
$({\bf\Theta}^{\mathrm{aux},(1),[N^{2}]}(N^{2}t_{2}+Nt_{1}))_{t_{1}>0}$, i.e.,
for $L(N)$ such that $\lim_{N\to\infty}L(N)=\infty$ and
$\lim_{N\to\infty}L(N)/N=0$, for all $\epsilon>0$,

(11.53) $\displaystyle\lim_{N\to\infty}\sup_{0\leq t\leq
L(N)}\left|\bar{\Theta}^{(1),[N^{2}]}(N^{2}t_{2}+Nt_{1})-\bar{\Theta}^{(1),[N^{2}]}(N^{2}t_{2}+Nt_{1}-t)\right|=0$
$\displaystyle\qquad\text{ in probability},$ (11.54)
$\displaystyle\lim_{N\to\infty}\sup_{0\leq t\leq
L(N)}\left|\Theta_{y_{1}}^{(1),[N^{2}]}(N^{2}t_{2}+Nt_{1})-\Theta_{y_{1}}^{(1),[N]}(N^{2}t_{2}+Nt_{1}-t)\right|=0$
$\displaystyle\qquad\text{ in probability},$ (11.55)
$\displaystyle\lim_{N\to\infty}\sup_{0\leq t\leq
L(N)}\left|\Theta_{y_{2}}^{(1),[N^{2}]}(N^{2}t_{2}+Nt_{1})-\Theta_{y_{2}}^{(1),[N]}(N^{2}t_{2}+Nt_{1}-t)\right|=0$
$\displaystyle\qquad\text{ in probability}.$

  5. 5.

Recall that there are $N$ 1-blocks in $[N^{2}]$. Since tightness of components
implies tightness of the process, step 3 implies that the full $1$-block
process

(11.56)
$\left(\left({\bf\Theta}_{i}^{\mathrm{aux},(1),[N^{2}]}(N^{2}t_{2}+Nt_{1})\right)_{t_{1}>0,\,i\in[N]}\right)_{N\in\mathbb{N}}$

is tight. From the tightness in steps 1 and 3 we can construct a subsequence
$(N_{k})_{k\in\mathbb{N}}$ along which

(11.57)
$\displaystyle\lim_{k\to\infty}\mathcal{L}\left[\left({\bf\Theta}^{\mathrm{eff},(2),[N_{k}^{2}]}(N_{k}^{2}t_{2})\right)_{t_{2}>0}\right],$
$\displaystyle\lim_{k\to\infty}\mathcal{L}\left[\left({\bf\Theta}_{i}^{\mathrm{aux},(1),[N_{k}^{2}]}(N_{k}^{2}t_{2}+N_{k}t_{1})\right)_{t_{1}>0,\,i\in[N_{k}]}\right]$

both exists. Define the measure

(11.58) $\nu^{(0)}(t_{2})=\prod_{i\in\mathbb{N}_{0}}\Gamma^{(0)}_{i}(t_{2}).$

Show that along the same subsequence the single components converge to the
infinite system, i.e.,

(11.59)
$\displaystyle\lim_{k\to\infty}\mathcal{L}\Big{[}\Big{(}Z^{[N_{k}^{2}]}(N_{k}^{2}t_{2}+N_{k}t_{1}+t_{0})\Big{)}_{t_{0}\geq
0}\Big{]}=\mathcal{L}\left[(Z^{\nu^{(0)}(t_{2})}(t_{0}))_{t_{0}\geq
0}\right].$

Here, $(Z^{\nu^{(0)}(t_{2})}(t_{0}))_{t_{0}\geq 0}$ is the process starting
from $\nu^{(0)}(t_{2})$ with components evolving according to (11.18), where
$\theta$ is now a random variable that inherits its law from

(11.60)
${}\lim_{k\to\infty}\mathcal{L}[({\bf\Theta}^{\mathrm{aux},(1),[N_{k}^{2}]}(N_{k}^{2}t_{2}+N_{k}t_{1}))_{i\in[N_{k}^{2}]}],$

and, similarly, the laws of $y_{1,0}$ and $y_{2,0}$ in the limiting process
$(Z^{\nu^{(0)}(t_{2})}(t_{0}))_{t_{0}\geq 0}$ are determined by

(11.61)
${}\lim_{k\to\infty}\mathcal{L}[({\bf\Theta}^{\mathrm{aux},(1),[N_{k}^{2}]}(N_{k}^{2}t_{2}+N_{k}t_{1}))_{i\in[N_{k}^{2}]}].$

  6. 6.

Use the limiting evolution of the single colonies obtained in step 5 to
identify the limiting $1$-block process along the same subsequence, i.e.,
identify the limit

(11.62)
$\lim_{k\to\infty}\mathcal{L}\left[\left({\bf\Theta}^{\mathrm{aux},(1),[N_{k}^{2}]}(N_{k}^{2}t_{2}+N_{k}t_{1})\right)_{t_{1}>0,i\in[N_{k}]}\right].$

  7. 7.

Identify the limit
$\lim_{k\to\infty}\mathcal{L}[({\bf\Theta}^{\mathrm{eff},(2),[N_{k}^{2}]}(N_{k}^{2}t_{2}))_{t_{2}>0}]$
with the help of the limiting evolution of the single colonies obtained in
step 5 and the limiting evolution of the full $1$-block process obtained in
step 6.

  8. 8.

Prove that the $1$-dormant single colonies at time $N^{2}t_{2}+Nt_{1}$ equal,
in the limit as $N\to\infty$, the $1$-dormant $1$-block averages. The proof of
this step shows how the evolution of the slow seed-banks must be analysed.

  9. 9.

Show that the convergence in step 8, step 7 and step 5 actually holds along
each subsequence. Therefore we obtain the limiting evolution of the single
colonies, the auxiliary $1$-block process and the effective $2$-block process.

  10. 10.

Use the Meyer-Zheng topology to describe the limiting evolution of

(11.63)
$\displaystyle\Big{(}\Theta_{x}^{(1),[N^{2}]}(N^{2}t_{2}+Nt_{1}),\Theta^{(1),[N^{2}]}_{y_{0}}(N^{2}t_{2}+Nt_{1}),\Theta^{(1),[N^{2}]}_{y_{1}}(N^{2}t_{2}+Nt_{1}),$
$\displaystyle\qquad\Theta^{(1),[N^{2}]}_{y_{2}}(N^{2}t_{2}+Nt_{1})\Big{)}_{t_{1}>0}$

and

(11.64)
$\Big{(}\Theta_{x}^{(2),[N^{2}]}(N^{2}t_{2}),\Theta^{(2),[N^{2}]}_{y_{0}}(N^{2}t_{2}),\Theta^{(2),[N^{2}]}_{y_{1}}(N^{2}t_{2}),\Theta^{(2),[N^{2}]}_{y_{2}}(N^{2}t_{2})\Big{)}_{t_{2}>0}.$

  11. 11.

Combine the above steps to complete the proof of Proposition 11.1.

####  11.3 Proof of two-level three-colour mean-field finite-systems scheme

In this section we prove the steps in the scheme given in Section 11.2.

#####  11.3.1 Tightness of the $2$-block estimators

In this section we prove step 1 of the scheme.

######  Lemma 11.4.

[Tightness of the 2-block estimator] Let

(11.65)
${\bf\Theta}^{\mathrm{eff},(2),[N^{2}]}(N^{2}t_{2})=(\bar{\Theta}^{(2),[N^{2}]}(N^{2}t_{2}),\Theta^{(2),[N^{2}]}_{y_{2}}(N^{2}t_{2}))$

be defined as in (11.22). Then
$(\mathcal{L}[({\bf\Theta}^{\mathrm{eff},(2),[N^{2}]}(N^{2}t_{2}))_{t_{2}>0}])_{N\in\mathbb{N}}$
is a tight sequence of probability measures on
$\mathcal{C}((0,\infty),[0,1]^{2})$.

###### Proof.

To prove the tightness of the $1$-blocks, we use [JM86, Proposition 3.2.3].
From (11.6) we find that $({\bf\Theta}^{\mathrm{eff},(2),[N^{2}]}(t))_{t>0}$
evolves according to

(11.66) $\displaystyle{\rm d}\bar{\Theta}^{(2),[N^{2}]}(t)$
$\displaystyle=\frac{1}{1+K_{0}+K_{1}}\frac{1}{N^{2}}\sum_{i\in[N^{2}]}\sqrt{g(x_{i}^{[N^{2}]}(t))}\,{\rm
d}w_{i}(t)$
$\displaystyle+\frac{1}{1+K_{0}+K_{1}}\frac{K_{2}e_{2}}{N^{2}}\left[\frac{1}{N^{2}}\sum_{i\in[N^{2}]}y_{i,2}^{[N^{2}]}(t)-\frac{1}{N^{2}}\sum_{i\in[N^{2}]}x_{i}^{[N^{2}]}(t)\right]\,{\rm
d}t,$ $\displaystyle{\rm d}\Theta^{(2),[N^{2}]}_{y_{2}}(t)$
$\displaystyle=\frac{e_{2}}{N^{2}}\left[\frac{1}{N^{2}}\sum_{i\in[N^{2}]}x_{i}^{[N^{2}]}(t)-\frac{1}{N^{2}}\sum_{i\in[N^{2}]}y_{i,2}^{[N^{2}]}(t)\right]\,{\rm
d}t.$

Therefore the process
$({\bf\Theta}^{\mathrm{eff},(2),[N^{2}]}(N^{2}t_{2}))_{t_{2}>0}$ evolves
according to

(11.67) $\displaystyle{\rm
d}\bar{\Theta}^{(2),[N^{2}]}(N^{2}t_{2})=\frac{1}{1+K_{0}+K_{1}}\sqrt{\frac{1}{N^{2}}\sum_{i\in[N^{2}]}g(x_{i}^{[N^{2}]}(N^{2}t_{2}))}\,{\rm
d}w_{i}(t_{2})$
$\displaystyle\qquad+\frac{1}{1+K_{0}+K_{1}}K_{2}e_{2}\left[\frac{1}{N^{2}}\sum_{i\in[N^{2}]}y_{i,2}^{[N^{2}]}(N^{2}t_{2})-\frac{1}{N^{2}}\sum_{i\in[N^{2}]}x_{i}^{[N^{2}]}(N^{2}t_{2})\right]\,{\rm
d}t_{2},$ $\displaystyle{\rm
d}\Theta^{(2),[N^{2}]}_{y_{2}}(N^{2}t_{2})=e_{2}\left[\frac{1}{N^{2}}\sum_{i\in[N^{2}]}x_{i}^{[N^{2}]}(N^{2}t_{2})-\frac{1}{N^{2}}\sum_{i\in[N^{2}]}y_{i,2}^{[N^{2}]}(N^{2}t_{2})\right]\,{\rm
d}t_{2}.$

To use [JM86, Proposition 3.2.3], we define $\mathcal{C}^{*}$ as the set of
polynomials on $([0,1]^{2})$. Since
$({\bf\Theta}^{\mathrm{eff},(2),[N^{2}]}(N^{2}t_{2}))_{t_{2}>0}$ is a semi-
martingale, by applying Itô’s formula we obtain that
$({\bf\Theta}^{\mathrm{eff},(2),[N^{2}]}(N^{2}t_{2}))_{t_{2}>0}$ is a
$\mathcal{D}$-semi-martingale with corresponding operator

(11.68) $\displaystyle G_{\dagger}^{(2),[N^{2}]}$
$\displaystyle\colon\,(\mathcal{C}^{*},[0,1]^{2},(0,\infty),\Omega)\to\mathbb{R},$
$\displaystyle G_{\dagger}^{(2),[N^{2}]}(f,(x,y),t,\omega)$
$\displaystyle=\frac{K_{2}e_{2}}{1+K_{0}+K_{1}}\left[y-\frac{1}{N^{2}}\sum_{i\in[N^{2}]}x^{[N^{2}]}_{i}(N^{2}t,\omega)\right]\,\frac{\partial
f}{\partial x}$
$\displaystyle\qquad+e_{2}\left[\frac{1}{N^{2}}\sum_{i\in[N^{2}]}x^{[N^{2}]}_{i}(N^{2}t,\omega)-y\right]\,\frac{\partial
f}{\partial y}$
$\displaystyle\qquad+\frac{1}{2(1+K_{0}+K_{1})^{2}}\frac{1}{N^{2}}\sum_{i\in[N^{2}]}g(x^{[N^{2}]}_{i}(N^{2}t,\omega))\,\frac{\partial^{2}f}{\partial
x^{2}}.$

The conditions $H_{1},\ H_{2},\ H_{3}$ in [JM86, Proposition 3.2.3] are
satisfied. Hence tightness follows from [JM86, Proposition 3.2.3]. ∎

#####  11.3.2 Stability of the $2$-block estimators

######  Lemma 11.5.

[Stability property of the $2$-block estimator] Let
$({\bf\Theta}^{\mathrm{eff},(2),[N^{2}]}(t))_{t>0}$ be defined as in (11.23).
For any $L(N)$ such that $\lim_{N\to\infty}L(N)=\infty$ and
$\lim_{N\to\infty}L(N)/N=0$,

(11.69) $\lim_{N\to\infty}\sup_{0\leq t\leq
L(N)}\left|\bar{\Theta}^{(2),[N^{2}]}(N^{2}t_{2})-\bar{\Theta}^{(2),[N^{2}]}(N^{2}t_{2}-Nt)\right|=0\text{
in probability}$

and

(11.70) $\lim_{N\to\infty}\sup_{0\leq t\leq
L(N)}\left|\Theta_{y_{2}}^{(2),[N^{2}]}(N^{2}t_{2})-\Theta_{y_{2}}^{(2),[N^{2}]}(N^{2}t_{2}-Nt)\right|=0\text{
in probability. }$

###### Proof.

Fix $\epsilon>0$. From the SSDE in (11.66) we obtain that, for $N$ large
enough,

(11.71) $\displaystyle\mathbb{P}\Bigg{[}\sup_{0\leq t\leq
L(N)}\Bigg{|}\bar{\Theta}^{(2),[N^{2}]}(N^{2}t_{2})-\bar{\Theta}^{(2),[N^{2}]}(N^{2}t_{2}-Nt)\Bigg{|}>\epsilon\Bigg{]}$
$\displaystyle=\mathbb{P}\Bigg{[}\sup_{0\leq t\leq
L(N)}\frac{1}{1+K_{0}+K_{1}}\Bigg{|}\int_{N^{2}t_{2}-Nt}^{N^{2}t_{2}}{\rm
d}w_{i}(r)\,\frac{1}{N^{2}}\sum_{i\in[N^{2}]}\sqrt{g(x^{[N^{2}]}_{i}(r))}$
$\displaystyle\qquad\qquad\qquad+\int_{N^{2}t_{2}-Nt}^{N^{2}t_{2}}{\rm
d}r\,\frac{K_{2}e_{2}}{N^{2}}\left[\Theta^{(2),[N^{2}]}_{y_{2}}(r)-\frac{1}{N^{2}}\sum_{i\in[N^{2}]}x^{[N^{2}]}_{i}(r)\right]\Bigg{|}>\epsilon\Bigg{]}$
$\displaystyle\leq\mathbb{P}\Bigg{[}\sup_{0\leq t\leq
L(N)}\,\frac{1}{1+K_{0}+K_{1}}\Bigg{|}\int_{N^{2}t_{2}-Nt}^{N^{2}t_{2}}{\rm
d}w_{i}(r)\,\frac{1}{N^{2}}\sum_{i\in[N^{2}]}\sqrt{g(x^{[N^{2}]}_{i}(r))}\,\Bigg{|}$
$\displaystyle\qquad\qquad\qquad\qquad>\epsilon-\frac{K_{2}e_{2}}{1+K_{0}+K_{1}}\frac{L(N)N}{N^{2}}\Bigg{]}$
$\displaystyle\leq\mathbb{P}\Bigg{[}\sup_{0\leq t\leq
L(N)}\frac{1}{1+K_{0}+K_{1}}\Bigg{|}\int_{N^{2}t_{2}-Nt}^{N^{2}t_{2}}{\rm
d}w_{i}(r)\frac{1}{N^{2}}\sum_{i\in[N^{2}]}\sqrt{g(x^{[N^{2}]}_{i}(r))}\,\Bigg{|}>\frac{\epsilon}{2}\Bigg{]}.$

By a similar optional stopping time argument as in the proof of Lemma 8.15,
the above computation shows that (11.69) holds. Equation (11.70) holds by a
similar argument as given in the proof of Lemma 10.5. ∎

#####  11.3.3 Tightness of the 1-block estimators

######  Lemma 11.6.

[Tightness of the 1-block estimator] Let

(11.72)
$\displaystyle{\bf\Theta}^{\mathrm{aux},(1),[N^{2}]}(N^{2}t_{2}+Nt_{1})$
$\displaystyle=(\bar{\Theta}^{(1),[N^{2}]}(N^{2}t_{2}+Nt_{1}),\Theta^{(1),[N^{2}]}_{y_{1}}(N^{2}t_{2}+Nt_{1}),\Theta^{(1),[N^{2}]}_{y_{2}}(N^{2}t_{2}+Nt_{1}))$

be defined as in (11.20). Then
$(\mathcal{L}[({\bf\Theta}^{\mathrm{aux},(1),[N]}(N^{2}t_{2}+Nt_{1}))_{t_{1}>0}])_{N\in\mathbb{N}}$
is a tight sequence of probability measures on
$\mathcal{C}((0,\infty),[0,1]^{3})$.

###### Proof.

To prove the tightness of the $1$-blocks, we again use [JM86, Proposition
3.2.3]. From (11.11) we find that the effective process
$({\bf\Theta}^{\mathrm{aux},(1),[N^{2}]}(N^{2}t_{2}+Nt_{1}))_{t_{1}>0}$
evolves according to

(11.73) $\displaystyle{\rm d}\bar{\Theta}^{(1),[N^{2}]}(Nt_{1})$
$\displaystyle=\frac{1}{1+K_{0}}c_{1}\left[\frac{1}{N^{2}}\sum_{j\in[N^{2}]}x^{[N^{2}]}_{j}(Nt_{1})-\frac{1}{N}\sum_{i\in[N]}x^{[N^{2}]}_{i}(Nt_{1})\right]\,{\rm
d}t_{1}$
$\displaystyle\qquad+\frac{1}{1+K_{0}}\sqrt{\frac{1}{N}\sum_{i\in[N]}g(x^{[N^{2}]}_{i}(Nt_{1}))}\,{\rm
d}w_{i}(t_{1})$
$\displaystyle\qquad+\frac{K_{1}e_{1}}{1+K_{0}}\left[\Theta_{y_{1}}^{(1),[N^{2}]}(Nt_{1})-\frac{1}{N}\sum_{i\in[N]}x^{[N^{2}]}_{i}(Nt_{1})\right]\,{\rm
d}t_{1}$
$\displaystyle\qquad+\frac{K_{2}e_{2}}{N(1+K_{0})}\left[\Theta^{(1),[N^{2}]}_{y_{2}}(Nt_{1})-\frac{1}{N}\sum_{i\in[N]}x^{[N^{2}]}_{i}(Nt_{1})\right]\,{\rm
d}t_{1},$ $\displaystyle{\rm d}\Theta^{(1),[N^{2}]}_{y_{1}}(Nt_{1})$
$\displaystyle=e_{1}\left[\frac{1}{N}\sum_{i\in[N]}x^{[N^{2}]}_{i}(Nt_{1})-\Theta^{(1),[N^{2}]}_{y_{1}}(Nt_{1})\right]\,{\rm
d}t_{1},$ $\displaystyle{\rm d}\Theta^{(1),[N^{2}]}_{y_{2}}(Nt_{1})$
$\displaystyle=\frac{e_{2}}{N}\left[\frac{1}{N}\sum_{i\in[N]}x^{[N^{2}]}_{i}(Nt_{1})-\Theta^{(1),[N^{2}]}_{y_{2}}(Nt_{1})\right]\,{\rm
d}t_{1}.$

To use [JM86, Proposition 3.2.3], we define $\mathcal{C}^{*}$ as the set of
polynomials on $([0,1]^{2})$. Since
$({\bf\Theta}^{\mathrm{aux},(1),[N^{2}]}(N^{2}t_{2}+Nt_{1}))_{t_{1}>0}$ is a
semi-martingale, by applying Itô’s formula we obtain that
$({\bf\Theta}^{\mathrm{aux},(1),[N^{2}]}(N^{2}t_{2}+Nt_{1}))_{t_{1}>0}$ is a
$\mathcal{D}$-semi-martingale with corresponding operator

(11.74) $\displaystyle G_{\dagger}^{(1),[N^{2}]}$
$\displaystyle\colon\,(\mathcal{C}^{*},[0,1]^{3},(0,\infty),\Omega)\to\mathbb{R},$
$\displaystyle G_{\dagger}^{(1),[N^{2}]}(f,(x,y_{1},y_{2}),t,\omega)$
$\displaystyle=\frac{c_{1}}{1+K_{0}}\left[\frac{1}{N^{2}}\sum_{j\in[N^{2}]}x^{[N^{2}]}_{j}(Nt,\omega)-\frac{1}{N}\sum_{i\in[N]}x^{[N^{2}]}_{i}(Nt,\omega)\right]\,\frac{\partial
f}{\partial x}$
$\displaystyle\qquad+\frac{K_{1}e_{1}}{1+K_{0}}\left[y_{1}-\frac{1}{N}\sum_{i\in[N]}x^{[N^{2}]}_{i}(Nt,\omega)\right]\,\frac{\partial
f}{\partial x}$
$\displaystyle\qquad+\frac{K_{2}e_{2}}{N(1+K_{0})}\left[y_{2}(Nt,\omega)-\frac{1}{N}\sum_{i\in[N]}x^{[N^{2}]}_{i}(Nt)\right]\,\frac{\partial
f}{\partial x}$
$\displaystyle\qquad+e_{1}\left[\frac{1}{N}\sum_{i\in[N]}x^{[N^{2}]}_{i}(Nt,\omega)-y_{1}\right]\,\frac{\partial
f}{\partial y_{1}}$
$\displaystyle\qquad+\frac{e_{2}}{N}\left[\frac{1}{N}\sum_{i\in[N]}x^{[N^{2}]}_{i}(Nt,\omega)-y_{2}(Nt,\omega)\right]\,\frac{\partial
f}{\partial y_{2}}$
$\displaystyle\qquad+\frac{1}{2(1+K_{0})^{2}}\frac{1}{N}\sum_{i\in[N]}g(x^{[N^{2}]}_{i}(Nt,\omega))\,\frac{\partial^{2}f}{\partial
x^{2}}.$

The conditions $H_{1},\ H_{2},\ H_{3}$ in [JM86, Proposition 3.2.3] are
satisfied as before. Hence we conclude that the sequence
$(\mathcal{L}[({\bf\Theta}^{\mathrm{aux},(1),[N^{2}]}(N^{2}t_{2}+Nt_{1}))_{t_{1}>0}])_{N\in\mathbb{N}}$
is tight. ∎

#####  11.3.4 Stability of the $1$-block estimators

######  Lemma 11.7.

[Stability property of the $1$-block estimator] Let
${\bf\Theta}^{\mathrm{aux},(1),[N^{2}]}(t)$ be defined as in (10.14). For any
$L(N)$ such that $\lim_{N\to\infty}L(N)=\infty$ and
$\lim_{N\to\infty}L(N)/N=0$,

(11.75) $\lim_{N\to\infty}\sup_{0\leq t\leq
L(N)}\left|\bar{\Theta}^{(1),[N^{2}]}(N^{2}t_{2}+Nt_{1})-\bar{\Theta}^{(1),[N^{2}]}(N^{2}t_{2}+Nt_{1}-t)\right|=0\text{
in probability},$ (11.76) $\lim_{N\to\infty}\sup_{0\leq t\leq
L(N)}\left|\Theta_{y_{1}}^{(1),[N^{2}]}(N^{2}t_{2}+Nt_{1})-\Theta_{y_{1}}^{(1),[N^{2}]}(N^{2}t_{2}+Nt_{1}-t)\right|=0\text{
in probability},$ (11.77) $\lim_{N\to\infty}\sup_{0\leq t\leq
L(N)}\left|\Theta_{y_{2}}^{(1),[N^{2}]}(N^{2}t_{2}+Nt_{1})-\Theta_{y_{2}}^{(1),[N^{2}]}(N^{2}t_{2}+Nt_{1}-t)\right|=0\text{
in probability}.$

###### Proof.

Define

(11.78) $u=N^{2}t_{2}+Nt_{1}.$

From the SSDE in (11.7) we obtain that

(11.79) $\displaystyle\mathbb{P}\Bigg{[}\sup_{0\leq t\leq
L(N)}\Bigg{|}\bar{\Theta}^{(1),[N^{2}]}(u)-\bar{\Theta}^{(1),[N^{2}]}(u-t)\Bigg{|}>\epsilon\Bigg{]}$
$\displaystyle=\mathbb{P}\Bigg{[}\sup_{0\leq t\leq
L(N)}\frac{1}{1+K_{0}}\Bigg{|}\int_{u-t}^{u}{\rm
d}r\,\frac{c_{1}}{N}\left[\frac{1}{N^{2}}\sum_{j\in[N^{2}]}x^{[N^{2}]}_{j}(r)-\frac{1}{N}\sum_{i\in[N]}x^{[N^{2}]}_{i}(r)\right]$
$\displaystyle\qquad\qquad\qquad+\int_{u-t}^{u}{\rm
d}r\,\frac{K_{1}e_{1}}{N}\left[\Theta^{(1),[N^{2}]}_{y_{1}}(r)-\frac{1}{N}\sum_{i\in[N]}x^{[N^{2}]}_{i}(r)\right]$
$\displaystyle\qquad\qquad\qquad+\frac{K_{2}e_{2}}{N^{2}}\left[\frac{1}{N}\sum_{i\in[N]}y^{[N^{2}]}_{i,2}(r)-\frac{1}{N}\sum_{i\in[N]}x^{[N^{2}]}_{i}(r)\right]$
$\displaystyle\qquad\qquad\qquad+\int_{u-t}^{u}{\rm
d}w_{i}(r)\,\frac{1}{N}\sum_{i\in[N]}\sqrt{g(x^{[N^{2}]}_{i}(r))}\,\Bigg{|}>\epsilon\Bigg{]}$
$\displaystyle\leq\mathbb{P}\Bigg{[}\sup_{0\leq t\leq
L(N)}\,\frac{1}{1+K_{0}}\Bigg{|}\int_{u-t}^{u}{\rm
d}w_{i}(r)\,\frac{1}{N}\sum_{i\in[N]}\sqrt{g(x^{[N^{2}]}_{i}(r))}\,\Bigg{|}$
$\displaystyle\qquad\qquad>\epsilon-\frac{L(N)2(c_{1}+K_{1}e_{1}+\frac{K_{2}e_{2}}{N})}{N(1+K_{0})}\Bigg{]}$
$\displaystyle\leq\mathbb{P}\Bigg{[}\sup_{0\leq t\leq
L(N)}\frac{1}{1+K_{0}}\Bigg{|}\int_{u-t}^{u}{\rm
d}w_{i}(r)\frac{1}{N}\sum_{i\in[N]}\sqrt{g(x^{[N^{2}]}_{i}(r))}\,\Bigg{|}>\frac{\epsilon}{2}\Bigg{]}.$

Via the same optional stopping time argument as in the proof of Lemma 8.15,
the above computation shows that (11.75) holds. Note that the extra drift term
$\frac{K_{2}e_{2}}{N}$ does not have any influence. Equations (11.76)–(11.77)
hold by a similar argument as given in the proof of Lemma 10.5. ∎

#####  11.3.5 Limiting evolution for the single components

######  Proposition 11.8.

[Equilibrium for the infinite system] Fix $t_{2},t_{1}>0$. Let
$(N_{k})_{k\in\mathbb{N}}\subset\mathbb{N}$ and let $L(N)$ be any sequence
satisfying $\lim_{N\to\infty}L(N)=\infty$ and $\lim_{N\to\infty}L(N)/N=0$ such
that

(11.80)
$\displaystyle\lim_{k\to\infty}\mathcal{L}\left[{\bf\Theta}^{\mathrm{aux},(1),[N_{k}^{2}]}(N^{2}_{k}t_{2}+N_{k}t_{1})\right]=P_{t_{1},t_{2}},$
$\displaystyle\lim_{k\to\infty}\mathcal{L}\left[\left(Y_{1,0}^{[N_{k}^{2}]}(N^{2}_{k}t_{2}+N_{k}t_{1}),Y_{2,0}^{[N_{k}^{2}]}(N^{2}_{k}t_{2})\right)\Big{|}{\bf\Theta}^{\mathrm{aux},(1),[N_{k}^{2}]}(N^{2}_{k}t_{2}+N_{k}t_{1})\right]$
$\displaystyle\qquad\qquad=P^{{z}_{1}^{\mathrm{eff}}(t_{1})},$
$\displaystyle\lim_{k\to\infty}\mathcal{L}\Bigl{[}\sup_{0\leq t\leq
L(N_{k})}\left|\bar{\Theta}^{[N^{2}_{k}]}(N^{2}_{k}t_{2}+N_{k}t_{1})-\bar{\Theta}^{[N^{2}_{k}]}(N^{2}_{k}t_{2}+N_{k}t_{1}-t)\right|$
$\displaystyle\qquad\qquad\qquad+\left|{\Theta}_{y_{1}}^{[N^{2}_{k}]}(N^{2}_{k}t_{2}+N_{k}t_{1})-{\Theta_{y_{1}}}^{[N^{2}_{k}]}(N^{2}_{k}t_{2}+N_{k}t_{1}-t)\right|$
$\displaystyle\qquad\qquad\qquad+\left|{\Theta}_{y_{2}}^{[N^{2}_{k}]}(N^{2}_{k}t_{2}+N_{k}t_{1})-{\Theta_{y_{2}}}^{[N^{2}_{k}]}(N^{2}_{k}t_{2}+N_{k}t_{1}-t)\right|\Bigr{]}=\delta_{0},$
$\displaystyle\lim_{k\to\infty}\mathcal{L}\Big{(}Z^{[N^{2}_{k}]}(N^{2}_{k}t_{2}+N_{k}t_{1}),\Big{)}=\nu(t_{1},t_{2}).$

Then $\nu(t_{1},t_{2})$ is of the form

(11.81) $\nu(t_{1},t_{2})=\int_{[0,1]^{2}}P_{t_{1},t_{2}}({\rm
d}\theta^{(1)},{\rm
d}\theta^{(1)}_{y})\,\int_{[0,1]^{\mathbb{N}_{0}}}P^{(\theta^{(1)},\theta^{(1)}_{y})}({\rm
d}{\bf y})\,\nu_{\theta,{\bf y}},$

where

(11.82) $\nu_{\theta,{\bf
y_{0}}}=\prod_{i\in\mathbb{N}_{0}}\Gamma_{(\theta,{\bf y_{0,i}})}$

with $\Gamma_{(\theta,{\bf y_{0,i}})}$ the equilibrium measure for the $i$’th
single colony defined in (11.24).

Note that by step 1 and step 3 we can find a subsequence
$(N_{k})_{k\in\mathbb{N}}$ such that the first and third line in (11.80) hold.
The second line in (11.80) follows from assumptions (11.37) and (11.38). To
prove Proposition 11.8 we proceed as in the proof of Proposition 10.6, but
with the finite system in (10.4) replaced by the system in (11.6) and the
infinite system in (10.11) replaced by the system in (11.18). Note that Lemma
10.7 holds also for the system in (11.18), after adding the non-interacting
component $y_{2,0}$ to the equilibrium. The equivalent of Lemma 10.8 will
again follow from the equivalent of Lemma 10.12. We will derive the analogue
of Lemmas 10.9 and 10.10 (see Lemma’s 11.9 and 11.10 below). Lemma 10.11 can
be extended with an extra colour-$2$ seed-bank estimator by using the same
proof. Since the infinite system for the single colonies in the two-layer
three-colour mean-field system (see (11.88)) equals the one for the one-layer
two-colour mean-field system, up to a non-interacting component, we obtain an
equivalent of Lemma 10.12. Finally, also the equivalent of Lemma 10.13 holds
under an additional assumption, see Lemma 11.11. Finally Corollary 11.12
states the equivalent of Corollary 10.14. With the help of the lemma’s and the
corollary, the proof of Proposition 11.8 follows from the same argument as
used in the proof of Proposition 10.6.

######  Lemma 11.9.

[Comparison of empirical averages]
Let $(\Theta^{(1),[N^{2}]}_{x}(t_{0}))_{t_{0}\geq 0}$ and
$(\Theta^{(1),[N^{2}]}_{y_{0}}(t_{0}))_{t_{0}\geq 0}$ be defined as in
(11.20). Then

(11.83)
$\displaystyle\mathbb{E}\left[\left|\Theta^{(1),[N]}_{x}(t)-\Theta^{(1),[N]}_{y_{0}}(t)\right|\right]$
$\displaystyle\leq\sqrt{\mathbb{E}\left[\left(\Theta^{(1),[N]}_{x}(0)-\Theta^{(1),[N]}_{y_{0}}(0)\right)^{2}\right]}{\rm
e}^{-(K_{0}e_{0}+e_{0})t}$
$\displaystyle\quad+\sqrt{\frac{1}{K_{0}e_{0}+e_{0}}\left[\frac{c_{1}}{N}+\frac{||g||}{N}+\frac{K_{1}e_{1}}{N}+\frac{K_{2}e_{2}}{N^{2}}\right]}.$

###### Proof.

The result follows by Itô-calculus on the SSDE in (11.6) and the same type of
argument as used in the proof of Lemma 10.9. ∎

Like for the mean-field system with one colour, we need to compare the finite
system in (11.6) with an infinite system. To derive the analogue of Lemma
10.10, let $L(N)$ satisfy $\lim_{N\to\infty}L(N)=\infty$ and
$\lim_{N\to\infty}L(N)/N=0$. Define $[N]_{i}$ to be the $1$-block that
contains site $i\in[N^{2}]$. Since we start our system in an exchangeable
measure and the dynamics are exchangeable, we will only consider the single
colonies in $[N]_{0}$, the $1$-block containing the site $0\in[N^{2}]$. In the
rest of the prove, we will suppress the $0$ from the notation i.e.,
$[N]_{0}=[N]$ and $\bar{\Theta}^{(1),[N^{2}]}_{0}=\bar{\Theta}^{(1),[N^{2}]}$.
Define

(11.84) $u=N^{2}t_{2}+Nt_{1}$

and let $\mu_{N}$ be the measure on $([0,1]^{3})^{\mathbb{N}_{0}}$ by
continuing the configuration of

(11.85) $\displaystyle\left(Z^{[N^{2}]}(u-L(N))\right)$
$\displaystyle=\left(X^{[N^{2}]}(u-L(N)),\left(Y_{0}^{[N^{2}]}(u-L(N)),Y_{1}^{[N^{2}]}(u-L(N)),Y_{2}^{[N^{2}]}(u-L(N))\right)\right)$

periodically to $([0,1]^{4})^{\mathbb{N}_{0}}$, i.e., we continue the
configuration of the single colonies in the first block to
$([0,1]^{4})^{\mathbb{N}_{0}}$. Let

(11.86)
$\bar{\Theta}^{(1),[N^{2}]}=\frac{1}{N}\sum_{i\in[N]}\frac{x^{[N^{2}]}_{i}(u-L(N))+K_{0}y^{[N^{2}]}_{i,0}(u-L(N))}{1+K_{0}}.$

Let

(11.87) $(Z^{\mu_{N}}(t))_{t\geq
0}=\bigl{(}X^{\mu_{N}}(t),\left(Y_{0}^{\mu_{N}}(t),Y_{1}^{\mu_{N}}(t),Y_{2}^{\mu_{N}}(t)\right)\bigr{)}_{t\geq
0}$

be the infinite system evolving according to

(11.88) $\displaystyle{\rm d}x^{\mu_{N}}_{i}(t)$
$\displaystyle=c_{0}\,[\bar{\Theta}^{(1),[N^{2}]}-x^{\mu_{N}}_{i}(t)]\,{\rm
d}t+\sqrt{g\big{(}x^{\mu_{N}}_{i}(t)\big{)}}\,{\rm
d}w_{i}(t)+K_{0}e_{0}\,[y^{\mu_{N}}_{i,0}(t)-x^{\mu_{N}}_{i}(t)]\,{\rm d}t,$
$\displaystyle{\rm d}y^{\mu_{N}}_{i,0}(t)$
$\displaystyle=e_{0}\,[x^{\mu_{N}}_{i}(t)-y^{\mu_{N}}_{i,0}(t)]\,{\rm d}t,$
$\displaystyle y^{\mu_{N}}_{i,1}(t)$ $\displaystyle=y^{\mu_{N}}_{i,1}(0),$
$\displaystyle y^{\mu_{N}}_{i,2}(t)$
$\displaystyle=y^{\mu_{N}}_{i,2}(0),\qquad i\in\mathbb{N}_{0},$

starting from initial distribution $\mu_{N}$. Then the following Lemma 11.10
is the equivalent of Lemma 10.10 for the three-colour two-layer mean-field
system. In particular, the infinite system considered in Lemma 11.10 is
similar to the infinite system in Lemma 10.10. The only difference is that
there is one more non-interacting component added in (11.88).

######  Lemma 11.10.

[Comparison of finite and infinite systems] Fix $t_{1},t_{2}>0$, and let
$u=N^{2}t_{2}+Nt_{1}$. Let $L(N)$ satisfy $\lim_{N\to\infty}L(N)=\infty$ and
$\lim_{N\to\infty}L(N)/N$. Suppose that

(11.89) $\displaystyle\lim_{N\to\infty}\sup_{0\leq t\leq
L(N)}\left|\bar{\Theta}^{(1),[N]}(u)-\bar{\Theta}^{(1),[N]}(u-t)\right|=0\
\text{ in probability}.$

Then, for all $t\geq 0$,

(11.90)
$\lim_{k\to\infty}\left|\mathbb{E}\left[f\big{(}Z^{\mu_{N}}(t)\big{)}-f\big{(}Z^{[N^{2}]}(u-L(N)+t)\big{)}\right]\right|=0\qquad\forall\,f\in\mathcal{C}\bigl{(}([0,1]^{3})^{\mathbb{N}_{0}},\mathbb{R}\bigr{)}.$

###### Proof.

We proceed as in the proof of Lemma 10.10 and couple the finite and infinite
systems by their Brownian motion, exactly as was done there. The single
components in the block around site $0$ of the finite process
$(Z^{[N^{2}]}(t))$ are evolving according to

(11.91) $\displaystyle{\rm
d}x^{[N^{2}]}_{i}(t)=c_{0}\,\left[\Theta^{(1),[N^{2}]}-x^{[N^{2}]}_{i}(t)\right]\,{\rm
d}t+c_{0}\,\left[\bar{\Theta}^{(1),[N^{2}]}(t)-\Theta^{(1),[N^{2}]}\right]\,{\rm
d}t$
$\displaystyle\qquad\qquad+c_{0}\,\left[\Theta^{(1),[N^{2}]}_{x}(t)-\bar{\Theta}^{(1),[N^{2}]}(t)\right]\,{\rm
d}t+\frac{c_{1}}{N}\left[\frac{1}{N^{2}}\sum_{i\in[N^{2}]}x_{j}^{[N^{2}]}(t)-x_{i}^{[N^{2}]}(t)\right]{\rm
d}t$ $\displaystyle\qquad\qquad+\sqrt{g(x^{[N^{2}]}_{i}(t))}\,{\rm
d}w_{i}(t)+K_{0}e_{0}\,[y^{[N^{2}]}_{i,0}(t)-x^{[N^{2}]}_{i}(t)]\,{\rm d}t$
$\displaystyle\qquad\qquad+\frac{K_{1}e_{1}}{N}\left[y^{[N^{2}]}_{i,1}(t)-x_{i}^{[N^{2}]}(t)\right]{\rm
d}t+\frac{K_{2}e_{2}}{N^{2}}\,[y^{[N^{2}]}_{i,0}(t)-x^{[N^{2}]}_{i}(t)]\,{\rm
d}t,$ $\displaystyle{\rm
d}y^{[N^{2}]}_{i,0}(t)=e_{0}\,[x^{[N^{2}]}_{i}(t)-y^{[N^{2}]}_{i,0}(t)]\,{\rm
d}t,$ $\displaystyle{\rm
d}y^{[N^{2}]}_{i,1}(t)=\frac{e_{1}}{N}\,[x^{[N^{2}]}_{i}(t)-y^{[N^{2}]}_{i,1}(t)]\,{\rm
d}t,$ $\displaystyle{\rm
d}y^{[N^{2}]}_{i,2}(t)=\frac{e_{2}}{N^{2}}\,[x^{[N^{2}]}_{i}(t)-y^{[N^{2}]}_{i,2}(t)]\,{\rm
d}t,\qquad i\in[N].$

Using this SSDE we can exactly proceed as in the proof of Lemma 10.10 to
obtain the result. Note that the colour-$2$ seed-bank can be treated just in
the same way as the colour-$1$ seed-bank in the proof of Lemma 10.10, since
its rate of interaction with the active population is even slower than the
rate of interaction of the colour-$1$ seed-bank. ∎