Finally, we state the equivalent of Lemma 10.13 for the three-colour two-layer
mean-field system.

######  Lemma 11.11.

[Coupling of finite systems] Let

(11.92)
$Z^{[N^{2}],1}=(X^{[N^{2}],1},Y_{0}^{[N^{2}],1},Y_{1}^{[N^{2}],1},Y_{2}^{[N^{2}],1})$

be the finite system evolving according to (11.6) starting from an
exchangeable initial measure. Let $\mu^{[N],1}$ be the measure obtain by
periodic continuation of the configuration of $Z^{[N^{2}],1}(0)$ in the
$1$-block around $0$. Similarly, let

(11.93)
$Z^{[N^{2}],2}=(X^{[N^{2}],2},Y_{0}^{[N^{2}],2},Y_{1}^{[N^{2}],2},Y_{2}^{[N^{2}],2})$

be the finite system evolving according to (11.6) starting from an
exchangeable initial measure. Let $\mu^{[N],2}$ be the measure obtained by
periodic continuation of the configuration of $Z^{[N^{2}],1}(0)$ in the
$1$-block around $0$. 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 variables $\bar{\Theta}^{[N],1}$ on
$(([0,1]^{4},\mu^{[N],1})^{\mathbb{N}_{0}})$, $\bar{\Theta}^{[N],2}$ on
$(([0,1]^{4})^{\mathbb{N}_{0}},\mu^{[N],2})$ and $\bar{\Theta}_{1}$ and
$\bar{\Theta}_{2}$ on
$(([0,1]^{4})^{\mathbb{N}_{0}}\times([0,1]^{4})^{\mathbb{N}_{0}},\mu)$ by

(11.94)
$\displaystyle\bar{\Theta}^{[N],1}=\frac{1}{N}\sum_{i\in[N]}\frac{x^{[N^{2}],1}_{i}+K_{0}y^{[N^{2}],1}_{i,0}}{1+K_{0}},\qquad\bar{\Theta}^{[N],2}=\frac{1}{N}\sum_{i\in[N]}\frac{x^{[N^{2}],2}_{i}+K_{0}y^{[N^{2}],2}_{i,0}}{1+K_{0}},$
$\displaystyle\bar{\Theta}^{1}=\lim_{n\to\infty}\frac{1}{n}\sum_{i\in[n]}\frac{x^{1}_{i}+K_{0}y^{1}_{i,0}}{1+K_{0}},\qquad\bar{\Theta}^{2}=\lim_{n\to\infty}\frac{1}{n}\sum_{i\in[n]}\frac{x^{2}_{i}+K_{0}y^{2}_{i,0}}{1+K_{0}},$

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
$Z^{[N^{2}],1}$, respectively, $Z^{[N^{2}],2}$. Suppose that

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

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

(11.96)
$\displaystyle\lim_{N\to\infty}\mathbb{E}\bigl{[}|x^{[N],1}_{i}(t(N))-x^{[N],2}_{i}(t(N))|+K_{0}|y^{[N],1}_{i,0}(t(N))-y^{[N],2}_{i,0}(t(N))|$
$\displaystyle\qquad+K_{1}|y^{[N],1}_{i,1}(t(N))-y^{[N],2}_{i,1}(t(N))|+K_{2}|y^{[N],1}_{i,2}(t(N))-y^{[N],2}_{i,2}(t(N))|\bigr{]}=0.$

###### Proof.

Like in the proof of Lemma 10.13, we can show with Itô calculus that the
function

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

is monotonically decreasing. Hence we can proceed as in the proof of Lemma
8.13 to show that (11.96) is true. ∎

From the above couplings we can derive the following corollary, which is the
analogue of Corollary 10.14 for the two-level three-colour mean-field system.

######  Corollary 11.12.

Fix $t_{1},t_{2}>0$ and set $u=N^{2}t_{2}+Nt_{1}$. Let $\mu_{N}$ be the
measure obtained by periodic continuation of

(11.98)
$Z^{[N^{2}]}(u-L(N))=\big{(}X^{[N^{2}]}(u-L(N)),Y_{0}^{[N^{2}]}(u-L(N)),Y_{1}^{[N^{2}]}(u-L(N)),Y_{2}^{[N^{2}]}(u-L(N))\big{)},$

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

(11.99)
$\Theta=\lim_{N\to\infty}\frac{1}{N}\sum_{i\in[N]}\frac{x_{i}^{\mu}+Ky_{i}^{\mu}}{1+K}\,\qquad\text{
in }L^{2}(\mu),$

and let
$(Z^{\nu_{\Theta}}(t))_{t>0}=(X^{\nu_{\Theta}}(t),Y_{0}^{\nu_{\Theta}}(t),Y_{1}^{\nu_{\Theta}}(t),Y_{2}^{\nu_{\Theta}}(t))_{t>0}$
be the infinite system with components evolving according to (11.18) with
$\theta=\Theta$ and $y_{i,1,0}$ and $y_{i,2,0}$ determined by assumption
(11.80) and starting from its equilibrium measure. Extend the finite system
$Z^{[N^{2}]}$ as a system on $([0,1]^{4})^{\mathbb{N}_{0}}$ by periodic
continuation. Construct $(Z^{[N^{2}]}(t))_{t>0}$ and
$(Z^{\nu_{\Theta}}(t))_{t>0}$ on one probability space. Then there exists a
sequence $(\bar{L}(N))_{N\in\mathbb{N}}$ such that
$\lim_{N\to\infty}\bar{L}(N)=\infty,$
$\lim_{N\to\infty}\frac{\bar{L(N)}}{N}=0$ and

(11.100)
$\displaystyle\lim_{N\to\infty}\mathbb{E}\left[\left|x_{i}^{[N^{2}]}(Ns)-x_{i}^{\nu_{\Theta}}(\bar{L}(N))\right|\right]+K_{0}\,\mathbb{E}\left[\left|y_{i,0}^{[N^{2}]}(Ns)-y_{i,0}^{\nu_{\Theta}}(\bar{L}(N))\right|\right]$
$\displaystyle+K_{1}\,\mathbb{E}\left[\left|y_{i,1}^{[N^{2}]}(Ns)-y_{i,1}^{\nu_{\Theta}}(\bar{L}(N))\right|\right]+K_{2}\,\mathbb{E}\left[\left|y_{i,2}^{[N^{2}]}(Ns)-y_{i,2}^{\nu_{\Theta}}(\bar{L}(N))\right|\right]=0,\,i\in[N].$

Note that Lemmas 11.10, 11.11 and Corollary 11.12 do not only hold for sites
$i$ in the $1$-block around $0$, but hold for for all sites $i\in[N^{2}]$,
after we replace $\Theta^{[N]_{0}}_{0}$ by $\Theta^{[N]_{i}}_{i}$ .

#####  11.3.6 Limiting evolution of the $1$-block estimator process

######  Proposition 11.13.

[Limiting evolution of the 1-blocks] Fix $t_{2}>0$. Let
$(L(N))_{N\in\mathbb{N}}$ satisfy $\lim_{N\to\infty}L(N)=\infty$ and
$\lim_{N\to\infty}L(N)/N=0$. Let $(N_{k})_{k\in\mathbb{N}}$ be a subsequence
such that

(11.101)
$\displaystyle\lim_{k\to\infty}\mathcal{L}\left[\left({\bf\Theta}^{\mathrm{eff},(2),[N_{k}^{2}]}(N_{k}^{2}t_{2})\right)\right]=P_{t_{2}}(\cdot),$
$\displaystyle\lim_{k\to\infty}\mathcal{L}\left[y_{2,1}^{[N_{k}^{2}]}(N_{k}t_{2})\Big{|}{\bf\Theta}^{(2),[N_{k}^{2}]}(N_{k}^{2}t_{2})\right]=P^{z_{2}(t_{2})},$
$\displaystyle\lim_{k\to\infty}\mathcal{L}\left[\left(Y_{1,0}^{[N_{k}^{2}]}(N^{2}_{k}t_{2}+N_{k}t_{1}),Y_{2,0}^{[N_{k}^{2}]}(N^{2}_{k}t_{2})\right)\Big{|}{\bf\Theta}^{\mathrm{aux},(1),[N_{k}^{2}]}(N^{2}_{k}t_{2}+N_{k}t_{1})\right]=P^{{z}_{1}^{\mathrm{eff}}(t_{1})},$
$\displaystyle\lim_{k\to\infty}\mathcal{L}\Bigg{[}\sup_{0\leq t\leq
L(N_{k})}\left|\bar{\Theta}^{(2),[N^{2}_{k}]}(N^{2}_{k}t_{2})-\bar{\Theta}^{(2),[N^{2}_{k}]}(N_{k}^{2}t_{2}-N_{k}t)\right|$
$\displaystyle\qquad\qquad+\left|{\Theta}_{y_{2}}^{(2),[N^{2}_{k}]}(N_{k}^{2}t_{2})-{\Theta_{y_{2}}}^{(2),[N^{2}_{k}]}(N^{2}_{k}t_{2}-Nt)\right|\Bigg{]}=\delta_{0}.$

Then, for the $1$-block around $0$,

(11.102)
$\displaystyle\lim_{k\to\infty}\mathcal{L}\left[{\bf\Theta}^{\mathrm{aux},(1),[N_{k}^{2}]}(N_{k}^{2}t_{2})\right]$
$\displaystyle=\int_{[0,1]^{2}}\int_{[0,1]}\Gamma^{\mathrm{aux},(1)}_{u,y_{2,1}}\,P^{(u,v)}({\rm
d}y_{2,1})\,P_{t_{2}}({\rm d}u,{\rm d}v),$

where $\Gamma^{\mathrm{aux},(1)}_{u,y_{2,1}}$ is the equilibrium measure of
(11.28) with $\theta$ replaced by $u$, and

(11.103)
$\lim_{k\to\infty}\mathcal{L}\left[\left({\bf\Theta}^{\mathrm{aux},(1),[N_{k}^{2}]}(N_{k}^{2}t_{2}+N_{k}t_{1})\right)_{t_{1}>0}\right]=\mathcal{L}\left[(z_{1}^{\mathrm{aux}}(t_{1}))_{t_{1}>0})\right],$

where $(z_{1}^{\mathrm{aux}}(t_{1}))_{t_{1}>0}$ is the process evolving
according to (11.28) with $\theta$ replaced by the random variable
$\bar{\Theta}^{(2)}(t_{2})$ and with initial measure
$\int_{[0,1]^{2}}\int_{[0,1]}\Gamma^{\mathrm{aux},(1)}_{u,y_{2,1}}\,P^{(u,v)}({\rm
d}y_{2,1})\,P_{t_{2}}({\rm d}u,{\rm d}v)$, and $y_{2,1}$ is a random variable.

Note that by tightness of the $2$-blocks and the assumptions in Proposition
11.1, we can always find a subsequence $(N_{k})_{k\in\mathbb{N}}$ such that
(11.101) holds and also (11.80) holds. To prepare for the proof of Proposition
11.13, we prove four lemmas: Lemma 11.14 shows that the limiting $1$-block
system has a unique equilibrium, Lemma 11.16 implies convergence of the active
$2$-block estimator and the combined $2$-block estimator, Lemma 11.17 gives a
regularity property for the $2$-block estimator, and Lemma 11.18 shows the
limiting evolution of the auxiliary $1$-block estimator process. Lemma 11.20
proves equation (11.102). After we derive these lemmas we prove Proposition
11.13.

######  Lemma 11.14.

[1-block equilibrium] For any initial distribution $\mu\in([0,1]^{3})$, the
process $(z_{1}^{\mathrm{aux}}(t_{1}))_{t_{1}>0}$ evolving according to
(11.28) is well defined and converges to a unique equilibrium measure

(11.104)
$\lim_{t_{1}\to\infty}\mathcal{L}[z^{\mathrm{aux}}_{1}(t_{1})]=\Gamma_{\theta,y_{2,1}}^{\mathrm{aux},(1)}.$

###### Proof.

By [YW71], the SSDE in (11.28) has a unique strong solution. By a similar
argument as in the proof of Lemma 10.7, the SSDE in (11.28) converges to a
unique equilibrium measure $\Gamma^{\mathrm{aux},(1)}_{\theta,y_{2,1}}$. ∎

######  Remark 11.15.

[Equilibrium measure] Note that Lemma 11.14 still holds when we allow $\theta$
and $y_{2,1}$ to be the random variables $\bar{\Theta}(t_{2})$ and $y_{2,1}$.
Assuming (11.101), we can derive the distributions of $\bar{\Theta}(t_{2})$
and $y_{2,1}$, and we can write the equilibrium as
$\int_{[0,1]^{2}}\int_{[0,1]}\Gamma^{\mathrm{aux},(1)}_{u,y_{2,1}}\,P^{(u,v)}({\rm
d}y_{2,1})\,P_{t_{2}}({\rm d}u,{\rm d}v)$. In what follows we abbreviate

(11.105)
$\Gamma^{(1)}_{\bar{\Theta}(t_{2}),y_{2,1,i}}=\int_{[0,1]^{2}}\int_{[0,1]}\Gamma^{\mathrm{aux},(1)}_{u,y_{2,1,i}}\,P^{(u,v)}({\rm
d}y_{2,1,i})\,P_{t_{2}}({\rm d}u,{\rm d}v).$

$\blacksquare$

######  Lemma 11.16.

[$2$-block averages] Define

(11.106)
$\displaystyle\Delta^{(2),[N^{2}]}_{\Sigma}(Nt_{1})=\frac{\Theta^{(2),[N^{2}]}_{x}(Nt_{1})+K_{0}\Theta^{(2),[N^{2}]}_{y_{0}}(Nt_{1})}{1+K_{0}}-\Theta^{(2),[N^{2}]}_{y_{1}}(Nt_{1}).$

Then

(11.107)
$\displaystyle\mathbb{E}\left[\left|\Delta^{(2),[N^{2}]}_{\Sigma}(Nt_{1})\right|\right]$
$\displaystyle\leq\sqrt{\mathbb{E}\left[\left(\Delta^{(2),[N^{2}]}_{\Sigma}(0)\right)^{2}\right]}{\rm
e}^{-e_{1}\left(\frac{1+K_{0}+K_{1}}{1+K_{0}}\right)t_{1}}$
$\displaystyle+\sqrt{\int_{0}^{t_{1}}{\rm
d}s\,2e_{1}\left(\frac{1+K_{0}+K_{1}}{1+K_{0}}\right){\rm
e}^{-2e_{1}\left(\frac{1+K_{0}+K_{1}}{1+K_{0}}\right)(t_{1}-s)}\mathbb{E}\left[\left|\bar{\Theta}^{(1),[N^{2}]}(Ns)-\Theta^{(1),[N^{2}]}_{x}(Ns)\right|\right]}$
$\displaystyle+\sqrt{\frac{1}{e_{1}}\left[\frac{K_{2}e_{2}}{N(1+K_{0}+K_{1})}+\frac{||g||}{2N(1+K_{0}+K_{1})}\right]}.$

###### Proof.

For the two-level mean-field system we have the following SSDE for the
$2$-block averages:

(11.108) $\displaystyle{\rm d}\Theta^{(2),[N^{2}]}_{x}(Nt_{1})$
$\displaystyle=\sqrt{\frac{1}{N^{3}}\sum_{i\in[N^{2}]}g(x^{[N^{2}]}_{i}(Nt_{1}))}\,{\rm
d}\tilde{w}(t_{1})$
$\displaystyle\qquad\qquad+NK_{0}e_{0}\,\left[\Theta^{(2),[N^{2}]}_{y_{0}}(Nt_{1})-\Theta^{(2),[N^{2}]}_{x}(Nt_{1})\right]\,{\rm
d}t_{1}$
$\displaystyle\qquad\qquad+K_{1}e_{1}\,\left[\Theta^{(2),[N^{2}]}_{y_{1}}(Nt_{1})-\Theta^{(2),[N^{2}]}_{x}(Nt_{1})\right]\,{\rm
d}t_{1}$
$\displaystyle\qquad\qquad+\frac{K_{2}e_{2}}{N}\,\left[\Theta^{(2),[N^{2}]}_{y_{2}}(Nt_{1})-\Theta^{(2),[N^{2}]}_{x}(Nt_{1})\right]\,{\rm
d}t_{1},$ $\displaystyle{\rm d}\Theta^{(2),[N^{2}]}_{y_{0}}(Nt_{1})$
$\displaystyle=Ne_{0}\,\left[\Theta^{(2),[N^{2}]}_{x}(Nt_{1})-\Theta^{(2),[N^{2}]}_{y_{0}}(Nt_{1})\right]\,{\rm
d}t_{1},$ $\displaystyle{\rm d}\Theta^{(2),[N^{2}]}_{y_{1}}(Nt_{1})$
$\displaystyle=e_{1}\left[\Theta^{(2),[N^{2}]}_{x}(Nt_{1})-\Theta^{(2),[N^{2}]}_{y_{1}}(Nt_{1})\right]\,{\rm
d}t_{1},$ $\displaystyle{\rm d}\Theta^{(2),[N^{2}]}_{y_{2}}(Nt_{1})$
$\displaystyle=\frac{e_{2}}{N}\left[\Theta^{(2),[N^{2}]}_{x}(Nt_{1})-\Theta^{(2),[N^{2}]}_{y_{2}}(Nt_{1})\right]\,{\rm
d}t_{1}.$

Therefore

(11.109) $\displaystyle{\rm
d}\left(\Delta^{(2),[N^{2}]}_{\Sigma}(Nt_{1})\right)^{2}=2\Delta^{(2),[N^{2}]}_{\Sigma}(Nt_{1})\,{\rm
d}\Delta^{(2),[N^{2}]}_{\Sigma}(Nt_{1})+{\rm
d}\left<\Delta^{(2),[N^{2}]}_{\Sigma}(Nt_{1})\right>$
$\displaystyle=2\Delta^{(2),[N^{2}]}_{\Sigma}(Nt_{1})\,\frac{1}{1+K_{0}}\sqrt{\frac{1}{N^{3}}\sum_{i\in[N^{2}]}g(x^{[N^{2}]}_{i}(Nt_{1}))}\,{\rm
d}\tilde{w}(t_{1})$
$\displaystyle\qquad+2\Delta^{(2),[N^{2}]}_{\Sigma}(Nt_{1})\,\frac{K_{1}e_{1}}{(1+K_{0})}\,\left[\Theta^{(2),[N^{2}]}_{y_{1}}(Nt_{1})-\Theta^{(2),[N^{2}]}_{x}(Nt_{1})\right]\,{\rm
d}t_{1},$
$\displaystyle\qquad+2\Delta^{(2),[N^{2}]}_{\Sigma}(Nt_{1})\,\frac{K_{2}e_{2}}{N(1+K_{0})}\,\left[\Theta^{(2),[N^{2}]}_{y_{2}}(Nt_{1})-\Theta^{(2),[N^{2}]}_{x}(Nt_{1})\right]\,{\rm
d}t_{1},$
$\displaystyle\qquad-2\Delta^{(2),[N^{2}]}_{\Sigma}(Nt_{1})\,e_{1}\left[\Theta^{(2),[N^{2}]}_{x}(Nt_{1})-\Theta^{(2),[N^{2}]}_{y_{1}}(Nt_{1})\right]\,{\rm
d}t_{1}$
$\displaystyle\qquad+\frac{1}{(1+K_{0})^{2}}\frac{1}{N^{3}}\sum_{i\in[N^{2}]}g(x^{[N^{2}]}_{i}(Nt_{1}))\,{\rm
d}t_{1}.$

Hence

(11.110) $\displaystyle\frac{{\rm d}}{{\rm
d}t}\mathbb{E}\left[\left(\Delta^{(2),[N^{2}]}_{\Sigma}(Nt_{1})\right)^{2}\right]$
$\displaystyle=-2e_{1}\left(\frac{1+K_{0}+K_{1}}{1+K_{0}}\right)\mathbb{E}\left[\left(\Delta^{(2),[N^{2}]}_{\Sigma}(Nt_{1})\right)^{2}\right]$
$\displaystyle\qquad+2e_{1}\left(\frac{1+K_{0}+K_{1}}{1+K_{0}}\right)$
$\displaystyle\qquad\qquad\times\mathbb{E}\left[\Delta^{(2),[N^{2}]}_{\Sigma}(Nt_{1})\left(\frac{\Theta^{(2),[N^{2}]}_{x}(Nt_{1})+K_{0}\Theta^{(2),[N^{2}]}_{y_{0}}(Nt_{1})}{1+K_{0}}-\Theta^{(2),[N^{2}]}_{x}(Nt_{1})\right)\right]\,$
$\displaystyle\qquad+\frac{K_{2}e_{2}}{N(1+K_{0})}\,2\mathbb{E}\left[\Delta^{(2),[N^{2}]}_{\Sigma}(Nt_{1})\,\left[\Theta^{(2),[N^{2}]}_{y_{2}}(Nt_{1})-\Theta^{(2),[N^{2}]}_{x}(Nt_{1})\right]\right]\,$
$\displaystyle\qquad+\frac{1}{(1+K_{0})^{2}}\mathbb{E}\left[\frac{1}{N^{3}}\sum_{i\in[N^{2}]}g(x^{[N^{2}]}_{i}(Nt_{1}))\right],$

and therefore

(11.111)
$\displaystyle\mathbb{E}\left[\left(\Delta^{(2),[N^{2}]}_{\Sigma}(Nt_{1})\right)^{2}\right]$
$\displaystyle=\mathbb{E}\left[\left(\Delta^{(2),[N^{2}]}_{\Sigma}(0)\right)^{2}\right]{\rm
e}^{-2e_{1}\left(\frac{1+K_{0}+K_{1}}{1+K_{0}}\right)t_{1}}+\int_{0}^{t_{1}}{\rm
d}s\,{\rm
e}^{-2e_{1}\left(\frac{1+K_{0}+K_{1}}{1+K_{0}}\right)(t_{1}-s)}h^{[N]}(s),$

where

(11.112) $\displaystyle
h^{[N]}(s)=2e_{1}\left(\frac{1+K_{0}+K_{1}}{1+K_{0}}\right)$
$\displaystyle\qquad\qquad\times\mathbb{E}\left[\Delta^{(2),[N^{2}]}_{\Sigma}(Ns)\,\left(\frac{\Theta^{(2),[N^{2}]}_{x}(Ns)+K_{0}\Theta^{(2),[N^{2}]}_{y_{0}}(Ns)}{1+K_{0}}-\Theta^{(2),[N^{2}]}_{x}(Ns)\right)\right]$
$\displaystyle\qquad\qquad+\frac{2K_{2}e_{2}}{N(1+K_{0})}\,\mathbb{E}\left[\Delta^{(2),[N^{2}]}_{\Sigma}(Ns)\,\left[\Theta^{(2),[N^{2}]}_{y_{2}}(Ns)-\Theta^{(2),[N^{2}]}_{x}(Ns)\right]\right]$
$\displaystyle\qquad\qquad+\frac{1}{(1+K_{0})^{2}}\mathbb{E}\left[\frac{1}{N^{3}}\sum_{i\in[N^{2}]}g(x^{[N^{2}]}_{i}(Ns))\right].$

Therefore

(11.113)
$\displaystyle\mathbb{E}\left[\left|\Delta^{(2),[N^{2}]}_{\Sigma}(Nt_{1})\right|\right]$
$\displaystyle\leq\sqrt{\mathbb{E}\left[\left(\Delta^{(2),[N^{2}]}_{\Sigma}(0)\right)^{2}\right]}{\rm
e}^{-e_{1}\left(\frac{1+K_{0}+K_{1}}{1+K_{0}}\right)t_{1}}$
$\displaystyle+\sqrt{\int_{0}^{t_{1}}{\rm
d}s\,2e_{1}\left(\frac{1+K_{0}+K_{1}}{1+K_{0}}\right){\rm
e}^{-2e_{1}\left(\frac{1+K_{0}+K_{1}}{1+K_{0}}\right)(t_{1}-s)}\mathbb{E}\left[\left|\bar{\Theta}^{(1),[N^{2}]}(Ns)-\Theta^{(1),[N^{2}]}_{x}(Ns)\right|\right]}$
$\displaystyle+\sqrt{\frac{1}{e_{1}}\left[\frac{K_{2}e_{2}}{N(1+K_{0}+K_{1})}+\frac{||g||}{2N(1+K_{0}+K_{1})}\right]}.$

∎

Let $\mu_{N_{k}}$ be the measure obtained by periodic continuation of the
configuration

(11.114) $\displaystyle Z^{[N^{2}]}(N_{k}^{2}t_{2}).$

Since the state space $([0,1]\times[0,1]^{3})^{\mathbb{N}_{0}}$ is compact, we
can pass to a further subsequence, to obtain

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

######  Lemma 11.17.

[Regularity for 2-block estimator] Let $\mu$ and $\mu_{N}$ be as defined
above. Let $(x_{i},y_{1,i},y_{2,i})_{i\in\mathbb{N}_{0}}$ be distributed
according to $\mu$. Define the random variable

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

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

(11.117)
$\displaystyle\phi^{[N^{2}]}=(\phi^{[N^{2}]}_{1},\phi^{[N^{2}]}_{2}),$
$\displaystyle\phi_{1}^{[N^{2}]}=\frac{1}{N^{2}}\sum_{i\in[N^{2}]}\frac{x_{i}^{[N^{2}]}+K_{0}y_{i,0}^{[N^{2}]}+K_{1}y_{i,1}^{[N^{2}]}}{1+K_{0}+K_{1}},\qquad\phi_{2}=\lim_{N\to\infty}\frac{1}{N^{2}}\sum_{i\in[N^{2}]}y^{[N^{2}]}_{i,2}.$

Then

(11.118) $\lim_{N\to\infty}\mathcal{L}[\phi^{[N^{2}]}]=\mathcal{L}[\phi].$

###### Proof.

Use a similar argument as in the proof of Lemma 10.11. ∎

We will first determine the limiting evolution of
$({\bf\Theta}^{\mathrm{aux},(1),[N_{k}^{2}]}(N_{k}^{2}t_{2}+Nt_{1}))_{t_{1}>0}$.
To do so we consider all the $N_{k}$ $1$-blocks in $[N_{k}^{2}]$. After that
we show that

(11.119)
${}\lim_{k\to\infty}\mathcal{L}\left[\left({\bf\Theta}_{i}^{\mathrm{aux},(1),[N_{k}^{2}]}(N_{k}^{2}t_{2})\right)_{i\in[N_{k}]}\right]=\prod_{i\in\mathbb{N}_{0}}\Gamma^{(1)}_{\bar{\Theta}(t_{2}),y_{2,1,i}},$

The limiting $1$-block process for the auxiliary estimator process (recall
(11.28)) is given by

(11.120) $\displaystyle({\bf z}_{1}^{\mathrm{aux}}(t))_{t>0}=({\bf
x}^{\mathrm{aux}}_{1}(t),{\bf y}^{\mathrm{aux}}_{1,1}(t),{\bf
y}^{\mathrm{aux}}_{2,1}(t))_{t>0},$ $\displaystyle{\bf
z}_{1}^{\mathrm{aux}}(t)=({z}_{1,i}^{\mathrm{aux}}(t))_{i\in\mathbb{N}_{0}},\qquad{\bf
x}_{1}^{\mathrm{aux}}(t)=({x}_{1,i}^{\mathrm{aux}}(t))_{i\in\mathbb{N}_{0}},$
$\displaystyle{\bf
y}_{1,1}^{\mathrm{aux}}(t)=({y}_{1,1,i}^{\mathrm{aux}}(t))_{i\in\mathbb{N}_{0}}\qquad{\bf
y}_{2,1}^{\mathrm{aux}}(t)=({y}_{2,i}^{\mathrm{aux}}(t))_{i\in\mathbb{N}_{0}}$

and its components evolve according to

(11.121) $\displaystyle{\rm d}x^{\mathrm{aux}}_{1,i}(t)$
$\displaystyle=\frac{1}{1+K_{0}}\Bigg{[}c_{1}[\bar{\Theta}^{(2)}(t_{2})-x^{\mathrm{aux}}_{1,i}(t)]\,{\rm
d}t+\sqrt{(\mathcal{F}^{(1)}g)(x^{\mathrm{aux}}_{1,i}(t))}\,{\rm d}w(t)$
$\displaystyle\qquad+K_{1}e_{1}\,[y^{\mathrm{aux}}_{1,1,i}(t)-x^{\mathrm{aux}}_{1,i}(t)]\,{\rm
d}t\Bigg{]}\,,$ $\displaystyle{\rm d}y^{\mathrm{aux}}_{1,1,i}(t)$
$\displaystyle=e_{1}\,[x^{\mathrm{aux}}_{1,i}(t)-y^{\mathrm{aux}}_{1,1,i}(t)]\,{\rm
d}t,$ $\displaystyle y^{\mathrm{aux}}_{2,1,i}(t)$
$\displaystyle=y_{2,1,i},\qquad i\in\mathbb{N}_{0},$

where

(11.122)
${}\bar{\Theta}^{(2)}(t_{2})=\lim_{N\to\infty}\sum_{i\in[N^{2}]}\frac{x_{i}^{[N^{2}]}+K_{0}y_{i,0}^{[N^{2}]}+K_{1}y_{i,1}^{[N^{2}]}}{1+K_{0}+K_{1}}\text{
in }L_{2}(\mu).$

Let $\mu^{(1)}_{N_{k}}$ be the law obtained by periodic continuation of
$({\bf\Theta}_{i}^{\mathrm{aux},(1),[N_{k}^{2}]}(N_{k}^{2}t_{2}))_{i\in[N_{k}]}$,
and let $\mu^{(1)}=\lim_{k\to\infty}\mu^{(1)}_{N_{k}}$ be any weak limit point
of the sequence $(\mu^{(1)}_{N_{k}})_{k\in\mathbb{N}}$.

######  Lemma 11.18.

[Limiting evolution of auxiliary $1$-block estimator] Let $\mathcal{L}[({\bf
z}_{1}^{\mathrm{aux}}(0))]=\mu^{(1)}$. Then the following hold.

  1. 1.

For all $t_{1}>0$ and $i\in[N_{k}]$,

(11.123) $\displaystyle\lim_{k\to\infty}$
$\displaystyle\mathbb{E}\Bigg{[}(1+K_{0})\left(x_{1,i}^{\mathrm{aux}}(t_{1})-\bar{\Theta}_{i}^{\mathrm{aux},(1),[N_{k}^{2}]}(N_{k}^{2}t_{2}+N_{k}t_{1})\right)^{2}$
$\displaystyle\qquad+K_{1}\left(y_{1,1,i}(t_{1})-\Theta_{y_{1},i}^{\mathrm{aux},(1),[N_{k}^{2}]}(N_{k}^{2}t_{2}+N_{k}t_{1})\right)^{2}$
$\displaystyle\qquad+K_{2}\left(y_{2,1,i}^{\mathrm{aux},(1),[N_{k}^{2}]}(t_{1})-\Theta_{y_{2},i}^{\mathrm{aux},(1),[N_{k}]}(N_{k}^{2}t_{2}+N_{k}t_{1})\right)^{2}\Bigg{]}=0.$

  2. 2.

For all $t_{2}>0$,

(11.124)
$\lim_{k\to\infty}\mathcal{L}\left[({\bf\Theta}^{\mathrm{aux},(1),[N_{k}^{2}]}(N_{k}^{2}t_{2}+Nt_{1}))_{t_{1}>0}\right]=\mathcal{L}[({\bf
z}_{1}^{\mathrm{aux}}(t_{1}))_{t_{1}>0}].$

###### Proof.

Abbreviate

(11.125) $\displaystyle\Delta_{i}^{(1),[N_{k}^{2}]}(N_{k}t_{1})$
$\displaystyle=x_{1,i}^{\mathrm{aux}}(t_{1})-\Theta_{i}^{\mathrm{aux},(1),[N_{k}^{2}]}(N_{k}^{2}t_{2}+N_{k}t_{1}),$
$\displaystyle\delta_{y_{1},i}^{(1),[N_{k}^{2}]}(N_{k}t_{1})$
$\displaystyle=y_{1,1,i}(t_{1})-\Theta_{y_{1},i}^{\mathrm{aux},(1),[N_{k}^{2}]}(N_{k}^{2}t_{2}+N_{k}t_{1}),$
$\displaystyle\delta_{y_{2},i}^{(1),[N_{k}^{2}]}(N_{k}t_{1})$
$\displaystyle=y_{2,1,i}^{\mathrm{aux}}(t_{1})-\Theta_{y_{2},i}^{\mathrm{aux},(1),[N_{k}^{2}]}(N_{k}^{2}t_{2}+N_{k}t_{1}).$

Extending $({\bf\Theta}^{\mathrm{aux},(1),[N_{k}^{2}]}(N_{k}t_{1}))_{t_{1}>0}$
as a process on $\mathbb{N}_{0}$ by periodic continuation, we can construct
$({\bf z}_{1}^{\mathrm{aux}}(t_{1}))_{t_{1}>0}$ and
$({\bf\Theta}^{\mathrm{aux},(1)}(N_{k}t_{1}))_{t_{1}>0}$ on one probability
space such that

(11.126)
$\lim_{k\to\infty}{\bf\Theta}^{\mathrm{aux},(1),[N_{k}^{2}]}(N_{k}^{2}t_{2})={\bf
z}_{1}^{\mathrm{aux},(1)}(0)\quad a.s.$

We couple the processes $({\bf z}_{1}^{\mathrm{aux}}(t_{1}))_{t_{1}>0}$ and
$({\bf\Theta}^{\mathrm{aux},(1),[N_{k}^{2}]}(N_{k}^{2}t_{2}+N_{k}t_{1}))_{t_{1}>0}$
by using the same Brownian motions for both processes. By Itô-calculus we
obtain for the coupled process (recall (11.73))

(11.127)
$\displaystyle\mathbb{E}\left[(1+K_{0})\left(\Delta^{(1),[N_{k}^{2}]}_{i}(N_{k}t_{1})\right)^{2}+K_{1}\left(\delta^{(1),[N_{k}^{2}]}_{y_{1},i}(N_{k}t_{1})\right)^{2}+K_{2}\left(\delta^{(1),[N_{k}^{2}]}_{y_{2},i}(N_{k}t_{1})\right)^{2}\right]$
$\displaystyle=$
$\displaystyle\mathbb{E}\left[(1+K_{0})\left(\Delta^{(1),[N_{k}^{2}]}_{i}(0)\right)^{2}+K_{1}\left(\delta^{(1),[N_{k}^{2}]}_{y_{1},i}(0)\right)^{2}+K_{2}\left(\delta^{(1),[N_{k}^{2}]}_{y_{2},i}(0)\right)^{2}\right]$
$\displaystyle-2c_{1}\int_{0}^{t_{1}}\mathbb{E}\left[\left(\Delta^{(1),[N_{k}^{2}]}_{i}(N_{k}s)\right)^{2}\right]{\rm
d}s$
$\displaystyle-2K_{1}e_{1}\int_{0}^{t_{1}}\mathbb{E}\left[\left(\Delta^{(1),[N_{k}^{2}]}_{i}(N_{k}s)-\delta^{(1),[N_{k}^{2}]}_{y_{1},i}(N_{k}s)\right)^{2}\right]{\rm
d}s$
$\displaystyle+2c_{1}\int_{0}^{t_{1}}\mathbb{E}\left[\Delta^{(1),[N_{k}^{2}]}_{i}(N_{k}s)\left(\Theta^{(2)}(t_{2})-\frac{1}{N_{k}^{2}}\sum_{j\in[N_{k}^{2}]}x^{[N_{k}^{2}]}_{j}(N^{2}_{k}t_{2}+N_{k}s)\right)\right]{\rm
d}s$
$\displaystyle+\left(K_{1}e_{1}+c_{1}\right)\int_{0}^{t_{1}}\mathbb{E}\left[\Delta^{(1),[N_{k}^{2}]}_{i}(N_{k}s)\right.$
$\displaystyle\qquad\qquad\qquad\left.\times\left[\frac{1}{N_{k}}\sum_{j\in[N_{k}]_{i}}x^{[N_{k}^{2}]}_{j}(N_{k}^{2}t_{2}+N_{k}s)-\bar{\Theta}^{\mathrm{aux},(1),[N_{k}^{2}]}_{i}(N_{k}^{2}t_{2}+N_{k}s)\right]\right]{\rm
d}s$
$\displaystyle+2K_{1}e_{1}\int_{0}^{t_{1}}\mathbb{E}\left[\delta^{(1),[N_{k}^{2}]}_{y_{1},i}(N_{k}s)\right.$
$\displaystyle\qquad\qquad\qquad\times\left.\left[\bar{\Theta}_{i}^{\mathrm{aux},(1),[N_{k}^{2}]}(N^{2}_{k}t_{2}+N_{k}s)-\frac{1}{N_{k}}\sum_{j\in[N_{k}]_{i}}x^{[N_{k}^{2}]}_{j}(N^{2}_{k}t_{2}+N_{k}s)\right]\right]{\rm
d}s$
$\displaystyle+2\frac{K_{2}e_{2}}{N_{k}}\int_{0}^{t_{1}}\mathbb{E}\Bigg{[}\left[\delta^{(1),[N_{k}^{2}]}_{y_{1},i}(N_{k}s)-\Delta^{(1),[N_{k}^{2}]}_{i}(N^{2}_{k}t_{2}+N_{k}s)\right]$
$\displaystyle\qquad\qquad\qquad\times\left[\frac{1}{N_{k}}\sum_{j\in[N_{k}]_{i}}x^{[N_{k}^{2}]}_{j}(N^{2}_{k}s)-\Theta^{\mathrm{aux},(1),[N_{k}^{2}]}_{y_{2},i}(N^{2}_{k}t_{2}+Nt_{1})\right]\Bigg{]}{\rm
d}s$
$\displaystyle+(1+K_{0})^{2}\int_{0}^{t_{1}}\mathbb{E}\left[\left(\sqrt{\left(\mathcal{F}g\right)(x_{1}^{\mathrm{aux}}(s))}-\sqrt{\frac{1}{N}\sum_{i\in[N]}g(x^{[N_{k}^{2}]}_{i}(N^{2}_{k}t_{2}+N_{k}s))}\right)^{2}\right]{\rm
d}s.$

Note that $|\Delta^{(1),[N_{k}^{2}]}_{i}|\leq 1$ and
$|\delta^{(1),[N_{k}^{2}]}_{y_{1},i}|\leq 1$. Note that the first term tends
to $0$ by (11.126). We show by dominated convergence that also all other
positive terms in the right-hand side of (11.127) tend to $0$ as $k\to\infty$.

For the third term, we can estimate

(11.128)
$\displaystyle\lim_{k\to\infty}\mathbb{E}\Bigg{[}\frac{c_{1}}{1+K_{0}}\Bigg{|}\frac{1}{N_{k}^{2}}\sum_{i\in[N_{k}^{2}]}x^{[N_{k}^{2}]}_{j}(N_{k}^{2}t_{2}+N_{k}s)-\bar{\Theta}^{(2)}(t_{2})\Bigg{|}\Bigg{]}$
$\displaystyle\leq\lim_{k\to\infty}\mathbb{E}\Bigg{[}\frac{c_{1}}{1+K_{0}}\Bigg{|}\Theta^{(2),[N_{k}^{2}]}_{x}(N_{k}^{2}t_{2}+N_{k}s)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad-\frac{\Theta^{(2),[N_{k}^{2}]}_{x}(N_{k}^{2}t_{2}+N_{k}s)+K_{0}\Theta^{(2),[N_{k}^{2}]}_{y_{0}}(N_{k}^{2}t_{2}+N_{k}s)}{1+K_{0}}\Bigg{|}\Bigg{]}$
$\displaystyle\quad+\mathbb{E}\Bigg{[}\frac{c_{1}}{1+K_{0}}\Bigg{|}\frac{\Theta^{(2),[N_{k}^{2}]}_{x}(N_{k}^{2}t_{2}+N_{k}s)+K_{0}\Theta^{(2),[N_{k}^{2}]}_{y_{0}}(N_{k}^{2}t_{2}+N_{k}s)}{1+K_{0}}$
$\displaystyle\qquad\,-\frac{\Theta^{(2),[N_{k}^{2}]}_{x}(N_{k}^{2}t_{2}+N_{k}s)+K_{0}\Theta^{(2),[N_{k}^{2}]}_{y_{0}}(N_{k}^{2}t_{2}+N_{k}s)+K_{1}\Theta^{(2),[N_{k}^{2}]}_{y_{1}}(N_{k}^{2}t_{2}+N_{k}s)}{1+K_{0}+K_{1}}\Bigg{|}\Bigg{]}$
$\displaystyle\quad+\mathbb{E}\Bigg{[}\frac{c_{1}}{1+K_{0}}\Bigg{|}\frac{\Theta^{(2),[N_{k}^{2}]}_{x}(N_{k}^{2}t_{2}+N_{k}s)+K_{0}\Theta^{(2),[N_{k}^{2}]}_{y_{0}}(N_{k}^{2}t_{2}+N_{k}s)+K_{1}\Theta^{(2),[N_{k}^{2}]}_{y_{1}}(N_{k}^{2}t_{2}+N_{k}s)}{1+K_{0}+K_{1}}$
$\displaystyle\qquad\qquad\qquad\qquad-\bar{\Theta}^{(2)}(t_{2})\Bigg{|}\Bigg{]}.$

The first term in (11.128) tends to zero by Lemma 11.9, the second term tends
to zero by Lemma 11.16, while the third term tends to zero by Lemma 11.17 and
Lemma 11.5, which is the third assumption in (11.101). Hence the third term in
(11.127) tends to zero by dominated convergence as $k\to\infty$.

The fourth and fifth term in (11.127) tend to zero by Lemma 11.9 and dominated
convergence. The sixth term in (11.127) tends to zero because the integral is
bounded by $t_{1}$ and there is a factor $\frac{1}{N_{k}}$ in front. To see
that the last term in the right-hand side in (11.127) tends to zero, recall
that the subsequence $N_{k}$ is chosen such that

(11.129)
${}\lim_{k\to\infty}\mathcal{L}\left[({\bf\Theta}^{\mathrm{aux},(1),[N_{k}^{2}]}(N_{k}^{2}t_{2}+Nt_{1}))_{t_{1}>0}\right]$

exists. Note that

(11.130)
$\displaystyle\mathbb{E}\left[\left(\sqrt{(\mathcal{F}g)(x_{1}^{\mathrm{aux}}(s))}-\sqrt{\frac{1}{N}\sum_{i\in[N]}g(x^{[N_{k}^{2}]}_{i}(N^{2}_{k}t_{2}+N_{k}s))}\right)^{2}\right]$
$\displaystyle\leq\mathbb{E}\left[\left|(\mathcal{F}g)(x_{1}^{\mathrm{aux}}(s))-\frac{1}{N}\sum_{i\in[N]}g(x^{[N_{k}^{2}]}_{i}(N^{2}_{k}t_{2}+N_{k}s))\right|\right],$

and hence we can apply a similar reasoning as in (9.97) to see that (11.130)
tends to zero as $k\to\infty$. Therefore we obtain

(11.131)
$\displaystyle\lim_{k\to\infty}\mathbb{E}\left[(1+K_{0})(\Delta^{(1),[N_{k}^{2}]}_{i}(N_{k}t_{1}))^{2}+K_{1}(\delta^{(1),[N_{k}^{2}]}_{y_{1},i}(N_{k}t_{1}))^{2}+K_{2}(\delta^{(1),[N_{k}^{2}]}_{y_{2},i}(N_{k}t_{1}))^{2}\right]$
$\displaystyle\qquad\qquad=0.$

To prove (11.124), note that (11.131) implies convergence of the finite-
dimensional distributions of
$({\bf\Theta}^{\mathrm{aux},(1),[N_{k}^{2}]}(N_{k}^{2}t_{2}+N_{k}t_{1}))_{t_{1}>0}$
by a similar argument as given below (9.36). By Lemma 11.6 we see that the
laws of the processes

(11.132)
${}\left(\mathcal{L}\left[({\bf\Theta}^{\mathrm{aux},(1),[N_{k}^{2}]}(N_{k}^{2}t_{2}+N_{k}t_{1}))_{t_{1}>0}\right]\right)_{k\in\mathbb{N}_{0}}$

are tight. Therefore (11.124) indeed holds. ∎

######  Remark 11.19.

Note that in the proof of Lemma 10.15 we could have proceeded as in the proof
of Lemma 11.18, instead of using the criterion in [JM86, Theorem 3.3.1].
$\blacksquare$

######  Lemma 11.20.

[Proof of (11.102)] Under the assumptions in Proposition 11.13,

(11.133)
$\displaystyle\lim_{k\to\infty}\mathcal{L}\left[{\bf\Theta}^{\mathrm{aux},(1),[N_{k}^{2}]}(N_{k}^{2}t_{2})\right]$
$\displaystyle=\int_{[0,1]^{2}}\int_{[0,1]}\Gamma^{\mathrm{aux},(1)}_{u,y_{2,1}}\,P^{(u,v)}({\rm
d}y_{2,1})\,P_{t_{2}}({\rm d}u,{\rm d}v).$

###### Proof.

For ease of notation, we drop the subsequence notation in this proof. Let
$(t_{n})_{n\in\mathbb{N}_{0}}$ be any sequence satisfying
$\lim_{n\to\infty}t_{n}=\infty$ and $\lim_{n\to\infty}t_{n}/n=0$. For each
$t_{n}$, let $\mu^{(1)}_{N,t_{n}}$ be the law obtained by periodic
continuation of the configuration of
$({\bf\Theta}_{i}^{\mathrm{aux},(1),[N^{2}]}(N^{2}t_{2}-Nt_{n}))_{i\in[N]}$.
Recall that, since our state space is compact, the sequence
$(\mu^{(1)}_{N,t_{n}})_{N\in\mathbb{N}}$ is tight. Let $\mu^{(1)}_{t_{n}}$ be
any weak limit point of the sequence $(\mu^{(1)}_{N,t_{n}})_{N\in\mathbb{N}}$.

Let $\mathcal{L}[{\bf z}_{1}^{\mathrm{aux},n}(0)]$ be the law obtained by
periodic continuation of
${\bf\Theta}^{\mathrm{aux},(1),[N^{2}]}(N^{2}t_{2}-Nt_{n})$. By Lemma 11.6 we
know that the sequence

(11.134)
${}\left(\mathcal{L}\left[\left({\bf\Theta}_{i}^{\mathrm{aux},(1),[N^{2}]}(N^{2}t_{2}-Nt_{n}+Nt_{1})\right)_{t_{1}>0,i\in[N]}\right]\right)_{N\in\mathbb{N}}$

is tight and hence for each $t_{n}$ we can pass to a subsequence such that

(11.135)
$\lim_{k\to\infty}\mathcal{L}\left[\left({\bf\Theta}_{i}^{\mathrm{aux},(1),[N_{k}^{2}]}(N_{k}^{2}t_{2}-N_{k}t_{n}+N_{k}t_{1})\right)_{t_{1}>0,i\in[N]}\right]$

exists. By Lemmas 11.5–11.17, we obtain for all $t_{n}$ that

(11.136)
$\lim_{N\to\infty}\bar{\Theta}^{(2),[N_{k}^{2}]}(N_{k}^{2}t_{2}-Nt_{n})=\bar{\Theta}^{(2)}(t_{2})\text{
in probability}.$

Then, by (11.127) in the proof of Lemma 11.18, for fixed $t_{n}$ and all
$i\in[N]$,

(11.137) $\displaystyle\lim_{k\to\infty}$
$\displaystyle\mathbb{E}\Bigg{[}(1+K_{0})\left(x_{1,i}^{\mathrm{aux},n}(t_{n})-\bar{\Theta}_{i}^{\mathrm{aux},(1),[N_{k}^{2}]}(N_{k}^{2}t_{2}-N_{k}t_{n}+N_{k}t_{n})\right)^{2}$
$\displaystyle\qquad+K_{1}\left(y_{1,1,i}^{\mathrm{aux},n}(t_{n})-\Theta_{y_{1},i}^{\mathrm{aux},(1),[N_{k}^{2}]}(N_{k}^{2}t_{2}-N_{k}t_{n}+N_{k}t_{n})\right)^{2}$
$\displaystyle\qquad+K_{2}\left(y_{2,1,i}^{\mathrm{aux},n}(t_{n})-\Theta_{y_{2},i}^{\mathrm{aux},(1),[N_{k}^{2}]}(N_{k}^{2}t_{2}-N_{k}t_{n}+N_{k}t_{n})\right)^{2}\Bigg{]}=0.$

By contradiction we can argue that

(11.138) $\displaystyle\lim_{N\to\infty}$
$\displaystyle\mathbb{E}\Bigg{[}(1+K_{0})\left(x_{1,i}^{\mathrm{aux},n}(t_{n})-\bar{\Theta}_{i}^{\mathrm{aux},(1),[N^{2}]}(N^{2}t_{2}-Nt_{n}+Nt_{n})\right)^{2}$
$\displaystyle\qquad+K_{1}\left(y_{1,1,i}^{\mathrm{aux},n}(t_{n})-\Theta_{y_{1},i}^{\mathrm{aux},(1),[N^{2}]}(N^{2}t_{2}-Nt_{n}+Nt_{n})\right)^{2}$
$\displaystyle\qquad+K_{2}\left(y_{2,1,i}^{\mathrm{aux},n}(t_{n})-\Theta_{y_{2},i}^{\mathrm{aux},(1),[N^{2}]}(N^{2}t_{2}-Nt_{n}+Nt_{n})\right)^{2}\Bigg{]}=0.$

To see why, suppose that (11.138) does not hold. Then for any $\delta>0$ we
can construct a sequence $(N_{l})_{l>0}$ such that, for $l\in\mathbb{N}$,

(11.139)
$\displaystyle\mathbb{E}\Bigg{[}(1+K_{0})\left(x_{1,i}^{\mathrm{aux},n}(t_{n})-\bar{\Theta}_{i}^{\mathrm{aux},(1),[N_{l}^{2}]}(N_{l}^{2}t_{2}-N_{l}t_{n}+N_{l}t_{n})\right)^{2}$
$\displaystyle\qquad+K_{1}\left(y_{1,1,i}^{\mathrm{aux},n}(t_{n})-\Theta_{y_{1},i}^{\mathrm{aux},(1),[N_{l}^{2}]}(N_{l}^{2}t_{2}-N_{l}t_{n}+N_{l}t_{n})\right)^{2}$
$\displaystyle\qquad+K_{2}\left(y_{2,1,i}^{\mathrm{aux},n}(t_{n})-\Theta_{y_{2},i}^{\mathrm{aux},(1),[N_{l}^{2}]}(N_{l}^{2}t_{2}-N_{l}t_{n}+N_{l}t_{n})\right)^{2}\Bigg{]}>\delta.$

However, also the sequence

(11.140)
$\left(\mathcal{L}\left[\left({\bf\Theta}_{i}^{\mathrm{aux},(1),[N_{l}^{2}]}(N_{l}^{2}t_{2}-N_{l}t_{n}+N_{l}t_{1})\right)_{t_{1}>0,i\in[N]}\right]\right)_{l\in\mathbb{N}}$

is tight. Hence we can pass to a further subsequence
$(N_{\tilde{l}})_{\tilde{l}\in\mathbb{N}}$ for which (11.137) holds. But this
contradicts (11.139). We conclude that (11.138) indeed holds. Moreover the
argument holds for all $t_{n}$, so that (11.138) holds for all $t_{n}$.

Hence for every $t_{n}$ there exists a $N_{n}$ such that, for all $N\geq
N_{n}$,

(11.141)
$\displaystyle\mathbb{E}\Bigg{[}(1+K_{0})\left(x_{1,i}^{\mathrm{aux},n}(t_{n})-\bar{\Theta}_{i}^{\mathrm{aux},(1),[N_{k}^{2}]}(N^{2}t_{2})\right)^{2}$
$\displaystyle\qquad+K_{1}\left(y_{1,1,i}^{\mathrm{aux},n}(t_{n})-\Theta_{y_{1},i}^{\mathrm{aux},(1),[N_{k}^{2}]}(N^{2}t_{2})\right)^{2}$
$\displaystyle\qquad+K_{2}\left(y_{2,1,i}^{\mathrm{aux},n}(t_{n})-\Theta_{y_{2},i}^{\mathrm{aux},(1),[N_{k}^{2}]}(N^{2}t_{2})\right)^{2}\Bigg{]}<\frac{1}{n}.$

In particular, we may require that $N_{n}>N_{n-1}$. Setting $N=N_{n}$, we
obtain

(11.142) $\displaystyle\lim_{n\to\infty}$
$\displaystyle\mathbb{E}\Bigg{[}(1+K_{0})\left(x_{1,i}^{\mathrm{aux},n}(t_{n})-\bar{\Theta}_{i}^{\mathrm{aux},(1),[N^{2}_{n}]}(N_{n}^{2}t_{2})\right)^{2}$
$\displaystyle\qquad+K_{1}\left(y_{1,1,i}^{\mathrm{aux},n}(t_{n})-\Theta_{y_{1},i}^{\mathrm{aux},(1),[N_{n}^{2}]}(N_{n}^{2}t_{2})\right)^{2}$
$\displaystyle\qquad+K_{2}\left(y_{2,1,i}^{\mathrm{aux},n}(t_{n})-\Theta_{y_{2},i}^{\mathrm{aux},(1),[N_{n}^{2}]}(N_{n}^{2}t_{2})\right)^{2}\Bigg{]}=0.$

If we can prove that

(11.143)
$\lim_{n\to\infty}\mathcal{L}[z_{i}^{\mathrm{aux},n}(t_{n})]=\Gamma^{(1)}_{\bar{\Theta}(t_{2})},$

then we are done. To see why, note that, for all
$f\in\mathcal{C}_{b}([0,1]\times[0,1]^{2})$, $f$ Lipschitz continuous

(11.144)
$\displaystyle\left|\mathbb{E}[f({\bf\Theta}_{i}^{\mathrm{aux},(1),[N^{2}]}(N^{2}t_{2}))]-\mathbb{E}^{\Gamma^{(1)}_{\bar{\Theta}(t_{2})}}[f]\right|$
$\displaystyle\leq$
$\displaystyle\left|\mathbb{E}[f({\bf\Theta}_{i}^{\mathrm{aux},(1),[N^{2}]}(N^{2}t_{2}))-f({z_{1,i}}^{\mathrm{aux},n}(t_{n}))]\right|+\left|\mathbb{E}[f({z_{1,i}}^{\mathrm{aux},n}(t_{n}))]-\mathbb{E}^{\Gamma^{(1)}_{\bar{\Theta}(t_{2})}}[f]\right|.$

Therefore if (11.143) holds, then for all $\epsilon>0$ we can choose $\bar{n}$
such that, for all $n>\bar{n}$,

(11.145)
$\left|\mathbb{E}[f({z_{1,i}}^{\mathrm{aux},n}(t_{n}))]-\mathbb{E}^{\Gamma^{(1)}_{\bar{\Theta}(t_{2})}}[f]\right|<\frac{\epsilon}{2}.$

By (11.142) we can find a $\hat{n}>\bar{n}$ such that for all $n>\hat{n}$

(11.146)
$\left|\mathbb{E}[f({\bf\Theta}_{i}^{\mathrm{aux}}(N_{n}^{2}t_{2}))-f({z_{1,i}}^{\mathrm{aux},n}(t_{\hat{n}}))]\right|<\frac{\epsilon}{2}.$

Using (11.143) and the fact that the Lipschitz functions are dense in
$f\in\mathcal{C}_{b}([0,1]\times[0,1]^{2})$, we obtain (11.133).

###### Proof of (11.143).

We use that any two systems $({\bf z}^{\mathrm{aux},1}(t_{1}))_{t_{1}>0}$ and
$({\bf z}^{\mathrm{aux},2}(t_{1}))_{t_{1}>0}$ evolving according to (11.28),
and having the same $y_{2,1}$-components and the same $\bar{\Theta}(t_{2})$,
can be constructed on one probability space and can coupled by their Brownian
motions. We obtain, for a component $i\in\mathbb{N}_{0}$,

(11.147)
$\displaystyle\mathbb{E}[|x_{1,i}^{\mathrm{aux},1}(t_{n})-x_{1,i}^{\mathrm{aux},2}(t_{n})|+K_{1}|y_{1,1,i}^{\mathrm{aux},1}(t_{n})-y_{1,1,i}^{\mathrm{aux},2}(t_{n})|+K_{2}|y_{2,1,i}^{\mathrm{aux},1}(t_{n})-y_{2,1,i}^{\mathrm{aux},2}(t_{n})|]$
$\displaystyle=\mathbb{E}[|x_{1,i}^{\mathrm{aux},1}(0)-x_{1,i}^{\mathrm{aux},2}(0)|+K_{1}|y_{1,1,i}^{\mathrm{aux},1}(0)-y_{1,1,i}^{\mathrm{aux},2}(0)|+K_{2}|y_{2,1,i}^{\mathrm{aux},1}(0)-y_{2,1,i}^{\mathrm{aux},2}(0)|]$
$\displaystyle-c\int_{0}^{t_{n}}\mathbb{E}[|x_{1,i}^{\mathrm{aux},1}(s)-x_{1,i}^{\mathrm{aux},2}(s)|]{\rm
d}s$
$\displaystyle-2K_{1}e_{1}\int_{0}^{t_{n}}\mathbb{E}\big{[}[|x_{1,i}^{\mathrm{aux},1}(s)-x_{1,i}^{\mathrm{aux},2}(s)|+K_{1}|y_{1,1,i}^{\mathrm{aux},1}(s)-y_{1,1,i}^{\mathrm{aux},2}(s)|]$
$\displaystyle\qquad\qquad\qquad\qquad\times 1_{\\{{\rm
sign}(x_{1,i}^{\mathrm{aux},1}(s)-x_{1,i}^{\mathrm{aux},2}(s))\neq{\rm
sign}(y_{1,1,i}^{\mathrm{aux},1}(s)-y_{1,1,i}^{\mathrm{aux},2}(s))\\}}\big{]}{\rm
d}s.$

Therefore the difference between these two systems monotonically decreases.

Since the state space $[0,1]\times[0,1]^{2}$ is compact, the sequence of laws

(11.148) ${}(\mathcal{L}[{z}_{i}^{\mathrm{aux},n}(0)])_{n\in\mathbb{N}}$

is tight. Therefore we can find converging subsequences such that

(11.149) $\lim_{k\to\infty}\mathcal{L}[{z_{i}}^{\mathrm{aux},n_{k}}(0)]=\mu$

for some probability measure $\mu$ on $[0,1]\times[0,1]^{2}$.

Let $(z^{\mathrm{aux},0}(t_{1}))_{t_{1}>0}$ be the limiting system evolving
according to (11.28) and starting from initial distribution $\mu$. By
Skorohod’s theorem, we can construct the sequence of limiting systems
$((z^{\mathrm{aux},n_{k}}(t_{1}))_{t_{1}>0})_{k\in\mathbb{N}}$ and
$(z^{\mathrm{aux},0}(t_{1}))_{t_{1}>0}$ on one probability space such that

(11.150)
$\lim_{k\to\infty}{z}^{\mathrm{aux},n_{k}}(0)=z^{\mathrm{aux},0}(0)\quad a.s.$

Use the coupling of Brownian motions to obtain

(11.151)
$\displaystyle\mathbb{E}[|x_{1,i}^{\mathrm{aux},n_{k}}(t_{n_{k}})-x_{1,i}^{\mathrm{aux},0}(t_{n_{k}})|+K_{1}|y_{1,1,i}^{\mathrm{aux},n_{k}}(t_{n_{k}})-y_{1,1,i}^{\mathrm{aux},0}(t_{n_{k}})|$
$\displaystyle\qquad+K_{2}|y_{2,1,i}^{\mathrm{aux},n_{k}}(t_{n_{k}})-y_{2,1,i}^{\mathrm{aux},0}(t_{n_{k}})|]$
$\displaystyle=\mathbb{E}[|x_{1,i}^{\mathrm{aux},n_{k}}(0)-x_{1,i}^{\mathrm{aux},0}(0)|+K_{1}|y_{1,1,i}^{\mathrm{aux},n_{k}}(0)-y_{1,1,i}^{\mathrm{aux},0}(0)|+K_{2}|y_{2,1,i}^{\mathrm{aux},n_{k}}(0)-y_{2,1,i}^{\mathrm{aux},0}(0)|]$
$\displaystyle-c\int_{0}^{t_{n_{k}}}\mathbb{E}[|x_{1,i}^{\mathrm{aux},n_{k}}(s)-x_{1,i}^{\mathrm{aux},0}(s)|]{\rm
d}s$
$\displaystyle-2K_{1}e_{1}\int_{0}^{t_{n_{k}}}\mathbb{E}\big{[}[|x_{1,i}^{\mathrm{aux},n_{k}}(s)-x_{1,i}^{\mathrm{aux},0}(s)|+K_{1}|y_{1,1,i}^{\mathrm{aux},n_{k}}(s)-y_{1,1,i}^{\mathrm{aux},0}(s)|]$
$\displaystyle\qquad\qquad\qquad\qquad\times 1_{\\{{\rm
sign}(x_{1,i}^{\mathrm{aux},n_{k}}(s)-x_{1,i}^{\mathrm{aux},0}(s))\neq{\rm
sign}(y_{1,1,i}^{\mathrm{aux},n_{k}}(s)-y_{1,1,i}^{\mathrm{aux},0}(s))\\}}\big{]}{\rm
d}s.$

Taking the limit $k\to\infty$ on both sides of (11.151), we obtain

(11.152)
$\displaystyle\lim_{k\to\infty}\mathbb{E}[|x_{1,i}^{\mathrm{aux},n_{k}}(t_{n_{k}})-x_{1,i}^{\mathrm{aux},0}(t_{n_{k}})|+K_{1}|y_{1,1,i}^{\mathrm{aux},n_{k}}(t_{n_{k}})-y_{1,1,i}^{\mathrm{aux},0}(t_{n_{k}})|$
$\displaystyle\qquad+K_{2}|y_{2,1,i}^{\mathrm{aux},n_{k}}(t_{n_{k}})-y_{2,1,i}^{\mathrm{aux},0}(t_{n_{k}})|]=0.$

Note that $\lim_{n\to\infty}t_{n}=\infty$ implies that
$\lim_{k\to\infty}t_{n_{k}}=\infty$, so $z_{i}^{\mathrm{aux},0}$ is the
limiting system in (11.28) with $\theta$ replaced by the random variable
$\bar{\Theta}(t_{2})$ and $y_{2,1,i}$. Therefore we can condition on
$\bar{\Theta}(t_{2})$ and $y_{2,1,i}$, and use the assumption in (11.80), to
obtain

(11.153)
$\lim_{k\to\infty}\mathcal{L}\left[z_{i}^{\mathrm{aux},0}(t_{n_{k}})\right]=\int_{[0,1]^{2}}\int_{[0,1]}\Gamma^{\mathrm{aux},(1)}_{u,y_{2,1}}\,P^{(u,v)}({\rm
d}y_{2,1})\,P_{t_{2}}({\rm d}u,{\rm d}v).$

Hence we conclude that

(11.154)
$\lim_{k\to\infty}\mathcal{L}[z_{i}^{\mathrm{aux},n_{k}}(t_{n_{k}})]=\int_{[0,1]^{2}}\int_{[0,1]}\Gamma^{\mathrm{aux},(1)}_{u,y_{2,1}}\,P^{(u,v)}({\rm
d}y_{2,1})\,P_{t_{2}}({\rm d}u,{\rm d}v).$

Equation (11.154) holds for all subsequences along which the initial
distribution converges,

(11.155)
$\lim_{k\to\infty}{z}_{i}^{\mathrm{aux},n_{k}}(0)=z_{i}^{\mathrm{aux},0}(0)\quad
a.s.$

We will show that this implies (11.143).

Suppose that

(11.156)
$\lim_{n\to\infty}\mathcal{L}[z_{i}^{\mathrm{aux},n}(t_{n})]\neq\int_{[0,1]^{2}}\int_{[0,1]}\Gamma^{\mathrm{aux},(1)}_{u,y_{2,1}}\,P^{(u,v)}({\rm
d}y_{2,1})\,P_{t_{2}}({\rm d}u,{\rm d}v).$

Then there exist $f\in\mathcal{C}_{b}([0,1]\times[0,1]^{3})$ and $\delta>0$
such that for all $N\in\mathbb{N}$ there exists an $n\in\mathbb{N}$, $n>N$
such that

(11.157)
$\left|\mathbb{E}[f(z_{i}^{\mathrm{aux},n}(t_{n}))]-\mathbb{E}^{\Gamma^{(1)}_{\bar{\Theta}(t_{2})}}[f]\right|>\delta.$

Hence we can construct a subsequence
$(z_{i}^{\mathrm{aux},n_{k}}(t_{1}))_{t_{1}>0,\,k\in\mathbb{N}}$ such that
(11.157) holds for each $k\in\mathbb{N}$. However, also for this sequence
$(\mathcal{L}[z_{i}^{\mathrm{aux},n_{k}}(0)])_{k\in\mathbb{N}}$ is tight.
Passing to a possibly further subsequence of converging initial distributions,
we argue like before to obtain that along this subsequence

(11.158)
$\lim_{k\to\infty}\left|\mathbb{E}[f(z_{i}^{\mathrm{aux},n_{k}}(t_{n_{k}}))]-\mathbb{E}^{\Gamma^{(1)}_{\bar{\Theta}(t_{2})}}[f]\right|=0.$

This contradicts (11.157) and so (11.143) is indeed true. ∎

###### Proof of Proposition 11.13

###### Proof.

Lemma 11.20 implies (11.102). Therefore Lemma 11.18 implies (11.103). ∎

#####  11.3.7 Convergence of 2-block process

In this section we derive the limiting evolution of the effective $2$-block
process.

######  Lemma 11.21.

[Convergence of the 2-block averages] Assume that
$(N_{k})_{k\in\mathbb{N}}\subset\mathbb{N}$ is a subsequence satisfying

(11.159)
$\displaystyle\lim_{k\to\infty}\mathcal{L}\left[y_{2,1}^{[N_{k}^{2}]}(N_{k}t_{2})\Big{|}{\bf\Theta}^{(2),[N_{k}^{2}]}(N_{k}^{2}t_{2})\right]=P^{z_{2}(t_{2})},$
$\displaystyle\lim_{k\to\infty}\mathcal{L}\left[\left(Y_{1,0}^{[N_{k}^{2}]}(N^{2}_{k}t_{2}+N_{k}t_{1}),Y_{2,0}^{[N_{k}^{2}]}(N^{2}_{k}t_{2})\right)\Big{|}{\bf\Theta}^{\mathrm{aux},(1),[N_{k}^{2}]}(N^{2}_{k}t_{2}+N_{k}t_{1})\right]$
$\displaystyle\qquad\qquad=P^{{z}_{1}^{\mathrm{eff}}(t_{1})}.$

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

(11.160)
$\lim_{k\to\infty}\mathcal{L}\left[\left({\bf\Theta}^{\mathrm{eff},(2),[N_{k}^{2}]}(N_{k}^{2}t_{2})\right)_{t_{2}>0}\right]=\mathcal{L}\left[\left(z_{2}^{\mathrm{eff}}(t_{2})\right)_{t_{2}>0}\right],$

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

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

###### Proof.

Again we use [JM86, Theorem 3.3.1]. By a similar argument as used in the proof
of Lemma 10.15 we can show that

(11.162) $\lim_{t_{2}\downarrow
0}\mathcal{L}\left[{\bf\Theta}^{\mathrm{eff},(2),[N_{k}^{2}]}(N_{k}^{2}t_{2})\right]=\delta_{(\vartheta_{1},\theta_{y_{2}})}.$

Note that by steps 1-4 of the scheme in Section 11.2 we can choose the
subsequence $(N_{k})_{k\in\mathbb{N}}$ such that both (11.80) and (11.101)
hold. Since we already established the tightness of the $2$-block in Lemma
11.4, we are left to show that, for all $t_{2}>0$,

(11.163) $\displaystyle\lim_{N\to\infty}$
$\displaystyle\mathbb{E}\Big{[}\Big{|}G_{\dagger}^{(2),[N_{k}^{2}]}(f,{\bf\Theta}^{\mathrm{eff},(2),[N_{k}^{2}]}(N_{k}^{2}t_{2}),t_{2},\omega)$
$\displaystyle\qquad\qquad\qquad-G^{(2)}f\left({\bf\Theta}^{\mathrm{eff},(2),[N_{k}^{2}]}(N_{k}^{2}t_{2})\right)\Big{|}\Big{]}=0,$

where $G_{\dagger}^{(2),[N_{k}^{2}]}$ is the $\mathcal{D}$-semi-martingale
operator defined in (11.68), $G^{(2)}$ is the generator of the process
$(z_{2}^{\mathrm{eff}}(t_{2}))_{t_{2}>0}$ defined in (11.35), and both
generators work on a probability space driven by one set of Brownian motions.
Note that, for all $t_{2}>0$,

(11.164)
$\displaystyle\mathbb{E}\Big{[}\Big{|}G_{\dagger}^{(2),[N_{k}^{2}]}\left(f,\left(\bar{\Theta}^{(2),[N_{k}^{2}]}(N_{k}^{2}t_{2}),\Theta^{(2),[N_{k}^{2}]}_{y_{2}}(N_{k}^{2}t_{2})\right),t_{2},\omega\right)$
$\displaystyle\qquad\qquad-G^{(2)}f\left(\bar{\Theta}^{(2),[N_{k}^{2}]}(N_{k}^{2}t_{2}),\Theta^{(2),[N_{k}^{2}]}_{y_{2}}(N_{k}^{2}t_{2})\right)\Big{|}\Big{]}$
$\displaystyle\leq\frac{K_{2}e_{2}}{1+K_{0}+K_{1}}\mathbb{E}\left[\left|\bar{\Theta}^{(2),[N_{k}^{2}]}(N_{k}^{2}t_{2})-\Theta_{x}^{(2),[N_{k}^{2}]}(N_{k}^{2}t_{2},\omega)\right|\left|\,\frac{\partial
f}{\partial x}\right|\right]$
$\displaystyle\qquad+e_{2}\mathbb{E}\left[\left|\Theta_{x}^{(2),[N_{k}^{2}]}(N_{k}^{2}t_{2},\omega)-\bar{\Theta}^{(2),[N_{k}^{2}]}(N_{k}^{2}t_{2})\right|\left|\,\frac{\partial
f}{\partial y}\right|\right]$
$\displaystyle\qquad+\frac{1}{(1+K_{0}+K_{1})^{2}}$
$\displaystyle\qquad\qquad\times\mathbb{E}\left[\left|\frac{1}{N^{2}_{k}}\sum_{i\in[N_{k}^{2}]}g(x_{i}^{[N^{2}_{k}]}(N_{k}^{2}t_{2},\omega))-(\mathcal{F}^{(2)}g)(\bar{\Theta}^{(2),[N_{k}^{2}]}(N_{k}^{2}t_{2}))\right|\left|\,\frac{\partial^{2}f}{\partial
x^{2}}\right|\right].$

The first and second term on the right-hand side tend to $0$ as $k\to\infty$
by a similar argument as used in (11.128) and below. For the third let
$[N]_{i}$ denote the $1$-block that contains site $i$ and let
$(z^{\nu_{\bar{\Theta}_{i}^{(1)}}}(t))_{t>0}$ be the limiting single colony
system, with drift towards the random variable $\bar{\Theta}_{i}$ and starting
from the equilibrium measure $\nu_{\bar{\Theta}_{i}^{(1)}}$. We construct the
single colony system $Z^{[N^{2}_{k}]}(N^{2}_{k}t_{2}-L(N)+t)_{t\geq 0}$ and
the limiting system $(z^{\nu_{\bar{\Theta}_{i}^{(1)}}}(t))_{t>0}$ on one
probability space, such that by Skohorod’s theorem, we can assume that the
convergence is almost surely. Note that $\bar{\Theta}_{i}$ is the limiting one
block. Then we can write

(11.165)
$\displaystyle\mathbb{E}\left[\left|\frac{1}{N^{2}_{k}}\sum_{i\in[N_{k}^{2}]}g(x_{i}^{[N_{k}^{2}]}(N_{k}^{2}t_{2},\omega))-(\mathcal{F}^{(2)}g)(\bar{\Theta}^{(2),[N_{k}^{2}]}(N_{k}^{2}t_{2}))\right|\left|\frac{\partial^{2}f}{\partial
x^{2}}\right|\right]$
$\displaystyle\qquad\leq\frac{1}{N_{k}}\sum_{i\in[N]}\mathbb{E}\left[\left|\frac{1}{N_{k}}\sum_{j\in[N]_{i}}g(x_{j}^{[N_{k}^{2}]}(N_{k}^{2}t_{2},\omega))-\frac{1}{N_{k}}\sum_{j\in[N]_{i}}g(x_{j}^{\nu_{\bar{\Theta}_{i}^{(1)}}}(L(N))\right|\left|\frac{\partial^{2}f}{\partial
x^{2}}\right|\right]$
$\displaystyle\qquad+\frac{1}{N_{k}}\sum_{i\in[N]}\mathbb{E}\left[\left|\frac{1}{N_{k}}\sum_{j\in[N]_{i}}g(x_{j}^{\nu_{\bar{\Theta}_{i}^{(1)}}}(L(N))-(\mathcal{F}^{(1)}g)(\bar{\Theta}_{i}^{(1)})\right|\left|\frac{\partial^{2}f}{\partial
x^{2}}\right|\right]$
$\displaystyle\qquad+\mathbb{E}\left[\left|\frac{1}{N_{k}}\sum_{i\in[N]}(\mathcal{F}^{(1)}g)(\bar{\Theta}_{i}^{(1)})-(\mathcal{F}^{(2)}g)(\bar{\Theta}_{i}^{(2)})\right|\left|\frac{\partial^{2}f}{\partial
x^{2}}\right|\right]$
$\displaystyle\qquad+\mathbb{E}\left[\left|(\mathcal{F}^{(2)}g)(\bar{\Theta}_{i}^{(2)})-(\mathcal{F}^{(2)}g)(\bar{\Theta}_{i}^{(2)}(N_{k}^{2}t_{2}))\right|\left|\frac{\partial^{2}f}{\partial
x^{2}}\right|\right].$

The first term on the right-hand side tends to zero by Lipschitz continuity
for $g$ and Corollary 11.12. The second term tends to zero by the law of large
numbers, since the limiting single colonies are i.i.d. given the value of the
random variable $\bar{\Theta}_{i}^{(1)}$. The third term tends to zero since
by Proposition 11.13 also the limiting $1$-blocks become independent given the
value of the $2$-block. Hence we can again apply the law of large numbers.
Finally, for the last term, note that since we construct the single components
and the limiting process on one probability space, we can argue like in the
proof of Lemma 10.11 that

(11.166)
$\lim_{k\to\infty}\mathbb{E}\left[\left|\bar{\Theta}_{i}^{(2)}-\bar{\Theta}_{i}^{(2)}(N_{k}^{2}t_{2})\right|\right]=0.$

Hence the last term tends to zero by the Lipschitz property of
$\mathcal{F}^{(2)}g$. ∎

######  Remark 11.22.

Instead of [JM86, Theorem 3.3.1] we could have used a similar strategy as in
the proof of Lemma 11.18 to obtain Lemma 11.21. $\blacksquare$

#####  11.3.8 State of the slow seed-banks

On time scale $t_{0}$, i.e., space-time scale $0$, the colour-$1$ seed-bank is
a “slow seed-bank,” since it does not move on this time scale. Because we
study the two-layer three-colour mean-field system from time $N^{2}t_{2}$
onwards, the $1$-block averages of the colour $1$-dormant population are
already in equilibrium. As a consequence we can exactly describe the single
$1$-dormant colonies, which turn out to be in a state that equals the current
$1$-block average of the dormant population of colour $1$. To obtain the
formal result we will first prove the following lemma.

######  Lemma 11.23.

[Slow seed-banks] Fix $t_{2},t_{1}>0$, for $i\in[N^{2}]$ and all $t_{0}\geq
0$,

(11.167)
$\lim_{N\to\infty}\left[y^{[N^{2}]}_{i,1}(N^{2}t_{2}+Nt_{1}+t_{0})-\Theta_{y_{1},i}^{(1),[N^{2}]}(N^{2}t_{2}+Nt_{1}+t_{0})\right]=0\qquad\text{a.s.},$

where $\Theta_{y_{1},i}^{(1),[N^{2}]}$ is the $1$-block average to which
$y^{[N^{2}]}_{i,1}$ contributes.

To prove Lemma 11.23, we need the kernel $b^{[N^{2}]}(\cdot,\cdot)$ defined in
2.30, which becomes in the current setting

(11.168)
$b^{[N^{2}]}((i,R_{i}),(j,R_{j}))=\left\\{\begin{array}[]{ll}1_{\\{d_{[N^{2}]}(i,j)\leq
1\\}}\frac{c_{0}}{N}+\frac{c_{1}}{N^{3}},&\text{ if }R_{i}=R_{j}=A,\\\
K_{m}\frac{e_{m}}{N^{m}},&\text{ if }i=j,\ R_{i}=A,\
R_{j}=D_{m},\,m\in\\{0,1,2\\},\\\ \frac{e_{m}}{N^{m}},&\text{ if }i=j,\
R_{i}=D_{m},\ R_{j}=A,\,m\in\\{0,1,2\\},\\\ 0,&\mbox{
otherwise}.\end{array}\right.$

The corresponding semigroup of the kernel $b^{[N^{2}]}(\cdot,\cdot)$ is
denoted by $b^{[N^{2}]}_{t}(\cdot,\cdot)$.

To prove Lemma 11.23 we will use the following lemma, which was proved in
[GHO20][Lemma 6.1] and for our setting reads as follows.

######  Lemma 11.24.

[First and second moment]
Let $\mathbb{E}_{z^{[N^{2}]}}$ the expectation if the process start from some
state $z^{[N^{2}]}\in([0,1]\times[0,1]^{3})^{[N^{2}]}$. For
$z^{[N^{2}]}\in([0,1]\times[0,1]^{2})^{[N^{2}]}$, $t\geq 0$ and
$(i,R_{i}),(j,R_{j})\in[N^{2}]\times\\{A,D_{0},D_{1},D_{2}\\}$,

(11.169)
$\mathbb{E}_{z^{[N^{2}]}}[z^{[N^{2}]}_{(i,R_{i})}(t)]=\sum_{\begin{subarray}{c}(k,R_{k})\in\\\
\Omega_{N}\times\\{A,D_{0},D_{1},D_{2}\\}\end{subarray}}b^{[N^{2}]}_{t}\big{(}(i,R_{i}),(k,R_{k})\big{)}\,z^{[N^{2}]}_{(k,R_{k})}$

and

(11.170)
$\displaystyle\mathbb{E}_{z^{[N^{2}]}}[z^{[N^{2}]}_{(i,R_{i})}(t)z^{[N^{2}]}_{(j,R_{j})}(t)]$
$\displaystyle=\sum_{\begin{subarray}{c}(k,R_{k}),(l,R_{l})\in\\\
\Omega_{N}\times\\{A,D_{0},D_{1},D_{2}\\}\end{subarray}}b^{[N^{2}]}_{t}\big{(}(i,R_{i}),(k,R_{k})\big{)}\,b^{[N^{2}]}_{t}\big{(}(j,R_{j}),(l,R_{l})\big{)}\,z^{[N^{2}]}_{(k,R_{k})}z^{[N^{2}]}_{(l,R_{l})}$
$\displaystyle\quad+\,2\int_{0}^{{t}}{\rm
d}s\sum_{k\in\Omega_{N}}b^{[N^{2}]}_{(t-s)}((i,R_{i}),(k,A))\,b^{[N^{2}]}_{(t-s)}((j,R_{j}),(k,A))\,\mathbb{E}_{z}^{[N^{2}]}[g(x^{[N^{2}]}_{k}(s))].$

###### Proof of Lemma 11.23.

The argument is given in such a way that it can easily be generalised to more
complicated systems, which we treat later. Let
$\bar{t}(N)=N^{2}t_{2}+Nt_{1}+t_{0}$. We will show that if $i,j\in[N]_{i}$,
i.e., $i$ and $j$ belong to the same $1$-block, then

(11.171)
$\lim_{N\to\infty}\mathbb{E}\left[\left(y^{[N^{2}]}_{i,1}(\bar{t}(N))-y^{[N^{2}]}_{j,1}(\bar{t}(N))\right)^{2}\right]=0.$

This implies (11.167). By Lemma 11.24, we can write

(11.172)
$\displaystyle\mathbb{E}\left[\left(y^{[N^{2}]}_{i,1}(\bar{t}(N))-y^{[N^{2}]}_{j,1}(\bar{t}(N))\right)^{2}\right]$
$\displaystyle=\sum_{\begin{subarray}{c}(k,R_{k}),(l,R_{l})\in\\\
[N^{2}]\times\\{A,D_{0},D_{1},D_{2}\\}\end{subarray}}\left(b^{[N^{2}]}_{\bar{t}(N)}\left((i,D_{1}),(k,R_{k})\right)-b^{[N^{2}]}_{\bar{t}(N)}\left((j,D_{1}),(k,R_{k})\right)\right)$
$\displaystyle\,\times\left(b^{[N^{2}]}_{\bar{t}(N)}\left((i,D_{1}),(l,R_{l})\right)-b^{[N^{2}]}_{\bar{t}(N)}\left((j,D_{1}),(l,R_{l})\right)\right)\mathbb{E}[z^{[N^{2}]}_{(k,R_{k})}z^{[N^{2}]}_{(l,R_{l})}]$
$\displaystyle+\,2\int_{0}^{\bar{t}(N)}{\rm
d}s\sum_{k\in[N^{2}]}\left(b^{[N^{2}]}_{(\bar{t}(N)-s)}((i,D_{1}),(k,A))-b^{[N^{2}]}_{(\bar{t}(N)-s)}((j,D_{1}),(k,A))\right)^{2}$
$\displaystyle\,\times\mathbb{E}[g(x^{[N^{2}]}_{k}(s))].$

Using a coupling argument, we show that both terms in (11.172) tend to $0$ as
$N\to\infty$. To prove that the first term tends to $0$, we will show that

(11.173) $\lim_{N\to\infty}\sum_{\begin{subarray}{c}(k,R_{k})\in\\\
[N^{2}]\times\\{A,D_{0},D_{1},D_{2}\\}\end{subarray}}\left|b^{[N^{2}]}_{\bar{t}(N)}\left((i,D_{1}),(k,R_{k})\right)-b^{[N^{2}]}_{\bar{t}(N)}\left((j,D_{1}),(k,R_{k})\right)\right|=0.$

To do so, let $(RW^{[N^{2}]}(t))_{t\geq 0}$ and
$(RW^{\prime[N^{2}]}(t))_{t\geq 0}$ be two independent random walks, starting
from $RW^{[N^{2}]}(0)=(i,D_{1})$ and $RW^{\prime[N^{2}]}(0)=(j,D_{1})$, where
$i$ and $j$ are in the same $1$-block. Let $RW^{[N^{2}]}$ and
$RW^{\prime[N^{2}]}$ both evolve according to the kernel
$b^{[N^{2}]}(\cdot,\cdot)$, so $b^{[N^{2}]}_{t}(\cdot,\cdot)$ is their
corresponding semigroup. Since $RW^{[N^{2}]}$ and $RW^{\prime[N^{2}]}$ both
start from the colour $1$-seed-bank, we can perfectly couple their switches
between $A$, $D_{0}$, $D_{1}$ and $D_{2}$. Since this implies that both
$RW^{[N^{2}]}$ and $RW^{\prime[N^{2}]}$ are always simultaneously active, we
can also couple the times when they jump due to migration and the distance
over which they migrate. However, we do not couple their migrations, i.e.
$RW^{[N^{2}]}$ and $RW^{\prime[N^{2}]}$ jump at the same time and over the
same distance, but they can jump to different sites. This implies that the
coupled process $(RW^{[N^{2}]}(t),RW^{\prime[N^{2}]}(t))_{t\geq 0}$ has
transition rates

(11.174) $((i,R_{i}),(j,R_{i}))\rightarrow\begin{cases}((k,A),(l,A))&\text{ if
}R_{i}=R_{j}=A\text{ and }d_{[N^{2}]}(i,k)=d_{[N^{2}]}(j,l)\\\ &\text{ at rate
}1_{\\{d_{[N^{2}]}(i,j)\leq 1\\}}\frac{c_{0}}{N}+\frac{c_{1}}{N^{3}},\\\
((i,D_{m}),(j,D_{m}))&\text{ if }R_{i}=R_{j}=A\text{ at rate
}\frac{K_{m}e_{m}}{N^{m}},\,m\in\\{0,1,2\\},\\\ ((i,A),(j,A))&\text{ if
}R_{i}=R_{j}=D_{m}\text{ at rate
}\frac{e_{m}}{N^{m}},\,m\in\\{0,1,2\\}.\end{cases}$

Define the event

(11.175) $H_{t}^{[N^{2}]}=\\{RW^{[N^{2}]}\text{ has migrated at least once up
to time }t\\}.$

Note that if $H_{t}^{[N^{2}]}$ has happened, then also $RW^{\prime[N^{2}]}$
has migrated. Hence

(11.176) $\displaystyle b^{[N^{2}]}_{\bar{t}(N)}$
$\displaystyle\left((i,D_{1}),(k,R_{k})\right)-b^{[N^{2}]}_{\bar{t}(N)}\left((j,D_{1}),(k,R_{k})\right)$
$\displaystyle=\mathbb{P}_{(i,D_{1})}\left(RW^{[N^{2}]}(\bar{t}(N))=(k,R_{k})\right)-\mathbb{P}_{(j,D_{1})}\left(RW^{\prime[N^{2}]}(\bar{t}(N))=(k,R_{k})\right)$
$\displaystyle=\tilde{\mathbb{P}}_{(i,D_{1}),(j,D_{1})}\left(RW^{[N^{2}]}(\bar{t}(N))=(k,R_{k}),\,H_{\bar{t}(N)}^{[N^{2}]}\right)$
$\displaystyle\qquad+\tilde{\mathbb{P}}_{(i,D_{1}),(j,D_{1})}\left(RW^{[N^{2}]}(\bar{t}(N))=(k,R_{k}),\,(H_{\bar{t}(N)}^{[N^{2}]})^{c}\right)$
$\displaystyle\qquad-\tilde{\mathbb{P}}_{(i,D_{1}),(j,D_{1})}\left(RW^{\prime[N^{2}]}(\bar{t}(N))=(k,R_{k}),\,H_{\bar{t}(N)}^{[N^{2}]}\right)$
$\displaystyle\qquad-\tilde{\mathbb{P}}_{(i,D_{1}),(j,D_{1})}\left(RW^{\prime[N^{2}]}(\bar{t}(N))=(k,R_{k}),\,(H_{\bar{t}(N)}^{[N^{2}]})^{c}\right)$
$\displaystyle=\tilde{\mathbb{P}}_{(i,D_{1}),(j,D_{1})}\left(RW^{[N^{2}]}(\bar{t}(N))=(k,R_{k}),\,(H_{\bar{t}(N)}^{[N^{2}]})^{c}\right)$
$\displaystyle\qquad-\tilde{\mathbb{P}}_{(i,D_{1}),(j,D_{1})}\left(RW^{\prime[N^{2}]}(\bar{t}(N))=(k,R_{k}),\,(H_{\bar{t}(N)}^{[N^{2}]})^{c}\right),$

where the last equality follows because, once the random walks have just
jumped once, $RW^{[N^{2}]}$ and $RW^{\prime[N^{2}]}$ are uniformly distributed
over $[N]\times A$ if their jump horizon was $1$ and they are uniformly
distributed of $[N^{2}]\times A$ if they jumped over distance $2$. Hence if
$H_{t}^{[N^{2}]}$ has occurred, then $RW^{[N^{2}]}$ and $RW^{\prime[N^{2}]}$
have the same distribution. Therefore

(11.177) $\sum_{\begin{subarray}{c}(k,R_{k})\in\\\
[N^{2}]\times\\{A,D_{0},D_{1},D_{2}\\}\end{subarray}}\left|b^{[N^{2}]}_{\bar{t}(N)}\left((i,D_{1}),(k,R_{k})\right)-b^{[N^{2}]}_{\bar{t}(N)}\left((j,D_{1}),(k,R_{k})\right)\right|\leq
2\tilde{\mathbb{P}}((H_{\bar{t}(N)}^{[N^{2}]})^{c})$

and we are left to show that

(11.178)
$\lim_{N\to\infty}\tilde{\mathbb{P}}((H_{\bar{t}(N)}^{[N^{2}]})^{c})=0.$

The event $(H_{\bar{t}(N)}^{[N^{2}]})^{c}$ occurs either when the random walks
do not wake up before time $\bar{t}(N)$ or when the random walks wake up
before time $\bar{t}(N)$ but do not migrate. By the coupling we only have to
consider one of the random walks. Therefore the probability that
$RW^{[N^{2}]}$ and $RW^{\prime[N^{2}]}$ do not wake up before time
$\bar{t}(N)$ is given by

(11.179) $\tilde{\mathbb{P}}_{(i,D_{1}),(j,D_{1})}\left(RW^{[N^{2}]}\text{
does not wake up before }\bar{t}(N)\right)={\rm
e}^{-\frac{e_{1}}{N}\bar{t}(N)}={\rm
e}^{-\frac{e_{1}(N^{2}t_{2}+Nt_{1}+t_{0})}{N}}$

and hence

(11.180)
$\lim_{N\to\infty}\mathbb{P}_{(i,D_{1}),(j,D_{1})}\left(RW^{[N^{2}]}\text{
does not wake up before }\bar{t}(N)\right)=0.$

The probability that the random walks do wake up, but do not migrate is a
little more complicated, since each time they wake up with positive
probability they go to sleep before they migrate. Define

(11.181) $\displaystyle C^{[N^{2}]}(t)$ $\displaystyle=\\{\\#\,\text{times
$RW^{[N^{2}]}$ gets active before time }t\\},$ $\displaystyle
T_{A}^{[N^{2}]}(t)$ $\displaystyle=\\{\text{total time $RW^{[N^{2}]}$ is
active up to time }t\\},$ $\displaystyle T_{D}^{[N^{2}]}(t)$
$\displaystyle=\\{\text{total time $RW^{[N^{2}]}$ is dormant up to time
}t\\}.$

Thus, $C^{[N^{2}]}(t)$ counts the number of active/dormant cycles. Define
$T^{[N^{2}]}_{A,n},\ T^{[N^{2}]}_{D,n}$ as the active respectively, dormant
time during the $n$th cycle. Define

(11.182) $\chi=K_{0}e_{0}+\frac{K_{1}e_{1}}{N}+\frac{K_{2}e_{2}}{N^{2}},$

so $\chi$ is the total rate at which $RW$ and $RW^{\prime}$ become dormant
when they are active. Define

(11.183) $c=c_{0}+\frac{c_{1}}{N},$

so $c$ is the total rate at which $RW$ and $RW^{\prime}$ migrate when they are
active. Then

(11.184)
$T^{[N^{2}]}_{A}(t)=\sum_{n=1}^{C^{[N^{2}]}(t)}T^{[N^{2}]}_{A,n},\qquad
T^{[N^{2}]}_{D}(t)=\sum_{n=1}^{C^{[N^{2}]}(t)}T^{[N^{2}]}_{D,n},$

where $T^{[N^{2}]}_{A,n}\,{\buildrel d\over{=}}\,\exp(\chi)$ and
$T^{[N^{2}]}_{D,n}\,{\buildrel
d\over{=}}\,\frac{1}{\chi}K_{0}e_{0}\,\exp(e_{0})+\frac{1}{\chi}\frac{K_{1}e_{1}}{N}\,\exp(\frac{e_{1}}{N})+\frac{1}{\chi}\,\frac{K_{2}e_{2}}{N^{2}}\,\exp(\frac{e_{2}}{N^{2}})$.
Once awake, $RW^{[N^{2}]}$ migrates at rate $c$ and hence the probability to
migrate before time $\bar{t}(N)$ is given by $1-{\rm
e}^{-cT^{[N^{2}]}_{A}(\bar{t}(N))}$. Therefore we are left to show that

(11.185)
$\lim_{N\to\infty}cT_{A}^{[N^{2}]}(\bar{t}(N))=\lim_{N\to\infty}c\sum_{n=1}^{C^{[N^{2}]}(\bar{t}(N))}T^{[N^{2}]}_{A,n}=\infty,\qquad
a.s.$

Since $T^{[N^{2}]}_{A,n}\,{\buildrel d\over{=}}\,\text{exp}(\chi)$, it is
enough to show that

(11.186) $\lim_{N\to\infty}C^{[N^{2}]}(\bar{t}(N))=\infty\qquad a.s.$

To do so, we assume the contrary, i.e., there exists an $R\in\mathbb{N}$ such
that for all $\bar{N}\in\mathbb{N}$ there exists an $N>\bar{N}$ such that

(11.187) $\mathbb{P}_{(i,D_{1})}(C^{[N^{2}]}(\bar{t}(N))\leq R)>0.$

Let $L(N)$ be such that $\lim_{N\to\infty}L(N)=\infty$ and
$\lim_{N\to\infty}L(N)/N=0$. Note that, by (11.180), we can condition on the
first wake-up time and estimate

(11.188) $\displaystyle\mathbb{P}_{(i,D_{1})}(C^{[N^{2}]}(\bar{t}(N))\leq R)$
$\displaystyle=\int_{0}^{\bar{t}(N)}{\rm
d}s\,\mathbb{P}_{(i,A)}(C^{[N^{2}]}(\bar{t}(N)-s)\leq R)\ \frac{e_{1}}{N}{\rm
e}^{-\frac{e_{1}}{N}s}$ $\displaystyle=\int_{0}^{\bar{t}(N)-L(N)}{\rm
d}s\,\mathbb{P}_{(i,A)}(C^{[N^{2}]}(\bar{t}(N)-s)\leq R)\ \frac{e_{1}}{N}{\rm
e}^{-\frac{e_{1}}{N}s}$
$\displaystyle\quad+\int_{\bar{t}(N)-L(N)}^{\bar{t}(N)}{\rm
d}s\,\mathbb{P}_{(i,A)}(C^{[N^{2}]}(\bar{t}(N)-s)\leq R)\ \frac{e_{1}}{N}{\rm
e}^{-\frac{e_{1}}{N}s}$
$\displaystyle\leq\mathbb{P}_{(i,A)}(C^{[N^{2}]}(L(N))\leq R)+{\rm
e}^{-\frac{e_{1}}{N}\bar{t}(N)}\left[{\rm e}^{\frac{e_{1}}{N}L(N)}-1\right].$

Note that the second term in the last inequality tends to $0$ as $N\to\infty$.
For the first term, note that we are now looking at time $L(N)$, i.e., time
scale $N^{0}$. Since

(11.189)
$\displaystyle\lim_{N\to\infty}\mathbb{P}_{i,A}\left(RW^{[N^{2}]}\text{ jumps
to $D_{1}$ or $D_{2}$ before time }L(N)\right)$
$\displaystyle\qquad=\lim_{N\to\infty}1-{\rm
e}^{-\left(\frac{K_{1}e_{1}}{N}+\frac{K_{2}e_{2}}{N^{2}}\right)L(N)}=0.$

we have
$\lim_{N\to\infty}\mathbb{P}_{i,A}\left(\\{RW^{[N^{2}]}(s)\in\\{A,D_{0}\\}\text{
for }s\in[0,L(N)]\\}\right)=1$. Hence, conditioned on the event
$\\{RW^{[N^{2}]}\in\\{A,D_{0}\\}\\}$, $T_{A,n}^{[N^{2}]}\,{\buildrel
d\over{=}}\,\exp(K_{0}e_{0})$ and $T_{D,n}^{[N^{2}]}\,{\buildrel
d\over{=}}\,\exp(e_{0})$. We therefore obtain

(11.190) $\displaystyle\mathbb{P}_{(i,A)}(C^{[N^{2}]}(L(N))\leq R)$
$\displaystyle=\mathbb{P}_{(i,A)}\left(\sum_{n=1}^{R}(T_{A,n}^{[N^{2}]}+T_{D,n}^{[N^{2}]})\geq
L(N)\right)$
$\displaystyle\leq\frac{R}{L(N)}\mathbb{E}_{(i,A)}\left[T_{A,n}^{[N^{2}]}+T_{D,n}^{[N^{2}]}\right]$
$\displaystyle=\frac{R}{L(N)}\left[\frac{1}{K_{0}e_{0}}+\frac{1}{e_{0}}\right].$

Taking the limit $N\to\infty$ in (11.190) and combining this with (11.188), we
conclude that (11.186) indeed holds. Hence also (11.178) and (11.173) hold.

We are left to show that

(11.191) $\displaystyle\lim_{N\to\infty}2\int_{0}^{\bar{t}(N)}\,{\rm
d}s\sum_{k\in[N^{2}]}\left(b^{[N^{2}]}_{(\bar{t}(N)-s)}((i,D_{1}),(k,A))-b^{[N^{2}]}_{(\bar{t}(N)-s)}((j,D_{1}),(k,A))\right)^{2}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times\mathbb{E}[g(x_{k}(s))]=0.$

Also here the idea is to make a similar coupling. As soon as the random walks
migrate, they are equally distributed. On time scale $N$, after waking up from
the colour $1$ seed-bank they will almost immediately migrate, since migration
happens on time scale $1$, i.e., by time $L(N)$ they have migrated with
probability tending to $1$. This will again be the key to show that (11.191)
tends to $0$ as $N\to\infty$.

Note that, by (11.172),

(11.192) $\displaystyle\Bigg{|}2\int_{0}^{\bar{t}(N)}{\rm
d}s\sum_{k\in[N^{2}]}\left(b^{[N^{2}]}_{(\bar{t}(N)-s)}((i,D_{1}),(k,A))-b^{[N^{2}]}_{(\bar{t}(N)-s)}((j,D_{1}),(k,A))\right)^{2}$
$\displaystyle\qquad\qquad\qquad\qquad\times\mathbb{E}[g(x_{k}^{[N^{2}]}(s))]\Bigg{|}\leq
2.$

We will again use the coupling in (11.175). Define

(11.193) $\tau^{[N^{2}]}=\inf\\{t\geq 0:RW^{[N^{2}]}(t)=(k,A)\text{ for some
}k\in[N^{2}]\\}.$

Then, for all $l\in[N^{2}]$,

(11.194) $\mathbb{P}_{(l,D_{1})}(\tau^{[N^{2}]}\leq t)=1-{\rm
e}^{-\frac{e_{1}}{N}t}.$

Setting $s=\bar{t}(N)-s$, we can rewrite the integral in (11.191) as

(11.195) $\displaystyle 2\int_{0}^{\bar{t}(N)}{\rm
d}s\sum_{k\in[N^{2}]}\Big{(}\tilde{\mathbb{P}}^{[N^{2}]}_{(i,D_{1}),(j,D_{1})}\left(RW^{[N^{2}]}(s)=(k,A)\right)$
$\displaystyle\qquad-\tilde{\mathbb{P}}^{[N^{2}]}_{(i,D_{1}),(j,D_{1})}\left(RW^{\prime[N^{2}]}(s)=(k,A),\right)\Big{)}^{2}\mathbb{E}[g(x_{k}^{[N^{2}]}(\bar{t}(N)-s))]$
$\displaystyle=2\int_{0}^{\bar{t}(N)}{\rm
d}s\sum_{k\in[N^{2}]}\Bigg{[}\int_{0}^{s}{\rm
d}r\,\tilde{\mathbb{P}}(\tau^{[N^{2}]}=r)$
$\displaystyle\qquad\left(\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{[N^{2}]}(s-r)=(k,A)\right)-\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{\prime[N^{2}]}(s-r)=(k,A)\right)\right)\Bigg{]}$
$\displaystyle\qquad\times\Big{(}\tilde{\mathbb{P}}^{[N^{2}]}_{(i,D_{1}),(j,D_{1})}\left(RW^{[N^{2}]}(s)=(k,A)\right)-\tilde{\mathbb{P}}^{[N^{2}]}_{(i,D_{1}),(j,D_{1})}\left(RW^{\prime[N^{2}]}(s)=(k,A)\right)\Big{)}$
$\displaystyle\qquad\times\mathbb{E}[g(x_{k}^{[N^{2}]}(\bar{t}(N)-s))].$

In what follows we will abbreviate

(11.196) $\displaystyle
P_{\Delta^{[N^{2}]}_{(i,A),(j,A)}}(t,((l,R_{l}),(j,R_{j})))$
$\displaystyle=\tilde{\mathbb{P}}_{(i,A),(j,A)}\left(RW^{[N^{2}]}(t)=(l,R_{l})\right)-\tilde{\mathbb{P}}_{(i,A),(j,A)}\left(RW^{\prime[N^{2}]}(t)=(j,R_{j})\right),$

and similarly, for $m\in\\{0,1,2\\}$,

(11.197) $\displaystyle
P_{\Delta^{[N^{2}]}_{(i,D_{m}),(j,D_{m})}}(t,((l,R_{l}),(j,R_{j})))$
$\displaystyle=\tilde{\mathbb{P}}_{(i,D_{m}),(j,D_{m})}\left(RW^{[N^{2}]}(t)=(l,R_{l})\right)-\tilde{\mathbb{P}}_{(i,D_{m}),(j,D_{m})}\left(RW^{\prime[N^{2}]}(t)=(j,R_{j})\right).$

By (11.192), we can use Fubini to swap the order of integration and
subsequently substitute $v=s-r$, to obtain

(11.198) $\displaystyle 2\int_{0}^{\bar{t}(N)}\,{\rm
d}r\,\tilde{\mathbb{P}}(\tau^{[N^{2}]}=r)\int_{0}^{\bar{t}(N)-r}{\rm
d}v\sum_{k\in[N^{2}]}P_{\Delta^{[N^{2}]}_{(i,A),(j,A)}}(v,((k,A),(k,A)))$
$\displaystyle\qquad\times
P_{\Delta^{[N^{2}]}_{(i,D_{1}),(j,D_{1})}}(v+r,((k,A),(k,A)))\mathbb{E}[g(x_{k}^{[N^{2}]}(\bar{t}(N)-r-v))]$
$\displaystyle=2\int_{0}^{\bar{t}(N)}{\rm
d}r\,\tilde{\mathbb{P}}(\tau^{[N^{2}]}=r)$
$\displaystyle\sum_{\begin{subarray}{c}(l,R_{l})\in\\\
[N^{2}]\times\\{A,D_{0},D_{1},D_{2}\\}\end{subarray}}\Big{(}{\mathbb{P}}_{(i,D_{1})}\left(RW^{[N^{2}]}(r)=(l,R_{l})\right)-{\mathbb{P}}_{(j,D_{1})}\left(RW^{\prime[N^{2}]}(r)=(l,R_{l})\right)\Big{)}$
$\displaystyle\quad\times\int_{0}^{\bar{t}(N)-r}{\rm
d}v\,\sum_{k\in[N^{2}]}P_{\Delta^{[N^{2}]}_{(i,A),(j,A)}}(v,((k,A),(k,A)))$
$\displaystyle\quad\times\mathbb{P}_{(l,R_{l})}\left(RW^{[N^{2}]}(v)=(k,A)\right)\mathbb{E}[g(x_{k}^{[N^{2}]}(\bar{t}(N)-r-v))],$

where in the last equality we use that the random walks move according to the
same kernel $b(\cdot,\cdot)$. We can continue by writing

(11.199) $\displaystyle 2\int_{0}^{\bar{t}(N)}{\rm
d}r\,\tilde{\mathbb{P}}(\tau^{[N^{2}]}=r)$
$\displaystyle\quad\sum_{\begin{subarray}{c}(l,R_{l})\in\\\
[N^{2}]\times\\{A,D_{0},D_{1},D_{2}\\}\end{subarray}}\Big{(}{\mathbb{P}}^{[N^{2}]}_{(i,D_{1})}\left(RW^{[N^{2}]}(r)=(l,R_{l})\right)-{\mathbb{P}}^{[N^{2}]}_{(j,D_{1})}\left(RW^{\prime[N^{2}]}(r)=(l,R_{l})\right)\Big{)}$
$\displaystyle\quad\times\int_{0}^{\bar{t}(N)-r}{\rm
d}v\,\sum_{k\in[N^{2}]}\bigg{(}\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{[N^{2}]}(v)=(k,A)\right)$
$\displaystyle\qquad\qquad\qquad-\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{\prime[N^{2}]}(v)=(k,A)\right)\bigg{)}$
$\displaystyle\quad\times\mathbb{P}^{[N^{2}]}_{(l,R_{l})}\left(RW^{[N^{2}]}(v)=(k,A)\right)\mathbb{E}[g(x_{k}^{[N^{2}]}(\bar{t}(N)-r-v))]$
$\displaystyle=2\int_{0}^{\bar{t}(N)}{\rm
d}r\,\tilde{\mathbb{P}}(\tau^{[N^{2}]}=r)\int_{0}^{r}{\rm
d}u\,\tilde{\mathbb{P}}(\tau^{[N^{2}]}=u)\sum_{\begin{subarray}{c}(l,R_{l})\in\\\
[N^{2}]\times\\{A,D_{0},D_{1},D_{2}\\}\end{subarray}}$
$\displaystyle\qquad\Big{(}\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{[N^{2}]}(r-u)=(l,R_{l})\right)-\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{\prime[N^{2}]}(r-u)=(l,R_{l})\right)\Big{)}$
$\displaystyle\quad\times\int_{0}^{\bar{t}(N)-r}{\rm
d}v\,\sum_{k\in[N^{2}]}\bigg{(}\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{[N^{2}]}(v)=(k,A)\right)$
$\displaystyle\qquad\qquad\qquad-\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{\prime[N^{2}]}(v)=(k,A)\right)\bigg{)}$
$\displaystyle\quad\times\mathbb{P}^{[N^{2}]}_{(l,R_{l})}\left(RW^{[N^{2}]}(v)=(k,A)\right)\mathbb{E}[g(x_{k}^{[N^{2}]}(\bar{t}(N)-r-v))]$
$\displaystyle+2\int_{0}^{\bar{t}(N)}{\rm
d}r\,\tilde{\mathbb{P}}(\tau^{[N^{2}]}=r)\tilde{\mathbb{P}}(\tau^{[N^{2}]}\geq
r)$ $\displaystyle\quad\times\int_{0}^{\bar{t}(N)-r}{\rm
d}v\,\sum_{k\in[N^{2}]}\bigg{(}\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{[N^{2}]}(v)=(k,A)\right)$
$\displaystyle\qquad\qquad\qquad-\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{\prime[N^{2}]}(v)=(k,A)\right)\bigg{)}$
$\displaystyle\quad\times\Big{(}\mathbb{P}^{[N^{2}]}_{(i,D_{1})}\left(RW^{[N^{2}]}(v)=(k,A)\right)-\mathbb{P}^{[N^{2}]}_{(j,D_{1})}\left(RW^{\prime[N^{2}]}(v)=(k,A)\right)\Big{)}$
$\displaystyle\quad\times\mathbb{E}[g(x_{k}^{[N^{2}]}(\bar{t}(N)-r-v))].$

We will show that both terms in the last equality of (11.199) tends to $0$ as
$N\to\infty$.

For the first term note that, by (11.170), (11.172) and (11.194), we have

(11.200) $\displaystyle 2\int_{0}^{\bar{t}(N)}{\rm
d}r\,\tilde{\mathbb{P}}(\tau^{[N^{2}]}=r)\int_{0}^{r}{\rm
d}u\,\tilde{\mathbb{P}}(\tau^{[N^{2}]}=u)$
$\displaystyle\quad\times\Bigg{[}\sum_{\begin{subarray}{c}(l,R_{l})\in\\\
[N^{2}]\times\\{A,D_{0},D_{1},D_{2}\\}\end{subarray}}\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{[N^{2}]}(r-u)=(l,R_{l})\right)$
$\displaystyle\qquad\qquad\qquad-\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{\prime[N^{2}]}(r-u)=(l,R_{l})\right)\Bigg{]}$
$\displaystyle\quad\times\int_{0}^{\bar{t}(N)-r}{\rm
d}v\,\sum_{k\in[N^{2}]}\bigg{(}\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{[N^{2}]}(v)=(k,A)\right)$
$\displaystyle\qquad\qquad\qquad-\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{\prime[N^{2}]}(v)=(k,A)\right)\bigg{)}$
$\displaystyle\quad\times\mathbb{P}^{[N^{2}]}_{(l,R_{l})}\left(RW^{[N^{2}]}(v)=(k,A)\right)\mathbb{E}[g(x_{k}(\bar{t}(N)-r-v))]$
$\displaystyle\leq 4\int_{0}^{\bar{t}(N)}{\rm
d}r\,\tilde{\mathbb{P}}(\tau^{[N^{2}]}=r)\int_{0}^{r}{\rm
d}u\,\tilde{\mathbb{P}}(\tau^{[N^{2}]}=u)$
$\displaystyle\quad\times\Bigg{|}\sum_{\begin{subarray}{c}(l,R_{l})\in\\\
[N^{2}]\times\\{A,D_{0},D_{1},D_{2}\\}\end{subarray}}\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{[N^{2}]}(r-u)=(l,R_{l})\right)$
$\displaystyle\qquad\qquad\qquad-\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{\prime[N^{2}]}(r-u)=(l,R_{l})\right)\Bigg{|}$
$\displaystyle\leq 4\int_{0}^{\bar{t}(N)}{\rm
d}r\,\tilde{\mathbb{P}}(\tau^{[N^{2}]}=r)\int_{0}^{r-L(N)}{\rm
d}u\,\tilde{\mathbb{P}}(\tau^{[N^{2}]}=u)$
$\displaystyle\quad\times\Bigg{|}\sum_{\begin{subarray}{c}(l,R_{l})\in\\\
[N^{2}]\times\\{A,D_{0},D_{1},D_{2}\\}\end{subarray}}\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{[N^{2}]}(r-u)=(l,R_{l})\right)$
$\displaystyle\qquad\qquad\qquad-\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{\prime[N^{2}]}(r-u)=(l,R_{l})\right)\Bigg{|}$
$\displaystyle+8\int_{L(N)}^{\bar{t}(N)}{\rm
d}r\,\tilde{\mathbb{P}}(\tau^{[N^{2}]}=r)\tilde{\mathbb{P}}(\tau^{[N^{2}]}\in[r-L(N),r])+8\mathbb{P}[\tau^{[N^{2}]}\in[0,L(N)]]$
$\displaystyle\leq 4\int_{0}^{\bar{t}(N)}{\rm
d}r\,\tilde{\mathbb{P}}(\tau^{[N^{2}]}=r)\int_{0}^{r-L(N)}{\rm
d}u\,\tilde{\mathbb{P}}(\tau^{[N^{2}]}=u)$
$\displaystyle\quad\times\Bigg{|}\sum_{\begin{subarray}{c}(l,R_{l})\in\\\
[N^{2}]\times\\{A,D_{0},D_{1},D_{2}\\}\end{subarray}}\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{[N^{2}]}(r-u)=(l,R_{l})\right)$
$\displaystyle\qquad\qquad\qquad-\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{\prime[N^{2}]}(r-u)=(l,R_{l})\right)\Bigg{|}$
$\displaystyle+16[1-{\rm e}^{-\frac{e_{1}}{N}L(N)}].$

Hence the last term in the last inequality tends to $0$ as $N\to\infty$.

To show that the first term in the last inequality tends to $0$ we use the
coupling again. Recall the definition of $H_{t}^{[N^{2}]}$ in (11.175). Note
that we can rewrite the sum as

(11.201) $\displaystyle\sum_{\begin{subarray}{c}(l,R_{l})\in\\\
[N^{2}]\times\\{A,D_{0},D_{1},D_{2}\\}\end{subarray}}\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{[N^{2}]}(r-u)=(l,R_{l})\right)-\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{\prime[N^{2}]}(r-u)=(l,R_{l})\right)$
$\displaystyle\leq\sum_{\begin{subarray}{c}(l,R_{l})\in\\\
[N^{2}]\times\\{A,D_{0},D_{1},D_{2}\\}\end{subarray}}\sum_{\begin{subarray}{c}(l^{\prime},R_{l^{\prime}})\in\\\
[N^{2}]\times\\{A,D_{0},D_{1},D_{2}\\}\end{subarray}}\Big{(}{\mathbb{P}}^{[N^{2}]}_{(i,A)}\left(RW^{[N^{2}]}(L(N))=(l^{\prime},R_{l}^{\prime})\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad-{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{\prime[N^{2}]}(L(N))=(l^{\prime},R^{\prime}_{l})\right)\Big{)}$
$\displaystyle\qquad\times\mathbb{P}_{(l^{\prime},R_{l}^{\prime})}^{[N^{2}]}\left(RW^{[N^{2}]}(r-u-L(N))=(l,R_{l})\right)$
$\displaystyle\leq\sum_{\begin{subarray}{c}(l^{\prime},R_{l^{\prime}})\in\\\
[N^{2}]\times\\{A,D_{0},D_{1},D_{2}\\}\end{subarray}}\left({\mathbb{P}}^{[N^{2}]}_{(i,A)}\left(RW^{[N^{2}]}(L(N))=(l^{\prime},R_{l}^{\prime})\right)-{\mathbb{P}}^{[N^{2}]}_{(j,A)}\left(RW^{\prime[N^{2}]}(L(N))=(l^{\prime},R^{\prime}_{l})\right)\right)$
$\displaystyle=\sum_{\begin{subarray}{c}(l^{\prime},R_{l^{\prime}})\in\\\
[N^{2}]\times\\{A,D_{0},D_{1},D_{2}\\}\end{subarray}}\Big{(}\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{[N^{2}]}(L(N))=(l^{\prime},R_{l}^{\prime}),(H_{t}^{[N^{2}]})^{c}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{\prime[N^{2}]}(L(N))=(l^{\prime},R^{\prime}_{l}),(H_{t}^{[N^{2}]})^{c}\right)\Big{)}$
$\displaystyle\leq
2\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left((H_{t}^{[N^{2}]})^{c}\right).$

To show that

(11.202)
$\lim_{n\to\infty}\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left((H_{t}^{[N^{2}]})^{c}\right)=0,$

we can use a similar strategy as between (11.188) and (11.190), but note that
we now start from two active sites instead of two $1$-dormant sites. Therefore
(11.186) directly follows from (11.189) and (11.190).

To show the second term in (11.199) tends to $0$, we write it as

(11.203) $\displaystyle 2\int_{0}^{\bar{t}(N)}{\rm
d}r\,\tilde{\mathbb{P}}(\tau^{[N^{2}]}=r)\tilde{\mathbb{P}}(\tau^{[N^{2}]}\geq
r)$ $\displaystyle\times\int_{0}^{\bar{t}(N)-r}{\rm
d}v\,\sum_{k\in[N^{2}]}\left(\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{[N^{2}]}(v)=(k,A)\right)-\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{\prime[N^{2}]}(v)=(k,A)\right)\right)$
$\displaystyle\times\left[\mathbb{P}^{[N^{2}]}_{(i,D_{1})}\left(RW^{[N^{2}]}(v)=(k,A)\right)-\mathbb{P}^{[N^{2}]}_{(j,D_{1})}\left(RW^{\prime[N^{2}]}(v)=(k,A)\right)\right]\mathbb{E}[g(x_{k}^{[N^{2}]}(\bar{t}(N)-r-v))]$
$\displaystyle=2\int_{0}^{\bar{t}(N)}{\rm
d}r\,\tilde{\mathbb{P}}(\tau^{[N^{2}]}=r)\tilde{\mathbb{P}}(\tau^{[N^{2}]}\geq
r)$ $\displaystyle\times\int_{0}^{\bar{t}(N)-r}{\rm
d}v\,\sum_{k\in[N^{2}]}\left(\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{[N^{2}]}(v)=(k,A)\right)-\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{\prime[N^{2}]}(v)=(k,A)\right)\right)$
$\displaystyle\times\int_{0}^{v}\,{\rm
d}u\,\tilde{\mathbb{P}}^{[N^{2}]}(\tau^{[N^{2}]}=u)$
$\displaystyle\quad\Big{(}\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{[N^{2}]}(v-u)=(k,A)\right)-\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{\prime[N^{2}]}(v-u)=(k,A)\right)\Big{)}$
$\displaystyle\times\mathbb{E}[g(x_{k}^{[N^{2}]}(\bar{t}(N)-r-v))].$

Changing the order of integration and setting $w=v-u$, we obtain

(11.204) $\displaystyle 2\int_{0}^{\bar{t}(N)}{\rm
d}r\,\tilde{\mathbb{P}}(\tau^{[N^{2}]}=r)\tilde{\mathbb{P}}(\tau^{[N^{2}]}\geq
r)$ $\displaystyle\times\int_{0}^{\bar{t}(N)-r}{\rm
d}u\,\tilde{\mathbb{P}}^{[N^{2}]}(\tau^{[N^{2}]}=u)\int_{u}^{\bar{t}(N)-r}\,{\rm
d}v\,$
$\displaystyle\sum_{k\in[N^{2}]}\left[\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{[N^{2}]}(v)=(k,A)\right)-\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{\prime[N^{2}]}(v)=(k,A)\right)\right]$
$\displaystyle\times\left[\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{[N^{2}]}(v-u)=(k,A)\right)-\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{\prime[N^{2}]}(v-u)=(k,A)\right)\right]$
$\displaystyle\times\mathbb{E}[g(x_{k}^{[N^{2}]}(\bar{t}(N)-r-v))]$
$\displaystyle=2\int_{0}^{\bar{t}(N)}{\rm
d}r\,\tilde{\mathbb{P}}(\tau^{[N^{2}]}=r)\tilde{\mathbb{P}}(\tau^{[N^{2}]}\geq
r)$ $\displaystyle\times\int_{0}^{\bar{t}(N)-r}{\rm
d}u\,\tilde{\mathbb{P}}^{[N^{2}]}(\tau^{[N^{2}]}=u)\int_{0}^{\bar{t}(N)-r-u}\,{\rm
d}w\,$
$\displaystyle\sum_{k\in[N^{2}]}\left[\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{[N^{2}]}(w+u)=(k,A)\right)-\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{\prime[N^{2}]}(w+u)=(k,A)\right)\right]$
$\displaystyle\times\left[\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{[N^{2}]}(w)=(k,A)\right)-\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{\prime[N^{2}]}(w)=(k,A)\right)\right]$
$\displaystyle\times\mathbb{E}[g(x_{k}^{[N^{2}]}(\bar{t}(N)-r-u-w))].$

This can be rewritten as

(11.205) $\displaystyle 2\int_{0}^{\bar{t}(N)}{\rm
d}r\,\tilde{\mathbb{P}}(\tau^{[N^{2}]}=r)\tilde{\mathbb{P}}(\tau^{[N^{2}]}\geq
r)$ $\displaystyle\int_{0}^{\bar{t}(N)-r}{\rm
d}u\,\tilde{\mathbb{P}}^{[N^{2}]}(\tau^{[N^{2}]}=u)$
$\displaystyle\quad\times\sum_{(l,R_{l})\in[N^{2}]}\left[\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{[N^{2}]}(u)=(l,R_{l})\right)-\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{\prime[N^{2}]}(u)=(l,R_{l})\right)\right]$
$\displaystyle\quad\times\int_{0}^{\bar{t}(N)-r-u}\,{\rm
d}w\,\sum_{k\in[N^{2}]}\left[\mathbb{P}^{[N^{2}]}_{(l,R_{l})}\left(RW^{[N^{2}]}(w)=(k,A)\right)\right]$
$\displaystyle\quad\times\left[\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{[N^{2}]}(w)=(k,A)\right)-\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{\prime[N^{2}]}(w)=(k,A)\right)\right]$
$\displaystyle\quad\times\mathbb{E}[g(x_{k}^{[N^{2}]}(\bar{t}(N)-r-u-w))]$
$\displaystyle\leq 8\int_{L(N)}^{\bar{t}(N)}{\rm
d}r\,\tilde{\mathbb{P}}(\tau^{[N^{2}]}=r)\tilde{\mathbb{P}}(\tau^{[N^{2}]}\geq
r)$ $\displaystyle\int_{L(N)}^{\bar{t}(N)-r}{\rm
d}u\,\tilde{\mathbb{P}}^{[N^{2}]}(\tau^{[N^{2}]}=u)$
$\displaystyle\quad\times\sum_{(l,R_{l})\in[N^{2}]}\left[\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{[N^{2}]}(u)=(l,R_{l})\right)-\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{\prime[N^{2}]}(u)=(l,R_{l})\right)\right]$
$\displaystyle+8\int_{L(N)}^{\bar{t}(N)}{\rm
d}r\,\tilde{\mathbb{P}}(\tau^{[N^{2}]}=r)\tilde{\mathbb{P}}(\tau^{[N^{2}]}\geq
r)$ $\displaystyle\int_{0}^{L(N)}{\rm
d}u\,\tilde{\mathbb{P}}^{[N^{2}]}(\tau^{[N^{2}]}=u)$
$\displaystyle\quad\times\sum_{(l,R_{l})\in[N^{2}]}\left[\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{[N^{2}]}(u)=(l,R_{l})\right)-\tilde{\mathbb{P}}^{[N^{2}]}_{(i,A),(j,A)}\left(RW^{\prime[N^{2}]}(u)=(l,R_{l})\right)\right]$
$\displaystyle+16\int_{0}^{L(N)}{\rm
d}r\,\tilde{\mathbb{P}}(\tau^{[N^{2}]}=r)\tilde{\mathbb{P}}(\tau^{[N^{2}]}\geq
r).$

This tends to $0$ by (11.201) and the reasoning below (11.202). ∎

#####  11.3.9 Limiting evolution of the estimator processes

In this section we show that the results along the subsequences used in steps
5-8 of the scheme for the two-level three-colour mean-field system actually
hold for all subsequences. Therefore the limiting evolution holds for
$N\to\infty$. Recall that Lemma 11.23 tells us that all single $1$-dormant
colonies equal the value of the $1$-dormant $1$-block average. Therefore the
second assumption in (11.80) in Proposition (11.8) can be replaced by

(11.206)
$\displaystyle\lim_{k\to\infty}\mathcal{L}\left[\left(Y_{2,0}^{[N_{k}^{2}]}(N^{2}_{k}t_{2})\right)\Big{|}{\bf\Theta}^{\mathrm{aux},(1),[N_{k}^{2}]}(N^{2}_{k}t_{2}+N_{k}t_{1})\right]=P^{{z}_{1}^{\mathrm{eff}}(t_{1})},$

since, by Lemma 11.23, the limiting law

(11.207)
$\lim_{k\to\infty}\mathcal{L}\left[\left(Y_{1,0}^{[N_{k}^{2}]}(N^{2}_{k}t_{2})\right)\right]$

is completely determined by the first line in (11.80). Hence, the assumptions
in Proposition 11.13 and Lemma 11.21 can be weakened in the same way. Using
that in Proposition 11.1 we assume (11.37) and (11.38), we find that the
$2$-block convergence stated in Lemma 11.21 holds along all subsequences we
choose in Step 5. We conclude that Proposition 11.1(a) is indeed true.
Combining Proposition 11.1(a) with steps 1-4 of the scheme and Lemma 11.23, we
find that the assumptions in Proposition 11.13 are true for all subsequences,
and we obtain the limiting evolution of the $1$-block estimator process.
Projecting this limiting evolution onto the active $1$-block average and the
$1$-dormant $1$-block average, we obtain Proposition 11.1(b). Finally,
combining Proposition 11.1(a), steps1-4, and the fact that Proposition 11.13
is true along all subsequences, we obtain Proposition 11.1(c) and (f).

#####  11.3.10 Convergence in the Meyer-Zheng topology

In this section we show how the results on the effective and estimator
processes can be use to show convergence of the full $1$\- and $2$-block
processes.

######  Lemma 11.25 ([Convergence of $1$ process in the Meyer-Zheng
topology]).

Assume that for the $1$-block estimator process defined in (11.21)

(11.208)
$\displaystyle\lim_{N\to\infty}\mathcal{L}\left[\left({\bf\Theta}^{\mathrm{aux},(1),[N^{2}]}(N^{2}t_{2}+Nt_{1})\right)_{t_{1}>0}\,\right]=\mathcal{L}\left[(z_{1}^{\mathrm{aux}}(t_{1}))_{t_{1}>0}\right],$

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 $\Gamma_{u}^{\mathrm{eff},(1)}$. Then

(11.209)
$\displaystyle\lim_{N\to\infty}\mathcal{L}\left[\left({\bf\Theta}^{(1),[N^{2}]}(N^{2}t_{2}+Nt_{1})\right)_{t_{1}>0}\right]=\mathcal{L}\left[(z_{1}^{\Gamma^{(1)}(t_{2})}(t_{1}))_{t_{1}>0}\right]$
$\displaystyle\qquad\qquad\qquad\text{ in the Meyer-Zheng topology},$

where $\Gamma^{(1)}(t_{2})$ is defined as in (11.45) and
$(z_{1}^{\Gamma^{(1)}(t_{2})}(t_{1}))_{t_{1}>0}$ is the process moving
according to (11.27) with initial measure $\Gamma^{(1)}(t_{2})$.

###### Proof.

By assumption 11.208 and Lemma 11.9, we can proceed as in the proof of
Proposition 10.16 to find (11.209). ∎

######  Lemma 11.26 ([Convergence of $2$-block process in the Meyer-Zheng
topology]).

Assume that for the effective $2$-block process defined in (11.23)

(11.210)
$\displaystyle\lim_{N\to\infty}\mathcal{L}\left[\left({\bf\Theta}^{\mathrm{eff},(2),[N^{2}]}(N^{2}t_{2})\right)_{t_{2}>0}\,\right]=\mathcal{L}\left[(z_{2}^{\mathrm{aux}}(t_{2}))_{t_{2}>0}\right],$

where $(z^{\mathrm{eff}}_{2}(t_{2}))_{t_{2}>0}$ is the process evolving
according to (11.35) and starting from $(\vartheta_{1},\theta_{y_{2}})$. Then
for the $2$-block estimator process defined in (11.23)

(11.211)
$\displaystyle\lim_{N\to\infty}\mathcal{L}\left[\left({\bf\Theta}^{(2),[N^{2}]}(N^{2}t_{2})\right)_{t_{2}>0}\right]=\mathcal{L}\left[\left(z_{2}(t_{2})\right)_{t_{2}>0}\right]$
$\displaystyle\text{in the Meyer-Zheng topology},$

where $\left(z_{2}(t_{2})\right)_{t_{2}>0}$ is the process evolving according
to (11.34) and starting in state
$(\vartheta_{1},\vartheta_{1},\vartheta_{1},\theta_{y_{2}})$.

###### Proof.

Combining Lemmas 11.9 and 11.16, we find for $t_{2}>0$

(11.212)
$\displaystyle\lim_{N\to\infty}\mathbb{E}\left[\left|\bar{\Theta}^{(2),[N^{2}]}(N^{2}t_{2})-\Theta_{x}^{(2),[N^{2}]}(N^{2}t_{2})\right|\right]$
$\displaystyle=0,$
$\displaystyle\lim_{N\to\infty}\mathbb{E}\left[\left|\bar{\Theta}^{(2),[N^{2}]}(N^{2}t_{2})-\Theta_{y_{0}}^{(2),[N^{2}]}(N^{2}t_{2})\right|\right]$
$\displaystyle=0,$
$\displaystyle\lim_{N\to\infty}\mathbb{E}\left[\left|\bar{\Theta}^{(2),[N^{2}]}(N^{2}t_{2})-\Theta_{y_{1}}^{(2),[N^{2}]}(N^{2}t_{2})\right|\right]$
$\displaystyle=0.$

Therefore we can again proceed as in the proof of Proposition 10.16 to find
(11.211). ∎

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

In Section 11.3.9 we already proved Proposition 11.1(a),(b),(c) and (f). The
proof of Proposition 11.1(d) follows from Proposition 11.1(a) by applying
Lemma 11.26. The proof of Proposition 11.1(e) follows from Proposition 11.1(b)
by applying Lemma 11.25. This completes the proof of Proposition 11.1.

###  12 Proofs: $N\to\infty$, hierarchical multi-scale limit theorems

In this section we prove the hierarchical multi-scale limit theorems stated in
Theorem 4.2 and Theorem 4.4. In Section 12.1 we first introduce the finite-
level mean-field finite-systems scheme. In Section 12.2 we given an outline of
how to prove the finite-level mean-field finite-systems scheme. In Section
12.3 we show how Theorems 4.2–4.4 can be obtained by a simple generalisation
of the finite-level mean-field finite-systems scheme. The proof of the finite-
level mean-field finite-systems scheme follows a similar lin3 of argument as
in Section 11.2 once we incorporate more levels. Since the proofs for the
finite-level mean-field systems scheme are similar as the proofs in Section
11.3, we will not write out the full proof, but only give an outline and a
sketch.

####  12.1 Finite-level mean-field finite-systems scheme and interaction chain

In this section we extend the two-level three-colour system to a $k$-level
$(k+1)$-colour system with an “outside world” for any $k\in\mathbb{N}$. This
outside world allows also the highest level, the $k$-block, to start from
equilibrium. It is also needed to generalize the results in this subsection to
the infinite hierarchical group.

######  $\blacktriangleright$ Definitions.

To set up the system, fix $k\in\mathbb{N}$ and consider the geographic space
obtained by truncating the hierarchical group $\Omega_{N}$ (recall (2.2))
after hierarchical level $k+1$, i.e., $\Omega_{N}^{k+1}=B_{k+1}(0)$ the
$(k+1)$-block centred at the origin (recall (2.2), (2.4) and Fig. 2). Note
that the $k+1$-block consists of $N$ $k$-blocks i.e.,
$B_{k+1}(0)=\bigcup_{i=0}^{N}B_{k}(i)$ and $B_{k+1}(0)=[N^{k+1}]$. The seed-
bank in this model consists of the $k+2$ layers corresponding to colours
$\\{0,\cdots,k\\}\cup\\{k+1\\}.$ On this space we again consider a restricted
version of the SSDE in (2.20) to the geographic space $\Omega_{N}^{k+1}$. The
migration kernel $a^{\Omega_{N}}(\cdot,\cdot)$ is restricted to
$\Omega_{N}^{k+1}$ by setting all migration outside $B_{k+1}(0)$ equal to 0,
i.e.,

(12.1)
$a^{[\Omega_{N}^{k+1}]}(\xi,\eta)=\sum_{l=1}^{k+1}1_{\big{\\{}d_{\Omega_{N}^{k+1}}(\xi,\eta)\leq
l\big{\\}}}\frac{c_{l}}{N^{l-1}}\frac{1}{N^{l}},$

where $d_{\Omega_{N}^{k+1}}$ is the hierarchical distance $d_{\Omega_{N}}$
restricted to the space $\Omega_{N}^{k+1}$. The colour-$l$ dormant population
exchanges individuals with the active population at rates
$\frac{e_{l}}{N^{l}}$, $\frac{K_{l}e_{l}}{N^{l}}$ for all $0\leq l\leq k$. We
set the interaction of the active population with the colour $(k+1)$-dormant
population equal to $0$. This seed-bank is only needed later, namely for the
“outside world”.

The state space of the finite-level mean-field system is

(12.2)
$S=(\mathfrak{s}^{k+1})^{\Omega_{N}^{k+1}},\quad\mathfrak{s}^{k+1}=[0,1]\times[0,1]^{k+2},$

and the system is denoted by

(12.3) $\displaystyle Z^{\Omega_{N}^{k+1}}=(Z^{\Omega_{N}^{k+1}}(t))_{t\geq
0},\qquad
Z^{\Omega_{N}^{k+1}}(t)=(z^{\Omega_{N}^{k+1}}_{\xi}(t))_{\xi\in\Omega_{N}^{k+1}},$
$\displaystyle
z^{\Omega_{N}^{k+1}}_{\xi}(t)=(x^{\Omega_{N}^{k+1}}_{\xi}(t),(y^{\Omega_{N}^{k+1}}_{\xi,m}(t))_{m=0}^{k+1}).$

The components evolve according to the SSDE

(12.4) $\displaystyle{\rm d}x^{\Omega_{N}^{k+1}}_{\xi}(t)$
$\displaystyle=\sum_{l=1}^{k}\,\frac{c_{l-1}}{N^{l-1}}\,\frac{1}{N^{l}}\sum_{\eta\in
B_{l}(\xi)}\,\left[x^{\Omega_{N}^{k+1}}_{\eta}(t)-x^{\Omega_{N}^{k+1}}_{\xi}(t)\right]\,{\rm
d}t$ $\displaystyle\qquad+\sqrt{g(x^{\Omega_{N}^{k+1}}_{\xi}(t))}\,{\rm
d}w_{\xi}(t)+\sum_{m=0}^{k}\frac{K_{m}e_{m}}{N^{m}}\,\left[y^{\Omega_{N}^{k+1}}_{\xi,m}(t)-x^{\Omega_{N}^{k+1}}_{\xi}(t)\right]\,{\rm
d}t,$ $\displaystyle{\rm d}y^{\Omega_{N}^{k+1}}_{\xi,m}(t)$
$\displaystyle=\frac{e_{m}}{N^{m}}\,\left[x^{\Omega_{N}^{k+1}}_{\xi}(t)-y^{\Omega_{N}^{k+1}}_{\xi,m}(t)\right]\,{\rm
d}t,\quad 0\leq m\leq k,$ $\displaystyle{\rm
d}y^{\Omega_{N}^{k+1}}_{\xi,m}(t)$
$\displaystyle=0,\qquad\xi\in\Omega_{N}^{k+1},$

with $B_{l}(\xi)$ the ball of radius $l$ around $\xi\in\Omega_{N}^{k+1}$.

Note that this system is the hierarchical SSDE in (2.20) with all interactions
at distance $>k$ switched off (i.e., $c_{l}=0$ for $l>k+1$), and also the
exchange with dormant populations of colour $m>k$ is switched off. As before,
by [SS80] the martingale problem associated with (12.4) is well-posed, and for
every initial state in $S$ the SSDE has a unique strong solution. We will
analyse (12.4) on time scales $1,N,N^{2},\cdots,N^{k}$. If for $0\leq l\leq k$
time runs on time scale $N^{l}$, then we write $N^{l}t_{l}$ with $t_{l}>0$.

To study the $k$-level mean-field system, we analyse the equivalent of the
block averages defined in (2.3). For the $k$-level mean-field system these are
given by

(12.5) $\displaystyle x^{\Omega^{k+1}_{N}}_{l}(t)$
$\displaystyle=\frac{1}{N^{l}}\sum_{\eta\in
B_{l}(0)}x^{\Omega_{N}^{k+1}}_{\eta}(N^{l}t),$ $\displaystyle
y^{\Omega^{k+1}_{N}}_{m,l}(t)$ $\displaystyle=\frac{1}{N^{l}}\sum_{\eta\in
B_{l}(0)}y^{\Omega_{N}^{k+1}}_{\eta,m}(N^{l}t),\qquad 0\leq m\leq k+1,\qquad
0\leq l\leq k.$

For $0\leq l\leq k$ these block averages evolve according to the SSDE

$\displaystyle{\rm d}{x}_{l}^{\Omega^{k+1}_{N}}(t)$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{k-(l-1)}\frac{c_{l+n-1}}{N^{n-1}}\big{[}{x}_{l+n}^{\Omega^{k+1}_{N}}(N^{-n}t)-{x}_{l}^{\Omega^{k+1}_{N}}(t)\big{]}\,{\rm
d}t$ $\displaystyle+\sqrt{\frac{1}{N^{l}}\sum_{i\in
B_{l}(0)}g(x_{i}(N^{l}t))}\,\,{\rm d}w_{l}(t)$
$\displaystyle\qquad+\sum_{m=0}^{k}N^{l}\frac{K_{m}e_{m}}{N^{m}}\big{[}{y}_{m,l}^{\Omega^{k+1}_{N}}(t)-{x}_{l}^{\Omega^{k+1}_{N}}(t)\big{]}\,{\rm
d}t,$ (12.7) $\displaystyle{\rm d}{y}_{m,l}^{\Omega^{k+1}_{N}}(t)$
$\displaystyle=$ $\displaystyle
N^{l}\frac{e_{m}}{N^{m}}\big{[}{x}_{l}^{\Omega^{k+1}_{N}}(t)-{y}_{m,l}^{\Omega^{k+1}_{N}}(t)\big{]}\,{\rm
d}t,\qquad 0\leq m\leq k,$ (12.8) $\displaystyle{\rm
d}y^{\Omega_{N}^{k+1}}_{k+1,l}(t)$ $\displaystyle=$ $\displaystyle 0,$

In the limit as $N\to\infty$, the active $l$-block average feels a drift
towards the active $(l+1)$-block average, which is not moving on time scale
$N^{l}$, at rate $c_{l}$. The diffusion term for the $l$-block average becomes
the average diffusion over the $l$-block. The drift of the active $l$-block
average towards the $l$-block average of $m$-dormant populations
$y_{m,l}^{\Omega_{N}^{k+1}}$ with $m>l$ vanishes in the limit as $N\to\infty$.
Therefore, the $m>l$ $m$-dormant populations are _slow seed-banks_ on space-
time scale $l$. The $l$-block average of the colour-$l$ dormant population
$y_{l,l}^{\Omega_{N}^{k+1}}$ has a non-trivial drift towards the active
$l$-block average, written $x_{l}^{\Omega_{N}^{k+1}}$. Therefore the
$l$-dormant population is the _effective seed-bank_ on space-time scale $l$.
For the colour $m$-dormant populations $y_{m,l}^{\Omega_{N}^{k+1}}$ with
$m<l$, we see that infinite rates appear. Therefore the $m$-dormant
populations with $m<l$ are _fast seed-banks_ on space-time scale $l$. We again
need the Meyer-Zheng topology to show that
$\lim_{N\to\infty}y_{m,l}^{\Omega_{N}^{k+1}}=\lim_{N\to\infty}x_{l}^{\Omega_{N}^{k+1}}$.
On space-time scale $l$, the colour-$l$ dormant population is the effective
seed-bank. To get rid of the infinite rates we again look at combinations.
From the above discussion and the SSDE in (12.1)–(12.1), we see that if we
consider the quantity

(12.9)
$\frac{x_{l}^{\Omega_{N}^{k+1}}(t)+\sum_{m=0}^{l-1}K_{m}y_{l,m}^{\Omega_{N}^{k+1}}(t)}{1+\sum_{m=0}^{l-1}K_{m}},$

then all infinite rates cancel out. Therefore

(12.10)
$\left(\frac{x_{l}^{\Omega_{N}^{k+1}}(t)+\sum_{m=0}^{l-1}K_{m}y_{l,m}^{\Omega_{N}^{k+1}}(t)}{1+\sum_{m=0}^{l-1}K_{m}},y_{l,l}^{\Omega_{N}^{k+1}}(t)\right)_{t>0}$

is called the _effective process_ on space-time scale $l$. Like in the simpler
mean-field finite-systems scheme, the effective process allows us to analyse
our system in path space.

An important difference between the finite-level mean-field system in
(12.1)–(12.7) and the two-level three-colour mean-field system in Section
(11.1) is that in the finite-level mean-field system also the highest level
$k$ has a drift towards the outside world. This outside world is the active
$k+1$-block average, which does not evolve on time scale $N^{k}$. This drift
allows the finite-level mean-field system to equilibrate to a non-trivial
equilibrium. In the two-level mean-field system, the highest level, i.e., the
active $2$-block average, does not feel a drift due to migration.
Consequently, the $2$-block averages will eventually cluster.

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

To state and prove the finite-level multi-scale limit, we need the following
three limiting processes. Recall (4.5) and (4.3). For $0\leq l\leq k$, let

(12.11) $(z_{l,(\theta,(y_{m,l})_{m=0}^{k+1})}(t))_{t\geq
0})=(x_{l}(t),(y_{m,l}(t))_{m=0}^{k+1})_{t\geq 0})$

be the process evolving according to

(12.12) $\displaystyle{\rm d}x_{l}(t)=E_{l}\Bigg{[}c_{l}[\theta-
x_{l}(t)]\,{\rm d}t+\sqrt{\mathcal{F}^{(l)}g(x_{1}(t))}\,{\rm
d}w(t)+K_{l}e_{l}\,[y_{l,l}(t)-x_{l}(t)]\,{\rm d}t\Bigg{]},$ $\displaystyle
y_{m,l}(t)=x_{l}(t),\qquad\text{ for }0\leq m<l$ $\displaystyle{\rm
d}y_{l,l}(t)=e_{l}\,[x_{l}(t)-y_{l,l}(t)]\,{\rm d}t,$ $\displaystyle
y_{m,l}(t)=y_{m,l},\qquad\text{ for }l<m\leq k+1,$

where $\theta\in[0,1]$ and $y_{m,l}\in[0,1]$ for $l<m\leq k+1$.

For $0\leq l\leq k$, let

(12.13) $(z_{l,(\theta,(y_{m,l})_{m=l+1}^{k+1})}^{\mathrm{aux}}(t))_{t\geq
0}=(x^{\mathrm{aux}}_{l}(t),(y^{\mathrm{aux}}_{m,l}(t))_{m=l+1}^{k+1})_{t\geq
0}$

be the process evolving according to

(12.14) $\displaystyle{\rm
d}x^{\mathrm{aux}}_{l}(t)=E_{l}\Bigg{[}c_{l}[\theta-x^{\mathrm{aux}}_{l}(t)]\,{\rm
d}t+\sqrt{\mathcal{F}^{(l)}g(x^{\mathrm{aux}}_{1}(t))}\,{\rm
d}w(t)+K_{l}e_{l}\,[y^{\mathrm{aux}}_{l,l}(t)-x^{\mathrm{aux}}_{l}(t)]\,{\rm
d}t\Bigg{]},$ $\displaystyle{\rm
d}y^{\mathrm{aux}}_{l,l}(t)=e_{l}\,[x^{\mathrm{aux}}_{l}(t)-y^{\mathrm{aux}}_{l,l}(t)]\,{\rm
d}t,$ $\displaystyle y^{\mathrm{aux}}_{m,l}(t)=y_{m,l},\qquad\text{ for
}l<m\leq k+1,$

where $\theta\in[0,1]$ and $y^{\mathrm{aux}}_{m,l}\in[0,1]$, for $l<m\leq
k+1$.

For $0\leq l\leq k$, let

(12.15) $(z^{\mathrm{eff}}_{l,\theta}(t))_{t\geq
0}=\left(x^{\mathrm{eff}}_{l}(t),y^{\mathrm{eff}}_{l,l}(t)\right)_{t\geq 0}$

be the _effective process_ evolving according to

(12.16) $\displaystyle{\rm
d}x^{\mathrm{eff}}_{l}(t)=E_{l}\left[c_{l}\,[\theta-x^{\mathrm{eff}}_{l}(t)]\,{\rm
d}t+\sqrt{(\mathcal{F}^{(l)}g)(x^{\mathrm{eff}}_{l}(t))}\,{\rm
d}w(t)+K_{l}e_{l}\,[y^{\mathrm{eff}}_{l,l}(t)-x^{\mathrm{eff}}_{l}(t)]\,{\rm
d}t\right],$ $\displaystyle{\rm
d}y^{\mathrm{eff}}_{l,l}(t)=e_{l}\,[x^{\mathrm{eff}}_{l}(t)-y^{\mathrm{eff}}_{l,l}(t)]\,{\rm
d}t.$

Comparing (12.12) with (12.16), we see that the effective process looks at the
non-trivial components of the full process. The auxiliary process in (12.14)
looks at the active population, the effective seed-bank and the slow seed-
banks.

To state and prove the finite-level multi-scale limit, we need the following
list of ingredients:

  1. 1.

For $t>0$ and for $0\leq l\leq k$, define the _$l$ -block estimators_

(12.17) $\displaystyle\bar{\Theta}^{(l),\Omega_{N}^{k+1}}(t)$
$\displaystyle=\frac{1}{N^{l}}\sum_{i\in
B_{l}}\frac{x^{\Omega_{N}^{k+1}}_{i}(t)+\sum_{m=0}^{l-1}K_{m}y^{\Omega_{N}^{k+1}}_{i,0}(t)}{1+K_{0}},$
$\displaystyle\Theta^{(l),\Omega_{N}^{k+1}}_{x}(t)$
$\displaystyle=\frac{1}{N^{l}}\sum_{i\in B_{l}}x^{\Omega_{N}^{k+1}}_{i}(t),$
$\displaystyle\Theta^{(l),\Omega_{N}^{k+1}}_{y_{m}}(t)$
$\displaystyle=\frac{1}{N^{l}}\sum_{i\in
B_{l}}y^{\Omega_{N}^{k+1}}_{i,m}(t),\qquad 0\leq m\leq k+1,$

and put

(12.18) $\displaystyle\boldsymbol{\Theta}^{(l),\Omega_{N}^{k+1}}(t),$
$\displaystyle=\left(\Theta^{(l),\Omega_{N}^{k+1}}_{x}(t),\big{(}\Theta^{(l),\Omega_{N}^{k+1}}_{y_{m}}(t)\big{)}_{m=0}^{k+1}\right),$
$\displaystyle=\left(\bar{\Theta}^{(l),\Omega_{N}^{k+1}}(t),\left(\Theta^{(l),\Omega_{N}^{k+1}}_{y_{l}}(t)\right)_{m=l}^{k+1}\right),$
$\displaystyle\boldsymbol{\Theta}^{\mathrm{eff},(l),\Omega_{N}^{k+1}}(t)$
$\displaystyle=\left(\bar{\Theta}^{(l),\Omega_{N}^{k+1}}(t),\Theta^{(l),\Omega_{N}^{k+1}}_{y_{l}}(t)\right).$

We call $(\boldsymbol{\Theta}^{(l),\Omega_{N}^{k+1}}(t))_{t>0}$ the _$l$
-block estimator process_,
$(\boldsymbol{\Theta}^{\mathrm{aux},(l),\Omega_{N}^{k+1}}(t))_{t>0}$ the
_auxiliary $l$-block estimator process_ and
$(\boldsymbol{\Theta}^{\mathrm{eff},(l),\Omega_{N}^{k+1}}(t))_{t>0}$ the
_effective $l$-block estimator process_.

  2. 2.

For $0\leq l\leq k$, define the _time scales_ $N^{l}$ such that

(12.19)
$\mathcal{L}[\bar{\Theta}^{(l),\Omega_{N}^{k+1}}(N^{l}t_{l}-L(N)N^{l-1})-\bar{\Theta}^{(l),\Omega_{N}^{k+1}}(N^{l}t_{l})]=\delta_{0}$

for all $L(N)$ such that $\lim_{N\to\infty}L(N)=infty$ and
$\lim_{N\to\infty}L(N)/N=0$, but not for $L(N)=N$. In words, $N$ is the time
scale on which $\bar{\Theta}^{(l),\Omega_{N}^{k+1}}(\cdot)$ starts evolving,
i.e.,
$\left(\bar{\Theta}^{(l),\Omega_{N}^{k+1}}(N^{l}t_{l})\right)_{t_{l}>0},$ is
no longer a fixed process.

  3. 3.

The _invariant measure_ for the evolution of the $l$-block average in (12.12),
denoted by

(12.20) $\Gamma_{\theta,y_{l}}^{(l)},\qquad y_{l}=(y_{m,l})_{m=0}^{k+1}.$

The _invariant measures_ of the auxiliary $l$-block process in (12.15) and the
effective $l$-block process in (12.16), denoted by, respectively,

(12.21) $\Gamma_{\theta,y_{l}}^{(l),\mathrm{aux}},\qquad
y_{l}=(y_{m,l})_{m=l+1}^{k+1}$

and

(12.22) $\Gamma_{\theta}^{(l),\mathrm{eff}}.$

  4. 4.

For $0\leq l\leq k$, let $\mathcal{F}^{E_{l},c_{l},K_{l},e_{l}}$ denote the
_renormalisation transformation_ acting on $\mathcal{G}$ defined by

(12.23)
$(\mathcal{F}^{E_{l},c_{l},K_{l},e_{l}}g)(\theta)=\int_{[0,1]^{2}}g(x)\,\Gamma_{\theta}^{(l)}({\rm
d}x,({\rm d}y)_{m=0}^{k+1}),\qquad\theta\in[0,1],$

and define the _iterates_ $\mathcal{F}^{(n)}$, $0\leq n\leq k$, of the
renormalisation transformation as the compositions

(12.24)
$\mathcal{F}^{(l)}=\mathcal{F}^{E_{n-1},c_{n-1},K_{n-1},e_{n-1}}\circ\cdots\circ\mathcal{F}^{E_{0},c_{0},K_{0},e_{0}},\qquad
0\leq l\leq k.$

(Recall (4.17).)

  5. 5.

To give a detailed description of the multi-scale behaviour of the SSDE in
(2.20), define the _interaction chain_

(12.25) $(M^{k}_{-l})_{-l=-(k+1),-k,\ldots,0}$

as the _time-inhomogeneous_ Markov chain on $[0,1]\times[0,1]^{k+1}$ with
initial state

(12.26)
$M^{k}_{-(k+1)}=(\vartheta_{k},\overbrace{\vartheta_{k},\cdots,\vartheta_{k}}^{k+1\text{
times}},\theta_{y,k+1})$

that evolves according to the transition kernel $Q^{[l]}$ from time $-(l+1)$
to time $-l$ given by

(12.27) $Q^{[l]}(u,{\rm d}v)=\Gamma_{u}^{(l)}({\rm d}v),\qquad 0\leq l\leq k.$

(Recall (4.18).)

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

######  Proposition 12.1.

[Finite-level mean-field: finite-systems scheme] Suppose that the initial
state of the system in (12.4) is given by
$\mu(0)=\mu^{\otimes[\Omega_{N}^{k+1}]}$ for some
$\mu\in\mathcal{P}([0,1]\times[0,1]^{k+2})$. Let $L(N)$ be such that
$\lim_{N\to\infty}L(N)=\infty$ and $\lim_{N\to\infty}L(N)/N=0$, and for
$t_{k},\ldots,t_{0}\in(0,\infty)$ set
$\bar{t}=L(N)N^{k}+\sum_{n=0}^{k}t_{n}N^{n}$.

  * (a)

For every $t_{k},\ldots,t_{0}\in(0,\infty)$,

(12.28)
$\displaystyle\lim_{N\to\infty}\mathcal{L}\Bigg{[}\Bigg{(}{\bf\Theta}^{(l),\Omega_{N}^{k+1}}\left(\bar{t}\right)\Bigg{)}_{l=k+1,k,\ldots,0}\Bigg{]}=\mathcal{L}\left[(M^{k}_{-l})_{-l=-(k+1),-k,\ldots,0}\right],$

where $(M^{k}_{-l})_{-l=-(k+1),-k,\ldots,0}$ is the interaction chain in
(12.25) starting from

(12.29)
$M^{k}_{-(k+1)}=(\vartheta_{k},\overbrace{\vartheta_{k},\cdots,\vartheta_{k}}^{k+1\text{
times}},\theta_{y,k+1}).$

  * (b)

For all $0\leq l\leq k$,

(12.30)
$\displaystyle\lim_{N\to\infty}\mathcal{L}\left[\left(\boldsymbol{\Theta}^{(l),\Omega_{N}^{k+1}}(\bar{t}+t_{l}N^{l})\right)_{t_{l}>0}\right]=\mathcal{L}\left[\left(z_{l,M^{k}_{-(l+1)}}(t_{l})\right)_{t_{l}>0}\right],$
$\displaystyle\text{in the Meyer-Zheng topology},$

where

(12.31) $(z_{l,M^{k}_{-(l+1)}}(t_{l}))_{t_{l}>0}$

is the processes defined (12.12) with $\theta$, $(y_{m},l)_{m=l+1}^{k+1}$
given by the corresponding components in $M^{k}_{-(l+1)}$ and with initial
measure

(12.32)
$\displaystyle\mathcal{L}\left[z_{l,M^{k}_{-(l+1)}}(0)\right]=\Gamma^{(l)}_{M^{k}_{-(l+1)}}$
$\displaystyle\Gamma^{(l)}_{M^{k}_{-(l+1)}}=\int_{\mathfrak{s}^{k+1}}\cdots\int_{\mathfrak{s}^{k+1}}\int_{\mathfrak{s}^{k+1}}\Gamma^{(k)}_{M^{k}_{-(k+1)}}({\rm
d}u_{k})\Gamma^{(k-1)}_{u_{k}}({\rm
d}u_{k-1})\cdots\Gamma^{(l+1)}_{u_{2}}({\rm d}u_{l+2})\Gamma^{(l)}_{u_{l+1}}.$

In Part (a), the limit does not depend on the choice of the times
$t_{k},\ldots,t_{0}$, since we let time start from a time larger than
$L(N)N^{k}$, so that in the limit as $N\to\infty$ all the $l$-block averages
with $l\leq k$ are already in quasi-equilibrium. In Part (b), for $l<k$ the
center of the drift for the active population is _random_ and is determined by
the first component of the interaction chain. Also the states of the
$m$-dormant populations with $l<m\leq k+1$ are determined by the interaction
chain.

######  Remark 12.2.

In contrast to Propositions 10.2–11.1, there are no assumptions on the seed-
bank behaviour in Proposition 12.1. This is because all the block-averages
that we consider are in equilibrium at time $\bar{t}$. Consequently on space-
time scales $l<m$ the $m$-dormant $l$-block average will equal the state of
the $m$-dormant $m$-block average at time $\bar{t}$. Therefore we say that the
state of the _slow seed-banks_ is determined by the space-time scaleon which
this seed-bank is effective. Hence the state of the slow seed-banks is
determined by the interaction chain. $\blacksquare$

The proof of Proposition 12.1 will be given in Section 12.2.

####  12.2 Proof of the mean-field finite-systems scheme: finite-level

We give a sketch of the proof Proposition 12.1. The proof uses a similar
scheme as the proof of Proposition 11.1. We state the scheme and indicate at
each step how it can be proved.

  1. 1.

Tightness of the auxiliary $l$-block estimator processes, for $0\leq l\leq k$,

(12.33)
$\left(\left({\bf\Theta}^{\mathrm{aux},(l),\Omega_{N}^{k+1}}(N^{l}t_{l})\right)_{t_{l}>0}\right)_{N\in\mathbb{N}}.$

###### Proof.

For each $0\leq l\leq k$ we use the tightness criterion in [JM86, Proposition
3.2.3.]. ∎

  2. 2.

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

(12.34) $\lim_{N\to\infty}\sup_{0\leq t\leq
L(N)}\left|\bar{\Theta}^{(l),\Omega_{N}^{k+1}}(N^{l}t_{l})-\bar{\Theta}^{(l),\Omega_{N}^{k+1}}(N^{l}t_{l}-N^{l-1}t)\right|=0\text{
in probability}$

and, for all $l\leq m\leq k+1$,

(12.35) $\lim_{N\to\infty}\sup_{0\leq t\leq
L(N)}\left|\Theta_{y_{m}}^{(l),\Omega_{N}^{k+1}}(N^{l}t_{l})-\Theta_{y_{m}}^{(l),\Omega_{N}^{k+1}}(N^{l}t_{l}-N^{l-1}t_{l})\right|=0\text{
in probability.}$

###### Proof.

Use a similar computation as in the proof of Lemma 11.7. ∎

  3. 3.

We analyse the behaviour of the slow seed-banks by proving the following
lemma.

######  Lemma 12.3.

[Slow seed-banks in the multi-level system] Let $\Theta^{(l)}_{y_{m},i}$
denote the $m$-dormant $l$-block average containing colony
$i\in\Omega_{N}^{k+1}$. Then for all $i\in\Omega_{N}^{k+1}$, $m<k+1$, $l<m$
and $t_{l}>0$,

(12.36)
$\lim_{N\to\infty}\left[y_{i,m}^{\Omega_{N}^{k+1}}(\bar{t}+N^{l}t_{l})-\Theta_{y_{m},i}^{(m),\Omega_{N}^{k+1}}(\bar{t}+N^{l}t_{l})\right]=0\quad
a.s.$

and hence

(12.37)
$\lim_{N\to\infty}\left[\Theta_{y_{m},i}^{(l),\Omega_{N}^{k+1}}(\bar{t}+N^{l}t_{l})-\Theta_{y_{m},i}^{(m),\Omega_{N}^{k+1}}(\bar{t}+N^{l}t_{l})\right]=0\quad
a.s.$

###### Proof.

We can proceed as in the proof of Lemma 11.23, after adapting the kernel
$b^{[N^{2}]}(\cdot,\cdot)$ to the kernel $b^{\Omega_{N}^{k+1}}(\cdot,\cdot)$.
Then we can use that, from each of the $m<k+1$ $m$-dormant populations,
individuals wake up before time $\bar{t}$ with probability $1$. For
individuals starting from an $m$-dormant state, we define the coupling event

(12.38) $H_{t}^{m,\Omega_{N}^{k+1}}=\\{RW^{\Omega_{N}^{k+1}}\text{ has
migrated over distance $m$ at least once up to time }t\\}.$

The migration over distance $m$ is needed because we need $m$-dormant
individuals to be uniformly distributed over the $m$-block in order to almost
surely equal the state of the $m$-block. ∎

  4. 4.

We prove the convergence of the single components. Recall that there are
$N^{k+1-l}$ $l$-blocks in $\Omega_{N}^{k+1}$. Since tightness of components
implies tightness of the process, step 1 implies that for $0\leq l\leq k$ the
full $l$-block processes

(12.39)
$\left(\left({\bf\Theta}_{i}^{\mathrm{aux},(l),\Omega_{N}^{k+1}}(\bar{t}+N^{l}t_{l})\right)_{t_{l}>0,\,i\in[N^{k+1-l}]}\right)_{N\in\mathbb{N}}$

are tight. From the tightness in steps 1 we can construct a subsequence
$(N_{n})_{n\in\mathbb{N}}$ along which, for all $0\leq l\leq k$,

(12.40)
$\displaystyle\lim_{n\to\infty}\mathcal{L}\left[\left({\bf\Theta}_{i}^{\mathrm{aux},(1),\Omega_{N_{n}}^{k+1}}(\bar{t}+N^{l}_{n}t_{l})\right)_{t_{l}>0,\,i\in[N_{n}^{k+1-l}]}\right]$

exists. Note that $\bar{t}$ depends on the subsequence. For example, along the
subsequence $(N_{\tilde{n}})_{\tilde{n}\in\mathbb{N}}$,

(12.41) $\bar{t}=L(N)N^{k}+\sum_{n=0}^{k}t_{n}N_{\tilde{n}}^{n}.$

We define the measure

(12.42)
$\nu^{(0)}_{\bf\Theta}=\prod_{i\in\mathbb{N}_{0}}\Gamma^{(0)}_{\Theta_{i}}(\bar{t}),$

where

(12.43) $\Theta_{i}\in\mathfrak{s}^{k+1}.$

In this step we show that along the same subsequence the single components
converge to the infinite system. We show that if

(12.44)
$\lim_{n\to\infty}\mathcal{L}[({\bf\Theta}^{\mathrm{aux},(1),\Omega_{N_{n}}^{k+1}}(\bar{t}))_{i\in[N_{n}^{k}]}]=P^{(1)},$

then

(12.45)
$\displaystyle\lim_{n\to\infty}\mathcal{L}\Big{[}\Big{(}Z^{\Omega_{N_{n}}^{k+1}}(\bar{t}+t_{0})\Big{)}_{t_{0}\geq
0}\Big{]}=\mathcal{L}\left[(Z^{\nu^{(0)}(\bar{t})}(t_{0}))_{t_{0}\geq
0}\right],$

where

(12.46) $\nu^{(0)}(\bar{t})=\int\nu^{(0)}_{u}P^{(1)}({\rm d}u).$

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

(12.47)
${}\lim_{n\to\infty}\mathcal{L}[({\bf\Theta}^{\mathrm{aux},(1),\Omega_{N_{n}}^{k+1}}(\bar{t}))_{i\in[N_{n}^{k}]}]$

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

(12.48)
${}\lim_{n\to\infty}\mathcal{L}[({\bf\Theta}^{\mathrm{aux},(1),\Omega_{N_{n}}^{k+1}}(\bar{t}))_{i\in[N_{n}^{k}]}].$

Note that we choose the subsequence $(N_{n})_{n\in\mathbb{N}}$ in such a way
that we know that the law $P^{(1)}$ in (12.44) exists.

###### Proof.

Proceed as in the proof of Proposition 11.8. Note that the assumptions on the
seed-banks in Proposition 11.8 follow from the choice of the subsequence and
Lemma 12.3. ∎

  5. 5.

Using the limiting evolution of the single colonies obtained in step 4, we can
identify the limiting $l$-block process along the same subsequence. For $1\leq
l\leq k$, we show that if

(12.49)
$\lim_{n\to\infty}\mathcal{L}[({\bf\Theta}^{\mathrm{aux},(l+1),\Omega_{N_{n}}^{k+1}}(\bar{t}))_{i\in[N_{n}^{k}]}]=P^{(l+1)},$

then

(12.50)
$\displaystyle\lim_{n\to\infty}\mathcal{L}\left[\left({\bf\Theta}^{\mathrm{aux},(l),\Omega_{N_{n}}^{k+1}}(\bar{t}+N^{l}_{n}t_{l})\right)_{t_{l}>0,\,i\in[N_{n}^{k+1-l}]}\right]=\mathcal{L}\left[(z_{l,{\bf\Theta}^{(l+1)}}^{\mathrm{aux}}(t))_{t\geq
0}\right],$

where
${\bf\Theta}^{(l+1)}=({\bf\Theta}^{(l+1)}_{x},({\bf\Theta}_{y_{m},l}^{(l+1)})_{m=l+1}^{k+1})\in[0,1]\times[0,1]^{k+2-(l+1)}$,

(12.51)
$\displaystyle\mathcal{L}\left[z_{l,{\bf\Theta}^{(l)}}^{\mathrm{aux}}(0)\right]$
$\displaystyle=\Gamma_{{\bf\Theta}^{(l+1)}}^{(l),\mathrm{aux}},$
$\displaystyle\Gamma_{{\bf\Theta}^{(l+1)}}^{(l),\mathrm{aux}}$
$\displaystyle=\int_{[0,1]\times[0,1]^{k+2-(l+1)}}\Gamma_{u}^{(l),\mathrm{aux}}P^{(l+1)}({\rm
d}u)$

and $(z_{l,{\bf\Theta}^{(l+1)}}^{\mathrm{aux}}(t))_{t\geq 0}$ is the process
evolving according to (12.14) with $\theta,(y_{m,l})_{m=l}^{k+1}$ replaced by
the random variables
${\bf\Theta}^{(l+1)}_{x},({\bf\Theta}^{(l+1)}_{y_{m},l})_{m=l}^{k+1}$. Note
that by the choice of the subsequence $(N_{n})_{n\in\mathbb{N}}$ we know that
for $1\leq l\leq k$ the limiting laws in (12.49) exist.

###### Proof.

The proof goes by induction. Using the convergence of the single components,
we can proceed as in the proof of Proposition 11.13 to prove the convergence
of the $1$-blocks averages

(12.52)
$\displaystyle\lim_{n\to\infty}\mathcal{L}\left[\left({\bf\Theta}_{i}^{\mathrm{aux},(1),\Omega_{N_{n}}^{k+1}}(\bar{t}+N_{n}t_{1})\right)_{t_{1}>0,\,i\in[N_{n}^{k+1-l}]}\right].$

Then, assuming that we have the convergence for all $0\leq l\leq L$, we get

(12.53)
$\displaystyle\lim_{n\to\infty}\mathcal{L}\left[\left({\bf\Theta}_{i}^{\mathrm{aux},(l),\Omega_{N_{n}}^{k+1}}(\bar{t}+N^{l}_{n}t_{l})\right)_{t_{l}>0,\,i\in[N_{n}^{k+1-l}]}\right],$

and we prove the convergence of

(12.54)
$\displaystyle\lim_{n\to\infty}\mathcal{L}\left[\left({\bf\Theta}_{i}^{\mathrm{aux},(L+1),\Omega_{N_{n}}^{k+1}}(\bar{t}+N^{(L+1)}_{n}t_{(L+1)})\right)_{t_{(L+1)}>0,\,i\in[N_{n}^{k+1-(L+1)}]}\right].$

This is done using a similar proof strategy as in the proof of Proposition
11.13. In particular, we need to derive the $l$-level equivalent of Lemma
11.16. Since this lemma is also key to proving convergence in the Meyer-Zheng
topology, we state it explicitly below. ∎

######  Lemma 12.4.

[$l$-block averages] Define

(12.55) $\displaystyle\Delta^{(l),\Omega_{N}^{k+1}}_{\Sigma}(N^{l-1}t_{l-1})$
$\displaystyle\,=\frac{\Theta^{(l),\Omega_{N}^{k+1}}_{x}(N^{l-1}t_{l-1})+\sum_{m=0}^{l-2}K_{m}\Theta^{(l),\Omega_{N}^{k+1}}_{y_{m}}(N^{l-1}t_{l-1})}{1+\sum_{m=0}^{l-2}K_{m}}-\Theta^{(l),\Omega_{N}^{k+1}}_{y_{l-1}}(N^{l-1}t_{l-1})$

and

(12.56) $R_{l}=\frac{1+\sum_{m=0}^{l-1}K_{m}}{1+\sum_{m=0}^{l-2}K_{m}}.$

For $t\geq 0$ set $\bar{\Theta}^{(0)}(t)=\Theta_{x}^{(0)}(t)=x_{0}(t)$. Then,
for $1\leq l\leq k$,

(12.57)
$\displaystyle\mathbb{E}\left[\left|\Delta^{(l),\Omega_{N}^{k+1}}_{\Sigma}(N^{l-1}t_{l-1})\right|\right]$
$\displaystyle\leq\sqrt{\mathbb{E}\left[\left(\Delta^{(l),\Omega_{N}^{k+1}}_{\Sigma}(0)\right)^{2}\right]}{\rm
e}^{-e_{l}R_{l}t_{l-1}}$ $\displaystyle\quad+\sqrt{\int_{0}^{t_{1}}{\rm
d}s\,2e_{l}R_{l}{\rm
e}^{-2e_{l}R_{l}(t_{1}-s)}\mathbb{E}\left[\left|\bar{\Theta}^{(l-1),\Omega_{N}^{k+1}}(N^{l-1}s)-\Theta^{(l-1),\Omega_{N}^{k+1}}_{x}(N^{l-1}s)\right|\right]}$
$\displaystyle\quad+\sqrt{\frac{1}{N}\frac{1}{2e_{l}(1+\sum_{m=0}^{l-1}K_{m})}\left[\sum_{n=l+1}^{k}\frac{c_{n-1}}{N^{n-(l+1)}}+\sum_{m=l}^{k}\frac{K_{m}e_{m}}{N^{m-l}}+E_{l-1}||g||\right]}.$

###### Proof.

Proceed like in the proof of Lemma 11.16, using the SSDE in (12.4) instead of
the SSDE in (11.6). ∎

We obtain the following useful corollary from Lemma 12.4.

######  Corollary 12.5.

For all $1\leq l\leq k$, $s>0$ and $\tilde{l}\geq l$,

(12.58)
$\lim_{N\to\infty}\mathbb{E}\left[\left|\bar{\Theta}^{(l-1),\Omega_{N}^{k+1}}(N^{\tilde{l}-1}s)-\Theta^{(l-1),\Omega_{N}^{k+1}}_{x}(N^{\tilde{l}-1}s)\right|\right]=0.$

###### Proof.

We proceed by induction. The result for $l=1$ is trivial. Suppose that the
result holds for $l=L$. Then for $l=L+1$ we obtain

(12.59)
$\displaystyle\mathbb{E}\left[\left|\bar{\Theta}^{(L),\Omega_{N}^{k+1}}(N^{L}s)-\Theta^{(L),\Omega_{N}^{k+1}}_{x}(N^{L}s)\right|\right]$
$\displaystyle\leq\mathbb{E}\left[\left|\bar{\Theta}^{(L),\Omega_{N}^{k+1}}(N^{L}s)-\frac{1}{N}\sum_{i=0}^{N-1}\bar{\Theta}^{(L-1),\Omega_{N}^{k+1}}_{i}(N^{L}s)\right|\right]$
$\displaystyle\qquad+\frac{1}{N}\sum_{i=0}^{N-1}\mathbb{E}\left[\left|\bar{\Theta}^{(L-1),\Omega_{N}^{k+1}}_{i}(N^{L}s)-{\Theta}^{(L-1),\Omega_{N}^{k+1}}_{x,i}(N^{L}s)\right|\right].$

Note that the second term in the right-hand side of (12.59) tends to $0$ as
$N\to\infty$ by the induction hypothesis. For the first term in the right-hand
side of (12.59), note that

(12.60)
$\displaystyle\mathbb{E}\left[\left|\bar{\Theta}^{(L),\Omega_{N}^{k+1}}(N^{L}s)-\frac{1}{N}\sum_{i=0}^{N-1}\bar{\Theta}^{(L-1),\Omega_{N}^{k+1}}_{i}(N^{L}s)\right|\right]$
$\displaystyle=\mathbb{E}\left[\left|\frac{\Theta^{(L),\Omega_{N}^{k+1}}_{x}(N^{L}s)+\sum_{m=0}^{L-1}K_{m}\Theta^{(L),\Omega_{N}^{k+1}}_{y_{m}}(N^{L}s)}{1+\sum_{m=0}^{L-1}K_{m}}\right.\right.$
$\displaystyle\qquad\qquad-\left.\left.\frac{\Theta^{(L),\Omega_{N}^{k+1}}_{x}(N^{L}s)+\sum_{m=0}^{L-2}K_{m}\Theta^{(L),\Omega_{N}^{k+1}}_{y_{m}}(N^{L}s)}{1+\sum_{m=0}^{L-2}K_{m}}\right|\right]$
$\displaystyle=\frac{K_{L-1}}{1+\sum_{m=0}^{L-1}K_{m}}\mathbb{E}\left[\left|\Theta^{(L),\Omega_{N}^{k+1}}_{y_{L-1}}(N^{L}s)-\bar{\Theta}^{(L),\Omega_{N}^{k+1}}(N^{L}s)\right|\right].$

Invoking Lemma 12.4 and using the induction hypothesis, we see that for $s>0$
and $\tilde{l}\geq L$ indeed

(12.61)
$\lim_{N\to\infty}\mathbb{E}\left[\left|\bar{\Theta}^{(L),\Omega_{N}^{k+1}}(N^{\tilde{l}}s)-\Theta^{(L),\Omega_{N}^{k+1}}_{x}(N^{\tilde{l}}s)\right|\right]=0.$

∎

  6. 6.

Show that the convergence in step 4 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.
This follows from the fact that the auxiliary $k$-estimator process converges
to the same limit along every subsequence. Consequently, the same holds for
the auxiliary $k-1$-estimator process. In this way we can traverse back
through the levels to obtain that all $l$-estimator process converges as
$N\to\infty$.

Define, for $0\leq l\leq k$,

(12.62) $\mathfrak{s}^{k+1}_{l}=[0,1]\times[0,1]^{k+2-l}.$

We obtain, for $0\leq l\leq k-1$,

(12.63)
$\displaystyle\lim_{N\to\infty}\mathcal{L}[({\bf\Theta}^{\mathrm{aux},(l+1),\Omega_{N}^{k+1}}(\bar{t}))]]=\Gamma_{{\bf\Theta}^{(l+2)}}^{(l+1),\mathrm{aux}},$
$\displaystyle\Gamma_{{\bf\Theta}^{(l+2)}}^{(l+1),\mathrm{aux}}=\int_{\mathfrak{s}^{k+1}_{l+2}}\cdots\int_{\mathfrak{s}^{k+1}_{k}}\Gamma^{(k),\mathrm{aux}}_{\vartheta_{k}}({\rm
d}u_{k})\cdots\Gamma^{(l+2),\mathrm{aux}}_{u_{l+3}}({\rm
d}u_{l+2})\Gamma^{(l+1),\mathrm{aux}}_{u_{l+2}}.$

Therefore by step 5

(12.64)
$\displaystyle\lim_{N\to\infty}\mathcal{L}\left[\left({\bf\Theta}^{\mathrm{aux},(l),\Omega_{N}^{k+1}}(\bar{t}+N^{l}_{n}t_{l})\right)_{t_{l}>0,\,i\in[N^{k+1-l}]}\right]=\mathcal{L}\left[(z_{l,{\bf\Theta}^{(l+1)}}^{\mathrm{aux}}(t))_{t\geq
0}\right],$

where
${\bf\Theta}^{(l+1)}=({\bf\Theta}^{(l+1)}_{x},({\bf\Theta}_{y_{m},l}^{(l+1)})_{m=l}^{k+1})\in\mathfrak{s}^{(k+1)}_{l}$
are random variables with law

(12.65)
$\mathcal{L}\left[{\bf\Theta}^{(l+1)}\right]=\Gamma_{{\bf\Theta}^{(l+2)}}^{(l+1),\mathrm{aux}}.$

The initial state of the limiting process in (12.64) is given by

(12.66)
$\displaystyle\mathcal{L}\left[z_{l,{\bf\Theta}^{(l)}}^{\mathrm{aux}}(0)\right]$
$\displaystyle=\Gamma_{{\bf\Theta}^{(l+1)}}^{(l),\mathrm{aux}},$
$\displaystyle\Gamma_{{\bf\Theta}^{(l+1)}}^{(l),\mathrm{aux}}$
$\displaystyle=\int_{\mathfrak{s}^{k+1}_{l+2}}\cdots\int_{\mathfrak{s}^{k+1}_{k}}\Gamma^{(k),\mathrm{aux}}_{\vartheta+k}({\rm
d}u_{k})\cdots\Gamma^{(l+2),\mathrm{aux}}_{u_{l+2}}({\rm
d}u_{l+1})\Gamma^{(l+1),\mathrm{aux}}_{u_{l+1}}$

and $(z_{l,{\bf\Theta}^{(l+1)}}^{\mathrm{aux}}(t))_{t\geq 0}$ is the process
evolving according to (12.14) with $\theta,(y_{m,l})_{m=l+1}^{k+1}$ replaced
by the random variables
${\bf\Theta}^{(l+1)}_{x},({\bf\Theta}^{(l+1)}_{y_{m}})_{m=l+1}^{k+1}$. Recall
that, by Lemma 12.3, we have, for $l+1\leq m\leq k+1$,

(12.67) ${\bf\Theta}^{(l+1)}_{y_{m}}={\bf\Theta}^{(m)}_{y_{m}}.$

  7. 7.

Use the Meyer-Zheng topology to obtain Proposition 12.1(b).

###### Proof.

Note that Lemma 12.4 and Corollary 12.5 together imply

(12.68)
$\displaystyle\lim_{N\to\infty}\mathbb{E}\left[\left|\bar{\Theta}^{(l),\Omega_{N}^{k+1}}(N^{l}t_{l})-\Theta_{x}^{(l),\Omega_{N}^{k+1}}(N^{l}t_{l})\right|\right]$
$\displaystyle=0,$
$\displaystyle\lim_{N\to\infty}\mathbb{E}\left[\left|\bar{\Theta}^{(l),\Omega_{N}^{k+1}}(N^{l}t_{l})-\Theta_{y_{m}}^{(l),\Omega_{N}^{k+1}}(N^{l}t_{l})\right|\right]$
$\displaystyle=0,\text{ for }0\leq m\leq l-1.$

Combining the result obtained in step 6 with the proof strategy followed in
Section 11.3.10, we get the claim. ∎

  8. 8.

Finally, we prove Proposition 12.1(a).

###### Proof.

Step 6 and step 7 yield the laws of the components $\mathcal{L}[M_{l}^{k}]$ of
the interaction chain $(M_{-l}^{k})_{-l=-(k+1)}^{0}$. Note that the state
space $([0,1]\times[0,1]^{k+2})^{k+2}$ is compact, and therefore the sequence
of random variables

(12.69)
$\left(\Bigg{(}{\bf\Theta}^{(l),\Omega_{N}^{k+1}}\left(\bar{t}\right)\Bigg{)}_{l=k+1,k,\ldots,0}\right)_{N\in\mathbb{N}}$

is tight. For any

(12.70) $\displaystyle f\colon\,([0,1]\times[0,1]^{k+2})^{n+2}\to\mathbb{R},$
$\displaystyle f(x)=\prod_{i=1}^{n}f_{i}(x_{i}),$ $\displaystyle
f_{i}\in\mathcal{C}_{b}([0,1]\to\mathbb{R}),$

we can use conditioning on the previous block average to obtain

(12.71)
$\lim_{N\to\infty}\mathbb{E}\left[f\left(\Bigg{(}{\bf\Theta}^{(l),\Omega_{N}^{k+1}}\left(\bar{t}\right)\Bigg{)}_{l=k+1,k,\ldots,0}\right)\right]=\mathbb{E}\left[f\left((M_{-l}^{k})_{-l=-(k+1)}^{0}\right)\right].$

Using that the set of functions of the form (12.70) is separating, we obtain
the claim. ∎

####  12.3 Proof: of the hierarchical multi-scale limit theorems.

In this section we prove Theorems 4.2 and 4.4. We start by proving Theorem
4.4. Theorem 4.2 will follow from Theorem 4.4 by projection onto the effective
components.

###### Proof of Theorem 4.2

###### Proof.

Recall the estimators in (4.11). Like for the finite-level hierarchical mean-
field system, we can define the auxiliary estimator process by

(12.72)
$\boldsymbol{\Theta}^{(l),\mathrm{aux},\Omega_{N}}(t)=\big{(}\bar{\Theta}^{(l),\Omega_{N}}(t),\big{(}\Theta^{(l),\Omega_{N}}_{y_{m}}(t)\big{)}_{m=l}^{\infty}\big{)}.$

For $l,k\in\mathbb{N}$ the processes
$({\bf\Theta}^{(l),\mathrm{aux},\Omega_{N}}(\bar{t}+N^{k}t))_{t>0}$ evolve
according to (recall, (4.54))

(12.73) $\displaystyle{\rm d}\bar{\Theta}^{(l),\Omega_{N}}(N^{k}t)$
$\displaystyle=E_{l}\sum_{n=l+1}^{\infty}\frac{c_{n-1}}{N^{n-1-k}}\left[\Theta_{x}^{(n),\Omega_{N}}(N^{k}t)-\Theta_{x}^{(l),\Omega_{N}}(N^{k}t)\right]\,{\rm
d}t$ $\displaystyle\qquad+E_{l}\sqrt{\frac{N^{k}}{N^{2l}}\sum_{\xi\in
B_{l}}g\big{(}x_{\xi}(N^{k}t)\big{)}}\,{\rm d}w(t)$
$\displaystyle\qquad+E_{l}\sum_{m=l}^{\infty}\frac{K_{m}e_{m}}{N^{m-k}}\left[\Theta^{(l),\Omega_{N}}_{y_{m}}(N^{k}t)-\Theta^{(l),\Omega_{N}}_{x}(N^{k}t)\right]\,{\rm
d}t,$ $\displaystyle{\rm d}\Theta^{(l),\Omega_{N}}_{y_{m}}(N^{k}t)$
$\displaystyle=\frac{e_{m}}{N^{m-k}}\left[\Theta^{(l),\Omega_{N}}_{x}(N^{k}t)-\Theta^{(l),\Omega_{N}}_{y_{m}}(N^{k}t)\right]\,{\rm
d}t,\qquad l\leq m\leq\infty.$

Therefore, for $l>k$ and all $\epsilon>0$,

(12.74) $\displaystyle\mathbb{P}\Bigg{[}\sup_{0\leq t\leq
L(N)}\Bigg{|}\bar{\Theta}^{(l),\Omega_{N}}(\bar{t})-\bar{\Theta}^{(l),\Omega_{N}}(\bar{t}+N^{k}t)\Bigg{|}>\epsilon\Bigg{]}$
$\displaystyle=\mathbb{P}\Bigg{[}\sup_{0\leq t\leq
L(N)}E_{l}\Bigg{|}\int_{\bar{t}}^{\bar{t}+N^{k}t}{\rm
d}r\,\sum_{n=l+1}^{\infty}\frac{c_{n-1}}{N^{n-1}}\left[\Theta_{x}^{(n),\Omega_{N}}(r)-\Theta_{x}^{(l),\Omega_{N}}(r)\right]$
$\displaystyle\qquad\qquad\qquad+\int_{\bar{t}}^{\bar{t}+N^{k}t}{\rm
d}r\,\sum_{m=l}^{\infty}\frac{K_{m}e_{m}}{N^{m}}\left[\Theta^{(l),\Omega_{N}}_{y_{m}}(r)-\Theta^{(l),\Omega_{N}}_{x}(r)\right]$
$\displaystyle\qquad\qquad\qquad+\int_{\bar{t}}^{\bar{t}+N^{k}t}{\rm
d}w_{i}(r)\,\sqrt{\frac{1}{N^{2l}}\sum_{\xi\in
B_{l}}g\big{(}x_{\xi}(r)\big{)}}\,\Bigg{|}>\epsilon\Bigg{]}$
$\displaystyle\leq\mathbb{P}\Bigg{[}\sup_{0\leq t\leq
L(N)}\,E_{l}\Bigg{|}\int_{\bar{t}}^{\bar{t}+N^{k}t}{\rm
d}w_{i}(r)\,\sqrt{\frac{1}{N^{2l}}\sum_{\xi\in
B_{l}}g\big{(}x_{\xi}(r)\big{)}}\,\,\Bigg{|}$
$\displaystyle\qquad\qquad>\epsilon-t\left[\sum_{n=l+1}^{\infty}\frac{c_{n-1}}{N^{n-1-k}}+\sum_{m=l}^{\infty}\frac{K_{m}e_{m}}{N^{m-k}}\right]\Bigg{]}.$

Note that, since $l>k$,

(12.75)
$\lim_{N\to\infty}t\left[\sum_{n=l+1}^{\infty}\frac{c_{n-1}}{N^{n-1-k}}+\sum_{m=l}^{\infty}\frac{K_{m}e_{m}}{N^{m-k}}\right]=0.$

Hence, like in the proof of Lemma 11.7, we can use an optional stopping
argument to obtain

(12.76) $\displaystyle\lim_{N\to\infty}\sup_{0\leq t\leq
L(N)}\Bigg{|}\bar{\Theta}^{(l),\Omega_{N}}(\bar{t})-\bar{\Theta}^{(l),\Omega_{N}}(\bar{t}+N^{k}t)\Bigg{|}=0\qquad\text{
in probability}.$

Using a similar computation as in (12.74), we can show

(12.77) $\displaystyle\lim_{N\to\infty}\sup_{0\leq t\leq
L(N)}\Bigg{|}\bar{\Theta}_{y_{m}}^{(l),\Omega_{N}}(\bar{t})-{\Theta}_{y_{m}}^{(l),\Omega_{N}}(\bar{t}+N^{k}t)\Bigg{|}=0\qquad\text{
in probability}.$

Hence we obtain that, on time scale $N^{k}$ as $N\to\infty$, the process
$({\bf\Theta}^{(l),\mathrm{aux},\Omega_{N}}(\bar{t}+N^{k}t))_{t>0}$ does not
evolve and therefore is still in its initial
state$({\bf\Theta}^{(l),\mathrm{aux},\Omega_{N}}(\bar{t}))$.

Using that the $l$-auxiliary estimator processes do not move for $l>k$, they
function like the “outside world” for the finite-level mean-field system in
Section 12.1. Therefore we can proceed as in the proof of Proposition 12.1 to
prove the second and third line in (4.29) in Theorem 4.2. The $l$-block
estimator process
$({\bf\Theta}^{(l),\mathrm{aux},\Omega_{N}}(\bar{t}+N^{l}t))_{t>0}$ evolves
according to (12.73) with $l=k$. Note that the extra interactions due to
migration over larger blocks $l>k$ and exchange with deeper seed-banks $m>k$
in (4.66) are of order $\mathcal{O}(1/N)$. Therefore these terms vanish as
$N\to\infty$, and we can just proceed as in the scheme of Section 12.2, to
obtain the second and third line in (4.29) in Theorem 4.2. Using these
results, we obtain that, for $l>k$,

(12.78)
${\bf\Theta}^{(l),\mathrm{aux},\Omega_{N}}(\bar{t})=\delta_{M^{k}_{-(k+1)}}.$

∎

###  13 Proofs: $N\to\infty$, orbit of renormalisation transformation

In this section we analyse the orbit of the renormalisation transformation and
show that it has the Fisher-Wright diffusion as a global attractor. In Section
13.1 we write down moment relations for the equilibrium defined in (4.8) for
single colonies (Proposition 13.1) and for block averages (Proposition 13.2).
In Section 13.2 we derive the iterates of these moment relations for single
colonies (Proposition 13.3) and for blocks (Proposition 13.4). In Section 13.3
we prove clustering (Propositions 13.5–13.6). In Section 13.4 we prove
Theorems 5.1 and 5.3, and work out the dichotomy of a finite seed-bank
($\rho<\infty$) versus infinite seed-bank ($\rho=\infty$).

####  13.1 Moment relations

We use Itô-calculus to compute the mixed moments. Recall
$\theta_{x},(\theta_{y_{m}})$ as defined in (2.21), $\vartheta_{k}$ as defined
in (4.3) and $\bar{\vartheta}^{(k)}$ (5.2). Also recall $E_{k}$ as defined in
(4.5). Abbreviate

(13.1)
$A_{0}^{n}=\frac{1}{2}\sum_{k=0}^{n}\frac{E_{k}}{c_{k}}\frac{(E_{k}c_{k}+e_{k})}{(E_{k}c_{k}+e_{k})+E_{k}K_{k}e_{k}},\qquad
n\in\mathbb{N},$

and

(13.2)
$B_{0}=\frac{1}{2}\frac{E_{0}^{2}}{(E_{0}c_{0}+e_{0})+E_{0}K_{0}e_{0}}.$

In the following proposition, the first five equations are first and second
moment relations, while the last equation is the definition of the
renormalisation transformation. Later we will see that this set of equations
can be iterated.

######  Proposition 13.1.

[Moment relations: single colonies]
Let $\vartheta_{0}$ be as defined in (4.3), and let
$\Gamma_{(\vartheta_{0},y_{l})}^{(0)}=\Gamma_{(\vartheta_{0},y_{l})}^{g,c_{0},E_{0},K_{0},e_{0}}$
be the equilibrium of (4.8) measure defined in (4.14) with $k=0$, with
$g\in\mathcal{G}$, $c_{0}\in(0,\infty)$, $E_{0}\in[0,1]$ and
$K_{0},e_{0}\in(0,\infty)$. Then the following moment relations hold:

(13.3)
$\displaystyle\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}x_{0}\;\Gamma_{(\vartheta_{0},y_{l})}^{g,E_{0},c_{0},K_{0},e_{0}}({\rm
d}z_{0})$ $\displaystyle=$ $\displaystyle\vartheta_{0},$ (13.4)
$\displaystyle\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}y_{0,0}\;\Gamma_{(\vartheta_{0},y_{l})}^{g,E_{0},c_{0},K_{0},e_{0}}({\rm
d}z_{0})$ $\displaystyle=$ $\displaystyle\vartheta_{0},$ (13.5)
$\displaystyle\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}x_{0}y_{0,0}\;\Gamma_{(\vartheta_{0},y_{l})}^{g,E_{0},c_{0},K_{0},e_{0}}({\rm
d}z_{0})$ $\displaystyle=$
$\displaystyle\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}y_{0,0}^{2}\;\Gamma_{(\vartheta_{0},y_{l})}^{g,E_{0},c_{0},K_{0},e_{0}}({\rm
d}z_{0}),$ (13.6)
$\displaystyle\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}x_{0}^{2}\;\Gamma_{(\vartheta_{0},y_{l})}^{g,E_{0},c_{0},K_{0},e_{0}}({\rm
d}z_{0})$ $\displaystyle=$
$\displaystyle\vartheta_{0}^{2}+A_{0}^{0}(\mathcal{F}g)(\vartheta_{0}),$
(13.7)
$\displaystyle\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}y_{0,0}^{2}\;\Gamma_{(\vartheta_{0},y_{l})}^{g,E_{0},c_{0},K_{0},e_{0}}({\rm
d}z_{0})$ $\displaystyle=$
$\displaystyle\vartheta_{0}^{2}+(A_{0}^{0}-B_{0})(\mathcal{F}g)(\vartheta_{0}),$
(13.8)
$\displaystyle\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}g(x_{0})\;\Gamma_{(\vartheta_{0},y_{l})}^{g,E_{0},c_{0},K_{0},e_{0}}({\rm
d}z_{0})$ $\displaystyle=$ $\displaystyle(\mathcal{F}g)(\vartheta_{0}).$

###### Proof.

For ease of notation we write $x,y_{0}$ instead of $x_{0},y_{0,0}$ for the
single colonies. We use Itô’s formula to compute the first and second moments,
and invoke the equilibrium condition to get the above formulas, except for the
last formula, which is the definition of $\mathcal{F}$ in (4.16).

1. We begin with the first moments of $x$ and $y_{0}$. For $k=0$, (4.8) becomes

(13.9) $\displaystyle{\rm d}x(t)$ $\displaystyle=$ $\displaystyle
E_{0}\left[c_{0}\big{[}\vartheta_{0}-x(t)\big{]}\,{\rm
d}t+\sqrt{g(x(t))}\,{\rm d}w_{0}(t)+K_{0}e_{0}\,[y_{0}(t)-x(t)]\,{\rm
d}t\right],$ (13.10) $\displaystyle{\rm d}y_{0}(t)$ $\displaystyle=$
$\displaystyle e_{0}\,[x(t)-y_{0}(t)]\,{\rm d}t,$ $\displaystyle{\rm
d}y_{m}(t)$ $\displaystyle=$ $\displaystyle 0.$

In equilibrium the distribution of $x(t)$ is constant in time, and so
$\frac{{\rm d}}{{\rm d}t}\mathbb{E}[x(t)]=0$, where $\mathbb{E}$ denotes
expectation w.r.t. $\Gamma_{\theta}^{g,c_{0},E_{0},K_{0},e_{0}}$. Integrating
(13.9) and taking the expectation, we get

(13.11) $\displaystyle\mathbb{E}[x(t)-x(0)]$ $\displaystyle=$ $\displaystyle
E_{0}\left[\mathbb{E}\left[\int_{0}^{t}{\rm
d}s\,c_{0}\big{[}\vartheta_{0}-x(s)\big{]}+K_{0}e_{0}\int_{0}^{t}{\rm
d}s\,[y_{0}(s)-x(s)]\right]\right],$ $\displaystyle=$ $\displaystyle
E_{0}\left\\{\int_{0}^{t}{\rm
d}s\,\mathbb{E}\Big{[}c_{0}\big{[}\vartheta_{0}-x(s)\big{]}+K_{0}e_{0}\big{[}y_{0}(s)-x(s)\big{]}\Big{]}\right\\},$

where in the second equation we use Fubini. Turning back to differential
notation, we see from (13.11) that

(13.12) $\frac{{\rm d}}{{\rm
d}t}\,\mathbb{E}[x(0)]=0=E_{0}\left\\{\mathbb{E}\Big{[}c_{0}\big{[}\vartheta_{0}-x(t)\big{]}+K_{0}e_{0}\big{[}y_{0}(t)-x(t)\big{]}\Big{]}\right\\},$

and it follows that

(13.13)
$\mathbb{E}\Big{[}c_{0}\big{[}\vartheta_{0}-x\big{]}+K_{0}e_{0}\big{[}y_{0}-x\big{]}\Big{]}=0.$

In the same way it follows from (13.10) that

(13.14) $\mathbb{E}\Big{[}e_{0}\big{[}x-y_{0}\big{]}\Big{]}=0.$

Therefore we obtain from (13.13)–(13.14) that

(13.15) $\mathbb{E}[x]=\mathbb{E}[y_{0}]=\vartheta_{0}.$

2. We next compute the second moments. By Itô’s formula,

(13.16) $\displaystyle{\rm d}(x(t))^{2}=2x(t)\,{\rm d}x(t)+({\rm d}x(t))^{2}$
$\displaystyle=2c_{0}x(t)E_{0}[\vartheta_{0}-x(t)]\,{\rm
d}t+2x(t)E_{0}\sqrt{g(x(t))}\,{\rm d}w_{0}(t)$
$\displaystyle\qquad+E_{0}\big{[}2K_{0}e_{0}\,x(t)y_{0}(t)-2K_{0}e_{0}\,x^{2}_{t}\big{]}\,{\rm
d}t+E_{0}^{2}g(x(t))\,{\rm d}t.$

Taking expectations and using that we are in equilibrium, we get

(13.17) $\displaystyle
0=2c_{0}\vartheta_{0}^{2}-2c_{0}\mathbb{E}[x^{2}]+2K_{0}e_{0}\mathbb{E}[xy_{0}]-2K_{0}e_{0}\mathbb{E}[x^{2}]+E_{0}\mathbb{E}[g(x)].$

Using $\mathbb{E}[g(x)]=(\mathcal{F}g)(\vartheta_{0})$, we find

(13.18)
$\mathbb{E}[x^{2}]=\frac{c_{0}}{(c_{0}+K_{0}e_{0})}\vartheta_{0}^{2}+\frac{K_{0}e_{0}}{(c_{0}+K_{0}e_{0})}\mathbb{E}[xy_{0}]+\frac{E_{0}}{2(c_{0}+K_{0}e_{0})}(\mathcal{F}g)(\vartheta_{0}).$

In the same way we find

(13.19) $\mathbb{E}[y_{0}^{2}]=\mathbb{E}[xy_{0}],$

and for the mixed second moment

(13.20)
$\mathbb{E}[xy_{0}]=\frac{E_{0}c_{0}}{(E_{0}c_{0}+e_{0})}\vartheta_{0}^{2}+\frac{e_{0}}{(E_{0}c_{0}+e_{0})}\mathbb{E}[x^{2}].$

Substituting (13.20) into (13.18), we find $\mathbb{E}[x^{2}]$ and hence also
$\mathbb{E}[y_{0}^{2}]$ and $\mathbb{E}[x_{0}y_{0}]$. This finishes the proof
of Proposition 13.1. ∎

Similar moment relations can be derived for the equilibrium measures of the
block averages. Define

(13.21)
$A_{m}^{n}=\frac{1}{2}\sum_{k=m}^{n}\frac{E_{k}}{c_{k}}\frac{(E_{k}c_{k}+e_{k})}{(E_{k}c_{k}+e_{k})+E_{k}K_{k}e_{k}},\qquad
m\in\mathbb{N}_{0},\,n\in\mathbb{N},$

and

(13.22)
$B_{m}=\frac{1}{2}\frac{E_{m}^{2}}{(E_{m}c_{m}+e_{m})+E_{m}K_{m}e_{m}},\qquad
m\in\mathbb{N}_{0}.$

Recall the definition of $\mathcal{F}^{(n)}$ in (4.17).

######  Proposition 13.2.

[Moment relations: blocks]
Let $\vartheta_{m}$ be as defined in (4.3), and let
$\Gamma_{(\vartheta_{m},y_{m})}^{(m)}=\Gamma_{(\vartheta_{m},y_{m})}^{\mathcal{F}^{(m)}g,c_{m},E_{m},K_{m},e_{m}}$
be the equilibrium measure of (4.8) with $k=m$, with $g\in\mathcal{G}$,
$c_{0}\in(0,\infty)$, $E_{0}\in[0,1]$ and $K_{0},e_{0}\in(0,\infty)$. Then the
following moment relations hold:

(13.23)
$\displaystyle\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}x_{m}\;\Gamma_{(\vartheta_{m},y_{m})}^{\mathcal{F}^{(m)}g,E_{m},c_{m},K_{m},e_{m}}({\rm
d}z_{m})$ $\displaystyle=$ $\displaystyle\vartheta_{m},$ (13.24)
$\displaystyle\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}y_{m,m}\;\Gamma_{(\vartheta_{m},y_{m})}^{\mathcal{F}^{(m)}g,E_{m},c_{m},K_{m},e_{m}}({\rm
d}z_{m})$ $\displaystyle=$ $\displaystyle\vartheta_{m},$ (13.25)
$\displaystyle\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}x_{m}y_{m,m}\;\Gamma_{(\vartheta_{m},y_{m})}^{\mathcal{F}^{(m)}g,E_{m},c_{m},K_{m},e_{m}}({\rm
d}z_{m})$
$\displaystyle=\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}y_{m,m}^{2}\;\Gamma_{(\vartheta_{m},y_{m})}^{\mathcal{F}^{(m)}g,E_{m},c_{m},K_{m},e_{m}}({\rm
d}z_{m}),$ (13.26)
$\displaystyle\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}x_{m}^{2}\;\Gamma_{(\vartheta_{m},y_{m})}^{\mathcal{F}^{(m)}g,E_{m},c_{m},K_{m},e_{m}}({\rm
d}z_{m})$ $\displaystyle=$
$\displaystyle\vartheta_{m}^{2}+A_{m}^{m}(\mathcal{F}^{(m+1)}g)(\vartheta_{m}),$
(13.27)
$\displaystyle\int_{[0,1]\times[0,1]^{\mathbb{N}_{0}}}y_{m,m}^{2}\;\Gamma_{(\vartheta_{m},y_{m})}^{\mathcal{F}^{(m)}g,E_{m},c_{m},K_{m},e_{m}}({\rm