A.3 Proof of lemma 1
We adapt a variant of the Ascoli–Arzelà theorem [Brezis and Brézis, 2011 ] .
We showed in Prop. 2 that there exists C>0 that only depends on the initialization such that, \forall t\geq 0,\forall i\in[N-1] ,

\|\theta^{N}_{n+1}(t)-\theta^{N}_{n}(t)\|\leq\frac{C}{N}.

This implies that

\|\theta^{N}_{j}(t)-\theta^{N}_{i}(t)\|\leq C\frac{|j-i|}{N}.

We also have that

\|\theta^{N}_{n}(t_{1})-\theta^{N}_{n}(t_{2})\|=\|N\int_{t_{1}}^{t_{2}} abla^{N}_{n}\|\leq C^{\prime}|t_{1}-t_{2}|

with C^{\prime}\geq 0 .

Its follows that \|\psi_{N}(s_{1},t_{1})-\psi_{N}(s_{2},t_{2})\|\leq\|\psi_{N}(s_{1},t_{1})-\psi_{N}(s_{1},t_{2})\|+\|\psi_{N}(s_{1},t_{2})-\psi_{N}(s_{2},t_{2})\| and thus

(i)\quad\|\psi_{N}(s_{1},t_{1})-\psi_{N}(s_{2},t_{2})\|\leq C^{\prime}|t_{1}-t_{2}|+C|s_{1}-s_{2}|+\frac{C}{N}.

We also have

(ii)\quad\forall N\in\mathbb{N},\quad\|\psi_{N}\|_{\infty}\leq\frac{1}{2}.

These two properties are essential to prove our lemma. We proceed as follows.

1) First, we denote ((s_{j},t_{j}))_{j\in\mathbb{N}}=(\mathbb{Q}\cap[0,1])\times\mathbb{Q}_{+} . Since we have the uniform bound (ii) , we extract using a diagonal extraction procedure a subsequence \psi_{\sigma(N)} such that \forall j\in\mathbb{N} ,

\psi_{\sigma(N)}(s_{j},t_{j})\to\psi(s_{j},t_{j})

(we denote the limit \psi(s_{j},t_{j}) ).

2) We show the convergence \forall s\in[0,1] and t\in\mathbb{R}_{+} .

Let \varepsilon>0 , s\in[0,1] and t\in\mathbb{R}_{+} . Since ((s_{j},t_{j}))_{j\in\mathbb{N}} is dense in [0,1]\times\mathbb{R}_{+} , there exists k\in\mathbb{N} such that |s_{k}-s|<\varepsilon and |t_{k}-t|<\varepsilon . Let N,M\in\mathbb{N} .

We have

\|\psi_{\sigma(N)}(s,t)-\psi_{\sigma(M)}(s,t)\|\leq\|\psi_{\sigma(N)}(s,t)-\psi_{\sigma(N)}(s_{k},t_{k})\|+\|\psi_{\sigma(N)}(s_{k},t_{k})-\psi_{\sigma(M)}(s_{k},t_{k})\|+\|\psi_{\sigma(M)}(s_{k},t_{k})-\psi_{\sigma(M)}(s,t)\|

so that

\|\psi_{\sigma(N)}(s,t)-\psi_{\sigma(M)}(s,t)\|\leq 2C\varepsilon+2C^{\prime}\varepsilon+\frac{C}{\sigma(N)}+\frac{C}{\sigma(M)}+\|\psi_{\sigma(N)}(s_{k},t_{k})-\psi_{\sigma(M)}(s_{k},t_{k})\|.

Since (\psi_{\sigma(N)}(s_{k},t_{k}))_{N\in\mathbb{N}} is a Cauchy sequence, this gives for N,M big enough that

\|\psi_{\sigma(N)}(s)-\psi_{\sigma(M)}(s,t)\|\leq(2(C+C^{\prime})+1)\varepsilon

and thus (\psi_{\sigma(N)}(s,t)) is a Cauchy sequence in \mathbb{R}^{d\times d} . As such, it converges and one has

\psi_{\sigma(N)}(s,t)\to\psi(s,t).

3) Recall that one has

\|\psi_{\sigma(N)}(s_{1},t_{1})-\psi_{\sigma(N)}(s_{2},t_{2})\|\leq C|s_{1}-s_{2}|+\frac{C}{\sigma(N)}+C^{\prime}|t_{1}-t_{2}|

so that letting N\to\infty gives

\|\psi(s_{1},t_{1})-\psi(s_{2},t_{2})\|\leq C|s_{1}-s_{2}|+C^{\prime}|t_{1}-t_{2}|

and \psi is Lipschitz continuous.

4) Let us finally show that the convergence is uniform in (s,t) . Let s\in[0,1] , \varepsilon>0 and \delta>0 such that if |s-u|<\delta , \forall t\in\mathbb{R}+ ,

\|\psi_{N}(s,t)-\psi_{N}(u,t)\|\leq\varepsilon+\frac{C}{N}

and \|\psi(s,t)-\psi(u,t)\|\leq\varepsilon . There exists a finite set of \{s_{j}\}_{j=1}^{k} such that

[0,1]\subset\cup_{j=1}^{k}]s_{j}-\frac{\delta}{2},s_{j}+\frac{\delta}{2}[.

For our s , there exists j\in\{1,\dots,k\} such that \|s-s_{j}\|\leq\delta .

There also exists t_{0}\geq 0 such that if t\geq t_{0} ,

\|\psi_{\sigma(N)}(s,t)-\psi_{\sigma(N)}(s,t_{0})\|\leq\sqrt{2eM}\int_{t_{0}}^{t}e^{-\frac{1}{e}mz}\sqrt{{\ell}^{N}(0)}dz\leq\varepsilon.

We have:

\|\psi_{\sigma(N)}(s,t_{0})-\psi(s,t_{0})\|\leq\|\psi_{\sigma(N)}(s,t_{0})-\psi_{\sigma(N)}(s_{j},t_{0})\|+\|\psi_{\sigma(N)}(s_{j},t_{0})-\psi(s_{j},t_{0})\|+\|\psi(s_{j},t_{0})-\psi(s,t_{0})\|

and thus:

\|\psi_{\sigma(N)}(s,t_{0})-\psi(s,t_{0})\|\leq 2\varepsilon+\frac{C}{\sigma(N)}+\max_{j\in\{1,\dots,k\}}\|\psi_{\sigma(N)}(s_{j},t_{0})-\psi(s_{j},t_{0})\|\leq 4\varepsilon for N big enough.

Finally, \|\psi_{\sigma(N)}(s,t)-\psi(s,t)\|\leq\|\psi_{\sigma(N)}(s,t)-\psi_{\sigma(N)}(s,t_{0})\|+\|\psi_{\sigma(N)}(s,t_{0})-\psi(s,t_{0})\|+\|\psi(s,t_{0})-\psi(s,t)\|\leq 6\varepsilon

for N big enough, independently of t and s . This concludes the proof.
∎