$\mathsf{Rep}(\rho ,j)$	The set of representatives of $\rho \in {\mathbb{R}}_{+}$, $j\in {\mathbb{R}}^d$: $\mathsf{Rep}(\rho ,j)=\mathsf{Rep}(\rho )\times \mathsf{Rep}(j)$.
$Q_{\epsilon }^z$	The cube of size $\epsilon >0$ centered in $\epsilon z\in {\mathbb{T}}^d$: for $z\in {\mathbb{Z}}_{\epsilon }^d$, $Q_{\epsilon }^z:={[0,\epsilon )}^d+\epsilon z$.
$S^{\overline{z}}$	Shift operator: $S_{\epsilon }^{\overline{z}}:\mathcal{X}\to \mathcal{X}$, $S_{\epsilon }^{\overline{z}}(x)=(\overline{z}+z,v)$ for $x=(z,v)\in \mathcal{X}$.
	Shift operator: $S_{\epsilon }^{\overline{z}}:\mathcal{E}\to \mathcal{E}$, $S_{\epsilon }^{\overline{z}}(x,y):=(S_{\epsilon }^{\overline{z}}(x),S_{\epsilon }^{\overline{z}}(y))$ for $(x,y)\in {\mathcal{E}}_{\epsilon }$
${\sigma }^z$	${\sigma }_{\epsilon }^{\overline{z}}\psi :{\mathcal{X}}_{\epsilon }\to \mathbb{R},({\sigma }_{\epsilon }^{\overline{z}}\psi )(x):=\psi (S_{\epsilon }^{\overline{z}}(x))\text{for}x\in {\mathcal{X}}_{\epsilon }.$
	${\sigma }_{\epsilon }^{\overline{z}}J:{\mathcal{E}}_{\epsilon }\to \mathbb{R},({\sigma }_{\epsilon }^{\overline{z}}J)(x,y):=J(S_{\epsilon }^{\overline{z}}(x,y))\text{for}(x,y)\in {\mathcal{E}}_{\epsilon }.$
$T_{\epsilon }^{\overline{z}}$	Rescaling operator: $T_{\epsilon }^{\overline{z}}:\mathcal{X}\to {\mathcal{X}}_{\epsilon }$: $T_{\epsilon }^{\overline{z}}(x)=(\epsilon (\overline{z}+z),v)$ for $x=(z,v)\in \mathcal{X}$.
${\tau }_{\epsilon }^z$	${\tau }_{\epsilon }^{\overline{z}}\psi :\mathcal{X}\to \mathbb{R},({\tau }_{\epsilon }^{\overline{z}}\psi )(x):=\psi (T_{\epsilon }^{\overline{z}}(x))\text{for}x\in \mathcal{X}$.
	${\tau }_{\epsilon }^{\overline{z}}J:\mathcal{E}\to \mathbb{R},({\tau }_{\epsilon }^{\overline{z}}J)(x,y):=J(T_{\epsilon }^{\overline{z}}(x),T_{\epsilon }^{\overline{z}}(y))\text{for}(x,y)\in \mathcal{E}$.
$\mathcal{C}\mathcal{E}$	Discrete continuity equation: $(\mathit{m},\mathit{J})\in \mathcal{C}\mathcal{E}$ iff ${\partial }_t{m}_t+\mathsf{\text{div}}J=0$ on $(\mathcal{X},\mathcal{E})$.
$\mathbb{C}\mathbb{E}$	Continuous continuity equation: $(\mathit{\mu },\mathit{\nu })\in \mathbb{C}\mathbb{E}$ iff ${\partial }_t{\mu }_t+\mathrm{\nabla }·\mathit{\nu }=0$ on ${\mathbb{T}}^d$.
$\text{BV}$	More precisely ${\text{BV}}_{\text{KR}}(\mathcal{I};{\mathcal{M}}_{+}({\mathbb{T}}^d))$: the space of time-dependent curves of
	(Positive) measures with bounded variation with respect to the $\text{KR}$ norm
	(Kantorovich–Rubenstein) on ${\mathcal{M}}_{+}({\mathbb{T}}^d)$.
$W^{1.1}$	More precisely $W_{\text{KR}}^{1,1}(\mathcal{I};{\mathcal{M}}_{+}({\mathbb{T}}^d))$: the space of time-dependent curves of
	(Positive) measures belonging to the Banach space $W^{1,1}(\mathcal{I};{({\mathcal{C}}^1({\mathbb{T}}^d))}^{\ast })$.
${\text{P}}_{\epsilon }\mu ,{\text{P}}_{\epsilon }\nu$	Discretisation of $\mu \in {\mathcal{M}}_{+}({\mathbb{T}}^d)$, $\nu \in {\mathcal{M}}^d({\mathbb{T}}^d)$: for $z\in {\mathbb{Z}}_{\epsilon }^d$, (${\text{P}}_{\epsilon }\mu (z),{\text{P}}_{\epsilon }\nu (z))\in$
	${\mathbb{R}}_{+}\times {\mathbb{R}}^d$, given by ${\text{P}}_{\epsilon }\mu (z)=\mu (Q_{\epsilon }^z)$, ${\text{P}}_{\epsilon }\nu (z)=((\nu ·e_i)(\partial Q_{\epsilon }^z\cap \partial Q_{\epsilon }^{z+e_i}){)}_i$.