to $0$.\
Given a solution $(u,m)$ of , we define two grid functions ${\tilde U}_{\sigma}$,${\tilde M}_{\sigma}$ on $({\mathbb{R}}^{N_h^2})^N$ by $$({\tilde U}_{\sigma})^n_{i,j}=u(x_{i,j},n{{\Delta t}}),\quad ({\tilde M}_{\sigma})^n_{i,j}=\frac 1 {h^2} \int_{|x-x_{i,j}|\le h/2} m(x,n{{\Delta t}}) dx .$$ Note that ${\tilde M}_{\sigma}\in {\mathcal{K}}_h^N$, see . Moreover, since $(u,m)$ is a smooth function, $({\tilde U}_{\sigma},{\tilde M}_{\sigma})$ is a solution of with the perturbation $(a^n_{\sigma},b^n_{\sigma})$ given by the consistency error of the scheme. Hence $$\label{eq:conv_consistency}
\max_{0\le n\le N}\{\|a^n_{\sigma}\|_\infty+\|b^n_{\sigma}\|_\infty\}\to 0 \quad \text{for ${\sigma}\to 0$}.$$ The first step of the proof is the convergence result $$\label{eq:conv_seminorm}
\lim_{|{\sigma}|\to 0}h^2{{\Delta t}}\sum_{n=0}^N\sum_{i,j}|[D_hU^n_{\sigma}]_{i,j}-[D_h({\tilde U}_{\sigma})^n]_{i,j}|^\beta=0.$$ The identity is obtained by means of the fundamental identity which does not involve the discrete fractional derivatives. Hence, for its proof, it is possible to repeat the same proof of Step 1 in [@acc Section 3]. Note that here it is employed that ${\tilde M}_{\sigma}>0$, which follows by the assumption $m_0\ge\delta>0$.\
The second step of the proof is the convergence of the solution of the Fokker-Planck equation. We denote with $M$, ${\tilde M}$ the grid functions corresponding to $m_{\sigma}$ and $m$, omitting the index ${\sigma}$ for simplicity. We set $E=M-{\tilde M}$ and $$\begin{aligned}
&\|[D_hE^{n+1}]\|^2_2=h^2\sum_{i,j} [D_hE^{n+1}]_{i,j}\cdot[D_hE^{n+1}]_{i,j},\\
&\|(E^{n+1},[D_hE^{n+1}])\|_2^2=\|E^{n+1}\|^2_2+\rho_\alpha\|[D_hE^{n+1}]\|^2_2.\end{aligned}$$ By subtracting the equations satisfied by $M$, ${\tilde M}$, we get $$E^{n+1}_{i,j}-\rho_\alpha (\Delta_h 