Convergence rates of particle approximation of forward-backward splitting algorithm for granular medium equations

We study the spatially homogeneous granular medium equation $${\mathrm{\partial }}_t\mu =\mathrm{d}\mathrm{i}\mathrm{v}(\mu \mathrm{\nabla }V)+\mathrm{d}\mathrm{i}\mathrm{v}(\mu (\mathrm{\nabla }W\ast \mu ))+\mathrm{\Delta }\mu ,$$ within a large and natural class of the confinement potentials $V$ and interaction potentials $W$. The considered problem do not need to assume that $\mathrm{\nabla }V$ or $\mathrm{\nabla }W$ are globally Lipschitz. With the aim of providing particle approximation of solutions, we design efficient forward-backward splitting algorithms. Sharp convergence rates in terms of the Wasserstein distance are provided.