Time Asymptotics and Scaling Limits for a Nonlocal Fokker-Planck Equation with Heavy-Tailed Kernel

We investigate the asymptotic behaviour of solutions of a class of nonlocal Fokker–Planck equations defined by nonsingular, heavy-tailed convolution kernels and characterised by a scaling parameter $\text{e}\in (0,1]$ and a fractional index $s\in (1\mathrm{/}2,1)$. By employing a suitable version of the generalised central limit for heavy-tailed distributions and the use of Harris’s theorem, we prove exponential convergence to the equilibrium with a rate that is independent of both $\text{e}$ and $s$. This allows us to show uniform–in–time convergence for both $\text{e}\to 0$ and $s\to 1$ recovering the limiting equations.