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 $\e\in(0,1]$ and a fractional index $s\in(1/2,1)$. 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 $\e$ and $s$. This allows us to show uniform--in--time convergence for both $\e\to 0$ and $s\to1$ recovering the limiting equations.