Construction of an eigen-solution for the Fokker-Planck operator with heavy tail equilibrium: an `a la Koch method in dimension 1

This paper is devoted to the construction of an \emph{eigen-solution} for the Fokker-Planck operator with heavy tail equilibrium. We propose an $\mathit{\text{alternative}}$ method in dimension 1, which will be generalizable in higher dimension. The later method is inspired by the work of H. Koch on non-linear KdV equation \cite{Koch}. As a consequence of this construction, we recover the result of G. Lebeau and M. Puel \cite{LebPu} on the fractional diffusion limit for the Fokker-Planck equation.