On the spectral problem and fractional diffusion limit for Fokker-Planck with(-out) drift and for a general heavy tail equilibrium

TL;DR

This paper investigates the spectral properties and fractional diffusion limits of a kinetic Fokker-Planck equation with heavy-tailed equilibria, extending previous results to more general, possibly non-symmetric cases.

Contribution

It establishes the existence of a unique eigenpair for the spectral problem with heavy-tailed equilibria and derives the fractional diffusion limit under broad conditions, including non-symmetric cases.

Findings

Existence of a unique eigenpair for the spectral problem with heavy-tailed equilibria.

Derivation of the fractional diffusion limit for the Fokker-Planck equation under general conditions.

Generalization of previous results to non-symmetric and non-centered equilibria.

Abstract

This paper is devoted to the study of a kinetic Fokker-Planck equation with general heavy-tailed equilibrium without an explicit formula, such as $C_\beta \langle v \rangle^{-\beta}$, in particular non-symmetric and non-centred. This work extends the results obtained in [Dechicha and Puel, 2023] and [Dechicha and Puel, Asymptot. Anal., 2024]. We prove that if the equilibrium behaves like $\langle v \rangle^{-\beta}$ at infinity with $\beta > d$, along with an other assumption, there exists a unique eigenpair solution to the spectral problem associated with the Fokker-Planck operator, taking into account the advection term. As a direct consequence of this construction, and under the hypothesis of the convergence of the rescaled equilibrium, we obtain the fractional diffusion limit for the kinetic Fokker-Planck equation, with or without drift, depending on the decay of the equilibrium and whether or not the first moment is finite. This latter result generalizes all previous results on the fractional diffusion limit for the Fokker-Planck equation and rigorously justifies the remarks and results mentioned in [Bouin and Mouhot, PMP, 2022, Section 9].