Spectral properties of L\'evy Fokker--Planck equations

TL;DR

This paper derives explicit formulas for fractional Hermite polynomials as eigenfunctions of Le9vy Fokker-Planck equations, providing spectral analysis and asymptotic solutions for fractional heat equations.

Contribution

It introduces explicit Euclidean formulas for fractional Hermite polynomials and characterizes the spectrum of Le9vy Fokker-Planck equations using Mellin transform techniques.

Findings

Spectral description of Le9vy Fokker-Planck operator

Explicit formulas for fractional Hermite polynomials

Asymptotic expansion for fractional heat equation solutions

Abstract

Hermite polynomials, which are associated to a Gaussian weight and solve the Laplace equation with a drift term of linear growth, are classical in analysis and well-understood via ODE techniques. Our main contribution is to give explicit Euclidean formulae of the fractional analogue of Hermite polynomials, which appear as eigenfunctions of a L\'evy Fokker-Planck equation. We will restrict, without loss of generality, to radially symmetric functions. A crucial tool in our analysis is the Mellin transform, which is essentially the Fourier transform in logarithmic variable and which turns weighted derivatives into multipliers. This allows to write the weighted space in the fractional case that replaces the usual $L_r^2(\mathbb R^n, e^{|x|^2/4})$. After proving compactness, we obtain a exhaustive description of the spectrum of the L\'evy Fokker--Planck equation and its dual, the fractional Ornstein--Uhlenbeck problem, which forms a basis thanks to the spectral theorem for self-adjoint operators. As a corollary, we obtain a full asymptotic expansion for solutions of the fractional heat equation.