Non-spectral modes and how to find them in the Ornstein-Uhlenbeck process with white {\mu}-stable noise

Abstract

We consider the Ornstein-Uhlenbeck process with a broad initial probability distribution (Levy distribution), which exhibits so-called non-spectral modes. The relaxation of such modes differs from those determined from the parameters of the corresponding Fokker-Planck equation. The first non-spectral mode is shown to govern the relaxation process and allows for estimation of the initial distribution's Levy index. A method based on continuous wavelet transformation is proposed to extract both (spectral and non-spectral) relaxation rates from a stochastic data sample.