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

TL;DR

This paper investigates non-spectral modes in the Ornstein-Uhlenbeck process driven by white {5}-stable noise, revealing their role in relaxation dynamics and proposing a wavelet-based method to extract relaxation rates from data.

Contribution

It introduces a novel approach to identify non-spectral modes in the Ornstein-Uhlenbeck process with Levy noise and estimates the Levy index from relaxation behavior.

Findings

Non-spectral modes influence relaxation in Levy-driven Ornstein-Uhlenbeck processes.

A wavelet-based method effectively extracts spectral and non-spectral relaxation rates.

The first non-spectral mode helps estimate the Levy index of the initial distribution.

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.